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1,108,101,563,215 | arxiv | \section{Conclusion}\label{Conclusion}
In this study, multiple NNs and extensions were trained on the southern Manitoba using the NALCMS dataset as labelled data. The results have shown that the best model variation is the VGGNet encoder with modified U-Net decoder and OS fixed at 4 that reached 92.4\% OA. Six different extensions were applied to the network, where most of them did not provide noticeable improvement both individually, and in combinations. Individual LULC class performance is highly dependent on the number of pixels representing that class; in general, the more pixels, the better the classification accuracy. Additionally, the best model variation was trained on the larger Landsat 5/7 Prairie datasets which resulted in a 80.66\% OA; an accuracy that is better than previously reported in the literature but equivalent to the performance of ArcGIS/Pro deep learning package \cite{arcgis:2020}. Also, the same model was trained on the Landsat 8 Lake Winnipeg dataset, where performance reached 88.04\% OA due to higher radiometric resolution. The Landsat 8 data increased the overall accuracy by 11.44\% compared to \cite{latifovic:2016} and is ~6-8\% higher than any other deep learning model present in the ArcGIS/PRo package that solves an equivalent problem on a similar satellite. And is on the level of the state-of-the-art DL models designed for datasets with a lower amount of classes. This shows that it is possible to produce fairly accurate NALCMS LULC maps from Landsat 5/7 and 8 sensors for 37 years of archived data and also future Landsat 8 data.
One of the limitations of DL is the amount of labelled data required to generated land cover map products. Thus, one of the most important directions of research in the future is to find a semi-supervised solution to produce accurate LULC maps, such as examination of unsupervised domain adaptation architectures \cite{UDAGAN:2017,UDA_GANRemoteSensing:2019}. Also, NNs can be trained on high-resolution satellite images to provide sharp and more detailed LULC maps. Furthermore, a multi-sensor model can be considered by training a network on multiple datasets from different sensors at once, making a more generic and versatile model, which can be used to develop LULC maps from the different sensors without prior training.
\section{Implementing and Training the Models}\label{section_Implementations}
\subsection{Data}
As the overall goal of this work was to develop models to automate LULC map production corresponding to the NALCMS labels; the models were developed using both the 2010 NALCMS labels with Landsat 5/7 data (2009-2011), as well as with 2015 NALCMS labels with Landsat 8 (2014-2016) data. In the Prairie (Figure \ref{fig:Big_Dataset}) and Lake Winnipeg/Churchill-Nelson watershed, 15 of the 19 NALCMS classes were present and a cloud label was added to make it 16 classes. Cloud labels were generated using Otsu thresholding techniques on the blue band \cite{Otsu:1979} for the Landsat 5/7 dataset; cloud identification was done using the Quality Assessment (QA) band for the Landsat 8 dataset.
Due to the associated training time required for an area the size of the watershed dataset, to allow for direct comparison to \cite{VictorAutomatedLULC:2019}, and to expand to larger regions of interest, we used three datasets in our analysis. The dataset were: (i) the southern Manitoba (148,800 $km^2$) and (ii) the three prairie provinces of Alberta, Saskatchewan and Manitoba ($\sim$ 1,960, 000 $km^2$) based on 2010 Landsat 5/7, and (iii) the Lake Winnipeg/ Churchill-Nelson watershed (most of the prairies, plus parts of eastern Ontario and North Dakota) using 2015 Landsat 8 ($\sim$ 1,400,000 $km^2$). Landsat 5/7 data was obtained for the southern Manitoba dataset (Figure \ref{fig:Small_Dataset}), which is the same data and area coverage used in \cite{VictorAutomatedLULC:2019} to see the impact of model-architecture experiments on results. Most of the experiments were performed on the Landsat 5/7 southern Manitoba dataset to reduce the training time required for the comparison of models and model extensions. Then, based on the results, the best architecture was chosen, and the models for Landsat 5/7 (Prairies) and Landsat 8 (Lake Winnipeg watershed) were generated.
\subsection{Data Preprocessing}
All satellite data and corresponding labels were preprocessed before training. A popular input-size for encoder-decoder networks is 224 x 224 pixel rows and columns, so the first step was to tile the larger dataset extents. For example, the mosaic forming the southern Manitoba dataset has 14,975 rows x 13,331 columns of pixels. By creating tiles of 224 x 224 pixels, there is a suitable number of images to train a DL model, and images of this size can be processed by our computer hardware. Consequently, we used the data augmentation technique called \textit{tiling}, which was used in \cite{StorieDeepLN:2018,HenryDeepLN:2018,VictorAutomatedLULC:2019,AlhassanDeepLN2020}, with some minor improvements.
\subsubsection{Tiling}
Tilling is the process of splitting a larger image into smaller-sized, square images (Figure \ref{fig:Preprocess_augmentation}), also referred to as tiles, where a tile is one 224 x 224 image (pixel row and column). The southern Manitoba dataset (14,975 x 13,331 pixels) generated 7,270 disjoint tiles. Moreover, increasing the number of tiles generated from the same region of interest extent can be used to improve feature detection. For example, tilling with 1/2 overlap (shifted tilling) was used to increase the size of the dataset by a factor of 4 (Figure \ref{fig:Preprocess_augmentation}), generating a dataset of 29,100 training tiles and 2,100 validation tiles. Additionally, the Landsat 5/7 Prairie dataset was generated using regular tiling and consisted of 85,400 training tiles and 8,500 validation tiles, while the Landsat 8 Lake Winnipeg watershed dataset was generated using shifted tiling and consisted of 107,900 training tiles and 12,000 validation tiles. Splitting tiles for training and validation was performed using a stratified random process to ensure that training and validation sets contain all LULC classes and tiles from all locations. The splitting ratio was 90\% training to 10\% validation, following \cite{StorieDeepLN:2018,HenryDeepLN:2018,VictorAutomatedLULC:2019,AlhassanDeepLN2020}. This splitting ratio allows effective training of the LULC classes with low presence or representation.
\begin{figure}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1\textwidth]{images/Tiling.PNG}
\end{subfigure}
\hspace{0.02\textwidth}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1\textwidth]{images/Shifting.PNG}
\end{subfigure}
\caption{Illustration of the tilling process\cite{VictorAutomatedLULC:2019}.}
\label{fig:Preprocess_augmentation}
\end{figure}
\subsubsection{Transformation}
Training DL models often results in either overfitting or underfitting \cite{OverfitUnderfit:2019} that is caused by a lack of variation in the dataset or inadequate model architecture. One of the simplest ways to resolve this issue is to shuffle data after every epoch and apply random data transformation on each image \cite{DataTransform1:2019,DataTransform2:2019}. Data transformation ensures that tiles are not represented in the same way during training by randomly applying rotation, flipping, or zoom cropping (Figure \ref{fig:Preprocess_Transformation}). Note that, unlike augmentation (shifted tiling), transformation is used during training and does not increase the number of generated tiles.
\begin{figure}
\centering
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=0.9\linewidth]{images/Trans_Regular_Tile.png}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=1\linewidth]{images/Trans_Rotated_Tile.png}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=1\linewidth]{images/Trans_Zoom_Tile.png}
\end{subfigure}
\begin{subfigure}{0.6\textwidth}
\includegraphics[width=1\linewidth]{images/Trans_Rotated_Flipped.png}
\end{subfigure}
\caption{Figure depicting the tile transformations used during training. This includes rotating, zoom cropping, and flipping both vertically and horizontally.}
\label{fig:Preprocess_Transformation}
\end{figure}
\subsection{Experimental Design}
Training of the networks was performed using the \textit{TensorFlow} \cite{abadi2016tensorflow} Python library with \textit{Docker} containers\footnote{\url{https://www.tensorflow.org/install/docker}}. Most of the results were generated using an NVIDIA Digits DevBox\footnote{\url{https://developer.nvidia.com/devbox}} containing four Titan X GPUs with 12GB of memory per GPU, 64 GB DDR4 RAM, and a Core i7-5930K 3.5 GHz processor. The training time on the southern Manitoba datasets varied between 5 and 7 days, while the Prairie and Lake Winnipeg watershed dataset took between 20 and 30 days. The prediction time of the network on the non-augmented southern Manitoba dataset took 8 minutes, while the prediction of the Prairie or Lake Winnipeg watershed datasets took close to 1.5 hours for both sensors. Most of the networks were trained with a batch size of 16 on a model that was distributed across 3 GPUs, leading to a global batch size of 48. However, some network variations used a lower batch size due to computational complexities. As per \cite{VictorAutomatedLULC:2019}, training was performed with a learning rate of $\eta=10^{-4}$ for 100 epochs, then with a learning rate of $\eta=10^{-5}$ for another 100 epochs using the Adam optimization algorithm \cite{Adam:2017}.
\section{Introduction}\label{section_Introduction}
\textit{Land Use and Land Cover} (LULC) maps are often the output products generated from raw satellite imagery where, in this case, each pixel of the image is assigned to a land cover class or label ({\em e.g.} water, forest, road, cropland, {\em etc.}). LULC map information is crucial for many applications such as natural resource management, wildlife habitat protection, urban expansion, hazard and damage delineation, {\em etc.} \cite{LULC:2004}. For many years, LULC maps were generated manually using semi-automated techniques such as supervised, unsupervised and hybrid algorithms ({\em e.g.}, K-means Clustering, Decision Tree and Maximum Likelihood Classier) \cite{RemoteSensingOld:2007}. However, these methods take a lot of time and effort to produce adequate accuracies but are rarely consistent over different production years or producers. Therefore, map automation is on the rise as it significantly increases the production frequency and consistency of LULC products. Methods of map automation include using advanced algorithms, machine learning tools \cite{RemoteSensingOld:2007}, or \textit{Deep Learning} (DL) \cite{DLinRemoteSensing:2019} started growing in terms of development and application in the last decade. DL has achieved success in the field of remote sensing by providing fast and consistent performance and solving problems such as image fusion, image registration, scene classification and object detection, LULC classification, {\em etc.} \cite{DLinRemoteSensing:2019}. Research on DL for remote sensing has resulted in groundbreaking papers published every few months in the fields of image processing \cite{ImageProcessing:2016}, computer vision \cite{ComputerVision:2016}, and object detection \cite{ObjectRecnog:2017}. Most issues in DL map applications are solved using Convolutional Neural Networks (CNN) as CNNs are considered the state-of-the-art method for computer vision tasks, such as segmentation, image classification and object recognition \cite{DLinRemoteSensing:2019,CNN_LitReview}.
To resolve the semantic segmentation task of LULC mapping, Long {\em et al.}\cite{FCN:2014} and Badrinarayanan{\em et al.}\cite{SegNet:2016} introduced an encoder-decoder architecture, which extracts features from the imagery using CNN in the form of an encoder and then decodes those features back to the original size using a reversed CNN. The implementation of \textit{encoder-decoder} networks in remote sensing \cite{DLLulcFirst:2015,DLLulcSecond:2015,StorieDeepLN:2018,HenryDeepLN:2018,AlhassanDeepLN2020}, has achieved great success with accurate and consistent results between 80\% and 90\% for LULC mapping. These successes have led to the implementation of DL models for mapping of LULC within commercial software, for example, ArcPro (Esri). However, in comparison to the published literature discussed above, commercially available DL models have lower overall classification accuracies, typically ranging from 78-88\% \cite{arcgis:2020} and have even lower performance with a larger number of LULC classes.
The goal of this work was to develop and improve the results of land cover mapping using DL techniques based on label or training data from the North American Land Change Monitoring System (NALCMS)\footnote{\url{http://www.cec.org/north-american-land-change-monitoring-system/}}. The NALCMS labels were chosen for training because of the large number of potential land cover categories (20 classes), ranging from arctic to tropical ecosystems, and because the NALCMS provides a much larger land area extent than other labelled/training datasets. The 30m NALCMS maps are currently only generated every five years, by the three-country consortium, due to cloud cover, technical and resource constraints \cite{latifovic:2016}. Other users are motivated to increase the frequency of production of the NALCMS products at accuracies that can only be obtained by developing automated DL approaches. For example, the overall classification accuracy of 82\% has been achieved for Mexico using MODIS data \cite{Colditz:2010} and 76.6\% accuracy for all of North America \cite{latifovic:2012}using Landsat \cite{latifovic:2016} both using 30m NALCMS.
To achieve the goal of improving overall accuracy, this work incorporated new and modified CNNs \cite{UNet:2015,ResnetV2:2016,Xception:2017,FeedbackwardDecoding:2019} to the existing architecture, and tested a variety of more advanced state-of-the-art model extensions and their combinations \cite{ProgressiveGAN:2018,Deeplabv3+:2018}. We experimented with CNNs on two different datasets, one that uses 8-bit data from Landsat 5/7, and the second uses 16-bit data from Landsat 8, both with a spatial resolution of 30m x 30m. Data from these two sensors have similar spectral resolutions with bands in the red, green, blue (RGB) and near- and shortwave-infrared regions. After determining the best model variation on the smaller datasets, two models were trained on the larger Landsat 5/7 Prairie dataset and the Landsat 8 Lake Winnipeg dataset. Landsat 5/7 Prairie dataset model resulted in a 80.66\% overall accuracy (OA), which is equivalent to the performance of ArcGIS/Pro deep learning package \cite{arcgis:2020}, while Landsat 8 Lake Winnipeg dataset model performance reached 88.04\% OA. The Landsat 8 data increased the OA by 11.44\% compared to \cite{latifovic:2016} and is ~6-8\%. Producing models for both Landsat 5/8 and Landsat 8 data means that historical and current LULC maps can be generated back to 1984 with more frequency, consistency and accuracy than ever before.
\section{Use of DL in Semantic Segmentation and Remote Sensing}\label{section_LiteratureReview}
\subsection{Deep Learning}
In the last two decades, the use of Neural Networks (NN) and DL has grown due to a large number of advances in many research areas \cite{DL_Field:2019} such as large datasets, improvements in the Graphics Processing Unit (GPU) technology, and models, which previously took weeks or months to train, now take just a few hours or days. In 2012, the AlexNet \cite{AlexNet:2012} model was presented and won the ImageNet Large Scale Visual Recognition Challenge. It was considered a revolutionary model that first showed NNs could outperform other algorithms in image recognition and classification tasks. This new architecture was only able to be trained due to the advent of general purpose computing using GPU. Advances in CNN architecture occurred in 2014, the VGGNet model \cite{VGG:2015} and GoogleNet, also known as the Inception model \cite{InceptionV1:2014}. Recently, Francois Chollet introduced an extension of the Inception architecture called Xception \cite{Xception:2017}, and He \cite{Resnet:2015} introduced groundbreaking architecture that resembles VGGNet, but is approximately eight times deeper.
CNNs were first successful at the recognition of handwritten digits using LeNet developed by LeCun in 1998 \cite{LeNet:1998} using CPUs. DL using GPUs advanced applications in speech recognition \cite{SpeechRecognition:2017}, natural language processing (NLP) \cite{NLP:2017}, image processing \cite{ImageProcessing:2016}, computer vision \cite{ComputerVision:2016}, object detection \cite{ObjectRecnog:2017}. Remote sensing applications was a natural extension of image and object recognition \cite{StorieDeepLN:2018,HenryDeepLN:2018,DLinRemoteSensing:2019,AlhassanDeepLN2020} especially in fields like image preprocessing and classification. This work focuses on mapping of LULC, within the branch of classification in DL for remote sensing.
\begin{table}
\centering
\caption{Comparison of popular remote sensing datasets and their state-of-the-art DL models, where most of the datasets are used as benchmarks. This table includes metrics for comparison such as Overall Accuracy (OA), mean Intersection over Union (mIoU), and F1-Score.}
\label{tab_dataset_comparison}
\begin{adjustbox}{width=\columnwidth,center}
\begin{tabular}{ccccccc}
\toprule
Dataset & Model & Classes & OA & F1-Score & mIoU & Reference \\
\midrule
INRIA & ICT-Net & 1 & 97.14 & - & 80.32 & \cite{chatterjee2019semantic}\\
AIRIS & ICT-Net & 1 & - & 95.7 & 91.7 & \cite{chatterjee2019semantic}\\
LandCover.ai & ResNet-50 & 4 & 85.56 & - & - & \cite{Boguszewski_2021_CVPR}\\
ISPRS Potsdam & ResUNet-a & 5 & 91.5 & 92.9 & - & \cite{diakogiannis2020resunet}\\
ISPRS Vaihingen & TreeUNet & 5 & 90.4 & 89.3 & - & \cite{yue2019treeunet}\\
GID-C & Dual Attention Deep Fusion & 5 & 85.49 & 85.37 & 74.45 & \cite{li2021dual}\\
DeepGlobe & Deep Aggregation Net & 6 & - & - & 52.72 & \cite{Kuo_2018_CVPR_Workshops}\\
GID-F & Dual Attention Deep Fusion & 15 & 83.93 & 83.23 & 71.29 & \cite{li2021dual}\\
CLC 2018 & U-Net & 15 & 82.41 & 77.27 & - & \cite{arcgis:2020}\\
NLCD 2016 & U-Net & 16 & 78.1 & 67.69 & - & \cite{arcgis:2020}\\
GeoManitoba & ResNet-FCN & 19 & 90.46 & - & 75.66 & \cite{VictorAutomatedLULC:2019}\\
\bottomrule
\end{tabular}
\end{adjustbox}
\end{table}
\subsection{Deep Convolutional Encoder-Decoders}\label{encoder_decoder_subsection}
Many computer vision tasks cannot be solved by conventional CNNs, such as object detection or semantic segmentation. However, with architectural structural changes to the networks, they can be used in these domains. To solve these tasks, the deep convolutional encoder-decoder architecture was first introduced in \cite{FCN:2014}. The concept of the encoder-decoder is to extract features from the input using a CNN and then decode those features back to the original size using a modified CNN before classification. This concept was later improved in \cite{UNet:2015,SegNet:2016} by slightly altering existing architecture and introducing more layers to the decoder. At the same time, Deeplab introduced in \cite{Deeplabv1:2016,Deeplabv3+:2018} focused more on modifying the encoder with the CNNs such as ResNet\cite{ResnetV2:2016}, and Xception \cite{Xception:2017}. Similarly, DeepLab expanded the architecture with network extensions, such as Atrous Spatial Pyramid Pooling (ASPP) and Output Stride (OS) to capture objects and context of the features at multiple scales. At the same time, researchers starting focusing on areas other than end-to-end network architectures. For instance, some researchers focused on network extensions, which are plug-in modules that aim to improve the performance of the deep convolutional encoder-decoders without significant changes to the existing architecture. In 2015, the context aggregation extension module \cite{Context:2016} was presented to increase the sharpness of predictions by aggregating multi-scale contextual information. Additionally, Goodfellow introduced a Generative Adversarial Network (GAN) \cite{GAN:2014,SemanticGAN:2016}. GAN consist of a generator and a discriminator, where the goal of the generator is to produce an image as close to the real one as possible, and the discriminator attempts to distinguish between original and generated images. \textit{Progressive GAN}, introduced in 2017 \cite{ProgressiveGAN:2018,SemanticProgressiveGAN:2019} improved the original GAN by gradually adding layers to both generator and discriminator throughout the training, which helps with the training stability.
\subsection{CNNs in Remote Sensing}
The success of DL in image processing encouraged researchers to use it in remote sensing. The development of CNNs was mainly focused on scene classification and LULC map production. One of the first studies classified LULC using high-resolution satellite images (UC Merced dataset) \cite{DLLulcFirst:2015,DLLulcSecond:2015} due to good structural information. At that time, conventional DL algorithms performed poorly on medium-resolution satellite images (10m-30m per pixels) because of a deficiency of such structural information. In 2017, Sharma presented a patch-based CNN \cite{patch-basedCNN:2017}, which performed effectively on medium-resolution satellite images. Also, Multi-Size/Scale ResNet Ensemble (MSRE) architecture was invented by \cite{latifovic:2019,Pouliot:2021} to perform a single-pixel prediction based on an input image cropped to a specific size.
Most of the modern architectures follow the encoder-decoder structure with some minor changes. One of the first to successfully use semantic segmentation architectures for remote sensing applications were \cite{StorieDeepLN:2018,HenryDeepLN:2018,VictorAutomatedLULC:2019,AlhassanDeepLN2020}. Their solution is based on an FCN structure with different encoders ({\em i.e.} VGGNet, GoogleNet, ResNet) used on Landsat 5/7 satellite images. Also, they implemented GAN and context module extensions to boost the performance of predictions. Additionally, work presented in 2018 \cite{ModifyCNNforLULC:2018} showed that architectures that were successful in semantic segmentation on the real-world images need to be modified to get their full potential in remote sensing. Many different modifications were made to improve the performance of the architectures on satellite imagery. For example, to keep most of the structural information, a no-downsampling encoder network that relies on atrous convolution was developed \cite{NoDownsmapling:2016}. Some works centred on modifying the decoder by adding symmetric unconvoluted layers \cite{unconvolutedCNN:2018,unconvolutedCNN2:2018}. These architectures significantly improved the accuracy of generated LULC maps.
In the field of remote sensing, there are a number of datasets and state-of-the-art DL models that focus on resolving object detection and semantic segmentation problems. Table \ref{tab_dataset_comparison} compares the most popular datasets and DL algorithms from literature and software. One trend that can be seen is that the lower the number of classes, the better the performance of the models. When the number of LULC classes exceeds 10, only ResNet-FCN was shown to obtain an OA higher than 85\%. Also, other models that provide accurate results follow the U-Net architecture \cite{UNet:2015}, Faster R-CNN \cite{FasterRCNN:2015}, and Mask R-CNN \cite{MaskRCNN:2017}. The last two derive from R-CNN \cite{RCNN:2014} architecture that relies on splitting images into segments then processing each by a DCNN.
\begin{figure}
\centering
\includegraphics[width=0.75\linewidth]{maps/2010_GT.pdf}
\caption{Southern Manitoba land cover based on NALCMS labels and Landsat 5/7 data.}
\label{fig:Small_Dataset}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{maps/2012_GT.pdf}
\caption{Prairie land cover based on NALCMS labels and Landsat 5/7 data.}
\label{fig:Big_Dataset}
\end{figure}
This work is a continuation of \cite{VictorAutomatedLULC:2019}, which was able to reach OA higher than 90\% on the dataset with 19 classes. While these results are good, the work was conducted on a relatively small geographical area ($\sim$ 148,800 $km^2$). In this paper, Landsat 5/7 data was used for mapping the geographic extent of both southern Manitoba (Figure \ref{fig:Small_Dataset}) and the Prairies (Figure \ref{fig:Big_Dataset}), while Landsat 8 was used for the entire Lake Winnipeg/Churchill-Nelson watershed (not shown).
\section{Experiments, Results, and Analysis}\label{section_Results}
The experiments evaluated and compared the performance of different model variations, which were trained on the Landsat 5/7 southern Manitoba. Then, based on the results, the best model combination was selected to train models using Landsat 5/7 Prairie and Landsat 8 Lake Winnipeg watershed datasets. The model trained on the Landsat 5/7 dataset will allow us to generate LULC maps from 1984 to 2017, and the model trained on the Landsat 8 dataset will allow us to generate LULC maps starting from 2013 and onwards. Lastly, the results of all generated products are discussed, assessed, and compared to existing state-of-the-art models. The overall training process is depicted in Figure \ref{fig:process_flow}.
\begin{figure}
\centering
\includegraphics[width=0.85\linewidth]{images/Flow.pdf}
\caption[Training flowchart.]{Training flowchart, where the blue shapes correspond to data, green shapes correspond to preprocessing, red shapes correspond to models and extensions, yellow shapes correspond to model assessment, and purple shapes correspond to final generated maps.}
\label{fig:process_flow}
\end{figure}
\subsection{Evaluation Metrics}
The experiment results were evaluated using OA, precision, recall, F1-score, and critical success index (CSI)\footnote{CSI is also known as a threat score, and identical to IoU} were used to evaluate the results. The OA is a metric that calculates the ratio between the number of correctly classified pixels and a total number of pixels in the image, thus providing a general assessment of the prediction, which can be easily compared and understood. Define $n_{ij}$ as the number of pixels in the image with ground truth label $i$ and corresponding predicted label $j$. Let $t_i=\sum_{j=1}^{C}{n_{ij}}$ denote the total number of pixels labelled with label $i$, $C$ is the number of classes, $n_{ii}$ are the number of pixels correctly predicted, and $n_{ji}$ are the number of incorrectly label pixels (with respect to label $i$). \cite{VictorAutomatedLULC:2019}. The OA is defined as
\begin{equation}\label{EQ_Pixel_Accuracy}
OA = \frac{\sum^{C}_{i=1}{n_{ii}}}{\sum^{C}_{i=1}{t_{i}}}.
\end{equation}
\begin{figure}
\begin{subfigure}{0.46\textwidth}
\centering
\includegraphics[width=1\linewidth]{images/Best_model.pdf}
\caption[Structure of the best model]{}
\label{fig:Best_model_1}
\end{subfigure}
\hfill
\begin{subfigure}{0.43\textwidth}
\centering
\includegraphics[width=1\linewidth]{images/Best_model_Inside.pdf}
\caption[Description of each block]{}
\label{fig:Best_model_2}
\end{subfigure}
\caption[Network architecture diagram for best model combination.]{Network architecture diagram of the structure for the best model combination: VGGNet encoder with modified U-Net decoder and OS 4 (a), and description of each block (b).}
\label{fig:Best_model}
\end{figure}
Using the OA, the best model combination was determined, which was later trained on larger datasets to save time for the experiments. Next, we will define, precision \cite{Precision_Recall:1968}, recall \cite{Precision_Recall:1968}, F1-score \cite{FScore:1986}, and CSI \cite{CSI:1990}, where the CSI score is intersection over union (IoU) \cite{CSI_IOU:2003}. Let class $i$ be labelled a \textit{positive class}, and every other class labelled a \textit{negative class}. Then, the true positive, $TP_i$, is defined as the number of correctly predicted pixels of the positive class, true negative $TN_i$ as the number of correctly predicted pixels of the negative class, false positive $FP_i$ as the number of incorrectly predicted pixels of the positive class, and false negative $FN_i$ as the number of incorrectly predicted pixels of the negative class. Given these definitions, we can then define the following metrics:
\begin{subequations}
\begin{equation} \label{EQ_Pixel_Precision}
Precision_i = \frac{TP_i}{TP_i + FP_i};
\end{equation}
\begin{equation} \label{EQ_Pixel_Recall}
Recall_i = \frac{TP_i}{TP_i + FN_i};
\end{equation}
\begin{equation} \label{EQ_Pixel_F1-Score}
F1_i = 2 * \frac{Precision_i * Recall_i}{Precision_i + Recall_i};
\end{equation}
\begin{equation} \label{EQ_Pixel_CSI}
CSI_i = \frac{TP_i}{TP_i + FP_i + FN_i}.
\end{equation}
\end{subequations}
Both the average and weighted average of each metric were calculated as
\begin{subequations}
\begin{equation} \label{EQ_Average}
Average = \frac{1}{l}\sum^{l}_{i=1}M_i,
\end{equation}
\begin{equation} \label{EQ_WAverage}
Weighted Average = \sum^{l}_{i=1}w_i*M_i.
\end{equation}
\end{subequations}
where $l$ is the number of classes, $M_{i}$ is the metric value of the class $i$, and $w_i$ is the pixel proportion of the class $i$. Note, all the metrics were calculated on a per tile basis, and to get global dataset evaluation, each class $i$ was calculated using a weighted average.
\subsection{Comparison of Model Variations}
All model combinations were trained and validated on the southern Manitoba dataset. The experiment included the comparison of encoders, such as VGGNet, GoogleNet, ResNet, Xception, and decoders, such as FCN-8 \cite{FCN:2014}, U-Net, Feedbackward decoder \cite{FeedbackwardDecoding:2019}. Table \ref{tab_decoders_results} shows the results of the trained decoders paired with the VGGNet encoder. Based on the results, the modified U-Net decoder was chosen as the main decoder network, as it is a U-Net decoder with a reduced number of convolutions to one per upsampling layer. The chosen decoder has 3 times fewer parameters than plain U-Net with a cost of 0.04\% of OA. Table \ref{tab_encoders_results} displays the results of all implemented encoders in combination with the modified U-Net decoder. VGGNet proved to have the highest OA of 90.11\%, followed by ResNet at 89.05\%.
\begin{table}
\centering
\caption{Results of trained VGGNet encoder in combination with different decoders on the Landsat 5/7 data from the southern Manitoba.}
\label{tab_decoders_results}
\begin{tabular}{lc}
\toprule
Network & OA \\
\midrule
FCN-8 \cite{FCN:2014} & 88.28 \\
U-Net \cite{UNet:2015} & \textbf{90.15} \\
Feedbackward \cite{FeedbackwardDecoding:2019} & 89.59 \\
Modified U-Net & \textbf{90.11} \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}
\centering
\caption{Results of trained modified U-Net decoder in combination with different encoders on the Landsat 5/7 data from the southern Manitoba.}
\label{tab_encoders_results}
\begin{tabular}{lc}
\toprule
Network & OA \\
\midrule
VGGNet \cite{VGG:2015} & \textbf{90.11} \\
GoogleNet \cite{InceptionV1:2014} & 83.14 \\
Xception \cite{Xception:2017} & 88.36 \\
ResNet \cite{ResnetV2:2016} & \textbf{89.05} \\
\bottomrule
\end{tabular}
\end{table}
VGGNet and ResNet CNNs with the modified U-Net decoder were chosen as backbone models for future experiments. During which both of these networks were trained with OS extension ranging from 32 to 2, where OS is 32 is the same as the models without extension. Additionally, networks were trained with different model and architectural extensions, such as ASPP, Context Module, GAN, Progressive GAN, and DeepLabv3+ (Table \ref{tab_Extensions}). Based on the results, adversarial training had lesser or no effect at all on the models that performed well on the given dataset. Also, the ResNet Deeplabv3+ architecture was trained on our dataset and showed low performance reaching 87.43\% of OA. The model with the best performance and lowest complexity was VGGNet with modified U-Net and OS 4, which reached 92.4\% on the southern extent of the Manitoba validation dataset. As the OA of the model approaches 100\% it becomes more difficult to improve the performance by introducing changes and enhancement to the model and architecture. In our experiments, we can see that extensions, like ASPP, context module and GAN can improve the performance of the model. However, if the model already performs well, implementing or even combining them will not provide further benefit, and in some cases even aggravate the results. The best performing network architecture is depicted in Figure \ref{fig:Best_model}. The best results generated by the model based on OA are shown in Figure \ref{fig:Tiles_Landsat5_Small_Good_Results}, where the model easily recognized edges of the water bodies, rivers and roads. Similarly, the worst results based on OA are shown in Figure \ref{fig:Tiles_Landsat5_Small_Bad_Results}, where the model did predict most of the area well but missed and misclassified some parts of the roads and water bodies.
\begin{figure}
\begin{subfigure}{0.49\textwidth}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Small/annotation/95_45_tile_00399_deg.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Small/annotation/98_25_tile_00326_deg.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Small/annotation/98_63_tile_00832_deg.png}
\end{subfigure}
\vspace{0.15cm}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Small/prediction/95_45_tile_00399_deg.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Small/prediction/98_25_tile_00326_deg.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Small/prediction/98_63_tile_00832_deg.png}
\end{subfigure}
\caption{}
\label{fig:Tiles_Landsat5_Small_Good_Results}
\end{subfigure}
\hfill
\begin{subfigure}{0.49\textwidth}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Small/annotation/53_88_tile_00406_deg.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Small/annotation/60_32_tile_00511_deg.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Small/annotation/61_59_tile_00703_deg.png}
\end{subfigure}
\vspace{0.15cm}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Small/prediction/53_88_tile_00406_deg.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Small/prediction/60_32_tile_00511_deg.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Small/prediction/61_59_tile_00703_deg.png}
\end{subfigure}
\caption{}
\label{fig:Tiles_Landsat5_Small_Bad_Results}
\end{subfigure}
\caption[Examples of (a) good and (b) poor results generated on the Landsat 5/7 data from the southern Manitoba.]{Examples of (a) good and (b) poor results generated by the best model trained on the Landsat 5/7 data from the southern Manitoba. The first row are ground truth tiles, and second row are predicted tiles.}
\label{fig:Tiles_Landsat5_Small_Results}
\end{figure}
\begin{table}
\centering
\caption{OA results of trained encoder-decoder models in combination with different extensions on the Landsat 5/7 data from the southern Manitoba.}
\begin{adjustbox}{width=\columnwidth,center}
\begin{tabular}{lcccccc}
\toprule
& \textbf{OS}& 32 & 16 & 8 & 4 & 2 \\
\textbf{Network + Extension} && & & & & \\
\midrule
VGGNet \cite{VGG:2015} && 90.11 & 90.66 & 91.25 & \textbf{92.4} & 91.01 \\
VGGNet + ASPP \cite{Deeplabv3+:2018} && 90.72 & - & - & 92.22 & - \\
VGGNet + Context Module \cite{Context:2016} && 90.18 & - & - & 92.04 & - \\
VGGNet + GAN \cite{SemanticGAN:2016} && 89.99 & - & - & 92.37 & - \\
VGGNet + Progressive GAN \cite{ProgressiveGAN:2018,SemanticProgressiveGAN:2019} && 90.09 & - & - & 92.41 & - \\
ResNet \cite{ResnetV2:2016} && 89.05 & 89.72 & 90.02 & \textbf{91.42} & - \\
ResNet + ASPP \cite{Deeplabv3+:2018} && - & - & - & 91.37 & - \\
ResNet + Context Module \cite{Context:2016} && - & - & - & 90.31 & - \\
ResNet + GAN \cite{SemanticGAN:2016} && 89.26 & - & - & 91.04 & - \\
ResNet + Deeplabv3+ \cite{Deeplabv3+:2018} && - & - & - & 87.43 & - \\
\bottomrule
\end{tabular}%
\end{adjustbox}
\label{tab_Extensions}%
\end{table}
\begin{table}
\centering
\caption{Pixel percentage of each class from the Landsat 5/7 Prairie and Landsat 8 Lake Winnipeg watershed datasets.}
\label{tab_pixel percentage}
\begin{adjustbox}{width=\columnwidth,center}
\begin{tabular}{lcccc}
\toprule
& \multicolumn{2}{c}{Pixel Percentage} & \multicolumn{2}{c}{Number of Pixels} \\
Name of the class & Landsat 5/7 & Landsat 8 & Landsat 5/7 & Landsat 8 \\
\midrule
No data & 32.78 & 50.88 & 1031407726 & 1208979489 \\
Temperate or sub-polar needleleaf forest & 15.52 & 4.26 & 488262071 & 101124573 \\
Sub-polar taiga needleleaf forest & 0.95 & 0.02 & 30020347 & 568110 \\
Temperate or sub-polar broadleaf deciduous forest & 5.42 & 4.47 & 170429106 & 106208424 \\
Mixed forest & 3.06 & 2.21 & 96116915 & 52604376 \\
Temperate or sub-polar shrubland & 5.84 & 1.52 & 183620158 & 36149337 \\
Temperate or sub-polar grassland & 5.00 & 4.14 & 157337200 & 98413645 \\
Sub-polar or polar shrubland-lichen-moss & 0.13 & 1.80E-04 & 4117649 & 4270 \\
Sub-polar or polar grassland-lichen-moss & 0.35 & 1.70E-03 & 11029343 & 40467 \\
Sub-polar or polar barren-lichen-moss & 0.07 & 7.49E-06 & 2302774 & 178 \\
Wetland & 4.97 & 3.76 & 156313523 & 89369338 \\
Cropland & 13.75 & 20.27 & 432590674 & 481744136 \\
Barren land & 1.10 & 0.51 & 34526845 & 12135657 \\
Urban and built-up & 0.85 & 1.12 & 26771188 & 26556386 \\
Water & 7.30 & 4.96 & 229553602 & 117838940\\
Snow and ice & 0.03 & 0.02 & 936557 & 534825 \\
Cloud & 2.89 & 1.85 & 90890767 & 43874765 \\
\bottomrule
\end{tabular}
\end{adjustbox}
\end{table}
\begin{table}
\centering
\caption{Per-class assessment of the models trained on the Landsat 5/7 Prairie and Landsat 8 data from the Lake Winnipeg watershed.}
\label{tab_per_class}
\begin{adjustbox}{width=\columnwidth,center}
\begin{tabular}{lcccc|cccc}
\toprule
& \multicolumn{4}{c}{Landsat 5/7} & \multicolumn{4}{c}{Landsat 8} \\
Name of the class & CSI & Precision & Recall & F1 & CSI & Precision & Recall & F1 \\
\midrule
No data & 99.76 & 99.78 & 99.98 & 99.88 & 99.99 & 99.99 & 99.99 & 99.99 \\
Temperate or sub-polar needleleaf forest & 74.99 & 84.15 & 87.33 & 85.71 & 76.21 & 84.65 & 88.44 & 86.5 \\
Sub-polar taiga needleleaf forest & 42.78 & 62.8 & 57.29 & 59.92 & 41.62 & 67.21 & 52.22 & 58.77 \\
Temperate or sub-polar broadleaf forest & 57.41 & 75.7 & 70.39 & 72.95 & 71.37 & 83.11 & 83.48 & 83.29 \\
Mixed forest & 40.78 & 63.27 & 53.42 & 57.93 & 56.03 & 74.1 & 69.67 & 71.82 \\
Temperate or sub-polar shrubland & 49.67 & 65.29 & 67.5 & 66.38 & 48.61 & 70.48 & 61.04 & 65.42 \\
Temperate or sub-polar grassland & 66.33 & 78.9 & 80.63 & 79.76 & 81.44 & 90.12 & 89.43 & 89.77 \\
Sub-polar or polar shrubland-lichen-moss & 21.68 & 55.47 & 26.24 & 35.63 & 17.62 & 54.06 & 20.73 & 29.97 \\
Sub-polar or polar grassland-lichen-moss & 42.36 & 57.97 & 61.13 & 59.51 & 21.25 & 59.3 & 24.87 & 35.04 \\
Sub-polar or polar barren-lichen-moss & 37.76 & 53.65 & 56.04 & 54.82 & 37.27 & 66.13 & 46.07 & 54.31 \\
Wetland & 58.08 & 73.75 & 73.21 & 73.48 & 75.76 & 85.78 & 86.64 & 86.21 \\
Cropland & 87.84 & 92.61 & 94.46 & 93.53 & 93.12 & 95.77 & 97.11 & 96.44 \\
Barren land & 72.2 & 82.01 & 85.79 & 83.85 & 70.79 & 84.13 & 81.7 & 82.9 \\
Urban and built-up & 43.61 & 63.08 & 58.56 & 60.74 & 57.79 & 74.52 & 72.01 & 73.24 \\
Water & 86.36 & 94.45 & 90.98 & 92.68 & 88.25 & 95.04 & 92.51 & 93.76 \\
Snow and ice & 58.75 & 67.31 & 82.21 & 74.02 & 69.57 & 86.6 & 77.96 & 82.05 \\
Cloud & 97.58 & 97.76 & 99.81 & 98.78 & 91.51 & 94.81 & 96.33 & 95.56 \\
\bottomrule
\end{tabular}
\end{adjustbox}
\end{table}
\subsection{Training Model on a Large Dataset}
\subsubsection{Landsat 5/7}
After determining that the VGGNet encoder with modified U-Net decoder and OS 4 was the best, this model combination was then used to train models on the Landsat 5/7 Prairie dataset. Table \ref{tab_pixel percentage} (left) shows pixel proportion of each class, Table \ref{tab_per_class} (left) shows the per-class assessment of the Landsat 5/7 Prairie dataset and Table \ref{tab_best_model_datasets} shows a global assessment of the predicted map. Based on these results, LULC classes that performed well were found to have a higher percentage of pixels or representation than other LULC classes which had a small number of pixels. It has been documented that deep learning models need a large number of class examples (pixels) to achieve good performance for that class \cite{johnson2019survey}. Forest-related classes with low presence were the classes most often misclassified. Temperate or sub-polar needleleaf forest class and the urban and built-up classes were misclassified with cropland. Moreover, prediction of the cloud class had extremely high accuracy because the clouds were discriminated based on labels generated using a thresholding method \cite{Otsu:1979}; the network did not have difficulties generating this cloud class due to it being a straightforward function with sufficient data examples. Overall, the performance of the model reached 80.66\% OA on the validation dataset, where only a few tiles had no data pixels. No data is the background class that covers empty areas on the image. The accuracy presented in this dataset is lower due to the size and amount of extractable features presented in this dataset; particularly the discrimination of clouds. It was noted that the cloud labels generated by the Otsu thresholding method did not classify some of the clouds, especially transparent ones, and misclassified bright urban areas as clouds. This model showed the performance to be slightly better than published by \cite{latifovic:2016} but equivalent to the models present in ArcGIS/Pro deep learning package \cite{arcgis:2020}. The best results generated by the model based on OA are shown in Figure \ref{fig:Tiles_Landsat5_Big_Good_Results}, where the model had no difficulty in predicting tiles with a low number of classes, and perfectly segmented water, forests and croplands. Similarly, the worst results based on OA are shown in Figure \ref{fig:Tiles_Landsat5_Big_Bad_Results}, where the model missed some of the classes on the tiles with a high number of them, especially grassland, lichen-moss related classes and roads.
\begin{figure}
\begin{subfigure}{0.49\textwidth}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Big/annotation/93_41_baseline__x=47040_y=35840.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Big/annotation/93_69_baseline__x=35840_y=7168.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Big/annotation/95_11_baseline_lr_dx=21504_dy=38528.png}
\end{subfigure}
\vspace{0.15cm}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Big/prediction/93_41_baseline__x=47040_y=35840.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Big/prediction/93_69_baseline__x=35840_y=7168.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Big/prediction/95_11_baseline_lr_dx=21504_dy=38528.png}
\end{subfigure}
\caption{}
\label{fig:Tiles_Landsat5_Big_Good_Results}
\end{subfigure}
\hfill
\begin{subfigure}{0.49\textwidth}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Big/annotation/44_98_baseline__x=15456_y=56672.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Big/annotation/53_13_baseline_lr_x=28896_y=25088.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Big/annotation/55_38_baseline__x=39648_y=7168.png}
\end{subfigure}
\vspace{0.15cm}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Big/prediction/44_98_baseline__x=15456_y=56672.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Big/prediction/53_13_baseline_lr_x=28896_y=25088.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_5_Big/prediction/55_38_baseline__x=39648_y=7168.png}
\end{subfigure}
\caption{}
\label{fig:Tiles_Landsat5_Big_Bad_Results}
\end{subfigure}
\caption{Examples of (a) good and (b) poor results generated by the best model trained on the Landsat 5/7 Prairie data. The first row are ground truth tiles, and second row are predicted tiles.}
\label{fig:Tiles_Landsat5_Big_Results}
\end{figure}
\subsubsection{Landsat 8}
Next, the same VGGNet encoder with modified U-Net decoder and OS 4 model was trained on the Landsat 8 Lake Winnipeg watershed which has higher radiometric resolutions (16-bit vs 8-bit of Landsat 5/7). The trend in the results provided by the model trained on the Landsat 8 dataset is similar to the Landsat 5/7 model and can be seen in Tables \ref{tab_per_class} and \ref{tab_best_model_datasets} (right). The OA of the predicted map increased by 8\%, and improvements in accuracy were noticed across most of the LULC classes. Since the land cover classes remained consistent, we speculate that these improvements are achieved due to the increased radiometric variability provided by the 16-bit instead of the 8-bit dataset, which made it easier to distinguish features due to a larger range of values. Recall, the cloud detection for Landsat 8 was trained using data generated from the Landsat 8 QA band provided by the sensor and not the thresholding method. Despite the results being lower than Landsat 7, the performance on the cloud class is still one of the best for this model. This model showed the performance higher than results reported by \cite{latifovic:2016} (11.44\% increase) and the models present in ArcGIS/Pro deep learning package \cite{arcgis:2020} by about ~6-8\%, and on the level of the state-of-the-art models designed for datasets with the lower amount of classes.
\begin{figure}
\begin{subfigure}{0.49\textwidth}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_8/annotation/96_56_shifted__x=18704_y=11536.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_8/annotation/96_63_shifted__x=18480_y=56336.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_8/annotation/98_32_shifted__x=30464_y=47488.png}
\end{subfigure}
\vspace{0.15cm}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_8/prediction/96_56_shifted__x=18704_y=11536.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_8/prediction/96_63_shifted__x=18480_y=56336.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_8/prediction/98_32_shifted__x=30464_y=47488.png}
\end{subfigure}
\caption{}
\label{fig:Tiles_Landsat8_Big_Good_Results}
\end{subfigure}
\hfill
\begin{subfigure}{0.49\textwidth}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_8/annotation/53_57_baseline_lr_x=27776_y=60480.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_8/annotation/61_02_baseline_lr_x=18144_y=39200.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_8/annotation/68_75_baseline_lr_x=20608_y=6272.png}
\end{subfigure}
\vspace{0.15cm}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_8/prediction/53_57_baseline_lr_x=27776_y=60480.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_8/prediction/61_02_baseline_lr_x=18144_y=39200.png}
\end{subfigure}
\begin{subfigure}{0.31\textwidth}
\includegraphics[width=1\linewidth]{images/Tiles/Landsat_8/prediction/68_75_baseline_lr_x=20608_y=6272.png}
\end{subfigure}
\caption{}
\label{fig:Tiles_Landsat8_Big_Bad_Results}
\end{subfigure}
\caption{Examples of (a) good and (b) poor results generated by the best model trained on the Landsat 8 data from the Lake Winnipeg watershed. The first row are ground truth tiles, and second row are predicted tiles.}
\label{fig:Tiles_Landsat8_Big_Results}
\end{figure}
\begin{table}
\centering
\caption{Global assessment of the models trained on the Landsat 5/7 Prairie and Landsat 8 data from the Lake Winnipeg watershed.}
\label{tab_best_model_datasets}
\begin{tabular}{lcc}
\toprule
Evaluation Metric & Landsat 5/7 & Landsat 8 \\
\midrule
OA & 80.66 & 88.04 \\
\midrule
\multicolumn{3}{c}{Average} \\
CSI & 55.19 & 62.39 \\
Precision & 68.72 & 79.11 \\
Recall & 67.35 & 71.26 \\
F1-Score & 67.63 & 74.07 \\
\midrule
\multicolumn{3}{c}{Weighted Average} \\
CSI & 71.51 & 82.68 \\
Precision & 82.32 & 89.92 \\
Recall & 82.44 & 90.06 \\
F1-Score & 82.33 & 89.97 \\
\bottomrule
\end{tabular}
\end{table}
\newpage
|
1,108,101,563,216 | arxiv | \section{}
In rare-earth compounds, the degrees of freedom of the 4$f$ ions are collectively involved at low temperature to form ordered states. The most common kind of order is the magnetic one that exists in a number of varieties. The orbital order is a less common situation, where only the orbital degeneracy of the 4$f$ ion is involved, without the stabilization of magnetic moments. It requires the orbital degeneracy to survive the crystal-field effect, which means the 4$f$ ions sites are of high symmetry. As the highest symmetry is that of the octaedra, cubic systems are the most favorable scene for the development of orbital related effects \cite{MorinSchmitt1990}. There, the lifting of the degeneracy is equivalent to the emergence of 4$f$ electric quadrupole moments and the associated order are referred to as "quadrupolar". By analogy with the ferromagnetism, the order that doesn't change the crystal periodicity is called ferroquadrupolar. One can also consider a state where the 4$f$ quadrupoles are changing from site to site, thus defining an "antiferroquadrupolar" order (AFQ), analog to the antiferromagnetism. For more than two decades, the most cited example of such an antiferroquadrupolar order has been CeB$_6$ phase II.\\
This system crystallizes within the CaB$_6$-type structure and displays complex magnetic phase diagrams \cite{Kawakami1980, Fujita1980}. CeB$_6$ orders antiferromagnetically at T$_N$ = 2.4 K (phase III) and its most intriguing feature is the occurrence of a non-magnetic state (phase II) below T$_Q$= 3.2 K, before entering the paramagnetic state (phase I). In phase II, a magnetic field induced $\langle\frac{1}{2}\frac{1}{2}\frac{1}{2} \rangle$ antiferromagnetism is detected via neutron diffraction \cite{Effantin1985}. Latter X-ray scattering experiments \cite{Nakao2001, Yakhou2001, Matsumura2009, Tanaka2004} have confirmed a non-magnetic $\langle\frac{1}{2}\frac{1}{2}\frac{1}{2}\rangle$ wave-vector, even in zero field.\\
Inelastic neutron scattering \cite{Zirngiebl1984} results are consistent with a crystal-field splitting of the Ce$^{3+}$, J = 5/2, multiplet resulting in a $\Gamma_8^+$ quadruplet ground state well separated (530 K) from the $\Gamma_7^+$ doublet. As this ground state has an orbital degeneracy, Effantin \textit{et al.} \cite{Effantin1985} interpreted the puzzling properties of phase II as resulting from an AFQ state.\\
Up to now, no solidly based model has been given for the microscopic organization of this phase. Some authors suggest a $\Gamma_3^+$ AFQ ordering \cite{Luthi1984} while, for others, it is of the $\Gamma_5^+$ type. The recurrent scheme is a $\Gamma_5^+$ AFQ state \cite{Sakai1997}, where a single quadrupolar component, $P_{xy}$, is propagated by the $[\frac{1}{2}\frac{1}{2}\frac{1}{2} ]$ wave-vector. This model doesn't preserve the cubic symmetry and, due to the coupling of the quadrupoles with the lattice, a $\Gamma_3^+$ ferroquadrupolar component and a macroscopic distortion should simultaneously develop. Facing such inconsistencies and a confusing mass of experimental data, we thought that a crucial clarification would come from an accurate investigation of phase II symmetry.\\
This can be achieved from macroscopic experiments on a single crystal. For a rare-earth based system, magnetic measurements are first to consider. In cubic crystals, an accurate determination of the first-order magnetic susceptibility can reveal a symmetry lowering : any deviation from the cubic symmetry will result in an anisotropic susceptibility. Due to the magnetoelastic coupling, the lattice necessarily reflects a symmetry lowering. The detection of the associated strain can be very sensitive, capacitance dilatometer being able to detect relative changes as small as 10$^{-8}$. The difficulty, when investigating a sample undergoing a symmetry lowering, is that it divides into domains, mixing their properties. This can be solved using a magnetic field of appropriate direction, in order to coerce the sample to a single domain state. If the anisotropy of the susceptibility has been previously determined, the optimal field direction is easily defined. All the reported experiments where made using single crystals in the shape of rectangular platelets (6x4x1 mm$^3$) previously used for neutron scattering experiments \cite{Plakhty2005}.\\
\begin{figure}
\includegraphics[width=\columnwidth]{FIG1.eps}
\caption{\label{chiplot} Magnetic susceptibilities of CeB$_6$ along the three high symmetry directions of the cube, as deduced from Arrott's plots in descending field.}
\end{figure}
The investigation of the magnetic susceptibility is based on magnetization measurements along the three main cubic directions ([001], [110] and [111]) in fields up to 8 T. The susceptibilities, $\chi_{001}$, $\chi_{110}$ and $\chi_{111}$, are extracted using the Arrott method \cite{Arrott1957} applied to the descending field data : only the field selected domains should contribute to the magnetization. Then, considering the tetragonal and trigonal symmetry lowerings \cite{Amara2010} cases, the measured susceptibility can be related to the normal susceptibilities \cite{Amara2010}, $\chi_{\|}$ and $\chi_{\bot}$, respectively parallel and perpendicular to the preserved, fourfold or threefold, axis of the system. One can thus identify the symmetry both via hierarchical and quantitative relations between $\chi_{001}$, $\chi_{110}$ and $\chi_{111}$.\\
Fig. \ref{chiplot} shows the thermal dependence of these three susceptibilities at temperatures below 4.5 K.
The transition between phases I and II is difficult to discern, both phases displaying a paramagnetic decrease of the susceptibility with the temperature. Within the precision of the measurements, typically 2\%, no difference between the three susceptibilities and, therefore, no deviation from the cubic symmetry can be detected.
In the antiferromagnetic state, the three susceptibilities are well separated, with a dominant $\chi_{001}$, an intermediate $\chi_{110}$ and a minimal $\chi_{111}$. This sequence is expected in case of a tetragonal symmetry lowering, with a maximum susceptibility along the fourfold axis ($\chi_{\|}>\chi_{\bot}$). In quantitative terms, this translates into $\chi_{110}=\frac{1}{4}\chi_{001}+\frac{3}{4}\chi_{111}$, which is precisely what is observed on Fig. \ref{chiplot}. This is consistent with the multiaxial antiferromagnetic models \cite{Effantin1985} with magnetic moments perpendicular to the fourfold axis.\\
\begin{figure}
\includegraphics[width=\columnwidth]{FIG2.eps}
\caption{\label{VarAngl} Relative changes in length along [001] (a and b) and [111] (c) while rotating a constant field in the (110) plane in the ranges of phase III (a), phase II (b and c) and phase I (b and c). The zero angle $\alpha$ correspond to a field aligned with the measured crystal direction. The vertical lines indicate high symmetry directions in the (110) plane.}
\end{figure}
Beyond the susceptibility analysis, one can further track a symmetry lowering using a high sensitivity dilatometer. For this purpose, we used the magnetostriction setup of the "Institut N\'eel", which is based on a capacitance cell that can be rotated in a 0-6 T horizontal magnetic field, in the 2-300 K temperature range. The sensitivity, in terms of relative change in length, is better than 10$^{-7}$. The alignment of the sample inside the setup is defined by the the choice of the crystal horizontal plane, which contains $[mnp]$, the probed crystal's direction, and $[ijk]$ the direction of the applied field. For a reference length $l_0$, a current length $l$, the relative change in length, and its geometrical conditions, can be noted as :
$ {}^{mnp}\lambda _{ijk} = \Delta l/l_0 = (l-l_0)/l_0$.\\
In the following, the default reference length $l_0$ is taken at T = 5 K and in zero field. All the striction phenoma are analyzed in terms of cubic normal strain modes \cite{Callen1963}. For the most common cases of a tetragonal or trigonal symmetry lowering, the strain ${}^{mnp}\lambda$ of a single domain depends on three quantities: $\varepsilon^\alpha$, the volume strain, $\varepsilon^\gamma$, the tetragonal strain, and the trigonal one $\varepsilon^\varepsilon$.\\
To detect a symmetry lowering, one can take advantage of the adjustable field direction : rotating the sample in a constant field will successively select different domains, inducing changes in the probed sample length. Fig. \ref{VarAngl} shows such measurements while rotating the field in the (110) plane, where all high symmetry axes of the cube are represented. In Fig. \ref{VarAngl} (a) and \ref{VarAngl} (b), the probed sample direction is [001], which is ideal for detecting a tetragonal mode. Fig. \ref{VarAngl} (a) shows what happens in the antiferromagnetic phase, under a 0.5 T applied field : spectacular, square like, changes in the sample length are observed. This is expected in case of a tetragonal symmetry lowering with maximum susceptibility along the fourfold axis : one goes from a single domain with [001] fourfold axis, for low ${\alpha}$ values, to a $\alpha = \pm 90 ^\circ $ state where the two other domains are selected, with fourfold axes [010] and [100]. The amplitude of the jump is directly related to the $\varepsilon^\gamma$ spontaneous strain of the antiferromagnetic state. Fig. \ref{VarAngl} (b) gives the angular dependence under the maximum field (6 T) in phases II (T = 3.2 K) and phase I (T = 6 K). These curves show no anomaly but a continuous, sine-type, change in the [001] length, which is typical of the paramagnetic state. The twisted phase II curve is an effect of higher-order susceptibilities in conjunction with a slight sample's misorientation. Fig. \ref{VarAngl} (c) shows the angular variations, in phases II and I, of the [111] length, which is adapted to detect a trigonal mode. Here again, the continuous, undulating curves give no evidence of domains. Higher-order effects are clearly present in phase II, with a resulting double bump around $\alpha = 90 ^\circ $.\\
The temperature dependence of the strain modes has been derived from a sequence of ${}^{mnp}\lambda _{ijk}$ measurements under a constant (0.5 T) field (Fig. \ref{Epsilon(T)}). It is only within the antiferromagnetic range that a spontaneous, tetragonal, strain is detected. The corresponding small accident on the trigonal curve is the result of an imperfect sample alignment : a small fraction of the tetragonal strain is detected. In absence of a trigonal mode, the volume effect $\varepsilon^\alpha$ (inset of Fig. \ref{Epsilon(T)}) can be safely derived from the length of a threefold axis : both transitions are visible, with a more pronounced thermal expansion in phase II, while phase III shows a substantial negative effect. This agrees with previous measurements by Sera \textit{et al.} \cite{Sera1988}.\\
\begin{figure}
\includegraphics[width=\columnwidth]{FIG3.eps}
\caption{\label{Epsilon(T)} Temperature variation of the $\gamma$ and $\varepsilon$ strains, as obtained from measurements under a 0.5 T magnetic field. The inset shows the volume $\alpha$ strain deduced from the length of [111], in zero field.}
\end{figure}
The magnetic and magnetostriction data are consistent with a cubic symmetry in phases II and I. In this cubic context, the magnetostriction setup can be used to probe the magnetoelastic susceptibilities of the system with respect to a field enforced tetragonal (field along a fourfold axis) or trigonal (field along a threefold axis) symmetry lowering. This is the parastriction analysis of the paramagnetic state magnetoelastic properties \cite{Morin1980A}. At the lowest order, the strains vary quadratically with the applied field : $\varepsilon^{\mu} = \chi^{\mu} H^2$, where $\chi^{\mu}$ is a magnetoelastic susceptibility and $\mu$ states for $\gamma$ or $\varepsilon$. In the harmonic treatment of 4$f$ magnetoelasticity, $\chi^\mu$ is proportional to the quadrupolar field-susceptibility $\chi^{\mu}_Q$ of the same symmetry. $\chi^\mu(T)$ can be deduced by linearizing the constant temperature strain as function of the squared applied field.
\begin{figure}
\includegraphics[width=\columnwidth]{FIG4.eps}
\caption{\label{Parastrict} Examples of parastriction curves in phases I and II, at T = 2.7 and 5 K, measured for the $\gamma$ and $\varepsilon$ symmetry lowering modes. Upper part : squared field dependence of the volume mode. Lower part : the tetragonal and trigonal strains.}
\end{figure}
Fig. \ref{Parastrict} shows examples of such curves in phase II (T = 2.7 K) and at the field crossing from phase I into phase II (T = 5 K). It appears that the latter has a surprisingly large volume dependence on the applied field (Fig. \ref{Parastrict}, upper part). On the $\varepsilon^\gamma$ parastriction curve, the transition from phase I to II is almost unnoticeable, whereas for the $\varepsilon$ mode, the negative slope is amplified in phase II at T = 5 K (Fig. \ref{Parastrict}, lower part). No hysteresis is observed in these curves, which confirms the absence of domain effects. In both phases, the absolute value of the susceptibility decreases with the temperature. This is expected if the degrees of freedom responsible for the field induced strains are not ordered. Quadrupoles are considered as the dominant actors in the parastriction of cubic rare-earth compounds, which results in a quadratic Curie-like behaviour for $\chi^\mu(T)$. To test this behaviour, $1/\sqrt{\chi^\mu}$ is represented as a function of T in Fig. \ref{CondParastrict}. In both phases, the curves seem indeed to vary linearly with the temperature. In phase II, the data for the $\gamma$ and $\varepsilon$ modes are almost superimposed. As these measurements mix elastic and magnetic properties, such an identity for different symmetry modes is fortuitous.\\
\begin{figure}
\includegraphics[width=\columnwidth]{FIG5.eps}
\caption{\label{CondParastrict} Thermal dependencies of $1/\sqrt{\chi^\gamma}$ and $1/\sqrt{\chi^\varepsilon}$ in the range of phase I an II. The lines are just guides to the eye and the inset details the data for phase II.}
\end{figure}
No symmetry lowering is detected in phase II, which discards non cubic models as the one of Ref. \cite{Sakai1997}. While a true AFQ state needs to be cubic, the parastriction results give evidence against the ordering of the quadrupoles. This is consistent with the elastic constants measurements \cite{Luthi1984}, probes of the quadrupoles stress-susceptibilities, which, at least for the trigonal mode, behave as if quadrupoles where not ordered. Nevertheless, we can deepen the analysis by specifying the characteristics of a cubic AFQ order. As the transition at T$_Q$ is second order, the ordered state should involve a single quadrupolar representation, $\Gamma_3^+$ ($\gamma$) or $\Gamma_5^+$ ($\varepsilon$), for a single wave-vector star. To obtain a high symmetry structure, stable down to 0 K (above 2.5 Tesla), all the rare-earth ions should be equivalent by symmetry.\\
Considering the $\Gamma_3^+$ representation, the emergence of a 4$f$ quadrupole is equivalent to a local tetragonal symmetry lowering : in a polar representation, the 4$f$ electronic distribution is stretched (or squeezed) along one of the fourfold axes of the cube. There are then three symmetry equivalent $\Gamma_3^+$ modes for the 4$f$ ion.
As for the $\Gamma_5^+$ representation, the quadrupoles develop with a local trigonal symmetry lowering, which means that there are four possible 4$f$ modes, one for each threefold axis. The cubic symmetry is preserved if all the modes, three for $\Gamma_3^+$ or four for $\Gamma_5^+$, are equally represented in the cubic AFQ cell. The smallest cell one can consider is built from the 8 sites at the vertexes of a cube, which is enough to describe structures with $\langle\frac{1}{2}00\rangle$, $\langle\frac{1}{2}\frac{1}{2}0\rangle$ and $\langle\frac{1}{2}\frac{1}{2}\frac{1}{2}\rangle$ wave-vectors. On this cell, it is clearly impossible to equally represent the three $\Gamma_3^+$ modes : there exists no cubic $\Gamma_3^+$ AFQ structure for these wave-vectors. This is not the case for $\Gamma_5^+$ and cubic AFQ states are possible for the $\langle\frac{1}{2}00\rangle$ and $\langle\frac{1}{2}\frac{1}{2}0\rangle$ wave-vectors. However, this is strictly impossible for $\langle\frac{1}{2}\frac{1}{2}\frac{1}{2}\rangle$, the only wave-vector that is detected in phase II. This star reduces to a single branch which defines only two 4$f$ modes, with opposite sign. An object that alternates in sign from site to site, while preserving the cubic symmetry, necessarily belongs to a unidimensional representation of the cube. Among the even $O_h$ group representations, only $\Gamma_2^+$ allows a change in sign. To find an electric multipole transforming according to $\Gamma_2^+$, one has to go up to 64 poles, which means a sixth-order operator in $\bm{J}$ components. For J = 5/2, these hexacontatetrapoles cancel : no 4$f$ electric multipole can describe the cubic $\langle\frac{1}{2}\frac{1}{2}\frac{1}{2}\rangle$ ordering of CeB$_6$.\\
To solve this puzzle, one has to consider alternative objects, transforming according to one of the unidimensional representations of $O_h$. In a cage system, as suggested by Kasuya \cite{Kasu1998}, one may consider the dynamic decentering of the 4$f$ ion, already present above T$_Q$ as a Jahn-Teller effect. Below T$_Q$ there might be a symmetry breaking in the spatial distribution of the Ce ion, that would alternate from site to site. In contrast with the 4$f$ electronic configuration, the related electric multipoles can be of odd parity : in addition to $\Gamma_2^+$, they may also transform according to the $\Gamma_1^-$ or $\Gamma_2^-$ representations.\\
The authors are indebted to Dr L.P. Regnault, who provided us with the sample and critically read the manuscript.
|
1,108,101,563,217 | arxiv | \section{Introduction}
\label{intro}
A prevalent view in science is that all phenomena in the universe can ``in principle'' be explained using fundamental laws of physics. This paradigm, called {\em reductionist hypothesis}, encouraged search for microscopic laws that led to fascinating discoveries in quantum mechanics and particle physics~\cite{Kane:book:Particle}. Buoyed by the success of these discoveries, some physicists are looking for a reductionist framework that can explain all the physical phenomena of the universe. This holy grail is referred to as {\em theory of everything (TOE), final theory, ultimate theory}, and {\em master theory}~\cite{Weinberg:book:dream,Hawking:book:TOE}. The aforementioned viewpoint has many champions and supporters, but it has also invited criticisms as descried below.
The degree of criticism and support to the reductionist paradigm vary. For example, Weinberg~\cite{Weinberg:book:dream} argues strongly in favour of reductionism, and claims that all scientists, including economists, practice reductionism. According to Weinberg, ``it saves scientists from wasting their ideas that are not worth pursuing'', and/or provides stronger theoretical basis for their hypothesis. Refer to \cite{Weinberg:book:dream,Hawking:book:TOE} for more references in support of reductionism.
In a somewhat sharp criticism, Anderson~\cite{Anderson:Science1972} argued that ``the reductionism hypothesis does not by any means imply a `constructionist' one: the ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe''. Further he agues that if the starting point of a field Y is field X, then it does not mean that all the laws of Y are ``just applied X''. He goes on the illustrate the above viewpoint by showing how the ideas of broken symmetries (apart from fundamental laws) help explain diverse phenomena of condensed matter physics.
In another article critical of reductionism, Laughlin and Pines~\cite{Laughlin:PNAS2000} write ``The emergent physical phenomena regulated by higher organizing principles have a property, namely their insensitivity to microscopics, that is directly relevant to the broad question of what is knowable in the deepest sense of the term.'' They further argue, ``Rather than a Theory of Everything we appear to face a hierarchy of Theories of Things, each emerging from its parent and evolving into its children as the energy scale is lowered.'' Also refer to Laughlin~\cite{Anderson:book:More,Laughlin:book}.
Another set of illustrations on limitations of reductionism are as follows. The letters of the book do not convey the story of a book. A combination of words, paragraphs, and chapters that describe subplots and plots makes the story. Similarly, music and paintings cannot be appreciated by just focussing on musical notes and photon packets; rather, they are complex hierarchical structures with notes and colours appearing at the bottom-most layer. The aesthetics and ecology of a building is impossible to derive from the properties of bricks and mortar. A complex computer programs is a hierarchical structure with program statements, functions, data structures, and their combinations (called {\em classes}); it is very difficult to decipher the functionality of a program if we focus only on the program statements. Carrying the analogy to physics, though every macroscopic physical system is made of electrons and protons, its macroscopic properties follow from the complex organization of different things. For the Earth, we need to focus on the macroscopic objects like atmosphere, oceans, lakes, land, life, etc., rather than electrons and protons that make them.
After so many discussion by eminent scientists, it appears futile to write more on this topic. However in the present article, I provide several interesting examples of hydrodynamic laws (a macroscopic description) that cannot be conveniently derived using the microscopic counterpart, for example, kinetic theory. These examples provide much simpler comparison between microscopic and macroscopic laws, in comparison to more complex ones involving stars, planets, biology, society, etc. The present article essentially advances the viewpoint that not all macroscopic phenomena can be explained from microscopic perspectives \cite{Anderson:Science1972,Laughlin:PNAS2000}.
\section{Kinetic theory and hydrodynamics}
\label{sec:KT}
In kinetic theory, we deal with a large number of particles (say $N$) that are specified by their position (${\bf r}$) and velocity (${\bf u}$). These particles are represented as a point in $6N$-dimensional phase space whose coordinates are $(x_a, y_a, z_a, p_{x,a}, p_{y,a}, p_{z,a})$, where $a$ is the particle label; or as $N$ points in a six-dimensional $\mu$-space whose coordinates are $(x, y, z, p_x, p_y, p_z)$. The density of these points in $\mu$-space is called {\em distribution function}, and it is denoted by $f({\bf r,u}, t)$~\cite{Choudhuri:book:Fluids}. The Boltzmann equation of kinetic theory describes the evolution of the distribution function, and it is the starting point for many works of statistical physics~\cite{Lifshitz:book:Physical_Kinetics,Choudhuri:book:Fluids,Liboff:book}. Kinetic theory successfully describes many phenomena---thermodynamics; phase transitions; observed properties of gas, liquids, polymers; etc.
On the other hand, hydrodynamic description involves real-space density $\rho({\bf r})$, velocity ${\bf u(r)}$, and internal energy $e({\bf r})$~\cite{Landau:book:Fluid}. The equations of these variables were derived in continuum framework by Euler, Navier, Stokes, and others. These equations are essentially Newton's laws of motion for fluid elements in the flow. Here, the field variables are averaged quantity over many microscopic particles. This is called {\em continuum approximation}. Note however that hydrodynamic equations can be derived using using kinetic theory. An averaging of the Boltzmann equation (with collision terms) and its various moments yields equations for $\rho({\bf r})$, ${\bf u(r)}$, and $e({\bf r})$~\cite{Lifshitz:book:Physical_Kinetics,Choudhuri:book:Fluids,Liboff:book}. Such derivations are popular among the astro- and plasma physicists.
In the following discussion we will describe several important hydrodynamic laws---Kolmogorov's theory of turbulence, irreversibility in turbulence, accelerated diffusion in turbulence, dynamic pressure, etc., which could be treated as macroscopic laws since they are derived using a multiscale description of hydrodynamic equations. We show in the next several sections that the above laws cannot be derived conveniently starting solely from kinetic theory. As far as we know, no one provided such derivations from the first principles. Note that even derivation of incompressible hydrodynamics from kinetic theory itself is quite difficult~\cite{Bisi:JPA2014}.
\section{Multiscale energy transfers and flux in hydrodynamic turbulence}
Many natural (astrophysical and geophysical) and engineering flows are turbulent. Generic features among them are---energy feed at the large scales, and energy flow to smaller and smaller scales that finally gets converted to heat. See Fig.~1 for an illustration. This multiscale feature has been propounded by Richardson, Taylor, Prandtl, Kolmogorov, and others~\cite{Kolmogorov:DANS1941Dissipation,Kolmogorov:DANS1941Structure,Frisch:book,Pope:book,Lesieur:book:Turbulence,McComb:book:Turbulence}. According to Kolmogorov, in incompressible hydrodynamic turbulence forced at large scales, the energy flux at the intermediate scale is constant ($\epsilon_u$), while the velocity fluctuations $u_l \sim (\epsilon_u l)^{1/3}$. The corresponding energy spectrum is $E_u(k) = K_\mathrm{Ko} \epsilon_u^{2/3} k^{-5/3}$, where $K_\mathrm{Ko}$ is Kolmogorov's constant, and $k$ is wavenumber. The multiscale energy transfer of Fig.~1 has been derived both in real space and Fourier space formulation of hydrodynamic turbulence~\cite{Kolmogorov:DANS1941Dissipation,Kolmogorov:DANS1941Structure,Frisch:book,Pope:book,Lesieur:book:Turbulence,McComb:book:Turbulence}.
\begin{figure
\centering
\includegraphics[width=0.8\linewidth]{turb.pdf}
\caption{Schematic diagrams illustrating energy transfers in three-dimensional hydrodynamic turbulence. The energy supplied at large scales cascades to the inertial range and then to the dissipative range. }
\label{fig:NS}
\end{figure}
Can we derive Kolmogorov's law in the framework of kinetic theory without going to hydrodynamic formulation? This remains a challenge. The multiscale flow structures (e.g., vortices within vortices) are natural in the hydrodynamic description of turbulence, but not very apparent in kinetic theory whose basic constituents are particles. Note however that we can obtain multiscale fluid structures by averaging or coarse-graining many times, as is often done in lattice hydrodynamics~\cite{Succi:book:Lattice}. {\em Yet, the derived structures follow the laws of hydrodynamics, and these laws are not transparent at the particle level. } Thus, macroscopic description provided by hydrodynamics is much more convenient for the description of turbulence.
Many natural flows involve more complex forces than those assumed in Kolmogorov's theory of turbulence (see Fig.~\ref{fig:NS}). For example, Ekman friction, which is of the form $-\alpha {\bf u}$ ($\alpha$ is a positive constant), induces dissipation of kinetic energy at all scales~\cite{Verma:EPL2012}. Consequently, the energy flux $\Pi_u(k)$ decreases with $k$. Hence, the kinetic energy in the flow at a given scale is lower than that for $\alpha =0$. This feature leads to a steeper spectrum for Ekman friction than that predicted by Kolmogorov's theory ($k^{-5/3}$). Similar steepening of kinetic energy spectrum is observed in buoyancy-driven turbulence~\cite{Obukhov:DANS1959} and in magnetohydrodynamic turbulence~\cite{Verma:ROPP2017}. A derivation of above variable energy flux is very easy in spectral description of hydrodynamics~\cite{Frisch:book,Lesieur:book:Turbulence,Verma:book:BDF}, but not in kinetic theory.
A cautionary remark is in order. In gas dynamics, kinetic theory is extensively employed to describe rarified gas for which hydrodynamic description breaks down~\cite{Succi:book:Lattice,Singh:PRE2016}. These ideas find applications in supernova explosions, supersonic rockets and jets, rarified plasma, etc.
\section{Dissipation, diffusion, and pressure in hydrodynamics}
In microscopic description of physical processes, the collisions or interactions among particles conserve energy. These processes also respect time reversal symmetry~\cite{Feynman:book:Character,Carroll:book:Time,Pathria:book}. Given this, it is very difficult to incorporate dissipation for an isolated system of particles. Hydrodynamic description bypasses this difficulty by postulating viscosity that sets up the energy cascade from large scales to small scales. The origin of such friction has been debated by researchers. In a multiscale hydrodynamic description, the viscosity converts coherent kinetic energy (related to the flow velocity) to incoherent heat energy of microscopic particles at the dissipation scale~\cite{Verma:arxiv:time}; note however that the total kinetic energy at the particle level is conserved.
Turbulence typically enhances diffusion. We illustrate this phenomena using an often-quoted example---heat diffusion from a heater. Since the thermal diffusion coefficient of air is $\kappa \approx 10^{-5}~\mathrm{m}^2/\mathrm{s}$, from kinetic theory or statistical mechanics, the time estimate for the heat diffusion by $L= 1$~m would be $L^2/\kappa \approx 10^5$ seconds. This estimate is clearly incorrect. In reality, heat is advected by the nonlinear term, hence the time scale is $L/U \approx 1/0.1 = 10$ seconds, where $U$ is the velocity of the large-scale structures~\cite{Verma:book:BDF}. A derivation of the aforementioned hydrodynamic diffusion from kinetic theory is not practical.
A related phenomena is {\em Taylor dispersion}~\cite{Taylor:PRSA1954} of particles in a turbulent flow. The distance between two particles in a turbulent flow increases as $t^{3/2}$, where $t$ is the elapsed time. Note that the Taylor dispersion is faster than ballistic dispersion ($\sim t$), which is the fastest dispersion for any particle in kinetic theory. The enhancement in Taylor dispersion is due to the advection of the particles by multiscale structures; the particles separated by a distance $r$ hop from vortices of size $r$ to larger vortices that move with even larger speeds. Again, Taylor dispersion would be hard to derive in kinetic theory.
As described in Section~\ref{sec:KT}, the hydrodynamic equations can be derived from kinetic theory. Such derivations yield equations for compressible flows for which the pressure is the {\em thermodynamic pressure} (that has origin in kinetic theory). However, there is another important pressure called {\em dynamic pressure} that appears in incompressible hydrodynamics. In Bernoulli's equation, $p+ \rho u^2/2 = \mathrm{constant}$, where $p$ is the dynamic pressure, which is distinct from the thermodynamic pressure. Note that the dynamic pressure can be derived easily in hydrodynamic framework~\cite{Frisch:book}, but it would be very hard to derive in kinetic theory (without going to coarse-grained picture of hydrodynamics). We remark that a compressible flow contains both dynamic and thermodynamic pressures~\cite{Zank:PF1991}, but their derivation in kinetic theory would be way too complex.
We conclude in the next section.
\section{Conclusions and Discussions}
In this article, we describe certain hydrodynamic (macroscopic) laws that are difficult to derive {\em directly} from microscopic framework such as kinetic theory. These laws include Kolmogorov's theory of turbulence, viscous dissipation and Taylor's dispersion in turbulent flows, and dynamic pressure. For these laws, the hydrodynamic description is more adequate than the kinetic theory. These observations are in the spirit of discussions by Anderson~\cite{Anderson:Science1972} and Laughlin~\cite{Laughlin:PNAS2000} where they argue in favour of hierarchical description of systems and laws.
We can go to a step (or hierarchy) further in the flow complexity. Planetary and stellar flows are quite complex; some of the leading problems in these fields are global warming, ice ages, magnetic field generation, corona heating, mantle and core dynamics of the Earth, land ocean coupling, monsoons, etc.~\cite{Fowler:book}. To address these problems, particle description is never employed. Further, it is impractical (in fact, impossible) to solve the relevant primitive equations---flow velocity, chemical constituents, moisture, ice---at all scales. For the Earth, the corresponding length scales range from $10^{-6}$ m to $4\times 10^6$ m. Hence, scientists often model these systems using relevant large-scale variables. For example, ice age is modelled using total solar radiation, carbon dioxide, and the mean temperature of the Earth. Similarly, the solar magnetic field is modelled using several magnetic modes in spherical harmonic basis~\cite{Jones:book_chapter}. There are other equally important tools like probability, filtering, and machine learning for describing aforementioned complex systems.
The next level of hierarchical structures are solar system, galaxy, and the universe. As we move up the hierarchy, the planetary and stellar atmosphere are ignored and newer sets of variables and equations are used. For example, Newton assumed the Sun and the Earth to be point particles for describing planetary motion; Millenium simulation of the universe treat the galaxies as point particles embedded in dark matter.
Thus, nature has hierarchical structures that have their own laws and relevant tools~\cite{Laughlin:PNAS2000}. However, the system descriptions and associated laws at different levels are connected to each other, most strongly among the neighbouring levels. For example, kinetic theory and hydrodynamics are intimately connected. Yet, the laws of the system at a given level are best derived using the equations and tools at that level. A possible hierarchical categorisation could be---nuclear and particle physics, atomic and molecular physics, condensed-matter physics, chemistry, biology, ecology, and so on. Another multiscale characterisation is---kinetic description of particles, hydrodynamic description of flows, planetary and stellar atmosphere and interiors, solar system, galaxies, galaxy clusters, universe. These structures help us identify the laws at each level, and derive relationships among them. It is important to keep in mind that the connections between the theories at different levels many involve many complications. Berry~\cite{Berry:PT2002} and Batterman~\cite{Batterman:book:Details} describe such issues, in particular {\em singular limits} encountered in such attempts.
Note that {\em holism}, considered to be the opposite philosophy of {\em reductionism}, advocates that the properties of a system are best understood as a {\em whole}, not as a sum of its parts~\cite{Auyang:book}. The hierarchical description however differs somewhat from holism, and it propounds that the universe is hierarchically structured, and it is best described by hierarchy of laws at different scales. These laws however may be interlinked, similar to the laws of kinetic theory and hydrodynamics.
The hierarchical framework is often invoked for describing emergent phenomena~\cite{Laughlin:PNAS2000,Laughlin:book,Anderson:Science1972,Anderson:book:More}. For example, chemists, biologists, and material scientists work tirelessly to discover new molecules and material with specific properties using ab-initio or first-principle calculations. However, centuries ago researchers used to rely on macroscopic properties of materials (such as, affinity to water, air, fire etc.). Although no one doubts the power of first-principle calculations, the former approaches too could be useful. A major component of climate research involves large-scale simulations of primitive variables on massive grids (say, with one billion grid points). In comparison, at present, much less attention goes into making a low-dimensional models based on large-scale or macroscopic variables, such as mean temperature, solar radiation, land-sea interactions, overall carbon dioxide content, etc. Many believe that a combination of both the approaches, microscopic and macroscopic, would yield richer dividends. These illustrations indicate that applications of hierarchical description may help address some of the complex problems we face today.
\section*{Acknowledgments}
I thank Anurag Gupta and Michael Berry for useful discussions.
|
1,108,101,563,218 | arxiv | \section{Introduction}\label{s-1}
The symmetric groups play two different roles in the representation
theory of general linear groups. On one side, let us call it the Weyl
side, the symmetric group $S_N$ of permutations of $N$ letters is the
subgroup of $\mathit{GL}_N$ of permutation matrices and thus acts on
any representation of $\mathit{GL}_N$. On the other side, say the
Schur side, $S_M$ is the group of permutation of factors of the $M$th
tensor power $U^{\otimes M}=U\otimes\cdots\otimes U$ of a
representation $U$ of $\mathit{GL}_N$ and the action of $S_M$ commutes
with the natural action of $\mathit{GL}_N$ on $U^{\otimes M}$. We will
focus on the case where $U$ is an exterior power of the vector
representation, but let us be general for the time being.
Both roles admit a $q$-deformed version but the symmetric group needs
to be replaced by the braid group. On the Weyl side one has an action
of the braid group $B_N$ on finite dimensional representations of the
quantum enveloping algebras $U_q\mathfrak{gl}_N$ deforming the above
action of the symmetric group. This is a special case of the action on
integrable representations of the Artin braid group corresponding to
the Weyl group of any semisimple Lie algebra \cite{Lusztig1993,
LevendorskiiSoibelman1990}. On the Schur side, the universal $R$-matrix
of $U_q\mathfrak gl_N$ defines a representation of the braid group
$B_M$ on the tensor power $U^{\otimes M}$ of a finite dimensional
representation $U$ of $U_q{\mathfrak{gl}}_N$. The action of the generator $s_i$,
braiding the $i$th with the $i+1$st strand is given by the action of
the $R$-matrix on the $i$th and $i+1$st factor.
Both sides are degenerate cases of versions with parameters, called
spectral (or equivariant) parameters on the Schur side and dynamical
(or Kähler) parameters on the Weyl side. In terms of representation
theory, this generalization amounts to replacing $U_q\mathfrak{gl}_N$
by its affine version $U_q\widehat{\mathfrak{gl}}_N$. A feature of
the versions with parameters, whose clarification is a goal of this
paper, is that the symmetric group plays the leading role, rather than
the braid group that is relevant in the degenerate limit. The
$R$-matrices of quantum groups first appeared in statistical mechanics
and quantum field theory as solutions
$R(z)\in\operatorname{End}(U\otimes U)$ of the Yang--Baxter equation,
depending meromorphically on a complex spectral parameter $z$. The
Yang--Baxter equation is equivalent to the braiding relation
\eqref{e-3} for the braiding matrix $\check R(z)=P\circ R(z)$
obtained by composing the $R$-matrix with the permutation
$P(u\otimes v)=v\otimes u$ of factors. Moreover one has the inversion
(or unitarity) relation $\check R(z^{-1})\check R(z)=\mathrm{id}$.
One obtains, on the Schur side, a representation of the symmetric
group $S_M$ on $U^{\otimes M}$-valued meromorphic functions
$f(z_1,\dots, z_M)$ of $M$ variables, such that the generator
$s_i=(i,i+1)$ is mapped to
\[
s_if(z_1,\dots,z_N) = \mathrm{id}^{\otimes (i-1)}\otimes \check
R(z_i/z_{i+1})\otimes \mathrm{id}^{\otimes (N-i-1)}
f(z_1,\dots,z_{i+1},z_i,\dots,z_N).
\]
Representations of the braid group on $U^{\otimes M}$ appear in
the limits $z_i/z_{i+1}\to 0$, $i=1,\dots,M-1$ or $\infty$. Indeed $R(z)$
converges in the limit $z\to\infty$ to the solution of the Yang--Baxter equation
without spectral parameter corresponding to $U_q{\mathfrak{gl}}_N$. The relation
$s_i^2=1$ is lost in this limit and one is left with a representation of the
braid group.
On the Weyl side, the discovery of a version of the quantum Weyl group
action with parameters is more recent
\cite{TarasovVarchenko2000,EtingofVarchenko2002} and is called dynamical Weyl
group. Let $T\subset SL_M(\mathbb C)$ be the Cartan torus of diagonal
matrices, $P_M=\operatorname{Hom}(T,\mathbb C^\times)$ the weight
lattice. Then finite dimensional representations $U$ of
$U_q\mathfrak{sl}_M$ (of type I with $q$ not a root of unity) have a
weight decomposition $U=\oplus_{\mu\in P_M}U[\mu]$. The Weyl group
$S_M$ acts on $T$ and $P_M$, defining a action of the braid group via
the canonical homomorphism $B_M\to S_M$. A dynamical action of the
braid group $B_M$ on an $\mathfrak{sl}_M$-module $U$ is a
representation $\rho$ of the braid group on rational $U$-valued
function on $T$ of the form
\[
(\rho(g)f)(t)=A_{g}(g^{-1}t)f(g^{-1}t),
\]
for some rational functions $t\mapsto A_g(t)\in\operatorname{End}(U)$,
compatible with the $T$-action on $U$ in the sense that
$A_g(t)(U[\mu])\subset U[g\mu]$. The theory of intertwining operators
developed in \cite{TarasovVarchenko2000, EtingofVarchenko2002}, which works
for arbitrary simple Lie algebras, gives rise to a dynamical action of
the braid group with many interesting properties, such as universal
formulas for $A_g(t)|_{U[\mu]}$ as the action of an element of
$U_q\mathfrak{sl}_M$ independent of $U$ and a compatibility with
inclusions of Lie subalgebras, implying that $A_{s_i}(t)$ for
generators $s_i$ is obtained as the image of the corresponding element
of $U_q(\mathfrak{sl}_2)$ by the inclusion associated with the $i$th
simple root.
In this paper we take $U=\bigwedge V$, the direct sum of exterior
powers of the vector representation $V=\mathbb C^N$ of $U_q{\mathfrak{gl}}_N$, and
relate the two actions of the symmetric group via quantum (skew) Howe
duality. The classical Howe duality \cite{Howe1992} is based on the
observation that on $\bigwedge(\mathbb C^N\otimes\mathbb C^M)$ we have
natural commuting actions of $\mathit{GL}_N$ and $\mathit{GL}_M$ whose
images span each other's commutant in the endomorphism ring. This
implies that endomorphisms of
$(\bigwedge V)^{\otimes M}=\bigwedge V\otimes\cdots\otimes\bigwedge V$
commuting with the $\mathit{GL_N}$ action on $V=\mathbb C^N$ are
spanned by a commuting action of $\mathit{GL}_M$. There is a also a
symmetric version of Howe duality, where exterior powers are replaced
by symmetric powers. Quantum version of Howe dualities are known. The
skew case was first considered in \cite{CautisKamnitzerLicata2010} and \cite{LehrerZhangZhang2011}.
See \cite{Stroppel2022} for recent developments of this story.
In Section \ref{s-3} we present an approach based on the action of the
Clifford algebra which is more elementary than the categorification
approach of \cite{CautisKamnitzerMorrison2014}, see Proposition \ref{p-5}.
In particular one obtains a $U_q{\mathfrak{gl}}_M$-action on
$(\bigwedge V)^{\otimes M}$ commuting with the diagonal
$U_q{\mathfrak{gl}}_N$-action. The upshot is that the action of the symmetric
group on $(\bigwedge V)^{\otimes M}$-valued functions by braiding
matrices with spectral parameter is realized by a dynamical Weyl group
action for the $U_q{\mathfrak{gl}}_M$-action, after a suitable identification of
spectral with dynamical parameters. Thus we connect the Schur with
the Weyl side. In the limit of infinite spectral/dynamical parameter
we recover the result of \cite{CautisKamnitzerLicata2010} relating
braid group action with quantum Weyl group action.
For $M=2$ we obtain an explicit formula for the $R$-matrix for pairs
of exterior powers of the vector representation in terms of
$U_q{\mathfrak{gl}}_2$, see Theorem \ref{t-1}. These $R$-matrices were calculated
in \cite{DateOkado1994} by the fusion procedure and are expressed in
terms of projections onto irreducible subrepresentations of the tensor
product. Our alternative formula does not rely on the decomposition into irreducible
subrepresentations. It is a sum of terms, each with a simple
pole in the spectral parameters, and gives an interpretation of the
residues at the poles. When written in terms of Clifford algebra
generators, our formula for the $R$-matrix of
$\bigwedge^kV\otimes\bigwedge^kV$ converges in the limit $q\to1,z\to 1$ to
the formula proposed by Smirnov \cite{Smirnov2016} in the case of the
Yangian. As in \cite{Smirnov2016}, it has the property that it also
applies to the limit $N\to\infty$, see Theorem \ref{t-3}. Finding a conceptual proof of
Smirnov's formula was an initial motivation of this work.
As an application of our result, we obtain in Theorem \ref{t-2} and
Corollary \ref{c-1} a dynamical action of the {\em symmetric group} on
integrable representations of $U_q{\mathfrak{gl}}_M$, rather than of the braid
group as in \cite{EtingofVarchenko2002}. In fact our dynamical action
differs from the dynamical action of the braid group defined in
\cite{EtingofVarchenko2002} by a transformation
$A_{g}(t)|_{U[\mu]}\mapsto A_{g}(t)|_{U[\mu]}f_{g,\mu}(t)$ for some
scalar meromorphic functions $f_{g,\mu}(t)$ obeying the cocycle
condition $f_{gh,\mu}(t)=f_{g,h\mu}(ht)f_{h,\mu}(t)$ for any braids
$g,h$, see Proposition \ref{p-3}. While in \cite{EtingofVarchenko2002}
the dynamical action of the braid group factors through the symmetric
group only when restricted to the zero weight space $U[0]$, our
modified action defines a representation of the symmetric group on the
whole space of $U$-valued functions. We expect that the same
transformation gives rise to a dynamical action of the Weyl group on
integrable representations of arbitrary reductive Lie algebras.
A summary of the results of this paper was announced in the Oberwolfach report
\cite{Felder2022}.
\subsection{Acknowledgments} This research was supported in part by
the National Centre of Competence in Research SwissMAP (grant numbers
182902, 205607) of the Swiss National Science Foundation. The second
author is also supported by the grant 196892 of the Swiss National
Science Foundation. This article was written in part while the second
author was at IHES, which he thanks for hospitality.
We thank Tommaso Botta for various useful discussions and suggestions.
The second author thanks the organizers of the mini-workshop on
``Recent developments in representation theory and mathematical physics'' at the
Mathematisches Forschungsinstitut Oberwolfach, where these results were presented,
and the participants, expecially Evgeny Mukhin and Catharina Stroppel for discussions.
We thank Andrey Smirnov and Huafeng Zhang for useful correspondence.
\section{Braiding matrices for exterior powers of the vector representation}\label{s-2}
\subsection{$q$-number notation}
We adopt the standard $q$-number notation: for $n\in\mathbb Z_{\geq0}$,
$[n]_q=\frac{q^n-q^{-n}}{q-q^{-1}}$. The $q$-factorial is
defined by $[0]_q!=1$, $[n]_q!=\prod_{j=1}^n[j]_q$, $n\geq 1$
and the $q$-binomial coefficient (or Gauss polynomial) is
\[
\qbinomial{n}{j}=\frac{[n]_q!}{[j]_q![n-j]_q!}.
\]
All these objects belong to $\mathbb Z[q,q^{-1}]$.
\subsection{Quantum enveloping algebras of type $A_{N-1}$ and
$A_{N-1}^{(1)}$}
Drinfeld--Jimbo quantum enveloping algebras are the subject of several textbooks,
for example \cite{Lusztig1993, ChariPressley1994}. Here we collect our
conventions.
Let $q$ be a non-zero complex number which is not a root of unity.
Let $(a_{ij})_{i,j=0}^{N-1}$ be the Cartan matrix of $A_{N-1}^{(1)}$:
$a_{ij}=2$ if $i=j$, $a_{ij}=-1$ if $j=i\pm 1\mod N$, and $a_{ij}=0$
otherwise. It is convenient to identify the index set $\{0,\dots,N-1\}$
with $\mathbb Z/N\mathbb Z$ in view of the symmetry $a_{i+1,j+1}=a_{ij}$.
The Hopf algebra $U'_q(\widehat{\mathfrak{sl}}_N)$ is the
unital algebra over $\mathbb C$ generated by $e_i,f_i,k_i^{\pm1}$,
$(i=0,\dots, N-1)$ and relations
\begin{align*}
k_ik_j&=k_jk_i,\quad k_ik_i^{-1}=k_i^{-1}k_i=1
\\
k_ie_jk_i^{-1}
&=q^{a_{ij}}e_j,\quad k_if_jk_i^{-1}=q^{-a_{ij}}f_j,
\\
e_if_j-f_je_i
&=\delta_{ij}\frac{k_i-k_i^{-1}}{q-q^{-1}}
\\
\sum_{r=0}^{1-a_{ij}} &\qbinomial {1-a_{ij}}r e_i^re_je_i^{1-a_{ij}-r}
=0,\quad \text{if $i\neq j$,}
\\
\sum_{r=0}^{1-a_{ij}}& \qbinomial {1-a_{ij}}r f_i^rf_jf_i^{1-a_{ij}-r}
=0,\quad \text{if $i\neq j$.}
\end{align*}
The coproduct is defined on generators as
$\Delta(e_i)=e_i\otimes 1+k_i\otimes e_i$,
$\Delta(f_i)=f_i\otimes k_i^{-1}+ 1\otimes f_i$ and
$\Delta(k^{\pm1}_i)=k^{\pm1}_i\otimes k^{\pm1}_i$. The element
$\prod_{i=0}^{N-1}k_i$ is central and group-like. For finite
dimensional representations of $U_q'(\widehat{\mathfrak{sl}}_N)$ this
central element acts by 1. The quantum loop algebra
$U_qL{\mathfrak{sl}}_N$ is the quotient of $U'_q(\widehat{\mathfrak{sl}}_N)$
by the relation $\prod_{i=0}^{N-1}k_i=1$. The Hopf algebra structure
descends to this quotient. It is a deformation of the universal
enveloping algebra of the Lie algebra $L\mathfrak{sl}_N={\mathfrak{sl}}_N[t,t^{-1}]$ of
loops in ${\mathfrak{sl}}_N$. The Hopf algebra
$U_qL{\mathfrak{sl}}_N$ contains the Hopf subalgebra
$U_q {\mathfrak{sl}}_N$ generated by $e_i,f_i,k_i$ with
$1\leq i\leq N-1$.
For $k\in\mathbb Z_{\geq0}$ the $k$th divided power of generators is
defined as
\[
e_i^{(k)}=\frac{e_i^k}{[k]_q!},\quad f_i^{(k)}=\frac{f_i^k}{[k]_q!},
\]
with the convention that $e_i^{(0)}=f_i^{(0)}=1$.
Let $\mathbb Z^N=\oplus_{i\in\mathbb Z/N\mathbb Z}\mathbb Z\epsilon_i$
be the (co)weight lattice of ${\mathfrak{gl}}_N$ with standard inner product
$(\ |\ )$, which identifies the Cartan subalgebra of diagonal matrix with its dual.
The root lattice of vectors of zero coordinate sum is
spanned by the simple roots $\alpha_i=\epsilon_i-\epsilon_{i+1}$,
$(i=1,\dots, N-1)$. We also set $\alpha_0=\epsilon_N-\epsilon_1$, the
highest root. The Hopf algebra $U_qL{\mathfrak{gl}}_N$ is obtained by adjoining
generators $t_\lambda$ for $\lambda\in \mathbb Z^N$ such that
$t_{\alpha_i}=k_i$ ($i=1,\dots,N-1$) with the relations
\[
t_\lambda t_\mu=t_{\lambda+\mu},\quad t_0=1,\quad t_\lambda
e_it_{-\lambda}=q^{(\lambda|\alpha_i)}e_i, \quad t_\lambda
f_it_{-\lambda}=q^{-(\lambda|\alpha_i)}e_i.
\]
It is a Hopf algebra with
$\epsilon(t_\lambda)=1,
S(t_\lambda)=t_{-\lambda},\Delta(t_\lambda)=t_\lambda\otimes
t_\lambda$. We set $t_i=t_{\epsilon_i}$ so that $k_i=t_it_{i+1}^{-1}$
for $i\in\mathbb Z/N\mathbb Z$. The Hopf subalgebra $U_q{\mathfrak{gl}}_N$ is
generated by $e_i,f_i$ with $1\leq i\leq N-1$ and all $t_i^{\pm1}$.
The multiplicative group $\mathbb C^\times $ acts on $U_qL{\mathfrak{gl}}_N$ by
algebra automorphisms, deforming the rescaling action of the loop
parameter: the automorphism $\varphi_z$ associated with
$z\in\mathbb C^\times$ acts as the identity on all generators except
$e_0$, $f_0$, for which
\[
\varphi_z(e_0)=ze_0,\quad \varphi_z(f_0)=z^{-1}f_0.
\]
The weight space of a $U_q{\mathfrak{gl}}_N$-module of weight $\mu\in \mathbb Z^N$
is the simultaneous eigen\-space of $t_\lambda$ with eigenvalue
$q^{(\mu|\lambda)}$. A $U_q{\mathfrak{gl}}_N$-module is of {\em type I} if it is the direct
sum of its finite dimensional weight spaces. All modules considered
in this paper are of type I.
\subsection{Exterior powers of the vector representation}
\label{ss-2.3}
Let $V=\mathbb C^N$ with basis $v_1,\dots,v_N$ and
$C_N$ be the Clifford algebra of $V\oplus V^*$ with quadratic form
$Q(v\oplus \alpha)=\alpha(v)$. It is the unital algebra generated by
the images of the standard basis and dual basis vectors
$\psi^*_i,\psi_i$, labeled by $i\in\{1,\dots,N\}$, which we often identify
with $\mathbb Z/N\mathbb Z$, subject to the relations
\begin{equation}\label{e-1}
\psi_i\psi_j=-\psi_j\psi_i,\quad\psi_i^*\psi_j^*=-\psi_j^*\psi_i^*,\quad
\psi_i\psi_j^*+\psi_j^*\psi_i=\delta_{ij}1.
\end{equation}
The Fock representation of $C_N$ is $\bigwedge V$ with $\psi^*_i$
acting as $v\mapsto v_i\wedge v$ and $\psi_i$ by contraction. It is
irreducible and generated by $|0\rangle=1$ such that
$\psi_i|0\rangle=0$ for all $i$. The commuting ``fermion number''
operators $n_i=\psi_i^*\psi_i$ have a common eigenbasis
$v_{i_1}\wedge\cdots\wedge v_{i_k}$ ($i_1<\dots<i_k$) and eigenvalue
$1$ if $i=i_j$ for some $j$ and $0$ otherwise. Thus
$t^{\pm1}_i:=q^{\pm n_i}=q^{\pm1}\psi_i^*\psi_i+\psi_i\psi_i^*$ obeys the
relations
\begin{equation}\label{e-2}
t_it_i^{-1}=1,\quad t_it_j=t_jt_i,\quad t_i\psi_j^*t_i^{-1}=q^{\delta_{ij}}\psi_j^*,\quad
t_i\psi_jt_i^{-1}=q^{-\delta_{ij}}\psi_j.
\end{equation}
for all $i,j$. Let $C_N(q)$ be the algebra with generators $t_i,\psi_i,\psi_i^*$,
($i=1,\dots,N$) and relations \eqref{e-1}, \eqref{e-2}. It is a $\mathbb Z/2\mathbb Z$-graded
algebra with $\psi,\psi_i^*$ odd and $t_i$ even, and $\bigwedge V$ is a $\mathbb Z/2\mathbb Z$-graded
module, with the convention that $|0\rangle$ is even.
\begin{prop}[Hayashi \cite{Hayashi1990}]\label{p-1}
There is a homomorphism $U_qL\mathfrak{gl}_N
\to C_N(q)$ such that
\[
e_i\mapsto \psi_i^*\psi_{i+1},\quad
f_i\mapsto\psi_{i+1}^*\psi_i,\quad
t_i^{\pm1}\mapsto t_i^{\pm 1},
\]
for $i\in\mathbb Z/N\mathbb Z$.
\end{prop}
Thus we have a representation of $U_qL{\mathfrak{gl}}_N$ on $\bigwedge V$. It is
the direct sum of the irreducible eigenspaces $\bigwedge^kV$ of the
central element $\prod_{i=1}^N t_i$. These irreducible representations
are generated by the highest weight vectors
\begin{equation}\label{e-hw}
v^k=\psi_1^*\cdots\psi_k^*|0\rangle
\end{equation}
of weight $\sum_{i=1}^k\epsilon_i$. In fact they are obtained from
the representation of the classical Lie algebra $L{\mathfrak{gl}}_N$ on the same
representation space and with the same action of the generators
$e_i,f_i$ by setting $t_\lambda=q^{\lambda}$ for Cartan elements
$\lambda$. The point is that, since the eigenvalues of $\epsilon_{i}$
are $1$ or $0$, the right-hand side $(q^{h_i}-q^{-h_i})/(q-q^{-1})$ of
the commutation relation of $e_i,f_i$ coincides with the action of its
classical counterpart $h_i=\epsilon_i-\epsilon_{i+1}$.
\begin{definition}\label{def-ext}
The {\em $j$-th exterior power of the vector representation} of $U_qL{\mathfrak{gl}}_N$ with spectral
parameter $z$ is the twist $\bigwedge^jV(z)$
of $\bigwedge^jV$ by the automorphism $\varphi_z$. The vector representation is
$V(z)=\bigwedge^1V(z)$.
\end{definition}
Thus the action of the generators is given by the formulae of
Proposition \ref{p-1} except that $e_0,f_0$ are mapped to
$z\psi_N^*\psi_{1}$ and $z^{-1}\psi_1^*\psi_N$, respectively.
\begin{remark} Our definition of the exterior power representation
has a slightly non-standard choice of signs. A more common convention is to
let $e_0$ and $f_0$ act as $(-1)^{j-1}z\psi_N^*\psi_1$, $(-1)^{j-1}z^{-1}\psi_1^*\psi_N$, respectively,
which in the limit $q\to 1$ correspond to the action of $zE_{N,1},z^{-1}E_{1,N}$, in terms of the
standard basis $(E_{i,j})$ of ${\mathfrak{gl}}_N$, so that $z$ has the
interpretation of an ``evaluation point.'' Our convention is better suited to the fermionic representation
and somewhat reduces the number of signs in the formulas. The relation of our definition of
the exterior power $\bigwedge^jV(z)$ to the definition $V^{(j)}(z)$ in \cite{DateOkado1994}
is $\bigwedge^jV(z)=V^{(j)}((-1)^{j-1}z)$,
\end{remark}
Another interpretation of the exterior powers, useful to compute $R$-matrices by the fusion procedure,
is in terms of $q$-deformed exterior products:
\begin{prop}[Jimbo \cite{Jimbo1986}]
Let $k\geq2$ and $S$ be the subspace of $V^{\otimes k}$ spanned by
$\alpha\otimes v_i\otimes v_i\otimes\beta$,
$\alpha\otimes (v_i\otimes v_j+q v_j\otimes v_i)\otimes \beta$ for
$i>j$, $\alpha\in V^{\otimes r},\beta\in V^{\otimes k-r-2}$,
$r=0,\dots,k-2$. Then $S$ is a $U_qL{\mathfrak{gl}}_N$ submodule of
\[
V^k(z):=V(q^{-k+1}z)\otimes\cdots\otimes V(q^{k-3}z)\otimes V(q^{k-1}z)
\]
and we have the isomorphism of $U_qL{\mathfrak{gl}}_N$-modules
$V^k(z)/S\to \bigwedge^k V(z)$ such that
$v_{i_1}\otimes \cdots \otimes v_{i_k}+S\mapsto
v_{i_1}\wedge\cdots\wedge v_{i_k}$ for $1\leq i_1<\cdots<i_k\leq N$.
\end{prop}
For generic $z_1,z_2$ the tensor products module
$\bigwedge^{k} V(z_1)\otimes \bigwedge^{k'} V(z_2)$ with action of
$U_q{\mathfrak{gl}}_N$ given by the coproduct are irreducible and do not depend
on the order of factors up to isomorphism. Thus there is an
isomorphism of $U_q\widehat{{\mathfrak{gl}}}_N$-modules
\[
\check R_{k,k'}(z_1/z_2)\colon\textstyle{\bigwedge^{k} V(z_1)\otimes
\bigwedge^{k'} V(z_2)\to \bigwedge^{k'} V(z_2)\otimes
\bigwedge^{k} V(z_1)},
\]
which, viewed as a linear map between the underlying vector spaces,
depends on the ratio of spectral parameters. We find it convenient to
normalize it in such a way that $v^k\otimes v^{k'}$ is mapped to
$(-1)^{kk'}v^{k'}\otimes v^k$. Then the $R$-matrix
\[
R_{k,k'}(z_1/z_2)=P\check R_{k,k'}(z_1/z_2)\in\operatorname{End}_{\mathbb C}(
\textstyle{\bigwedge^{k} V(z_1)\otimes
\bigwedge^{k'} V(z_2)}),
\]
obtained by the composition with the graded flip
$P\colon v\otimes w\to (-1)^{kk'}w\otimes v$, restricts to the identity
on the product of highest weight spaces.
Then by the irreducibility of tensor products
at generic values of the spectral parameter we have the braiding relation
\begin{align}\label{e-3}
(\check R_{k_2k_3}(z_2/z_3)
&\otimes\mathrm{id})
(\mathrm{id}\otimes\check R_{k_1k_3}(z_1/z_3))
(\check R_{k_1k_2}(z_1/z_2)\otimes\mathrm{id})\notag
\\
&=
(\mathrm{id}\otimes\check R_{k_1k_2}(z_1/z_2))
(\check R_{k_1k_3}(z_1/z_3)\otimes\mathrm{id})
(\mathrm{id}\otimes\check R_{k_2k_3}(z_2/z_3)),
\end{align}
on $
\bigwedge^{k_1}V(z_1)\otimes \bigwedge^{k_2}V(z_2)\otimes \bigwedge^{k_3}V(z_3)$,
equivalent to the Yang--Baxter equation for the $R$-matrices.
Moreover we have the inversion (or unitarity) relation
\begin{equation}\label{e-4}
\check R_{k'k}(z^{-1})\check R_{kk'}(z)=\mathrm{id}.
\end{equation}
These $R$-matrices were computed
by Date and Okado \cite{DateOkado1994} in terms of projections onto irreducible
submodules for the action of $U_q{\mathfrak{gl}}_N$. The coefficients were computed using
the fusion procedure \cite{KulishReshetikhinSklyanin1981}.
Here we give an alternative fermionic formula.
Let $\bigwedge V\otimes\bigwedge V$ be the tensor product of
$\mathbb Z/2\mathbb Z$-graded vector spaces. The action of tensor
products of linear maps is given by the sign rule:
$(f\otimes g)(v\otimes w)=(-1)^{|g||v|}fv\otimes gw$ for $g$ of degree
$|g|$ and $v$ of degree $|v|$. We introduce the following even
endomorphisms of $\bigwedge V\otimes\bigwedge V$:
\begin{equation}\label{e-EFK}
E=\sum_{i=1}^N
(\psi^{*}_i\otimes {\psi}_i)\prod_{j>i}{K}_j
\qquad
F=-\sum_{i=1}^N
\prod_{j<i}{K}^{{-1}}_j({\psi}_i\otimes\psi^{*}_i)
\qquad
{K}=\prod_{i=1}^N{{K}}_i,
\end{equation}
with $K_i=t_i\otimes t_i^{-1}$ ($i=1,\dots,N$).
\begin{lemma}\label{l-1}
The endomorphisms $E,F,K$ commute with the action of $U_q{\mathfrak{gl}}_N$ and
obey the commutation relations of $U_q\mathfrak{sl}_2$:
\[
[E,F]=\frac{K-K^{-1}}{q-q^{-1}},\quad KEK^{-1}=q^2E,\quad
KFK^{-1}=q^{-2}F.
\]
\end{lemma}
This is an instance of Howe duality and is discussed in the next
section. It is convenient to extend this action to an action of
$U_q{\mathfrak{gl}}_2$ by setting $T_1=\prod_{i=1}^N t_i\otimes 1$,
$T_2=\prod_{i=1}^N1\otimes t_i$ so that
$\bigwedge^{k}V\otimes \bigwedge^{k'}V$ is the weight space of weight
$(k,k')$.
\begin{theorem}\label{t-1}
The braiding matrix
\[
\check
R_{k,k'}(z_1/z_2)\colon\textstyle{\bigwedge^kV(z_1)\otimes\bigwedge^{k'}V(z_2)\to
\bigwedge^{k'}V(z_1)\otimes\bigwedge^{k}V(z_1)},
\]
normalized so that
$\check R_{k,k'}(z_1/z_2)\,v^k\otimes v^{k'}=(-1)^{kk'}v^{k'}\otimes v^{k}$, is
given by
\[
\check R_{k,k'}(z_1/z_2)=(-1)^{\min(k,k')}A_{k-k'}(z_1/z_2)
\]
where
$A_{k-k'}(z_1/z_2)\in U_q\mathfrak{gl}_2$ is defined by
\[
A_m(z)=\sum_{j=0}^n
(-q)^{j}\frac{1-q^{|m|}z}{1-q^{2j+|m|}z}
\begin{cases}
E^{(j)}F^{(j+m)},& \text{if $m\geq0$,}
\\
E^{(j-m)}F^{(j)},& \text{if $m\leq 0$,}
\end{cases}
\]
with the divided power notation $E^{(l)}=E^l/[l]_q!$, $F^{(l)}=F^l/[l]_q!$ and
\[
n=\min(k,k',N-k,N-k').
\]
\end{theorem}
The proof relies on Howe duality, which we proceed to explain, and the computation
\cite{DateOkado1994} of the $R$-matrices by the fusion procedure.
\section{Quantum skew Howe duality}\label{s-3}
The classical skew-symmetric version of Howe duality \cite[Section 4]{Howe1992}
states that the natural actions of $\mathit{GL}_N$ and
$\mathit{GL}_M$ on $\bigwedge(\mathbb C^N\otimes\mathbb C^M)$ generate
each other's commutant and one has a decomposition into irreducibles
of $\mathit{GL}_N\times \mathit{GL}_M$
\[
\bigwedge (\mathbb C^N\otimes\mathbb C^M)\cong
\bigoplus_{\lambda}V^N_\lambda\otimes V^M_{\lambda^t}.
\]
The sum is over all Young diagrams $\lambda$ with at most $N$ rows
such that the transposed diagrams $\lambda^t$ has at most $M$ rows
(equivalently, the length of the first row of $\lambda$ is at most
$M$) and $V^N_\lambda$ denotes the irreducible representation of
$\mathit{GL}_N$ of highest weight $\lambda$. The following nice
pictorial description of the corresponding highest weight vectors is
due to Howe in {\it loc.~cit.}: write the tensor products of basis vectors
$v_{i,r}=v_i\otimes w_r$ $(i=1,\dots,N, r=1,\dots, M)$ in the boxes of an
$N\times M$ grid (the tensors $v_{1,r}$ go in the first row,
$v_{2,r}$ in the second row, etc.). The Young diagrams
occuring in the decomposition fit in this rectangle. The highest
weight vector corresponding to $\lambda$ is obtained by taking the
wedge product of the basis vectors in the boxes of the Young diagram
$\lambda$. For example, if $\lambda=(3,1)$,
we take $v_{1,1}\wedge v_{1,2}\wedge v_{1,2}\wedge v_{2,1}$.
It is indeed of weight $\lambda$ for $\mathit{GL}_N$ and of
weight $\lambda^t$ for $\mathit{GL}_M$.
The quantum group version of Howe duality is known: in the analogous
case of the symmetric algebra it is discussed in \cite{Baumann1999},
\cite{ToledanoLaredo2002} and \cite{Zhang2002}.
The skew-symmetric case we use here is
explained in \cite{CautisKamnitzerLicata2010} (for $M=2$), \cite{LehrerZhangZhang2011} and
\cite{CautisKamnitzerMorrison2014}.
Here we present
an equivalent variant based on the Clifford algebra action.
Let $V=\mathbb C^N$ with basis $v_1,\dots,v_N$ be the vector
representation of $U_q\mathfrak {gl}_N$ and $W=\mathbb C^M$ with basis
$w_1,\dots,w_M$ be the vector representation of
$U_q\mathfrak{gl}_M$. Then we have an isomorphism of graded
commutative algebras
\[
\bigwedge(V\otimes W)\cong\bigwedge(V\oplus\cdots\oplus V) \to
\left(\bigwedge V\right)^{\otimes M} =\bigwedge
V\otimes\cdots\otimes \bigwedge V
\]
sending $v_i\otimes w_j$ to
$1\otimes\cdots\otimes 1\otimes v_i\otimes 1\otimes \cdots\otimes 1$
with $v_i$ placed in the $j$-th factor. The right-hand side is a
tensor product of $U_q\mathfrak{gl}_N$-modules and thus
$U_q\mathfrak{gl}_N$ acts on it by the iterated coproduct. Similarly
we have an isomorphism
$\bigwedge(V\otimes W)\cong \bigwedge W\otimes\cdots \otimes\bigwedge
W$ sending $v_i\otimes w_j$ to
$1\otimes \cdots\otimes 1\otimes w_j\otimes 1\otimes \cdots \otimes 1$
with $i$-th factor $w_j$. Here we let $U_q\mathfrak{gl}_M$ act via
the {\em opposite} coproduct $\Delta'=\sigma\circ\Delta$, with
$\sigma(a\otimes b)=b\otimes a$.
\begin{prop}\label{p-5}
The actions of $(U_q\mathfrak{gl}_N,\Delta)$ and
$(U_q\mathfrak{gl}_M,\Delta')$ on $\bigwedge(V\otimes W)$ pulled
back by the isomorphisms
\[
\left(\bigwedge W\right)^{\otimes N}\leftarrow\bigwedge(V\otimes W)
\rightarrow\left(\bigwedge V\right)^{\otimes M}
\]
commute. If $q$ is not a root of unity the images of
$U_q\mathfrak{gl}_N$ and $U_q\mathfrak {gl}_M$ in
$\operatorname{End}\bigwedge(V\otimes W)$ are commutants of each
other and we have a decomposition
\begin{equation}\label{e-dec}
\bigwedge(V\otimes W)\cong\bigoplus_\lambda V^N_\lambda\otimes V^M_{\lambda^t}
\end{equation}
into simple $U_q\mathfrak{gl}_N\otimes U_q\mathfrak{gl}_M$-modules,
where $\lambda$ runs over Young diagrams with at most $N$ rows and
at most $M$ columns.
\end{prop}
For the proof we view $\bigwedge(V\otimes W)$ as the Fock module of
the Clifford algebra of $V\otimes W\oplus (V\otimes W)^*$. It has
generators $\psi^*_{i,r},\psi_{i,r}$ corresponding to the tensor basis
$v_i\otimes w_r$ and its dual basis. As in Section \ref{ss-2.3} we introduce the
operators
$t_{i,r}^{\pm1}=q^{\pm1}\psi_{i,r}^*\psi_{i,r}+\psi_{i,r}\psi_{i,r}^*$.
Via the isomorphism to $(\bigwedge V)^{\otimes M}$, $\psi_{i,r}^*$
acts as
$\mathrm{id}^{\otimes(r-1)}\otimes \psi_i^*\otimes\mathrm{id}^{\otimes
(N-r)}$ and $\psi_{i,r}$ acts as
$\mathrm{id}^{\otimes(r-1)}\otimes \psi_i\otimes\mathrm{id}^{\otimes
(N-r)}$. Thus the pull-back of the action of $U_q\mathfrak{gl}_N$ on
$\bigwedge(V\otimes W)$ is given on generators by
\begin{align*}
e_i&\mapsto\sum_{r=1}^M\prod_{s=1}^{r-1} k_{i,s}
\psi^*_{i,r}\psi_{i+1,r}, \quad k_{i,s}=t_{i,s}t_{i+1,s}^{-1}\\
f_i&\mapsto\sum_{r=1}^M
\psi^*_{i+1,r}\psi_{i,r}\prod_{s={r+1}}^M k_{i,s}^{-1}
\\
t_i^{\pm1}&\mapsto \prod_{r=1}^Mt_{i,r}^{\pm1}.
\end{align*}
Similary we can pull-back the action of $U_q\mathfrak{gl}_M$ on
$(\bigwedge W)^{\otimes N}$ with the opposite coproduct. The action
of generators (denoted by capital letters to distinguish them)
\begin{align*}
E_r&\mapsto\sum_{i=1}^N\psi^*_{i,r}\psi_{i,r+1}
\prod_{j=i+1}^{N} K_{j,r},\quad K_{j,r}=t_{j,r}t_{j,r+1}^{-1},
\\
F_r&\mapsto\sum_{i=1}^N
\prod_{j=1}^{i-1} K_{j,r}^{-1}\psi^*_{i,r+1}\psi_{i,r},
\\
T_r^{\pm1}&\mapsto \prod_{i=1}^Nt_{i,r}^{\pm1}.
\end{align*}
In the case $M=2$ this reduces to the formulae of the previous section, by remembering the sign rule
in the definition of $F$. By construction, we obtain actions of $U_q{\mathfrak{gl}}_N$ and $U_q{\mathfrak{gl}}_M$ and we need
to check that they commute:
\begin{lemma}\label{l-2}
The actions of $U_q{\mathfrak{gl}}_N$ and $U_q{\mathfrak{gl}}_M$ on $\bigwedge(V\otimes W)$
commute.
\end{lemma}
\begin{proof} It is clear that $t_i^{\pm1}$ commutes with the action
of $U_q{\mathfrak{gl}}_M$ and that $T_r^{\pm1}$ commutes with the action of
$U_q{\mathfrak{gl}}_N$. Let us check that $[e_i,E_r]=0$. The cases of the other
generators are dealt with in the same way.
The only terms in the sum
defining $e_i$ and $E_r$ that contribute non-trivially to the commutator are
\begin{align*}
[e_i,E_r]&=\prod_{s<r}k_{i,s} [
\psi_{i,r}^*\psi_{i+1,r}+k_{i,r}\psi_{i,r+1}^*\psi_{i+1,r+1},
\\
&\quad
\psi_{i,r}^*\psi_{i,r+1}K_{i+1,r}+
\psi_{i+1,r}^*\psi_{i+1,r+1}] \prod_{j>i+1}K_{j,r}.
\end{align*}
The bracket on the right-hand side is
\begin{align*}
&[\psi_{i,r}^*\psi_{i+1,r}, \psi_{i+1,r}^*\psi_{i+1,r+1}]
+[k_{i,r}\psi_{i,r+1}^*\psi_{i+1,r+1}, \psi_{i,r}^*\psi_{i,r+1}K_{i+1,r}]
\\
&=\psi_{i,r}^*\psi_{i+1,r+1}
+k_{i,r}\psi_{i+1,r+1}\psi_{i,r}^*K_{i+1,r}
\\
&=\psi_{i,r}^*\psi_{i+1,r+1}+t_{i,r}\psi_{i+1,r+1}\psi_{i,r}^*t_{i+1,r+1}^{-1}
\end{align*}
We have the relations $t_{i,r}\psi_{i,r}^*=q\psi_{i,r}^*$ and
$\psi_{i+1,r+1}t_{i+1,r+1}^{-1}=q^{-1}\psi_{i+1,r+1}$ and we are left
with $\psi_{i,r}^*\psi_{i+1,r+1}+\psi_{i+1,r+1}\psi_{i,r}^*=0$.
\end{proof}
To complete the proof of Proposition \ref{p-5}, we observe that for
$q$ not a root of unity the decomposition into irreducible
representations in the classical case deforms to the quantum group
case. In fact the highest weight vectors generating irreducible
$U_q{\mathfrak{gl}}_N\otimes U_q{\mathfrak{gl}}_M$-modules are still given by Howe's
construction.
\begin{proof}[Proof of Theorem \ref{t-1}]
We use the formula of Date and Okado \cite{DateOkado1994} for the action of the $R$-matrix on highest weight vectors
of irreducible $U_q{\mathfrak{gl}}_N$-subrepresentations in
$\bigwedge^kV\otimes \bigwedge^{k'}V$. The decomposition into
irreducibles is best understood in terms of Howe duality: we consider
the case $M=2$ of the decomposition \eqref{e-dec}
of Proposition \ref{p-5}. The left-hand side is
\[
\textstyle{
\bigwedge(V\otimes\mathbb C^2)=\bigwedge(V\oplus V)=
\bigwedge V\otimes\bigwedge V=\oplus_{k,k'=0}^N\bigwedge^kV\otimes
\bigwedge^{k'} V
}
\]
and the summand labeled by $(k,k')\in\{0,\dots,N\}^2$ is the weight
space of weight $(k,k')$ for the $U_q{\mathfrak{gl}}_2$-action. The right-hand
side is a sum over Young diagram $\lambda$ with at most two columns
and at most $N$ rows. Thus
$\lambda=(2,\dots,2,1,\dots,1)=2^{\ell'}1^{\ell-\ell'}$ ($\ell'$
rows of length 2, $\ell-\ell'$ rows of length 1) and
$\lambda^t=(\ell,\ell')$ with $N\geq \ell\geq\ell'\geq0$. The
irreducible $U_q{\mathfrak{gl}}_2$-module $V^2_{(\ell,\ell')}$ has a non-trivial
weight space of weight $(k,k')$ if and only if $k+k'=\ell+\ell'$
for some $s\geq0$ and $\ell'-\ell\leq k-k'\leq \ell-\ell'$.
In this case the weight space is one-dimensional.
If $k\geq k'$ the corresponding highest weights are
\[
(\ell,\ell')=(k,k'),(k+1,k'-1),\dots,(k+n,k'-n),
\]
where $n=\min(k',N-k)$, whereas if $k\leq k'$ they are
\[
(\ell,\ell')=(k',k),(k'+1,k-1),\ldots,(k'+n,k-n),
\]
where $n=\min(k,N-k')$. In both cases we can write
\[
n=\min(k,k',N-k,N-k').
\]
With this notation, the weight $(k,k')$ subspace in the decomposition \eqref{e-dec}
gives the decomposition
\[
{\textstyle \bigwedge^{k}V\otimes\bigwedge^{k'}V}\cong\oplus_{s=0}^{n}
V^N_{(\overline k+s,\underline k-s)^t},\quad
\text{with $\overline k=\max(k,k')$ and
$\underline k=\min(k,k')$},
\]
as an $U_q{\mathfrak{gl}}_N$-module. The $s=0$ component is generated by the
highest weight vector $v^{(\overline{k},\underline{k})}_0=v^{\overline{k}}
\otimes v^{\underline{k}}$, see \eqref{e-hw}, and
the highest weight vector $v^{(k,k')}_s$ of $V^N_{(\overline k+s,\underline k-s)^t}$
is proportional to $F^{(s)}v_0^{(k+s,k'-s)}$ if $k\geq k'$ and to
$F^{(k'-k+s)}v_0^{(k'+s,k-s)}$ if $k\leq k'$.
We normalize it as in \cite{DateOkado1994}
by fixing the coefficient of
the basis vector $\psi^*_k\cdots\psi^*_1|0\rangle$ in the first factor. Namely,
\[
v_s^{(k,k')}=\psi_k^*\cdots\psi_1^*|0\rangle\otimes\psi^*_{\overline
k+s}\cdots \psi^*_{\overline k+1}\psi^*_{\underline
k-s}\cdots\psi_1^*|0\rangle+\cdots.
\]
Let
\[
\textstyle Q_s\colon \bigwedge^k
V\otimes\bigwedge^{k'}V\to\bigwedge^{k'} V\otimes\bigwedge^{k}V
\]
be the unique $U_q{\mathfrak{gl}}_N$-linear map sending $v^{(k,k')}_s$ to the
vector of the same weight $v^{(k',k)}_{s'}$, $s'=k-k'+s$ and vanishing on the
other irreducible components. In our
notation and with the normalization of Theorem \ref{t-1}, the formula of
\cite{DateOkado1994}, obtained by the fusion procedure, is
\[
\check R_{k,k'}(z)
=\sum_{s=0}^{n}(-1)^{kk'+(k-k')s}\prod_{j=1}^s\frac{z-q^{|k-k'|+2j}}{1-z
q^{|k-k'|+2j}}Q_s.
\]
To deduce a formula in terms of the $U_q{\mathfrak{gl}}_2$-action we need
to compare the normalizations of the highest weight vectors.
If $k\geq k'$ a straightforward computation using the formula for $F$
in \eqref{e-EFK} shows that
\[
v^{(k,k')}_s=(-1)^{sk}q^{ss'}F^{(s)}v^{(k+s,k'-s)}_0,
\quad
v^{(k',k)}_{s'}=(-1)^{s'k'}q^{ss'}F^{(s')}v^{(k+s,k'-s)}_0.
\]
It follows that for $k\geq k'$
\[
\check R_{k,k'}(z)F^{(s)}v_0^{(k+s,k'-s)}
=(-1)^{k'}\prod_{j=1}^s\frac{z-q^{|k-k'|+2j}}{1-z
q^{|k-k'|+2j}}F^{(s')}v_0^{(k+s,k'-s)}
\]
To prove the formula for $\check R$ given in the Theorem
we need to compute the action of $A_{k-k'}(z)$ on the irreducible
representation generated by the highest weight vector
$v^{(k+s,k'-s)}_0$ of $\mathfrak{sl}_2$-weight
$\ell=k-k'+2s\in\mathbb Z_{\geq0}$. Comparing with the formula for
$A_m(z)$ from the Appendix (see Lemma \ref{l-A1}) we see that
$\check R_{k,k'}(z)$ acts by the same factor on $F^{(s)}v_0^{(k+s,k'-s)}$
up to the sign $(-1)^{k'}=(-1)^{\min(k,k')}$.
If $k\leq k'$ the reasoning is similar: in this case the
highest weight vector $v^{(k,k')}_s$ is proportional
to $F^{(s')}v_0^{(k'+s,k-s)}$ with $s'=k'-k+s$ and
\[
v^{(k,k')}_s=(-1)^{sk'}q^{ss'}F^{(s')}v^{(k'+s,k-s)}_0, \quad
v^{(k',k)}_{s'}=(-1)^{s'k}q^{ss'}F^{(s)}v^{(k'+s,k-s)}_0.
\]
From this we obtain
\[
\check R_{k,k'}(z)F^{(s')}v_0^{(k'+s,k-s)}
=(-1)^{k}\prod_{j=1}^s\frac{z-q^{|k-k'|+2j}}{1-z
q^{|k-k'|+2j}}F^{(s)}v_0^{(k'+s,k-s)},
\]
which is the same as the action of $A_{k-k'}(z)$ up to the sign
$(-1)^{k}=(-1)^{\min(k,k')}$.
\end{proof}
\section{Dynamical Weyl group}\label{s-4}
In Theorem \ref{t-1} the braiding matrix for the tensor product of two
representations of $U_qL\mathfrak{gl}_N$ obtained from exterior powers
of the vector representation of $U_q\mathfrak{gl}_N$ are expressed
in terms of the action of $U_q\mathfrak{gl}_2$ via Howe duality.
In this section we consider braiding matrices on the tensor product of $M$
exterior powers and notice that they are correspondingly expressed
in terms of the action of $U_q\mathfrak{gl}_M$ via the simple root embeddings of
$U_q{\mathfrak{gl}}_2$.
The braid and inversion relations of the $\check R$-matrices acting on
a $M$-fold tensor product is then translated by Howe duality to braid and inversion
relations in $U_q{\mathfrak{gl}}_M$.
The result is then that these are relations of the
dynamical Weyl group.
Recall that the (ordinary) Weyl group of $\mathit{GL}_M$ is
the symmetric group $S_M$. It acts on $\mathbb C^M$ and
the weight lattice $\mathbb Z^M$ by permutations. It is generated by the transpositions $s_i\in S_M$ of
$i$ and $i+1$ ($i=1,\dots,M-1$) with the relations $s_i^2=1$, $s_is_j=s_js_i$ for
$|i-j|\geq 2$ and $s_is_{i+1}s_i=s_{i+1}s_is_{i+1}$ for
$1\leq i\leq M-2$; the dynamical Weyl group relations are a version
with parameters.
Let us define formal series $A_{i,\mu}(z)$ of rational functions of
$M$-variables $z=(z_1,\dots,z_M)$ with values in
$U_q{\mathfrak{sl}}_M\subset U_q{\mathfrak{gl}}_M$ for $1\leq i\leq M-1$ and
$\mu\in P_M=\mathbb Z^M$:
\[
A_{i,\mu}(z)=\sum_{j=0}^{\infty}
(-q)^{j}
\frac{z_{i+1}-q^{|\mu_i-\mu_{i+1}|}z_i}
{z_{i+1}-q^{2j+|\mu_i-\mu_{i+1}|}z_i}
E_i^{(j+(\mu_{i+1}-\mu_i)_+)}F_i^{(j+(\mu_i-\mu_{i+1})_+)}.
\]
Here we set $(x)_+=\max(x,0)$ and, as before, we denote the generators
by capital letters to distinguish them from the generators of
$U_q{\mathfrak{gl}}_N$. For any finite dimensional $U_q{\mathfrak{gl}}_N$-module $U$ of type I
only finitely many terms contribute non-trivially to the sum and
$A_{i,\mu}(z)$ maps the weight space $U[\mu]$ of weight $\mu$ to
$U[s_i\mu]$. Notice that $A_{i,\mu}(z)$ depends on $\mu$ through its class
in the weight lattice $P_M=\mathbb Z^M/\mathbb Z(1,\dots,1)$ of $\mathfrak{sl}_M$.
For $M=2$, the map $\mathbb Z^2\to\mathbb Z$, $\mu\mapsto \mu_1-\mu_2$ induces
a bijection $P_2\cong \mathbb Z$. With this identification, $A_{1,m}(z_1,z_2)=A_m(z_1/z_2)$ where
\begin{equation}\label{e-5}
A_m(z)= \sum_{j\geq0} (-q)^{j}\frac{1-q^{|m|}z}{1-q^{2j+|m|}z}
\begin{cases}
E^{(j)}F^{(j+m)},& \text{if $m\geq 0$,}
\\
E^{(j+|m|)}F^{(j)},& \text{if $m\leq 0$.} \end{cases}
\end{equation}
The general series $A_{i,\mu}(z)$ reduces to this case via the
embedding $j_i\colon U_q\mathfrak{sl}_2\hookrightarrow
U_q\mathfrak{sl}_M$ sending $E,F,K^{\pm1}$ to $E_i,F_i,K_i^{\pm1}$:
\[ A_{i,\mu}(z)=j_i\left(A_{\mu(h_i)}(z_i/z_{i+1})\right).
\] By construction, see Theorem \ref{t-1}, the action of
$A_{i,\mu}(z)$ on the tensor product of exterior powers
$\bigwedge^{\mu_i}V(z_i)$ via Howe duality is the action of the
braiding matrix $\check R_{\mu_i,\mu_{i+1}}(z_i/z_{i+1})$ on the $i$th
and $i+1$st factors (up to an irrelevant sign). Note that this tensor
product is the weight space of weight $\mu$ for the action of
$U_q{\mathfrak{gl}}_M$ on the tensor product of $M$ copies of $\bigwedge V$.
\begin{theorem}\label{t-2} Let $U$ be any finite dimensional
$U_q{\mathfrak{gl}}_M$-module of type I. Then the restriction of $A_{i,\mu}(z)$ to
the weight space $U[\mu]$ of weight $\mu$ obeys the following
relations.
\begin{enumerate}
\item[(i)] For $1\leq i,j\leq M-1$ such that $|i-j|\geq 2$,
\[ A_{i,s_i\mu}(s_jz)A_{j,\mu}(z) =
A_{j,s_j\mu}(s_iz)A_{i,\mu}(z),
\] in $\operatorname{Hom}(U[\mu],U[s_is_j\mu])$.
\item[(ii)] For all $1\leq i\leq M-1$,
\[ A_{i,s_i\mu}(s_iz)A_{i,\mu}(z) =\mathrm{id}
\] in $\operatorname{End}(U[\mu])$.
\item[(iii)] For $1\leq i\leq M-2$,
\begin{align*} A_{i,s_{i+1}s_i\mu}&(s_{i+1}s_iz)
A_{i+1,s_i\mu}(s_iz)A_{i,\mu}(z) \\ &=
A_{i+1,s_is_{i+1}\mu}(s_is_{i+1}z)
A_{i,s_{i+1}\mu}(s_{i+1}z)A_{i+1,\mu}(z),
\end{align*} in $\operatorname{Hom}(U[\mu],U[s_is_{i+1}s_i\mu])$.
\end{enumerate}
\end{theorem}
\begin{proof} By complete reducibility we may assume that $U$ is
irreducible. The relations hold for the action of $A_{i,\mu}$ on
$\oplus_{i=1}^M\bigwedge^{\mu_i}V(z_i)$ via Howe duality: (i) follows
from the fact that the corresponding $\check R$-matrices act on
different factors of the tensor product, (ii) is the inversion
relation \eqref{e-4} and (iii) is the braiding relation \eqref{e-3}.
Thus the relations hold for any irreducible $U_q{\mathfrak{gl}}_M$-module
occurring in the decomposition of the tensor product. But from
Proposition \ref{p-5} we see that any partition does occurs as a highest weight if we
take $N$ sufficiently large. This covers all type I irreducible
representations of $U_q\mathfrak{sl}_M$. For $U_q{\mathfrak{gl}}_M$ we can obtain
all representations by tensoring these with a power of the one
dimensional representations on which $U_q\mathfrak{sl}_M$ acts
trivially (i.e. by the counit) and $t_i$ act by $q^{-1}$ for all
$i$. This has the effect of shifting the $\mu_i$s by a common amount
and does not affect the validity of the claim, since $A_{i,\mu}(z)$
depends on differences of $\mu_i$s.
\end{proof} This result can be reformulated as the construction of a
representation of the Weyl group on $U$-valued functions. Let $U$ be a
finite-dimensional type I $U_q{\mathfrak{gl}}_M$-module with weight decomposition
$U=\oplus_\mu U[\mu]$. Let $A_{i,U}(z)$ be the
$\operatorname{End}U$-valued rational function such that
\begin{equation}\label{e-6}
A_{i,U}(z)|_{U[\mu]}=A_{i,\mu}(z)\colon U[\mu]\to U[s_i\mu]
\end{equation}
for all weights $\mu$.
\begin{cor}\label{c-1} Let $U$ be a finite-dimensional $U_q{\mathfrak{gl}}_M$-module of type I.
The endomorphisms $s_1,\dots, s_{M-1}$ of the vector space $U(z)$ of
$U$-valued rational functions in $M$ variables $z$ given by
\[ (s_if)(z)=A_{i,U}(s_iz)f(s_iz),\quad i=1,\dots,M-1,
\] define a representation of the symmetric group $S_M$.
\end{cor}
\begin{proof} The proof is a straightforward verification that the
relations of the symmetric group are equivalent to the identities
(i)--(iii) of Theorem \ref{t-2}. For example, to check the
relation $s_is_{i+1}s_i=s_{i+1}s_is_{i+1}$ we write
\begin{align*} (s_is_{i+1}s_if)(z) &=A_{i,U}(s_iz)(s_{i+1}s_if)(s_iz)
\\ &=A_{i,U}(s_iz)A_{i+1,U}(s_{i+1}s_iz)(s_if)(s_{i+1}s_iz) \\
&=A_{i,U}(s_iz)A_{i+1,U}(s_{i+1}s_iz)A_{i,U}(s_is_{i+1}s_iz)f(s_{i}s_{i+1}s_iz).
\end{align*}
We get the left-hand side of (iii) if we replace $z$ by $ s_is_{i+1}s_iz$ and
restrict to the weight space $U[\mu]$.
\end{proof}
We notice that the operators $A_{i,U}(z)$ have similar properties as
the operators of the dynamical Weyl group of $U$ introduced and
studied in \cite{TarasovVarchenko2000,EtingofVarchenko2002} in the
framework of the theory of intertwiners. As in the case of the
dynamical Weyl group, their restriction to weight spaces are the image
of elements of $U_q\mathfrak{sl}_2$ by embeddings
$U_q\mathfrak{sl}_2\hookrightarrow U_q\mathfrak{sl}_M$ corresponding
to simple roots \footnote{In \cite{EtingofVarchenko2002} the dynamical
Weyl group is defined for any semisimple Lie algebra.}. They obey
relations (i) and (iii) of Theorem \ref{t-2} but not (ii). In
other words they define a representation of the braid group $B_M$ on
rational $U$-valued functions. In this representation the squares
$s_i^2$ of the generators act by multiplication by a rational function on each weight
space.
Let us compare our operators $A_{i,U}(z)$ with the operators
$\mathscr A_{s_i,U}(\lambda)$ of \cite[Section
4.2]{EtingofVarchenko2002} defining the dynamical action of generators,
in the case of $U_q\mathfrak{sl}_M$. The
latter operators are initially defined as functions of sufficiently
large antidominant weights $\lambda$. They extend (upon choosing a
logarithm of $q$) to meromorphic functions of
$\lambda\in \mathfrak h^*$ in the dual of the Cartan subalgebra
$\mathfrak h=\{x\in\mathbb C^M\,|\,\sum_i x_i=0\}$ of
$\mathfrak{sl}_M$. Let $h_i=(0,\dots,0,1,-1,0,\dots,0)$,
with $1$ in the position $i=1,\dots,M-1$, be the basis of coroots. To relate $z$ to $\lambda$
we identify $\mathfrak h^*$ with $\mathbb C^M/\mathbb C(1,\dots, 1)$
and write $z=q^{2\lambda}=(q^{2\lambda_1},\dots,q^{2\lambda_M})$. This
is defined up to simultaneous scaling of the variables
$z_i$. Since $A_{i,U}(z)$ is a function of the ratios of $z_j$
$A_{i,U}(q^{2\lambda})$ is well-defined.
\begin{prop}\label{p-3}
\[
A_{i,U}(q^{2\lambda})|_{U[\mu]}=
\begin{cases}
q^{-\mu(h_i)}\mathscr{A}_{s_i,U}(\lambda), &\text{if $\mu(h_i)\geq0$,}
\\
\displaystyle (-1)^{\mu(h_i)} q^{-\mu(h_i)}\frac{[\lambda(h_i)-\mu(h_i)/2]_q}{[\lambda(h_i)+\mu(h_i)/2]_q}
\mathscr{A}_{s_i,U}(\lambda),&\text{if
$\mu(h_i)\leq 0$}.
\end{cases}
\]
\end{prop}
\begin{proof}
Since $A_{i,\mu}$ are images of elements of $U_q\mathfrak{sl}_2$ by
embeddings into $U_q\mathfrak{sl}_M$, it is sufficient to consider
the case $M=2$ and we set $\lambda=\lambda_1-\lambda_2$ and $z=q^{2\lambda}=z_1/z_2$.
By complete reducibility we may also assume that
$U=L_\ell$ is the irreducible representation of $U_q\mathfrak{sl}_2$
of highest weight $\ell$ and dimension $\ell+1$. It has
one-dimensional weight spaces $L_\ell[\ell-2k]$ with basis
$v_{\ell-2k}=F^{(\ell)}v_\ell$, ($k=0,\dots,\ell$), in terms of the
highest weight vector $v_\ell$. In this case we have a unique
generator $s_1$ sending $x\in\mathfrak h^*$ to $-x$. The following
formula for $\mathscr A_{s_1,L_\ell}(\lambda)$ can be extracted from
\cite{EtingofVarchenko2002}, see Corollary 8 (iii), Proposition 12
and the definition of the dynamical action of the braid group at the
end of Section 4.2:
\[
\mathscr A_{s_1,L_{\ell}}(\lambda)v_{m}= (-1)^kq^{m}\prod_{j=1}^{k}
\frac{[\lambda+k-j+\ell/2]_q} {[\lambda-k+j-\ell/2]_q}v_{-m},\quad
m=\ell-2k.
\]
By setting $z=q^{2\lambda}$, this can be rewritten as
\[
\mathscr A_{s_1,L_{\ell}}(\lambda)v_{\ell-2k}=
(-1)^kq^{m+k(\ell-k+1)}\prod_{j=0}^{k-1}\frac{1-z
q^{-\ell+2j}}{1-zq^{\ell-2j}} v_{2k-\ell}.
\]
On the other hand, $A_{1,L_\lambda}(z)$ acts on the weight $m$
subspace of any representation as $A_m(z)$, see \eqref{e-5}.
It follows from Lemma \ref{l-A1} that
$A_m(z)=\varphi_m(z)\mathscr A_{s,L_\ell}(q^{2\lambda})$ with
$\varphi_m(z)=q^{-m}$ for $m\geq0$. If $m\leq0$, \dontprint{
\begin{align*}
\varphi_m(z)
&=
q^{-m+\frac14(\ell-|m|)(\ell+|m|+2)-\frac14(\ell+|m|)(\ell-|m|+2)}
\frac{
\prod_{j=0}^{\frac{\ell-|m|}2-1}\frac{1-zq^{-\ell+2j}}{1-zq^{\ell-2j}}
}
{
\prod_{j=0}^{\frac{\ell+|m|}2-1}\frac{1-zq^{-\ell+2j}}{1-zq^{\ell-2j}}
}
\\
&=
q^{-m+\frac14(\ell-|m|)(\ell+|m|+2)-\frac14(\ell+|m|)(\ell-|m|+2)}
\prod_{j=\frac{\ell-|m|}2}^{\frac{\ell+|m|}2-1}\frac{1-zq^{\ell-2j}}{1-zq^{-\ell+2j}}
\\
&=\frac{1-zq^{-m}}{1-zq^{m}}.
\end{align*}
}
\[
\varphi_m(z) =(-1)^m
\prod_{j=\frac{\ell-|m|}2}^{\frac{\ell+|m|}2-1}\frac{1-zq^{\ell-2j}}{1-zq^{-\ell+2j}}
=(-1)^m\frac{1-zq^{-m}}{1-zq^{m}}, \quad m\leq 0.
\]
so that
$\varphi_m(q^{2\lambda})=(-1)^mq^{-m}\frac{[\lambda-m/2]_q}
{[\lambda+m/2]_q}$.
\end{proof}
\subsection{$B$-operators, quantum Weyl group, Yangian limit}
The limiting values of $R$-matrices as the spectral parameter tends to $0$ or $\infty$ are
solutions of the Yang--Baxter equations without spectral parameter. They are $R$-matrices
for quantum enveloping algebras of finite dimensional Lie algebras.
By dividing the $A$-operators by their limiting values we obtain weight zero ``$B$-operators''
\cite{EtingofVarchenko2002}.
Define a formal series $B_m(z)$ by the formula
\begin{equation}\label{e-7}
A_m(z)=A_{m}(0)B_m(z).
\end{equation}
The series $B_m(z)$ is well-defined as an endomorphism of the
$m$-weight space of any finite dimensional representation and one has
a universal formula:
\begin{prop}\label{p-6}\
\begin{enumerate}
\item[(i)]
For $m\geq0$,
\[
B_m(z)=\sum_{j=0}^\infty
q^{\frac{j(j-3)}2}\frac{(-z)^j}{[j]_q!}F^jE^j\prod_{i=1}^j\frac{1-q^2}{1-z
q^{m+2i}}
\]
\item[(ii)]
For $m\leq0$,
\[
B_m(z)=\sum_{j=0}^\infty
q^{\frac{j(j-3)}2}\frac{(-z)^j}{[j]_q!}E^jF^j\prod_{i=1}^j\frac{1-q^2}{1-z
q^{-m+2i}}
\]
\end{enumerate}
\end{prop}
\begin{proof}
The first formula can be checked directly or deduced from the remark
after Corollary 40 in \cite{EtingofVarchenko2002}, by using
Proposition \ref{p-3}. There is a similar formula in {\it
loc.~cit.}, Proposition 14, but probably the powers of $q$ need to
be adjusted there. The second formula follows by applying the Cartan involution, see
Corollary \ref{c-A1}
\end{proof}
For $m=k-k'$, these operators define via Howe duality
$U_q{\mathfrak{gl}}_N$-linear endomorphisms of
$\bigwedge^{k} V(z_1)\otimes\bigwedge^{k'}V(z_2)$. Their
representation theoretical meaning in terms of $U_qL{\mathfrak{gl}}_N$ is not
clear but in the limit $q\to 1$, $A_{m}(0)$ tends up to signs to the
permutation $P$ and in a suitable $z\to 1$ limit, $B_m(z)$ converges
to a rational $R$ matrix, intertwining the coproduct and the opposite
coproduct of the Yangian. More precisely we let
$q=\exp(\epsilon\hbar)$, $z=\exp(2\epsilon u)$ and consider the limit
$\epsilon\to0$:
\begin{theorem}\label{t-3}
Let $\check R_{k,k'}(z,q)$ be the family of braiding matrices of
Theorem \ref{t-1}, viewed as a linear map on the underlying vector
space $\bigwedge^kV\otimes \bigwedge^{k'}V$. Let
\[
R_{k,k'}(z,q)=P\check R_{k,k'}(z,q)\in\textstyle{\operatorname{End}_{\mathbb
C}(\bigwedge^kV\otimes \bigwedge^{k'}V)},\text{ with
$Pv'\otimes v=(-1)^{kk'}v\otimes v'$,}
\]
be the corresponding
$R$-matrix, normalized to be the identity on the product of highest
weight spaces. Then the limit
\[ R^{\mathrm{rat}}_{k,k'}(u,\hbar)=\lim_{\epsilon\to
0}R_{k,k'}(e^{\hbar \epsilon},e^{2\epsilon u})
\] exists and is a ${\mathfrak{gl}}_N$-invariant solution of the Yang--Baxter
equation. Moreover
\[ R^{\mathrm{rat}}_{k,k'}(u,\hbar)=
\sum_{j\geq0}\frac{(-\hbar)^j}{j!}\prod_{i=1}^j\frac1{u+\hbar(i+|k-k'|/2)}
\begin{cases} F^jE^j&\text{if $k\geq k'$,} \\ E^jF^j&\text{if
$k\leq k'$.}
\end{cases}
\] Here $E=\sum_{i=1}^N\psi^*_i\otimes\psi_i$,
$F=-\sum_{i=1}^N\psi_i\otimes\psi_i^*$ are the $q\to 1$ limits of the
$U_q\mathfrak{sl}_2$-generators of Section \ref{s-2}.
\end{theorem}
\begin{proof} The limit exists since both $E, F$ and the coefficients
in the formula for $\check R_{k,k'}$ have a limit as $\epsilon\to
0$. The Yang--Baxter equation then follows from the braiding relation \eqref{e-3}
of $\check R_{k,k'}$. The action of the subalgebra $U_q{\mathfrak{gl}}_N$ does
not involve the spectral parameter and becomes an action of $U{\mathfrak{gl}}_N$,
so that the limit is ${\mathfrak{gl}}_N$-invariant.
In a neighbourhood of $q=1$, $\check R_{k,k'}(z,q)$ is holomorphic in
the disk $|z|<1/2$. Its value at $q=1$ is independent of $z$ and is a
homomorphism of $L{\mathfrak{gl}}_N$-modules, which are irreducible for
$z=z_1/ z_2\neq1$. By Schur's lemma
$\check R_{k,k'}(z,1)=\check R_{k,k'}(0,1)$ must be proportional to
the homomorphism $P\colon u\otimes v\mapsto (-1)^{kk'}v\otimes
u$. With the normalization of Theorem \ref{t-1}, the factor of proportionality is 1.
It follows that
\[
R^{\mathrm{rat}}_{k,k'}(u,\hbar) = \lim_{\epsilon\to 0}\check
R_{k,k'}(0,e^{\epsilon \hbar})^{-1} \check R_{k,k'}(e^{2\epsilon
u},e^{\epsilon\hbar}).
\]
By Theorem \ref{t-1} this can be expressed in terms of
$A_m(z)=A_m(z,q)$ for $m=k-k'$:
\[
R^{\mathrm{rat}}_{k,k'}(u,\hbar) = \lim_{\epsilon\to
0}A_m(0,e^{\epsilon\hbar})^{-1}A_m(e^{2\epsilon
u},e^{\epsilon\hbar}) = \lim_{\epsilon\to 0}B_m(e^{2\epsilon
u},e^{\epsilon\hbar}).
\]
Then, in the limit $\epsilon\to 0$, $E=\sum\psi_i\otimes\psi_i^*$ and
$F=-\sum\psi_i^*\otimes\psi_i$ obey $\mathfrak{sl}_2$ commutation
relations and, by Proposition \ref{p-6}, $B_m$ tends to
\[
\sum_{j=0}^\infty\frac{(-\hbar)^j}{j!}F^jE^j\prod_{i=1}^j\frac1{u+\hbar(i+m/2)}
\]
for $m\geq0$ and
\[
\sum_{j=0}^\infty\frac{(-\hbar)^j}{j!}E^jF^j\prod_{i=1}^j\frac1{u+\hbar(i-m/2)}
\]
for $m\leq 0$.
\end{proof}
\begin{remark} If $k=k'$ we recover the formula proposed by Smirnov, see \cite{Smirnov2016}, eq.~(115).
Understanding this formula was part of the motivation of the present work. For general $k,k'$ our formulas seem
to be related to the formula in \cite{Smirnov2016} by a shift of spectral parameter which could be partly explained by
a redefinition of exterior powers.
\end{remark}
|
1,108,101,563,219 | arxiv | \section{Introduction}
The $SO(10)$ grand unified theory (GUT)~\cite{Georgi:1974my,Fritzsch:1974nn} is a viable extension of the Standard Model (SM)
for its attractive features such as the embedding of the SM gauge groups into an anomaly-free group, the unification of one generation of the matter fields into a {\bf 16} representation field,
and the automatic realization of the seesaw mechanism~\cite{Gell-Mann:1979vob,Yanagida:1979as,Yanagida:1979gs,seesaw4} that naturally explains the tiny neutrino mass.
The supersymmetric (SUSY) $SO(10)$ GUT can further alleviate the gauge hierarchy problem,
and achieve the gauge coupling unification without intermediate scale.
A drawback of the SUSY $SO(10)$ GUT is that proton decay mediated by colored Higgsinos (dimension-five proton decay)~\cite{Weinberg:1981wj,Sakai:1981pk}
is too rapid to be consistent with the current experimental bounds on proton partial lifetimes.
In particular, since the unification of the top and bottom quark Yukawa couplings implies $\tan\beta\sim50$, the contribution of $E^cU^cU^cD^c$ operators~\cite{Goto:1998qg} to the $p\to K^+ \bar{\nu}_\tau$ decay is enhanced, and since simultaneous cancellations of $E^cU^cU^cD^c$ and $QQQL$ operators' contributions to $p\to K^+ \bar{\nu}_\tau$, and $QQQL$ operators' contributions to $p\to K^+ \bar{\nu}_\mu$ are difficult to realize, the SUSY $SO(10)$ GUT is in severe tension with the experimental bound on the $p\to K^+ \bar{\nu}$ partial lifetime~\cite{Super-Kamiokande:2014otb}.
However, there is a possibility that a texture of the Yukawa coupling matrices
suppresses the troublesome dimension-five proton decay, because the decay amplitudes are proportional to bi-products of Yukawa couplings.
In this paper, we pursue the above possibility and find conditions for a texture of the Yukawa coupling matrices suppressing dimension-five proton decay.
We further obtain specific Yukawa coupling matrices that satisfy the conditions.
We focus on a general renormalizable $SO(10)$ GUT model which contains ${\bf 10}+{\bf 126}+{\bf \overline{126}}+{\bf 120}$ representation fields from which the Higgs fields of the minimal SUSY SM (MSSM) originate
(a broader class of renormalizable SUSY $SO(10)$ GUT models have been studied in Refs.~\cite{Matsuda:2000zp}-\cite{Haba:2020ebr}).
In the model, the {\bf 16} matter fields have Yukawa couplings with the ${\bf 10},{\bf \overline{126}},{\bf 120}$ fields,
which provide the quark and lepton Yukawa couplings and Majorana mass of the singlet neutrinos.
Note that the Yukawa couplings with the ${\bf 10},{\bf \overline{126}},{\bf 120}$ fields are most general,
since ${\bf 16}\times {\bf 16}={\bf 10}+{\bf 126}+{\bf 120}$ and they are the only allowed renormalizable couplings involving a pair of {\bf 16} matter fields.
In the main body of the paper, we identify those components of the Yukawa coupling matrices that are involved in dimension-five proton decay and that can be on the order of the up quark Yukawa coupling without contradicting that they give the realistic quark and lepton Yukawa couplings.
Here the up quark Yukawa coupling is considered as the smallest scale of the components of the Yukawa coupling matrices
because it is a specially small Yukawa coupling in the SUSY $SO(10)$ GUT where $\tan\beta\sim50$.
That the components identified above be on the order of the up quark Yukawa coupling, is the desired
conditions for a texture suppressing dimension-five proton decay.
Next, we obtain specific Yukawa coupling matrices satisfying these conditions,
by fitting the experimental data of quark and lepton masses and flavor mixings
with the Yukawa coupling matrices of the ${\bf 10},{\bf \overline{126}},{\bf 120}$ fields
under the constraint that the components identified above be on the order of the up quark Yukawa coupling.
Then we calculate proton partial lifetimes from these Yukawa coupling matrices, and compare them with those calculated from Yukawa coupling matrices that do not necessarily satisfy the conditions. Thereby we evaluate how dimension-five proton decay is suppressed owing to the conditions.
Previously, suppression of dimension-five proton decay by a texture of the Yukawa coupling matrices in the SUSY $SO(10)$ GUT
has been studied in Refs.~\cite{Dutta:2004zh,Dutta:2005ni,Haba:2020ebr}.
Those papers deal with the case when the active neutrino mass is dominated by the contribution of the Type-2 seesaw mechanism
coming from a tiny VEV of the $SU(2)_L$-triplet component of the ${\bf \overline{126}}$ field.
However, the dominance of the Type-2 seesaw contribution is not a general situation,
since it requires a fine-tuning of a mass term, coupling constants
and VEVs of GUT-breaking fields~\cite{Haba:2020ebr,Goh:2004fy} so as not to spoil the gauge coupling unification.
Thus, the present paper considers the case when the active neutrino mass is generated solely by the Type-1 seesaw mechanism,
with singlet neutrinos coming from the {\bf 16} matter fields and their Majorana mass from the GUT-breaking VEV of the ${\bf \overline{126}}$ field.
This paper is organized as follows:
In Section~\ref{section-model}, we review the general renormalizable SUSY $SO(10)$ GUT model containing ${\bf 10}+{\bf 126}+{\bf \overline{126}}+{\bf 120}$ fields,
and write the formulas for partial widths of dimension-five proton decay.
In Section~\ref{section-identification}, we derive conditions for a texture of the Yukawa coupling matrices suppressing dimension-five proton decay.
It will turn out that not only the texture of the Yukawa coupling matrices, but also a certain texture of the colored Higgs mass matrix is needed to suppress dimension-five proton decay. The latter texture is studied in Section~\ref{section-coloredhiggs}.
In Section~\ref{section-fitting}, we numerically obtain specific Yukawa coupling matrices satisfying the conditions found in Section~\ref{section-identification}.
We further calculate proton partial lifetimes from them and evaluate how dimension-five proton decay is suppressed when these conditions are met.
Section~\ref{summary} summarizes the paper.
\\
\section{General Renormalizable SUSY $SO(10)$ GUT Model}
\label{section-model}
\subsection{Model Description}
We consider a SUSY $SO(10)$ GUT model that contains
single {\bf 10}, single pair of ${\bf 126}+{\bf \overline{126}}$, single {\bf 120} fields, denoted by $H,\Delta+\overline{\Delta},\Sigma$, respectively.
The matter fields of MSSM and a singlet neutrino of each generation are unified into a ${\bf 16}$ representation field, denoted by ${\bf 16}^i$ with $i$ being the flavor index.
The Yukawa couplings are given by
\begin{eqnarray}
W\ =\ (\tilde{Y}_{10})_{ij}\,{\bf 16}^i H{\bf 16}^j+(\tilde{Y}_{126})_{ij}\,{\bf 16}^i\overline{\Delta}{\bf 16}^j+(\tilde{Y}_{120})_{ij}\,{\bf 16}^i\Sigma{\bf 16}^j,
\label{gutyukawa}
\end{eqnarray}
where $\tilde{Y}_{10}$, $\tilde{Y}_{126}$, $\tilde{Y}_{120}$ are Yukawa coupling matrices in the flavor space,
and $\tilde{Y}_{10}$, $\tilde{Y}_{126}$ are complex symmetric and $\tilde{Y}_{120}$ is complex antisymmetric.
The quark and lepton Yukawa couplings are assumed to arise solely from Eq.~(\ref{gutyukawa}).
Additionally, we introduce single {\bf 210} and single {\bf 45} fields, denoted by $\Phi,A$, respectively.
The $\Phi,A$ develop vacuum expectation values (VEVs) to break $SU(5)$ subgroup of the $SO(10)$ while $\Delta+\overline{\Delta}$ develop VEVs to break $U(1)$ subgroup.
When the $SO(10)$ is broken into the SM gauge groups $SU(3)_C\times SU(2)_L\times U(1)_Y$, the $({\bf1},{\bf2},\pm\frac{1}{2})$ components of $H,\Delta,\overline{\Delta},\Sigma,\Phi$ yield the Higgs fields of MSSM.
Accordingly, $\tilde{Y}_{10},\tilde{Y}_{126},\tilde{Y}_{120}$ give
the up-type quark, down-type quark, charged lepton and neutrino Dirac Yukawa coupling matrices, $Y_u,Y_d,Y_e,Y_D$, as
\begin{eqnarray}
W_{\rm eff}\ =\ (Y_u)_{ij}\,Q_i H_u U_j^c+(Y_d)_{ij}\,Q_i H_d D_j^c+(Y_e)_{ij}\,L_i H_d E_j^c+(Y_D)_{ij}\,L_i H_u N_j^c,
\label{mssmyukawa}
\end{eqnarray}
where the Yukawa coupling matrices satisfy at a GUT breaking scale $\mu_{\rm GUT}$ the following relations:
\begin{eqnarray}
&&Y_u\ =\ Y_{10}+r_2\,Y_{126}+r_3\,Y_{120},
\label{yu}\\
&&Y_d\ =\ r_1\left(Y_{10}+Y_{126}+Y_{120}\right),
\label{yd}\\
&&Y_e\ =\ r_1\left(Y_{10}-3Y_{126}+r_e\,Y_{120}\right),
\label{ye}\\
&&Y_D\ =\ Y_{10}-3r_2\,Y_{126}+r_D\,Y_{120},
\label{ydirac}
\end{eqnarray}
where $Y_{10}\propto\tilde{Y}_{10}$, $Y_{126}\propto\tilde{Y}_{126}$, $Y_{120}\propto\tilde{Y}_{120}$,
and $r_1,r_2,r_3,r_e,r_D$ are complex numbers determined from the mass matrix of the $({\bf1},{\bf2},\pm\frac{1}{2})$ components.
Hereafter we perform a phase redefinition of fields to make $r_1$ real positive.
The GUT-breaking VEV of $\overline{\Delta}$, denoted by $\overline{v}_R$, provides the singlet neutrinos with Majorana mass as
\begin{eqnarray}
W_{\rm Majorana} \ =\ \frac{1}{2}(M_N)_{ij} \,N_i^c\,N_j^c, \ \ \ \ \ M_N \propto \tilde{Y}_{126}\,\overline{v}_R.
\label{majorana}
\end{eqnarray}
Integrating out $N_i^c$'s, we get the Weinberg operator
\begin{eqnarray}
W_{\rm eff} \ =\ \frac{1}{2}(C_\nu)_{ij}\, L_i H_u L_j H_u,
\label{weinberg}
\end{eqnarray}
where the Wilson coefficient $C_\nu$ satisfies at the scale of the singular values of $M_N$,
\begin{eqnarray}
C_\nu \ =\ -Y_DM_N^{-1}Y_D^T.
\label{cnu}
\end{eqnarray}
Eq.~(\ref{weinberg}) gives rise to the Type-1 seesaw contribution to the active neutrino mass.
In this paper, we assume that the VEV of the $({\bf 1},{\bf 3},1)$ component of $\overline{\Delta}$ is so small that the Type-2 seesaw contribution is negligible compared to the Type-1 seesaw one.
\\
\subsection{Dimension-five Proton Decay}
The $({\bf 3},{\bf1},-\frac{1}{3})+({\bf \overline{3}}, {\bf1},\frac{1}{3})$ components of $H$, $\Delta$, $\overline{\Delta}$, $\Sigma$, $\Phi$,
which we call colored Higgs fields and denote by $H_C^A,\overline{H}_C^B$ ($A,B$ are labels), induce dimension-five operators responsible for proton decay.
After the GUT breaking, the colored Higgs fields have GUT-scale mass ${\cal M}_{H_C}$ and Yukawa couplings with matter fields as
\begin{align}
W_{\rm colored \, Higgs}
&=\sum_{A,B} \ \overline{H}_C^A ({\cal M}_{H_C})_{AB} H_C^B
\nonumber\\
&+\sum_{A}\ \frac{1}{2}(Y_L^A)_{ij}Q_i H_C^A Q_j + (\overline{Y}^A_L)_{ij}Q_i \overline{H}^A_C L_j + (Y_R^A)_{ij}E^c_i H_C^A U^c_j + (\overline{Y}_R^A)_{ij}U^c_i \overline{H}_C^A D^c_j.
\label{coloredHiggs}
\end{align}
Here $Y_L^A,\overline{Y}_L^A,\,Y_R^A,\,\overline{Y}^A_R \propto Y_{10}$ when $H_C^A,\overline{H}^A_C$ are components of $H$,
and they are proportional to $Y_{126}$ when $H_C^A,\overline{H}^A_C$ are components of $\overline{\Delta}$.
When $H_C^A,\overline{H}^A_C$ are components of $\Sigma$, Yukawa couplings $\overline{Y}_L^A,\,Y_R^A,\,\overline{Y}^A_R$ are proportional to $Y_{120}$ while $Y_L^A$ vanishes because $Y_L^A$ must be symmetric.
In the other cases, all the Yukawa couplings vanish.
By integrating out the colored Higgs fields, we obtain dimension-five operators responsible for proton decay
\begin{eqnarray}
W_5 \ = \ -\frac{1}{2}C_{5L}^{ijkl}\,(Q_k Q_{l})(Q_i L_{j}) - C_{5R}^{ijkl}\,E^c_k U^c_{l} U^c_i D^c_{j},
\label{w5}
\end{eqnarray}
where isospin indices are summed in each bracket in the first term.
The Wilson coefficients satisfy at the scale of the singular values of ${\cal M}_{H_C}$,
\begin{align}
C_{5L}^{ijkl} \ &=\ \sum_{A,B}({\cal M}_{H_C}^{-1})_{AB}
\left\{(Y_L^A)_{kl}(\overline{Y}^B_L)_{ij}
-\frac{1}{2}(Y_L^A)_{li}(\overline{Y}^B_L)_{kj}-\frac{1}{2}(Y_L^A)_{ik}(\overline{Y}^B_L)_{lj}\right\},
\label{c5lgeneral}\\
C_{5R}^{ijkl} \ &=\ \sum_{A,B}({\cal M}_{H_C}^{-1})_{AB}
\left\{(Y_R^A)_{kl}(\overline{Y}_R^B)_{ij}-(Y_R^A)_{ki}(\overline{Y}_R^B)_{lj}\right\}.
\label{c5rgeneral}
\end{align}
We write the partial widths of the $p\to K^+ \bar{\nu}_\tau$, $p\to K^+ \bar{\nu}_\mu$, $p\to K^+ \bar{\nu}_e$, $p\to K^0 \mu^+$, $p\to K^0 e^+$ decays induced by dimension-five operators (other decay modes will be commented on in the last paragraph of Section~\ref{section-identification}).
The partial widths read, for $\beta=e,\mu$,
\begin{align}
\Gamma(p\to K^+\bar{\nu}_\tau)
= &\frac{m_N}{64\pi}\left(1-\frac{m_K^2}{m_N^2}\right)^2
\left\vert \beta_H(\mu_{\rm had})\frac{1}{f_\pi}\left\{
\left(1+\frac{D}{3}+F\right)C_{LL}^{s\tau \,ud}(\mu_{\rm had})+\frac{2D}{3}C_{LL}^{d\tau \,us}(\mu_{\rm had})\right\}
\right.
\nonumber\\
&\left. \ \ \ \ \ \ \ \ \ \ \ \ +\alpha_H(\mu_{\rm had})\frac{1}{f_\pi}\left\{
\left(1+\frac{D}{3}+F\right)C_{RL}^{ud \,\tau s}(\mu_{\rm had})+\frac{2D}{3}C_{RL}^{us \,\tau d}(\mu_{\rm had})
\right\}\right\vert^2,
\label{ptoknutau}
\\
\Gamma(p\to K^+\bar{\nu}_{\beta})
= &\frac{m_N}{64\pi}\left(1-\frac{m_K^2}{m_N^2}\right)^2
\left\vert \beta_H(\mu_{\rm had})\frac{1}{f_\pi}\left\{
\left(1+\frac{D}{3}+F\right)C_{LL}^{s\beta \,ud}(\mu_{\rm had})+\frac{2D}{3}C_{LL}^{d\beta \,us}(\mu_{\rm had})\right\}
\right\vert^2,
\label{ptoknumu}
\\
\Gamma(p\to K^0\beta^+)
= &\frac{m_N}{64\pi}\left(1-\frac{m_K^2}{m_N^2}\right)^2
\left\vert \beta_H(\mu_{\rm had})\frac{1}{f_\pi}
\left(1-D+F\right)\overline{C}_{LL}^{u\beta \,us}(\mu_{\rm had})
\right\vert^2,
\label{ptok0mu}
\end{align}
where the Wilson coefficients of dimension-six operators $C_{RL},C_{LL},\overline{C}_{LL}$ satisfy\footnote{
By writing $C_{5L}^{s\alpha\, ud}$, we mean that $Q_i$ is in the flavor basis where the down-type quark Yukawa coupling is
diagonalized and that the down-type quark component of $Q_i$ is exactly $s$ quark.
Likewise, $Q_k$ is in the flavor basis where the up-type quark Yukawa coupling is
diagonalized and its up-type component is exactly $u$ quark,
and $Q_l$ is in the flavor basis where the down-type quark Yukawa coupling is
diagonalized and its down-type quark component is exactly $d$ quark.
The same rule applies to other Wilson coefficients.
}, for $\alpha=e,\mu,\tau$,
\begin{align}
&C_{RL}^{ud\,\tau s}(\mu_{\rm had}) = A_{RL}(\mu_{\rm had},\mu_{\rm SUSY})
\frac{\mu_H}{m_{\tilde{t}_R}^2}
\frac{1}{16\pi^2}{\cal F}'(V^{\rm ckm}_{ts})^*\, y_t y_\tau
\,C_{5R}^{ud\tau t}\vert_{\mu=\mu_{\rm SUSY}},
\label{crl1}
\\
&C_{RL}^{us\,\tau d}(\mu_{\rm had}) = A_{RL}(\mu_{\rm had},\mu_{\rm SUSY})
\frac{\mu_H}{m_{\tilde{t}_R}^2}
\frac{1}{16\pi^2}{\cal F}'(V^{\rm ckm}_{td})^*\, y_t y_\tau
\,C_{5R}^{us\tau t}\vert_{\mu=\mu_{\rm SUSY}}
\label{crl2}
\\
&C_{LL}^{s\alpha\,ud}(\mu_{\rm had}) = A_{LL}(\mu_{\rm had},\mu_{\rm SUSY})
\frac{M_{\widetilde{W}}}{m_{\tilde{q}}^2}
\frac{1}{16\pi^2}{\cal F}\,
g_2^2\left(C_{5L}^{s\alpha\,ud}-C_{5L}^{u\alpha\, ds}\right)\vert_{\mu=\mu_{\rm SUSY}},
\label{cll1}
\\
&C_{LL}^{d\alpha\,us}(\mu_{\rm had}) = A_{LL}(\mu_{\rm had},\mu_{\rm SUSY})
\frac{M_{\widetilde{W}}}{m_{\tilde{q}}^2}
\frac{1}{16\pi^2}{\cal F}\,
g_2^2\left(C_{5L}^{d\alpha\, us}-C_{5L}^{u\alpha\, ds}\right)\vert_{\mu=\mu_{\rm SUSY}},
\label{cll2}
\\
&\overline{C}_{LL}^{u\beta\,us}(\mu_{\rm had}) = A_{LL}(\mu_{\rm had},\mu_{\rm SUSY})
\frac{M_{\widetilde{W}}}{m_{\tilde{q}}^2}
\frac{1}{16\pi^2}{\cal F}\,
g_2^2\left(-C_{5L}^{u\beta\, us}+C_{5L}^{s\beta\, uu}\right)\vert_{\mu=\mu_{\rm SUSY}},
\label{cll3}
\end{align}
and the Wilson coefficients of dimension-five operators satisfy
\begin{align}
&C_{5R}^{ud\tau t}(\mu_{\rm SUSY}) =
A_R^{\tau t}(\mu_{\rm SUSY},\mu_{\rm GUT})\sum_{A,B}({\cal M}_{H_C}^{-1})_{AB}\left.
\left\{(Y_R^A)_{\tau t}(\overline{Y}^B_R)_{ud}-(Y_R^A)_{\tau u}(\overline{Y}^B_R)_{td}\right\}\right|_{\mu=\mu_{\rm GUT}},
\label{c5r1}
\\
&C_{5R}^{us\tau t}(\mu_{\rm SUSY}) =
A_R^{\tau t}(\mu_{\rm SUSY},\mu_{\rm GUT})\sum_{A,B}({\cal M}_{H_C}^{-1})_{AB}\left.
\left\{(Y_R^A)_{\tau t}(\overline{Y}^B_R)_{us}-(Y_R^A)_{\tau u}(\overline{Y}_R^B)_{ts}\right\}\right|_{\mu=\mu_{\rm GUT}},
\label{c5r2}
\\
&C_{5L}^{s\alpha\, ud}(\mu_{\rm SUSY})-C_{5L}^{u\alpha\, ds}(\mu_{\rm SUSY})
\nonumber\\
&=A_L^\alpha(\mu_{\rm SUSY},\mu_{\rm GUT})\sum_{A,B}({\cal M}_{H_C}^{-1})_{AB}\,\left.
\frac{3}{2}\left\{(Y_L^A)_{ud}(\overline{Y}_L^B)_{s\alpha}-(Y_L^A)_{ds}(\overline{Y}_L^B)_{u\alpha}\right\}\right|_{\mu=\mu_{\rm GUT}},
\label{c5l1}\\
&C_{5L}^{d\alpha\, us}(\mu_{\rm SUSY})-C_{5L}^{u\alpha\, ds}(\mu_{\rm SUSY})
\nonumber\\
&=A_L^\alpha(\mu_{\rm SUSY},\mu_{\rm GUT})\sum_{A,B}({\cal M}_{H_C}^{-1})_{AB}\,\left.
\frac{3}{2}\left\{(Y_L^A)_{us}(\overline{Y}_L^B)_{d\alpha}-(Y_L^A)_{ds}(\overline{Y}_L^B)_{u\alpha}\right\}\right|_{\mu=\mu_{\rm GUT}},
\label{c5l2}
\\
&C_{5L}^{u\beta \,us}(\mu_{\rm SUSY})-C_{5L}^{s\beta \,uu}(\mu_{\rm SUSY})
\nonumber\\
&=A_L^\beta(\mu_{\rm SUSY},\mu_{\rm GUT})\sum_{A,B}({\cal M}_{H_C}^{-1})_{AB}\,\left.
\frac{3}{2}
\left\{(Y_L^A)_{us}(\overline{Y}_L^B)_{u\beta}-(Y_L^A)_{uu}(\overline{Y}_L^B)_{s\beta}\right\}\right|_{\mu=\mu_{\rm GUT}},
\label{c5l3}
\end{align}
where $\mu_{\rm had}$ denotes a hadronic scale, $\mu_{\rm SUSY}$ a soft SUSY breaking scale and $\mu_{\rm GUT}$ a GUT- breaking scale.
Here $\alpha_H,\beta_H$ denote the hadronic matrix elements, $D,F$ are parameters of the baryon chiral Lagrangian,
and $C_{LL},\overline{C}_{LL},C_{RL}$ are the Wilson coefficients of the effective Lagrangian where the SUSY particles are integrated out,
$-{\cal L}_6 = C_{LL}^{ijkl}(\psi_{u_Lk}\psi_{d_Ll})(\psi_{d_Li}\psi_{\nu_Lj})+\overline{C}_{LL}^{ijkl}(\psi_{u_Lk}\psi_{d_Ll})(\psi_{u_Li}\psi_{e_Lj})+C_{RL}^{ijkl}(\psi_{\nu_Lk}\psi_{d_Ll})(\psi_{u_R^ci}\psi_{d_R^cj})$
($\psi$ denote SM Weyl spinors, and spinor indices are summed in each bracket).
In Eqs.~(\ref{crl1})-(\ref{cll3}), $y_t,y_\tau,g_2$ denote the top quark Yukawa, tau lepton Yukawa and weak gauge couplings in MSSM, respectively, and $V^{\rm ckm}_{ij}$ denotes $(i,j)$-component of CKM matrix.
${\cal F}',{\cal F}$ are loop functions defined as ${\cal F}' =\frac{1}{x-y}(\frac{x}{1-x}\log x - \frac{y}{1-y}\log y)$
and ${\cal F} =\frac{1}{z-w}(\frac{z}{1-z}\log z - \frac{w}{1-w}\log w) + \frac{1}{z-1}(\frac{z}{1-z}\log z+1)$,
where $x=|\mu_H|^2/m_{\tilde{t}_R}^2$, $y=m_{\tilde{\tau}_R}^2/m_{\tilde{t}_R}^2$, $z=|M_{\widetilde{W}}|^2/m_{\tilde{q}}^2$, $w=m_{\tilde{\ell}_\alpha}^2/m_{\tilde{q}}^2$,
and $\mu_H,m_{\tilde{t}_R},m_{\tilde{\tau}_R},M_{\widetilde{W}},m_{\tilde{\ell}_\alpha},m_{\tilde{q}}$ denote the pole masses of Higgsinos, isospin-singlet top squark, isospin-singlet tau slepton,
Winos, isospin-doublet slepton of flavor $\alpha$, and 1st and 2nd generation isospin-doublet squarks, respectively.
$A_{LL},A_{RL}$ account for corrections from renormalization group (RG) evolutions in SM from scale $\mu_{\rm SUSY}$ to $\mu_{\rm had}$.
Here RG corrections involving SM Yukawa couplings other than the top quark one are neglected and thus $A_{LL},A_{RL}$ are flavor-universal.
$A_R^{\tau t},A_L^\alpha$ account for corrections from RG evolutions in MSSM from scale $\mu_{\rm GUT}$ to $\mu_{\rm SUSY}$.
We rewrite the flavor-dependent part of Eqs.~(\ref{c5r1})-(\ref{c5l3}) with the GUT Yukawa coupling matrices $Y_{10},Y_{126},Y_{120}$ as
($\alpha=e,\mu,\tau$; $\beta=e,\mu$)
\begin{align}
&\sum_{A,B}({\cal M}_{H_C}^{-1})_{AB}\left\{(Y_R^A)_{\tau t}(\overline{Y}^B_R)_{ud}-(Y_R^A)_{\tau u}(\overline{Y}^B_R)_{td}\right\}
\nonumber\\
&=\frac{1}{M_{H_C}}
\left[
a\left\{(Y_{10})_{\tau_R t_R}(Y_{10})_{u_Rd_R}-(Y_{10})_{\tau_R u_R}(Y_{10})_{t_Rd_R}\right\}
+b\left\{(Y_{10})_{\tau_R t_R}(Y_{126})_{u_Rd_R}-(Y_{10})_{\tau_R u_R}(Y_{126})_{t_Rd_R}\right\}\right.
\nonumber\\
&+c\left\{(Y_{10})_{\tau_R t_R}(Y_{120})_{u_Rd_R}-(Y_{10})_{\tau_R u_R}(Y_{120})_{t_Rd_R}\right\}
\nonumber\\
&+d\left\{(Y_{126})_{\tau_R t_R}(Y_{10})_{u_Rd_R}-(Y_{126})_{\tau_R u_R}(Y_{10})_{t_Rd_R}\right\}
+e\left\{(Y_{126})_{\tau_R t_R}(Y_{126})_{u_Rd_R}-(Y_{126})_{\tau_R u_R}(Y_{126})_{t_Rd_R}\right\}
\nonumber\\
&+f\left\{(Y_{126})_{\tau_R t_R}(Y_{120})_{u_Rd_R}-(Y_{126})_{\tau_R u_R}(Y_{120})_{t_Rd_R}\right\}
\nonumber\\
&+g\left\{(Y_{120})_{\tau_R t_R}(Y_{10})_{u_Rd_R}-(Y_{120})_{\tau_R u_R}(Y_{10})_{t_Rd_R}\right\}
+h\left\{(Y_{120})_{\tau_R t_R}(Y_{126})_{u_Rd_R}-(Y_{120})_{\tau_R u_R}(Y_{126})_{t_Rd_R}\right\}
\nonumber\\
&\left.+j\left\{(Y_{120})_{\tau_R t_R}(Y_{120})_{u_Rd_R}-(Y_{120})_{\tau_R u_R}(Y_{120})_{t_Rd_R}\right\}\right],
\nonumber\\
\label{yyr1}
\\
&\sum_{A,B}({\cal M}_{H_C}^{-1})_{AB}\left\{(Y_R^A)_{\tau t}(\overline{Y}^B_R)_{us}-(Y_R^A)_{\tau u}(\overline{Y}^B_R)_{ts}\right\}
={\rm (Above \ expression \ with \ exchange} \ d_R\leftrightarrow s_R),
\label{yyr2}
\\
&\sum_{A,B}({\cal M}_{H_C}^{-1})_{AB}\left\{(Y_L^A)_{ud}(\overline{Y}^B_L)_{s\alpha}-(Y_L^A)_{ds}(\overline{Y}^B_L)_{u\alpha}\right\}
\nonumber\\
&=\frac{1}{M_{H_C}}
\left[
a\left\{(Y_{10})_{u_L d_L}(Y_{10})_{s_L\alpha_L}-(Y_{10})_{d_L s_L}(Y_{10})_{u_L\alpha_L}\right\}
+b\left\{(Y_{10})_{u_L d_L}(Y_{126})_{s_L\alpha_L}-(Y_{10})_{d_L s_L}(Y_{126})_{u_L\alpha_L}\right\}\right.
\nonumber\\
&+c\left\{(Y_{10})_{u_L d_L}(Y_{120})_{s_L\alpha_L}-(Y_{10})_{d_L s_L}(Y_{120})_{u_L\alpha_L}\right\}
\nonumber\\
&+d\left\{(Y_{126})_{u_L d_L}(Y_{10})_{s_L\alpha_L}-(Y_{126})_{d_L s_L}(Y_{10})_{u_L\alpha_L}\right\}
+e\left\{(Y_{126})_{u_L d_L}(Y_{126})_{s_L\alpha_L}-(Y_{126})_{d_L s_L}(Y_{126})_{u_L\alpha_L}\right\}
\nonumber\\
&\left.+f\left\{(Y_{126})_{u_L d_L}(Y_{120})_{s_L\alpha_L}-(Y_{126})_{d_L s_L}(Y_{120})_{u_L\alpha_L}\right\}\right],
\nonumber\\
\label{yyl1}
\\
&\sum_{A,B}({\cal M}_{H_C}^{-1})_{AB}\left\{(Y_L^A)_{us}(\overline{Y}^B_L)_{d\alpha}-(Y_L^A)_{ds}(\overline{Y}^B_L)_{u\alpha}\right\}
={\rm (Above \ expression \ with \ exchange} \ d_L\leftrightarrow s_L),
\label{yyl2}
\\
&\sum_{A,B}({\cal M}_{H_C}^{-1})_{AB}\left\{(Y_L^A)_{us}(\overline{Y}^B_L)_{u\beta}-(Y_L^A)_{uu}(\overline{Y}^B_L)_{s\beta}\right\}
\nonumber\\
&=\frac{1}{M_{H_C}}
\left[
a\left\{(Y_{10})_{u_L s_L}(Y_{10})_{u_L\beta_L}-(Y_{10})_{u_L u_L}(Y_{10})_{s_L\beta_L}\right\}
+b\left\{(Y_{10})_{u_L s_L}(Y_{126})_{u_L\beta_L}-(Y_{10})_{u_L u_L}(Y_{126})_{s_L\beta_L}\right\}\right.
\nonumber\\
&+c\left\{(Y_{10})_{u_L s_L}(Y_{120})_{u_L\beta_L}-(Y_{10})_{u_L u_L}(Y_{120})_{s_L\beta_L}\right\}
\nonumber\\
&+d\left\{(Y_{126})_{u_L s_L}(Y_{10})_{u_L\beta_L}-(Y_{126})_{u_L u_L}(Y_{10})_{s_L\beta_L}\right\}
+e\left\{(Y_{126})_{u_L s_L}(Y_{126})_{u_L\beta_L}-(Y_{126})_{u_L u_L}(Y_{126})_{s_L\beta_L}\right\}
\nonumber\\
&\left.+f\left\{(Y_{126})_{u_L s_L}(Y_{120})_{u_L\beta_L}-(Y_{126})_{u_L u_L}(Y_{120})_{s_L\beta_L}\right\}\right],
\nonumber\\
\label{yyl3}
\end{align}
where $M_{H_C}$ is the scale of the singular values of ${\cal M}_{H_C}$, and $a,b,c,d,e,f,g,h,j$ are $O(1)$ numbers
determined from ${\cal M}_{H_C}$ as Ref.~\cite{Fukuyama:2004xs}-\cite{Bajc:2005qe}.
Here $(Y_{10})_{\tau_Rt_R}$ denotes the component of $Y_{10}$ in the term $(Y_{10})_{ij}\,\Psi_i H \Psi_j$
that involves the right-handed tau lepton component of $\Psi_i$ and the right-handed top quark component of $\Psi_j$.
Other symbols are defined analogously.
\\
\section{Conditions for a Texture of the Yukawa Coupling Matrices suppressing Dimension-five Proton Decay}
\label{section-identification}
We identify those components of the Yukawa coupling matrices $Y_{10},Y_{126},Y_{120}$
that are involved in dimension-five proton decay, namely, appear in Eqs.~(\ref{yyr1})-(\ref{yyl3}),
and that can be on the order of the up quark Yukawa coupling $y_u$.
Here the up quark Yukawa coupling is considered as the smallest scale of the components of the Yukawa coupling matrices
because it is a specially small Yukawa coupling in the SUSY $SO(10)$ GUT where $\tan\beta\sim50$.
That the components identified above be on the order of the up quark Yukawa coupling, is the desired conditions for a texture suppressing dimension-five proton decay.
\\
\begin{itemize}
\item
We focus on the first term in each \{...\} of Eqs.~(\ref{yyr1}),(\ref{yyr2}).
Components $(Y_{10})_{\tau_R t_R}$ and $(Y_{126})_{\tau_R t_R}$ are almost $(3,3)$-components of the symmetric matrices $Y_{10},Y_{126}$
and hence {\bf always} on the order of the top quark Yukawa coupling $y_t$.
For other components, at most two of $(Y_{10})_{u_R d_R}$, $(Y_{126})_{u_R d_R}$, $(Y_{120})_{u_R d_R}$
and at most two of $(Y_{10})_{u_R s_R}$, $(Y_{126})_{u_R s_R}$, $(Y_{120})_{u_R s_R}$
can be on the order of the up quark Yukawa coupling $y_u$.
However, all of them cannot be so because of the following equalities that result from Eq.~(\ref{yd}):
\begin{align}
(Y_{10})_{u_R d_R}+(Y_{126})_{u_R d_R}+(Y_{120})_{u_R d_R} \ &= \ \frac{1}{r_1}(Y_d)_{u_R d_R}
\nonumber\\
&\simeq \ \frac{y_t}{y_b}y_d\times (d_L\mathchar`-u_R \ {\rm part \ of \ the \ mixing \ matrix}),
\label{urdr}
\\
(Y_{10})_{u_R s_R}+(Y_{126})_{u_R s_R}+(Y_{120})_{u_R s_R} \ &= \ \frac{1}{r_1}(Y_d)_{u_R s_R}
\nonumber\\
&\simeq \ \frac{y_t}{y_b}y_s\times (s_L\mathchar`-u_R \ {\rm part \ of \ the \ mixing \ matrix}),
\label{ursr}
\end{align}
where $r_1$ is estimated to be $y_b/y_t$ so that the ratio of the top and bottom quark Yukawa couplings is reproduced.
$d_L\mathchar`-u_R \ {\rm part \ of \ the \ mixing \ matrix}$ is estimated to be about 1 and we get
$(Y_{10})_{u_R d_R}+(Y_{126})_{u_R d_R}+(Y_{120})_{u_R d_R}\simeq\frac{y_t}{y_b}y_d$, which is much greater than $y_u$.
Also, $s_L\mathchar`-u_R \ {\rm part \ of \ the \ mixing \ matrix}$
is estimated to be the Cabibbo angle $\lambda=0.22$ and hence
we get $(Y_{10})_{u_R s_R}+(Y_{126})_{u_R s_R}+(Y_{120})_{u_R s_R}\simeq0.22\times\frac{y_t}{y_b}y_s$,
which is much greater than $y_u$.
Therefore, to make the entire Eqs.~(\ref{yyr1}),(\ref{yyr2}) proportional to $y_u$,
we have to tune the colored Higgs mass matrix such that some of $a,b,c,d,e,f,g,h,j$ are much smaller than 1.
The most economical choice is to tune the colored Higgs mass matrix to make
\begin{eqnarray}
c=f=0
\label{coloredHiggsTexture}
\end{eqnarray}
and at the same time consider the following texture:
\begin{eqnarray}
&&(Y_{10})_{u_R d_R}=O(y_u), \ \ \ \ \ (Y_{126})_{u_R d_R}=O(y_u), \ \ \ \ \ (Y_{120})_{u_R d_R}\simeq\frac{y_t}{y_b}y_d,
\label{texture1}
\\
&&(Y_{10})_{u_R s_R}=O(y_u), \ \ \ \ \ (Y_{126})_{u_R s_R}=O(y_u), \ \ \ \ \ (Y_{120})_{u_R s_R}\simeq0.22\times\frac{y_t}{y_b}y_s.
\label{texture2}
\end{eqnarray}
The terms $j(Y_{120})_{\tau_R t_R}(Y_{120})_{u_R d_R}$ and $j(Y_{120})_{\tau_R t_R}(Y_{120})_{u_R s_R}$ appear to be not proportional to $y_u$.
However, since $Y_{120}$ is antisymmetric in the flavor space, $(Y_{120})_{\tau_R t_R}$, being a nearly diagonal component, is so small that
the above terms are suppressed compared to the other remaining terms.
\item
We focus on the second term in each \{...\} of Eq.~(\ref{yyr1}).
Each term is estimated to be $\sin^2\theta^{\rm ckm}_{13}\, y_t^2$ ($\theta^{\rm ckm}_{13}$ is the $(1,3)$-mixing angle of CKM matrix),
since each term contains two 1st-3rd generation flavor mixings.
The value of $\sin^2\theta^{\rm ckm}_{13}\, y_t^2$ is numerically close to $y_u\,y_t$.
Thus, these terms are {\bf always} on the same order as first terms
$(Y_{10})_{\tau_R t_R}(Y_{10})_{u_R d_R}$, $(Y_{10})_{\tau_R t_R}(Y_{126})_{u_R s_R}$, $(Y_{126})_{\tau_R t_R}(Y_{10})_{u_R d_R}$, $(Y_{126})_{\tau_R t_R}(Y_{126})_{u_R s_R}$
with the texture of Eq.~(\ref{texture1}).
Likewise the second term in each \{...\} of Eq.~(\ref{yyr2}) is estimated to be $\sin\theta^{\rm ckm}_{13}\sin\theta^{\rm ckm}_{23}\,y_t^2$.
These terms contribute to the proton decay amplitude
{\bf always} by a similar amount to the second terms in \{...\}'s of Eq.~(\ref{yyr1})
because they enter the proton decay amplitude in the form
$V_{ts}^{\rm ckm}(Y_A)_{\tau_R u_R}(Y_B)_{t_R d_R}$ and $V_{td}^{\rm ckm}(Y_A)_{\tau_R u_R}(Y_B)_{t_R s_R}$
and the CKM matrix components satisfy $|V_{ts}^{\rm ckm}|\sim\sin\theta^{\rm ckm}_{23}$ and
$|V_{td}^{\rm ckm}|\sim\sin\theta^{\rm ckm}_{13}$.
\item
We proceed to Eqs.~(\ref{yyl1}),(\ref{yyl2}).
It is impossible to suppress $(Y_{A})_{s_L \alpha_L}$, $(Y_{A})_{d_L \alpha_L}$, $(Y_{A})_{u_L \alpha_L}$ for {\bf all flavors} $\alpha=e,\mu,\tau$ to the order of $y_u$.
Therefore, we do not consider a texture where some of them are $O(y_u)$.
For other components, at least one of $(Y_{10})_{u_L s_L}$, $(Y_{10})_{d_L s_L}$, $(Y_{126})_{u_L s_L}$, $(Y_{126})_{d_L s_L}$ is on the order of
$|V^{\rm ckm}_{cd}|\frac{y_t}{y_b}y_s$ because of two equalities below,
\begin{align}
&(Y_{10})_{s_L c_L}+(Y_{126})_{s_L c_L}+(Y_{120})_{s_L c_L} \ \simeq \ \frac{y_t}{y_b}y_s
\times (c_L\mathchar`-s_R \ {\rm part \ of \ the \ mixing \ matrix)},
\label{clsl}
\\
&(Y_{10})_{d_L s_L}+(Y_{126})_{d_L s_L}-V^{\rm ckm}_{ud} \left\{ (Y_{10})_{u_L s_L}+(Y_{126})_{u_L s_L} \right\}
\nonumber\\
&\ \ \ \ \ \ \ \ =\ V^{\rm ckm}_{cd} \left\{ (Y_{10})_{c_L s_L}+(Y_{126})_{c_L s_L} \right\}
+V^{\rm ckm}_{td} \left\{ (Y_{10})_{t_L s_L}+(Y_{126})_{t_L s_L} \right\},
\label{dlsl}
\end{align}
and by the facts that $c_L\mathchar`-s_R \ {\rm part \ of \ the \ mixing \ matrix}$ is about 1 because $c_L$ and $s_R$ are 2nd generation flavors,
and that $(Y_{120})_{s_L c_L}$ is suppressed compared to $(Y_{10})_{c_L s_L},(Y_{126})_{c_L s_L}$ because it is nearly $(2,2)$-component of the antisymmetric matrix $Y_{120}$.\footnote{
The situation that the term $V^{\rm ckm}_{td} \left\{ (Y_{10})_{t_L s_L}+(Y_{126})_{t_L s_L} \right\}$
cancels the term $V^{\rm ckm}_{cd} \left\{ (Y_{10})_{c_L s_L}+(Y_{126})_{c_L s_L} \right\}$ is incompatible with the correct quark Yukawa couplings.}
As a result, at least one of $(Y_{10})_{u_L s_L}$, $(Y_{10})_{d_L s_L}$, $(Y_{126})_{u_L s_L}$, $(Y_{126})_{d_L s_L}$ cannot be on the order of $y_u$.
Still, it is possible to make the entire Eqs.~(\ref{yyl1}),(\ref{yyl2}) proportional to $y_u$
by tuning the colored Higgs mass matrix such that $a,b,d,e$ satisfy
\begin{eqnarray}
a\, (Y_{10})_{d_L s_L}+d\, (Y_{126})_{d_L s_L}=0, \ \ \ \ \ \ \ b\, (Y_{10})_{d_L s_L}+e\, (Y_{126})_{d_L s_L}=0
\end{eqnarray}
and at the same time considering the following texture:
\begin{eqnarray}
(Y_{10})_{u_L s_L}=O(y_u), \ \ \ \ \ (Y_{126})_{u_L s_L}=O(y_u).
\end{eqnarray}
\item
Finally, we focus on Eq.~(\ref{yyl3}).
The only components that do not appear in Eqs.~(\ref{yyl1}),(\ref{yyl2}) are $(Y_{10})_{u_L u_L}$, $(Y_{126})_{u_L u_L}$,
and there is no obstacle in considering the following texture:
\begin{eqnarray}
(Y_{10})_{u_L u_L}=O(y_u), \ \ \ \ \ (Y_{126})_{u_L u_L}=O(y_u).
\end{eqnarray}
\end{itemize}
To summarize, dimension-five proton decay is suppressed if the components below are all on the order of the up quark Yukawa coupling $y_u$,
\begin{eqnarray}
&&(Y_{10})_{u_R d_R}, \ (Y_{126})_{u_R d_R}, \ (Y_{10})_{u_R s_R}, \ (Y_{126})_{u_R s_R},
\nonumber\\
&&(Y_{10})_{u_L d_L}, \ (Y_{126})_{u_L d_L}, \ (Y_{10})_{u_L s_L}, \ (Y_{126})_{u_L s_L}, \ (Y_{10})_{u_L u_L}, \ (Y_{126})_{u_L u_L},
\label{tobeyu}
\end{eqnarray}
and at the same time the colored Higgs mass matrix is tuned such that $a,b,c,d,e,f,g,h,j$ in Eqs.~(\ref{yyr1})-(\ref{yyl3}) satisfy
\begin{eqnarray}
&&c=f=0,
\label{coloredhiggstex1}\\
&&a\, (Y_{10})_{d_L s_L}+d\, (Y_{126})_{d_L s_L}=0, \ \ \ \ \ \ b\, (Y_{10})_{d_L s_L}+e\, (Y_{126})_{d_L s_L}=0.
\label{coloredhiggstex2}
\end{eqnarray}
That the components of Eq.~(\ref{tobeyu}) be on the order of the up quark Yukawa coupling, is the desired conditions for a texture of the Yukawa coupling matrices suppressing dimension-five proton decay.
We comment on nucleon decay modes other than Eqs.~(\ref{ptoknutau})-(\ref{ptok0mu}).
The partial widths of the $N\to \pi \beta^+$ and $p\to \eta \beta^+$ decays ($\beta=e,\mu$)
involve the same Yukawa coupling components as the $p\to K^0\beta^+$ except that $s_L$ is replaced by $d_L$.
Hence, once we consider the texture where $(Y_{10})_{u_L u_L}$, $(Y_{126})_{u_L u_L}$ are on the order of the up quark Yukawa coupling,
the $N\to \pi \beta^+$ and $p\to \eta \beta^+$ decays are also suppressed.
Constraints on the rest of the decay modes are relatively weak~\cite{ParticleDataGroup:2022pth} and are not in tension with the SUSY $SO(10)$ GUT.
\\
\section{Texture of the Colored Higgs Mass Matrix}
\label{section-coloredhiggs}
We present a texture of the colored Higgs mass matrix that gives $c=f=0$ and $a/d=b/e$.
Here $a/d=b/e$ is a necessary condition for Eq.~(\ref{coloredhiggstex2}).
We utilize the result of Ref.~\cite{Fukuyama:2004ps}, and use the same notation of fields, coupling constants and VEVs except that {\bf 120} field is written as $\Sigma$ in our paper.
The definitions of the couplings, coupling constants and masses are summarized in Appendix~A.
The desired texture of the colored Higgs mass matrix is obtained from the following relations of the coupling constants and VEVs:
\begin{align}
\lambda_{18} \ &= \ 0, \ \ \ \ \
\lambda_{20} \ = \ 0, \ \ \ \ \
\frac{\lambda_{21}}{\lambda_{19}}\ = \ 3\frac{\lambda_{17}}{\lambda_{16}},
\nonumber\\
i\,A_1 \ &= \ -\frac{1}{6}\frac{\lambda_{21}}{\lambda_{19}}\Phi_3, \ \ \ \ \
i\,A_2 \ = \ -\frac{\sqrt{3}}{6}\frac{\lambda_{21}}{\lambda_{19}}\Phi_2.
\label{texture}
\end{align}
The above relations are obtained by fine-tuning, which is natural at the quantum level thanks to the non-renormalization theorem.
The VEV configuration of Eq.~(\ref{texture}) can be consistent with the $F$-flatness conditions of the six SM-gauge-singlet components $\Phi_1,\Phi_2,\Phi_3,A_1,A_2,v_R$
when six model parameters, for example $\lambda_5/\lambda_2,\,\lambda_6/\lambda_2,\,\lambda_7/\lambda_2,\,m_1,m_2,m_4$, are tuned appropriately.
When Eq.~(\ref{texture}) holds, the colored Higgs mass matrix has the following texture:
\begin{align}
&W_{\rm colored\,Higgs} \supset \begin{pmatrix}
H^{(\overline{3},1,\frac{1}{3})} & \overline{\Delta}^{(\overline{3},1,\frac{1}{3})}_{(6,1,1)} & \Delta^{(\overline{3},1,\frac{1}{3})}_{(6,1,1)} & \Delta^{(\overline{3},1,\frac{1}{3})}_{(\overline{10},1,3)}
& \Phi^{(\overline{3},1,\frac{1}{3})} & \Sigma^{(\overline{3},1,\frac{1}{3})}_{(6,1,3)} & \Sigma^{(\overline{3},1,\frac{1}{3})}_{(\overline{10},1,1)}
\end{pmatrix}{\cal M}_{H_C}
\begin{pmatrix}
H^{(3,1,-\frac{1}{3})} \\ \Delta^{(3,1,-\frac{1}{3})}_{(6,1,1)} \\ \overline{\Delta}^{(3,1,-\frac{1}{3})}_{(6,1,1)} \\ \overline{\Delta}^{(3,1,-\frac{1}{3})}_{(10,1,3)}
\\ \Phi^{(3,1,-\frac{1}{3})} \\ \Sigma^{(3,1,-\frac{1}{3})}_{(6,1,3)} \\ \Sigma^{(3,1,-\frac{1}{3})}_{(10,1,1)}
\end{pmatrix},
\end{align}
where
\begin{align}
&{\cal M}_{H_C}
=\begin{pmatrix}
m_3 &\frac{ \lambda_3\Phi_2}{\sqrt{30}}-\frac{ \lambda_3\Phi_1}{\sqrt{10}} & -\frac{\lambda_4\Phi_1}{\sqrt{10}}-\frac{\lambda_4\Phi_2}{\sqrt{30}}
& -\sqrt{\frac{2}{15}}\lambda_4\Phi_3 & \frac{\lambda_4\overline{v}_R}{\sqrt{5}} & 0 & 0 \\
\frac{\lambda_4\Phi_2}{\sqrt{30}}-\frac{\lambda_4\Phi_1}{\sqrt{10}} & m_2+i\frac{\lambda_{21}}{\lambda_{19}}\frac{\lambda_6\Phi_2}{30\sqrt{2}} & 0 & 0 &0 &0 & 0 \\
-\frac{\lambda_3\Phi_1}{\sqrt{10}}-\frac{\lambda_3\Phi_2}{\sqrt{30}} & 0 & m_2-i\frac{\lambda_{21}}{\lambda_{19}}\frac{\lambda_6\Phi_2}{30\sqrt{2}} &\frac{\lambda_2\Phi_3}{15\sqrt{2}} & -\frac{\lambda_2\overline{v}_R}{10\sqrt{3}} & 0 & 0 \\
-\sqrt{\frac{2}{15}}\lambda_3\Phi_3 & 0 & \frac{\lambda_2\Phi_3}{15\sqrt{2}} & m_{66} & -\frac{\lambda_2\overline{v}_R}{5\sqrt{6}} & 0 & 0 \\
\frac{\lambda_3v_R}{\sqrt{5}} & 0 & -\frac{\lambda_2v_R}{10\sqrt{3}} & -\frac{\lambda_2v_R}{5\sqrt{6}} & m_{77} & 0 & 0 \\
-\lambda_{17}\frac{\Phi_3}{\sqrt{3}} & 0 & \lambda_{21}\frac{\Phi_3}{6\sqrt{5}} & \lambda_{21}\frac{\Phi_2}{3\sqrt{5}} & \frac{\lambda_{21}\overline{v}_R}{2\sqrt{15}} & m_{22} & \frac{2\lambda_{15}\Phi_3}{9} \\
-\sqrt{\frac{2}{3}}\lambda_{17}\Phi_2 & 0 & \lambda_{21}\frac{\Phi_2}{3\sqrt{10}} & \lambda_{21}\frac{\Phi_3}{3\sqrt{10}} &
\frac{\lambda_{21}\overline{v}_R}{2\sqrt{15}} & \frac{2\lambda_{15}\Phi_3}{9} & m_{33}
\end{pmatrix}
\label{texture-coloredhiggs}
\\
&m_{66}= m_2+\lambda_2(\frac{\Phi_1}{10\sqrt{6}}+\frac{\Phi_2}{30\sqrt{2}})-i\frac{\lambda_{21}}{\lambda_{19}}\frac{\lambda_6\Phi_2}{30\sqrt{2}}
\\
&m_{77} = m_1+\lambda_1(\frac{\Phi_1}{\sqrt{6}}+\frac{\Phi_2}{3\sqrt{2}}+\frac{2\Phi_3}{3})-i\frac{\lambda_{21}}{\lambda_{19}}\frac{\sqrt{2}\lambda_7\Phi_2}{15}
\\
&m_{22} = m_6+\frac{1}{3}\sqrt{\frac{2}{3}}\lambda_{15}\Phi_1
\\
&m_{33} = m_6+\frac{\sqrt{2}}{9}\lambda_{15}\Phi_2.
\end{align}
The dimension-five operators responsible for proton decay Eq.~(\ref{w5}) satisfy at the scale of the colored Higgs mass,
\begin{align}
W_5=&-\begin{pmatrix}
(\tilde{Y}_{10})_{ij} & 0 & (\tilde{Y}_{126})_{ij} & (\tilde{Y}_{126})_{ij}
& 0 & (\tilde{Y}_{120})_{ij} & (\tilde{Y}_{120})_{ij}
\end{pmatrix}{\cal M}_{H_C}^{-1}
\begin{pmatrix}
(\tilde{Y}_{10})_{kl} \\ (\tilde{Y}_{126})_{kl} \\ 0 \\ 0 \\ 0 \\ (\tilde{Y}_{120})_{kl} \\ (\tilde{Y}_{120})_{kl}
\end{pmatrix}E_i^cU_j^cU_k^cD_l^c
\nonumber\\
&-\begin{pmatrix}
(\tilde{Y}_{10})_{ij} & 0 & (\tilde{Y}_{126})_{ij} & (\tilde{Y}_{126})_{ij}
& 0 & 0 & 0
\end{pmatrix}{\cal M}_{H_C}^{-1}
\begin{pmatrix}
(\tilde{Y}_{10})_{kl} \\ (\tilde{Y}_{126})_{kl} \\ 0 \\ 0 \\ 0 \\ (\tilde{Y}_{120})_{kl} \\ (\tilde{Y}_{120})_{kl}
\end{pmatrix}(Q_iQ_j)(Q_kL_l),
\end{align}
where the inverse of the colored Higgs mass ${\cal M}_{H_C}^{-1}$ has the following properties resulting from Eq.~(\ref{texture-coloredhiggs}):
\begin{itemize}
\item
The upper-right $5\times 2$ part of ${\cal M}_{H_C}^{-1}$ is zero because the upper-right $5\times 2$ part of ${\cal M}_{H_C}$ is zero.
Thus, the coefficients proportional to $(\tilde{Y}_{10})_{ij}(\tilde{Y}_{120})_{kl}$ or $(\tilde{Y}_{126})_{ij}(\tilde{Y}_{120})_{kl}$ are zero, namely,
we get $c=f=0$ in Eqs.~(\ref{yyr1})-(\ref{yyl3}).
\item
The upper-left $5\times5$ part of ${\cal M}_{H_C}^{-1}$ is given by the inverse of the upper-left $5\times5$ part of ${\cal M}_{H_C}$, since the upper-right $5\times 2$ part of ${\cal M}_{H_C}$ is zero.
Then the equalities $({\cal M}_{H_C})_{32}=({\cal M}_{H_C})_{42}=({\cal M}_{H_C})_{52}=0$
lead to the relation $({\cal M}_{H_C}^{-1})_{11}:({\cal M}_{H_C}^{-1})_{31}:({\cal M}_{H_C}^{-1})_{41}=
({\cal M}_{H_C}^{-1})_{12}:({\cal M}_{H_C}^{-1})_{32}:({\cal M}_{H_C}^{-1})_{42}$.
Since the coefficients proportional to $(\tilde{Y}_{10})_{ij}(\tilde{Y}_{10})_{kl}$, $(\tilde{Y}_{10})_{ij}(\tilde{Y}_{126})_{kl}$, $(\tilde{Y}_{126})_{ij}(\tilde{Y}_{10})_{kl}$ or $(\tilde{Y}_{126})_{ij}(\tilde{Y}_{126})_{kl}$ are given by
\begin{align}
&({\cal M}_{H_C}^{-1})_{11}\ (\tilde{Y}_{10})_{ij}(\tilde{Y}_{10})_{kl}+\left\{({\cal M}_{H_C}^{-1})_{31}+({\cal M}_{H_C}^{-1})_{41}\right\}(\tilde{Y}_{126})_{ij}(\tilde{Y}_{10})_{kl}
\nonumber\\
&+({\cal M}_{H_C}^{-1})_{12}\ (\tilde{Y}_{10})_{ij}(\tilde{Y}_{126})_{kl}+\left\{({\cal M}_{H_C}^{-1})_{32}+({\cal M}_{H_C}^{-1})_{42}\right\}(\tilde{Y}_{126})_{ij}(\tilde{Y}_{126})_{kl},
\end{align}
the equality $({\cal M}_{H_C}^{-1})_{11}/\left\{({\cal M}_{H_C}^{-1})_{31}+({\cal M}_{H_C}^{-1})_{41}\right\}
=({\cal M}_{H_C}^{-1})_{12}/\left\{({\cal M}_{H_C}^{-1})_{32}+({\cal M}_{H_C}^{-1})_{42}\right\}$
leads to the desired relation $a/d=b/e$ in Eqs.~(\ref{yyr1})-(\ref{yyl3}).
\end{itemize}
We still have to check that Eq.~(\ref{texture}) is consistent with the situation that $a/d$ satisfies $a\, (Y_{10})_{d_L s_L}+d\, (Y_{126})_{d_L s_L}=0$,
that only one pair of (${\bf 1}$, ${\bf2}$, $\pm\frac{1}{2}$) fields (corresponding to the MSSM Higgs fields) have almost zero mass compared to the GUT scale,
and that $r_1,r_2,r_3,r_e,r_D$ take values that reproduce the realistic quark and lepton masses and flavor mixings.
With Eq.~(\ref{texture}), and with the tuning of $\lambda_5/\lambda_2,\,\lambda_6/\lambda_2,\,\lambda_7/\lambda_2,\,m_1,m_2,m_4$ to satisfy the $F$-flatness conditions,
the colored Higgs mass matrix and the mass matrix of the $({\bf 1},{\bf 2},\pm\frac{1}{2})$ components
still have free parameters $\Phi_1,\Phi_2,\Phi_3,v_R$, $m_3$, $\lambda_{21}/\lambda_{19},\,\lambda_{21}/\lambda_2$, $\lambda_1/\lambda_2,\,\lambda_3/\lambda_2,\,\lambda_4/\lambda_2,\,\lambda_{15}/\lambda_2,\,\lambda_{17}/\lambda_2$.
These free parameters are sufficient to make one pair of (${\bf 1}$, ${\bf2}$, $\pm\frac{1}{2}$) fields nearly massless and realize
any values of $a/d$, $r_1,r_2,r_3,r_e,r_D$.
\\
\section{Yukawa Coupling Matrices satisfying the Conditions}
\label{section-fitting}
\subsection{Procedures of the Analysis}
\label{section-fitting1}
We will obtain specific Yukawa coupling matrices $Y_{10}$, $Y_{126}$, $Y_{120}$ which give the realistic quark and lepton masses and flavor mixings and which satisfy the conditions found in Section~\ref{section-identification} that
the components of Eq.~(\ref{tobeyu}) be on the order of the up quark Yukawa coupling $y_u$.
To this end, we fit the experimental values of the quark and lepton masses and flavor mixings
with $Y_{10}$, $Y_{126}$, $Y_{120}$ and numbers $r_1,r_2,r_3,r_e,r_D$ based on Eqs.~(\ref{yu})-(\ref{cnu})
and at the same time minimize the following quantity:
\begin{align}
&\frac{1}{y_u^2}\sum_{A=10,126} \
\left|(Y_A)_{u_R d_R}\right|^2+\left|(Y_A)_{u_R s_R}\right|^2+\left|(Y_A)_{u_L d_L}\right|^2+\left|(Y_A)_{u_L s_L}\right|^2+\left|(Y_A)_{u_L u_L}\right|^2.
\label{tobeminimized}
\end{align}
The procedures are as follows:
\\
We adopt the following experimental values of the quark and charged lepton masses and quark flavor mixings:
We use the results of lattice calculations of the isospin-averaged quark mass and the strange quark mass in $\overline{{\rm MS}}$ scheme
in Refs.~\cite{lattice1}-\cite{lattice6}, which read
$\frac{1}{2}(m_u+m_d)(2~{\rm GeV})=3.373(80)~{\rm MeV}$ and $m_s(2~{\rm GeV})=92.0(2.1)~{\rm MeV}$,
and the estimate on the up/down quark mass ratio in Ref.~\cite{latticereview}, $m_u/m_d=0.46(3)$.
We use the results of QCD sum rule calculations of the charm and bottom quark masses in $\overline{{\rm MS}}$ scheme in Ref.~\cite{cb},
which read $m_c(3~{\rm GeV})=0.986-9(\alpha_s^{(5)}(M_Z)-0.1189)/0.002\pm0.010~{\rm GeV}$
and $m_b(m_b)=4.163+7(\alpha_s^{(5)}(M_Z)-0.1189)/0.002\pm0.014~{\rm GeV}$.
We use the top quark pole mass measured by CMS in Ref.~\cite{Sirunyan:2019zvx}, which reads $M_t=170.5\pm0.8$~GeV.
We calculate the CKM mixing angles and CP phase from the Wolfenstein parameters in Ref.~\cite{ckmfitter}.
The lepton pole masses and $W$, $Z$, Higgs boson pole masses are taken from Particle Data Group~\cite{ParticleDataGroup:2022pth},
and the QCD and QED gauge couplings are set as $\alpha_s^{(5)}(M_Z)=0.1181$ and $\alpha^{(5)}(M_Z)=1/127.95$.
The above data are translated into the values of the quark and lepton Yukawa coupling matrices and the gauge couplings
at scale $\mu=M_Z$ in $\overline{{\rm MS}}$ scheme
with the help of the code~\cite{code} based on Refs.~\cite{Jegerlehner:2001fb}-\cite{threshold}.
\\
We calculate the two-loop RG equations~\cite{Machacek:1983tz}-\cite{Machacek:1984zw} of SM from scale $\mu=M_Z$ to the soft SUSY breaking scale $\mu_{\rm SUSY}$.
The results are matched to the Yukawa coupling matrices and gauge couplings of MSSM in $\overline{{\rm DR}}$ scheme.
Then we calculate the two-loop RG equations of MSSM from scale $\mu_{\rm SUSY}$ to the GUT scale $\mu_{\rm GUT}$.
We assume a degenerate SUSY particle mass spectrum where the pole masses of SUSY particles and $\tan\beta$ are given by
\begin{eqnarray}
&&m_{\rm sfermion}=m_{H^0}=m_{H^\pm}=m_A=1500~{\rm TeV},
\nonumber\\
&&|M_{\widetilde{g}}|=|M_{\widetilde{W}}|=|\mu_H|=1500~{\rm TeV},
\ \ \ \ \ \ \tan\beta = 50.
\label{massspectrum}
\end{eqnarray}
We set $\mu_{\rm SUSY}=1500$~TeV and $\mu_{\rm GUT}=2\cdot10^{16}$~GeV.
The values of the Yukawa coupling matrices at scale $\mu=\mu_{\rm GUT}$ are shown in Table~\ref{values},
in the form of the singular values of the matrices and the parameters of the CKM matrix at this scale.
The errors of the quark Yukawa couplings, propagated from the experimental errors of the corresponding masses,
and the maximal errors of the CKM parameters, obtained by assuming maximal correlation of the experimental errors of the Wolfenstein parameters, are also displayed.
\begin{table}[H]
\begin{center}
\caption{The singular values of the Yukawa coupling matrices and the CKM mixing angles and CP phase in MSSM
at $\mu=\mu_{\rm GUT}=2\cdot10^{16}$~GeV.
Also shown are the errors of the quark Yukawa couplings, propagated from the experimental errors of the corresponding masses,
and the maximal errors of the CKM parameters, obtained by assuming maximal correlation of the experimental errors of the Wolfenstein parameters.}
\begin{tabular}{|c||c|} \hline
& Value with Eq.~(\ref{massspectrum}) \\ \hline
$y_u$ &2.79(14)$\times10^{-6}$ \\
$y_c$ &0.001435(15) \\
$y_t$ &0.4718(58) \\ \hline
$y_d$ &0.0003081(97) \\
$y_s$ &0.00614(14) \\
$y_b$ &0.3497(20) \\ \hline
$y_e$ &0.0001258 \\
$y_\mu$ &0.02655 \\
$y_\tau$ &0.5081 \\ \hline
$\cos\theta_{13}^{\rm ckm}\sin\theta_{12}^{\rm ckm}$ & 0.22475(25) \\
$\cos\theta_{13}^{\rm ckm}\sin\theta_{23}^{\rm ckm}$ & 0.0430(11) \\
$\sin\theta_{13}^{\rm ckm}$ & 0.00380(23) \\
$\delta_{\rm km}$~(rad) &1.147(33) \\ \hline
\end{tabular}
\label{values}
\end{center}
\end{table}
\noindent
Also, we evaluate one-loop RG corrections to the Wilson coefficient of the Weinberg operator.
We write the Weinberg operator in MSSM as Eq.~(\ref{weinberg}) and that in SM as $-{\cal L}=\frac{1}{2}(C'_\nu)_{ij}\,\psi_{L_i}\psi_{L_j}H H$
where $\psi_{L_i}$ denote the lepton doublets and $H$ the Higgs field.
We express the one-loop RG corrections in MSSM and SM as $C_\nu(\mu)=R(\mu)C_\nu(\mu_{\rm SUSY})R^T(\mu)$, $C'_\nu(\mu)=R'(\mu)C'_\nu(M_Z)R'^T(\mu)$, respectively,
and perform the matching as $C_\nu(\mu_{\rm SUSY})=C'_\nu(\mu_{\rm SUSY})$, since $\tan\beta\gg1$.
We solve the one-loop RG equations and calculate the product of $R(\mu_{\rm GUT})$ and $R'(\mu_{\rm SUSY})$.
Here we approximate the scale of the Majorana mass $(M_N)_{ij}$ to be $\mu_{\rm GUT}$.
The product of $R(\mu_{\rm GUT})$ and $R'(\mu_{\rm SUSY})$ in the flavor basis where the lepton doublets have a diagonal Yukawa coupling matrix, is calculated as
\begin{eqnarray}
R(\mu_{\rm GUT})R'(\mu_{\rm SUSY}) = \begin{pmatrix}
1.09 & 0 & 0 \\
0 & 1.09 & 0 \\
0 & 0 & 1.14 \\
\end{pmatrix}.
\label{rr}
\end{eqnarray}
\\
We fit the quark and charged lepton Yukawa couplings and the CKM parameters at $\mu=\mu_{\rm GUT}$ in Table~\ref{values}
with the Yukawa coupling matrices $Y_{10},Y_{126},Y_{120}$ and numbers $r_1,r_2,r_3,r_e$ based on Eqs.~(\ref{yu})-(\ref{ye}).
Also, we calculate the neutrino mass matrix up to the overall constant, which is proportional to $C'_\nu(M_Z)$,
from $Y_{10},Y_{126},Y_{120}$ and $r_2,r_D$ using Eqs.~(\ref{ydirac})-(\ref{cnu}),(\ref{rr}), and with it we fit the neutrino oscillation data in NuFIT~5.1~\cite{Esteban:2020cvm,nufit}.
Meanwhile, we minimize Eq.~(\ref{tobeminimized}).
We restrict the parameter space to the region with $r_3=0$, since it is easier to minimize Eq.~(\ref{tobeminimized}) when $r_3=0$.
This is because when $r_3=0$, $Y_u=Y_{10}+r_2Y_{126}$ holds and we get
$|(Y_{10})_{u_Li}+r_2(Y_{126})_{u_Li}|=|(Y_u)_{u_L i}|\leq y_u$ and $|(Y_{10})_{u_Ri}+r_2(Y_{126})_{u_Ri}|=|(Y_u)_{u_R i}|\leq y_u$ for any flavor index $i$.
Under the restriction of $r_3=0$, we parametrize the Yukawa coupling matrices as follows:
We go to the flavor basis where the isospin-doublet down-type quark components of ${\bf 16}^i$ matter fields have a diagonal Yukawa coupling matrix.
Since $Y_u=Y_{10}+r_2Y_{126}$ is symmetric, $Y_u$ in this basis can be written as
\begin{eqnarray}
Y_u=V_{\rm CKM}^T
\begin{pmatrix}
y_u & 0 & 0 \\
0 & y_c \,e^{2i \,d_2} & 0 \\
0 & 0 & y_t \,e^{2i \,d_3}\\
\end{pmatrix}
V_{\rm CKM},
\label{yu3}
\end{eqnarray}
where $V_{\rm CKM}$ denotes the CKM matrix and $d_2,d_3$ are undetermined phases.
In the same flavor basis, $Y_d$ can be written as
\begin{eqnarray}
Y_d=\begin{pmatrix}
y_d & 0 & 0 \\
0 & y_s & 0 \\
0 & 0 & y_b \\
\end{pmatrix}
V_{dR},
\label{yd3}
\end{eqnarray}
where $V_{dR}$ is an undetermined unitary matrix.
From Eqs.~(\ref{yu})-(\ref{ydirac}), $Y_e$ and $Y_D$ are written in terms of $Y_u,Y_d$ as
\begin{eqnarray}
&&\frac{1}{r_1}Y_e=Y_u-(3+r_2)Y_{126}+r_e\frac{1}{r_1}\frac{1}{2}\left(Y_d-Y_d^T\right),
\\
&&Y_D=Y_u-4r_2Y_{126}+r_D\frac{1}{r_1}\frac{1}{2}\left(Y_d-Y_d^T\right)
\end{eqnarray}
with
\begin{eqnarray}
Y_{126} = \frac{1}{1-r_2}\left\{\frac{1}{r_1}\frac{1}{2}\left(Y_d+Y_d^T\right)-Y_u\right\}.
\end{eqnarray}
The Majorana mass Eq.~(\ref{majorana}) is found to satisfy
\begin{eqnarray}
M_N \propto Y_{126}.
\end{eqnarray}
The analysis of fitting and minimization proceeds with the above parameterization as follows:
We fix $y_u,y_c,y_t$ and the parameters of the CKM matrix at the central values in Table~\ref{values}.
Then we randomly generate $y_d/r_1,\,y_s/r_1,\,y_b/r_1$, phases $d_2,d_3$, unitary matrix $V_{dR}$,
and complex numbers $r_2,r_e,r_D$, and calculate the singular values of $\frac{1}{r_1}Y_e$.
We determine $r_1$ by requiring that the smallest singular value of $Y_e$ equal the value of $y_e$ in Table~\ref{values}.
Then we require that the values of $y_d,y_s,y_b$ be within the 3$\sigma$ ranges in Table~\ref{values}, and the first and second largest singular values of $Y_e$
respectively be within $\pm$0.1\% ranges of the values of $y_\tau,y_\mu$
\footnote{Since the experimental errors of the charged lepton Yukawa couplings are negligibly small,
here we loosen the fitting criteria.}.
Because the active neutrino mass matrix $M_\nu$ is proportional to $C'_\nu(M_Z)$, we calculate $M_\nu$ up to the overall constant as
\begin{eqnarray}
M_\nu \ \propto \ \left(R(\mu_{\rm GUT})R'(\mu_{\rm SUSY})\right)^{-1} Y_D Y_{126}^{-1} Y_D^T \left(R(\mu_{\rm GUT})R'(\mu_{\rm SUSY})\right)^{T\,-1}.
\label{neutrinomassmatrix}
\end{eqnarray}
Then we calculate from Eq.~(\ref{neutrinomassmatrix}) the three neutrino mixing angles
$\sin^2\theta_{12}$, $\sin^2\theta_{23}$, $\sin^2\theta_{13}$ and the ratio of the neutrino mass squared differences $\Delta m_{21}^2/\Delta m_{31}^2$,
and require them to be within the 3$\sigma$ ranges of NuFIT5.1 results (with SK atmospheric data).
Here we assume the normal mass hierarchy, because it is almost impossible to realize the inverted mass hierarchy from $Y_D,Y_{126}$ as these matrices
have hierarchical structures.
Finally, we select sets of values of $y_d/r_1,\,y_s/r_1,\,y_b/r_1$, $d_2,d_3$, $V_{dR}$, $r_2,r_e,r_D$ that meet the above fitting criteria,
calculate Eq.~(\ref{tobeminimized}) from the sets, and minimize the value of Eq.~(\ref{tobeminimized}).
\\
\subsection{Result}
\label{section-fitting2}
From the analysis of Section~\ref{section-fitting1}, we have obtained the values of the Yukawa coupling matrices $Y_{10},Y_{126},Y_{120}$ and numbers $r_1,r_2,r_e,r_D$ in Appendix~B. There, components of $Y_{10},Y_{126}$ satisfy
\begin{align}
&|(Y_{10})_{u_R d_R}|/y_u=0.77, \ \ \ |(Y_{126})_{u_R d_R}|/y_u=1.0, \ \ \ |(Y_{10})_{u_R s_R}|/y_u=0.28, \ \ \ |(Y_{126})_{u_R s_R}|/y_u=0.43,
\nonumber\\
&|(Y_{10})_{u_L d_L}|/y_u=1.7, \ \ \ |(Y_{126})_{u_L d_L}|/y_u=1.5, \ \ \ |(Y_{10})_{u_L s_L}|/y_u=0.42, \ \ \ |(Y_{126})_{u_L s_L}|/y_u=0.49,
\nonumber\\
&|(Y_{10})_{u_L u_L}|/y_u=0.85, \ \ \ |(Y_{126})_{u_L u_L}|/y_u=0.90.
\end{align}
Clearly, the conditions that the components of Eq.~(\ref{tobeyu}) be on the order of the up quark Yukawa coupling are satisfied.
We evaluate how dimension-five proton decay is suppressed when the conditions are met.
To this end, we compare ``minimal proton partial lifetimes" calculated from the Yukawa coupling matrices of Appendix~B,
with those calculated from results of ``fitting without minimizing Eq.~(\ref{tobeminimized})" where we only
fit the quark and charged lepton Yukawa couplings, CKM parameters and neutrino oscillation data as Section~\ref{section-fitting1} but do not minimize Eq.~(\ref{tobeminimized}) so that the conditions are not necessarily satisfied.
Here the ``minimal proton partial lifetimes", $1/\Gamma_{\rm max}(p\to K^+ \bar{\nu})$, $1/\Gamma_{\rm max}(p\to K^0 \mu^+)$, $1/\Gamma_{\rm max}(p\to K^0 e^+)$,
are defined as ($\beta=e,\mu$)
\begin{align}
&\Gamma_{\rm max}(p\to K^+ \bar{\nu}) = \frac{m_N}{64\pi}\left(1-\frac{m_K^2}{m_N^2}\right)^2
\nonumber\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times\left(|A_{\rm max}(p\to K^+ \bar{\nu}_\tau)|^2+|A_{\rm max}(p\to K^+ \bar{\nu}_\mu)|^2+|A_{\rm max}(p\to K^+ \bar{\nu}_e)|^2\right),
\\
&\Gamma_{\rm max}(p\to K^0 \beta^+) = \frac{m_N}{64\pi}\left(1-\frac{m_K^2}{m_N^2}\right)^2|A_{\rm max}(p\to K^0 \beta^+)|^2,
\end{align}
where
\begin{align}
A_{\rm max}(p\to K^+ \bar{\nu}_\tau) &= |A_{\rm max}(p\to K^+ \bar{\nu}_\tau)_{{\rm from} \ C_{5R}}|+|A_{\rm max}(p\to K^+ \bar{\nu}_\tau)_{{\rm from} \ C_{5L}}|
\label{ataufromc5randc5l}
\end{align}
and
\begin{align}
&A_{\rm max}(p\to K^+ \bar{\nu}_\tau)_{{\rm from} \ C_{5R}}
=
\alpha_H(\mu_{\rm had})\frac{1}{f_\pi}A_{RL}(\mu_{\rm had},\mu_{\rm SUSY}) \frac{\mu_H}{m_{\tilde{t}_R}^2}
\frac{1}{16\pi^2}{\cal F}'\, y_t y_\tau\,
A_R^{\tau t}(\mu_{\rm SUSY},\mu_{\rm GUT})\frac{1}{M_{H_C}}
\nonumber\\
&\ \ \ \ \ \ \ \ \times \max_{A,B}\left\{ \left|
\left(1+\frac{D}{3}+F\right)(V^{\rm ckm}_{ts})^* \left((Y_{A})_{\tau_R t_R}(Y_{B})_{u_Rd_R}-(Y_{A})_{\tau_R u_R}(Y_{B})_{t_Rd_R}\right) \right.\right.
\nonumber\\
&\left.\left.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \frac{2D}{3}(V^{\rm ckm}_{td})^* \left((Y_{A})_{\tau_R t_R}(Y_{B})_{u_Rs_R}-(Y_{A})_{\tau_R u_R}(Y_{B})_{t_Rs_R}\right) \right| \right\},
\label{ataufromc5r}\\
&A_{\rm max}(p\to K^+ \bar{\nu}_\tau)_{{\rm from} \ C_{5L}}
=\beta_H(\mu_{\rm had})\frac{1}{f_\pi}A_{LL}(\mu_{\rm had},\mu_{\rm SUSY}) \frac{M_{\widetilde{W}}}{m_{\tilde{q}}^2}\frac{1}{16\pi^2}{\cal F} \, g_2^2 \, A_L^\tau(\mu_{\rm SUSY},\mu_{\rm GUT})\frac{1}{M_{H_C}}
\nonumber\\
&\ \ \ \ \ \ \times \max_{A',B'}\left\{ \left|
\left(1+\frac{D}{3}+F\right)(Y_{A'})_{u_Ld_L}(Y_{B'})_{s_L\tau_L} + \frac{2D}{3}(Y_{A'})_{u_Ls_L}(Y_{B'})_{d_L\tau_L}\right| \right\},
\label{ataufromc5l}\\
&A_{\rm max}(p\to K^+ \bar{\nu}_\beta)
=\beta_H(\mu_{\rm had})\frac{1}{f_\pi}A_{LL}(\mu_{\rm had},\mu_{\rm SUSY}) \frac{M_{\widetilde{W}}}{m_{\tilde{q}}^2}\frac{1}{16\pi^2}{\cal F} \, g_2^2 \, A_L^\beta(\mu_{\rm SUSY},\mu_{\rm GUT})\frac{1}{M_{H_C}}
\nonumber\\
&\ \ \ \ \ \ \times \max_{A',B'}\left\{ \left|
\left(1+\frac{D}{3}+F\right)(Y_{A'})_{u_Ld_L}(Y_{B'})_{s_L\beta_L} + \frac{2D}{3}(Y_{A'})_{u_Ls_L}(Y_{B'})_{d_L\beta_L}\right| \right\},
\label{abeta}\\
&A_{\rm max}(p\to K^0 \beta^+)=\beta_H(\mu_{\rm had})\frac{1}{f_\pi}A_{LL}(\mu_{\rm had},\mu_{\rm SUSY}) \frac{M_{\widetilde{W}}}{m_{\tilde{q}}^2}\frac{1}{16\pi^2}{\cal F} \, g_2^2 \, A_L^\beta(\mu_{\rm SUSY},\mu_{\rm GUT})\frac{1}{M_{H_C}}
\nonumber\\
&\ \ \ \ \ \ \times (1-D+F) \ \max_{A',B'}\left\{ \left| (Y_{A'})_{u_Ls_L}(Y_{B'})_{u_L\beta_L}-(Y_{A'})_{u_Lu_L}(Y_{B'})_{s_L\beta_L} \right| \right\},
\label{abetaprime}
\end{align}
where $A,B$ in Eq.~(\ref{ataufromc5r}) run as $(A,B)=(10,10)$, (10,126), (126,10), (126,126), (120,10), (120,126), (120,120),
and $A',B'$ in Eqs.~(\ref{ataufromc5l})-(\ref{abetaprime}) run as $(A',B')=(10,10)$, (10,126), (126,10), (126,126).
We assume the SUSY particle spectrum of Eq.~(\ref{massspectrum}) and take $M_{H_C}=2\cdot10^{16}$~GeV in the calculation.
Note that Eqs.~(\ref{ataufromc5r})-(\ref{abetaprime}) take into account the texture of the colored Higgs mass matrix satisfying Eqs.~(\ref{coloredhiggstex1}),(\ref{coloredhiggstex2}).
An implication of Eqs.~(\ref{ataufromc5randc5l})-(\ref{abetaprime}) is that we estimate the maximal values of the amplitudes without specifying the $O(1)$ numbers $a,b,d,e,g,h,j$ in Eqs.~(\ref{yyr1})-(\ref{yyl3}) and the relative phase between the Higgsino mass and Wino mass.
The ``minimal proton partial lifetimes" calculated from the Yukawa coupling matrices of Appendix~B are
\footnote{
These values are consistent with the current experimental bounds on the $p\to K^+ \bar{\nu}$ partial lifetime~\cite{Super-Kamiokande:2014otb}
and on the $p\to K^0 \mu^+/e^+$ partial lifetimes~\cite{Super-Kamiokande:2022egr,Super-Kamiokande:2005lev},
which justifies our choice of the benchmark SUSY particle mass spectrum~Eq.~(\ref{massspectrum}).
}
\begin{eqnarray}
1/\Gamma_{\rm max}(p\to K^+ \bar{\nu}) = 6.3\times10^{33}~{\rm years},
\label{knunum}\\
1/\Gamma_{\rm max}(p\to K^0 \mu^+) = 7.7\times10^{36}~{\rm years},
\label{kmunum}\\
1/\Gamma_{\rm max}(p\to K^0 e^+) = 2.4\times10^{39}~{\rm years}.
\label{kenum}
\end{eqnarray}
On the other hand, the ``minimal proton partial lifetimes" calculated from multiple results of ``fitting without minimizing Eq.~(\ref{tobeminimized})" are distributed as Fig.~\ref{histogram}.
\begin{figure}[H]
\begin{center}
\includegraphics[width=100mm]{knu.eps}
\\
\includegraphics[width=80mm]{kmu.eps}
\includegraphics[width=80mm]{ke.eps}
\caption{
Distributions of the ``minimal proton partial lifetimes" calculated from multiple results of ``fitting without minimizing Eq.~(\ref{tobeminimized})".
The upper, lower-left and lower-right panels correspond to the distributions of $1/\Gamma_{\rm max}(p\to K^+ \bar{\nu})$,
$1/\Gamma_{\rm max}(p\to K^0 \mu^+)$ and $1/\Gamma_{\rm max}(p\to K^0 e^+)$, respectively.
In the upper panel, the blue dot-dashed line indicates the current bound on the $p\to K^+ \bar{\nu}$ partial lifetime~\cite{Super-Kamiokande:2014otb}, the red solid line the value of $1/\Gamma_{\rm max}(p\to K^+ \bar{\nu})$ calculated from the Yukawa coupling matrices of Appendix~B in Eq.~(\ref{knunum}), and the blue dashed line the $3\sigma$ discovery reach of {\bf 2 years} running of Hyper-Kamiokande~\cite{Hyper-Kamiokande:2018ofw}.
In the lower-left panel, the blue dot-dashed line indicates the current bound on the $p\to K^0 \mu^+$ partial lifetime~\cite{Super-Kamiokande:2022egr} (part of the line is hidden behind the histogram),
and the red solid line the value of $1/\Gamma_{\rm max}(p\to K^0 \mu^+)$ calculated from the Yukawa coupling matrices of Appendix~B in Eq.~(\ref{kmunum}).
In the lower-right panel, the blue dot-dashed line indicates the current bound on the $p\to K^0 e^+$ partial lifetime~\cite{Super-Kamiokande:2005lev},
and the red solid line the value of $1/\Gamma_{\rm max}(p\to K^0 e^+)$ calculated from the Yukawa coupling matrices of Appendix~B in Eq.~(\ref{kenum}).
}
\label{histogram}
\end{center}
\end{figure}
\noindent
In Fig.~\ref{histogram}, we overlay the values calculated from the Yukawa coupling matrices of Appendix~B in Eqs.~(\ref{knunum})-(\ref{kenum}). Also, the current bounds on the $p\to K^+ \bar{\nu}$~\cite{Super-Kamiokande:2014otb}, $p\to K^0 \mu^+$~\cite{Super-Kamiokande:2022egr}, $p\to K^0 e^+$~\cite{Super-Kamiokande:2005lev} partial lifetimes, and the $3\sigma$ discovery reach of 2 years running of Hyper-Kamiokande~\cite{Hyper-Kamiokande:2018ofw} are shown.
Comparing Eqs.~(\ref{knunum})-(\ref{kenum}) with Fig.~\ref{histogram}, we see that the proton partial lifetimes calculated from the Yukawa coupling matrices of Appendix~B are on the upper edge of the distributions of proton partial lifetimes calculated from results of ``fitting without minimizing Eq.~(\ref{tobeminimized})".
This confirms that the texture of the Yukawa coupling matrices satisfying the conditions that {\it the components of Eq.~(\ref{tobeyu}) be on the order of the up quark Yukawa coupling}, contributes to suppressing dimension-five proton decay.
Specifically, the benchmark SUSY particle mass spectrum Eq.~(\ref{massspectrum}), where the SUSY particle masses are all at 1500~TeV and $\tan\beta=50$, is consistent with all the experimental bounds on proton partial lifetimes if the above conditions are satisfied.
On the other hand, if these conditions are not met, this benchmark almost always violates the bound on the $p\to K^+ \bar{\nu}$ partial lifetime.
We also see that for this benchmark mass spectrum, when the above conditions are satisfied,
we expect to discover the $p\to K^+ \bar{\nu}$ decay with 2 years running of Hyper-Kamiokande.
\\
We comment that the Yukawa coupling matrices and coefficients in Appendix~B
give a prediction on poorly or not measured neutrino parameters, which are the Dirac CP phase of the neutrino mixing matrix, the sum of the neutrino mass, and the effective neutrino mass for neutrinoless double $\beta$ decay.
The prediction is shown in Appendix~C.
\\
\section{Summary}
\label{summary}
We have pursued the possibility that dimension-five proton decay is suppressed by a texture of the Yukawa coupling matrices
in the general renormalizable SUSY $SO(10)$ GUT model where Yukawa coupling matrices of ${\bf 16}$ representation matter fields with ${\bf 10},{\bf \overline{126}},{\bf 120}$ fields $Y_{10},Y_{126},Y_{120}$ give the quark and lepton Yukawa couplings and Majorana mass of the singlet neutrinos.
We have derived conditions for a texture of the Yukawa coupling matrices suppressing dimension-five proton decay, which state that
{\it components
$(Y_{10})_{u_R d_R}$, $(Y_{126})_{u_R d_R}$,
$(Y_{10})_{u_R s_R}$, $(Y_{126})_{u_R s_R}$, $(Y_{10})_{u_L d_L}$, $(Y_{126})_{u_L d_L}$, $(Y_{10})_{u_L u_L}$,
$(Y_{126})_{u_L u_L}$, $(Y_{10})_{u_L s_L}$, $(Y_{126})_{u_L s_L}$
should be all on the order of the up quark Yukawa coupling} $y_u$.
Additionally, the colored Higgs mass matrix should satisfy Eqs.~(\ref{coloredhiggstex1}),(\ref{coloredhiggstex2}).
We have obtained the values of the Yukawa coupling matrices that satisfy the above conditions and that are consistent with
the experimental data of quark and lepton masses and flavor mixings.
By comparing the ``minimal proton partial lifetimes" calculated from the Yukawa coupling matrices that meet the conditions and those that do not necessarily so, we have confirmed that the texture of the Yukawa coupling matrices satisfying the conditions contributes to suppressing dimension-five proton decay.
Specifically, we have found that a SUSY particle mass spectrum where the SUSY particle masses are all at 1500~TeV and $\tan\beta=50$
is consistent with all the experimental bounds on proton decay if the above conditions are satisfied.
Also, for this mass spectrum, when the conditions are met, we expect to discover the $p\to K^+ \bar{\nu}$ decay with 2 years running of Hyper-Kamiokande.
\\
\section*{Acknowledgement}
This work is partially supported by Scientific Grants by the Ministry of Education, Culture, Sports, Science and Technology of Japan,
No.~21H00076 (NH) and No.~19K147101 (TY).
\\
\section*{Appendix~A: Superpotential}
We review our definition of the coupling constants and masses for
$H$, $\Delta$, $\overline{\Delta}$, $\Sigma$, $\Phi$, $A$ fields in
${\bf 10}$, ${\bf 126}$, ${\bf \overline{126}}$, ${\bf 120}$, ${\bf 210}$, ${\bf 45}$ representations,
which follows Eq.~(2) of Ref.~\cite{Fukuyama:2004ps}.
The couplings are defined in the same way as Eq.~(3) of Ref.~\cite{Fukuyama:2004ps}.
Note that ${\bf 120}$ representation field is written as $D$ in Ref.~\cite{Fukuyama:2004ps}, while we write it as $\Sigma$.
The coupling constants are defined as
\begin{eqnarray}
W &=& \frac{1}{2}m_1\Phi^2 + m_2 \overline{\Delta}\Delta + \frac{1}{2}m_3H^2
\nonumber\\ &+& \frac{1}{2}m_4A^2 + \frac{1}{2}m_6\Sigma^2
\nonumber\\ &+& \lambda_1\Phi^3 + \lambda_2\Phi\overline{\Delta}\Delta + (\lambda_3\Delta+\lambda_4\overline{\Delta})H\Phi
\nonumber\\ &+& \lambda_5 A^2\Phi - i\lambda_6A\overline{\Delta}\Delta + \frac{\lambda_7}{120}\varepsilon A\Phi^2
\nonumber\\ &+& \lambda_{15}\Sigma^2\Phi
\nonumber\\ &+& \Sigma\{\lambda_{16}HA + \lambda_{17}H\Phi + (\lambda_{18}\Delta+\lambda_{19}\overline{\Delta})A + (\lambda_{20}\Delta+\lambda_{21}\overline{\Delta})\Phi\}
\label{gutsuperpotential}
\end{eqnarray}
where $\varepsilon$ denotes the antisymmetric tensor in $SO(10)$ space.
\\
\section*{Appendix~B: Values of $Y_{10}$, $Y_{126}$, $Y_{120}$ and $r_1,r_2,r_e,r_D$}
We present the values of the Yukawa coupling matrices $Y_{10}$, $Y_{126}$, $Y_{120}$ and numbers $r_1,r_2,r_e,r_D$ obtained from the analysis of Section~\ref{section-fitting1}.
$Y_{10}$, $Y_{126}$, $Y_{120}$ are shown repeatedly in three different flavor bases.
For reference, the central value of the up quark Yukawa coupling at scale $\mu=\mu_{\rm GUT}=2\cdot10^{16}$~GeV in $\overline{{\rm DR}}$ scheme is $y_u=2.79\cdot10^{-6}$.
\begin{align}
\begin{pmatrix}
(Y_{10})_{u_Rd_R} & (Y_{10})_{u_Rs_R} & (Y_{10})_{u_Rb_R}\\
(Y_{10})_{c_Rd_R} & (Y_{10})_{c_Rs_R} & (Y_{10})_{c_Rb_R}\\
(Y_{10})_{t_Rd_R} & (Y_{10})_{t_Rs_R} & (Y_{10})_{t_Rb_R}\\
\end{pmatrix}&=\begin{pmatrix}
2.14\cdot10^{-6}\,e^{3.14\,i} & 0.793\cdot10^{-6}\,e^{-0.78\,i} & 0.00164\,e^{-0.17\,i} \\
0.00195\,e^{-1.39\,i} & 0.00841\,e^{-1.41\,i} & 0.0227\,e^{-1.20\,i} \\
0.00659\,e^{-2.64\,i} & 0.0245\,e^{-2.89\,i} & 0.230\,e^{0.45\,i} \\
\end{pmatrix}
\\
\begin{pmatrix}
(Y_{126})_{u_Rd_R} & (Y_{126})_{u_Rs_R} & (Y_{126})_{u_Rb_R}\\
(Y_{126})_{c_Rd_R} & (Y_{126})_{c_Rs_R} & (Y_{126})_{c_Rb_R}\\
(Y_{126})_{t_Rd_R} & (Y_{126})_{t_Rs_R} & (Y_{126})_{t_Rb_R}\\
\end{pmatrix}&=\begin{pmatrix}
2.78\cdot10^{-6}\,e^{-2.03\,i} & 1.21\cdot10^{-6}\,e^{0.99\,i} & 0.00171\,e^{2.14\,i} \\
0.00233\,e^{0.98\,i} & 0.0100\,e^{0.97\,i} & 0.236\,e^{1.11\,i} \\
0.00686\,e^{0.49\,i} & 0.0241\,e^{0.25\,i} & 0.251\,e^{-0.41\,i} \\
\end{pmatrix}
\\
\begin{pmatrix}
(Y_{120})_{u_Rd_R} & (Y_{120})_{u_Rs_R} & (Y_{120})_{u_Rb_R}\\
(Y_{120})_{c_Rd_R} & (Y_{120})_{c_Rs_R} & (Y_{120})_{c_Rb_R}\\
(Y_{120})_{t_Rd_R} & (Y_{120})_{t_Rs_R} & (Y_{120})_{t_Rb_R}\\
\end{pmatrix}&=\begin{pmatrix}
0.000396\,e^{0.01\,i} & 0.00172 & 0.000353\,e^{1.61\,i} \\
0.00173\,e^{-3.13\,i} & 0.000397\,e^{0.02\,i} & 0.000145\,e^{-0.07\,i} \\
0.000272\,e^{-2.82\,i} & 0.000169\,e^{1.45\,i} & 6.78\cdot10^{-6}\,e^{-3.11\,i} \\
\end{pmatrix}
\end{align}
\begin{align}
\begin{pmatrix}
(Y_{10})_{u_Ld_L} & (Y_{10})_{u_Ls_L} & (Y_{10})_{u_Lb_L}\\
(Y_{10})_{c_Ld_L} & (Y_{10})_{c_Ls_L} & (Y_{10})_{c_Lb_L}\\
(Y_{10})_{t_Ld_L} & (Y_{10})_{t_Ls_L} & (Y_{10})_{t_Lb_L}\\
\end{pmatrix}&=\begin{pmatrix}
4.70\cdot10^{-6}\,e^{-1.04\,i} & 1.17\cdot10^{-6}\,e^{-1.52\,i} & 0.00164\,e^{0.58\,i} \\
0.00195\,e^{-2.53\,i} & 0.00840\,e^{0.61\,i} & 0.0227\,e^{-0.45\,i} \\
0.00493\,e^{1.94\,i} & 0.0250\,e^{-0.85\,i} & 0.230\,e^{1.20\,i} \\
\end{pmatrix}
\\
\begin{pmatrix}
(Y_{126})_{u_Ld_L} & (Y_{126})_{u_Ls_L} & (Y_{126})_{u_Lb_L}\\
(Y_{126})_{c_Ld_L} & (Y_{126})_{c_Ls_L} & (Y_{126})_{c_Lb_L}\\
(Y_{126})_{t_Ld_L} & (Y_{126})_{t_Ls_L} & (Y_{126})_{t_Lb_L}\\
\end{pmatrix}&=\begin{pmatrix}
4.23\cdot10^{-6}\,e^{0.65\,i} & 1.35\cdot10^{-6}\,e^{0.28\,i} & 0.00171\,e^{2.89\,i} \\
0.00233\,e^{-0.15\,i} & 0.0100\,e^{2.99\,i} & 0.0236\,e^{1.86\,i} \\
0.00517\,e^{-1.17\,i} & 0.0246\,e^{2.30\,i} & 0.251\,e^{0.345\,i} \\
\end{pmatrix}
\\
\begin{pmatrix}
(Y_{120})_{u_Ld_L} & (Y_{120})_{u_Ls_L} & (Y_{120})_{u_Lb_L}\\
(Y_{120})_{c_Ld_L} & (Y_{120})_{c_Ls_L} & (Y_{120})_{c_Lb_L}\\
(Y_{120})_{t_Ld_L} & (Y_{120})_{t_Ls_L} & (Y_{120})_{t_Lb_L}\\
\end{pmatrix}&=\begin{pmatrix}
0.000396\,e^{-1.12\,i} & 0.00172\,e^{2.01\,i} & 0.000354\,e^{2.36\,i} \\
0.00173\,e^{-1.12\,i} & 0.000401\,e^{-1.14\,i} & 0.000143\,e^{0.69\,i} \\
0.000284\,e^{-0.58\,i} & 0.000148\,e^{-2.01\,i} & 7.33\cdot10^{-6}\,e^{-2.36\,i} \\
\end{pmatrix}
\end{align}
\begin{align}
\begin{pmatrix}
(Y_{10})_{u_Lu_L} & (Y_{10})_{u_Lc_L} & (Y_{10})_{u_Lt_L}\\
& (Y_{10})_{c_Lc_L} & (Y_{10})_{c_Lt_L}\\
& & (Y_{10})_{t_Lt_L}\\
\end{pmatrix}&=\begin{pmatrix}
2.36\cdot10^{-6}\,e^{0.97\,i} & 0.0000700\,e^{0.59\,i} & 0.00164\,e^{0.58\,i} \\
& 0.00914\,e^{0.52\,i} & 0.0225\,e^{-0.46\,i} \\
& & 0.231\,e^{1.20\,i} \\
\end{pmatrix}
\label{y10ulul}\\
\begin{pmatrix}
(Y_{126})_{u_Lu_L} & (Y_{126})_{u_Lc_L} & (Y_{126})_{u_Lt_L}\\
& (Y_{126})_{c_Lc_L} & (Y_{126})_{c_Lt_L}\\
& & (Y_{126})_{t_Lt_L}\\
\end{pmatrix}&=\begin{pmatrix}
2.52\cdot10^{-6}\,e^{-1.77\,i} & 0.0000729\,e^{2.89\,i} & 0.00171\,e^{2.89\,i} \\
& 0.0108\,e^{2.90\,i} & 0.0234\,e^{1.84\,i} \\
& & 0.251\,e^{0.34\,i} \\
\end{pmatrix}
\label{y126ulul}\\
\begin{pmatrix}
(Y_{120})_{u_Lu_L} & (Y_{120})_{u_Lc_L} & (Y_{120})_{u_Lt_L}\\
& (Y_{120})_{c_Lc_L} & (Y_{120})_{c_Lt_L}\\
& & (Y_{120})_{t_Lt_L}\\
\end{pmatrix}&=\begin{pmatrix}
0 & 0.00177\,e^{2.02\,i} & 0.000283\,e^{2.45\,i} \\
& 0 & 0.000149\,e^{0.70\,i} \\
& & 0 \\
\end{pmatrix}
\label{y120ulul}
\end{align}
\begin{eqnarray}
r_1=0.794, \ \ \ r_2=0.960\,e^{0.84\,i}, \ \ \ r_e=1.00\,e^{1.48\,i}, \ \ \ r_D=0.488\,e^{-2.93\,i}.
\end{eqnarray}
In Eqs.~(\ref{y10ulul})-(\ref{y120ulul}), we do not display some off-diagonal components because in this flavor basis, $Y_{10},Y_{126}$ are symmetric and $Y_{120}$ is antisymmetric.
\\
\section*{Appendix~C: Prediction on Neutrino Parameters}
The result of the analysis of Section~\ref{section-fitting1}, shown in Appendix~B, gives the following prediction on
the Dirac CP phase of the neutrino mixing matrix, $\delta_{\rm CP}$, the sum of the neutrino mass, $\sum_{i=1}^3 m_i$, and the effective neutrino mass for neutrinoless double $\beta$ decay, $|m_{ee}|$:
\begin{align}
\delta_{\rm CP}&=-1.42~{\rm rad},
\label{prediction_delta}\\
\sum_{i=1}^3 m_i &= 0.0631~{\rm eV},
\\
|m_{ee}|&=0.000237~{\rm eV}.
\end{align}
We caution that the prediction on $\delta_{\rm CP}$ is not stable in the analysis, namely,
results of the analysis of Section~\ref{section-fitting1} with slightly higher values of Eq.~(\ref{tobeminimized})
yield significantly different predictions on $\delta_{\rm CP}$.
Therefore, Eq.~(\ref{prediction_delta}) should not be fully trusted.
\\
|
1,108,101,563,220 | arxiv | \section{Introduction}
\label{sec:introduction}
A great amount of progress has been made in the area of automated fact-checking.
This includes more accurate machine learning models for veracity prediction and datasets of both naturally occurring \cite{wang-2017-liar,augenstein-etal-2019-multifc,hanselowski-etal-2019-richly} and human-crafted \cite{thorne-etal-2018-fever} fact-checking claims, against which
the models can be evaluated. However, a few blind spots exist in the state-of-the-art. In this work we address specifically two
shortcomings: the narrow focus on political claims, and the paucity of explainable systems.
One subject area which we believe could benefit from expertise-based fact-checking is public health -- including the study of epidemiology, disease prevention in a population, and the formulation of public policies \cite{turnock2012public}.
Recent events, including the COVID-19 pandemic, demonstrate the significant potential harm of misinformation in the public health setting, and the importance in accurately fact-checking claims. Unlike political and general misinformation, specific expertise is required in order to fact check claims in this domain. Oftentimes this expertise
may be limited, and
thus claims which surround public health may be inaccessible (e.g., because of the use of jargon and biomedical terminology) in a way political claims are not.
Nonetheless, like political misinformation, the public health variety is also potentially very dangerous, because it can put people in imminent danger and risk lives.
Typically, statements which are candidates for fact-checking
originate in the political domain \cite{vlachos-riedel-2014-fact,ferreira-vlachos-2016-emergent,wang-2017-liar}, and
tend to surround more general topics or be non-subject specific
\cite{thorne-etal-2018-fever}. This follows the trend of the rising interest in political fact-checking in the last decade \cite{graves2018boundaries}. There are on-going efforts with respect to fact-checking scientific claims \cite{grabitz2017science}. Fact-checking in domains where specific subject expertise is required presents an interesting challenge because general purpose fact-checking systems will not necessarily adapt well to these domains.
The second shortcoming we look to address is the paucity of explainable models for fact-checking (of any kind). Explanations have a particularly important role to play in the task of automated fact-checking. The efficacy of journalistic fact-checking hinges on the credibility and reliability of the fact-check, and explanations (e.g., provided by model agnostic tools such as LIME \cite{ribeiro2016should}) can strengthen this by communicating fidelity in predictive models.
Explainable models can also aid the end users' understanding as they further elucidate claims and their context.
In this study we explore the novel case of explainable automated fact-checking for claims for which specialised expertise or in-domain knowledge is essential. For our case study we examine the the public health (biomedical) context.
The system for veracity prediction we aim to produce must fulfil two requirements: (1) it should provide a human-understandable explanation (i.e., judgment) for the fact-checking prediction, and (2) that judgement should be understandable for people who do not have expertise in the subject domain.
We list the following as our \textbf{three main contributions} in this paper:
\begin{enumerate}
\item We present a novel dataset for explainable fact-checking with gold standard fact-checking explanations by journalists. To the best of our knowledge, this is the first dataset specifically for fact-checking in the public health setting.
\item We introduce a framework for generating explanations and veracity prediction specific to public health fact-checking. We show that gains can be made through the use of in-domain data.
\item In order to evaluate the quality of our fact-checking explanations, we define three coherence properties. These can be evaluated by humans as well as computationally, as approximations for human evaluations of fact-checking explanations.
\end{enumerate}
The explanation model trained on in-domain data outperforms the general purpose model on
summarization evaluation and also when evaluated for explanation quality.
\section{Related Work}
\label{sec:related-work}
A number of recent works in automated fact-checking look at various formulations of fact-checking and its analogous tasks \cite{ferreira-vlachos-2016-emergent,hassan2017toward,zlatkova-etal-2019-fact}. In this paper, we choose to focus on the two specific aspects of concern to us, which have not been thoroughly explored in the literature. These are
domain-specific and expertise-based claim verification and explainability for automated fact-checking{} predictions.
\subsection{Language Representations for Health}
Fewer language resources exist for medical and scientific applications of NLP
compared with other NLP application settings, e.g., social media analysis, NLP for law, and computational journalism and fact-checking. We consider the former below.
There are a number of open source pre-trained language models for NLP applications in the scientific and biomedical domains. The most recent of these pre-trained models are based on the BERT language model \cite{devlin-etal-2019-bert}.
One example is \textsc{BioBERT}, which is fine-tuned for the biomedical setting \cite{lee2020biobert}. \textsc{BioBERT} is trained on abstracts from PubMed and full article texts from PubMed Central. \textsc{BioBERT} demonstrates higher accuracies when compared to BERT for named entity recognition, relation extraction and question answering in the biomedical domain.
\textsc{SciBERT} is another BERT-based pre-trained model \cite{beltagy-etal-2019-scibert}. \textsc{SciBERT} is trained on 1.14M Semantic Scholar articles relating to computer science and biomedical sciences. Similar to \textsc{BioBERT}, \textsc{SciBERT} also shows improvements on original BERT for in-domain tasks. \textsc{SciBERT} outperforms BERT in five NLP tasks including named entity recognition and text classification.
Given that models for applications of NLP tasks in the biomedical domain, e.g., question answering, show marked improvement when domain-specific, we hypothesize that public health fact-checking could also benefit from the language representations suited for that specific domain. We will make use of both \textsc{SciBERT} and \textsc{BioBERT} in our framework.
\subsection{Explainable Fact-Checking.}
A number of in-roads have been made in developing models to extract explanations from automated fact-checking systems. To our knowledge, the current state of the art in explainable fact-checking mostly looks to produce extractive explanations, i.e., explanations for veracity predictions in relation to inputs to the system.
Instead, our focus in this paper is on abstractive explanations. We choose this approach, which aims to distill the explanation into the most salient components which form it, as more amenable to users with limited domain expertise, as we discuss below.
Various methods have been applied to the explainable fact-checking task. These methods span the gamut form logic-based approaches such as probabilistic answer set programming \cite{Ahmadi:2019} and reasoning with Horn rules \cite{Ahmadi:2019,gad2019exfakt} to deep learning and attention-based approaches, e.g., leveraging co-attention networks and human annotations in the form of news article comments \cite{Shu:2019}. The outputs of these systems also take a number of forms including Horn rules \cite{Ahmadi:2019},
saliency maps \cite{Shu:2019, Popat:2018}, and natural language generation \cite{Atanasova:2020}.
All approaches produce explanations which are a distillation of the most relevant portion of the system input. In this paper we expand on the work by \citeauthor{Atanasova:2020} as we formulate explanation generation as a summarization exercise. However, our work differs from the
existing literature as we construct a framework for joint extractive and \emph{abstractive} explanation generation, as opposed to a purely extractive model. We choose an abstractive approach as we hypothesize that particularly in the case of public health claims, where specific expertise is required to understand the context, abstractive explanations can make the explanation more accessible, particularly for those with little knowledge of the subject matter. In this way we take into account the nature of the claims, something other explainable fact-checking systems do not consider.
\subsection{Evaluation of Explanation Quality}
Only a few explainable fact-checking systems employ thorough evaluation in order to assess the quality of explanations produced. In the cases where evaluations are provided, these primarily take the form of human evaluation, e.g., enlisting annotators to score the quality of explanations with respect to some properties \cite{Atanasova:2020,gad2019exfakt} or through the use of an established evaluation metric in the case where explanation generation is modelled as another task \cite{Atanasova:2020}.
There is also work on the evaluation of explanation quality more broadly, independently of the task for which explanations are sought. Notably, \citeauthor{sokol2019desiderata} (\citeyear{sokol2019desiderata}) present explainability factsheets for evaluating (machine learning) explanations along five axes, including usability. One of the usability criteria discussed by \citeauthor{sokol2019desiderata} is \emph{coherence}, which we use to develop our three explanation quality properties (see Section \ref{sec:explanation-evaluation}). Whereas \citeauthor{sokol2019desiderata} discuss coherence in general, we provide concrete definitions and use them for evaluating our methods for explaining veracity predictions for public health claims.
\section{The \textsc{PubHealth} dataset}
\label{sec:data}
We constructed a dataset of 11,832 claims for fact-checking{}, which are related a range of health topics including biomedical subjects (e.g., infectious diseases, stem cell research), government healthcare policy (e.g., abortion, mental health, women's health), and other public health-related stories (see \textit{unproven}, \textit{false} and \textit{mixture} examples in Table \ref{tab:example}), along with explanations offered by
journalists to support veracity labelling of these claims. The claims were collected from two sources: fact-checking websites and news/news review websites. An example dataset entry is shown in Table \ref{tab:example}.
To the best of our knowledge, this is the first fact-checking dataset to explicitly include gold standard texts provided by journalists specifically as explanation of the fact-checking judgment.
We describe below how the data was collected and processed to obtain the final \textsc{PubHealth} dataset, and provide an analysis of the dataset.
\begin{table*}
\centering
\small
\begin{tabular}{p{5cm}cp{8.3cm}}
\toprule
\textbf{Claim} & \textbf{Label} & \textbf{Explanation}\\
\midrule
Blue Buffalo pet food contains unsafe and higher-than-average levels of lead. & \textcolor{blue}{\textbf{UNPROVEN}} & Aside from a single claimant’s lawsuit against Blue Buffalo and an unrelated recall on one variety of Blue Buffalo product in March 2017, we found no credible information suggesting that Blue Buffalo dog food was tested and found to have abnormally high levels of lead.\\
\addlinespace[0.1cm]
\hdashline
\addlinespace[0.1cm]
Children who watch at least 30 minutes of “Peppa Pig” per day have a 56 percent higher probability of developing \textbf{autism}. & \textcolor{alizarin}{\textbf{FALSE}} & Talk of a Harvard study linking the popular British children’s show “Peppa Pig” to \textbf{autism} went viral, but neither the study nor the scientist who allegedly published it {exists}.\\
\addlinespace[0.1cm]
\hdashline
\addlinespace[0.1cm]
Expired boxes of cake and pancake mix are dangerously toxic. & \textcolor{brown}{\textbf{MIXTURE}} & What's true: Pancake and cake mixes that contain {mold} can cause \textbf{life-threatening allergic reactions}.\newline What's false: Pancake and cake mixes that have passed their expiration dates are not inherently dangerous to ordinarily \textbf{healthy} people, and the yeast in packaged baking products does not ``over time develops spores."\\
\addlinespace[0.1cm]
\hdashline
\addlinespace[0.1cm]
Families tell U.S. lawmakers of \textbf{heparin} deaths. & \textcolor{darkgreen}{\textbf{TRUE}} & A man who said he lost his wife and a son to reactions from tainted \textbf{heparin} made with ingredients from China urged U.S. lawmakers on Tuesday to protect \textbf{patients} from other unsafe \textbf{drugs}.\\
\bottomrule
\end{tabular}
\caption{Example of claims and explanations for \textsc{PubHealth} dataset entries. Vocabulary from the public health glossary which are contained in the claims and explanations are in \textbf{bold}.}
\label{tab:example}
\end{table*}
\subsection{Data collection}
Initially, we scraped 39,301 claims, amounting to: 27,578 fact-checked claims from five fact-checking websites (Snopes\footnote{\scriptsize\url{https://www.snopes.com/}}, Politifact\footnote{\scriptsize\url{https://www.politifact.com/}}, TruthorFiction\footnote{\scriptsize\url{https://www.truthorfiction.com/}}, FactCheck\footnote{\scriptsize\url{https://www.factcheck.org/}}, and FullFact\footnote{\scriptsize\url{https://fullfact.org/}}); 9,023 news headline claims from the health section and health tags of Associated Press\footnote{\scriptsize\url{https://apnews.com/}} and Reuters News\footnote{\scriptsize\url{https://uk.reuters.com/news/health}} websites; and 2,700 claims from the news review site Health News Review (HNR)\footnote{\scriptsize{\url{ https://www.healthnewsreview.org/}}}.
We scraped data for two text fields which are essential for fact-checking: 1) the full text of the fact-checking or news article
discussing the veracity of the claim,
and 2)
the fact-checking justification or news summary as explanation for the veracity label of the claim. We also collected
the URLs of sources cited by the journalists in the fact-checking and news articles.
For each URL, in the case where the referenced sources could be accessed and read, we also scraped the source texts.
All claims make reference to articles published between October 19 1995 and May 14 2020. In addition to the claim, article texts, explanation texts, and the date on which the fact-check or news article was published, we scraped meta-data related to each claim. These meta-data include the tags (single or multiple tokens) which may, for example, categorize the topics of the claim or indicate the source of the claim (see Appendix \ref{appendix:labels}), and the names of the fact-checkers and news reporters who contributed to the article.
\subsection{Data processing and analysis}
The data processing involved three tasks: standardizing the veracity labels, filtering out non-biomedical claims from the dataset, and finally removing claims with incomplete and brief explanations.
Labels for news headline claims did not require standardization, as we assumed all news headline claims (coming from reputable sources as they were) to be verified and thus labelled these \textit{true},
but filtered out from the dataset news entries with the headline prefixes ``AP EXCLUSIVE", ``Correction", ``AP Interview", and ``AP FACT CHECK". Indeed, it would be difficult to label the veracity of the claim in this type of entries. On the other hand, fact-check and news claims, which were associated with 141 different veracity labels, did require compression. We standardized the original labels for 4-way classification (see Appendix \ref{appendix:labels}). The chosen 4 labels are \textit{true}, \textit{false}, \textit{mixture}, and \textit{unproven}. We discounted claims with labels that cannot be reduced to one of these 4 labels. The distribution of labels in the final \textsc{PubHealth} is shown in Table \ref{tab:data}. The dataset consists of a majority false claims. Unproven claims are the least common in the dataset.
\begin{table}[!htb]
\centering
\small
\begin{tabular}{lrrrrr}
\toprule
\textbf{Website} & \textbf{tru.} & \textbf{fal.} & \textbf{mix.} & \textbf{unp.} & \textbf{total}\\
\midrule
AP News & 2,132 & 0 & 0 & 0 & 2,132 \\
FactCheck & 0 & 50 & 29 & 8 & 87 \\
FullFact & 65 & 39 & 16 & 48 & 168 \\
HNR & 819 & 839 & 745 & 0 & 2,403 \\
Politifact & 671 & 1,339 & 423 & 0 & 2,433\\
Reuters & 1,971 & 0& 0 & 0 & 1,971 \\
Snopes & 386 & 1,131 & 405 & 220 & 2,142\\
TruthOrFict. & 132 & 172 & 120 & 72 & 496 \\
\midrule
Total & 6,176 & 3,570 & 1,526 & 299 & 11,832 \\
\bottomrule
\end{tabular}
\caption{Summary of the distribution of true (\textbf{tru.}), false (\textbf{fal.}), mixture (\textbf{mix.}) and unproven (\textbf{unp.}) veracity labels in \textsc{PubHealth}, across the original sources from which data originated.}
\label{tab:data}
\end{table}
The second step in processing the data was to remove claims with no biomedical context. This step was especially crucial for the claims which originated from fact-checking websites where the bulk of fact-checks concern political and economic claims. Health claims are easier to acquire from news websites, such as Reuters, as they can be quickly identified by the section of the website in which they were located during the data collection process.
Although we mentioned that a sizeable number of claims from fact-checking sources are related to political events, some are connected to both political and health events or other mixed health context, and we collected claims whose subject matter intersects other topics in order to obtain a subject-rich dataset (see Appendix \ref{appendix:labels}).
Claims in the larger dataset were filtered according to a lexicon of 7,000 unique public health and health policy terms scraped from five health information websites (See Appendix \ref{appendix:labels}).
Furthermore, we manually added 65 more public health terms that were not retrieved during the initial scraping, but which we determined would positively contribute to the lexicon because of their relevance to the COVID-19 pandemic (see Appendix \ref{appendix:labels}). These claims were identified through exploratory data analysis of bigram and trigram collocations in \textsc{PubHealth}.
In order to filter out the entries which are not health-related, we kept only claims with main article texts that mentioned more than three unique terms in our lexicon. Specifically, let $L$ be our lexicon, and $A_c$ and $T_c$, respectively, be the article text and claim text accompanying a candidate dataset entry $c$. Then, we included in \textsc{PubHealth} only the following set $C$ of claim entries, with accompanying information:
\begin{equation*}
\label{eq:1}
\begin{aligned}
C_{A} = \{c \mid
\{l_1,...,l_n\}= A_{c}\cap L,
n > 3 \}
\end{aligned}
\end{equation*}
\begin{equation*}
\label{eq:2}
\begin{aligned}
C_{T} =\{c \mid
\{l_1,...,l_n\}= T_{c}\cap L
n >
\}
\end{aligned}
\end{equation*}
\begin{equation}
\label{eq:3}
\begin{aligned}
C = C_{A}\cup C_{T}
\end{aligned}
\end{equation}
As we already knew that all Reuters health news claims qualify for our dataset, we used the lower bound frequency of words from our lexicon present in these article texts to determine our lower bound of three unique terms.
We acknowledge that there might be disparities in the amount of medical information present in entries. However, analysis of the dataset shows, quite promisingly, that on average claims' accompanying article texts have $8.92\pm5.54$ unique health lexicon terms and claim texts carry $4.45\pm0.88$ unique terms
from the health lexicon.
Claims and explanations in the entries in the dataset were
also cleaned. Specifically, we also ensured all claims are between 25 and 400 characters in length. We removed explanations less than 25 characters long as we determined that very few claims shorter than this length contained fully formed claims; we removed claims longer than 400 characters to avoid the complexities of dealing with texts containing multiple claims. We also omitted claims and explanations ending in a question mark to ensure that all claims are statements, i.e., clearly defined.
Note that one aspect of the explanations' quality which we chose not to control, was the intended purpose of the text we labelled as the explanation: as shown in Table \ref{tab:explanation-sources} in Appendix~\ref{appendix:labels}, there was a wide variation across the websites we crawled.
Table \ref{tab:comparisonfcdatasets} shows the Flesch-Kincaid~\cite{kincaid1975derivation} and Dale-Chall~\cite{chall1995readability} readability evaluations of claims from our fact-checking{} dataset when compared to four other fact-checking datasets.
The results show that \textsc{PubHealth} claims are, on average, the most challenging to read. Claims from our dataset have a mean Flesch-Kincaid reading ease score of 59.1, which corresponds to a 10th-12th grade reading level and fairly difficult to read. The other fact-checking{} datasets have reading levels which fit into the 6th, 7th and 8th grade categories.
Similarly for the Dale-Chall readability metric, on average our claims are more difficult to understand. Our claims have a mean score of 9.5 which is equivalent to the reading age of college student, whereas all other datasets' claims have an average score which indicates that they are readable by 10th to 12th grade students.
Both these results support our earlier assertion about the complexity of public health claims relative to political and more general claims.
\begin{table}[H]
\centering
\small
\begin{tabular}{lccccl}
\toprule
\textbf{Dataset} & \multicolumn{2}{c}{\textbf{Flesch-Kincaid}} & \multicolumn{2}{c}{\textbf{Dale-Chall}} \\
\midrule
& $\mu$ & $\sigma$ & $\mu$ & $\sigma$ \\
\cmidrule(lr) {2-3}\cmidrule(lr) {4-5}
\footnotesize{\citeauthor{wang-2017-liar} (\citeyear{wang-2017-liar})} & 61.9 & 20.2 & 8.4 & 2.2 \\
\footnotesize{\citeauthor{shu2019fakenewsnet} (\citeyear{shu2019fakenewsnet})} & 67.1 & 24.3 & 8.9 & 3.0 \\
\footnotesize{\citeauthor{thorne-etal-2018-fever} (\citeyear{thorne-etal-2018-fever})} & 71.7 & 24.9 & 8.2 & 3.3 \\
\footnotesize{\citeauthor{augenstein-etal-2019-multifc} (\citeyear{augenstein-etal-2019-multifc})} & 60.8 & 22.1 & 8.9 & 2.5 \\
\midrule
Our dataset & 59.1 & 23.3 & 9.5 & 2.6 \\
\bottomrule
\end{tabular}
\caption{Comparison of readability of claims presented
in large fact-checking datasets (i.e., those with $> 10$K claims). We compute the mean and standard deviation for Flesch-Kincaid and Dale-Chall scores of claims for LIAR \cite{wang-2017-liar}, FEVER \cite{thorne-etal-2018-fever}, MultiFC \cite{augenstein-etal-2019-multifc}, FAKENEWSNET \cite{shu2019fakenewsnet}, and also our own fact-checking{} dataset. The sample sizes used for evaluation for each dataset are as follows, LIAR: 12,791, MultiFC: 34,842, FAKENEWSNET: 23,196, FEVER: 145,449, and 11,832 for our dataset.}
\label{tab:comparisonfcdatasets}.
\end{table}
\section{Methods}\label{sec:methods}
In this section we describe in detail the methods we employed for devising automated fact-checking models.
We trained two fact-checking{} models: a classifier for veracity prediction, and a second summarization model for generating fact-checking explanations. The former returns the probability of an input claim text belonging to one of four classes: true, false, unproven, mixture. The latter uses a form of joint extractive and abstractive summarization to generate explanations for the veracity of claims from article text about the claims. Full details of hyperparameters chosen and computer infrastructure which was employed can be found in Appendix \ref{appendix:reprodubility}.
\subsection{Veracity Prediction}
\begin{figure}[H]
\centering
\includegraphics[ width=0.48\textwidth,
trim = 0.2cm 0cm 0cm 0.2cm, clip]{predictor}
\caption{Architecture of veracity prediction.}
\label{fig:pipeline}
\end{figure}
Veracity prediction is composed of two parts: evidence selection and label prediction (see Figure~\ref{fig:pipeline}).
For evidence selection, within fact-checking and news articles, we employ Sentence-BERT (S-BERT) \cite{reimers-gurevych-2019-sentence}. SBERT is a model for sentence-pair regression tasks which is based on the BERT language model \cite{devlin-etal-2019-bert}, to encode contextualized representations for each of the evidence sentences and then rank these sentences according to their cosine similarity with respect to the contextualized representation of the claim sentence. We then
select the top $k$ sentences for veracity prediction. As with sentence selection approaches from the fact-checking literature \cite{nie2019combining,zhong2019reasoning}, we choose $k=5$.
The claim and selected evidence sentences form the inputs for the label prediction part of our model (see Figure~\ref{fig:pipeline}).
We fine-tuned, on the \textsc{PubHealth} dataset, pre-trained models for the downstream task of fact-checking label prediction. We employed four pre-trained models: original BERT uncased, \textsc{SciBERT}, \textsc{BioBERT} v1.0, and also \textsc{BioBERT} v1.1.
The two versions of \textsc{BioBERT} differ slightly in that the earlier version is trained for 470K steps on PubMed abstracts and PubMed Central (PMC) full article texts, whereas \textsc{BioBERT} v1.1 is trained for 1M steps on PubMed abstracts.
\subsection{Explanation Generation as Abstractive Summarization}
We make use of extractive-abstractive summarization \cite{Liu-Lapata:2019} in developing the explanation model. We choose this architecture because explanations for claims which concern a specific topic area having a highly complex lexicon can benefit from the ability to articulate judgment in simpler terms. In order to deploy the model proposed by \cite{Liu-Lapata:2019}
we also implemented an explanation generation model.
Just as is the case for the predictor model, the explanation model is fine-tuned for the task on evidence sentences ranked by S-BERT.
However, for the explanation model we use all article sentences as well as the claim sentence to fine-tune a BERT-based summarization model pre-trained on the Dailymail/CNN news article and summaries dataset \cite{hermann2015teaching}. One of our models, \textsc{ExplainerFC}, is fine-tuned using non-public health data, which we extract from the portion of the 39.3K originally crawled fact-checks, news reviews, and news articles not included in \textsc{PubHealth}. For fairness, we ensure these data have the same proportion of claims from each website and the number of examples is the same as \textsc{PubHealth}. The second model, \textsc{ExplainerFC-Expert}, is fine-tuned on \textsc{PubHealth}. Also, we evaluate both models on \textsc{PubHealth} test data. Table \ref{tab:generated_explanations} shows an example of the explanations generated by the two methods.
\section{Results}
We conducted experiments to evaluate the performance of both predictor(s) and explainer(s). The performance of the (various incarnations of the) prediction model is evaluated using an automatic approach, whereas the performance of the (two incarnations of the) explainer is assessed using both automatic and human evaluation.
\begin{figure}[ht]
\centering
\small
\begin{tikzpicture}
[
auto,
block/.style = { rectangle,
draw=black,
fill=white, text width=17em, text centered,
minimum height=2em },
block1/.style = { rectangle, draw=white,
fill=white, text width=17em, text centered,
minimum height=2em },
line/.style = { draw, thick, ->, shorten >=2pt },
]
\matrix [column sep=5mm, row sep=0.5mm] {
& \node [text centered] (x) {\textbf{Claim} \hspace{0.75cm} \textbf{Label:} \textcolor{alizarin}{\textbf{FALSE}}}; \\
& \node [block1] (doa) {Under Obamacare, patients 76 and older must be admitted to the hospital by their primary care physicians in order to be covered by Medicare.}; \\
& \node [text centered] (x) {}; \\
& \node [text centered] (x) {\textbf{Gold explanation}}; \\
& \node [block] (track) {Obamacare does not require that patients 76 and older must be admitted to the hospital by their primary care physicians in order to be covered by Medicare.}; & \\
& \node [text centered] (x) {}; \\
& \node [text centered] (y) {\textbf{EXPLAINERFC explanation}}; \\
& \node [block] (xfc)
{What's true: nothing in the Affordable Care Act requires that a primary care physician admit patients 76 or older to a hospital in order for their hospital care to be treated under Medicare. What's false: none of the provisions or rules put an upper age limit on medicare coverage.}; & \\
& \node [text centered] (x) {}; \\
& \node [text centered] (y) {\textbf{EXPLAINERFC-EXPERT explanation}}; \\
& \node [block] (xfc-expert)
{The Affordable Care Act does not require Medicare to admit patients to a hospital after paying the Part B deductible. It’s not the same age limit on medicare coverage. But the evidence doesn't specifically set an upper age limit.}; & \\
};
\end{tikzpicture}
\caption{Example of model-generated explanations as compared to the gold standard from our fact-checking{} dataset.
}
\label{tab:generated_explanations}
\end{figure}
\subsection{Prediction}
We split the \textsc{PubHealth} dataset as follows: 9,466 training examples, 1,183 examples for validation and 1,183 examples for testing.
We evaluated veracity prediction using macro-F1, precision, recall and accuracy metrics as shown in Table \ref{tab:veracity_results}. We employ two baselines: a randomized sentence selection approach with BERT (bert-base-uncased) classifier, and lastly a BERT model, also using pre-trained uncased BERT, which does not make use of sentence selection and instead makes use of the entire article text to fine-tune for the fact-checking task.
Out of the four BERT-derived models, \textsc{SciBERT} achieves the highest macro F1, precision and accuracy scores on the test set. \textsc{BioBERT} v1.1 achieves the second highest scores for F1, precision and accuracy. As expected, \textsc{BioBERT} v1.1 outperforms \textsc{BioBERT} v1.0 on all four metrics. The standard BERT model achieves the highest precision score of the four models, however it also achieves the lowest recall and F1 scores. This supports the argument we presented in Section \ref{sec:introduction} that subject-specific fact-checking can benefit from training on in-domain models.
\begin{table}[ht]
\centering
\small
\begin{tabular}{lcccc}
\toprule
\textbf{Model} & \textbf{Pr.} & \textbf{Rc.} & \textbf{F1} & \textbf{Acc.}\\
\midrule
BERT (rand. sents.) & 38.97 & 39.38 & 39.16 & 20.99\\
BERT (all sents.) & 56.50 & 56.50 & 56.50 & 55.40\\
\midrule
\midrule
\textsc{BERT} (top k sents.) & \textbf{77.39} & 54.77 & 63.93 & 66.02\\
\textsc{SciBERT} & 75.69 & \textbf{66.20} & \textbf{70.52} & \textbf{69.73}\\
\textsc{BioBERT} 1.0 & 73.93 & 57.57 & 64.57 & 65.18\\
\textsc{BioBERT} 1.1 & 75.04 & 61.68 & 67.48 & 68.89\\
\bottomrule
\end{tabular}
\caption{Veracity prediction results for the two baselines and four BERT-based models on the test set. Model performance is assessed against precision (Pr.), recall (Rc.), macro F1, and accuracy (Acc.) metrics.}
\label{tab:veracity_results}
\end{table}
\subsection{Explanations}
We use two methods for evaluating the quality of explanations generated by our methods: automated evaluation and qualitative evaluation, in turn amounting to human and computational evaluation of explanation properties.
\subsubsection{Automated Evaluation}
We make use of ROUGE summarization evaluation metrics \cite{lin-2004-rouge}. Specifically we use the F1 values for ROUGE-1, ROUGE-2, and ROUGE-L, to evaluate the explanations generated by the \textsc{ExplainerFC} and \textsc{ExplainerFC-Expert} models.
As in the setup employed by \citeauthor{Liu-Lapata:2019} (\citeyear{Liu-Lapata:2019}), we compare our explanation models to two other methods: a \textsc{Lead-3} baseline, which constructs a summary out of the first three sentences of an article, and an extractive summarization-based \textsc{Oracle} upper bound.
The results of this evaluation are shown in Table \ref{tab:rouge}. The \textsc{ExplainerFC-Expert} explanation model outperforms \textsc{ExplainerFC}. \textsc{ExplainerFC-Expert} achieves higher scores than \textsc{ExplainerFC} for R1, R2, and RL metrics.
\begin{table}[ht]
\centering
\small
\begin{tabular}{lccc}
\toprule
\multirow{2}{*}{Model} & \multicolumn{3}{c}{ROUGE-F}
\\
& R1 & R2 & RL \\
\midrule
\textsc{Oracle} & 39.24 & 14.89 & 32.78 \\
\textsc{Lead-3} & 29.01 & 10.24 & 24.18\\
\midrule
\midrule
\textsc{ExplainerFC} & 31.42 & 12.38 & 26.27 \\
\textsc{ExplainerFC-Expert} & \textbf{32.30} & \textbf{13.46} & \textbf{26.99}\\
\bottomrule
\end{tabular}
\caption{ROUGE-1 (R1), ROUGE-2 (R2) and ROUGE-L (RL) F1 scores for explanations generated via our two explanation models.}
\label{tab:rouge}
\end{table}
\subsection{Evaluation of Explanation Quality}
\label{sec:explanation-evaluation}
As the explanations we generate are from heterogeneous sources (and therefore not directly comparable), evaluation using ROUGE does not present us with a complete picture of the usefulness or quality of these explanations. For this reason, we adapt to the task of explainable fact-checking three of the desirable usability properties for machine learning explanations offered by \citeauthor{sokol2019desiderata} (\citeyear{sokol2019desiderata}). We define these properties formally and evaluate the quality of the generated explanations against them. These same properties are also used for our human evaluations and for a comparison between human and computational evaluation of the quality of our explanations. To the best of our knowledge, ours is the first systematic evaluation of the quality of explanations for fact-checking in terms of formal properties. We define the three explanation properties as (two forms of) global coherence and (a form of) local coherence,
as follows.
\paragraph{Global Coherence} refers to the suitability of fact-checking explanations with respect to both the claim and label to which it is associated. We consider two incarnations of global coherence:
\begin{itemize}
\item \emph{Strong global coherence}. Let $E$ be an explanation of the veracity label $l$ for claim $C$, where
e_{1},\dots, e_{N
$ are all the individual sentences which make up $E$. Then, $E$ satisfies strong global coherence iff $\forall e_{i}\in E$, $e_{i} \models C$. Put simply, for this property to hold for a generated fact-checking explanation, every sentence in the explanatory text must entail ($\models$) the claim.
\item \emph{Weak global coherence}. Let $E$ be an explanation of the veracity label $l$ for claim $C$, where
e_{1},\dots, e_{N
$ are all the individual sentences which make up $E$. Then, $E$ satisfies weak global coherence iff $\forall e_{i}\in E$,
$e_{i} \not\models \neg C$. For this property to hold for a generated fact-checking explanation, no sentence in the explanatory text should contradict the claim (by entailing its negation); from a natural language inference (NLI) perspective, for weak global coherence to hold all explanatory sentences should entail or have a neutral relation with respect to the claim.
\end{itemize}
When measuring coherence, we treat as \textit{neutral} claims originally labelled as \textit{false} if their claim is \textit{contradicted} by its explanation. Note that if the \textit{false} claim is \emph{entailed} by its explanation we do not reassign the label, because doing so would impose too strong an assumption that the entailment is related to the veracity which we cannot verify.
\paragraph{Local Coherence.} Let $E$ be an explanation of the veracity label $l$ for claim $C$, where
e_{1},\dots, e_{N}
$ are all the individual sentences which make up $E$. Then, $E$ satisfies local coherence iff $\forall e_{i}, e_j \in E$,
$e_{i} \not\models \neg e_{j}$.
Local coherence is a measure of how cohesive sentences in an explanation are. For local coherence to hold any two sentences in an explanation must not contradict each other, i.e., there is no pairwise disagreement between sentences which make up the explanation.
Note that all three coherence properties relate to the usability property of coherence discussed by \citeauthor{sokol2019desiderata} (\citeyear{sokol2019desiderata}). Local coherence draws specifically on the idea of avoiding internal inconsistencies in explanations.
Figure~\ref{fig:text-example} shows an example of evaluation of the three properties, for a specific claim-explanation pair.
Schematic examples of explanations and evidence sentence relations which satisfy these coherence properties are shown in Appendix \ref{appendix:coherence}.
\subsubsection{Human \& Computational Evaluations}
We employ human evaluation in order to assess the quality of the gold and generated explanations with respect to these properties. Also, we conduct a computational evaluation of the three coherence properties using NLI.
For human evaluation, we randomly sampled 25 entries from the test set of \textsc{PubHealth}, and enlisted 5 annotators to evaluate the quality of the gold explanations and explanations generated by \textsc{ExplainerFC} and \textsc{ExplainerFC-Expert} for these entries.
We asked participants to annotate explanations according to the following criteria: 1) the agreement and disagreement between sentences in the explanation, and 2) relevance of the explanation to the claim. Further information, including an example from the questionnaire, can be found in Appendix \ref{appendix:questionnaire}.
We conducted the computational evaluation on three pretrained NLI models: 1) a decomposable attention model \cite{parikh-etal-2016-decomposable} using ELMo embeddings \cite{peters-etal-2018-deep} trained on
the Stanford Natural Language Inference (SNLI) corpus \cite{Bowman:2015}, 2) RoBERTa \cite{liu2019roberta} trained on SNLI, and 3) RoBERTa trained on the Multi-Genre Natural Language Inference (MNLI) corpus \cite{williams-etal-2018-broad}. We implemented these evaluation methods using the AllenNLP platform \cite{gardner-etal-2018-allennlp}.
For the human evaluation we computed Randolph's free-marginal $\kappa$ \cite{randolph2005free} and overall agreement (O.A.) for all multiple choice questions. For the gold explanations, we computed $\kappa$ (and O.A.) of 0.24 (62\%), 0.48 (65.6\%), and 0.39 (54.4\%) for 2-, 3-, and 4-ary questions respectively. For \textsc{ExplainerFC}, 0.06 (53.2\%), 0.17 (44.8\%), and 0.12 (34\%) for 2-, 3-, and 4-ary questions respectively. Lastly for \textsc{ExplainerFC-Expert}, we computed $\kappa$ and O.A. of 0.36 (68\%), 0.44 (62.73\%), and 0.20 (40\%) for 2-, 3-, and 4-ary questions. The computational evaluation was conducted on all examples from the test set. The results of both the human and computational evaluation of the three coherence measures are shown in Table~\ref{tab:human}. Our results suggest that the NLI approximation is a reliable approximation for weak global coherence and local coherence properties. However, entailment appears to be a poor approximation for strong global coherence. Further, a larger human evaluation study would be required in order to verify these results.
\begin{table}[ht]
\centering
\small
\begin{tabular}{p{2.5cm}p{1cm}p{1cm}p{1cm}}
\toprule
\multirow{3}{*}{Evaluation Method} & \multirow{2}{*}{\textbf{SGC}} & \multirow{2}{*}{\textbf{WGC}} &
\multirow{2}{*}{\textbf{LC}}\\
& & \\
\cmidrule{2-4}
& \multicolumn{3}{c}{Gold explanations}\\
\midrule
Human & 76.80 & 98.40 & 65.60\\
DA+ELMo; SNLI & 8.72 & 87.61 & 55.20\\
RoBERTa; SNLI & 1.28 & 75.87 & 52.12\\
RoBERTa; MNLI & 2.66 & 87.52 & 54.84\\
\midrule
\multicolumn{4}{c}{\textsc{ExplainerFC} generated explanations}\\
\midrule
Human & 53.60 & 88.80 & 58.10\\
DA+ELMo; SNLI & 8.26 & 89.45 & 51.32\\
RoBERTa; SNLI & 0.46 & 76.42 & 48.01\\
RoBERTa; MNLI & 0.73 & 84.59 & 50.20\\
\midrule
\multicolumn{4}{c}{\textsc{ExplainerFC-Expert} generated explanations}\\
\midrule
Human & 60.4 & 76.80 & 59.30\\
DA+ELMo; SNLI & 7.61 & 89.72 & 64.60\\
RoBERTa; SNLI & 0.64 & 76.15 & 60.07\\
RoBERTa; MNLI & 2.48 & 84.04 & 62.43\\
\bottomrule
\end{tabular}
\caption{\% of explanations which satisfy strong global coherence (SGC), weak global coherence (WGC) and local coherence (LC) properties.}
\label{tab:human}
\end{table}
\begin{figure}[ht]
\begin{tabular}{|p{7.3cm}|}
\hline
\textbf{Claim} \\
A list of chemicals, written as if they were ingredients on a food label, accurately depicts the chemical composition of a banana. \\
\textbf{Label: } \textcolor{darkgreen}{\textbf{TRUE}} \\
\textbf{Explanation} \\
In sum, this graphic accurately depicts the chemicals that comprise a banana, using a variety of tactics to make that completely natural food appear to be full of “chemicals” — something originally created by a high school chemistry teacher as part of a lesson on chemophobia. \\
\hline
\end{tabular}
\caption{Example of
explanation which satisfies all three coherence properties.}
\label{fig:text-example}
\end{figure}
\section{Conclusion and Future work}
In this paper, we explored fact-checking for claims for which specific expertise is required to produce a veracity prediction and explanations (i.e., judgments used for awarding the label/veracity prediction). To support this exploration we constructed \textsc{PubHealth}, a sizeable dataset for public health fact-checking and the first fact-checking dataset to include explanations as annotations. Our results show that training veracity prediction and explanation generation models on in-domain data improves the accuracy of veracity prediction and the quality of generated explanations compared to training on generic language models without explanation.
We hope to explore the topics of explainable fact-checking and specialist fact-checking further. In order to do this, we hope to explore other subjects, in addition to public health, for which fact-checking requires a level of expertise in the subject area. Furthermore, we hope to explore the quality of fact-checking explanations with respect to properties other than coherence, e.g., actionability and impartiality. lastly, we plan to explore congruity between veracity prediction and explanation generation tasks, i.e., generating explanations which are compatible with the predicted label and vice versa.
\section*{Acknowledgements}
We would like to thank all those who participated in the explanation evaluation study for their valuable contributions.
The first author is supported by a doctoral training grant from the UK Engineering and Physical Sciences Research Council (EPSRC).
|
1,108,101,563,221 | arxiv | \section{Deep $Q$-Learning}
$Q$-learning is a reinforcement learning (RL) algorithm to teach an agent what action to take in an environment under what circumstances. It is model-free (does not need require a model of the environment).
For a finite Markov decision process, Q-learning finds a policy that maximizes the expectation value of the cumulative rewards: the reward over all successive steps starting from the current state.
Formally, RL involves a set of states $S$ and a set of actions $A$. By taking an action $a_t \in A$ at step $t$, the agent makes the environment transition from a state $s_t$ to another state $s_{t+1}$. Performing an action $a_t$ in a given state $s_t$ provides the agent with a reward $r_t$ (a scalar). For Markov decision processes, the state $s_{t+1}$ and the rewards $r_{t}$ only depend on the action $a_t$ and state $s_t$, and not on the previous ones. The episodic RL problem can be represented by a trajectory
\begin{equation}
s_0 \rightarrow a_0 \rightarrow r_0 \rightarrow s_1 \rightarrow \dots \rightarrow a_{n-1} \rightarrow r_{n-1} \rightarrow s_{n},
\end{equation}
where $n$ is the number of steps in an episode.
As explained in the main text, $t=0, \dots, n$, corresponding to the steps of the DQS. At $t=0$, the quantum wave function is in a given initial state $|\psi_0\rangle$ and $n$ is the number of entangling gates in the DQS. The actions $a_t = (\theta^{xx}_t, \theta_t^{z, 1}, \theta_t^{x, 1}, \dots, \theta_t^{z, N}, \theta_t^{x, N}) \in A$ define unitary operators $U_t$ [see Eq.~\eqref{eq:action_step}], and the states $s_t$ represent the quantum wave functions of the qubits $|\psi_t\rangle = U_t \cdots U_2 U_1 |\psi_0\rangle$. However, we internally represent the state as $[t, U_t]$. It has the advantage of having a fixed dimension, and even though it does not contain the full information of the wave function, the efficiency of the algorithm the RL algorithm is not affected because of the small number of steps in the episode. Finally, the reward $r_t \in [0, 1]$ is chosen as
\begin{align}
r_t = \left\{
\begin{aligned}
0\,&, \quad {\rm if}\, t < n \\
f(|\psi_{\rm target}\rangle , |\psi_{\rm DQS} \rangle)\,&, \quad {\rm if}\, t = n.
\end{aligned}
\right. ,
\end{align}
where the function $f$ is either the fidelity reward of the local reward defined in the main text. At the end of the episode, $t=n$, the reward characterizes how close the final state $|\psi_{\rm DQS}\rangle$ is to the target state $|\psi_{\rm target} \rangle = e^{-i\mathcal{H} \tau} |\psi_0 \rangle $. For intermediate steps, the rewards is set to 0 as we do not constraint the specific evolution of the quantum wave function between the initial and target state.
An essential part of any RL algorithm is to define a policy -- how to choose the next action $a_t$ given that the environment is in the state $s_t$.
The central objects in $Q$-learning are the action-value functions, or $Q$ functions, which are used to define the policy. The optimal action-value function $q_*(s, a)$ is defined as the expected total return $\sum_{t=1}^{n} r_t$, given that the environment is in the state $s$ and that the agent takes the action $a$ and acts optimally afterwards. Once the $q_*$ is known, the optimization problem becomes trivial, and the optimal policy is $a_t = {\rm argmax}_a q_*(s_t, a)$. The goal of Q-learning is to build an approximation $Q(s, a)$ through clever exploration of the environment. The optimal $Q$ function satisfies the Bellman optimality equation \cite{sutton2018reinforcement}, which is solved numerically using temporal difference learning through the update rule
\begin{equation}
\label{eq:update}
Q(s_t, a_t) \leftarrow Q(s_t, a_t) + \alpha[r_t + \max_a Q(s_{t+1}, a) - Q(s_t, a_t)],
\end{equation}
where $\alpha$ is a learning rate. In Q-learning the actions are chosen using current estimations of the $Q$ function, $a_t = {\rm argmax}_a Q(s_t, a)$, on top of which noise is added in order to increase exploration of the environment (the algorithm is thus off-policy as the behavior policy differs from the target policy). After meaning episodes, the algorithm converges and the policy obtained using $Q(s, a)$ (i.e. $a_t = {\rm argmax}_a Q(s_t, a)$) results in near-optimal sequences of gates.
In the vanilla $Q$-learning algorithm, $s$ and $a$ only take a finite number of discrete values and $Q(s, a)$ is a matrix. To cope with complex environements (with continuous-valued states or highly-dimensional states, or both), function approximators are used for $Q$.
In Deep $Q$-learning, the $Q$ function is approximated with a neural network. For instance, in the original papers of DeepMind \cite{mnih2013playing,mnih2015human}, convolutional networks are used to process the state of the environment made of pixels on a screen. In our algorithm, we use three-layer dense neural networks because the dimensionality is not so high.
Typically, the neural network has a state $s$ as an input, and outputs different values $Q(s, a)$ for all the possible discrete values of $a$.
In our problem, the actions themselves are also continuous. In this case, there are Actor-Critic RL algorithms that solve the continuous action problem, such as deterministic policy gradient \cite{lillicrap2015continuous}. We tried using deterministic policy gradient, but it turned out that a more simple modification of the deep $Q$-network worked better. The neural networks used to represent $Q(s, a)$ have both actions $a$ and states $s$ as inputs, and a single scalar as output for the value of $Q(s, a)$. The only non-trivial difference from the usual deep $Q$ network is how to calculate ${\rm argmax}_a Q(s, a)$: we use gradient ascent (with Nesterov accelerated gradient) to maximize $Q(s, a)$ with respect to the action inputs and with fixed state inputs.
In addition, RL is known to be unstable or divergent when a nonlinear function approximators such as a neural networks are used \cite{mnih2015human}.
The instability comes from the correlations present in the sequence of data used to train the neural network, the fact that small updates to $Q$ can significantly change the policy, and the correlations between $Q$ and the target values. Indeed, the $Q$ function is theoretically both used to choose the behavior $a_t$ and the target in Eq.~\eqref{eq:update}. We use common techniques developed to solve these problems:
experience replay, and iterative updates of the target $Q$ network (a separate network only periodically updated in order to reduce correlations with the target)
In the following, we present the details of the algorithm used with pseudocode. The hyperparameters are given as we used them after tuning (through grid search).
\begin{algorithm}[H]
\centering
\caption{Deep Q Networks}\label{algorithm}
\begin{algorithmic}[1]
\State $N \gets$ number of sites, $n \gets$ number of steps (typically $n=3$)
\State Initialize replay memory $\mathcal{D}$ with capacity of $50$ episodes.
\State $d_a \gets 2 N + 1$ (dimension of actions)
\State Initialize $Q_{\rm behavior}$ with random weights (dense NN with architecture [150: tanh, 40: relu, 1: $\sigma$] and input dim. $n + 2d_a$).
\State $Q_{\rm target} \gets Q_{\rm behavior}$ (copy the NN weights)
\For{episode $=1$ to $5 \cdot 10^4$}
\State Set value of $\epsilon$ for exploration according to some schedule (from $1$ to $0.005$ exponentially)
\State $s_0 \gets [onehot(t=0)] + [0]*d_a$
\State every
\For{$t=0$ to $n-1$}
\State Select action $a_t \gets \Call{choose\_action}{s_t}$.
\State Add Gaussian noise with standard deviation $\epsilon/2$ to $a_t$.
\State $r_t \gets 0$, $s_{t+1} \gets [onehot(t), a_t]$
\EndFor
\State $r_{n-1} \gets \Call{Reward}{a_0, \dots, a_{n-1}}$, scalar reward calculated from the target and final states (fidelity or local reward).
\State Push all ($s_t, a_t, r_t, s_{t+1}$) transitions of the episode to memory $\mathcal{D}$.
\For{$(s_t, a_t, r_t, s_{t+1})$ in $\mathcal{D}$}
\State set target $y \gets r_t + \max_a Q_{\rm target}(s_{t+1}, a)$
\State Perform backpropagation on $Q_{\rm behavior}(s_t, a_t)$ with target $y$ (with adam optimizer and logcosh loss)
\EndFor
\EndFor
\State \textbf{Return} sequence ($a_0, \dots, a_{n-1}$) with max reward $r_{n-1}$.
\\
\Function{choose\_action}{$s_t$}
\State Find ${\rm argmax}_a Q_{\rm behavior}(s_t, a)$ by performing gradient ascent (with Nesterov accelerated gradient) on $Q_{\rm behavior}(s_t, \cdot)$.
\State Start from $15$ different $a$ (with all components between $-1$ and $1$). momentum $= 0.9$, learning rate $=0.6$.
\State Return best $a$.
\EndFunction
\\
\Function{reward}{$a_0, \dots, a_{n-1}$}
\For{$t=0$ to $n-1$}
\State Obtain physical parameters by rescaling the components of $a_t$:
\State $\theta^{xx}_t \gets a_t[0] \cdot 0.2$
\State $(\theta^{z, 1}_t ,\dots, \theta^{x, N}_t) \gets a_t[1:-1] \cdot 0.4$
\EndFor
\State Compute $|\psi_{\rm DQS}\rangle = U_{n} \cdots U_2 U_1 |\psi_0\rangle$
\State Return reward calculated from $|\psi_{\rm DQS}\rangle$ and $|\psi_{\rm target}\rangle$.
\EndFunction
\end{algorithmic}
\end{algorithm}
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{2636_gates.eps}
\caption{\small Example of a resulting DQS quantum circuit obtained from our algorithm (16-qubit DQS using local reward for the Schwinger model).}
\label{fig:gates_drawing}
\end{figure}
\end{document}
|
1,108,101,563,222 | arxiv | \section{Introduction}\label{sec:intro}
A key feature of many living systems, such as bacteria and eukaryotic cells, is their ability to actively self-propel. Indeed, cellular motility underlies numerous critical biological processes, such as infection and biofilm formation by bacteria, cancer metastasis, tissue repair and wound healing, and morphogenesis of new tissues and organs. As a result, motility behaviors have been extensively investigated for diverse classes of cells, both in isolation and in groups, in uniform environments such as liquid cultures or flat Petri dishes. These studies have provided tremendous insight into the biological and physicochemical mechanisms that engender and regulate cellular motility.
However, cells often inhabit complex and non-uniform spaces---e.g., gels and tissues in the body, or soils and sediments in the environment---that impose additional geometric constraints, mechanical cues, and other external stimuli such as chemical gradients and fluid flow. These factors can fundamentally alter cellular motility, hindering or promoting active transport in unexpected ways, and giving rise to fascinating behaviors such as directed migration and large-scale clustering. These phenomena are not only biologically important, but have consequences for industrial and environmental processes ranging from chemical sensing/delivery to sustaining plant growth in agriculture. Moreover, studies of these phenomena motivate new ways of thinking about transport processes---in many cases, challenging current understanding based on studies of either active matter in uniform settings or ``passive'', thermally-equilibrated matter in complex environments. Biological systems are structurally messy, rheologically complex, and are inherently multicomponent, disordered, and out of equilibrium, and yet, they robustly self-assemble and function in a coordinated manner. Thus, studies of active transport in complex environments could yield new insights for the control of soft and active materials more broadly. It could also inspire the development of new forms of adaptive and multifunctional matter that are capable of recapitulating and even going beyond the capabilities of living systems.
In this chapter, we describe recent progress in studies of active transport in complex environments. We focus on two prominent biological systems---bacteria and eukaryotic cells---as archetypes of active matter; however, where relevant, we also reference works on other forms of active matter as well as more general theoretical/computational models. Our goal is not to present a comprehensive overview of all the literature in this field, but rather, to highlight some areas of research whose growth has been particularly rapid. Along with describing research findings, within each section, we describe specific open questions and promising avenues for future research. Finally, motivated by the diverse forms of active matter---ranging from enzymes and driven biopolymer assemblies, to microorganisms and synthetic microswimmers, to larger animals and even robots---we conclude this chapter by highlighting general principles connecting active systems across vastly different length and time scales.
\section{Bacteria}\label{sec:sec2}
Bacteria are an archetype of active microswimmers that can move using a variety of different mechanisms. Indeed, the ability of many bacteria to self-propel was noted in their initial discovery in 1676: upon peering into a drop of pond water with a microscope, Antony van Leeuwenhoek remarked \textit{``that no more pleasant sight has ever yet come before my eye than these many thousands of living creatures, seen all alive in a little drop of water, moving among one another, each several creature having its own proper motion."}~\citep{bergsciam} Over the centuries since, researchers have extensively characterized the motility of diverse species of bacteria, typically focusing on cells in uniform environments such as in liquid cultures or near flat surfaces.
A common mechanism by which many bacteria self-propel in fluid is using one or more flagella---slender, actively-moving $\sim5~\mu$m-long appendages attached to the cell body surface~\citep{berg}. For the canonical examples of \textit{Escherichia coli} and \textit{Bacillus subtilis}, as well as many other species of bacteria, multiple helical flagella rotate synchronously as a bundle, enabling the cell to ``push'' surrounding fluid~\citep{lauga2009hydrodynamics} and swim along directed, ballistic ``runs'' of mean speed $\bar{v}_{\rm{r}}\sim25~\mu$m/s. Runs do not proceed indefinitely; instead, they are punctuated by reorientations (``tumbles'') caused by one or more of the flagella rotating in the opposite direction, which forces the flagellar bundle to splay out and reorient the cell body, before rapidly coming back together and pushing the cell along another run in a different direction. Runs therefore have a mean length $\bar{\ell}_{\rm{r}}\sim50~\mu$m. Compared to runs, which last on average $\bar{\tau}_{\rm{r}}\sim2$ s, tumbles are effectively instantaneous, lasting on average $\bar{\tau}_{\rm{t}}\sim0.1$ s. Hence, over large length and time scales, this run-and-tumble motion can be modeled as a random walk with an active translational diffusion coefficient given by $D_{\rm{b}}\sim\bar{\ell}_{\rm{r}}^{2}/\bar{\tau}_{\rm{r}}\sim1000~\mu$m$^{2}$/s. While other species of bacteria self-propel using other motility mechanisms, resulting in e.g., run-reverse or run-reverse-flick motion instead, these dynamics can again be modeled as random walks with active translation diffusion coefficients that are determined by the features of single-cell propulsion ~\citep{Xie:2011,Taktikos:2013,theves2013bacterial,Taute:2015}.
While this description enables bacterial transport to be predicted in uniform liquids, in many real-world settings, bacteria must navigate complex environments such as gels and tissues in the body, micro- and meso-porous foods, and soils, sediments, and subsurface formations in the environment. In some cases, bacterial transport can be harmful to humans: for example, during an infection, pathogens squeeze through pores in tissues and gels and thereby spread through the body~\citep{balzan,chaban,pnas,harman,ribet,siitonen,lux,oneil}---potentially leading to the formation of virulent biofilm communities that are recalcitrant to treatment. Similarly, during meat spoilage, pathogenic bacteria squeeze through pores in tissue and spread in contaminated meat~\citep{Gill1977,Shirai2017}.
Bacterial spreading can also be beneficial: for example, a promising route toward cancer treatment relies on engineered bacteria penetrating into tumors and delivering anticancer agents ~\citep{thornlow,Toley:2012}. In agriculture, rhizosphere bacteria move through soils to help sustain and protect plant roots, which impacts crop growth and productivity ~\citep{dechesne,souza,turnbull,watt,babalola}. Furthermore, in environmental settings, bioremediation efforts often seek to apply motile bacteria to migrate towards and degrade contaminants trapped in porous groundwater aquifers ~\citep{roseanne18,Adadevoh:2016,ford07,wang08}. Moreover, motility enables bacteria to adopt symbiotic relationships with other organisms in natural environments, helping to promote ecological stability and even shaping evolutionary processes ~\citep{raina2019role}.
However, despite their potentially harmful or beneficial consequences, there is an incomplete understanding of how bacteria move in such complex environments---which inherently have heterogeneous structures that impose physical obstructions to the cells, are often deformable or viscoelastic, and expose cells to stimuli such as chemical gradients and fluid flow. This gap in knowledge hinders attempts to predict the spread of infections, design new bacterial therapies, develop accurate ecological models, and implement effective agricultural and bioremediation strategies. Elucidating how complexities such as physical confinement, environmental deformability and viscoelasticity, and external stimuli alter bacterial motility and spreading is therefore an important frontier of current research that bridges biology, physics, and engineering.
Moreover, bacteria are an attractive system to use in fundamental studies of active matter due to the diversity and tunability of their transport behavior. For example, the run-and-tumble dynamics exhibited by \textit{E. coli}, \textit{B. subtilis}, and many other bacteria are also observed in diverse other active systems, including self-propelled colloids, algae, eukaryotic cells, and animals ~\citep{karani2019tuning,lozano2018run,polin2009chlamydomonas,heuze2013migration,benichou2011intermittent}. As a result, a growing number of studies describe active particles using run-and-tumble dynamics, and results obtained with bacteria may be applicable to other systems as well. Furthermore, the use of different bacterial species and genetic mutants enables the study of cells of diverse other motility behaviors, shapes, sizes, and surface chemistries, which provides a means to explore a broad range of microswimmer properties ~\citep{tokarova2021patterns,harman,theves2013bacterial,Xie:2011,packer1997rhodobacter,kinosita2018unforeseen,kuhn2017bacteria,jin2011bacteria}. Thus, studies of bacterial transport in complex environments can provide useful fundamental insights into the dynamics of active matter more broadly.
In this section, we review recent studies of the single- and multi-cellular transport of bacteria---focusing on rod-shaped run-and-tumble bacteria as a model system---in complex environments characterized by physical confinement, viscoelasticity, and forcing by external stimuli such as chemical gradients and fluid flow. Because our focus is on active transport, we only focus on the dynamics of planktonic, non surface-attached bacteria. However, studying the formation of surface-attached bacterial communities in complex environments is another active area of current research that has been reviewed recently in e.g., Refs.~\citenum{wong2021roadmap} and \citenum{conrad2018confined}. Furthermore, we note that many bacteria self-propel through other mechanisms, such as by swarming or pulling themselves on surfaces ~\citep{jin2011bacteria}---which potentially enables them to sense mechanical cues on surfaces ~\citep{Persat:2015} and thereby respond to a broader range of stimuli. Investigating these other forms of active transport in complex environments is another important area of current research ~\citep{tokarova2021patterns}. Finally, motivated by rapid progress in the synthesis of colloidal microswimmers capable of recapitulating many of the features of bacteria ~\citep{Bechinger2016b}, we also integrate references to studies of synthetic microswimmers throughout this section where relevant. Such systems enable swimmer shape, deformability, and propulsion mechanism to be tuned even more broadly, and investigating the influence of these factors on active transport will continue to be a useful direction for future work.
\subsection{Complex geometries and obstacles}\label{bac-geometry}
\textbf{Swimming near flat and curved surfaces.} The simplest form of physical confinement is being forced to interact with a solid surface. In this case, hydrodynamic and physicochemical interactions retain cells near the surface ~\citep{Takagi:2014,Brown:2016,Frymier:1995,Lauga:2006,molaei,wensink2008aggregation,elgeti2009self,schaar2015detention,spagnolie2012hydrodynamics,berke2008hydrodynamic,vigeant2002reversible,li2009accumulation,sipos2015hydrodynamic}, although additional corrugations on the surface can suppress this effect ~\citep{kurzthaler2021microswimmers,mok2019geometric}. Similar behavior arises at an immiscible liquid interface ~\citep{vladescu2014filling,desai2018hydrodynamics,ahmadzadegan2019hydrodynamic}, although additional effects arising from interfacial flows can also play a role ~\citep{deng2020motile}. When the surface is flat, or its radius of curvature is much larger than the cell size, an \textit{E. coli} cell continues to swim in planar clockwise (when viewed from above) circles along the surface (Fig.~\ref{fig:bact-figure1}\textbf{A})---reflecting the clockwise (when viewed from behind the cell) counter-rotation of the cell body in response to the counter-clockwise rotation of the flagellar bundle that propels the cell ~\citep{Frymier:1995,Lauga:2006}. More generally, the direction of this circular motion presumably depends on which direction the flagella rotate; further details of the underlying physics are provided in the chapter \textit{Hydrodynamics of Cell Swimming}. Random fluctuations or tumbles then enable the cell to escape. Conversely, when the radius of curvature of the surface of an obstacle is comparable to or smaller than the cell size, the obstacle only rectifies the direction of swimming, causing the cell to be transiently retained at the surface before continuing to move beyond it ~\citep{spagnolie2015geometric,Takagi:2014} (Fig.~\ref{fig:bact-figure1}\textbf{B-C}).\\
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{chapter/figs/bact-fig1.png}
\caption{\begin{small}\textbf{Microswimmers moving near flat and curved surfaces.} \textbf{A}. Superposition of phase-contrast video microscopy images showing \textit{E. coli} cells swimming in circular trajectories near a flat glass surface. Reprinted with permission from Ref.~\citenum{Lauga:2006} (Elsevier, 2006). \textbf{B}. Trajectory of a chemically propelled $2~\mu$m-long micro-rod orbiting around a passive sphere of diameter $\sim6~\mu$m. Reprinted with permission from Ref.~\citenum{Takagi:2014} (The Royal Society of Chemistry, 2014). \textbf{C}. Twenty instances of computed trajectories for a pusher microswimmer incident upon a large sphere for smaller (left) and larger (right) microswimmer translational diffusion constants $D$, which have been made dimensionless as detailed further in Ref.~\citenum{spagnolie2015geometric}. The left panel shows an example in which microswimmers remain hydrodynamically bound to the sphere surface; color coding/shading indicates the final position in the normal direction with darker swimmers coming out of the page and lighter swimmers going into the page. The right shows an example in which microswimmer motion is rectified, but they do not remain bound. Reprinted with permission from Ref.~\citenum{spagnolie2015geometric} (The Royal Society of Chemistry, 2015).\end{small}
\label{fig:bact-figure1}}
\end{center}
\end{figure}
\noindent \textbf{Transport in structured 2D environments.} This phenomenon can have important consequences for individual bacteria moving in crowded and porous media, in which they are forced to interact with many surrounding solid surfaces. Most three-dimensional (3D) porous media are opaque, typically precluding direct observation of bacteria and other microswimmers within the pore space. MRI measurements provide quantification of macroscopic spreading ~\citep{sherwood2003analysis}; however, they do not have sufficient sensitivity to probe single-cell behavior. As a result, experiments typically focus on simplified, two-dimensional (2D) models of porous media made by planar confinement of arrays of cylindrical obstacles. These studies, corroborated and extended by simulations, have catalogued a diversity of transport behaviors. Intuitively, one expects that the presence of obstacles hinders transport---which is indeed the case for microswimmers in disordered media with densely-packed obstacles ~\citep{Mokhtari:2019,morin,sosa2017motility,raatz2015swimming,volpe2011microswimmers,martinez2018collective,martinez2020trapping}, even causing surface-mediated localization in sufficiently dense media ~\citep{morin,hofling2013anomalous,van1982transport,Bertrand:2018,zeitz2017active,jakuszeit2019diffusion,chamolly2017active,aragones2018diffusion,paoluzzi2014run,guccione2017diffusivity,Reichhardt:2014,reichhardt2018,martinez2020trapping} (Fig.~\ref{fig:bact-figure2}\textbf{A-B}). In ordered media, however, surface interactions can help guide swimmers through the straight interstitial channels of the obstacle lattice, promoting size-dependent swimmer transport ~\citep{Brown:2016,dietrich2018active,chopra2020geometric}. Moreover, at low densities of small obstacles, bacterial transport can be unexpectedly \textit{enhanced} beyond the case of obstacle-free environments due to rectification of the circular planar motion of the cells as they encounter obstacles ~\citep{Makarchuk:2019}.\\
\noindent \textbf{Transport in structured 3D environments.} While these 2D studies provide important intuition, how such effects translate to 3D porous media is unclear. For example, the pore space of a 3D medium is more connected, and the threshold solid fraction at which pores can percolate through the medium is lower than in a 2D medium ~\citep{sahini2003applications}. As a result, bacteria have more paths and degrees of freedom available to them in 3D. In addition, the flow field generated by a microswimmer is fundamentally different in 3D compared to 2D media ~\citep{lauga2009hydrodynamics}. Simulations provide clues that these effects strongly impact microswimmer behavior, suggesting that active transport in 2D and 3D media may be fundamentally different ~\citep{morin,stenhammar2014phase}. Thus, there is growing interest in experiments capable of probing active transport in 3D porous media.
\begin{figure*}[htp]
\begin{center}
\includegraphics[width=0.7\textwidth]{chapter/figs/bact-fig2.png}
\caption{
\begin{small}\textbf{Hindered motility of microswimmers in crowded environments.} \textbf{A}. Colloidal rollers in random obstacle lattices of different densities. Trajectories of colloidal rollers (red and yellow) are superimposed on the pictures of the obstacle lattices. Scale bar is $500~\mu$m. Obstacle density increases from left to right. At the lowest density, the trajectories form a single percolating cluster; at the intermediate density, the trajectories form disconnected clusters (yellow), while the largest cluster (in red) still percolates through the observation region; at the highest obstacle density, none of the disconnected clusters percolate, and no macroscopic transport is observed. The largest cluster of maximal dimension $l_c$ is colored in red. Reprinted with permission from Ref.~\citenum{morin} (American Physical Society, 2017). \textbf{B}. Left shows the computed trajectory of a stiff, elongated, active filament in a random obstacle lattice; right shows the computed trajectory of a flexible active filament, which curls up and intermittently jumps to nearby sites. Reprinted with permission from Ref.~\citenum{Mokhtari:2019} (American Physical Society, 2019). \textbf{C}. Left shows a schematic of a transparent, 3D porous medium comprised of a jammed packing of hydrogel particles swollen in liquid cell culture medium (gray circles). Fluorescent \textit{E. coli} (green rods) swim through the pores between particles; inset shows a micrograph of a GFP-labeled bacterium. Scale bar represents $2~\mu$m. Right shows trajectories of cells moving through the pore space \textit{via} hopping-and-trapping motility; in a medium with smaller pores, hopping lengths are smaller and trapping durations are longer, while for a cell with lower activity, hopping lengths are unchanged but trapping durations are longer. Inset images show selected superimposed frames of cellular motion, equally spaced in time, showing that the cell is localized and randomly oriented during trapping, but moving and directed during hopping. Scale bar is $20~\mu$m. Adapted with permission from Refs.~\citenum{tapa19a} and \citenum{tapa19b} (Springer Nature and The Royal Society of Chemistry, 2019).\label{fig:bact-figure2}\end{small}}
\end{center}
\end{figure*}
One approach is to study cells inoculated in viscoelastic ``semisolid'' agar---a common microbiology assay for bacterial motility ~\citep{tittsler}. When the cells are sufficiently dense, their macroscopic spreading through the porous 3D agar matrix can be visualized using a conventional camera. In a seminal study, Wolfe and Berg~~\citep{Wolfe:1989} used this approach to study the spreading of mutants of \textit{E. coli}, including those whose tumbling rate could be controlled chemically. Intriguingly, they found that the extent of spreading in the agar matrix was non-monotonic with the cell tumbling rate: while incessantly tumbling cells did not appreciably spread, as one may expect, smooth swimming cells that do not tumble at all also did not appreciably spread. Instead, these cells became entrapped in the agar matrix---revealing that tumbling is essential for cells to spread in complex, porous environments. Theory and simulations also capture this non-monotonic behavior and have demonstrated it more generally for diverse microswimmers in media with different obstacle geometries ~\citep{licata,Bertrand:2018,Volpe:2017,christinageometric2021}. Related work has also shown that bacterial spreading is similarly non-monotonic in the number of flagella used by the cells, which tunes the amount by which their cell bodies reorient during tumbles ~\citep{Najafieaar6425}. Thus, bacterial spreading in crowded and porous environments likely depends on the frequency and amplitude of reorientations as well as the geometry of the environment, which influences how cells move and collide with surrounding obstacles. Further exploration of this interplay, both for bacteria and other microswimmers more broadly, will be an interesting direction for future research.
While experiments using semisolid agar provides powerful insights into the ability of cells to macroscopically spread, they are limited in their ability to elucidate single-cell behavior; such media are turbid, precluding high-fidelity and long-time visualization of individual cells. Moreover, there is limited control over the properties of the agar matrix, and thus, such media often do not have well-defined pore structures. Hence, to overcome these limitations, researchers have developed 3D porous media made of densely-packed hydrogel particles swollen in defined liquids containing the salts and nutrients required to sustain cellular functioning ~\citep{tapa19b} (Fig.~\ref{fig:bact-figure2}\textbf{C}, left). The individual cells cannot penetrate into the particles; instead, cells move by swimming through the pores formed between adjacent hydrogel particles. The stress exerted by cellular swimming is insufficient to deform the solid matrix, and thus, the packings act as rigid, static matrices. Tuning the hydrogel particle packing density provides a straightforward way to tune the pore size distribution of the overall packing, and thereby modulate cellular confinement. Moreover, because the particles are highly swollen, these packings are transparent, enabling cellular transport to be interrogated \textit{in situ} at single cell resolution using confocal microscopy.
Experiments using this platform have revealed fundamental differences in the transport of \textit{E. coli} in 3D porous media compared to in bulk liquid ~\citep{tapa19a,tapa19b}. In particular, analysis of the trajectories of single cells demonstrated that the paradigm of run-and-tumble motility does not apply to cells in tight porous media with mean pore sizes between $\sim1$ to $10~\mu$m---characteristic of many bacterial habitats ~\citep{dullien,fatin,lang,zalc,lindquist}, and comparable to the overall size of a single cell. Instead, the cells exhibit a distinct mode of motility known as ``hopping-and-trapping'' (Fig.~\ref{fig:bact-figure2}\textbf{C}, right). To move through the pore space, a cell must perform a run along a straight line path; however, it collides with an obstacle well before it completes its run. Thus, runs are truncated (termed ``hops''), with the hopping lengths set by the lengths of the straight line paths that fit in the pore space (known as ``chords'' ~\citep{torquato,lu92}). Collisions with the surrounding matrix do not act as tumbles that rapidly reorient the cell, as is often assumed; instead, steric interactions between flagella and the surrounding solid matrix cause the cell to become transiently trapped, as first observed by Wolfe and Berg ~\citep{Wolfe:1989} and corroborated by experiments in 2D microfluidic models as well ~\citep{tokarova2021patterns}. When a cell is trapped, its flagella remain bundled and continue to rotate---in some cases, over durations much longer than the duration of runs in unconfined liquid, indicating that confinement suppresses unbundling ~\citep{tapa19a}. The torque generated causes the cell body to constantly reorient until the flagella have room to unbundle and rebundle on the opposing pole of the cell body, enabling the cell to escape---similar to the swim-pause-reverse behavior seen for bacteria in model wedge-shaped geometries ~\citep{Cisneros:2006}. After it escapes, the cell then continues to move through the pore space along another hop until it again encounters an obstacle and becomes trapped. Thus, trapping of cells is an active matter analog of the intermittent trapping of passive species that arises during thermally-activated transport in diverse disordered systems ~\citep{bouchaud}, such as colloidal particles in dense suspensions or polymer networks, adsorbing solutes in porous media, macromolecules inside cells, molecules at membranes, and even charges in amorphous electronic materials ~\citep{scher,weeks02,wong,drazer,akimoto,yamamoto13,yamamoto14}.
Indeed, by analogy to the process by which large polymers thermally escape from tight pores in a disordered porous medium ~\citep{muthu87,muthu89a,muthu89b}, tight spots of a crowded 3D environment can be thought of as entropic traps for cells ~\citep{tapa19b}, within which only certain configurations of the cell enable it to escape ~\citep{han}. The duration of trapping is then determined by a competition between cellular activity ~\citep{Woillez2019,Geiseler2016,Bi2016,takatori2015towards} and confinement (Fig.~\ref{fig:bact-figure2}\textbf{C}, right). Unlike conventional tumbles, trapping lasts $\tau_{\rm{t}}\sim1$ to $10$ s, longer than the hopping duration $\tau_{\rm{h}}\sim0.1$ to $1$ s. Hence, over large length and time scales, each cell performs a random walk with pauses between successive steps, reminiscent of a L\'{e}vy walk with rests ~\citep{Zaburdaev:2015}---suggesting that bacteria swimming in a porous medium have unexpected similarities to moving mammalian cells ~\citep{harris2012generalized}, robots searching for a target ~\citep{nurzaman2009yuragi}, and tracers in a chaotic flow ~\citep{weeks1997experimental}. This process can therefore be modeled using a stochastic two-state random walk approach ~\citep{lazaro}. The characteristic step length is given by the mean hopping length, $\bar{\ell}_{\rm{h}}\sim1$ to $10~\mu$m, which is set by the geometry of the porous medium, and the characteristic step time is primarily determined by the mean trapping duration, $\bar{\tau}_{\rm{t}}\sim1$ to $10$ s, which is set by both medium geometry and cellular activity. These quantities then yield a translational diffusion coefficient $D_{\rm{b}}\sim\bar{\ell}_{\rm{h}}^{2}/\bar{\tau}_{\rm{t}}\sim0.1$ to $100~\mu$m$^{2}$/s, much smaller than the unconfined case, that can quantitatively describe bacterial spreading in 3D porous media over large length and time scales ~\citep{tapa19a}. \\
\begin{figure}
\includegraphics[width=\columnwidth]{chapter/figs/bact-fig3.pdf}
\caption{\begin{small}\textbf{An example of active transport in a deformable medium.} Giant unilamellar vesicle containing motile \textit{Bacillus subtilis}. The schematic shows how the 3D system (A) is imaged at a single equatorial cross section (B) to generate the experimental images in (C)-(F). The dashed blue outline in (C) shows the undeformed spherical shape of the membrane when bacteria are non-motile, while (D)–(F) show how motile bacteria (dotted red outlines) generate large membrane deformations at different times. Reprinted with permission from Ref.~\citenum{PhysRevLett.124.158102} (American Physical Society, 2020). (G) shows results from simulations of self-propelled particles enclosed by a deformable membrane. This diagram maps the different states that arise for different particle P\'{e}clet numbers (Pe), which indirectly quantifies the propulsion force, and volume fractions $\phi$. Three regimes emerge: tethering (blue symbols), fluctuating (red symbols) and bola/prolate (green symbols) vesicle shapes. The points corresponding to the snapshots have black outlines. Further details are provided in the chapter \textit{Computational Physics of Active Matter}. Reprinted with permission from Ref.~\citenum{vutukuri2020active} (Springer Nature, 2020).\end{small}\label{fig:bact-figure3}}
\end{figure}
\noindent \textbf{Additional open questions.} While these studies provide a revised view of motility in crowded and porous environments, and draw intriguing connections between out-of-equilibrium, active matter and passive systems, they also motivate many fascinating questions to be addressed by future work. For example: How do the characteristics of hopping and trapping motility depend on cellular morphology and surface properties, as well as the geometry of the surrounding crowded environment? How do these behaviors change for other forms of active matter that employ other motility behaviors in unconfined environments distinct from run-and-tumble dynamics? How do chemical interactions (e.g., surface adhesion) and mechanical interactions (e.g., deformations of cells as well as their surroundings), which arise in many complex environments, influence active transport? One example of how mechanical interactions, in which cellular swimming deforms a surrounding boundary, give rise to intriguing dynamics is shown in Fig. \ref{fig:bact-figure3}; other examples are described in Section \ref{bac-polymer} below.
Furthermore, how do these behaviors change for more concentrated bacterial populations? In bulk fluid, short-range interactions between active particles, such as bacteria, enable them to coordinate motion and spontaneously ``flock'' along the same direction or coherently ``swirl'' ~\citep{Driscoll2017a,martin2018collective,ginelli2010large,dunkel2013fluid,dombrowski2004self,cisneros2010fluid,ishikawa2011energy,wensink2012meso,colin2019chemotactic}. At higher concentrations, they can even form dense clusters or completely phase separate, depending on the interactions between them ~\citep{takatori2015towards,cates2015motility,theurkauff2012dynamic,buttinoni2013dynamical,pohl2014dynamic,palacci2013living,aragones2019aggregation,chen2015dynamic,petroff2015fast,ginot2015nonequilibrium,redner2013reentrant,fily2012athermal,digregorio2018full,thutupalli2018flow,geyer2019freezing}. When the cells are globally confined (e.g., in a thin cylindrical chamber), such collective behaviors can be promoted \cite{wioland2013confinement,lushi2014fluid} and potentially even harnessed to drive larger-scale autonomous motion \cite{gao2017self,young2021many,sanchez2012spontaneous}. However, when cells are surrounded by a disordered array of obstacles, this solid matrix suppresses these short-range interactions and thus can suppress such collective behaviors ~\citep{martin2018collective,Chepizko:2013,altenberger1986hydrodynamic,chepizhko2015active,chepizhko2013optimal,sandor2017dynamic,sandor2017dewetting,sese2018velocity}. Nevertheless, other forms of coordination that rely on the coupling of cellular motion to external stimuli that can extend over longer distances, such as chemical fields, can still persist; these are described further in Section \ref{bac-stimuli}.
\subsection{Viscoelastic polymer solutions}\label{bac-polymer}
Many bacterial habitats are polymeric fluids. Prominent examples include mucus in the lungs and digestive tract ~\citep{lai2009micro,celli2009helicobacter,steinberg2019high,pnas}, exopolymers in the ocean ~\citep{decho1990microbial}, and cell-secreted extracellular polymeric substances (EPS) that encapsulate surface-attached biofilms ~\citep{dorken2012aggregation}. One may expect that, for a given bacterial motor torque, the increased fluid viscosity caused by polymers would lead to slower cellular swimming. Surprisingly, however, experiments show that increasing polymer concentration increases swimming speed, sometimes followed by a decrease in speed at large concentrations ~\citep{schneider1974effect,berg1979movement}. In some cases, this behavior may reflect the metabolism of polymers by bacteria; however, it is also observed with non-metabolizable polymers, suggesting a physicochemical role played by polymers as well. While fully elucidating the relative influence of fluid viscosity, viscoelasticity, and shear thinning on this behavior is still an outstanding challenge, recent work has started to shed light on the physics of bacterial swimming in polymer solutions. \\
\begin{figure}[htp]
\begin{center}
\includegraphics[width=\columnwidth]{chapter/figs/bact-fig4.pdf}
\caption{\begin{small}\textbf{Bacterial swimming in viscoelastic polymer solutions.} \textbf{A}. Kinematics of swimming \textit{E. coli} cells in both Newtonian and viscoelastic fluids. Left and right panels show trajectories of cells in Newtonian buffer and in viscoelastic carboxy-methyl cellulose solution. Cells in polymer solution move remarkably straighter compared to cells in buffer. Sample cell trajectories in the magnified view show run-and-tumble motility i.e. nearly straight lines connected at random angles (tumbles denoted by arrows). Further magnified views show that the cell body orientation oscillates or `wobbles' in the buffer solution, but wobbles diminish in the polymer solution. Reprinted with permission from Ref.~\citenum{patteson2015running} (Springer Nature, 2015). \textbf{B}. Typical simulation snapshots of bacterial dynamics in fluids consisting of a Newtonian background fluid and including semiflexible polymers at low (top) and high (bottom) volume fractions. The starting position of the cell body is indicated by the black dashed line, showing that the bacterium swims faster in the more concentrated suspension. The colours of individual polymers are to aid visualization. Reprinted with permission from Ref.~\citenum{zottl2019enhanced} (Springer Nature, 2019).\end{small}\label{fig:bact-figure4}}
\end{center}
\end{figure}
\noindent\textbf{Transport in dilute polymer solutions.} In dilute solutions of long and flexible polymers, \textit{E. coli} exhibit three notable changes in their swimming kinematics ~\citep{patteson2015running} (Fig. \ref{fig:bact-figure4}\textbf{A}): (i) the cells tumble less frequently, (ii) the cells ``wobble" less during runs, and (iii) the swimming speed is increased. By tracking cells in solutions of polymers of different molecular weights, viscosities, and fluid elasticities, the authors of Ref. \citenum{patteson2015running} proposed that the first effect reflects the increased viscosity of the solution due to polymers; as viscous stresses increase, the mechanical load on the flagellar motor increases, which increases the effective energy barrier for the cell motor to switch and thereby decreases the tumbling rate ~\citep{qu2018changes,fahrner2003bacterial,yuan2009switching}. By contrast, the second and third effects are instead thought to reflect the increased elasticity of the solution due to polymers. In polymer-free solutions, individual cells appear to wobble during their runs, due to the projection of the cell's helical trajectory in three dimensions that is produced by the counter-rotating cell body and flagellar bundle ~\citep{watari2010hydrodynamics}. The curvature associated with this helical trajectory stretches surrounding polymers ~\citep{pakdel1996elastic}, producing elastic hoop stresses that stabilize wobbling and direct more of the thrust in the forward direction, enabling the cell to swim faster ~\citep{PhysRevFluids.6.053301,housiadas2021squirmers,binagia2020swimming}. Because these effects all depend closely on the specific mechanics of flagellar propulsion, understanding how they manifest for other microswimmers that employ other methods of self-propulsion is another area of active inquiry ~\citep{zhu2012self,qin2015flagellar}. Indeed, analysis of the fluid dynamics of diverse other microswimmers indicates that elastic stresses can induce faster swimming even for swimmers that do not wobble \cite{liu2011force,castillo2019drag,pak2014generalized,puente2019viscoelastic,rogowski2021symmetry,binagia2020swimming}.
Elastic stresses may not be the only---or even the dominant---cause, however, of observations (i)--(iii) noted above; indeed, recent experiments studying \textit{E. coli} swimming in colloidal suspensions reported similar behavior \cite{kamdar2021}. Thus, the authors suggested that the hydrodynamic interaction between individual bacteria and the colloidal component of complex fluids in general, mediated by the background Newtonian fluid, is instead the primary cause of these swimming kinematics. Further work disentangling the relative influence of elastic stresses and other hydrodynamic interactions on active transport will therefore be a useful direction for future research.\\
\noindent \textbf{Transport in more concentrated polymer solutions.} When the solution is semidilute, interactions between polymer molecules play an appreciable role as well. Physical entanglements or chemical crosslinks can cause the polymers to form a rigid network that obstructs cells, leading to effects such as those described in Section \ref{bac-geometry}. In the absence of such network formation, or over time scales long enough for the network to relax, the solution rheology and microstructure regulate cellular swimming. At the scale of a single cell, the fast-rotating flagellar bundle generates strong shear locally near the `tail' of the cell body, causing neighboring polymer molecules to align and locally reduce the solution viscosity. For a given motor torque, this viscosity reduction at the flagella is thought to enable the cell to swim faster ~\citep{martinez2014flagellated,qu2020effects,zhang2018reduced}. Simulations also suggest that the polymer molecules themselves also distribute non-uniformly in the vicinity of the bacterium, with polymer being depleted inside and around the flagella (Fig. \ref{fig:bact-figure4}\textbf{B}). This depletion leads to an apparent slip that, again, is thought to also enable the cell to swim faster ~\citep{zottl2019enhanced}. \\
\noindent \textbf{Additional behaviors that arise in polymeric environments.} How do these behaviors change for more concentrated bacterial populations? As noted in Section \ref{bac-geometry}, in bulk fluid, short-range interactions enable bacteria and other forms of active matter to coordinate their motion over large scales. In a polymeric fluid, the elastic stresses generated by swimming are thought to drive bacteria together, enhancing their local aggregation and, in some cases, suppressing large-scale coherent motions and morphological instabilities ~\citep{li2016collective,emmanuel2020active} (Fig. \ref{fig:bact-figure5}). However, when concentrated suspensions are physically confined as well, polymer stresses can conversely \textit{promote} coherent motion over length scales much larger than in bulk fluids---for example, driving the formation of uniformly rotating `giant vortices' ~\citep{liu2021viscoelastic}. Another effect arises when cells are sufficiently close to each other; in a polymer solution, if two cells are closer than the polymer size, they deplete the polymer from the space in between them. As a result, the unbalanced osmotic pressure of the solution forces the cells together. This attractive depletion interaction can drive the aggregation of both non-motile and motile cells in a manner similar to passive colloidal particles ~\citep{dorken2012aggregation,schwarz2010polymer}, although the swimming stress generated by motile cells suppresses aggregation ~\citep{porter2019interplay}. Intriguingly, for the case of motile cells, the finite size aggregates that form \textit{via} depletion interactions show unidirectional rotation, driven by the torques exerted by the bacteria at the outer periphery of each aggregate ~\citep{schwarz2012phase}. Disentangling the influence of these different behaviors, as well as other behaviors that can arise for microswimmers with other swimming characteristics \cite{elfring2010two,elfring2015theory,datt2015squirming,datt2017active,pietrzyk2019flow}, will be an important direction for future research.
\begin{figure}[htp]
\begin{center}
\includegraphics[width=\columnwidth]{chapter/figs/bact-fig5.png}
\caption{\begin{small}\textbf{Viscoelasticity can suppress coherent motion in concentrated bacterial populations.} Computed flow field and distribution of `pusher' microswimmers in a \textbf{A} Newtonian and \textbf{B} viscoelastic fluid. Contours in the main frame and the right-bottom quarter of \textbf{B} show the dimensionless horizontal velocity component $u$ and the dimensionless elastic energy given by the trace of the polymer contribution to the stress tensor tr$(\tau^p)/2$, respectively. Reprinted with permission from Ref.~\citenum{li2016collective} (American Physical Society, 2016).\end{small}\label{fig:bact-figure5}}
\end{center}
\end{figure}
\subsection{External stimuli}\label{bac-stimuli}
\textbf{Transport in response to chemical stimuli.} An important feature of many bacteria, as well as other forms of active matter ~\citep{saha2014clusters,liebchen2019interactions,Liebchen2018a,popescu2018chemotaxis,singh2019competing,marsden2014chemotactic,agudo2019active,Illien2017,Stark2018}, is their ability to sense chemical stimuli and respond by modulating motility. As the cells swim in bulk liquid, they sense varying concentrations of different chemicals---e.g., sugars, amino acids, and oxygen---that continually bind and unbind to and from cell-surface receptor complexes. This binding triggers a network of subcellular processes that primarily modulate the frequency of tumbling and bias run length. Cells thus perform longer or shorter runs when moving toward regions of higher or lower concentration of chemoattractants, respectively, and exhibit the opposite behavior in their response to chemorepellents. As a result, the random walk determined by run-and-tumble motion is biased ~\citep{berg}; the cell's intrinsic ability to bias its motion is described by a parameter, having the same units as the diffusion coefficient, that is known as the chemotactic coefficient $\chi$ ~\citep{lauffenburger1991quantitative,Keller1971a}. The flux due to chemotaxis is then given by $bv_{c}$, where $b$ is the number density of cells and $v_{c}\equiv\chi\boldsymbol{\nabla} f(c)$ is known as the chemotactic velocity; the function $f(c)$ describes the ability of the bacteria to logarithmically sense nutrient at a concentration $c$ ~\citep{Cremer2019,fu2018spatial,dufour,yang2015relation}.
At the multicellular level, this process of chemotaxis can mediate directed collective motion, or migration. A particularly striking example arises when the cells continually consume a surrounding attractant, such as a nutrient or oxygen: the cells collectively generate a local gradient that they in turn bias their motion along, leading to the formation of a coherent band of cells that continually propagates ~\citep{adler1966science,adler1966effect,Cremer2019,fu2018spatial,saragosti2011directional,douarche2009coli,tuval2005bacterial}, sustained by its continued consumption of the surrounding attractant. This phenomenon was first demonstrated by Adler in 1966 through an elegant experiment in which he placed a dense plug of \textit{E. coli} at one end of a capillary tube containing nutrient-rich liquid ~\citep{adler1966science}; the cells spectacularly migrated through the tube as bands visible to the naked eye. Since then, continuum-scale models coupling attractant dynamics to bacterial fluxes have been developed ~\citep{lauffenburger1991quantitative,Keller1971a} and can successfully capture the key features of chemotactic migration in bulk liquid ~\citep{Keller1971a,fu2018spatial,Seyrich2019} (Fig. \ref{fig:bact-figure6}\textbf{A}) and in viscoelastic agar ~\citep{croze,Cremer2019}. Additional studies using a range of different chemoattractants have shown that this phenomenon of chemotactic migration can enable populations to escape from harmful environments or to colonize new terrain ~\citep{Cremer2019,PhysRevX.10.031017}.
\begin{figure*}[htp]
\begin{center}
\includegraphics[width=0.7\textwidth]{chapter/figs/bact-fig6.pdf}
\caption{\begin{small}\textbf{Chemotactic migration of bacterial populations.} \textbf{A}. Images taken at different time points (vertical axis) showing \textit{E. coli} performing chemotactic migration in a straight microfluidic channel. Red and cyan dots show cells with low and high tumble bias---the fraction of time a cell spends tumbling---indicating that cells self-organize within a traveling band. Scale bar shows 0.6 mm. Reprinted with permission from Ref.~\citenum{fu2018spatial} (Springer Nature, 2018). \textbf{B}. Magnified
bottom-up fluorescence intensity projection of a chemotactic front propagating within a 3D porous medium, showing individual cells of \textit{E. coli}. Arrow indicates direction of overall propagation. Scale bar denotes $200~\mu$m. Reprinted with permission from Ref.~\citenum{tapabiophys} (Elsevier, 2021). \textbf{C}. Microscopy images showing the outward spreading of chemotactic bacteria in 2D fractal microfluidics; time stamp indicates time elapsed after initial inoculation in the center. Reprinted with permission from Ref.~\citenum{PhysRevX.10.031017} (American Physical Society, 2020).\end{small}\label{fig:bact-figure6}}
\end{center}
\end{figure*}
Moreover, because this form of collective migration relies on the coupling between a population-generated chemical gradient---which can extend over long distances spanning hundreds of cells---and biased cellular motion along this gradient, different bacteria can collectively influence and coordinate each other's motion across long distances, solely through chemoattractant consumption. This principle was recently demonstrated using 3D-printed cylindrical regions of \textit{E. coli} separated by different distances ~\citep{tapabiophys}. When the separation between the two cylinders was smaller than the length scale $\approx500~\mu$m over which nutrient is depleted by consumption, the cells were able to ``smell'' each other; as a result, chemotactic bands did not propagate between the two cylinders. However, when the separation between the two cylinders was much larger, cells migrated in chemotactic bands both outward and towards each other, in between the two cylinders. This observation suggests that, simply by tuning the geometry of a bacterial population, chemotactic migration can be directed and controlled.\\
\noindent \textbf{Chemotaxis in structured environments.} Chemotaxis can also enable collective migration to persist after a population is confronted with perturbations---either external ~\citep{sandor2017dynamic,Morin2017,Wong2014,Chepizhko2013,chepizhko2013optimal,chepizhko2015active,Toner2018,Maitra2020a}, stemming from heterogeneities in the environment, or internal, stemming from differences in the behavior of individual cells ~\citep{Yllanes2017,Bera2020,Alirezaeizanjani2020}. Such perturbations are usually thought to \textit{disrupt} collective migration. For example, as described in Section \ref{bac-geometry}, collisions with surrounding obstacles disrupt short-range coordination between cells ~\citep{sandor2017dynamic,Morin2017,Yllanes2017,Bera2020,Chepizhko2013,chepizhko2013optimal,chepizhko2015active,Toner2018}. However, as revealed through experiments on \textit{E. coli}, by generating large-scale nutrient gradients through consumption, bacterial populations can drive chemotactic migration, even in highly-confining, disordered, porous environments ~\citep{tapabiophys} (Fig. \ref{fig:bact-figure6}\textbf{B}). Nevertheless, several key differences arise in this case compared to that of migration in bulk liquid. First, the motility parameters $D_{\rm{b}}$ and $\chi$ are strongly reduced in a confinement-dependent manner, reflecting the hindered motion of individual cells in the tight pore space, as well as the increased occurrence of cell-cell collisions in the tight pore space---fundamentally altering the dynamics and morphology of a spreading population, as further explored theoretically in Ref. \citenum{amchin2021confinement}. Second, the cells use a different primary mechanism to perform chemotaxis. Instead of modulating the frequency of reorientations, which is not possible in tight environments, the cells primarily modulate the amount by which their cell bodies reorient by modulating the number of flagella that become unbundled during reorientations---thereby biasing their subsequent motion toward regions of higher nutrient concentration ~\citep{Berg:1972,vladimirov2010predicted,saragosti2011directional}. Thus, by employing multiple different mechanisms to bias their motion, cells can direct their transport in a range of different environments---motivating further studies of active matter transport in a variety of complex settings. Another recent example demonstrating how chemotaxis enables bacteria to escape from fractal mazes is shown in Fig. \ref{fig:bact-figure6}\textbf{C}.\\
\noindent\textbf{Morphological consequences of chemotaxis.} Perturbations are also typically thought to produce large-scale disruptions to velocity correlations between cells and the overall morphology of a cellular population ~\citep{Wong2014,Alert2020,Alert2019,Driscoll2017a,Doostmohammadi2016,williamson2018stability,saverio,bonilla2019contrarian} due to hydrodynamic ~\citep{koch,kela1,kela2} or chemical ~\citep{BenAmar2016,BenAmar2016b,funaki,brenner,mimura,stark} effects. However, recent calculations for suspensions of self-propelled particles have suggested that chemotaxis can stabilize such hydrodynamic instabilities ~\citep{Nejad2019}. Furthermore, experiments using \textit{E. coli} in straight channels have revealed that, even when cells have substantial differences in individual chemotactic abilities, they are able to migrate together with a common drift velocity by spontaneously sorting themselves to different positions with different local nutrient gradients (Fig. \ref{fig:bact-figure6}\textbf{B})---resulting in the formation of a chemotactic band that continues to migrate coherently ~\citep{fu2018spatial}.
Other work studying \textit{E. coli} in 3D environments has shown that chemotaxis enables large-scale perturbations to the overall morphology of a population to be smoothed out as cells migrate, enabling them to continue to migrate together as a coherent band ~\citep{smoothing}. This population-scale smoothing unexpectedly reflects the manner in which individual cells transduce external signals during chemotaxis. In particular, it arises from limitations in the ability of cells to sense nutrient at high concentrations, resulting in spatial variations in cellular response to the local nutrient gradient. This behavior can be quantitatively described by spatial variations in the chemotactic velocity $\textbf{v}_{c}\propto\boldsymbol{\nabla} f(c)=f'(c)\boldsymbol{\nabla} c$; as in linear response theory, this velocity can be viewed as the bacterial response to the driving force given by the nutrient gradient, $\boldsymbol{\nabla}c$, modulated by the chemotactic response function $\chi f'(c)$. Thus, there are two distinct factors that influence how quickly cells move at different locations of a chemotactic band: the magnitude of the local nutrient gradient, and the magnitude of the local nutrient concentration itself, which sets the magnitude of the cellular response. Importantly, because the sensing function $f(c)$ is bounded, the response $f'(c)$ decreases at high nutrient concentrations, at which the cell surface receptors become increasingly saturated. As a result, regions of the population that are exposed to \textit{either} a weaker nutrient gradient $\boldsymbol{\nabla}c$ or more nutrient $c$, which leads to a weaker response $f'(c)$, can move slower than other regions. In the experiments of Ref. \citenum{smoothing}, outward-protruding regions of the population were exposed to more nutrient, causing the cells to respond more weakly and migrate slower, enabling the other regions to catch up and ultimately smoothing out morphological perturbations. Because migration in diverse other active matter systems also involves some form of limited sensing---e.g., chemotaxis by other bacteria ~\citep{menolascina2017logarithmic}, enzymes ~\citep{granick,Agudo-Canalejo2018a,Mohajerani2018}, amoeba ~\citep{keller1970initiation}, and mammalian cells ~\citep{Camley2018,iglesias,theveneau2010collective,mainikulesa,Malet-Engra2015,Puliafito2015,Tweedy2020}, durotaxis by eukaryotic cells ~\citep{roca2013mechanical,sunyer2016collective,alert2018role}, phoresis by active colloids ~\citep{Illien2017,Liebchen2018a,Stark2018}, and phototaxis by robots ~\citep{Mijalkov2016,palagireview}---exploring such effects more broadly will be a fascinating direction for future work.
In follow-up work, the authors used a linear stability analysis of the equations governing chemotactic migration to establish the conditions under which small-amplitude morphological perturbations die away or instead grow \cite{alert2021sensing}. This analysis again reveals that the morphological stability of such chemotactic bands is determined by limitations in the
ability of individual cells to sense and thereby respond to the chemical gradient. Furthermore, it predicts that when sensing is limited at too low chemical concentrations, chemotactic bands are instead destabilized, leading to a chemotactic fingering instability that will be interesting to search for in future experiments. Intriguingly, as detailed in Ref. \citenum{alert2021sensing}, experimental data on \textit{E. coli} chemotaxis seem to suggest that the cells' sensory machinery might have evolved to ensure stable front propagation; experiments focused on
further testing this hypothesis for diverse other cell types will be useful in establishing its generality.\\
\begin{figure}[t!]
\begin{center}
\includegraphics[width=\columnwidth]{chapter/figs/bact-fig7.pdf}
\caption{\begin{small}\textbf{Examples of fluid flow influencing active transport.} \textbf{A}. Lattice-Boltzmann simulation of a rectangular fluid-filled channel with a cylindrical obstacle placed at the center. Inside the channel are run-and-tumble microswimmers. The fluid is driven by an external force density. The stream lines and the colormap represent the driven flow-field projected on to the $xy$-plane normalized by the swimming speed $U_s$. Reprinted with permission from Ref.~\citenum{lee2021influence} (The Royal Society of Chemistry, 2021). \textbf{B}. Spatial distribution of \textit{Bacillus subtilis} bacteria (blue dots) in response to a chemorepellent injection, superposed on the map of fluid velocity $u$ rescaled by its average value, $u_{\text{av}}$. As a result of pore disorder, the velocity field exhibits high-velocity channels (dark red) and low-velocity micropockets (light yellow). Reprinted with permission from Ref.~\citenum{de2021chemotaxis} (Springer Nature, 2021).\end{small}\label{fig:bact-figure7}}
\end{center}
\end{figure}
\noindent\textbf{Additional open questions.} Most studies of chemotaxis only employ a single chemoattractant, for simplicity. However, in natural habitats, bacteria are often exposed to multiple, often competing, chemical stimuli. In this case, does cellular behavior simply reflect a sum of behaviors caused by each stimulus? Building on an 1888 study by Pfeffer, Adler and Tso tested this hypothesis in 1974 by exposing \textit{E. coli} to capillary tubes containing both attractant and repellant---and concluded that the cells do indeed sum the competing behaviors ~\citep{adler1974decision}. Subsequent work, however, has suggested that this simple picture may not be complete ~\citep{strauss1995analysis}; instead, bacterial decision making in response to multiple stimuli also reflects the differing abundances of cell-surface receptors for different stimuli ~\citep{kalinin2010responses}, and thus may depend on the particular combination of stimuli presented ~\citep{zhang2019escape}. Additionally, nutrient and chemoattractant availability can strongly fluctuate, both spatially and temporally, in many bacterial habitats ~\citep{Clausznitzer:2014:10.1371/journal.pcbi.1003870}---due to fluctuations in exogenous sources and sinks, as well as due to the secretion or degradation of chemical cues by other cells. Understanding how bacteria sense stimuli and make decisions in such complex environments, and unraveling how rich population-scale dynamics emerge from these single-cell behaviors, will be a useful direction for future work. Furthermore, it will be interesting to develop synthetic forms of active matter that can similarly sense stimuli, process information, and respond by modulating transport in complex environments.
Many bacterial habitats also have fluid flow ~\citep{Conrad2018}, which can further alter oxygen, nutrient, and cellular profiles in interesting ways. Near surfaces, the torque from fluid shear causes the cells to rotate and swim along periodic helical-like trajectories near the surface ~\citep{rusconi2014bacterial,zottl2012nonlinear,zottl2013periodic,junot2019swimming} or even swim upstream near the surface ~\citep{figueroa2020coli,hill2007hydrodynamic,kaya2012direct,mathijssen2019oscillatory,nash2010run,tung2015emergence}, possibly promoting ~\citep{lee2021influence,secchi2020effect} or alternatively suppressing ~\citep{molaei2016succeed} attachment at specific locations. In 2D porous media, these effects can lead to retention of cells at the solid surfaces ~\citep{creppy2019effect}, but enhanced spreading of cells through the spaces between ~\citep{creppy2019effect,Alonso:2019} (Fig. \ref{fig:bact-figure7}\textbf{A}). In addition, the coupling between fluid flow and solute spreading can lead to the persistence of pore-scale chemical gradients that can lead to the retention of bacteria \textit{via} chemotaxis in low-flow regions of the pore space ~\citep{de2021chemotaxis} (Fig. \ref{fig:bact-figure7}\textbf{B}). Exploring such couplings between fluid flow, chemical transport, and bacterial transport will be an important direction for future work. Finally, we note that interactions with other stimuli, such as viscosity gradients, gravity, and light, can often play a role as well---potentially giving rise to other behaviors that motivate the development of new mathematical models ~\cite{desai2017modeling,datt2019active}.
\section{Eukaryotic cells}\label{sec:sec3}
Anyone who has observed the synchronous motion of embryonic cells or the coordinated behavior of an epithelial monolayer closing a wound can understand why the migration of eukaryotic cells has fascinated scientists for centuries. The interactions between the cells and their environment give rise to large-scale dynamics that are both fundamentally interesting, and enable the coherent migration of the group to serve a biological purpose. For example, in embryogenesis, morphogenesis, and other forms of development, cells grow, proliferate and migrate as a group in a fluid fashion through a complex environment until they reach their final shape---at which point they can collectively self-stiffen to hold their shape and withstand subsequent mechanical stresses~\citep{mongera2018fluid}, as suggested a century ago in a celebrated monograph by D'Arcy Thompson~\citep{thompson1942growth}. In addition to regulating the growth and form of living creatures, cellular migration also underlies critical pathological processes, such as tumor invasion, in which clusters of malignant cells navigate complex environments to invade healthy tissues~\citep{friedl2009collective}.
Therefore, unraveling the role of environment-dependent physical forces and chemical signals in coupling different cells and regulating group dynamics is of fundamental importance to understanding the emergent behaviors underlying key \textit{in vivo} processes. Additionally, progress in the ability to control cell–cell interactions, tissue architecture and properties, and environmental stimuli has generated exciting possibilities to generate and control functional tissue cultures by using different guidance cues---a promising avenue that could help to develop clinically useful applications, or to engineer living meta-materials at our whim.
In this section, we review recent studies of the transport of eukaryotic cells in environments characterized by complex geometries and external stimuli, such as electric fields, chemical gradients, and changes in mechanical properties. For other aspects and perspectives of eukaryotic cell migration, the reader is referred to the excellent reviews of Refs.~\citenum{lecuit2007cell,friedl2009collective,lecuit2011force,mayor2016front,stramer2017mechanisms,ladoux2017mechanobiology,Alert2020}.
\subsection{Complex geometries and obstacles}\label{subsec:euk-geom}
Single-cell and collective migration are strongly affected by geometric constraints. For instance, at early embryonic stages, the geometry confining the embryo is essential to build the specific cell arrangements that lead to the desired final morphology~\citep{stooke2018physical,goodwin2020mechanics,giammona2021physical}. During metastatic colonization, path-finder malignant cells outwardly migrate from the tumour colony squeezing through a microstructured extracellular matrix~\citep{kim2009tumor,ben2012bacterial,Wong2014,massague2021metastasis}. Similarly, immune cells such as macrophages, neutrophils, and dendritic cells migrate through a complex external medium, i.e. lymphoid tissues, to trap and kill pathogens~\citep{friedl2008interstitial,hampton2019lymphatic}. It is thus of great importance to understand and control how the structurally complex environments where eukaryotic cells thrive affect their main migrating mechanisms and survival strategies---a challenge that will help to understand a large number of biological and physiological processes. In this section, we highlight some of the complex structured habitats that have been explored through \textit{in vitro} experiments employing confined and intricate geometries, microchannels, and complex patterned substrata that force the cells to adopt a certain shape.\\
\noindent \textbf{Single-cell migration in structured 2D environments.} The motion of a single cell is a well-studied process that is dictated by intra/extracellular signals and mechanical forces~\citep{ridley2003cell}---yet many aspects involving cell-cell interactions and the role of a complex confining environment on their dynamics remain poorly understood. Concerning 2D configurations, several works have explored the effect of structured and intricate geometries, together with solid obstacles on the behavior and migration of single eukaryotic cells. Fig.~\ref{fig:figure1} highlights some of these studies.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.7\textwidth]{chapter/figs/Euk_Fig1.pdf}
\caption{\begin{small}\textbf{Examples of single-cell migration in structured 2D environments}. \textbf{A}. Time-lapse snapshots showing the transient dynamics of two eukaryotic cells (MDA-MB-231 cells) interacting inside a confined geometry. Arrows indicate regions of high actin activity. Scale bar is $25~\mu$m. Reprinted with permission from Ref.~\citenum{bruckner2021learning} (National Academy of Sciences, 2021). \textbf{B}. Left: Dendritic cell choosing the path of least resistance within an confined geometry with pores of different sizes. Right: nucleus-free cytoplasts (outlined in yellow) migrating through a pillar array of large (orange) and small (green) pores without choosing any preferred path. Time is indicated as h:min:s. Adapted with permission from Ref.~\citenum{renkawitz2019nuclear} (Springer Nature, 2019).\label{fig:figure1}\end{small}
\end{center}
\end{figure*}
One simple way of imposing geometric constraints is using ``two-state" micropatterns consisting of two square adhesive sites connected by a thin gap. Ref.~\citenum{bruckner2019stochastic} studied the single-cell migration dynamics of cancerous (MDA-MB-231) and a non-cancerous (MCF10A) cells in such geometries. The authors found that the motions of both type of cells exhibit qualitatively similar nonlinear migratory dynamics \com{in the position-velocity cellular phase space resembling a nonlinear oscillator}, which can indeed be decomposed into deterministic and stochastic contributions. In particular, they showed that these two cells exhibit different deterministic dynamics---namely, cancerous cells reached a limit cycle, \com{i.e. the position and velocity amplitudes of the self-sustained oscillations reach a fixed value at long time,} and non-cancerous cells displayed excitable bistable dynamics. \com{In this latter scenario, the migratory dynamics does not exhibit self-sustained oscillations, and therefore instead of a limit cycle the authors found that two fixed points for the position and velocity appeared on either side of the bridge. They found that the basins of attraction of these two fixed points occupied all the micro-environment, thus any small perturbation can trigger a switch from one state to the other, indicating the excitable bistable dynamics of MCF10A cells}. More recently, Ref.~\citenum{bruckner2021learning} studied cell-cell interactions and cell coordination using the same confined micro-environment (Fig.~\ref{fig:figure1}\textbf{A}), which the authors referred to as an \textit{experimental cell collider}. They observed \com{several} collision behaviors depending on the interaction between different types of cells. In particular, repulsion and friction interactions dominate between non-cancerous cells, whereas attractive and anti-friction interactions arise between cancerous cells. In the former scenario, cells slow down as they move past each other, which corresponds to a reversal behavior usually termed as `contact inhibition' of locomotion. In the latter, the so-called antifriction produces a sliding behavior that the authors termed `contact sliding' locomotion, where cancerous cells accelerate as they move side by side with increasing velocity, becoming weakly repulsive at long distances. Both distinguished dynamics are in fairly good agreement with their theoretical results based on an interacting stochastic equation of motion. With this theory, the authors also found another interaction dynamics where cells follow each other, a behavior usually referred to as `contact following' locomotion. This work exemplifies how analyzing cell-cell interactions in engineered micro-environments can help shed light on the mechanisms by which cancerous and non-cancerous cells interact during \textit{in vivo} processes---such as the migratory dynamics of malignant cells during tumor invasion, in which the interactions between cancerous and non-cancerous cells underlie the competition between both populations.
The migration of macrophages, neutrophils, and dendritic cells through complex lymphoid tissues plays a crucial role during immune surveillance and immune responses. To unravel the behavior of immune cells in such complex environments, Ref.~\citenum{renkawitz2019nuclear} studied their migration dynamics in different 2D and 3D geometries. Concerning the former, the authors analyzed the migration of individual dendritic cells moving within intricate branching microchannels consisting of a main channel splitting into four different sized smaller channels (see Fig.~\ref{fig:figure1}\textbf{B}). The authors observed that the cells migrate through the main channel until the branching point. Strikingly, the cells then spread their bodies through the different branches, using their nucleus as an indicator to choose the best path, thus being able to move through the path of least resistance (Fig.~\ref{fig:figure1}\textbf{B}, left); nucleus-free cytoplasts, on the other hand,
do not select any particular path when navigating through arrays of cylindrical pillars (Fig.~\ref{fig:figure1}\textbf{B}, right). By highlighting the role of the nucleus as a sensor of the cellular microenvironment, this research thus helps to shed light on the process by which immune cells help navigate through the body and disseminate malignant tumor cells.\\
\begin{figure*}[htp]
\begin{center}
\includegraphics[width=0.70\textwidth]{chapter/figs/Euk_Fig3.pdf}
\caption{\begin{small}\textbf{Collective cell migration in structured 2D environments}. \textbf{A}. Substrate patterning shows that eukaryotic cells (bovine pulmonary artery endothelial cells) experience different proliferation patterns at the edges than in the bulk. Scale bars are $10~\mu$m. Reprinted with permission from Ref.~\citenum{nelson2005emergent} (Copyright National Academy of Sciences, 2005). \textbf{B}. Left: sketch of the experimental setup, and two snapshots displaying the collective migration of MDCK cells in micro-printed stripes. Scale bar is $100~\mu$m. Right: bulk instabilities, namely swirls and vortices, disappear under strong confinement, i.e. in the narrow stripes. Scale bars are $50~\mu$m. Reprinted with permission from Ref.~\citenum{vedula2012emerging} (National Academy of Sciences, 2012). \textbf{C}. Left: sketch of the experimental setup consisting of an array of micropillars where uniform populations of mammary epithelial cells (MCF-10A) and breast adenocarcinoma cells (MDA-MB-231) were cultured. Right and bottom: EMT-activated cells migrate collectively advancing as a front where individual cells detached and move individually. Below the front, cells exhibited phenotypic plasticity undergoing a MET transition. Color bars indicate the number of epithelial and mesenchymal cells as functions of time and the distance from the initial migration front. Adapted with permission from Ref.~\citenum{Wong2014} (Springer Nature, 2014). \end{small}\label{fig:figure3}}
\end{center}
\end{figure*}
\noindent \textbf{Collective migration in structured 2D environments.} In most \textit{in vivo} situations cells grow and migrate as a group. Hence, a growing area of research focuses on the collective behavior of groups of cells in complex 2D environments mimicking those arising in fundamental biological processes, such as morphogenesis, development, wound healing, and cancer progression. In Fig.~\ref{fig:figure3}, we highlight different works that have explored the collective behavior of eukaryotic cells in structured 2D geometries.
One way of creating complex 2D environments is using substratum patterning using adhesive islands coated with extracellular matrix~\citep{chen1997geometric}. A collective of eukaryotic cells can then grow and adapt their overall shape to that of the adhesive islands. For isolated cells, the striking results of Ref.~\citenum{chen1997geometric} showed that cell shape directly determines life and death, and proliferation and quiescence, although the exact mechanisms by which cells transduce changes in their geometry to biochemical responses are still the subject of active inquiry. Then, using aggregates of bovine pulmonary artery endothelial cells and normal rat kidney epithelial cells on microprinted substrata of different shapes and sizes, the authors of Ref.~\citenum{nelson2005emergent} showed that the geometry of the substratum pattern tunes the mechanical forces associated with a growing monolayer, which in turn determines the resulting patterns of cellular proliferation (\com{Fig.~\ref{fig:figure3}\textbf{A}}). Further increasing the complexity of the 2D environment, the authors of Ref.~\citenum{vedula2012emerging} then cultured 2D monolayers of Madin-Darby Canine Kidney (MDCK) epithelial cells in micro-printed stripes, as shown in \com{Fig.~\ref{fig:figure3}\textbf{B}}. The authors found that, under strong confinement in narrow stripes, collective motions such as multicellular swirls and vortices that typically arise in monolayers in uniform, unstructured environments disappear. Instead, the monolayer only migrates unidirectionally \textit{via} a contraction-relaxation mechanism.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.7\textwidth]{chapter/figs/Euk_Fig2.pdf}
\caption{\begin{small}\textbf{Examples of individual eukaryotic cells interacting with complex 3D environments}. \textbf{A}. Dendritic cells migrating through a complex 3D collagen matrix. Cells move towards the region of higher collagen density \textit{via} chemotaxis. The cells that encountered high density regions tangentially circumvented them, thus choosing the path of least resistance. Reprinted with permission from Ref.~\citenum{renkawitz2019nuclear} (Springer Nature, 2019). Scale bar on the left is $100~\mu$m and on the enlarged regions on the right $10~\mu$m.
\textbf{B}. Deformation of a hydrogel induced by an encapsulated fibroblast. In the color bar the bead displacement trajectories are represented. Scale bar is $50~\mu$m. Reprinted with permission from Ref.~\citenum{legant2010measurement} (Springer Nature, 2010). \textbf{C}. An individual MDA-MB-231 cell in blue contracting a 3D collagen network in green. Scale bar is $10~\mu$m. Reprinted with permission from Ref.~\citenum{han2018cell} (National Academy of Sciences, 2018).\end{small} \label{fig:figure2}}
\end{center}
\end{figure*}
Individual and collective cell migration also plays a crucial role during pathological processes such as tumor invasion and metastasis. In cancer progression, it is known that malignant cells undergo different phenotypic transitions and acquire characteristics of embryonic mesenchymal cells, thus becoming loosely packed and less organized, and exhibiting a greater migratory capacity that enables invasion. This phenomenon is usually referred to as the epithelial-mesenchymal transition (EMT)~\citep{yang2004twist,polyak2009transitions,thiery2009epithelial,kalluri2009basics,hanahan2011hallmarks,francou2020epithelial}. When these malignant cells detach from the tumor community, they adapt their migratory path to the complex medium they inhabit and invade healthy tissues and organs far from the primary tumor. To understand this change in behaviors, the authors of Ref.~\citenum{Wong2014} studied the migration of \com{uniform cultures of} epithelial (\com{mammary epithelial MCF-10A}) and mesenchymal (\com{breast adenocarcinoma MDA-MB-23}) cells in a micro-fabricated 2D array of pillars \com{(Fig.~\ref{fig:figure3}\textbf{C})}. Intriguingly, they found that uniform cultures of breast adenocarcinoma MDA-MB-231 mesenchymal cells eventually \com{exhibit a phenotypic plasticity in the loading area with no pillars (region $y < 0$ in Fig.~\ref{fig:figure3}\textbf{C}), thus} acquiring epithelial \com{characteristics} associated with a mesenchymal-epithelial transition (MET); however, the moving interfacial front scattered EMT-activated cells that migrate individually through the pillars with a larger velocity than the collective migration. \com{The authors showed that the migration front propagates as the square root of time, and proposed an interesting analogy comparing such emergent dynamics with a model inspired by phase transitions during binary-mixture solidification. In their experiments, the solidification advancing front is associated with the MET in the loading region. The authors clearly showed that the scattering behavior at the collectively advancing front is enhanced by the regularly spaced micro-pillars, since they disrupt cell-cell contact stimulating individual cell migration, both characteristics associated with EMT. This work evidences that analyzing the collective dynamics of cells in two-dimensional structured environments can provide quantitative measurements and new insights into tumor dissemination during metastasis. Additionlly, this research direction could pave the way to designing new therapeutic strategies that inhibit individual migration during tumor invasion by enhancing MET.}\\
\noindent\textbf{Single-cell migration in structured 3D environments.} The healthy and pathological processes described above, namely immune surveillance, wound healing, or tumor invasion, typically occur within a heterogeneous 3D environment crowded by cells and extracellular matrix. Although several studies have unraveled the molecular and biophysical mechanisms involved in single and collective cell migration in 2D configurations, as briefly highlighted above, how cells navigate through complex 3D tissue structures remains poorly understood~\citep{wu2018biophysics,yamada2019mechanisms,Alert2020}. This gap in knowledge arises partly due to the new modes of motility that may emerge in structured 3D environments, since they could be a combination of the different migration modes employed by eukaryotic cells, e.g., amoeboid (fast migration mode in which a rounded cell with a leading-edge protrusion undergoes deformation to squeeze through gaps), mesenchymal (slower migration mode in which a more polarized elongated cell develops actin-rich leading-edge protrusions that produce adhesive interactions with the medium), or collective (cells migrating coherently as a group), depending on the context~\citep{friedl2009collective,wirtz2011physics,roussos2011chemotaxis,paul2016engineered,yamada2019mechanisms}. Additionally, the interaction with a compliant 3D environment is considerably different compared from the case of a rigid 2D substrate. For instance, in 3D, focal adhesions to collagen fibers are smaller and are known to regulate the cellular migration speed; moreover, the subcellular cytoskeleton appears to be organized differently, and thus cell-exerted tractions are also different~\citep{bloom2008mapping,fraley2010distinctive,legant2010measurement,han2018cell,yamada2019mechanisms}. Exploring first the migration of single eukaryotic cells within complex 3D environments is thus crucial for quantitative insights into the biological processes cited above.
In the context of immune cells, the authors of Ref.~\citenum{renkawitz2019nuclear} explored the migration of dendritic cells through heterogenous hydrogels with pore sizes much smaller than the size of a single cell. In particular, the authors investigated the migration of immune cells in collagen networks with graded pore sizes ranging from 1 $\mu$m to 5 $\mu$m arising from gradients in the density of collagen fibers. To this end, they forced the cells to migrate in response to chemotactic gradients (detailed further in the next section) directed towards regions of smaller pores size/higher collagen density. As shown in \com{Fig.~\ref{fig:figure2}\textbf{A}}, cells facing head-on regions of smaller pore size/higher density continue moving aligned with the chemotactic cue, whereas cells encountering those regions more tangentially circumnavigated the smaller pore size/high-density region by deviating from the chemotactic path. Together with the findings in 2D confined geometries described previously, these results suggest that cells have the ability to distinguish between different pore sizes, migrating preferentially towards the path of least resistance.
Additionally, several works have performed quantitative force measurements of isolated cells embedded in 3D matrices to discover how eukaryotic cells mechanically interact with their environment~\citep{bloom2008mapping,fraley2010distinctive,legant2010measurement,han2018cell}. In Ref.~\citenum{legant2010measurement} the authors described a technique that measures the traction stresses exerted by cells in 3D elastic hydrogels by tracking the displacement of fluorescent beads in the vicinity of each cell (\com{Fig.~\ref{fig:figure2}\textbf{B}}). Using this methodology, the authors showed that isolated cells probe their surrounding environment by exerting strong inward tractions using long and slender extensions. More recently, building on this work, the authors of Ref.~\citenum{han2018cell} used a combination of theory and numerical simulations to demonstrate that a related measurement technique based on nonlinear stress inference microscopy captures the cell-induced stress field. Experiments on isolated cells embedded in different 3D extracellular matrices---collagen, fibrin,
and Matrigel---revealed that cells interact with their surrounding network by pulling on it, producing large stresses that in turn generate large stiffness gradients (\com{Fig.~\ref{fig:figure2}\textbf{C}}). \\
\begin{figure*}[htp]
\begin{center}
\includegraphics[width=0.7\textwidth]{chapter/figs/Euk_Fig4.pdf}
\caption{\begin{small}\textbf{Examples of collective cell migration and growth in structured 3D environments}. \textbf{A}. Time-lapse images showing the growth and branching of mesenchyme-free mouse lung embryonic epithelium embedded in a 3D Matrigel. Reprinted with permission from Ref.~\citenum{varner2015mechanically} (National Academy of Sciences, 2015). Scale bar is $200~\mu$m.
\textbf{B}. \textit{In vitro} multicellular invasion through a 3D collagen matrix by HT-MT1 cells (fibrosarcoma). Cells migrate collectively by degrading the collagen matrix. The arrow indicates the migration direction, and scale bar is $20~\mu$m. Reprinted with permission from Ref.~\citenum{wolf2007multi} (Springer Nature, 2007). \textbf{C}. Experimental setup and time-lapse images of a tumor spheroid (HT1080 human fibroscarcoma) invading a type I collagen matrix at 0, 3 and 6 days. Scale bar is $500~\mu$m. Reprinted with permission from Ref.~\citenum{valencia2015collective} (Impact Journals, 2015).
\textbf{D}. 3D microtumor embedded in collagen gels of different concentration. At low-density collagen gels (left column), MDA-MB-231 breast cancer cells invade the cavity faster than in high-density collagen matrices (right column). At low-density gels the invasion started even before interstitial flow was applied and single-cell and collective migration coexisted. Reprinted with permission from Ref.~\citenum{tien2020matrix} (Elsevier, 2020).\end{small}
\label{fig:figure4}}
\end{center}
\end{figure*}
\noindent\textbf{Collective migration in structured 3D environments.}
Molecular signals and resulting biochemical patterns are typically thought to dictate the spatial patterns that emerge during embryonic development and morphogenesis. Nevertheless, several works have shown that physical forces in structured 3D environments can also drive spatial patterning of cellular groups during developmental processes, as detailed further in Refs.~\citenum{stooke2018physical,goodwin2020mechanics}. For example, the authors of Ref.~\citenum{varner2015mechanically} showed that even in the absence of mesenchyme, \com{a loosely organized embryonic connective tissue of undifferentiated cells that surrounds the epithelium and helps the formation of spatial patterns}, purely physical mechanisms are able to determine the spatial patterning of an embryonic airway epithelium \com{(see Fig.~\ref{fig:figure4}\textbf{A})}. In particular, the authors analyzed the branching morphogenesis of a mesenchyme-free embryonic lung embedded in a viscoelastic 3D gel matrix (Matrigel). They showed that the dominant wavelength of the branching patterns varies with the gel concentration, observing that explants cultured within higher concentrations of Matrigel displayed shorter branch wavelengths. Since the rate of proliferation also increased with the gel concentration, due to the growth factors present at Matrigel, the authors disentangled the role of the gel mechanical properties and the epithelial growth on the branching patterns. In particular, they showed that the experimental results were independent of the gel stiffness, i.e. altering the matrix stiffness with methylcellulose while keeping constant the ligand density did not affect the branch wavelengths, thus suggesting that a purely elastic instability is not responsible for the pattern formation. However, increasing the fibroblast growth factors while keeping the mechanical properties of Matrigel constant, increased the epithelial expansion and decreased the dominant wavelength, suggesting that a growth-induced viscoelastic instability is the responsible for the spatial patterning, in agreement with a simplified viscoelastic Maxwell model showing that the dominant wavelength is independent of the mechanical properties of the surroundings.
Concerning tumor invasion and dissemination of malignant cells, several works have analyzed how a complex 3D environment, such as a collagen matrix, affects the motility and invasion of malignant cells. For example, Ref.~\citenum{wolf2007multi} studied the collective 3D migration of fibrosarcoma (HT-MT1) and breast cancer (MDA-MB-231) cells through a 3D collagen matrix, as shown in \com{Fig.~\ref{fig:figure4}\textbf{B}}. Notably, invasive migration through these complex 3D media exhibits significantly different features to that of migration on 2D substrata. In 3D matrices, collective cell migration requires pericellular remodelling of the extracellular matrix, in which cells \com{located at the edge of the colony} locally degrade the matrix by secreting metalloproteases to eventually migrate collectively in cellular strands. Similarly, the authors of Ref.~\citenum{valencia2015collective} performed experiments on migrating HT1080 human fibroscarcoma spheroids invading a 3D collagen matrix (\com{Fig.~\ref{fig:figure4}\textbf{C})}. They showed that tumor cells at the periphery \com{of the spheroidal tumor colony} become polarized, adopting an aligned and elongated morphology, which enables them to migrate more effectively than cells located closer to the core where cell density is higer. Their results suggest that this directed cell dissemination at the \com{border of the tumor community} is a general feature of migration in 3D collagen matrices, which requires integrin-based cell-collagen adhesion and myosin-based contractility. Taken together, these works show that finger-like instabilities at the edge of the invasive front not only arise in 2D~\cite{omelchenko2003,poujade2007,sepulveda2013,basan2013,tarle2015modeling,mayor2016front,begnaud2016mechanics,nesbitt2017edge,vishwakarma2018mechanical,williamson2018stability,Alert2019,xi2019material,yang2020leader,bonilla2020tracking,trenado2021cellsubstrate}, but also in 3D environments. More recently, by using a 3D microfluidic culture system, the authors of Ref.~\citenum{tien2020matrix} showed that low-density collagen gels hasten the invasion of MDA-MB-231 breast cancer cells---a phenomenon that only depends on the pore size, thus being independent of the interstitial flow speed and the elastic properties of the surrounding medium. Furthermore, in gels of low collagen density, cellular invasion towards a lymphatic-like cavity started even in the absence of an applied interstitial flow, with both single and collective cell migration operating simultaneously (\com{see Fig.~\ref{fig:figure4}\textbf{D})}.
Thus, interactions with a structured 3D environment regulate both single-cell and collective migration and growth. Extending existing theoretical frameworks, and developing new ones, to treat these interactions will be an important direction for future work. For example, although challenging, discrete models, such as agent-based or active-vertex models accounting for sub- and supra-cellular features~\citep{marchetti2013hydrodynamics,hakim2017collective,barton2017active,Alert2020}, can be extended to account for a structured environment, namely intricate and/or compliant geometries. Additionally, despite the complexity of these systems, continuum theories based on liquid crystal~\citep{de1993physics} and active-gel physics~\citep{marchetti2013hydrodynamics,prost2015active} not only have advanced our understanding of single-cell behavior, but have been also employed to describe the physics of collective eukaryotic cells by coarse graining different supra-cellular mechanisms~\citep{marchetti2013hydrodynamics,hakim2017collective,Alert2020}. Incorporating features of complex and structured environments into such frameworks could provide a way to predict cellular behavior more generally.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.7\textwidth]{chapter/figs/Euk_Fig5.pdf}
\caption{\begin{small}\textbf{Examples of galvanotaxis influencing the collective migration of eukaryotic cells}. \textbf{A}. Left: sketch of the monolayer model of wound healing and the electric current as a function of time during healing for human skin and rat cornea cells. The red arrow indicates the endogenous lateral electric field directed towards the wound due to the trans-epithelial potential difference. Right: electric-field induced cell migration. The wound opens or closes depending on the polarity of the electric field. Scale bar is $20~\mu$m. Reprinted with permission from Ref.~\citenum{zhao2006electrical} (Springer Nature, 2006). \textbf{B}. Left: top view of the experimental chamber and a color map indicating the magnitude of the divergent electric field. Right: Epithelial monolayer behavior in terms of the horizontal ($x$) velocity after the application of an electric field. Scale bar is $800~\mu$m. Reprinted with permission from Ref.~\citenum{cohen2014galvanotactic} (Springer Nature, 2014).\end{small}\label{fig:figure5}}
\end{center}
\end{figure*}
\subsection{External stimuli}
A strong scientific effort has been devoted to unravel the emergent behaviors of eukaryotic cells when they are exposed to diverse external stimuli. In this section, we review some of the most studied external stimuli and guidance cues applied to eukaryotic cells---namely electric fields (galvanotaxis), chemical gradients (chemotaxis), or gradients in the substrate stiffness (durotaxis). Other types of taxis and guidance cues are described in e.g., Refs.~\citenum{roca2013mechanical,cortese2014influence,haeger2015collective}.\\
\noindent\textbf{Galvanotaxis.} It is well known that, under normal physiological conditions, eukaryotic cells generate a voltage across their plasma membrane, usually referred to as the membrane potential. This voltage is used for several intracellular functions including transport and signaling. Moreover, at the collective cell level, tissues, organs, and embryos are surrounded by an epithelial layer responsible for producing trans-epithelial potentials. Hence, electric fields are endogenous and carry out fundamental functions during several biological processes such as tissue repair, wound healing, and embryonic development~\citep{jaffe1977electrical,nuccitelli2003role,mccaig2005controlling,zhao2006electrical,cortese2014influence,ross2017use}.
Furthermore, for over a century, it has been known that eukaryotic cells exposed to an external d.c. electric field similar in magnitude to those detected endogenously respond by reorienting and migrating in the direction dictated by the electric potential---a process known as \textit{galvanotaxis}. Indeed, this research direction dates back to the pioneering works of L. Galvani, C. Mateucci, and E. duBois-Reymond~\citep{du1843vorlaufiger}, who measured the electric field in injuries and during wound healing, during the 18th and 19th centuries. In particular, they found that when a wound occurs, the disruption in multicellular integrity triggers an electric field of $~$0.1-1 V cm$^{-1}$ to be directed towards the wound. In his seminal work in 1891~\citep{dineur1891note}, E. Dineur subsequently found that leukocytes exhibit a directed motion guided by the electric current, a phenomenon that he coined as galvanotaxis in honor of L. Galvani. Despite these early discoveries, the precise mechanisms underlying this form of taxis are still under debate~\citep{allen2013electrophoresis,cortese2014influence,ross2017use} in large part due to its dependence on the cell type, medium conditions, and on different signaling cascades. However, researchers are converging on the idea that electrotactic and chemotatic cues share common downstream motility pathways~\citep{zhao2006electrical,allen2013electrophoresis,sun2013keratocyte}. Thus, it is now well established that electric cues have great potential in directing collective cell migration and controlling emergent group behaviors~\citep{zhao2006electrical,mccaig2005controlling,cortese2014influence,cohen2014galvanotactic,ross2017use}, which could be used eventually for beneficial purposes in clinical applications. Indeed, galvanotaxis provides more precise control and faster responses than other external stimuli, such as chemical signals.
Concerning spreading epithelial monolayers and wound healing, the experiments of Ref.~\citenum{zhao2006electrical} showed that electric fields are able to override other guidance cues. The authors showed that an external d.c. electric field is able to alter wound healing, enhancing healing when is applied in the same direction of the trans-epithelial potential, or opening the wound when is directed in the opposite direction (Fig.~\ref{fig:figure5}\textbf{A}). Their results also suggest that electrotactic and chemotatic cues activate common signaling pathways. More recently, the work of Ref.~\citenum{cohen2014galvanotactic} studied the collective cell migration of a 2D epithelial monolayer undergoing galvanotaxis (Fig.~\ref{fig:figure5} \textbf{B}). The authors found that, after applying a divergent electric field for one hour, the migration of the whole epithelial monolayer was guided by the electric current density. Strikingly, the group of cells moved collectively in the direction aligned with the electric-field lines while maintaining the epithelial integrity, as show in Fig.~\ref{fig:figure5}\textbf{B}. By contrast, due to the characteristic behavior of the leader cells located at the edge of the monolayer, the electric field was not able to polarize them and guide their migration. Building on these results, the authors of ~\citep{zajdel2020scheepdog,zajdel2020come,shim2021cellular} developed an experimental device to analyze the behavior of large tissues undergoing galvanotaxis. Using this device, they found that wound healing is accelerated when an external electric field is applied, the healing process being twice faster than under normal conditions. Additionally, strongly-adhesive tissues are more complex to control \textit{via} galvanotaxis; nevertheless, a sufficiently strong electric field can eventually damage the epithelium.
\begin{figure*}
\begin{center}
\includegraphics[width=0.7\textwidth]{chapter/figs/Euk_Fig6.pdf}
\caption{\begin{small}\textbf{Examples of chemotactic guidance in both individual and collective cell migration within structured environments}. \textbf{A}. Chemotactic reorientation of a granulocyte towards a stimulus (pipette). Reprinted with permission from Ref.~\citenum{gerisch1981chemotactic} (The Company of Biologists, 1981). \textbf{B}. Single cells and cluster of cells migrating toward the chemoattractant source in a very directional manner. Reprinted with permission from Ref.~\citenum{malet2015collective} (Elsevier, 2015). \textbf{C}. Upper panel: Long-lived behavior of coordinated movement in carcinoma cell streams. Scale bars are $50~\mu$m. Middle panel: coordinated motion of cells through the tumor position (indicated with the arrows). Scale bars are $25~\mu$m. Lower panel: Random motion of cells in presence of a tumor. Scale bars are $25~\mu$m. Reprinted with permission from Ref.~\citenum{roussos2011mena} (The Company of Biologists, 2011).\end{small}\label{fig:figure6}}
\end{center}
\end{figure*}
From a theoretical perspective, the coupling between an external d.c. electric field and the stresses involved during eukaryotic collective migration, such as active traction stresses, contractile stresses, and viscous/elastic stresses~\citep{Alert2020}, remains underexplored. A systematic comparison between experiments and new discrete and continuum models incorporating electric stresses and current-dependent polarization will be essential to unravel the physics involved in galvanotaxis and the collective behaviors that result.\\
\noindent\textbf{Chemotaxis.} Eukaryotic cells also have the ability to sense and respond to extracellular chemical stimuli, although the underlying mechanisms employed are distinct from chemotaxis in prokaryotic organisms~\citep{ridley2003cell}, discussed previously in Section~\ref{bac-stimuli}. Within the context of eukaryotic organisms, chemotaxis is the most extensively studied directional cue, mainly due to its fundamental role in a large variety of biological processes. For instance, chemotaxis plays a key role in driving individual and collective cell migration in developmental processes. It is also important for the trafficking of immune cells and in tissue regeneration/wound healing. Moreover, chemotaxis is not only relevant in healthy physiological processes, but it also plays key roles in the progression cancer~\citep{muller2001involvement,kedrin2007cell,roussos2011chemotaxis}, as well as in the pathogenesis of inflammatory diseases such as asthma, allergies, and atherosclerosis~\citep{luster1998chemokines}.
The first reported observations of chemotaxis in eukaryotic cells date back to the 19th century by the ophthalmologist T. Leber~\citep{leber}, who showed the attraction of leukocytes by chemical gradients, although the precise extracellular mechanism responsible for guiding leukocytes was only identified several decades later~\citep{zigmond1977ability,devreotes1988chemotaxis,jin2008chemotaxis}. In 1947, J.T. Bonner and L.J. Savage found that the soil amoeba \textit{Dictyostelium discoideum} responds to the gradient of a chemical by deforming and migrating in the direction of the gradient, thus clearly showing that chemical signals are able to polarize and organize the cellular movement~\citep{bonner1947evidence}. This work paved the way for subsequent work utilizing \textit{D. discoideum} as a model organism to study chemotaxis~\citep{gerisch1968cell,gerisch1981chemotactic,gerisch1981chemotactic,devreotes1988chemotaxis,king2009chemotaxis,insall2010understanding,nichols2015chemotaxis}. A tremendous amount of research has built on this body of work, as reviewed in Refs.~\citenum{devreotes2003eukaryotic,jin2006moving,jin2008chemotaxis,swaney2010eukaryotic,insall2010understanding,roussos2011chemotaxis,levine2013physics}, to elucidate chemotaxis in diverse eukaryotes. For example, studies of human leukocytes have demonstrated that---similar to \textit{D. discoideum}---these cells respond to extracellular chemical stimuli by deforming and extending lamellipodia before migrating ~\citep{gerisch1981chemotactic,parent1998g}, a process driven by actin polymerization (Fig.~\ref{fig:figure6}\textbf{A}). Additionally, these cells are able to reorient and perform U-turns in response to an external chemical gradient. This behavior is distinct from chemotaxis of bacteria and other prokaryotic organisms, which instead perform biased random walks~\citep{Frymier:1995,Lauga:2006} as described in Section~\ref{bac-stimuli}.
Indeed, unlike prokaryotes, eukaryotic cells sense chemical gradients by measuring how the chemoattractant varies with respect to their size---a feat accomplished by triggering complex intracellular signals that drive cell polarization, deformation, and locomotion~\citep{jin2008chemotaxis,insall2010understanding,levine2013physics,thomas2018decoding}. In particular, chemotaxis in eukaryotic cells arises in three distinct steps---directional signaling, polarization, and motility---in which cells interact with the chemoattractant \com{or chemorepellent} and generate a response that determines the direction of motion \textit{via} intracellular signals. Several theoretical models have been developed to describe these signal transduction dynamics in \textit{D. discoideum} cells~\citep{meinhardt1999orientation,levchenko2002models,levine2006directional,levine2013physics}, resulting in local excitation-global inhibition (LEGI) equations that quantify the competition between the global chemical signal and the production of a local membrane-bound activator~\citep{parent1999cell}.
As described above, individual eukaryotic organisms employ chemotaxis for tracking food, finding the best path in complex environments such as soil, and escaping from predators~\citep{jin2006moving,jin2008chemotaxis,swaney2010eukaryotic,roussos2011chemotaxis,levine2013physics}. But chemotaxis is also used to guide collective cell migration, namely in \textit{in vivo} processes including metastasis~\citep{roussos2011chemotaxis}, tissue self-organization during morphogenesis~\citep{haas2006chemokine}, embryogenesis, immune surveillance, and wound healing. For example, chemotaxis enables immune cells such as T lymphocytes and neutrophils, which typically move randomly, to transition to a directed swarm behavior when inflammation occurs, in response to chemokine gradients~\citep{malet2015collective}. Fig.~\ref{fig:figure6}\textbf{B} exemplifies this process: cells initially move randomly, but after a few hours, form multicellular clusters that sense a difference in chemokine concentrations across them and migrate toward a chemokine source. Similar behavior has been observed during tumor invasion of 3D hydrogels~\citep{liu2013minimization}. In particular, experiments~\citep{theveneau2010collective,malet2015collective} and theoretical modeling (e.g. agent-based, phase field, and cellular Potts models)~\citep{devreotes2003eukaryotic,camley2016emergent,najem2016phase} have shown that cells located at the \com{periphery of the colony} use the local information of the chemokine concentration to regulate their contact inhibition of locomotion, enabling the cluster directed migration.
More recently, Ref.~\citenum{Tweedy2020} reported a combination of experiments and mathematical modeling showing that both \textit{D. discoideum} and metastatic cancer cells are able to solve complex mazes by using a self-generated chemotaxis mechanism that allows them to perform long-range navigation. This mechanism is in many ways similar to that by which bacterial populations perform chemotactic migration in response to a self-generated nutrient gradient, as described in Sec.~\ref{bac-stimuli}. In particular, Ref.~\citenum{Tweedy2020} showed that cells produce local chemical gradients in their vicinity, driving directed migration at essential decision-making points in the maze such as corners, junctions, shortcuts, and dead-end pores---suggesting a fundamental role of chemotaxis in regulating migration in complex environments during both healthy and pathological \textit{in vivo} processes. Also within the context of tumor invasion, Ref.~\citenum{roussos2011mena} explored the different types of cell motility that can arise during tumor cell invasion in a complex tissue environment in the presence of chemoattractants. In particular, the authors examined the actin and cell-migration regulatory protein called Mena and its isoforms Mena11a and Mena$^{\mbox{\tiny{INV}}}$, which are associated with breast cancer cell discohesion, invasion, and intravasation. They showed that Mena$^{\mbox{\tiny{INV}}}$ has higher sensitivity to epidermal growth factor than Mena11a, promoting coordinated and directed carcinoma cell streaming through the primary tumor, as shown in Fig.~\ref{fig:figure6}\textbf{C} (bottom). Moreover, they showed that such coordinated streaming migration is a long-lived phenomenon, as observed in Fig.~\ref{fig:figure6}\textbf{C} (top), persisting for longer than 24 hours.
As highlighted above, several theoretical works have incorporated chemotaxis into variants of agent-based and discrete models. Similarly accounting for chemotaxis in active-vertex models~\citep{barton2017active} and in continuum theories based on liquid crystal theory and active-gel physics~\citep{de1993physics,marchetti2013hydrodynamics,prost2015active,Alert2020} would also be interesting.\\
\begin{figure*}
\begin{center}
\includegraphics[width=.7\textwidth]{chapter/figs/Euk_Fig7.pdf}
\caption{\begin{small}\textbf{Examples of durotaxis influencing both single and collective migration of eukaryotic cells}. \textbf{A}. Eukaryotic cell (REF52 fibroblast) spreading between on top of a surface of micropillars with varying rigidity, and the corresponding applied stress as a function of the substrate stiffness. Scale bar is $10~\mu$m. Reprinted with permission from Ref.~\citenum{trichet2012evidence} (National Academy of Sciences, 2012). \textbf{B}. Monolayer of human skin keratinocyte cells deforming a flexible carbon fibre. Scale bar is $1~$mm. Reprinted with permission from Ref.~\citenum{valencia2021interaction} (Elsevier, 2021).\end{small}\label{fig:figure7}}
\end{center}
\end{figure*}
\noindent \textbf{Durotaxis.} As recently as 20 years ago, the authors of Refs.~\citenum{pelham1997cell,lo2000cell} discovered that eukaryotic cells are able to sense gradients in the stiffness of an underlying substratum. In particular, fibroblasts adapt their shape and move in the direction of increasing substrate stiffness, a phenomenon termed \textit{durotaxis}---derived from the Latin word \textit{durus}, meaning hard, and the Greek word \textit{taxis}, meaning arrangement. Since the original finding, research on durotaxis has remained active due to its potential implications in processes such as tissue regeneration, development, and tumor invasion. For example, studies have shown that diverse eukaryotic cells sense and respond to the rigidity of underlying substrata by changing shape and applying mechanical forces to probe the stiffness of their environment~\citep{discher2005tissue,saez2005mechanical,isenberg2009vascular,trichet2012evidence,vincent2013mesenchymal,roca2013mechanical,sunyer2020durotaxis}. The local mechano-sensitivity mechanism is usually explained in terms of force-induced conformational changes of constituent proteins at focal adhesion sites, which enhances the growth of local contacts by enabling the binding of new proteins~\citep{nicolas2004cell}.
Other mechanisms by which cells can sense differences in substrate stiffness have also been suggested. Some authors, based on rearward actin cytoskeleton flow, have proposed a rigidity-dependent motor-clutch mechanism in which the strength of the molecular clutches linking F-actin to the substrate, and consequently the transmitted traction forces, depend on the substrate stiffness~\citep{chan2008traction,harland2011adhesion,bangasser2013determinants}. Other studies have shown that cells apply mechanical forces that keep the deformation of the substrate constant, thus maintaining larger traction forces in stiffer regions, thereby enabling migration in the direction of increasing substrate rigidity~\citep{saez2005mechanical,ghibaudo2008traction,ghassemi2012cells,trichet2012evidence} (Fig.~\ref{fig:figure7}\textbf{A}). This process has been proposed to be achieved either locally \textit{via} actin-myosin interactions at integrin-mediated focal adhesion sites~\citep{sheetz1998cell,fouchard2011acto,ghassemi2012cells,plotnikov2012force}, or at the global cell scale by cytoskeleton actin stress fibers pulling in focal adhesions~\citep{trichet2012evidence}. Intriguingly, there is an optimal stiffness range (3-10 nN/$\mu$m) in which durotaxis is most effective, with about 70\% of cells moving toward the stiffer region. Durotaxis can also be negative, with migration directed towards softer regions, for instance in retinal ganglion cell axons in \textit{Xenopus} developing brain~\citep{koser2016mechanosensing}. Indeed, the authors of Ref.~\citenum{isomursu2020negative} recently observed negative durotaxis in human glioblastoma (brain tumor) cells and in breast cancer cells (switched \textit{via} talin downregulation)---suggesting that solely integrin-mediated adhesion and motor-clutch dynamics are sufficient to explain both positive and negative durotaxis.
While initial research efforts focused on durotaxis of isolated cells, recent work has shown that groups of eukaryotic cells also sense substrate rigidity and migrate collectively in a preferred direction imposed by the stiffness gradient---a phenomenon termed \textit{tissue durotaxis}~\citep{sunyer2016collective}. In this work, the authors showed that 2D rectangular clusters of cells expand asymmetrically faster toward the stiffer region of the substrate. Their findings indicated that proliferation does not contribute to this asymmetric migration; instead, cell-cell adhesion is required for collective cell durotaxis. Thus, durotaxis is not only a consequence of local sensing of substrate rigidity, but also can involve a long-range mechanism mediated by adhesion between cells. This work also demonstrates that durotaxis is a robust guidance mechanism that can be used for tuning the directed collective motion of eukaryotic cells.
Durotaxis was firstly discovered in healthy cells, but it can also drive the 2D migration of cancerous cells, as shown in Refs.~\citenum{duchez2019durotaxis,sunyer2020durotaxis}. Nevertheless, to the best of our knowledge, collective durotaxis has not yet been demonstrated \textit{in vivo} in healthy or malignant cells~\citep{sunyer2020durotaxis}. Furthermore, it may be that cells use durotaxis as a migration cue to navigate through complex 3D tissue environments, such as when malignant cells leave a tumor and migrate through the surrounding extracellular matrix, a behavior associated with dissemination and metastasis. Indeed, there is a strong relationship between the stiffness of the environment and cancer cell spreading, which is mainly achieved \textit{via} cancer‐associated fibroblasts~\citep{bonnevie2018physiology}. Exploring these possibilities will be a useful direction for future work.
More generally, an important direction for future experiments is to explore other examples of cellular durotaxis in response to 2D and 3D deformable boundaries of complex shapes. However, fabricating substrata with well-defined shapes and stiffness gradients remains a challenge. Some experimental techniques that are often used involve preparing substrata by coalescing unpolymerized gel droplets of different concentrations and then polymerizing them; using microfluidic technologies to prepare substrata of different chemistries; and preparing substrata using photopolymerizable polymer mixtures~\citep{sunyer2020durotaxis}. In another approach, in recent work, the authors of Ref.~\citenum{valencia2021interaction} described a way to study the interaction of an epithelial monolayer of human skin keratinocyte cells with a deformable carbon fiber (Fig.~\ref{fig:figure7} \textbf{B}). This approach could pave the way to new experimental techniques able to measure the force exerted by a developing tissue, or to explain the interaction between groups of cells and external compliant bodies. Developing other ways to experimentally explore durotaxis will be a useful direction for future work.
Moreover, given that durotaxis is still an emerging area of research, there are only a few theoretical and computational studies that examine this mode of directed migration. As mentioned previously, some works have employed molecular clutch models to explain the underlying physics during durotaxis~\citep{chan2008traction,harland2011adhesion,bangasser2013determinants,sunyer2020durotaxis}. However, extending these clutch models to the case of collective cell durotaxis remains to be done. Within the context of continuum theories, the authors of Ref.~\citenum{alert2018role} recently analyzed the influence of durotaxis on the wetting/dewetting transition of a circular spreading tissue monolayer. The epithelial monolayer was modeled as a compressible polar fluid, taking into account cell-substrate friction, contact-active traction, and tissue contractility. The authors incorporated the substrate stiffness gradient by assuming that the cell-substrate friction and contact-active traction depend on the substrate elasticity. Their results suggest that tissue durotaxis can emerge from different wetting/dewetting states of the two moving edges of a tissue induced by the substrate stiffness gradient. For sufficiently large contractile stresses, the edge on the stiffer region spreads (wetting) and drags the other edge, causing it to retract (dewetting)---in agreement with the asymmetric migration towards the stiffer region observed in Ref.~\citenum{sunyer2016collective}. For small contractility, both edges wet the substrate, but the edge on the softer region slows down since moving towards decreasing stiffness decreases its active traction. More recently, by means of theory and numerical simulations within the same continuum framework as in Refs.~\citenum{blanch2017effective,alert2018role,Alert2020}, the work of Ref.~\citenum{trenado2021cellsubstrate} showed that solely cell-substrate friction is responsible for triggering complex secondary fingering instabilities at the edges of a 2D spreading monolayer, qualitatively similar to those observed in experiments. To test these predictions~\citep{alert2018role,Alert2019,trenado2021cellsubstrate}, new experiments must be conducted to systematically examine the ability of such active continuum models to describe experimental observations.
As a final note, we emphasize that eukaryotic cells respond to many other stimuli---for example through haptotaxis~\citep{carter1967haptotaxis,sengupta2021principles} (directed transport up a gradient in cellular adhesion sites or substrate-bound chemoattractants), phototaxis~\citep{foster1984rhodopsin,arrieta2019light} (directed transport in response to light), and topotaxis~\citep{park2018topotaxis,sengupta2021principles} (directed transport in response to spatial variations of topography). Indeed, multiple external stimuli often coexist in \textit{in vivo} processes together with a structured 3D environment. Hence, an important direction for future research is to elucidate the influence of multiple simultaneous stimuli on cellular migration. Although challenging, new experiments and both discrete and continuum theoretical models accounting for coupled external stimuli will shed light on different aspects of the \textit{in vivo} processes discussed throughout this chapter.
\section{Outlook}\label{sec:sec4}
This chapter has reviewed the transport of two prominent forms of biological active matter---bacteria and eukaryotic cells---in complex environments. Due to space constraints, we have not presented a fully comprehensive review of all the studies concerning it; instead, we have highlighted several sub-topics concerning which research has been particularly active. Within each section, we have described specific open questions and prospects for future research. Therefore, we conclude this chapter by highlighting general principles and areas for future inquiry.
We have described ways in which---across both bacteria and eukaryotic cells---geometric constraints, mechanical properties, and external stimuli fundamentally alter active transport, modulating spreading and directed migration in often unexpected ways. Scientifically, these findings motivate the development of new models of transport. Furthermore, they challenge and expand current understanding of collective behavior and emergent phenomena in complex systems. Practically, these findings help predict and control a broad range of applications. For example, they underlie significant biomedical processes such as the progression of infection, drug delivery, tissue repair/wound healing, morphogenesis and development, and the progression of cancer. They also help inform the development of active systems that can rationally direct the transport of cargos, localize chemical transformations, or detect and respond to stimuli in environmental and biological settings \citep{Guix:2018,Alapan:2019,Erkoc:2019,Toley:2012,Duong:2019, Sedighi:2019,Soler:2013,Gao:2014,Adadevoh:2016,Ren:2019,Mano:2017,Gomez:2017,Vutukuri:2020,Nelson2010,Patra2013,Abdelmohsen2014,Wang2012a,Ebbens2016,soria2020predictive}.
While we have focused on the transport of bacteria and eukaryotic cells, extensive research has and continues to explore the ideas and phenomena described in this chapter for diverse other forms of active matter---ranging from other bacteria and eukaryotic cell types, as well as other microscopic organisms such as algae and protists, archaea, and sperm, subcellular active matter such as enzymes and driven biopolymer assemblies, and more macroscopic forms of active matter such as driven granular media, birds, fish, large mammals, and robots \citep{Bechinger2016b,Lang:2018,Deblais:2020phase,Deblais:2020rheology,Volpe:2017,soria2020predictive}. While these different forms of active matter vary across many orders of magnitude in size and speed, and have vastly different sources of activity, many of them share common features and therefore can often be described using more general models (e.g., Ref. \citenum{alert2021active}). An outstanding challenge is to develop a broad framework capable of quantitatively describing these non-equilibrium, yet universal, features---but also, to explore the limitations of such general descriptions and more deeply understand the rich phenomena that are specific to each system.
\section*{Acknowledgements}
A.M.-C. acknowledges support from the Human Frontier Science Program (LT000035/2021-C), from the Princeton Center for Theoretical Science, and from the FEDER/Ministerio de Ciencia, Innovaci\'{o}n y Universidades – Agencia Estatal de Investigaci\'{o}n through the project DPI2017-88201-C3-3-R, and the Red Nacional para el Desarrollo de la Microflu\'{i}dica, RED2018-102829-T. C.T. acknowledges financial support by the FEDER/Ministerio de Ciencia, Innovaci\'{o}n y Universidades – Agencia Estatal de Investigaci\'{o}n grant MTM2017-84446-C2-2-R, by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation). S.S.D. acknowledges financial support by NSF grant CBET-1941716 and EF-2124863, the Project X Innovation Fund, the Eric and Wendy Schmidt Transformative Technology Fund at Princeton, the Princeton Catalysis Initiative, and the Princeton Center for Complex Materials, a Materials Research Science and Engineering Center supported by NSF grant DMR-2011750. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
|
1,108,101,563,223 | arxiv | \section{Introduction}
The difficulty to justify the validity of the probabilistic solutions
of the American options is well-known. This is in fact due to the lack
of regularity of the classical solutions of the variational
inequalities (cf.\cite{B-L}) which are satisfied by the value function which
characterizes the Snell envelope (cf. \cite{Lam} for a recent survey
about this subject). In particular the value function is not twice
differentiable hence the It\^o formula is not applicable to apply the
usual probabilistic techniques. In the case of Black and Scholes model, there are
some results using extensions of the It\^o formula for the Brownian
motion, which, however, are of limited utility for more general cases.
In this note we give hopefully more general results in the
sense that the option is constructed by the assets which obey to a
general, finite dimensional stochastic differential equation with
deterministic coefficients, i.e., a diffusion process. The basic hypothesis used is the
nondegeneracy of this diffusion in the sense of the Malliavin
calculus (cf. \cite{Mal}): recall that an ${\rm I\!R}^d$-valued random variable $F=(F_1,\ldots,F_d)$,
defined on a Wiener space is called nondegenerate
(cf.\cite{Mal,ASU,ASU-1}) if it is infinitely
Sobolev differentiable with respect to the Wiener measure and if the
determinant of the inverse of the matrix $((\nabla F_i,\nabla
F_j)_H\,:i,\,j\leq d)$, where $\nabla$ denotes the Sobolev derivative
on the Wiener space, is in all the $L^p$-spaces w.r. to the Wiener
measure. In this case, the mapping $f\to f\circ F$, defined from the smooth functions
on ${\rm I\!R}^d$ to the space of smooth functions on the Wiener space extends
continuously to a linear mapping, denoted as $T\to
T(F),\,T\in{\mathcal S}'({\rm I\!R}^d)$, from the tempered distributions
${\mathcal S}'({\rm I\!R}^d)$ to the space of Meyer distributions on the Wiener space
(cf.\cite{Mal,ASU,ASU-1}). Similarly, if $F$ is replaced with an It\^o
process whose components satisfy similar regularity properties, we obtain an
It\^o formula for $T(F_t)-T(F_s)$, $0<s\leq t$, where the stochastic integral should
be treated as a distribution-valued Gaussian divergence and the
absolutely continuous term is a Bochner integral concentrated in some
negatively indexed Sobolev space. Moreover, if this
latter term is a positive distribution, then the resulting integral is a Radon measure on
the Wiener space due to a well-known result about the positive Meyer
distributions on the Wiener space (cf.\cite{ASU-0,ASU-00,ASU,ASU-1}).
Having summarized the technical tools that we use, let us
explain now the main results of the paper: for the uniqueness result we treat two different
situations; namely the first one where the coefficients are time
dependent and the variational inequality is
interpreted as an evolutionary variational inequality in
${\mathcal S}'({\rm I\!R}^d)$. The second one concerns the case where the
coefficients are time-independent and we interpret it as an inequality
in the space ${\mathcal D}'(0,T)\otimes {\mathcal S}'({\rm I\!R}^d)$ with a boundary
condition, which is of course
more general than the first one. In both cases the operators are
supposed only to be hypoelliptic; a hypothesis which is far more general
than the ellipticity hypothesis used in \cite{B-L}. The homogeneity in
time permits us more generality since, in this case the time-component
regularization by the
mollifiers of the solution candidates preserve their property of
being negative distributions, hence measures. The existence is studied
in the last section using the similar techniques and we obtain as a by
product some regularity results about the solution of the variational
inequality. In particular, we realize there that even if the density
of the underlying diffusion has zeros, there is still a solution on
the open set which corresponds to the region of $[0,T]\times {\rm I\!R}^d$
where the density is strictly positive.
\section{Preliminaries and notation}
\label{preliminaries}
\noindent
Let $W$ be the classical Wiener space $C([0,T],{\rm I\!R}^n)$ with the Wiener
measure $\mu$. The
corresponding Cameron-Martin space is denoted by $H$. Recall that the
injection $H\hookrightarrow W$ is compact and its adjoint is the
natural injection $W^\star\hookrightarrow H^\star\subset
L^2(\mu)$.
Since the translations of $\mu$ with the elements of $H$ induce measures
equivalent to $\mu$, the G\^ateaux derivative in $H$ direction of the
random variables is a closable operator on $L^p(\mu)$-spaces and this
closure will be denoted by $\nabla$ cf., for example
\cite{F-P},\cite{ASU, ASU-1}. The corresponding Sobolev spaces
(the equivalence classes) of the real random variables
will be denoted as ${\rm I\!D}_{p,k}$, where $k\in {\rm I\!N}$ is the order of
differentiability and $p>1$ is the order of integrability. If the
random variables are with values in some separable Hilbert space, say
$\Phi$, then we shall define similarly the corresponding Sobolev
spaces and they are denoted as ${\rm I\!D}_{p,k}(\Phi)$, $p>1,\,k\in
{\rm I\!N}$. Since $\nabla:{\rm I\!D}_{p,k}\to{\rm I\!D}_{p,k-1}(H)$ is a continuous and
linear operator its adjoint is a well-defined operator which we
represent by $\delta$. $\delta$ coincides with the It\^o
integral of the Lebesgue density of the adapted elements of
${\rm I\!D}_{p,k}(H)$ (cf.\cite{ASU,ASU-1}).
For any $t\geq 0$ and measurable $f:W\to {\rm I\!R}_+$, we note by
$$
P_tf(x)=\int_Wf\left(e^{-t}x+\sqrt{1-e^{-2t}}y\right)\mu(dy)\,,
$$
it is well-known that $(P_t,t\in {\rm I\!R}_+)$ is a hypercontractive
semigroup on $L^p(\mu),p>1$, which is called the Ornstein-Uhlenbeck
semigroup (cf.\cite{F-P,ASU,ASU-1}). Its infinitesimal generator is denoted
by $-{\mathcal L}$ and we call ${\mathcal L}$ the Ornstein-Uhlenbeck operator
(sometimes called the number operator by the physicists). The
norms defined by
\begin{equation}
\label{norm}
\|\phi\|_{p,k}=\|(I+{\mathcal L})^{k/2}\phi\|_{L^p(\mu)}
\end{equation}
are equivalent to the norms defined by the iterates of the Sobolev
derivative $\nabla$. This observation permits us to identify the duals
of the space ${\rm I\!D}_{p,k}(\Phi);p>1,\,k\in{\rm I\!N}$ by ${\rm I\!D}_{q,-k}(\Phi')$,
with $q^{-1}=1-p^{-1}$,
where the latter space is defined by replacing $k$ in (\ref{norm}) by
$-k$, this gives us the distribution spaces on the Wiener space $W$
(in fact we can take as $k$ any real number). An easy calculation
shows that, formally, $\delta\circ \nabla={\mathcal L}$, and this permits us
to extend the divergence and the derivative operators to the
distributions as linear, continuous operators. In fact
$\delta:{\rm I\!D}_{q,k}(H\otimes \Phi)\to {\rm I\!D}_{q,k-1}(\Phi)$ and
$\nabla:{\rm I\!D}_{q,k}(\Phi)\to{\rm I\!D}_{q,k-1}(H\otimes \Phi)$ continuously, for
any $q>1$ and $k\in {\rm I\!R}$, where $H\otimes \Phi$ denotes the
completed Hilbert-Schmidt tensor product (cf., for instance
\cite{Mal,ASU,ASU-1}). We shall denote by ${\rm I\!D}(\Phi)$ and ${\rm I\!D}'(\Phi)$
respectively the sets
$$
{\rm I\!D}(\Phi)=\bigcap_{p>1,k\in {\Bbb N}}{\rm I\!D}_{p,k}(\Phi)\,,
$$
and
$$
{\rm I\!D}'(\Phi)=\bigcup_{p>1,k\in {\Bbb N}}{\rm I\!D}_{p,-k}(\Phi)\,,
$$
where the former is equipped with the projective and the latter is
equipped with the inductive limit topologies. A map $F\in{\rm I\!D}({\rm I\!R}^d)$ is
called nondegenerate if $\det\gamma\in\cap_pL^p(\mu)$, where $\gamma$ is
the inverse of the matrix $((\nabla F_i,\nabla F_j)_H,i,j\leq d)$ and
$(\cdot,\cdot)_H$ denotes the scalar product in $H$. For
such a map, it is well-known that (\cite{Mal,ASU,ASU-1}) the map $f\to f\circ F$ from
${\mathcal S}({\rm I\!R}^d)\to {\rm I\!D}$ has a linear, continuous extension to
${\mathcal S}'({\rm I\!R}^d)\to {\rm I\!D}'$, where ${\mathcal S}({\rm I\!R}^d)$ and ${\mathcal S}'({\rm I\!R}^d)$ denote the space of
rapidly decreasing functions and tempered distributions on ${\rm I\!R}^d$,
respectively. In fact, due to the ``polynomially increasing''
character of the tempered distributions, the range of this extension
is much smaller than ${\rm I\!D}'$, in fact it is included in
$$
\tilde{{\rm I\!D}}'=\bigcap_{p>1}\bigcup_{k\in {\rm I\!N}}{\rm I\!D}_{p,-k}\,.
$$
This notion has been extended in \cite{ASU-0} and used
to give an extension of the It\^o formula as follows:
\begin{theorem}
\label{ito-thm}
Assume that $(X_t,t\in [0,T])$ is an ${\rm I\!R}^d$-valued non-degenerate It\^o process with
the decomposition
$$
dX_t=b_tdt+\sigma_t dW_t
$$
where $b\in{\rm I\!D}^a(L^2([0,T])\otimes {\rm I\!R}^d)$ and
$\sigma\in{\rm I\!D}^a(L^2([0,T])\otimes {\rm I\!R}^d\otimes {\rm I\!R}^n)$, where the upper index
$^a$ means adapted to the Brownian filtration. Assume further that
\begin{equation}
\label{non-deg-cond}
\int_\varepsilon^1(\det\gamma_s)^pds\in L^1(\mu)\,,
\end{equation}
for any $p>1$, where $\gamma_s$ is the inverse of the matrix $((\nabla
X^i_t,\nabla X^j_t)_H;\,i,j\leq d)$. Then, for any $T\in {\mathcal S}'({\rm I\!R}^d)$ and $0<s<t\leq 1$, we
have
$$
T(X_t)-T(X_s)=\int_s^t A_u T(X_u)du+\int_s^t(\partial
T(X_u),\sigma_udW_u)\,,
$$
where $A_u=\frac{1}{2}a_{i,j}(u)\partial_{i,j}+b_i(u)\partial_i$, the
first integral is a Bochner integral in $\tilde{{\rm I\!D}}'$ and the second one is
the extended divergence operator explained above.
\end{theorem}
\begin{remarkk}
The conditions under which the hypothesis (\ref{non-deg-cond}) holds
are extremely well-studied in the literature, cf. \cite{K-S-1,K-S-2}.
\end{remarkk}
\begin{remarkk}
The divergence operator acts
as an isomorphism between the spaces ${\rm I\!D}_{p,k}^a(H)$ and ${\rm I\!D}_{p,k}$
for any $p>1,\,k\in{\rm I\!R}$, cf. \cite{ASU-00}.
\end{remarkk}
\begin{remarkk}
We can extend the above result easily to the case where $t\to T_t$ is
a continuous map of finite total variation from $[0,T]$ to ${\mathcal S}'({\rm I\!R}^d)$ in the sense that,
the mapping
$t\to \langle T_t,g\rangle$ is of finite total variation on $[0,T]$ for any
$g\in{\mathcal S}({\rm I\!R}^d)$. In fact, the kernel theorem of A. Grothendieck
implies that $T_t$ can be represented as
$$
T_t=\sum_{i=1}^\infty \lambda_i\alpha_i(t)\,F_i\,,
$$
where $(\lambda_i)\in l^1$, $(\alpha_i)$ is bounded in the total variation
norm and $(F_i)$ is bounded in ${\mathcal S}'({\rm I\!R}^d)$. Using this
decomposition, it is straightforward to show that
$$
T(t,X_t)-T(s,X_s)=\int_s^t A_u T(u,X_u)du+ \int_s^tT(du,X_u) +\int_s^t(\partial
T(u,X_u),\sigma_udW_u)\,.
$$
where the second integral is defined as
$$
\int_s^tT(du,X_u)=\sum_{i=1}^\infty \lambda_i\int_s^t\,F_i(X_u)
d\alpha_i(u)
$$
and the right hand side is independent of any particular representation of $T_t$.
integrals are concentrated in ${\rm I\!D}'$
\end{remarkk}
\noindent
We can prove easily the following result using the technique
described in \cite{ASU-0,ASU-1}:
\begin{theorem}
\label{ito-1}
Assume that $(l_t,t\in [0,1])$ is an It\^o process
$$
dl_t=m_tdt+\sum_iz^i_tdW^i_t\,,
$$
with $m,z^i\in {\rm I\!D}^a(L^2[0,T])$, then we have
\begin{eqnarray*}
l_tT(t,X_t)-l_s T(s,X_s)&=&\int_s^t l_uA_u T(u,X_u) du +\int_s^t l_u T(du,X_u)\\
&&+\int_s^tl_u(\partial
T(u,X_u),\sigma_udW_u)+\int_s^tT(u,X_u)m_u du\\
&&+\int_s^tT(u,X_u)\sum_iz^i_udW_u^i+\int_s^t(\partial
T(u,X_u),\sigma_u z_u)du
\end{eqnarray*}
where $z=(z^1,\ldots,z^n)$.
\end{theorem}
An important feature of the distributions on the Wiener space is the
notion of positivity: we say that $S\in{\rm I\!D}'$ is positive if for any positive
$\varphi\in{\rm I\!D}$, we have $S(\varphi)=\langle S,\varphi\rangle\geq
0$. An important result about the positive distributions is the
following (cf. \cite{A-M,Mal,Sug,ASU,ASU-1}):
\begin{theorem}
Assume that $S$ is a positive distribution in ${\rm I\!D}'$, then there
exists a positive Radon measure $\nu_S$ on $W$ such that
$$
\langle S,\varphi\rangle=\int_W \varphi d\nu_S\,,
$$
for any $\varphi\in {\rm I\!D}\cap C_b(W)$. In particular, if a sequence $(S_n)$
of positive distributions converge to $S$ weakly in ${\rm I\!D}'$, then
$(\nu_{S_n})$ converges to $\nu_S$ in the weak topology of measures.
\end{theorem}
\begin{remarkk}
In fact we can write, for any $\varphi\in {\rm I\!D}$
$$
\langle S,\varphi\rangle=\int_W\tilde{\varphi} d\nu_S\,,
$$
where $\tilde{\varphi}$ denotes a redefinition of $\varphi$ which is
constructed using the
capacities associated to the scale of Sobolev spaces
$({\rm I\!D}_{p,k},\,p>1,k\in{\Bbb N})$, cf. \cite{F-P}.
\end{remarkk}
\section{Uniqueness of the solution of parabolic variational inequality}
Assume that $(X^s_t(x), \,0\leq s\leq t\leq T)$ is a diffusion process
governed by an ${\rm I\!R}^n$-valued Wiener process $(W_t,t\in [0,T])$.
We assume that the diffusion has smooth, bounded drift and diffusion coefficients $b(t,x),\sigma(t,x)$
defined on $[0,T]\times {\rm I\!R}^d$, with values in ${\rm I\!R}^d$ and ${\rm I\!R}^d\otimes
{\rm I\!R}^n$ respectively and we denote by $A_t$ its infinitesimal
generator. We shall assume that $X_t^s$ is nondegenerate for any
$0\leq s<t\leq T$, $\partial/\partial t+A_t$ is hypoelliptic
and
$$
\int_{s+\varepsilon}^t (\det\gamma^s_v)^pdv\in L^1(\mu)
$$
for any $0<s<t \leq T$ and $\varepsilon>0$, where $\gamma_v$ is the inverse of
the matrix $((\nabla X^{s,i}_v,\nabla X^{s,j}_v)_H:\,i,j\leq d)$.
Suppose that $f\in C_b({\rm I\!R}^d)$ and we shall study the
following partial differential inequality whose solution will be
denoted by $u(t,x)$:
\begin{theorem}
\label{ineq-th-1}
Assume that $u\in C_b([0,T]\times {\rm I\!R}^d)$ such that $t\to
\langle u(t,\cdot),g\rangle $ is of finite total variation on $[0,T]$
for any $g\in{\mathcal S}({\rm I\!R}^d)$ and that it satisfies the following properties:
\begin{eqnarray}
\frac{\partial u}{\partial t}+A_tu-ru&\leq& 0,\,\,u\geq
f\,{\mbox{in}}\,[0,T]\times {\rm I\!R}^d\label{rel-1}\\
(\frac{\partial u}{\partial t}+A_tu-ru)(f-u)&=& 0,\,\label{rel-2}\\
U(T,x)&=&f(x)\,,\label{rel-3}
\end{eqnarray}
where all the derivatives are taken in the sense of distributions, in
particular the derivative w.r. to $t$ is taken using the
$C^\infty$-functions of compact support in $(0,T)$.
Then
$$
u(t,x)=\sup_{\tau\in{\mathcal Z}_{t,T}}E\left[f(X^{t}_\tau(x))\exp-\int_t^\tau r(s,X_s^t(x))ds\right]\,,
$$
where ${\mathcal Z}_{t,T}$ denotes the set of all the stopping times with
values in $[t,T]$ and $r$ is a smooth function on $[0,T]\times {\rm I\!R}^d$.
\end{theorem}
\noindent {\bf Proof:\ }
We shall prove the case $t=0$. Let us denote by $l$ the process
defined as $l_t=\exp-\int_0^tr(s,X_s)ds$. From Theorem \ref{ito-1}, we
have, for any $\varepsilon>0$,
\begin{equation}
\label{ito-dev}
l_tu(t,X_t)-l_\varepsilon u(\varepsilon,X_\varepsilon)-\int_\varepsilon^t l_s (A_su(s,X_s)ds-(ru)(s,X_s)ds+u(ds,X_s))=M^\varepsilon_t
\end{equation}
where $M^\varepsilon_t$ is a
${\rm I\!D}'$-valued martingale difference, i.e., denoting by
$E[\cdot|{\mathcal F}_s]$ the extension of the conditional expectation
operator to ${\rm I\!D}'$ {\footnote{Such an extension is licit since the
conditional expectation
operator commutes with the Ornstein-Uhlenbeck semigroup.}}, we
have $E[M^\varepsilon_t|{\mathcal F}_s]=M^\varepsilon_s$ for any $\varepsilon\leq s\leq t$. Note
also that $K_tu=\frac{\partial u}{\partial t}+A_tu-ru)\leq 0$ hence its composition with $X_t$ is a negative
measure and this implies that the integral at the l.h.s. of
(\ref{ito-dev}) is a negative distribution on the Wiener space. Consequently we have
\begin{equation}
\label{eqn-1}
M^\varepsilon_t\geq l_tu(t,X_t)-l_\varepsilon u(\varepsilon,X_\varepsilon)
\end{equation}
in ${\rm I\!D}'$. For $\alpha>0$, let $P_\alpha$ be the Ornstein-Uhlenbeck
semigroup and define $M^{\alpha,\varepsilon}_t$ as
$$
M^{\alpha,\varepsilon}_t=P_\alpha M^\varepsilon_t\,.
$$
Then $(M^{\alpha,\varepsilon}_t,t\geq \varepsilon)$ is a continuous martingale (in
the ordinary sense). From the inequality (\ref{eqn-1}), we have, for
any $\tau\in{\mathcal Z}_{\varepsilon,T}$,
$$
M^{\alpha,\varepsilon}_\tau\geq P_\alpha(l_tu(t,X_t)-l_\varepsilon
u(\varepsilon,X_\varepsilon))|_{t=\tau}\,.
$$
Taking the expectation of both sides, we get
$$
E[l_\varepsilon u(\varepsilon,X_\varepsilon)]\geq E[ P_\alpha(l_tu(t,X_t)|_{t=\tau}]
$$
for any $\alpha>0$, hence we also have
$$
E[l_\varepsilon u(\varepsilon,X_\varepsilon)]\geq E[ l_\tau u(\tau,X_\tau)]
$$
for any $\varepsilon>0$ which is arbitrary and finally we obtain
$$
u(0,x)\geq E[ l_\tau u(\tau,X_\tau)]
$$
for any $\tau\in {\mathcal Z}_{0,T}$.
\noindent
To show the reverse inequality let $D=\{(s,x):\,u(s,x)\neq f(x)\}$ and
define
$$
\tau_x=\inf(s:\,(s,X^{0,x}_s)\in D^c)\,.
$$
Since $K_t$ is
hypoelliptic, and since $K_tu=0$ on the set $D$, $u$ is smooth in
$D$. If $\mu\{\tau_x=0\}=1$, from the continuity of $u$, we have
$$
u(0,x)=f(x)=E[l_{\tau_x}u(\tau_x,X^{0,x}_{\tau_x})]\,,
$$
hence the supremum is attained in this case. If $\mu\{\tau_x\neq
0\}>0$, then from the $0-1$-law $\mu\{\tau_x\neq 0\}=1$ and $\tau_x$
is predictable. Let $(\tau_n,n\geq 1)$ a sequence of stopping times
announcing $\tau_x$. From the classical It\^o formula, we have
$$
l_{\tau_n}u(\tau_n,X_{\tau_n})-u(0,x)=\int_0^{\tau_n}l_s(\sigma^\star\partial
u)(s,X_s)\cdot dW_s\,.
$$
By the hypothesis the l.h.s. is uniformly integrable with respect to
$n\in{\rm I\!N}$, consequently we obtain
$$
u(0,x)=\lim_nE[l_{\tau_n}u(\tau_n,X_{\tau_n})]=E[l_{\tau}u(\tau,X_{\tau})]
$$
hence $\tau_x$ realizes the supremum.
\hspace*{\fill
\noindent
In the homogeneous case the finite variation property of the solution
follows directly from the quasi-variational inequality:
\begin{theorem}
Suppose that the infinitesimal generator $A_t$ of the process $(X_t)$ is
independent of $t\in [0,T]$ and denote it by $A$. In other words the process is homogeneous in
time. Assume that $u\in C_b([0,T]\times {\rm I\!R}^d)$ satisfies the following properties:
\begin{eqnarray}
\frac{\partial u}{\partial t}+Au-ru&\leq& 0,\,\,u\geq
f\,{\mbox{in}}\,[0,T]\times {\rm I\!R}^d\label{rel-11}\\
(\frac{\partial u}{\partial t}+Au-ru)(f-u)&=& 0\,\,{\mbox{in}}\,[0,T]\times {\rm I\!R}^d\label{rel-12}\\
U(T,x)&=&f(x)\,.\label{rel-13}
\end{eqnarray}
Then
$$
u(t,x)=\sup_{\tau\in{\mathcal Z}_{t,T}}E\left[f(X^{t}_\tau(x))\exp-\int_t^\tau r(s,X_s^t(x))ds\right]\,,
$$
where ${\mathcal Z}_{t,T}$ denotes the set of all the stopping times with
values in $[t,T]$ and $r$ is a smooth function on $[0,T]\times {\rm I\!R}^d$.
\end{theorem}
\begin{remarkk}
The relations (\ref{rel-11}) and (\ref{rel-12}) are to be understood
in the weak sense. This means that for any $g$ a $C^\infty$ function
of support in $(0,T)$ and $\gamma\in{\mathcal S}({\rm I\!R}^d)$, both of which are
positive, we have
$$
\left\langle \frac{\partial u}{\partial t}+Au-ru,g\otimes\gamma\right\rangle \leq 0
$$
and
$$
\left\langle (\frac{\partial u}{\partial t}+Au-ru)(f-u),g\otimes\gamma\right\rangle= 0\,.
$$
\end{remarkk}
\noindent {\bf Proof:\ }
As in the proof of the preceding theorem, we shall prove the equality
for $t=0$, then the general case follows easily.
Let $\rho_\delta$ be a mollifier on ${\rm I\!R}$ and let $\eta_\varepsilon$ be a
family of positive smooth functions on $(0,T)$, equal to unity on the
interval $[\varepsilon,T-\varepsilon]$, converging to the indicator function of
$[0,T]$ pointwise. Define $u^{\delta,\varepsilon}$ as
$$
u^{\delta,\varepsilon}=\rho_\delta\star(\eta_\varepsilon u)\,.
$$
From the hypothesis the distribution $\nu$ defined by
$$
\nu=\frac{\partial u}{\partial t}+Au-ru
$$
is a negative measure on $(0,T)\times {\rm I\!R}^d$. A simple calculation
gives
$$
\frac{\partial u^{\delta,\varepsilon}}{\partial
t}+Au^{\delta,\varepsilon}-ru^{\delta,\varepsilon}=\rho_\delta\star(\eta'_\varepsilon
u)+\rho_\delta\star(\eta_\varepsilon\nu)+\rho_\delta\star(\eta_\varepsilon r
u)-ru^{\delta,\varepsilon}\,.
$$
As in the preceding theorem, we have from Theorem \ref{ito-1}
$$
l_tu^{\delta,\varepsilon}(t,X_t)-l_au^{\delta,\varepsilon}(a,X_a)-\int_a^tl_sK_su^{\delta,\varepsilon}(s,X_s)ds=M_t^{\delta,\varepsilon,a}\,,
$$
where $M^{\delta,\varepsilon,a}$ is a ${\rm I\!D}'$-martingale difference. Since
$\nu$ is a negative measure, we get the following inequality in
${\rm I\!D}'$:
$$
M_t^{\delta,\varepsilon,a}\geq l_tu^{\delta,\varepsilon}(t,X_t)-l_au^{\delta,\varepsilon}(a,X_a)-\int_a^t[\rho_\delta\star(\eta'_\varepsilon
u)+\rho_\delta\star(\eta_\varepsilon r u)-ru^{\delta,\varepsilon}](s,X_s)ds
$$
Let now $(P_\alpha,\alpha\geq 0)$ be the Ornstein-Uhlenbeck
semigroup. Then $(P_\alpha M_t^{\delta,\varepsilon,a},\,a\leq t\leq T)$ is a
real valued martingale difference, consequently, we have
\begin{eqnarray*}
0&=&E[(P_\alpha M_t^{\delta,\varepsilon,a})_{t=\tau}]\\
&\geq&
E\left[P_\alpha\left(l_t
u^{\delta,\varepsilon}(t,X_t)-l_au^{\delta,\varepsilon}(a,X_a)-
\int_a^t[\rho_\delta\star(\eta'_\varepsilon
u)+\rho_\delta\star(\eta_\varepsilon r
u)-ru^{\delta,\varepsilon}](s,X_s)ds\right)_{t=\tau}\right]\,,
\end{eqnarray*}
for any stopping time $\tau$ with values in $[\varepsilon,T-\varepsilon]$. By
letting $\alpha\to 0$, we get by continuity
$$
0\geq
E\left[l_\tau
u^{\delta,\varepsilon}(\tau,X_\tau)-l_a u^{\delta,\varepsilon}(a,X_a)-
\int_a^\tau[\rho_\delta\star(\eta'_\varepsilon
u)+\rho_\delta\star(\eta_\varepsilon r
u)-ru^{\delta,\varepsilon}](s,X_s)ds\right]\,.
$$
Let us
choose $a>0$ and let then $\varepsilon,\delta\to 0$. Note that
$\eta'_\varepsilon \to \delta_0-\delta_T$ (i.e., the Dirac measures at $0$
and at $T$), by the choice of $a$ and by the weak convergence of
measures and by the dominated convergence theorem, we obtain
$$
\lim_{\varepsilon,\delta\to 0}E\int_a^\tau \left(\rho_\delta\star(\eta'_\varepsilon
u)\right)(s,X_s)ds=0\,.
$$
Again from the dominated convergence theorem we have
$$
\lim_{\varepsilon,\delta\to 0}E\int_a^\tau [\rho_\delta\star(\eta_\varepsilon r
u)-ru^{\delta,\varepsilon}](s,X_s)ds=0\,.
$$
Consequently
$$
E[l_a u(a,X_a)]\geq E[l_\tau u(\tau,X_\tau)]\geq E[l_\tau f(X_\tau)]\,,
$$
for any stopping time $\tau$ with values in $[a,T-a]$, since $a>0$ is
arbitrary, the same inequality holds also for any stopping time with
values in $[0,T]$; hence
$$
u(0,x)]\geq E[l_\tau f(X_\tau)]
$$
for any stopping time $\tau\in {\mathcal Z}_{0,T}$ and we obtain the first
inequality:
$$
u(0,x)]\geq\sup_{\tau\in{\mathcal Z}_{0,T}} E[l_\tau f(X_\tau)]\,.
$$
The proof of the reverse inequality is exactly the same that of
Theorem \ref{ineq-th-1} due to the
hypoellipticity hypothesis.
\hspace*{\fill
\section{Existence of the solutions}
\noindent
In this section, under the hypothesis of the preceding section, we shall prove that the function defined by the
Snell envelope (cf. \cite{Karoui}) of the American option satisfies the variational
inequality (\ref{rel-11}) and the equality (\ref{rel-12}).
We start with a lemma:
\begin{lemma}
\label{lemma-1}
Assume that $Z=(Z_t,t\in[0,T])$ is a uniformly integrable, real-valued martingale on
the Wiener space. Let $Z^\kappa=(Z^\kappa_t,t\in[0,T])$ be defined as
$Z_t^\kappa=P_\kappa Z_t$, where $P_\kappa$ is the Ornstein-Uhlenbeck
semigroup at the instant $\kappa>0$. Then $(Z_t^\kappa,t\in[0,T])$ is
a uniformly integrable martingale with
\begin{equation}
\label{Davis}
E[\langle Z^\kappa, Z^\kappa\rangle_T^{1/2}]\leq c\,E[\langle Z,
Z\rangle_T^{1/2}]\,,
\end{equation}
where $c$ is a constant independent of $Z$ and $\kappa$. In
particular, if $Z$ has the representation
$$
Z_T=\int_0^T(m_s,dW_s)\,,
$$
with $m\in L^1(\mu,H)$ optional, then
$$
P_\kappa Z_T=\int_0^Te^{-\kappa}(P_\kappa m_s,dW_s)\,.
$$
\end{lemma}
\noindent {\bf Proof:\ }
From Davis' inequality (cf. \cite{Mey}), we have
\begin{eqnarray*}
E[\langle Z^\kappa, Z^\kappa\rangle_T^{1/2}]&\leq&c_1 E[\sup_{t\in [0,T]}|Z_t^\kappa|]\\
&\leq&c_1 E[P_\kappa(\sup_{t\in [0,T]}|Z_t|)]\\
&=&c_1 E[\sup_{t\in [0,T]}|Z_t|]\\
&\leq& c\,E[\langle Z,Z\rangle_T^{1/2}]\,.
\end{eqnarray*}
The second part is obvious from the inequality (\ref{Davis}).
\hspace*{\fill
\begin{theorem}
\label{existence-thm}
Assume that $(X_t^s)$ is a hypoelliptic diffusion such that, for any
$\varepsilon>0$,
$$
\int_{s+\varepsilon}^T (\det\gamma_v^s)^p dv\in L^1(\mu)
$$
for any $p>1$. Let $p(s,t;x,y),\,s<t,\,x,y\in {\rm I\!R}^d$ be the density of
the law of $X^s_t(x)$ and denote by $S_{0,z}$ the open set
$$
S_{0,z}=\{(s,y)\in (0,T)\times {\rm I\!R}^d:\,s>0,\,p(0,s;z,y)>0\}\,.
$$
Then, for any $z\in {\rm I\!R}^d$, $u$ is a solution of the variational inequality
(\ref{rel-11},\ref{rel-12},\ref{rel-13}) in ${\mathcal D}'(S_{0,z})$. If
$S_{0,z}=(0,T)\times {\rm I\!R}^d$ for any $z\in {\rm I\!R}^d$, then $u$ is a solution of the variational inequality
(\ref{rel-11},\ref{rel-12},\ref{rel-13}) in ${\mathcal D}'(0,T)\otimes {\mathcal D}'({\rm I\!R}^d)$.
\end{theorem}
\noindent {\bf Proof:\ }
From the optimal stopping results, we know that $u$ is a bounded,
continuous function and $t\to u(t,x)$ is monotone, decreasing
(cf.\cite{Karoui}). Moreover
$$
u(t,X_t)l_t-u(0,x)=M_t+B_t
$$
is a supermartingale where $X_t=X^0_t(x)$ and we denoted by $M$ its
martingale part and by $B$ its continuous, decreasing process part. In
particular $dB\times d\mu$ defines a negative measure $\gamma$ on
$[0,T]\times C([0,T],{\rm I\!R}^d)$. We can write $u(ds,x)$ as the sum
$u_{ac}(s,x)ds+u_{sing}(ds,x)$ where $u_{ac}$ is defined as the absolutely
continuous part of $u$ and $u_{sing}$ is the singular part.
We have, from the extended It\^o formula,
\begin{eqnarray*}
u(t,X_t)l_t-u(\varepsilon,X_\varepsilon)&=&\int_\varepsilon^t
\left(A_su-ru+u_{ac})(s,X_s)\right)ds+u_{sing}(ds,X_s)+\int_\varepsilon^t((\sigma\partial
u)(s,X_s),dW_s)\\
&=&M^\varepsilon_t+B^\varepsilon_t\,,
\end{eqnarray*}
hence regularizing both parts by the Ornstein-Uhlenbeck semigroup,
from Lemma \ref{lemma-1}, we get
\begin{eqnarray*}
B^\varepsilon_t&=&\int_\varepsilon^t \left(A_su-ru+u_{ac})(s,X_s)\right)ds+u_{sing}(ds,X_s)\\
M^\varepsilon_t&=&\int_\varepsilon^t((\sigma\partial u)(s,X_s),dW_s)\,.
\end{eqnarray*}
Consequently, for any $\alpha\in {\mathcal D}(0,T)$ and $\phi\in {\rm I\!D}$
\begin{eqnarray*}
E\left[\phi\int_0^T\alpha(s) dB_s\right]&=&\int\alpha\otimes \phi d\gamma\\
&=&\int_{(0,T)}\alpha(s)\langle
\left(A_su-ru+u_{ac})(s,X_s)\right)ds+u_{sing}(ds,X_s),\phi\rangle
\end{eqnarray*}
and this quantity is negative for any $\alpha\in {\mathcal D}_+(0,T)$ and
$\phi\in {\rm I\!D}_+$. Let now $0\leq g\in{\mathcal S}({\rm I\!R}^d)$ and assume that
$(t_i,\,i\leq m)$ is a partition of $[0,T]$. Define $\xi_m$ as
$$
\xi_m(t,w)=\sum_i 1_{[t_i,t_{i+1}]}(t)\,g(X_{t_i})\,.
$$
Then it is immediate from the hypothesis about the diffusion process
$(X_t)$ that $(\xi_m,\,m\geq 1)$ converges to
$(g(X_s)1_{[0,T]}(s),\,s\in [0,T])$ in ${\rm I\!D}(L^p([0,T]))$ for any
$p\geq 1$ and $(\xi_m(s,\cdot)$ converges to $g(X_s)$ in ${\rm I\!D}$ for
any fixed $s\in [0,T]$ as the partition pace tends to zero. Let us
represent $u(ds,\cdot)$, using the kernel theorem (cf.\cite{Gro,Sch}), as
$$
u(ds,\cdot)=\sum_{k=1}^\infty \lambda_i T_k\otimes \alpha_k\,,
$$
where $(\lambda_k)\in l^1$, $(T_k)\subset {\mathcal S}'({\rm I\!R}^d)$ is bounded and
$(\alpha_k)$ is a sequence of measures on $[0,T]$, bounded in total
variation norm. It follows then
$$
u(ds,X_s)=\sum_{k=1}^\infty \lambda_i T_k(X_s) \alpha_k(ds)
$$
and this some is convergent in $V([0,T])\tilde{\otimes}{\rm I\!D}_{p,-k}$ for
some $k\in{\Bbb N}$ and $p>1$, in the projective topology, where $V([0,T])$ denotes the
Banach space of measures on $[0,T]$ under the total variation norm.
Since
$$
\sup_{s\in [0,T]}\|\xi_m(s,X_s)\|_{p,l}\leq \sup_{s\in [0,T]}\|g\circ
X_s\|_{p,l} \,,
$$
uniformly in $m\in {\rm I\!N}$, for any $p,\,l$ and since
$$
\|\xi_m(s,\cdot)-\xi_n(s,\cdot)\|_{p,l}\to 0
$$
as $m,n\to\infty$ for any $p,\,l$ and $s\in [0,T]$, we obtain
$$
\lim_{m\to\infty}\int_{(0,T)}\delta(s)\langle
\xi_m(s,\cdot)-g(X_s),u(ds,X_s)\rangle=0
$$
for any $\delta\in{\mathcal D}(0,T)$ from the dominated convergence theorem.
The above relation implies in particular
that we have
$$
\int_{(0,T)}\alpha(s)\langle (A_su-ru+u_{ac})(s,\cdot),p_{0,s}g\rangle
ds+\int_{(0,T)}\alpha(s)\langle u_{sing}(ds,\cdot),g\,p_{0,s}\rangle\leq 0\,,
$$
with smooth, positive $\alpha$ and $g$, where the brackets in the integral correspond to the duality between
${\mathcal D}({\rm I\!R}^d)$ and ${\mathcal D}'({\rm I\!R}^d)$. For the functions of support in
$(0,T)$, we can replace the term
$$
\int_{(0,T)} \alpha(s)\left\langle u(ds,X_s),g(X_s)\right\rangle
$$
by
$$
\int_{(0,T)} \alpha(s)\langle\frac{\partial}{\partial s} u(s,X_s),g(X_s)\rangle
$$
where $\partial/\partial s$ denotes the derivative in ${\mathcal D}'(0,T)$.
Since $\alpha$ and $g$ are
arbitrary, we obtain the inequality (\ref{rel-11}) in
${\mathcal D}'(S_{0,x})$. If $S_{0,x}=(0,T)\times {\rm I\!R}^d$, then we have the
inequality in the sense of distributions on $(0,T)\times {\rm I\!R}^d$.
To complete the proof, let $D$ be the set defined as
$$
D=\{(s,x)\in (0,T)\times {\rm I\!R}^d: \,u(s,x)=f(x)\}\,.
$$
Then we have
$$
\int_0^T 1_{D^c}(s,X_s) dB_s=0
$$
almost surely (cf.\cite{Karoui}). Let $C=-B$, then for any smooth
function $\eta\in {\mathcal D}(0,T)\otimes {\mathcal S}({\rm I\!R}^d)$ such that $\eta\leq
1_{D^c}$, we have
\begin{eqnarray*}
0&=&E\int_0^T1_{D^c}(s,X_s)dC_s\\
&\geq&E\int_0^T\eta(s,X_s)dC_s\\
&=&-\int_{(0,T)}\langle (A_s u-r u)(s,X_s)ds+u(ds,X_s),\eta(s,X_s)\rangle\\
&\geq&0\,,
\end{eqnarray*}
where, the second equality follows from the estimates above. Hence
$$
A_su-ru+\frac{\partial}{\partial s}u=0
$$
as a distribution on the set $S_{0,x}\cap D^c$, by the
hypoellipticity, the equality is everywhere on this set. If
$S_{0,x}=(0,T)\times {\rm I\!R}^d$, then we obtain the relation
(\ref{rel-12}).
\hspace*{\fill
\begin{remarkk}
From the general theory, we can express the martingale part of
$(l_tu(t,X_t),\,t\in [0,T])$ as
$$
\int_0^T(H_s,dW_s)
$$
where $H$ is an adapted process which is locally integrable. On the
other hand we have
$$
M^\varepsilon_t=\int_\varepsilon^t (\sigma(s,X_s)\partial u(s,X_s),dW_s)
$$
where the r.h.s. is to be interpreted in a negatively indexed Sobolev
space on the Wiener space. Using Lemma \ref{lemma-1}, we obtain the
identity
$$
H_s=\sigma(s,X_s)\partial u(s,X_s)
$$
$ds\times d\mu$-a.s., in particular we have
$$
E\left[\left(\int_0^T|\sigma(s,X_s)\partial u(s,X_s)|^2ds\right)^{1/2}\right]<\infty\,.
$$
\end{remarkk}
|
1,108,101,563,224 | arxiv | \section{Introduction} While spin phenomena in two dimensions have been subjected to intense scrutiny for decades, only recently have vdW magnets emerged as a concrete platform for the exploration of two-dimensional (2$d$) magnetism~\cite{burch2018,gong2017,huang2017,park2016}. In most of these compounds, a long-range order is stabilized by an in-plane or out-of-plane magnetic anisotropy that circumvents the restrictions of the Mermin-Wagner theorem~\cite{mermin1966,hohenberg1967,huang2017,lee2016,wang2016,gong2017,bonilla2018,ohara2018}.
Monolayers of chromium halides $\text{CrX}_{3}$ (X=Cl,Br,I) have been proposed as testbed for the Berezinskii-Kosterlitz-Thouless universality class that has been long sought in magnetic systems~\cite{berezinskii1971,kosterlitz1973,kosterlitz1974,kim2021,troncoso2020}.
Their honeycomb lattice structure has opened up opportunities to investigate Dirac bosons, whose statistics and interactions drastically differed from their far more scrutinized electronic counterpart~\cite{pershoguba2018}. With strong spin-orbit coupling (SOC) and an edge-sharing octahedra structure, vdW ferromagnets can display a bond-directional anisotropic exchange interaction, i.e. the Kitaev interaction~\cite{kitaev2006,xu2018,lee2020}, providing a route for the investigation of spin liquid states with spin $S=3/2$~\cite{xu2020:kitaev}.
Furthermore, the lattice structure symmetry allows for next-to-nearest neighbor (NNN) out-of-plane Dzyaloshinskii-Moriya (DM) interactions. NNN DM interactions on a honeycomb ferromagnetic lattice play a role analogous to SOC in graphene: magnons accumulate an additional phase upon propagation between NNN sites and topologically nontrivial edge states can emerge~\cite{chen2018,kim2016,ruckriegel2018}.
The variety of magnetic regimes displayed by vdW magnets can be further enriched by tuning their properties through electric fields, proximity effects or chemical doping~\cite{wang2018, behera2019, lu2017,liu2018,zhong2017,hellman2017,abramchuk2018,kondo2020d}. Recently, Tartaglia \textit{et al.}~\cite{tartaglia2020} have shown that the magnetic anisotropy of chromium halides can be continuously tuned by adjusting the relative composition of halides. Importantly, varying the ratio of ligands not only affects the overall anisotropy, but also leads to a crystalline structure with a lower symmetry group than its stochiometric counterpart.
\begin{figure}[b!]
\centering
\includegraphics[width=0.5 \textwidth]{fig1.png}
\caption{Lattice structure of a chromium trihalide monolayer. The magnetic atoms (Cr) are arranged in a honeycomb lattice. The Cr-Cr coupling is mediated by I, Cl and Br ligands. Solid colored dots refer to atoms above the Cr plane and open dots refer to atoms below the Cr plane. In this work, we explore the emergence of topologically protected magnon edge states that yield a thermal Hall flow, transverse with respect to the direction of an applied temperature gradient $\Delta T$.}
\label{fig:crystal}
\end{figure}
Motivated by these experimental advances, in this work we investigate the magnetic properties of a chromium trihalide CrClBrI layer, shown in Fig.~\ref{fig:crystal}. We show that the richness of spin-spin interactions can lead, depending on the parameters, to topological magnon phases and to a wide array of noncollinear spin states and magnetic defects.
This work is organized as follows:
In section II, we establish a Hamiltonian spin model for a chromium trihalide CrClBrI layer. In section III, we explore a set of system parameters corresponding to a two-sublattice ground state. In this regime, we show that the spin-wave bands can have nonvanishing Chern number, which signals the presence of topologically protected edge states. We investigate the contribution of these edge states to the magnon thermal Hall effect~\cite{katsura2010,matsumoto2011,murakami2017,onose2010}. Finally, in section IV, we demonstrate using Monte Carlo techniques that our model can support exotic noncollinear ground states such as spin cycloids and Bloch and N\'{e}el skyrmions.
\section{Model}
Let us consider a monolayer of chromium trihalide CrClBrI. The magnetic Cr atoms are arranged on a honeycomb lattice and each $i$th site $\mathbf{r}_{i}$ carries a spin moment $\textbf{S}_i = (S_i^x, S_i^y, S_i^z)$. The spin-spin interactions between Cr atoms are mediated by the nonmagnetic ligands (Cl, I, and Br) lying out of the Cr plane, as shown in Fig.~\ref{fig:structure}(a). The distribution of ligands breaks the $C_3$ symmetry of the honeycomb lattice and allows interactions to be bond-dependent. The nearest-neighbor (NN) Heisenberg exchange term can be generally written as
\begin{figure}
\centering
\includegraphics[width=250px]{fig2.pdf}
\caption{
(a) The bond geometry is shown. $A$ and $B$ label the two magnetic sublattices of the honeycomb lattice, while $\boldsymbol\alpha_n$ and $\boldsymbol\beta_n$ label, respectively, the NN and NNN bond vectors, with $n=1,2,3$. The length of the NNN bond is $a$, i.e. $| \boldsymbol \beta_n | = a$. On the $\boldsymbol \alpha_1$ bond, the mirror plane is shown in purple, along with the NN DM vector (red arrow). The NNN DM vector is shown in red on the $\boldsymbol \beta_2$ vector.
(b) The NN bond geometry along the hopping direction defined by $\boldsymbol\alpha_1$, mediated by a Cl below the plane and Br above the plane. The purple plane containing both Cr atoms and the two mediating halides is a mirror plane: by Moriya's rules, the DM vector $\textbf{D}_{1}$ (red arrow) is constrained to be perpendicular to this plane.
(c) The NNN bond geometry along the hopping direction defined by $\boldsymbol\beta_2$. The red arrow represents the NNN DM vector $\textbf{D}'_{2,A}$. }
\label{fig:structure}
\end{figure}
\begin{equation} \label{jterm}
\mathcal{H}_{J}^{NN} = -\sum\limits_{\langle i,j \rangle} J_{ij} \textbf{S}_i \cdot \textbf{S}_j\,,
\end{equation}
where $\langle .. \rangle$ denotes summation over the nearest neighbors and $J_{ij}$ is the bond-dependent ferromagnetic exchange coupling. Here, $J_{ij}$ takes the values $J_1$, $J_2$, or $J_3$ for the NN bond along $\boldsymbol \alpha_1$, $\boldsymbol \alpha_2$ and $\boldsymbol \alpha_3$, respectively.
The bond geometry is shown in Fig.~\ref{fig:structure}(a).
In addition, the SOC allows for an antisymmetric exchange, i.e. a Dzyaloshinskii-Moriya (DM) interaction between both NN and NNN atoms. The NN DM interaction contribution to the Hamiltonian reads
\begin{equation} \label{dterm}
\mathcal{H}_{DM}^{NN} = -\sum\limits_{\langle i,j \rangle} \textbf{D}_{ij} \cdot (\textbf{S}_i \times \textbf{S}_j).
\end{equation}
The DM vectors are determined by Moriya's rules \cite{moriya1960} according to the local symmetry of the bond. Similar to the NN Heisenberg interaction~\eqref{jterm}, the DM strength is bond-dependent, i.e. $\textbf{D}_{ij} = \textbf{D}_n$, with $n=1,2,3$. On the $\boldsymbol \alpha_n$ bond, the plane containing the Cr atoms and mediating ligands is a mirror plane of the bond; thus, $\textbf{D}_n$ is perpendicular to this mirror plane:
\begin{equation} \label{dnn}
\textbf{D}_{n} = D_{n} \hat{\gamma}_n,
\end{equation}
where $\hat{\gamma}_{1(2)} = \left(-\frac{1}{\sqrt{6}}, \pm \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{3}}\right)$, and $\hat{\gamma}_{3} = \left(\sqrt{\frac{2}{3}}, 0, \frac{1}{\sqrt{3}}\right)$ are the unit vectors perpendicular to the mirror plane, depicted in Fig.~\ref{fig:structure}(b).
The SOC also allows for a NN Kitaev interaction~\cite{aguilera2020}, which can be written as
\begin{equation} \label{kterm}
\mathcal{H}_{K}^{NN} = -\sum\limits_{\langle i,j \rangle} K_{ij} S_i^{\gamma_n} S_j^{\gamma_n},
\end{equation}
where $S_i^{\gamma_n}$ = $\textbf{S}_i \cdot \hat{\gamma}_n$ and $K_{ij} = K_n$. We can combine Eqs.~\eqref{jterm}, \eqref{dterm}, and \eqref{kterm} by writing
\begin{equation} \label{comb1}
\mathcal{H}^{NN} = \mathcal{H}_{J}^{NN} + \mathcal{H}_{DM}^{NN} + \mathcal{H}_{K}^{NN} = \sum\limits_{\langle i,j \rangle} \textbf{S}_i^T \Lambda_{n} \textbf{S}_j,
\end{equation}
where
\begin{equation} \label{lambdamat}
\Lambda_{n} =
\begin{bmatrix}
-J_{n} & -D_{n}^z & D_{n}^y \\
D_{n}^z & -J_{n} & -D_{n}^x \\
-D_{n}^y & D_{n}^x & -J_{n}
\end{bmatrix}
- K_n \hat{\gamma}_n \otimes \hat{\gamma}_n,
\end{equation}
is the NN interaction matrix, and $n$ is understood to index the $\langle i, j \rangle$ bond type.
The NNN Heisenberg and DM interactions can be included as
\begin{equation} \label{NNNjdterm}
\mathcal{H}^{NNN} = -\sum\limits_{\langle \langle i,j \rangle \rangle} J'_{ij} \textbf{S}_i \cdot \textbf{S}_j - \sum\limits_{\langle \langle i,j \rangle \rangle} \textbf{D}'_{ij} \cdot (\textbf{S}_i \times \textbf{S}_j)\,,
\end{equation}
where $\langle \langle .. \rangle \rangle$ denotes summation over next-to-nearest neighbors. Here, $J'_{ij}$ and $D'_{ij}$ are, respectively, the bond-dependent NNN Heisenberg and DM interaction strength.
There are three distinct NNN bonds on each of the two sublattices for a total of six possible NNN exchange parameters. For the sublattice $s=A,B$, the bond along the hopping direction $\pm \boldsymbol
\beta_n$, sketched in Fig.~\ref{fig:structure}(a), mediates a Heisenberg exchange $J'_{ij} = J'_{n,s}$ and a DM interaction $\textbf{D}'_{ij} = \pm \textbf{D}'_{n,s}$.
The lack of point-group symmetries provides no restriction on the NNN DM vectors according to Moriya's rules. Thus, the NNN DM vector $\textbf{D}'_{n,s}$ can be generally written in terms of the local bond geometry as
\begin{equation} \label{dnnn}
\begin{split}
\textbf{D}'_{n,s} &= \left(D'_{n,s} \sin{\tilde{\theta}_{n,s}}\right) R_z\left(\tilde{\phi}_{n,s}\right) \hat{\boldsymbol \beta}_{n} \\
&+ \tau_{s}\left(D'_{n,s} \cos{\tilde{\theta}_{n,s}}\right) \hat{\textbf{z}},
\end{split}
\end{equation}
where $ \hat{\boldsymbol \beta}_{n} = \boldsymbol \beta_{n}/| \boldsymbol \beta_{n} |$, $R_z (\tilde{\phi}_{n,s})$ describes a right-handed rotation by an angle $\tilde{\phi}_{n,s}$ about the $\hat{\mathbf{z}}$ axis and $\tau_{A(B)} = \pm 1$. The angles $\tilde{\theta}_{n,s}$ and $\tilde{\phi}_{n,s}$ are the spherical coordinates of $\textbf{D}'_{n,s}$ with azimuthal angle measured relative to the $\boldsymbol \beta_n$ bond on the $s$ sublattice; this geometry is shown in Fig.~\ref{fig:structure}(c). When the mediating halides are of the same type, the axis bisecting the bond vector through the mediating Cr is a two-way rotation axis, which constrains $\tilde{\phi}_{n,B} = 0$.
We can rewrite Eq.~(\ref{NNNjdterm}) in a compact form as
\begin{equation} \label{NNNterms}
\mathcal{H}^{NNN} = \sum\limits_{\langle \langle i,j \rangle \rangle} \textbf{S}_i^T \Xi_{n,s} \textbf{S}_j,
\end{equation}
with
\begin{equation}\label{ximat}
\Xi_{n,s} =
\begin{bmatrix}
-J'_{n,s} & -D_{n,s}^{\prime z} & D_{n,s}^{\prime y} \\
D_{n,s}^{\prime z} & -J'_{n,s} & -D_{n,s}^{\prime x} \\
-D_{n,s}^{\prime y} & D_{n,s}^{\prime x} & -J'_{n,s}
\end{bmatrix}.
\end{equation}
Further, we include a single-ion anisotropy term, $\mathcal{H}_{A}$, and a Zeeman interaction, $\mathcal{H}_{B}$, due to a uniform external magnetic field $\mathbf{B}=B \hat{\mathbf{z}}$ as
\begin{equation} \label{kbterm}
\mathcal{H}_A + \mathcal{H}_B = -A\sum\limits_i (S_i^z)^2 - g \mu_B B\sum\limits_i S_i^z.
\end{equation}
where $A>0$ parametrizes the strength of the easy-axis anisotropy~\cite{tartaglia2020}, $g$ is the g-factor and $\mu_{B}$ is the Bohr magneton.
At each magnetic site, we can orient a spin-space Cartesian coordinate system such that the new $\hat{\mathbf{z}}$ axis locally lies along the classical orientation of the onsite spin operator $\tilde{\textbf{S}}_i$. The latter can be related to the spin operator $\textbf{S}_i $ in the global frame of reference via the transformation
\begin{equation} \label{spinvec}
\textbf{S}_i = R_i(\theta_i, \phi_i) \tilde{\textbf{S}}_i.
\end{equation}
Here, $R_i(\theta_i,\phi_i) = R_z(\phi_i)R_y(\theta_i)$, where $R_{z(y)} (\zeta)$ describes a right-handed rotation by an angle $\zeta$ about the global $\hat{\mathbf{z}}$ $(\hat{\mathbf{y}})$ axis, and $\theta_i$ and $\phi_i$ are, respectively, the polar and azimuthal angles of the classical orientation of the spin $\mathbf{S}_{i}$. Equations \eqref{comb1}, \eqref{NNNterms} and \eqref{kbterm} can be combined into the full Hamiltonian in local coordinates as
\begin{equation} \label{HL}
\begin{split}
\mathcal{H} &= \sum\limits_{\langle i, j \rangle} \tilde{\textbf{S}}_i^T \tilde{\Lambda}_{n}\tilde{\textbf{S}}_j + \sum\limits_{\langle \langle i, j \rangle \rangle} \tilde{\textbf{S}}_i^T \tilde{\Xi}_{n,s} \tilde{\textbf{S}}_j \\
& - A\sum\limits_i \left( R_i \tilde{\textbf{S}}_i \right)^2_z -
\mu_B B\sum\limits_i (R_i \tilde{\textbf{S}}_i)_z, \\
\end{split}
\end{equation}
where $\left( \cdot \right)_\mu$ is the $\mu$ component of a vector. Here, we have introduced the rotated interaction matrices $\tilde{\Lambda}_n = R_i^T \Lambda_n R_j$ and $\tilde{\Xi}_{n,s} = R_i^T \Xi_{n,s} R_j$.
\subsection{BdG Hamiltonian}
Far below the magnetic ordering temperature $T_{c}$, i.e. for $T \ll T_c$, we can access the magnon spectrum by linearizing the Holstein-Primakoff transformation \cite{holstein1940} in the local frame of reference, i.e.
\begin{equation} \label{hptransform}
\begin{split}
\tilde{S}_i^+ &= \tilde{S}_i^x + i \tilde{S}_i^y = \sqrt{2S}\sqrt{1 - \frac{d_i^\dagger d_i}{2S}} d_i \approx \sqrt{2S} d_i, \\
\tilde{S}_i^z &= S - d_i^\dagger d_i\,,
\end{split}
\end{equation}
where $S$ is the classical spin (in units of $\hbar$) and $d_i$ ($d^{\dagger}_{i}$) the magnon annihilation (creation) operator at the $i$th site, obeying the bosonic commutation relation ${[d_i, d_j^\dagger] = \delta_{ij}}$. We plug Eq.~(\ref{hptransform}) into Eq.~\eqref{HL} and truncate the Hamiltonian beyond the quadratic terms in the Holstein-Primakoff boson operators since interactions between magnons can be neglected in the temperature regime of interest. We group terms constant in magnon operators in the classical energy term $E_{Cl}(\{\theta_i, \phi_i\} \vert_i)$ \footnote{This is equivalent to regarding $\textbf{S}_i$ as classical spin vectors and equating the $\mathcal{H}$ with $E_{Cl}$.}. Minimization of $E_{Cl}$ with respect to $\{ \theta_i, \phi_i \} \vert_i$ gives the ground-state spin configuration. Here, we focus on a ground state with two-sublattice translational symmetry, i.e.
\begin{equation}\label{spinAB}
\textbf{S}_i = S (\cos \phi_s \sin \theta_s, \sin \phi_s \sin \theta_s, \cos \theta_s),
\end{equation}
where $s=A,B$.
The classical energy then takes the form
\begin{equation} \label{eclassical}
\begin{split}
& E_{Cl}(\{\theta_i, \phi_i\} \vert_i)/N = E_{Cl}(\theta_A, \phi_A, \theta_B, \phi_B)/N \\
&= -g \mu_B B S (\cos \theta_A + \cos \theta_B) - A S^2 (\cos^2 \theta_A + \cos^2 \theta_B). \\
&+ S \sum\limits_{n=1}^3 \tilde{\Lambda}_n^{zz},
\end{split}
\end{equation}
where $N$ is the total number of Cr atoms in the sample. Equation ~(\ref{eclassical}) can be minimized by gradient descent or Monte Carlo methods.
In what follows, we relabel the operator $d_i$ as $a_i$ ($b_i$) on the $A$ $(B)$ sublattice. We can introduce the magnon operators in momentum space, i.e. $a_\textbf{k}$ and $b_\textbf{k}$, by performing a Fourier transformation:
\begin{equation} \label{fouriertransform}
a_i = \sqrt{\frac{2}{N}}\sum\limits_{\textbf{k}} e^{i \textbf{k} \cdot \textbf{r}_i} a_\textbf{k}, \qquad b_i =\sqrt{\frac{2}{N}} \sum\limits_{\textbf{k}} e^{i \textbf{k} \cdot \textbf{r}_i} b_\textbf{k}\,,
\end{equation}
where $\mathbf{k}=(k_{x},k_{y})$ is the 2$d$ wavevector and the summation is taken over the first Brillouin zone.
Substituting Eq.~\eqref{fouriertransform} into the Hamiltonian~(\ref{HL}) yields
\begin{equation} \label{HBdG}
\mathcal{H} = \frac{1}{2}\sum\limits_{\textbf{k}} \psi_\textbf{k}^\dagger \mathcal{H}_{\text{BdG}}(\textbf{k}) \psi_\textbf{k},
\end{equation}
where $\psi_{\textbf{k}}^\dagger =
\begin{bmatrix}
a_\textbf{k}^\dagger, & b_\textbf{k}^\dagger, & a_{-\textbf{k}}, & b_{-\textbf{k}} \\
\end{bmatrix}$ and
\begin{equation} \label{HBdGG}
\mathcal{H}_{\text{BdG}}(\textbf{k}) =
\begin{bmatrix}
h(\textbf{k}) & \Delta(\textbf{k}) \\
\Delta^*(-\textbf{k}) & h^*(-\textbf{k}) \\
\end{bmatrix},
\end{equation}
is a $4 \times 4$ Bogoliubov de Gennes (BdG) Hamiltonian.
Here, $h(\textbf{k})$ and $\Delta(\textbf{k})$ are $2\times2$ matrices satisfying $h^\dagger(\textbf{k}) = h(\textbf{k})$ and $\Delta^T(\textbf{k}) = \Delta(-\textbf{k})$. Introducing
\begin{equation}\label{pmmats}
\begin{split}
\tilde{\Lambda}_n^\pm &= \tilde{\Lambda}_n^{xx} \pm \tilde{\Lambda}_n^{yy} + i (\tilde{\Lambda}_n^{yx} \mp \tilde{\Lambda}_n^{xy}), \\
\tilde{\Xi}_{n,s}^\pm &= \tilde{\Xi}_{n,s}^{xx} \pm \tilde{\Xi}_{n,s}^{yy} + i (\tilde{\Xi}_{n,s}^{yx} \mp \tilde{\Xi}_{n,s}^{xy}),
\end{split}
\end{equation}
the submatrices $h$ and $\Delta$ can be written explicitly as
\begin{equation}\label{hmatrix}
\begin{split}
h_{11}(\textbf{k}) &= g \mu_B B \cos \theta_A + \frac{6 A S}{2} \cos^2 \theta_A - A S \\
& - S\sum\limits_{n=1}^3[ \tilde{\Lambda}_n^{zz} + 2\tilde{\Xi}_{n,A}^{zz} - \text{Re} (\tilde{\Xi}_{n,A}^{+} e^{i \textbf{k} \cdot \boldsymbol \beta_n})], \\
h_{22}(\textbf{k}) &= g \mu_B B \cos \theta_B + \frac{6 A S}{2} \cos^2 \theta_B - A S \\
& - S\sum\limits_{n=1}^3[ \tilde{\Lambda}_n^{zz} + 2\tilde{\Xi}_{n,B}^{zz} - \text{Re} (\tilde{\Xi}_{n,B}^{+} e^{i \textbf{k} \cdot \boldsymbol \beta_n})], \\
h_{12}(\textbf{k}) &= \frac{S}{2}\sum\limits_{n=1}^3 \tilde{\Lambda}_n^+ e^{-i \textbf{k} \cdot \boldsymbol\alpha_n}, \; \; \; \; h_{21}(\textbf{k}) = h_{12}^*(\textbf{k}), \\
\end{split}
\end{equation}
and
\begin{align}\label{dmatrix}
\Delta_{11}(\textbf{k}) &= -AS \sin^2 \theta_A, \; \; \; \; \Delta_{22}(\textbf{k}) = -AS \sin^2 \theta_B\,, \\
\Delta_{12}(\textbf{k}) &= \frac{S}{2}\sum\limits_{n=1}^3 \tilde{\Lambda}_n^- e^{-i \textbf{k} \cdot \boldsymbol\alpha_n}, \; \; \;
\Delta_{21}(\textbf{k}) = \Delta_{12}(-\textbf{k})\,.
\end{align}
Since the system is bosonic, the Hamiltonian $\mathcal{H}_{\text{BdG}}(\textbf{k})$ must be diagonalized by a paraunitary BdG transformation~\cite{kohei2019,shindou2013,kondo2020h}. In other words, one should diagonalize the effective Hamiltonian
\begin{align}
\tilde{\mathcal{H}}(\textbf{k}) = \Sigma_z \mathcal{H}_{\text{BdG}}(\textbf{k})\,, \; \; \; \Sigma_z = \sigma_z \otimes 1_{2 \times 2}\,,
\label{effH}
\end{align}
where we have introduced the third Pauli matrix $\sigma_z$ and the $2 \times 2$ identity matrix $1_{2 \times 2}$. We label the ${M=2}$ positive eigenvalues and associated eigenvectors of $\tilde{\mathcal{H}}(\textbf{k})$ as, respectively, $\mathcal{E}_m (\textbf{k})$ and $\ket{m(\textbf{k})}$. The remaining $M$ states with negative eigenvalues $-\mathcal{E}_{m}(-\textbf{k})$ are an artifact of doubling the degrees of freedom and can be discarded.
\section{Topological magnons}
\subsection{Topological classification}
The topological classification of the Hermitian matrix $\mathcal{H}_{\text{BdG}}(\textbf{k})$ reduces to the classification of the effective Hamiltonian $\tilde{\mathcal{H}}(\textbf{k})$, which is generally non-Hermitian~\cite{kohei2019, lieu2018}. However, the Hermiticity of the physical system guarantees that the effective matrix $\tilde{\mathcal{H}}(\textbf{k})$ has a built-in pseudo-Hermiticity symmetry, i.e.
\begin{equation} \label{PHH}
\eta^{-1} \tilde{\mathcal{H}}^{\dagger}(\textbf{k}) \eta = \tilde{\mathcal{H}}(\textbf{k}), \\ \quad \eta = \Sigma_z.
\end{equation}
\begin{table*}
\centering
\begin{tabularx}{\textwidth}{| Z | Z | Z | Z | Z | Z |}
\hline\hline
$S = 3/2$ & $g \mu_B B = 0.25$ & $A = 0.22$ & $J_1 = 1.2$ & $J_2 = 1.5$ & $J_3 = 1.8$ \\
\hline
$K_1 = 0.7$ & $K_2 = 0.5$ & $K_3 = 1.1$ & $D_1 = 0.2$ & $D_2 = 0.3$ & $D_3 = 0.6$ \\
\hline
$J'_{1,A} = 0.2$ & $J'_{2,A} = 0.4$ & $J'_{3,A} = 0.2$ & $J'_{1,B} = 0.1$ & $J'_{2,B} = 0.3$ & $J'_{3,B} = 0.4$ \\
\hline
$D'_{1,A} = 0.4$ & $D'_{2,A} = 0.2$ & $D'_{3,A} = 0.25$ & $D'_{1,B} = 0.5$ & $D'_{2,B} = 0.15$ & $D'_{3,B} = 0.05$ \\
\hline
$\tilde{\theta}_{1,A} = -0.17$ & $\tilde{\theta}_{2,A} = -0.07$ & $\tilde{\theta}_{3,A} = 0.22$ & $\tilde{\theta}_{1,B} = -0.37$ & $\tilde{\theta}_{2,B} = -0.47$ & $\tilde{\theta}_{3,B} = -0.57$ \\
\hline
$\tilde{\phi}_{1,A} = 0.3$ & $\tilde{\phi}_{2,A} = -0.8$ & $\tilde{\phi}_{3,A} = 0.2$ & $\tilde{\phi}_{1,B} = 0$ & $\tilde{\phi}_{2,B} = 0$ & $\tilde{\phi}_{3,B} = 0$ \\
\hline\hline
\end{tabularx}
\caption{Parameters used in the numerical diagonalization of $\tilde{\mathcal{H}}(\textbf{k})$~(\ref{effH}). All energy scales are in meV and angles are in radians.}
\label{table:values}
\end{table*}
Furthermore, the Hamiltonian $\tilde{\mathcal{H}}(\textbf{k})$ obeys particle-hole symmetry (PHS), i.e.
\begin{equation} \label{PHS}
\mathcal{C} \tilde{\mathcal{H}}^T(\textbf{k}) \mathcal{C}^{-1} = -\tilde{\mathcal{H}}(-\textbf{k}), \\ \quad \mathcal{C} = \sigma_y \otimes 1_{2\times 2}\,.
\end{equation}
However, as discussed in detail by Refs.~\cite{kohei2019,lieu2018,lein2019}, for free bosons, particle-hole symmetry should be regarded as a built-in constraint of the Bogoliubov-de-Gennes Hamiltonian~\eqref{HBdGG}, rather than as a physical symmetry that can be selectively broken. Thus, the topological classification of $\tilde{\mathcal{H}}(\textbf{k})$ should effectively neglect Eq.~(\ref{PHS}).
When $\tilde{\Xi}^{xy}_{n,s} = \tilde{\Xi}^{yx}_{n,s} = 0$ and $\tilde{\Lambda}_n^{xy} = \tilde{\Lambda}_n^{yx} = 0$, the magnon Hamiltonian obeys time-reversal symmetry, i.e.
\begin{equation} \label{TRS}
\mathcal{T} \tilde{\mathcal{H}}^*(\textbf{k}) \mathcal{T}^{-1} = \tilde{\mathcal{H}}(-\textbf{k}), \\ \quad \mathcal{T} = 1_{4 \times 4}\,.
\end{equation}
Generally, Eq.~(\ref{TRS}) holds in the absence of Kitaev or DM interactions, i.e. when $D'_{n,s} = D_{n} = K_{n} = 0$ for each $n$.
In this case, the Hamiltonian belongs to the symmetry class $AI + \eta_{+}$~\cite{kohei2019}, which corresponds to a topologically trivial phase.
In the presence of finite Kitaev or DM interaction, the relevant symmetry class is $A + \eta$~\cite{kohei2019}, which supports a topologically nontrivial phase characterized by a nonvanishing Chern number~\cite{shindou2013}. The (bosonic) Chern number of the $m$th band can be written as
\begin{equation} \label{chernc}
c_m = \frac{1}{2\pi}\int\limits_{BZ} d^2 \mathbf{k} \; \Omega_m^z (\textbf{k}),
\end{equation}
where
\begin{equation} \label{berrycurvature}
\boldsymbol\Omega_m (\textbf{k}) = \nabla_\mathbf{k} \times i \bra{m(\textbf{k})} \nabla_\mathbf{k} \ket{m(\textbf{k})},
\end{equation}
is the Berry curvature on the $m$th band.
\begin{figure*}[ht]
\centering
\includegraphics[width=1.0\linewidth]{fig3.pdf}
\caption{
(a) Spin-wave dispersion.
(b) The $z$-component $\Omega^{z}_{1}(\mathbf{k})$ of the Berry curvature~(\ref{berrycurvature}). The 1st Brillouin zone is indicated by a white hexagon. The local maxima of the Berry curvature are shifted off of the high symmetry points $\textbf{K}$ and $\textbf{K}'$ of the Brillouin zone due to $C_{3}$ symmetry breaking.
(c-d) Exact diagonalization of Eq.~(\ref{HBdGG}) in a ribbon geometry with zigzag and armchair terminations, respectively, and 30 unit cells width. In both cases, the spectrum displays two topologically-protected edge states (blue and red line). Two bulk states are also highlighted in orange.
(e-f) The eigenstates of the highlighted modes in (c-d) are shown. The edge states are exponentially confined to the top and bottom of the sample, whereas the bulk states are delocalized Bloch states.
}
\label{fig:edgestates}
\end{figure*}
\begin{figure*}[ht!]
\centering
\includegraphics[width=1.0\textwidth]{fig4.pdf}
\caption{
(a) Thermal Hall conductivity~(\ref{thermalhalleq}) as a function of temperature.
(b) The contribution of the lower band to the thermal Hall conductivity along a path of high symmetry in the BZ at various temperatures. In the subplots (c-f) the temperature is set at $T = 0.86$ meV and the $x$-axis of each subplot is a ratio of a parameter value to its initial value obtained in Table~\ref{table:values}, indicated by the superscript (0). The spins equilibrium positions are recalculated for each data point. Dependence of the thermal Hall conductivity~(\ref{thermalhalleq}) on the (c) magnetic field $B$;
(d) NN DM magnitude $D_{n}$ ($D_{n}$ is increased for $n=1,2,3$, i.e. $D_{n}/D^{(0)}_{n}$ is equal for each bond);
(e) NNN DM magnitude $D'_{n,s}$;
(f) NN Heisenberg exchange strength $J_{1}$.
}
\label{fig:thermalhall}
\end{figure*}
\subsection{Topological edge states}
Using the values in Table \ref{table:values}, the minimization of Eq.~\eqref{eclassical} by direct gradient descent yields the spin equilibrium positions $\theta_A \approx 0.41$, $\theta_B \approx 0.39$, $\phi_A \approx 0.18$, and $\phi_B \approx 0.18$. The bands acquire a nonzero Chern number, i.e. $c_{m}=\pm 1$ for $m=1(2)$.
We find that NN, NNN DM and Kitaev interactions can break time-reversal symmetry and open Chern-insulating gaps in the magnon spectrum. Figure \ref{fig:edgestates}(a) shows the gapped spectrum for the parameters of Table \ref{table:values}. Due to the lack of $C_{3}$ rotation symmetry, the Dirac nodes are not globally stable and the local maxima of the Berry curvature are shifted off the high symmetry point $\textbf{K}$ and $\textbf{K}'$, as shown in Fig.~\ref{fig:edgestates}(b). By varying the anisotropy of our parameters, we find that the two Dirac nodes can meet up and annihilate at the $\textbf{M}$ point.
The open boundary condition spectrum that results from exact diagonalization of Eq.~(\ref{effH}) in a ribbon geometry with zig-zag and armchair edges are presented in Fig.~\ref{fig:edgestates}(c-d). Two topologically-protected dispersive magnon modes, localized at the edges of the ribbon (see Fig.~\ref{fig:edgestates}(e-f)), emerge as consequence of the topologically nontrivial character of the magnon bands.
\setlength{\extrarowheight}{2pt}
\subsection{Thermal Hall effect}
It is well known that a temperature gradient can induce a magnon transverse heat current in systems with topologically nontrivial magnon bands~\cite{katsura2010,onose2010,laurell2018,owerre2016,matsumoto2014,mook2014,moulsdale2019}. The (intrinsic) magnon thermal Hall conductivity can be calculated as~\cite{murakami2017}
\begin{equation} \label{thermalhalleq}
\kappa_{xy}(T) = - \frac{T}{4 \pi^2} \sum\limits_{m = 1}^2 \int\limits_{BZ} d^2 \mathbf{k} \; \Omega_m^z (\textbf{k}) c_2 \left[ g_T(\mathcal{E}_m (\textbf{k})) \right],
\end{equation}
where $k_{B} = \hbar = 1$, $g_T(x) = (e^{x/T}-1)^{-1}$ is the Bose-Einstein distribution function and
\begin{equation} \label{c2}
c_2(x) = (1 + x) \left[ \log{\left( \frac{1+x}{x} \right) } \right]^2 - \left( \log{x} \right)^2 - 2 \text{Li}_2 (-x).
\end{equation}
Here, $\text{Li}_s (z)$ is the polylogarithm of order $s$ and argument $z$.
Figure~\ref{fig:thermalhall}(a) shows that, at low temperature, $\kappa_{xy}(T)$ displays a surprising change of sign. The sign change can be understood by rewriting Eq.~\eqref{thermalhalleq} as
\begin{equation}
\begin{split}
\kappa_{xy}(T) &= - \frac{T}{4 \pi^2} \sum\limits_{m = 1}^2 \int\limits_{BZ} d^2 \mathbf{k} \; \tilde{\kappa}_{m}(\textbf{k}), \\
\tilde{\kappa}_{m}(\textbf{k}) &= \Omega_m^z(\textbf{k})c_2\left[g_T(\mathcal{E}_n(\textbf{k}))\right].
\end{split}
\end{equation}
Here, $\tilde{\kappa}_m(\textbf{k})$ is proportional to the contribution to $\kappa_{xy}(T)$ from the $m$th band at the momentum $\textbf{k}$. Since $c_2$ is positive and monotonically increasing, the sign of $\tilde{\kappa}_m(\textbf{k})$ depends only on $\Omega_m^z(\textbf{k})$.
For the lower magnon band, the Berry curvature $\Omega_1^z(\textbf{k})$ has negative sign in the neighborhood of the $\boldsymbol \Gamma$ point, while it is positive around the gap-closing points near $\textbf{K}$ and $\textbf{K}'$. At lower temperatures, only states in the lower band in the vicinity of the $\boldsymbol \Gamma$ point are
populated. The factor of $c_2[g_T(\mathcal{E}_n(\textbf{k}))]$ suppresses finite contribution to $\tilde{\kappa}_1(\textbf{k})$ at reciprocal lattice points except those close to $
\boldsymbol \Gamma$. As $T$ increases, the states at the gap-closing points near $\textbf{K}$ and $\textbf{K}'$ become populated and, due to their large negative Berry curvature, come to dominate $\tilde{\kappa}_1(\textbf{k})$. This leads to the sign change of the thermal Hall conductivity $\kappa_{xy}(T)$ at $T \approx 0.7$ meV, shown in Fig.~\ref{fig:thermalhall}(b).
Another sign change in the thermal Hall conductivity $\kappa_{xy}$ occurs when the magnitude of the magnetic field is increased, as depicted in Fig.~\ref{fig:thermalhall}(c). Increasing the magnetic field yields to an overall shift of the bands to higher energies. As a result, states that once populated the region near $\textbf{K}$ become energetically unfavorable while states near $\boldsymbol \Gamma$ remain populated, thus causing the sign of $\kappa_{xy}$ to change.
The influence of the NNN and NN DM interaction on the thermal Hall flow is depicted, respectively, in Fig.~\ref{fig:thermalhall}(d) and Fig.~\ref{fig:thermalhall}(e). The NN (NNN) DM interaction change both the matrix elements of $\Lambda_n$ ($\Xi_{n,s}$) as well as the ground state configuration, which in turn modifies the overall structure of $\boldsymbol \Omega_m$ and $\mathcal{E}_m$ . The result is that $\tilde{\kappa}_1$ near $\boldsymbol \Gamma$, which is the primary contribution to $\kappa_{xy}$, increases with $D'_{n,s}/D^{\prime (0)}_{n,s}$ and decreases with $D_{n}/D^{(0)}_{n}$. Increasing either DM magnitude further causes the ground state to leave the uniform regime and our earlier assumption of two-sublattice translational symmetry breaks down.
In Fig.~\ref{fig:thermalhall}(f), the NN Heisenberg exchange along $\boldsymbol \alpha_1$ is increased. Initially, this leads to $\kappa_{xy}$ increasing, but around $J_1/J_1^{(0)} \approx 3$, the anisotropy becomes high enough to push the Dirac nodes together at $\textbf{M}$, where they annihilate, and the system enters a topologically trivial phase.
\begin{figure}[t!]
\includegraphics[width=0.5\textwidth]{fig5.pdf}
\caption{
Ground state spin textures obtained by MCMC. Each plot shows the classical spin moments projected onto the $xy$ plane, where blue lines indicate a positive $z$-component and red lines indicate a negative $z$-component.
(a) The ground state for the values given in Table~\ref{table:values}.
(b) A spin cycloid.
(c-d) Bloch and N\'{e}el skyrmions, respectively. }
\label{fig:montecarlo}
\end{figure}
\section{Monte Carlo simulations}
Throughout our discussion, we have focused on a ground state with a two-sublattice translational symmetry and we have shown that the symmetry-breaking interactions, i.e., NN and NNN DM and Kitaev, can give rise to topologically nontrivial spin-wave bands. In this last section, we show that changing the strength and/or the anisotropy of the symmetry-breaking spin interactions can yield spin textures that have a nontrivial real-space topology.
The large parameter space allows for a wide variety of noncollinear ground states that can be accessed by Markov-Chain Monte Carlo (MCMC)~\cite{xu2020, liang2020}, which we have used to verify that the values given in Table~\ref{table:values} correspond to a two-sublattice ground state.
Taking a 20$\times$20 lattice subject to periodic boundary conditions, we perform annealed Metropolis MCMC followed by gradient descent, guaranteeing that the solution is at least a local minima (metastable state), if not the true ground state. In Fig.~\ref{fig:montecarlo}(a), we show the ground state using values obtained in Table~\ref{table:values} has a two-sublattice periodicity; the polar and azimuthal angles of the spin moments agree to within 1\% of those obtained by gradient descent.
In the remaining figures, we explore other parameter regimes. Fig~\ref{fig:montecarlo}(b) shows a spin cycloid, while Fig~\ref{fig:montecarlo}(c-d) show Bloch and N\'{e}el skyrmions, which emerge when there is a strong enough NNN or NN DMI, respectively \cite{yu2010, heinze2011, rler2006, kzsmrki2015}.
\section{Conclusions}
In this work, we have constructed a model for a CrClBrI monolayer, though an appropriate choice of parameters reduces our model to a generic two-sublattice translationally symmetric CrCl$_{3-x-y}$Br$_x$I$_y$ monolayer. Focusing on a linear spin-wave regime and on a ground state with a sublattice unit cell, we have shown that (both NN and NNN) DMI and the Kitaev interactions can drive the system into a magnon Chern insulating phase. The topologically-protected magnon edge states associated with nonvanishing Chern numbers yield a thermal Hall effect. We find that the sign of the thermal Hall conductivity can be controlled by tuning temperature and external magnetic fields.
Finally, we show that our spin model can support a variety of ground states depending on the choice of parameters, including magnetic topological defects. Chromium trihalides have been also proposed as possible hosts of quantum spin liquids (QSL) \cite{xu2018,xu2020:kitaev}. However, the experimental results of Tartaglia \textit{et al.} \cite{tartaglia2020} show that CrClBrI has a frustration index of $f \sim 2$, which suggests that the magnetic interactions are not sufficiently frustrated to support a QSL ground state. It is also worth noting that, while it may be possible in principle for the model presented in the present work to support a QSL ground state, our Monte-Carlo simulations show that -- for the parameters considered in this analysis -- the ground state spin arrangement is not frustrated.
We hope that our results will stimulate systematic \textit{ab initio} and experimental investigations of the coupling strengths introduced in our model.
\section{Acknowledgments}
The authors thank F. Tafti for insightful discussions.
\bibliographystyle{prsty}
|
1,108,101,563,225 | arxiv | \section{Introduction}
\label{sec:intro}
As a rare example of a class of exactly-solvable, nonequilibrium interacting
particle systems, asymmetric simple exclusion processes of various types have found
favour with researchers in statistical mechanics, mathematical physics
and probability theory \cite{Liggett99,Derrida98,GM06,BE07}. These systems comprise
hard-core particles hopping in a preferred direction on a one-dimensional lattice and have been
used to describe systems as diverse as traffic flow \cite{Popkov01},
the dynamics of ribosomes \cite{Macdonald68} and molecular motors
\cite{AGP99,KL03},
the flow of hydrocarbons through a zeolite pore \cite{Chatterjee09},
the growth of a fungal filament \cite{Sugden07}, and in queueing
theory \cite{Martin07,EFM09}. The physical interest lies mainly in the rich
phase behaviour that arises as a direct consequence of being driven
away from equilibrium.
Of particular interest are the open boundary cases. Here the
system can be thought of as being placed between two boundary
reservoirs, generally of different densities. The two reservoirs
enforce a steady current across the lattice, and therewith a
nonequilibrium steady state. It was first argued by Krug
\cite{Krug91} that as the boundary densities are varied nonequilibrium
phase transitions may occur in which steady state bulk
quantities---such as the mean current or density---exhibit
nonanalyticities. Such phase transitions were seen explicitly in
first a mean-field approximation \cite{Macdonald68,DDM92} and then in
the exact solution \cite{DEHP93,Schutz93} of the totally asymmetric exclusion process with open boundaries,
hereafter abbreviated as TASEP, and of related processes \cite{BE07}. Since then arguments have been developed
to predict the phase diagram of more general one-dimensional driven
diffusive systems \cite{Popkov99,HKPS01,MB05}.
Our interest in this work is in the distinction between two phase diagrams for the TASEP:
one of which characterises the steady-state behaviour, and the other the dynamics. The static phase diagram \cite{DEHP93,Schutz93} is shown in
Fig.~\ref{fig:spd} where the left boundary density is $\alpha$ and the right is $1-\beta$.
There are three possible phases in the steady state: a low density (LD) phase where the bulk density is
controlled by the left boundary and is equal to $\alpha$; a high
density (HD) phase where the bulk density is controlled by the right
boundary and is equal to $1-\beta$; a maximal current (MC) phase where the
bulk density is $\frac{1}{2}$. The high and low density phases are
further divided into subphases (e.g. LD1, LDII) according to the functional form of
the spatial decay of the density profile to the bulk value near the
non-controlling boundary. In these subphases the lengthscale over which the decay is
observable remains finite as the system size is increased. This change
at the subphase boundaries in the form of the decay
is thus not a true phase transition in the thermodynamic sense.
\begin{figure}
\centering
\subfigure[]{\label{fig:spd}%
\includegraphics[width=0.45\linewidth]{StationaryPhaseDiagram}}
\subfigure[]{\label{fig:dynpd}%
\includegraphics[width=0.45\linewidth]{DynamicalPhaseDiagram}}
\caption{ \subref{fig:spd} Static and \subref{fig:dynpd} dynamic phase diagrams for the TASEP. Solid lines indicate thermodynamic phase transitions at which the current and bulk density are nonanalytic. Dotted lines indicate subphase boundaries. In the static case, the density profile a finite distance from the boundary is nonanalytic. In the dynamic case, the longest relaxation time exhibits a nonanalyticity.}
\end{figure}
More recently, de~Gier and Essler have performed an exact
analysis of the ASEP's dynamics \cite{deGier05, deGier06, deGier08}
and derived the the longest relaxation times of the system by
calculating the gap in the spectrum using the Bethe ansatz.
The analysis builds on directly related work on the XXZ spin chain with nondiagonal
boundary fields \cite{Neopmechiel03a, Neopmechiel03b}.
The dynamical phase diagram they obtain, in which phases are associated
with different functional forms of the longest relaxation time, is illustrated in figure
\ref{fig:dynpd}. The same phases (high density, low density and
maximal current) are found as in the static phase diagram, however, a
hitherto unexpected \emph{dynamical transition} line which subdivides the
low density and high density phases is now apparent. This line which
we shall refer to as the de Gier--Essler (dGE) line replaces the subphase
boundaries at $\alpha=1/2, \beta <1/2$ $\beta=1/2, \alpha <1/2$ in the
static phase diagram and, for example, subdivides the low density region into LDI$'$, LDII$'$.
Although there is nothing to rule out the prediction of dGE of a
dynamical transition at a distinct location to any static transition,
the result came as something of a surprise. This is because the static
phase diagram had been successfully interpreted in terms of an
effective, dynamical theory thought to be relevant for late-time
dynamics, referred to as domain wall theory (DWT). We shall review
DWT more fully in Section~\ref{subsec:DWT}. For the moment we note
that DWT correctly predicts the static subphase boundary and the associated
change in the decay of the density profile, but does not predict the
dGE line. Therefore, it was previously thought that a dynamical transition also
occurred at the subphase boundary and initial calculations of
relaxation times appeared to be consistent with this
\cite{DS00,Nagy02}.
Our primary goal in this work is to establish numerically that a dynamical
transition does indeed occur along the dGE line rather than at the
static subphase boundary. We used two distinct approaches. First, we carried
out direct Monte Carlo simulations of the TASEP dynamics and identified
the dominant transient in three time-dependent observables. We find that these
simulations are consistent with a dynamical transition at the dGE line
but are not sufficiently accurate to rule out the scenario of the
dynamical transition occurring at the subphase boundary.
We instead turn to a Density Matrix Renormalisation Group (DMRG) approach
\cite{DMRGBook} to acquire convincing evidence that a dynamical transition
occurs at the dGE line. The DMRG is an approximate means to obtain the
lowest-lying energies and eigenstates of model Hamiltonians with short-range
interactions. Although originally developed in the context of the Hubbard
and Heisenberg models \cite{White92} (see also \cite{Schollwock05} for a
comprehensive review), it has also been applied to nonequilibrium processes
such as reaction-diffusion models \cite{Carlon99}. In particular, it has
also been used to investigate the spectrum of the TASEP \cite{Nagy02}.
However, the dynamical transition was not evident from the data presented in
that work---perhaps because it had not been predicted at that time. By
revisiting this approach, we obtain estimates of the longest relaxation time
that allow us to rule out the domain wall theory prediction for the
dynamical transition.
The paper is organised as follows. We begin in Section~\ref{sec:moddef}
by recalling the definition of the TASEP, and by reviewing in more detail the
static and dynamic phase diagrams discussed above. Then, in Section~\ref{sec:sims},
we present our Monte Carlo simulation data, followed by the DMRG results in Section~\ref{sec:dmrg}. In Section~\ref{sec:discuss} we return to more general questions about the distinction between the static and dynamic phase diagram with particular reference to different theoretical approaches to the TASEP. We conclude in Section~\ref{sec:conclude} with some open questions.
\section{Model definition and phase diagrams}
\label{sec:moddef}
Although we have alluded to the basic properties of the totally asymmetric simple exclusion
process (TASEP) in the introduction, in
the interest of a self-contained presentation we provide in this section a precise model
definition and full details of the static and dynamic phase diagrams.
\subsection{Model definition}
The TASEP describes the biased diffusion of particles on a
one-dimensional lattice with $L$ sites. No more than one particle can
occupy a given site, and overtaking is prohibited. The stochastic
dynamics are expressed in terms of transition rates, for example a
particle on a site $i$ in the bulk hops to the right as a Poisson
process with rate $1$, but only if site $i+1$ is unoccupied:
At the left boundary only particle influx takes
place, with rate $\alpha$, and at the right boundary only particle
outflux takes place, with rate $\beta$, as shown in Fig.~\ref{fig:asep}. The corresponding
reservoir densities are $\alpha$ at the left and $1-\beta$ at the right.
\begin{figure}
\begin{center}
\includegraphics[width=0.55\linewidth]{asep}
\end{center}
\caption{\label{fig:asep} The dynamics of the totally asymmetric simple exclusion process (TASEP) with open boundaries. Arrows show moves that may take place, and the labels indicate the corresponding stochastic rates.}
\end{figure}
\subsection{Static phase diagram}
As noted in the introduction, the TASEP static phase diagram can be
determined from, for example, exact expressions for the current and density
profile in the steady state \cite{DEHP93,Schutz93}. The current takes
three functional forms according to whether it is limited by a slow insertion
rate (LD: $\alpha < \beta, \alpha < \frac{1}{2}$), by a slow exit rate
(HD: $\beta < \alpha, \beta < \frac{1}{2}$) or by the exclusion interaction
in the bulk (MC: $\alpha>\frac{1}{2}, \beta>\frac{1}{2}$). This behaviour of the
current leads to the identification of the thermodynamic phases separated by
solid lines in Fig.~\ref{fig:spd}. Along these lines there are nonanalyticities
in both the current and the density far from either boundary. The relevant expressions for these quantities are shown on Fig.~\ref{fig:spd}.
The density \emph{near} one of the boundaries is nonanalytic along the lines
$\alpha=\frac{1}{2}$ and $\beta=\frac{1}{2}$ in the HD and LD phases respectively. These
subphases are shown dotted in Fig.~\ref{fig:spd}. Inspection of the functional form
of the density profile reveals that these are not true thermodynamic phase transitions,
in the sense that the deviation from the bulk value extends only a finite distance into the bulk,
and thus contributes only subextensively to the nonequilibrium analogue of a free energy (see e.g., \cite{BE02} for the definition of such a quantity).
To be more explicit, consider for example the behaviour
near the right boundary in the low-density phase. Here the bulk density is $\rho=\alpha$. The
mean occupancy of the lattice site positioned a distance $j$ from the
right boundary approaches in the thermodynamic limit $L\to\infty$ the
form \cite{Schutz93, DEHP93, BE07}
\begin{equation}
\label{eq:rhoLD}
\rho_{L-j} \sim \left\{ \begin{array}{l@{\qquad}l@{\qquad}l}
\alpha + c_{\rm I}(\beta) \left( \frac{\alpha(1-\alpha)}{\beta(1-\beta)} \right)^{j} & \alpha < \beta < \frac{1}{2} & \mbox{(LDI)}\\
\alpha + c_{\rm II}(\alpha, \beta) \frac{ [ 4 \alpha(1-\alpha) ]^{j} }{j^{3/2}} & \frac{1}{2} < \beta & \mbox{(LDII)}
\end{array}\right. \;.
\end{equation}
In these expressions, $c_{\rm I}$ and $c_{\rm II}$ are functions of the boundary
parameters that we leave unspecified here so as to focus on the lengthscale of the
exponential decay from the right boundary. As $\beta$ is increased from zero, the decay length increases until
$\beta=\frac{1}{2}$. Then the decay length is constant, and the exponential is modulated by a power law. The
density profile at the left boundary exhibits the same kind of nonanalyticity
in the high-density phase as $\alpha$ is increased through
$\frac{1}{2}$ as a consequence of the particle-hole symmetry,
$\rho_{i-1}(\alpha,\beta) = 1-\rho_{L-i}(\beta,\alpha)$, exhibited by
the model.
\subsection{Dynamical phase diagram}
\label{sec:dynpd}
The dynamic phase diagram is obtained by examining a quantity referred to as the \emph{gap} by de~Gier and Essler \cite{deGier05,deGier06,deGier08}. This is simply the largest nonzero eigenvalue of the transition matrix governing the continuous-time stochastic dynamics of the TASEP. More formally, one starts with the master equation
\begin{equation}
\label{eq:me}
\frac{{\rm d}}{{\rm d}t} |P(t) \rangle = M |P(t) \rangle \;,
\end{equation}
where the matrix $M$ encodes the transition rates and $|P(t)\rangle$ is the vector of probabilities for each configuration. Because $M$ is a stochastic matrix it is guaranteed \cite{Altenberg02} by the Perron-Frobenius theorem to have eigenvalues satisfying
\begin{equation}
\lambda_0 = 0 \, > \, {\rm Re}(\lambda_1) \,\geq\, {\rm Re}(\lambda_2) \geq \ldots \;.
\end{equation}
The spectrum corresponds to a set of relaxation times $\tau_i = -({\rm Re}\, \lambda_i)^{-1}$. The longest (non-infinite) relaxation time is $\tau_1$, and the associated eigenvalue $\lambda_1$ is the gap which we will henceforth denote by the symbol $\varepsilon$. At long times the relaxation of any observable is expected to decay exponentially with a characteristic timescale $\tau_1$, a fact we will later use to estimate the gap from Monte Carlo simulations.
The thermodynamic phase boundaries are found by identifying lines along which the gap vanishes, indicating the coexistence of two stationary eigenstates (phases). The exact calculations of de~Gier and Essler \cite{deGier05,deGier06} show that, in the thermodynamic limit $L\to\infty$, the gap vanishes along the
line $\alpha=\beta<\frac{1}{2}$ that separates the HD and LD phases. The gap also vanishes in the entirety of the MC phase, reflecting the generic long-range (power-law) correlations seen in this phase \cite{DEHP93}. Thus at this level, the static and dynamic phase diagrams coincide.
Where they differ is in the subdivision of the high- and low-density phases\footnote{de~Gier and Essler \cite{deGier05,deGier06,deGier08} refer to these as ``massive'' phases by analogy with the quantum spin chains; we shall only use the terminology associated with the static phase diagram to avoid confusion.}, in which the gap remains finite in the limit $L\to\infty$. There is a region, marked LDI$'$ and HDI$'$ on Fig.~\ref{fig:dynpd}, within which the gap assumes the asymptotic expression
\begin{equation}
\label{eq:gap_MI}
\varepsilon(L) = -\alpha - \beta + \frac{2}{(ab)^{\frac{1}{2}} + 1} - \frac{\pi^2}{ (ab)^{\frac{1}{2}} - (ab)^{-\frac{1}{2}}} L^{-2} + \mathcal{O}(L^{-3})
\end{equation}
in which
\begin{equation}
a = \frac{1-\alpha}{\alpha} \quad \mbox{and}\quad b = \frac{1-\beta}{\beta} \;.
\end{equation}
Within the low-density phase ($\alpha<\beta, \alpha<\frac{1}{2}$), this form of the gap applies for values of $\beta < \beta_c$ where
\begin{equation}
\label{eq:beta_c}
\beta_c(\alpha) = \left[1 +\left(\frac{\alpha}{1-\alpha}\right)^{\frac{1}{3}}\right]^{-1} \;.
\end{equation}
Likewise, in the high-density phase ($\beta<\alpha, \beta<\frac{1}{2}$), the region within which the gap is given by (\ref{eq:gap_MI}) is bounded by $\alpha<\alpha_c$ where
\begin{equation}
\label{eq:alpha_c}
\alpha_c(\beta) = \left[1 +\left(\frac{\beta}{1-\beta}\right)^{\frac{1}{3}}\right]^{-1} \;.
\end{equation}
In the remainder of the low-density phase, $\alpha<\frac{1}{2}, \beta>\beta_c$ the gap takes the form
\begin{equation}
\label{eq:gap_MIIa}
\varepsilon(L) = -\alpha - \beta_c + \frac{2}{(ab_c)^{\frac{1}{2}} + 1} - \frac{4\pi^2}{ (ab_c)^{\frac{1}{2}} - (ab_c)^{-\frac{1}{2}}} L^{-2} + \mathcal{O}(L^{-3})\;.
\end{equation}
Finally, we have by symmetry that when $\beta<\frac{1}{2}, \alpha>\alpha_c$,
\begin{equation}
\label{eq:gap_MIIb}
\varepsilon(L) = -\alpha_c - \beta + \frac{2}{(a_cb)^{\frac{1}{2}} + 1} - \frac{4\pi^2}{ (a_cb)^{\frac{1}{2}} - (a_cb)^{-\frac{1}{2}}} L^{-2} + \mathcal{O}(L^{-3}) \;.
\end{equation}
In these expressions,
\begin{equation}
a_c = \frac{1-\alpha_c}{\alpha_c} \quad\mbox{and}\quad b_c = \frac{1-\beta_c}{\beta_c} \;.
\end{equation}
The boundaries between the dynamic subphases---the de~Gier--Essler (dGE) lines---are shown dotted in Fig.~\ref{fig:dynpd}. Note that the coefficient of the $L^{-2}$ term in this asymptotic expansion is discontinuous across the dynamical transition line.
In order to illustrate the behaviour of the gap along the static and dynamic transition lines, we plot it as a
function of $\beta$ at some $\alpha < \frac{1}{2}$ in Fig.~\ref{fig:deGierGapInf}. The most striking feature is
the constancy of the gap above the nontrivial critical point $\beta_c$. Later in this work, we will use the constancy of the gap above a critical threshold as an empirical means to identify the dynamical transition point.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{deGierGapInf}
\caption{ \label{fig:deGierGapInf} Exact $L\to\infty$ gap (black, dashed) as a function of $\beta$ for $\alpha = 0.3$, described by the HDI$'$ equation~(\ref{eq:gap_MI}) (red) for $\beta < \beta_c\,$ ($\beta_c \approx 0.57$, indicated), and by equation~(\ref{eq:gap_MIIa}) (green) for $\beta > \beta_c$.}
\end{figure}
\subsection{Domain wall theory}
\label{subsec:DWT}
An alternative approach to study the TASEP dynamics is domain wall theory (DWT)~\cite{Kolomeisky98}. Although much simpler than the Bethe ansatz approach of de~Gier and Essler, it correctly predicts the static phase boundaries and the analytical form of the gap in a region of the phase diagram. However, it predicts dynamic subphases that are distinct from those found by de~Gier and Essler, and that correspond to the static subphases. In the numerical study that follows, it will be important to be able to distinguish between these two sets of predictions, and so we summarise DWT here.
The basis of this approach is to assume that collective relaxational dynamics
are effectively reducible to a single coordinate describing the position of an interface, or domain
wall. The wall separates a domain of density $\rho^-$ and current $j^-$ to
the left from a domain of density $\rho^+$ and current $j^+ =
\rho^+(1-\rho^+)$ to the right. Each domain is taken to possess the
steady state characteristics imposed by the boundary reservoir on
that particular side of the wall, with $\rho^+$ and $\rho^-$
therefore as in Fig.~\ref{fig:DWTPhaseDiagram}.
The effective theory is expected to be exact along the line $\alpha= \beta <1/2$ where
the exact properties of the system such as
density profile \cite{DEHP93,Schutz93} and current fluctuations \cite{DEM95} are recovered.
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{DWTPhaseDiagram}
\caption{\label{fig:DWTPhaseDiagram} The phase diagram obtained from domain wall theory. $\rho^{+}$ and $\rho^{-}$ indicate the densities of the left and right domains. In the domain wall theory, static and dynamic transitions coincide along the dashed lines.}
\end{figure}
Let us consider the case $\alpha <1/2$ and $\beta <1/2$. The motion of the wall is then described microscopically as a random walker with left and right hopping rates $D^-$ and $D^+$ given respectively by imposing mass conservation on the fluxes into and out of the wall \cite{DS00}:
\begin{equation}
\label{D+D-}
D^- = \frac{j^-}{\rho^+ - \rho^-}= \frac{\alpha(1-\alpha)}{1-\alpha-\beta} \qquad D^+ = \frac{j^+}{\rho^+ - \rho^-}
=\frac{\beta(1-\beta)}{1-\alpha-\beta},
\end{equation}
For $\alpha > \beta$ the random walk is biased to the left
and in the stationary state the domain wall is localised at the left boundary
and the bulk density is given by $\rho^+=1-\beta$.
For $\alpha < \beta$ the random walk is biased to the right
and in the stationary state the domain wall is localised at the right boundary
and the bulk density is given by $\rho^-=\alpha$. Thus the first order transition at $\alpha= \beta$ is correctly predicted.
Moreover, the gap in the resulting spectrum for large $L$ given \cite{DS00} is given by
\begin{equation}
\label{eq:DWTGap}
\varepsilon_{DWT}= -D^+ - D^- + 2\sqrt{D^+D^-}\left(1-\frac{\pi^2}{2L^2} + O(L^{-3}) \right).
\end{equation}
Remarkably this expression is identical to (\ref{eq:gap_MI})
to order $1/L^2$. Thus in the region $\alpha<1/2, \beta <1/2$ DWT correctly predicts the gap.
When $\beta >1/2$ so that $\rho^+ = 1/2$, one can take $j^+$ in (\ref{D+D-}) to depend on the size $\ell$ of the right-hand domain. That is, one puts $j^+ \to j^+(\ell)$ equal to the stationary current in a TASEP of size $\ell$ in the maximal current phase. This implies the large $\ell$ behaviour
\begin{equation}
j^+ \simeq \frac{1}{4}(1 + \frac{3}{2\ell})
\label{lpmc}
\end{equation}
This results in a modification of the density profile to an exponential spatial decay modulated by a power law with power 3/2, similar to Eq. \ref{eq:rhoLD} (LDII).
In brief, the DWT is remarkably successful, correctly predicting the static phase diagram (including subphases), and the
exact thermodynamic gap function found by de~Gier and Essler in the region $\alpha<1/2$ and $\beta <1/2$. It differs in that the dynamic subphases are not given by the dGE lines, but the static subphase transition lines. Thus, the region within which the gap is constant is different in the two theories, as indicated by Fig.~\ref{fig:GapComparisonDWTExact}.
\begin{figure}
\centering
\includegraphics[width=0.66\linewidth]{GapComparisonDWTExact}
\caption{\label{fig:GapComparisonDWTExact} Comparison of the gap for $L \to \infty$ from domain wall theory (dashed line) with the exact gap (solid line) at $\alpha = 0.1$ (for which $\beta_c \approx 0.675$).}
\end{figure}
\section{Evidence for the dynamical transition from Monte Carlo simulations}
\label{sec:sims}
We now describe attempts to measure the gap in Monte Carlo simulations of the TASEP, with a particular interest in distinguishing between the dGE and DWT predictions. As noted previously, the ensemble average of any time-dependent observable ${\cal O}(t)$ should be dominated at late times by an exponential decay ${\rm e}^{-|\lambda_1| t}$ to its stationary value $\langle {\cal O} \rangle$. Recall that $\lambda_1$ is the largest nonzero eigenvalue of the matrix $M$ appearing in the master equation (\ref{eq:me}), and that all nonstationary eigenvalues have negative real part. In principle, therefore, all one needs to do is pick an observable, and measure its late-time decay rate. In practice, this is made difficult by the fact that \emph{all} decay modes may contribute to $\langle {\cal O}(t)\rangle$, and that by the time the higher modes have decayed away, the residual signal $\langle{\cal O}(t)\rangle - \langle {\cal O} \rangle$ may be extremely small and swamped by the noise.
Since we are interested in the time-dependence of an observable, we employ a \emph{continuous-time} (Gillespie) algorithm \cite{Gillespie77} to simulate the TASEP dynamics. More precisely, we maintain a list of events (i.e., a particle hopping to the next site, or entering or leaving the system) that can take place given the current lattice configuration. A particular event $i$ is then chosen with a probability proportional to its rate $\omega_i$ as specified in Section~\ref{sec:moddef}. A time variable is then incremented by an amount chosen from an exponential distribution with mean $( \sum_i \omega_i )^{-1}$. In this way, a member of the ensemble of all continuous-time trajectories of the TASEP dynamics from some prescribed initial condition is generated with the appropriate probability.
\subsection{Decay of total occupancy to stationarity}
\label{sec:decay}
We consider first the decay of the total number of particles on the lattice, $N(t)$, to its stationary value as measured by the function
\begin{equation}
\label{Rt}
R(t) = \frac{\langle N(t) \rangle - \langle N \rangle}{\langle N(0) \rangle - \langle N \rangle} \;.
\end{equation}
In this expression, angle brackets denote an average over an ensemble of initial conditions and over stochastic trajectories. In each simulation run, an initial condition was constructed in which each site was independently occupied with a probability $p=1-\beta$ (in the LD phase) or $p=\alpha$ (in the HD phase). In the bulk, these densities then relax to the steady-state values displayed on Fig.~\ref{fig:spd}.
Once $R(t)$ has been sampled over multiple trajectories, the task is to identify a time window over which one can fit an exponential decay and therewith estimate a gap. The start and end points of this windows are both crucial. If it starts too early, then one may expect contributions from subdominant transients (i.e., the decays at rate $\lambda_2, \lambda_3, \ldots$) to systematically skew the estimate of the gap. If it ends too late, noise may instead dominate the estimate.
The noise at the top end we handle by examining the behaviour of local decay rates
\begin{equation}
\mu_i = \frac{\ln R(t_{i+i}) - \ln R(t_{i})}{t_{i+1} - t_i}
\end{equation}
where $t_i$ and $t_{i+1}$ are successive time points at which $R(t)$ was sampled. At late times, one should find $\mu_{i+1}/\mu_{i} \to 1$. Strong deviations from unity indicate the dominance of noise, and we rejected points after which the magnitude of this ratio exceeded a critical value. For our data sets, we found that $5$ was a suitable choice for this value: see Fig.~\ref{fig:mu} for an example.
\begin{figure}
\centering
\subfigure[]{\label{fig:mu}%
\includegraphics[height=4.2cm]{DataConvergence}}
\subfigure[]{\label{fig:R2mc}%
\includegraphics[height=4.2cm]{RsqL50}}
\caption{Criteria for choosing the range of data over which to fit a single exponential to the decay of the density to stationarity. \subref{fig:mu} The ratio of gradients $\mu_{i+1}/\mu_i$ which should be close to unity where a single exponential is a good fit. At large times, this ratio becomes noisy; the presence of extreme values (here, a magnitude larger than 5) is used to identify the onset of noise. \subref{fig:R2mc} The behaviour of ${\cal R}^2$ as a function of the start of the window. We choose this such that ${\cal R}^2$ is maximised. In both cases, $L=50$, $\alpha=0.3$ and $\beta=0.64$.}
\end{figure}
The bottom end of the window was chosen by maximising a goodness-of-fit measure to a fit of the exponential $f(t) = a{\rm e}^{-\lambda t}$ to data points within the window. We adopted the \emph{adjusted coefficient of determination} \cite{Ryan1997}
\begin{equation}
\label{R2}
{\cal R}^2 = 1 - \frac{\frac{1}{n-k}\sum_{i} (R(t_i) - f(t_i))^2}{\frac{1}{n-1}\sum_{i} (R(t_i) - \bar{R})^2} \;.
\end{equation}
as our goodness-of-fit measure, varying the start of the window until this quantity was maximised. In this expression $\bar{R}$ is the arithmetic mean of $R(t)$ over the set of $n$ times $t_i$ falling within the window, and $k$ is the number of free parameters in the fit function $f(t)$, i.e., $k=2$. This goodness-of-fit measure trades off the increased quality of the fit obtained by discarding data points against the increasing uncertainty in the fitting parameters that comes with the noisier data at the top end of the window. We show in Fig.~\ref{fig:R2mc} an example of how the goodness-of-fit varies with the size of the window. We remark that the relative flatness of ${\cal R}^2$ above the optimal starting time can be ascribed to the fact that only a single decay mode remains in this region. Although this procedure is a little cumbersome to apply to each data set, we found it preferable to fitting multiple exponentials to the data for $R(t)$, which leads to large uncertainties in the measured gap.
We show results in Fig.~\ref{fig:mcgapfinite}, in which the gap is plotted as a function of $\beta$ for fixed $\alpha=0.3$ for a range of system sizes. At the smallest system sizes ($L=10$) data from Monte Carlo simulation are in agreement with results obtained through exact diagonalisation of the matrix $M$. As the system size is increased beyond the sizes for which exact techniques remain tractable, the curves approach those predicted by dGE and DWT. However, it is not possible to discern from the largest system simulated, $L=150$, which of these theories is more appropriate.
\begin{figure}
\begin{center}
\includegraphics[width=0.66\linewidth]{RgapsL40-150}
\end{center}
\caption{\label{fig:mcgapfinite} The gap as a function of $\beta$ at $\alpha=0.3$ and system sizes $L$ ranging from $L=40$ to $L=150$ (from bottom to top) obtained from Monte Carlo simulations. The $L\to\infty$ dGE (solid line) and DWT (dashed line) predictions are shown for comparison: even with the largest system $L=150$ simulated, one cannot distinguish between them numerically.}
\end{figure}
We therefore attempt to extrapolate to the limit $L\to\infty$ by fitting a function of the form $\varepsilon + a_2 L^{-2} + a_3 L^{-3} + a_4 L^{-4}$ to estimates of the gap at finite system sizes $L$ but fixed $\alpha$ and $\beta$. This particular fitting function was found to have the best goodness-of-fit (taking into account the varying number of parameters) out of those that were tried, and furthermore had the most uniform goodness-of-fit as a function of $\beta$. The results of this extrapolation procedure are shown in Fig.~\ref{fig:mcgap}. These data appear to exclude the functional form for the domain wall theory, although the uncertainties in the estimated gaps are such that further evidence is needed in order to confidently rule it out.
\begin{figure}
\begin{center}
\includegraphics[width=0.66\linewidth]{ExtrapolatedGap}
\end{center}
\caption{\label{fig:mcgap} The gap extrapolated to $L=\infty$ from the data shown in Fig.~\ref{fig:mcgapfinite}. Again the dGE (solid line) and DWT (dashed line) predictions are shown for comparison: here the data are potentially more compatible with the former than the latter.}
\end{figure}
\subsection{Decay of other observables}
One possible means to reduce the uncertainty in the measured gap is to try different observables ${\cal O}(t)$ in the hope that contributions from subdominant decay modes---and in particular the second longest-lived mode that decays at rate $\lambda_2$---are reduced and thereby allow a more accurate estimate of $\lambda_1$. We considered two candidates.
The first was the autocorrelation function of the total occupancy, ${\cal O}(t) = N(t_0) N(t_0+t)$, where $t_0$ is some fixed time point in the steady state. The analogue of the function $R(t)$ is
\begin{equation}
\chi(t) = \frac{\langle N(t_0+t)N(t_0) \rangle - \langle N \rangle^2}{\langle N^2 \rangle - \langle N \rangle^2}
\label{chi}
\end{equation}
where here all averages are taken in the steady state. Numerically, this quantity is slightly more convenient than $N(t)$, since one can obtain $\langle N(t_0+t)N(t_0) \rangle$ by sampling at different $t_0$ and $t$ from a single simulation run that has reached the steady state.
\begin{figure}
\begin{center}
\includegraphics[width=0.55\linewidth]{asep2cp}
\end{center}
\caption{\label{fig:2cp} Interactions between first class (circles) and second class (hexagons) particles. Second class particles hop can forward into empty sites and exchange places with first-class particles behind them. Note that in our simulations, at most one second-class particle was present at any instant, and that the behaviour at the boundaries was chosen so as not to affect the motion of first-class particles (see text).}
\end{figure}
The other observable we investigated was the position of a second class particle \cite{BCFG89,Ferrari91,DJLS93}. Like a first-class particle, a second-class particle hops to the right into vacant sites. However, it may also exchange places with a particle occupying the site to its left. Both of these processes take place at unit rate, and first-class hop to the right at the same rate, irrespective of whether the site in front of them is empty or occupied by a second-class particle---see Fig.~\ref{fig:2cp}. So as not to affect the entry of first-class particles into the system, a second-class particle on the left-boundary site is forced to exit if a first-class particle attempts to enter. It is reinserted as soon as the left-boundary site becomes vacant again. Second-class particles are prevented from exiting at the right boundary. In these simulations one can measure the position of the second class particle, $x(t)$, starting from an initial condition where each site is occupied by a first-class particle with density $\rho$ (as described above) except for the left boundary site, which is always initially occupied by the second-class particle. Here, the analogue of $R(t)$ is
\begin{equation}
\xi(t) = \frac{\langle x(t) \rangle - \langle x \rangle}{x(0) - \langle x \rangle} \;.
\label{scp}
\end{equation}
Gaps obtained from these two functions $\chi(t)$ and $\xi(t)$ are compared with those obtained from $R(t)$ in Fig.~\ref{fig:observables} at various system sizes. We see that there are no systematic differences between these functions at the system sizes studied, and therefore that they are unlikely to provide improved estimates of the gap in the thermodynamic limit $L\to\infty$. We therefore conclude that whilst Monte Carlo simulation data is possibly more consistent with the de~Gier--Essler prediction for the gap than the domain wall theory, the effect of the changing gap on the relaxation of a typical observable is very small and hard to disentangle from the noise.
\begin{figure}
\begin{center}
\includegraphics[width=0.66\linewidth]{ThreeObservables}
\end{center}
\caption{\label{fig:observables} The gap as a function of $\beta$ as obtained by consideration of the relaxation of three different observables. Solid lines show data obtained from the relaxation of the density for system sizes $L=10,20,30,50,150$ (bottom to top). Alongside the lower three curves are plotted data from the stationary density auto-correlation function (\ref{chi}) (dashed lines), and alongside the upper two data from the position of a single second-class particle (\ref{scp}) (large symbols). All three observables yield consistent results.}
\end{figure}
\section{Evidence for the dynamical transition from density matrix renormalisation}
\label{sec:dmrg}
Given the difficulties in measuring the gap from Monte Carlo simulations, we now turn to a fundamentally different numerical approach, namely the density matrix renormalisation group (DMRG) \cite{DMRGBook} that was briefly discussed in the introduction. We begin by recapitulating the essential features of the DMRG approach before demonstrating that it does indeed show a dynamical transition along the de~Gier--Essler line.
\subsection{DMRG algorithm for the TASEP gap}
\label{subsec:dmrg_algorithm}
The density matrix renormalisation group (DMRG) procedure builds an
approximation to the transition matrix $M$ that appears in the master
equation (\ref{eq:me}). The basic idea is to repeatedly add sites to the
system and renormalise the enlarged transition matrix so that its
dimensionality remains constant as the system size grows. In this
way, the approximated transition matrix remains sufficiently small that its
spectrum (and in particular the gap) can be determined using standard
numerical diagonalisation methods. The success of this procedure relies on
finding a reduced basis set at each renormalisation step that allows the
lowest-lying eigenstates to remain well approximated as sites are added.
Since the TASEP has open boundaries, the extension of the lattice is achieved in this instance by adding sites to the \emph{middle} of the system rather than at one of the ends \cite{Nagy02}. The system is thus divided into two `blocks': the left and right halves of the lattice. It is extended in each iteration by adding a site at the right-hand end of the left block, and the left-hand end of the right block. In order to understand the results that come out of the procedure, it is worth explaining in a little more detail how the renormalisation is actually achieved---full details are provided in \ref{app:dmrg}.
We outline the procedure in terms of the transformation applied to the left block. Suppose that at the start of the iteration, the block comprises $\ell$ lattice sites. The set of states the block can be in is specified by two `quantum' numbers, $p = 1,2 \ldots, m$ and $\sigma=0,1$. The quantity $p$ indexes the configuration of the first $\ell-1$ sites, and $\sigma$ the occupancy of the rightmost site. The first step of the renormalisation is to construct some basis in the $2m$-dimensional space spanned by $p$ and $\sigma$ and, crucially, retain only $m$ of these basis vectors. This defines a renormalised index $\tilde{p}=1,2,\ldots,m$ for the \emph{entire} $\ell$-site block. A new site is then added to the right hand side of the lattice. The configuration of this additional site is specified by the renormalised coordinate $\tilde{\sigma}$. See Fig.~\ref{fig:dmrgstep}. The reason for keeping the rightmost site in the `physical' (site occupancy) basis is that one needs to project the transition matrix for hops from site $\ell$ to $\ell+1$ onto the space spanned by $\tilde{p}$ and $\tilde{\sigma}$. These transition rates are initially only known in this physical basis. Thus through this procedure, we obtain a (truncated) representation of the transition matrix for an $\ell+1$-site system in a space of the same dimensionality as that used to describe the $\ell$-site system.
\begin{figure}
\centering
\includegraphics[width=0.4\linewidth]{renorm}
\caption{\label{fig:dmrgstep} Illustration of the DMRG procedure applied to the left block. Before the transformation, the state of the block is specified by $1\le p\le m$ and $\sigma=0,1$. After adding a new site to the end of the block, the original $\ell$ sites are renormalised so that they are specified by the two numbers $1\le \tilde{p} \le m$ and $\tilde{\sigma}=0,1$. The right block is a mirror image of the left, and treated the same way.}
\end{figure}
The same transformation is applied to the right block, the only difference being that it is the leftmost site that is expressed in the physical basis. Interactions between the two blocks enter through the construction of the renormalised indices $\tilde{p}$ as we now describe. First, a transition matrix $M'$ for the entire system of $2\ell$ sites is constructed by combining the matrices for the two halves, and by adding an interaction term that allows a particle in the left block to hop to the right block. Again, having the internal two sites specified in the physical basis helps here. This transition matrix is then diagonalised. However, it is not these eigenstates that are used to perform the renormalisation. Rather, a \emph{density matrix}, which is a symmetric combination of the two longest-lived eigenstates of $M'$, is constructed, and eigenvectors of this density matrix are instead used for the renormalisation. The idea is that this form of the density matrix allows the stationary and longest-lived transient state to be accurately represented in large systems. The specific form of the density matrix, and the prescription for obtaining the truncated basis set, is given in \ref{app:dmrg}. For further details about the principles behind DMRG and other applications we refer to \cite{DMRGBook,White98,Noack05}, and especially \cite{Schollwock05}.
\subsection{DMRG results}
\label{subsec:dmrg_results}
We used the DMRG algorithm outlined above to estimate the gap for a given combination of $\alpha$ and $\beta$ by starting with an exact diagonalisation of the $L=8$ system, and keeping $m=16$ eigenvectors of the density matrix in each renormalisation step. In principle, one ought to be able to access arbitrarily large system sizes with this method. In practice---and as was also noted in \cite{Nagy02}---the algorithm eventually goes unstable, which is typically manifested through the gap acquiring an imaginary part. We simply ignore data for system sizes where the instability is judged to have kicked in.
Although we can access larger system sizes with the DMRG approach than was possible with Monte Carlo (e.g., $L$ up to about 250, as shown in Fig.~\ref{fig:dmrgRawData}), it is still necessary to extrapolate to the thermodynamic limit, $L\to\infty$. We follow a similar procedure to that described in Section~\ref{sec:decay}. That is, we specify a finite-size fitting function of the form $f(L) = \varepsilon + a_2 L^{-2}$, and adjust the smallest value of $L$ used for the fit. Again, we use ${\cal R}^2$ as a goodness-of-fit measure, i.e., Eq.~(\ref{R2}) with $t$ replaced by $L$. We show in Fig.~\ref{fig:dmrgRsq} how the goodness of fit varies with the minimum value of $L$ used in the fit; choosing the optimal (largest) value yields an estimate of the gap that we observe from Fig.~\ref{fig:dmrgFitGaps} that appears more consistent with the de~Gier--Essler prediction than domain wall theory.
\begin{figure}
\centering
\includegraphics[width=0.66\linewidth]{dmrgRawData}
\caption{Stable DMRG data for the gap (red crosses) and the exact thermodynamic gap (black line) at $\alpha = 0.4$ and $\beta = 0.75$.}
\label{fig:dmrgRawData}
\end{figure}
\begin{figure}
\centering
\subfigure[]{\label{fig:dmrgRsq}%
\includegraphics[width=0.45\linewidth]{dmrgRsq} }
\subfigure[]{\label{fig:dmrgFitGaps}%
\includegraphics[width=0.45\linewidth]{dmrgFitGaps} }
\caption{ \subref{fig:dmrgRsq} Coefficient of determination $R^2$ and \subref{fig:dmrgFitGaps} extrapolated gap, as a function of $L_{\rm min}$ for DMRG data corresponding to $\alpha = 0.4$, $\beta = 0.75$. The optimal choice of $L_{\rm min}$ lies just below $200$, for which the extrapolated gap very closely matches the de~Gier-Essler result and clearly differs from the DWT gap. }
\end{figure}
We show the DMRG estimate of the gap obtained in this way as a function of $\beta$ in Fig.~\ref{fig:dmrggapacr}. The agreement with the analytical prediction common to dGE and DWT below $\beta < \frac{1}{2}$ is very good. For larger $\beta$ the data are scattered around an apparently constant value, indicating the presence of inaccuracies in the DMRG algorithm and/or the extrapolation of $L\to\infty$. If we regard these data as independent measurements of the same value, we can obtain an estimate of the uncertainty in the constant value of the gap above some critical point by simply calculating the standard deviation. We find the DMRG gap to approach the constant value $\varepsilon = -0.00125(9)$ when $\alpha=0.4$. For comparison, de Gier and Essler predict a constant value for the gap of $-0.00121\ldots$ above the dynamical transition, while domain wall theory predicts $-0.00102\ldots$. The DMRG measurement is then clearly consistent with the de~Gier--Essler prediction, whilst the difference from the domain wall theory value is significant. Taking this result together with the Monte Carlo data we conclude that a dynamical transition does indeed occur at the dGE line, and not where DWT predicts.
\begin{figure}
\centering
\includegraphics[width=0.66\linewidth]{dmrggapacr}
\caption{DMRG estimates of the gap (red crosses), exact thermodynamic gap (black curve) and DWT gap (green curve), all as a function of $\beta$ at $\alpha = 0.4$ (for which $\beta_c \approx 0.53$).}
\label{fig:dmrggapacr}
\end{figure}
\section{Discussion of the distinction between static and dynamic subphases}
\label{sec:discuss}
In this paper we have presented numerical evidence, the most convincing of which came from DMRG
calculations, that a dynamical transition occurs in the TASEP at the dGE line rather than at the subphase boundary. However, this finding in turn raises a number of open questions. For example, does the dGE line---and in particular its
departure from the static subphase boundary---have any physical significance? How is the static subphase boundary manifested in the eigenvalue spectrum? We discuss the distinction between the two phase diagrams first in general theoretical terms, and then with reference to different treatments of the TASEP.
\subsection{General theory of nonequilibrium phase transitions}
One way to gain a general understanding of nonequilibrium phase transitions is through the analogue of partition function for a system of size $L$, $Z_L$ \cite{BE02,Altenberg02}. It has been shown that for any system governed by Markovian dynamics, it may be written as
\begin{equation}
Z_L= \prod_{j\neq0} (-\lambda_j)
\label{Z}
\end{equation}
where $(-\lambda_j)$ are the eigenvalues of the Markov transition matrix which defines the dynamics \cite{Blythe01phd,Altenberg02}. For a finite, irreducible configuration space there is a unique stationary state corresponding to
the largest eigenvalue $\lambda_0=0$. By the Perron-Frobenious theorem \cite{Senata}, $\lambda_0$
is non-degenerate, therefore the gap is given by the second largest eigenvalue $\lambda_1$
and the largest relaxation time is $\tau_1 = -1/\lambda_1$.
From our knowledge of equilibrium phase transitions, we expect a
static phase transition to occur when $\lim_{L\to\infty}\frac{1}{L}\ln
Z_L$ exhibits nonanalyticities. We see from (\ref{Z}) that to obtain a
static phase transition we require a major restructuring of the
eigenvalue spectrum. To be general and explicit let us consider the
nonequilibrium analogue of the free energy density,
\begin{equation}
h_L = \frac{\ln Z_L}{L} = \frac{1}{L} \sum_{\lambda_i \ne 0} \ln (- \lambda_i ) \;.
\label{h}
\end{equation}
Consider an arbitrary eigenvalue $\lambda_i$ in
(\ref{h}), possessing a nonanalyticity at a
critical value of some parameter. Unless a significant
number of other eigenvalues in the spectrum
converge onto the same nonanalyticity in the thermodynamic limit, the effect of the
$\lambda_i$ nonanalyticity is lost as $L \to \infty$ and
(\ref{h}) and hence $h_L$ remain analytic.
On the other hand a dynamical phase transition, as defined above, is
only concerned with the eigenvalue $\lambda_1$. Therefore a dynamical
phase transition does not necessarily coincide with a static phase
transition. One very simple example would be if the two leading
subdominant eigenvalues $\lambda_1$ and $\lambda_2$ crossed at some
values of a control parameter. Then the gap would behave in a nonanalytic
way but $Z$ would be analytic.
Let us consider more explicitly the TASEP. From the exact solution of the TASEP \cite{DEHP93,BE07}
the expression for $Z_L$, as defined by (\ref{Z}), is
\begin{equation}
Z_L = \frac{(\alpha \beta)^{L+1}}{\beta-\alpha}\left[ \Phi_L(\alpha) - \Phi_L(\beta)\right]
\label{Z2}
\end{equation}
where
\begin{equation}
\Phi_L(x)= \sum_{p=1}^L \frac{p (2L-p-1)!}{L! (L-p)!} x^{-(p+1)}
\end{equation}
and for $L$ large
\begin{equation}
\Phi_L(x)
\simeq \left\{ \begin{array}{ll}
\displaystyle \frac{x (1-2x)}{[x(1-x)]^{L+1}} & x < \frac{1}{2} \\[2ex]
\displaystyle \frac{4^L}{\sqrt{\pi L}} & x = \frac{1}{2} \\
\displaystyle \frac{x\, 4^L}{\sqrt{\pi} (2x-1)^2 L^{3/2}} & x >
\frac{1}{2}
\end{array}\right. \;.
\label{Phi}
\end{equation}
Now consider, for example, the case $\alpha <1/2$. As we increase
$\beta$ a first-order static phase transition occurs on the line
$\beta =\alpha $ where the dominant contribution to (\ref{Z2}) changes
from $\Phi_L(\beta)$ to $\Phi_L(\alpha)$.
Thus at the phase transition $h_L$ defined in (\ref{h}) is non-analytic.
The subphase boundary is at
$\beta = 1/2$ where the asymptotic behaviour of the subdominant
contribution, $\Phi_L(\beta)$ changes.
As this is a subdominant contribution to $\ln Z_L$,
$h_L$ remains analytic at the sub-phase boundary.
At the dGE line there is no noticeable
change in the asymptotic behaviour of $Z_L$
and $h_L$ is analytic.
\subsection{Static vs dynamic transitions within various treatments of the TASEP}
We illustrate further the distinctions between the static and dynamics phase diagrams
by collecting exact results for the TASEP together with the predictions of
various level of approximations such as mean field theory and domain
wall theory.
\paragraph{Burgers Equation (Mean Field theory)}
We begin by considering a mean field description of the system
that is given by the Burgers equation,
\begin{equation}
\label{eq:burg}
\partial_t\rho(x,t) = -(1-2\rho)\partial_x\rho + \frac{1}{2}\partial^2_x\rho
\;,
\end{equation}
where $\rho(x,t)$ is a density field.
Although the Burgers equation has been
studied extensively in the literature, we are not aware of any
treatment of the case with prescribed boundary reservoir densities.
In the appendix we give the solution of (\ref{eq:burg}) subject to the
boundary conditions $\rho(0) = \alpha$ and $\rho(L) = 1-\beta$, and
arbitrary initial condition.
The results for the Burgers gap $\varepsilon_B$ may be summarised as follows
(restricting ourselves to the case $\alpha <1/2$):
\begin{eqnarray}
\mbox{For}\quad \beta <1/2\quad
\varepsilon_B &=& -2 | \alpha(1-\alpha) - \beta(1-\beta) | \\
\mbox{For}\quad \beta >1/2\quad
\varepsilon_B &=& - \frac{(1-2\alpha)^2}{2}
\end{eqnarray}
The behaviour of the gap is plotted in Fig.~\ref{fig:bgap}. Note that
a dynamical transition occurs at $\beta = 1/2$. Interestingly, it turns out
that there is no change in the density profile at this value so, in
fact, the mean field theory predicts a dynamical transition instead of
a subphase boundary at $\beta = 1/2$.
Thus even at the level of mean-field theory, the TASEP exhibits the phenomenon of a dynamic transition that is not accompanied by a static transition.
\begin{figure}
\centering
\includegraphics[width=0.6\linewidth]{GapComparisonBurgersExact}
\caption{Comparison of the gap for $L \to \infty$ from the spectrum of the viscous Burgers equation (\ref{eq:burg}) to the exact thermodynamic gap, for $\alpha = 0.1$ and $\beta_c \approx 0.675$.}
\label{fig:GapComparisonBurgersExact}
\label{fig:bgap}
\end{figure}
\paragraph{Domain wall theory} The predictions of domain wall theory were reviewed in Section~\ref{subsec:DWT}. As we pointed out there, DWT gives a subphase boundary at $\beta=1/2$ along with a dynamical transition: the distinction between the two phase diagrams is lost. However, in the subphases LDI and HDI, the DWT gives the correct dynamical gap to order $1/L^2$.
\paragraph{Exact results} As was discussed in Section~\ref{sec:moddef}, the
exact results for the stationary state \cite{DEHP93,Schutz93}
and the spectrum \cite{deGier05,deGier06}
reveal behaviour that is different from both mean-field and domain wall theory. In common
with the latter, there is a static subphase boundary at $\beta = \frac{1}{2}$. Like the mean field
theory, the location of the dynamic transition occurs at a distinct location to the static phase boundary. However, this location $\alpha_c(\beta)$ is nontrivial and is not predicted by either of the simpler theories. Moreover, the behaviour of the gap at the static transition point $\beta=\alpha$ differs between the mean-field and exact theories: in the former, the gap has a cusp, whereas in the latter it varies analytically.
\section{Conclusion}
\label{sec:conclude}
In this work we have provided numerical evidence to
confirm the existence of the dynamical transition line
predicted by de Gier and Essler.
Ultimately the evidence came
from the approximate, but quantitatively reliable, DMRG technique.
To conclude, we now return to the main open question resulting from this work,
namely that of the physical significance of the dynamical transition that
takes place along the de~Gier--Essler line. We remark again that, as was
seen in Section~\ref{sec:sims}, direct measurement of the gap in Monte Carlo
simulations was very difficult. The nonanalyticity in the leading eigenvalue
was barely detectable in the stochastic simulation data. We were therefore
unable to identify, for example, whether the system relaxed to stationarity
in a fundamentally different way either side of the transition line.
Given the success of the domain wall theory in predicting the gap in part of the phase diagram,
it is tempting to believe that with some refinement it may
be able to explain the dynamical transition and furthermore reveal the physics associated
with it. For example, it has been noted in\cite{deGier06}, that DWT may be modified to
give the correct gap throughout the LD1$'$ region by simply imposing that the right
domain density be $\rho^+ = 1-\beta_c$ in this region.
However DWT then no longer predicts a static subphase boundary at $\beta=1/2$
and also the transition line $\alpha_c(\beta)$ is not predicted.
It may be that some knowledge of the form of the eigenvectors associated to the low lying states will allow an understanding of what occurs at the dynamical transition.
It would be of interest to extend the DMRG studies to investigate the
form of the eigenfunctions. Also recent analytic work by Simon \cite{Simon09},
in which eigenvectors are constructed for the partially asymmetric exclusion process
through the co-ordinate Bethe ansatz, may help to shed light on the matter.
\ack
R.A.B.\ thanks the RCUK for the support of an Academic Fellowship. AP would like to thank Fabian Essler for hospitality and discussion,
and acknowledges financial support from the EPSRC.\\
|
1,108,101,563,226 | arxiv | \section{Introduction}
In nature, the animal kingdom has developed the ability to detect or reflect circular polarized light (CPL) as an added modality
for enhanced vision, better navigation or ``private communications'' \cite{Rossel1986,Cronin2003,Tsyr2008}.
The characterization of light polarization states is also essential in a variety of reasearch and industry fields ranging from
pharmaceutics drugs circular dichroism~\cite{Purdie1996}, polarimetric enhanced vision and imaging~\cite{Khorasaninejad2006}, spectroscopic material characterization, medical and scientific diagnosis~\cite{Greenfield2006,Sparks7816} and optical
communications~\cite{Farshchi2011}. The development of a simple, efficient and integrable chiral
photodetector could offer improvements in actual uses but also favour novel device design and applications inspired by natural
vision systems.\\
The inherent difficulty in discriminating the circular polarization states of light resides in the lack of chirality in
materials commonly employed in optoelectronics. Common (non-chiral) photodetector are thus used in combination with a set of
moving, and often bulky optics such as quarter waveplates and polarizers~\cite{Berry1977}.
Thus, conventional systems make it a challenge to construct robust, miniature and integrated devices. A direct detection of the
CPL state is a desirable improvement offering advantages in term of robustness and integrability in addition to a simplified
technology. Albeit ultra-compact optical elements for manipulating circularly polarized are actively
investigated~\cite{Yu2012,Gansel2009}, the realization of a simple,
room temperature ``single-measurement'' circular polarization degree photo-detector remains elusive.
\begin{figure}[!h]
\centering
\includegraphics[width=\linewidth]{figure_1}
\caption{ a) Sketch of the experimental configuration used to simultaneously measure the photoluminescence (PL) and the photoconductivity (PC). $B_{\rm N}$ and $B_{\rm ext}$ indicate respectively the induced nuclear magnetic field and the external magnetic field created by a permanent magnet in the configuration when the magnetic fields are parallel (see text). b) Variation of PL and PC signals as a function of the external magnetic field for different light polarizations. $\lambda_{\mathrm{exc}}$=852 nm, $P_{\mathrm{exc}}$=20 mW.}
\label{fig:Fig1}
\end{figure}
Different approaches have put forward plasmonic or CMOS-like nanostructures~\cite{meta_1,meta_2,CMOS} or organic chiral
molecules~\cite{Schulz2019,Yang2013}. These devices often suffer from limited acceptance angle and with the use of multiple optical elements
it makes technically challenging to realise miniature and integrated CPL detectors.
Recent measurements on quantum-dots-based hybrid devices have shown a photocurrent helicity asymmetry of 0.4\% at
4 K~\cite{Cadiz2020}, 2\% ($T$=4 K) in a similarly designed quantum wells hybrid ferromagnetic/semiconductor structures or again 5.9\% in a
Ge-based device ($T$=4 K)~\cite{Rinaldi2012}. Better results have been reported in chiral plasmonic metamaterials. However, these are absorption devices lacking the direct electrical detection~\cite{Hovel2008}. Very recently, 2D WTe$_{2}$ has been used by exploiting the spin photogalvanic effect~\cite{Ji2020}.
Each of these approaches tackles one or more of the desired characteristics, but suffers some drawbacks such as a narrow spectral range or they do require multiple, complex technological processing steps to be fabricated.\\
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figure_2}
\caption{a) to c): Schematic view of the SDR capture under different excitation polarizations when a positive (same orientation as the $\sigma^{-}$ photo-generated spins) external magnetic field is applied to the epilayer in Faraday geometry. d) Representation of the corresponding conduction electron population at dynamical equilibrium.}
\label{fig:Fig2}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figure_3}
\caption{Schematic of the hyperfine and Zeeman splitted levels of the electron-defect system as a function of the total magnetic field ($B_{\rm N}+B_{\rm ext}$). The vertical dotted line schematically indicates the value of $B_{\rm N}$ when the external magnetic field is zero. If $B_{\rm ext}$ is decreased or increased from $0$ the defect capture rate will respectively increase or decrease leading respectively to lower or higher PC. The shaded regions near the origin and for high total magnetic field schematically indicate respectively the regions where the nuclear and electron spin states are fully mixed or substantially independent.}
\label{fig:Fig3}
\end{figure}
Here we implement the simple solution to this challenge recently proposed in~\cite{kunold-theory} which makes use of a dilute nitride
GaAsN semiconductor epilayer as the spin photoconductive device capable of determining directly, by a voltage or current
measurement, the handedness of an incident light. The device operation relies on the spin dependent recombination of conduction electrons and on the hyperfine interaction (HFI) between bound electrons and nuclei on paramagnetic centers
to provide the device with respectively a sensitivity to the degree of circular polarization and its handedness.\\
This semiconductor belongs to the family of dilute nitride materials consisting of a GaAs-based semiconductor where a
small percentage of N is introduced to induce the formation of Ga$^{2+}$ interstitial paramagnetic defects which display both an
exceptionally large spin dependent recombination (SDR) for conduction band electrons and a defect nuclear hyperpolarization
at room temperature~\cite{sdr1,Lombez2005,wang_room-temperature_2009,Kalevich2012}. A very efficient defect recombination is usually an undesirable feature for optoelectronics devices.
However, as the capture rate is here
dependent on the spin polarization of the conduction band
electrons, it provides a way of controlling the conduction band population and thus the conductivity via the spin polarization.
Key to this device workings is the fact that this spin polarization is likewise transferred not only to the defect-bound
electron but also to the surrounding nuclei via a concomitant action of the hyperfine interaction and dipole-dipole spin
relaxation~\cite{kunold2020}. This yields a sizeable effective nuclear magnetic field $B_N$ whose
direction is parallel to the conduction electron spin polarization and thus to the photon helicity.
This phenomenon is however equally efficient for spin up or spin down conduction electron polarization allowing only the
determination of a circular polarization degree without its handedness. By exploiting in addition the interplay between
the HFI and a small external magnetic field, the GaAsN epilayer acquires a chiral photoconductivity (PC) capable of discriminating the
handedness of an incident light in addition to a linearly polarized one as demonstrated below.\\
It is important to note that spin dependent recombination and nuclear polarization can occur on many defects and in various
materials but typically with a weak efficiency or at cryogenic temperatures. In contrast, in GaAsN-based semiconductors they
manifest with macroscopic record-high values at room temperature, paving the way to a practical exploitation of the
phenomenon.
In addition, these paramagnetic defects can be induced to the whole range of GaAs-based alloys from AlGaAsN to
InGaAsN~\cite{sdr1}, thus allowing a wide range of wavelength operation from the visible to the infra-red spectral region.
Nitrogen-free gallium-ion implanted semiconductor
systems have also been shown to display this giant spin-dependent recombination, allowing dissociating the spin dependent
recombination efficiency and the defect density from the material gap and to offer extended
device configurations and applicabilities~\cite{Nguyen2013}. Finally, as the the working principle relies on the creation of a spin polarized conduction-band
population, the device's working range can in principle cover a wide spectrum of excitation wavelength as long as a minimum optical orientation in the conduction band can be achieved.
\section{Results and Discussion}
\textbf{Figure 1b} presents the simultaneous measurement of the PL and PC of a GaAsN (2.1\% N) sample for different light polarization states as
a function of the applied external magnetic field $B_{\rm ext}$ in Faraday geometry for a 20 mW excitation at 852 nm.
We can see a very similar behaviour for PL and PC as a function of the magnetic field and light polarization,
confirming that the same physical phenomena underpin both measurements.
From these data two features can be noted: (i) The conduction electron population, as revealed by the PL or PC intensities, depends on
the incident light helicity as soon as $B_{\rm ext}\neq$ 0 and for a very small magnetic field strength. This weak field rules out any
macroscopic Zeeman effect on the conduction or defect electron states capable of producing a thermal spin population.
(ii) The asymmetry persists for a large range of magnetic fields (the PL and PC intensities under circular polarization
are expected to eventually become identical for $B_{\rm ext}> 300$ mT~\cite{kunold-theory}).
These data identify a broad range in external magnetic field where the discrimination of the light polarization can be achieved
and used in a single-reading electrical detection.
\\
The particular dependence of the PL and PC signals as a function of the external magnetic field for different polarization
has been successfully interpreted in the framework of an interplay between the hyperfine interaction $\hat{H}_{HFI}=\mathcal{A} \hat{I} \cdot \hat{S}$ (where $\mathcal{A}$
is the hyperfine interaction constant)
between the Ga$^{2+}$ nuclei (spin $I$=3/2) and the bound
electrons (spin $S$=1/2), and the Zeeman interaction induced by the external magnetic field. The hyperfine interaction reduces the SDR efficiency due to the
mixing of the electron's and nucleus's spins~\cite{Ivchenko2015}, but allows for the flow of angular
momentum transfer from the conduction band electron to the defect nucleus in combination with the SDR~\cite{ibarra-Sierra2017}.
The way this interplay is exploited can be clarified with the help of \textbf{figure 2} and \textbf{3}: When the spin of the conduction electrons
shows a preferential orientation (such as under an at least partially circularly polarized light, figure 2a) the defect resident electron
acquires the same spin orientation, through the spin dependent recombination, in a few tens of picoseconds~\cite{sdr1}.
This spin polarization is likewise transferred to the defect nucleus and neighbouring nuclei through a combination of the
hyperfine interaction and dipole relaxation, yielding an effective nuclear magnetic field $B_{\rm N}$. This spin polarization transfer
from the conduction band to the defect electron and nuclear spins has a fundamental role: (i)
by forcing the same spin orientation onto the defect resident electron, it prevents any further conduction electron capture thanks
to the Pauli's principle. The capture of the conduction electrons is greatly reduced giving the sample a ``{\bf Large}'' conduction
population. Without an external magnetic field this happens with the same efficiency for a right or left circular excitation.
Instead, under a linearly-polarized excitation (figure 2c) an equivalent density of spin-up and spin-down conduction population is
photo-generated and no net transfer of spin polarization to the defects can occur.
The conduction capture is no more spin-dependent and the
defect can efficiently capture the conduction electrons leading to a ``{\bf Low}" conduction population.
(ii) Under a weak external magnetic field in Faraday geometry, the PL (or PC) intensity under a circularly polarized light
takes the form of an inverted Lorentzian (figure 2d)~\cite{ibarra-Sierra2017} displaced from the origin in opposite
direction for a right or left circular incident
polarization due to the presence of the nuclear field $B_{\rm N}$ which adds up to the external field.
Now the system is effectively chiral: When $B_{\rm N}$ is parallel or antiparallel to an applied external magnetic field of comparable
strength, the HFI in the defect system is proportionally weakened or strengthened with respect to the electron Zeeman effect
offering a way to proportionally control the SDR efficiency (figure 3). The conduction carrier capture time (figure 2a and b) becomes
dependent on the total magnetic field $B_{\rm tot}=B_{\rm N}+B_{\rm ext}$. As the sign of $B_{\rm N}$ is directly linked to the
average conduction electrons spin orientation, the sample conductivity becomes now dependent on the average conduction
electrons spin orientation and therefore on the light polarization state incident on the sample.
\\
In more details, the device works on the following principle: A permanent magnet with a field of 30 to 50 mT is placed on the substrate
side of the GaAsN epilayer to have the field parallel (or anti-parallel) to the optical axis of the incident light. The magnetic
field is chosen such that it falls in the optimum region for the discrimination between a circular
right or left incident polarization~\cite{kunold-theory}, which is of the same order of magnitude as $B_{\rm N}$.
A constant voltage in the range 0.1 to 4 V is applied to the sample contacts. At this point, the
reading of the photoconductivity value reflects the polarization state of the incident light. \textbf{Figure 4a} presents such a
measurement in the case of a 30 mW incident light at 852 nm under the 50 mT magnetic field. We observe a clear difference
between the signal measured for a right, left or linear incident polarization, which allows for the determination of
the light helicity with a simple PC reading. Figure 4b shows how the signal varies when the incident polarization is continuously
modified from right
circular to linear and to left circular by turning the angle of the quarter-wave plate in the incident optical path. This shows in addition that, albeit for a small polarization interval, the device can also determine the
polarization degree of an incident light in a single measurement. Considering that the root mean square error $RMS=0.0068$ mV as measured from a linear fit of the first plateau signal for a $\sigma^-$ excitation in figure 4a, and the signal difference between a linear and $\sigma^-$ excitation, $\delta=$0.36 mV, we can estimate a measurement of a circular polarization degree with a precision of about 2\% in this proof of concept device.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figure_4}
\caption{(a) Photoconductivity signal measured at 50 mT for different polarizations of the incident light. b) Photoconductivity
signal measured as a function of the quarter-wave plate angle with respect to the incoming linearly polarized light excitation.}
\label{fig:device-working}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figure_5}
\caption{a) The PC contrast $\Delta_{PC}$ as a function of the excitation power at 852 nm excitation. b) The PC contrast as a function of the excitation wavelength.}
\label{fig:Fig3}
\end{figure}
We have characterized the PC contrast $\Delta_{PC}=\frac{PC^{-}-PC^+}{PC^{-}+PC^+}$ ($PC^{-/+}$ indicates respectively the
PC under a $\sigma^-$ or $\sigma^+$ excitation) as a function of the incident wavelength and power. \textbf{Figure 5a} reports the data
obtained as a function of the SDR ratio at $B_{\rm N}$=0, SDR$_{r}(0)=100\frac{PC^{\sigma}}{PC^{\pi}}$, by varying the incident
power. The contrast reaches a maximum for a relatively weak powers and
then monotonically decreases as the power is increased. This behaviour is well accounted for~\cite{kunold-theory} at low power
by the power threshold for the build-up of the nuclear magnetic field. Increasing the excitation power beyond the optimum leads to a decrease of the SDR, leading
in return to a decrease of the photocurrent contrast.
Figure 5b presents the PC contrast as a function of the excitation wavelength. A constant incident power $P_{\rm exc}$= 20 mW
has been used here for the whole spectral range explored. We can deduce that for this Nitrogen content and for this simple
photoconductivity configuration, the device can be used over 100 nm range. The hight wavelength limit ($\lambda$= 870 nm, $E$= 1.424 eV) is due to
the change in the absorption coefficient of the device once the excitation energy is smaller than the GaAs gap energy. Exciting with photon energies above
the GaAs gap allows instead for a larger absorption of the incoming light as this can occur over the thicker GaAs buffer layer and not only on the thin 100 nm GaAsN one, while preserving a substantial conduction electron spin polarization during the thermalisation and
diffusion into the GaAsN layer~\cite{sipe}. No higher excitation wavelength has been tested. However,
thanks to the optical selection rules, a PC contrast should in principle be observable for wavelengths up to the GaAsN gap
as the conduction spin polarization will increase for excitations closer to the gap, albeit higher powers will be necessary to compensate for the lower absorption in the thin layer. The low wavelength limit ($\lambda$= 704 nm, $E$= 1.76 eV) is instead linked to the onset of the spin-orbit
band where no or very weak conduction spin polarization can be achieved. It is important also to consider here that both the
absorption coefficient and the photogenerated conduction spin polarization are wavelength dependent. As the SDR depends
on the absorbed power and the spin polarization achieved at the bottom of the conduction band, the relation between the PC
contrast and wavelength is non-linear.
As suggested in~\cite{kunold-theory} an improved version of the device
could be envisaged in the form of either a balanced photodetector composed of two slabs of GaAsN each subject to opposite
magnetic fields $\bf{B_1}$ and $\mathbf{B_2}=-\mathbf{B_1}$. A common voltage source is used to
drive in both GaAsN slabs and the measured PCs from the two photodetectors are subtracted. In this configuration,
the sign and not the magnitude of the resulting current will reveal the polarization state of the incident light. An alternative solution
is also proposed where the differential measurement is obtained in a single GaAsN layer subject to an alternating magnetic field
produced by a micro-coil powered by an alternating current (\textbf{figure 6a}).
In order to make a proof of concept, we have tested a modified version of the second device design where the single GaAsN
is here subject to a constant magnetic field while the incident light has a periodically varying polarization state (\textbf{figure 6b}).
The polarization modulation is obtained by rotating a quarter-wave
plate at a frequency $f_{\mathrm{\rm rot}}=\omega_{\rm rot}/2\pi$. In this configuration, the polarization state after the waveplate will periodically change
from linear to circular right, linear, circular left and back to linear, twice per rotation.
By demodulating the measured photocurrent at a frequency $2 f_{\rm rot}$ using a lock-in amplifier, the resulting signal is proportional to $(i^{\sigma-} -i^{\sigma+})$, i.e. the difference in the photocurrent signal between a circular right and circular left polarized excitation.
\begin{figure}
\centering
\includegraphics[width=\columnwidth,angle=0]{figure_6}
\caption{a) A representation of one of the differential measurement photodetector designs proposed in~\cite{kunold-theory}.
A micro-coil provides an alternating positive and negative magnetic field and the light polarization is determined by
the resulting sign of the measured current by subtracting the measurements obtained under opposite magnetic fields.
b) The scheme used here for the proof of concept where a constant magnetic field is used and the device is illuminated with an alternating circular polarization.}
\label{fig:rotating_scheme}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figure_7}
\caption{The PC measured using the polarization-modulation technique as a function of the incident power. The curves have been vertically displaced for clarity.}
\label{fig:rotating_scheme}
\end{figure}
\textbf{Figure~\ref{fig:rotating_scheme}} reports the direct measurements of the difference between a circular right and circular left
incident polarization using the set-up presented above, as a function of the excitation power at 852 nm for 25 - 40 mW and
$f_{\rm rot}$= 21 Hz.
We observed that a sizeable differential signal can be obtained for a large range of external magnetic fields.
In the differential configuration the signal will be different than zero only for circularly polarized light and the limit to the minimum measurable circular polarization degree will be set mainly by the noise equivalent signal of the reading.
This is expected to be greatly improved with a photodiode design instead of a photoconductor one.
A linear or unpolarized light will give instead a zero reading.\\
\section{Conclusions}
In conclusion, we have shown that leveraging the giant spin dependent recombination and the hyperpolarization of Ga$^{2+}$ interstitial defect in dilute nitrides epilayers allows for the realization of a chiral III-V dilute nitride semiconductor useable for the
direct electrical measurement of the incident light helicity at room temperature. As the working principle relies on the optical orientation of conduction electrons, the working spectral range could be easily extended for wavelengths up to the material band gap and as long a non zero spin polarization of photoexcited electrons can be achieved.
This paradigm, enabled by the presence of Ga paramagnetic defects in dilute nitrides, can in principle be extended to the whole family of (In)(Al)GaAsN alloys with gaps ranging from the visible to the infrared. The possibility of producing interstitial defects by implantation in N-free alloys~\cite{Nguyen2013} could also provide further application possibilities.
\section{Methods}
The sample studied for the proof of principle demonstration consists of a $100$ nm thick
GaAs$_{1-x}$N$_x$ epilayer ($x=0.021$) grown by molecular beam epitaxy on a (001) semi-insulating GaAs substrate and
capped with $10$ nm GaAs. The room temperature gap is at 1080 nm. The excitation light was provided by either a 852 nm laser diode or a continuous wave Ti:Sa laser as the
tunable wavelength source.
The lasers were focused to a $\approx$ 100 $\mu$m diameter spot (FWHM), in between two Ag electrodes deposited onto the
sample surface about 1 mm apart ({figure 1a}). The laser light polarization was controlled by a Glan-Taylor polarizer followed by a quarter
wave plate. The laser intensity was modulated by a mechanical chopper at 170 Hz. The sample photoconductivity
(PC) was measured synchronously using a lock-in amplifier from the voltage drop at the terminals of a 1 M$\Omega$ load
resistor placed in series with the sample. We have carefully focused the laser in order to avoid any spurious effects related to
the contact illumination. A constant voltage has been applied between the sample electrodes to perform the PC reading.
The photoluminescence (PL) intensity was simultaneously measured with the same lock-in technique by recording its total
intensity, filtered of the laser scattered light and substrate contribution using a series of long-pass optical filters, and integrated by
an InGaAs photodiode. A permanent Neodymium magnet has been used to apply a magnetic field in the Faraday geometry and
the field strength on the sample surface has been varied by changing the magnet's distance from the sample. All the experiments
were performed at room temperature.\\
\medskip
\bibliographystyle{MSP}
|
1,108,101,563,227 | arxiv | \section{Introduction}
The concept of synthetic frequency lattice in photonics~\cite{Yuan:2018-1396:OPT, Ozawa:2019-349:NRP}, where dynamic modulation induces cross-talks between discrete spectral lines to produce lattice-like behavior~\cite{Bell:2017-1433:OPT, Dutt:2020-59:SCI}, is stimulating rapidly growing research in recent years. The development of synthetic lattices not only provides a new means for the reshaping of discrete spectral lines, but also enables tailored transformations of optical spectra underpinned by the remarkable capability of inducing multiple arbitrary short- and long-range couplings in frequency domain~\cite{Bell:2017-1433:OPT}, beyond the physical constraints of spatial lattices~\cite{Szameit:2009-2838:OL}. So far, synthetic lattices were extensively investigated in their capacity for creating multiple artificial dimensions~\cite{Yuan:2016-741:OL, Ozawa:2016-43827:PRA, Lustig:2019-356:NAT, Maczewsky:2020-76:NPHOT, Wang:2002.08591:ARXIV}, yet their other potential applications remain largely undeveloped. This motivates our current research on employing the fundamental advances of synthetic lattices to the practically important task of fast and efficient characterisation of frequency combs.
The full characterization of ultra-short optical signals, including both their phase and coherence properties, is crucial for the development and understanding of novel engineered light sources, such as optical frequency combs~\cite{Walmsley:2009-308:ADOP, Kippenberg:2011-555:SCI, Schmeissner:2014-263906:PRL}, frequency encoded quantum states \cite{Kues:2017-622:NAT}, and optical soliton molecules~\cite{Herink:2017-50:SCI}. Furthermore, full optical signal characterization is important for the communication of timing information over fiber networks \cite{Foreman:2007-21101:RSI} and wavelength division multiplexing transmission formats where the relative phases between individual carriers are important~\cite{Ellis:2005-504:IPTL}.
The most commonly used methods for measuring optical pulses, frequency-resolved optical gating (FROG) \cite{Scott:2007-9977:OE} and Spectral Phase Interferometry for Direct Electric-field Reconstruction (SPIDER) \cite{Walmsley:2009-308:ADOP,Dorrer:2003-477:OL}, require complex multimode optical setups in order to reconstruct the amplitude and degree of coherence.
Techniques with the capability to recover spectral phase information with just a single spatial mode are being actively developed. This includes
ultra-fast signal conversion methods like stretch transform spectroscopy~\cite{Solli:2008-48:NPHOT}, which could allow real time measurement of pulse to pulse variations in the spectral phase profile, without a need to use an interferometric setup~\cite{Xu:2016-27937:SRP}. The technique of electro-optic spectral shear interferometry (EOSI)~\cite{Wong:1994-287:OL, Dorrer:2003-477:OL, Bromage:2006-3523:OL} can also be implemented for determining the spectral phase and amplitude of frequency combs using a single mode setup, by employing an electro-optic modulator (EOM)~\cite{Supradeepa:2010-18171:OE}. The electro-optic phase modulation enables particularly effective characterization of weak coherent frequency combs,
even in quantum regime at the single- and few-photon levels~\cite{Kues:2017-622:NAT, Imany:2018-13813:PRA}. However, EOSI necessarily requires a sequence of different EOM shears to be applied to the signal to fully determine its state, which significantly limits the data acquisition speed.
Furthermore, the temporal resolution of EOSI is limited by the free spectral range of the EOM, while
the steady growth of telecommunication bandwidth calls for fast and efficient methods of optical spectral characterisation.
Here we propose and experimentally demonstrate that specially designed synthetic lattices can facilitate a new method for recovering the full spectral amplitude, phase and coherence information of an optical frequency comb from a single-shot measurement of the spectral intensities after an input signal is transformed in a nonlinear fiber. This is mediated by a co-propagating pump, which is tailored to realize an effective
synthetic lattice for the signal through the energy-conserving process of four-wave mixing Bragg scattering (FWM-BS)~\cite{Bell:2017-1433:OPT}.
The presented method can be much faster than FROG, SPIDER or EOM based approaches, since it only needs a single measurement and minimal post-processing, while allowing for a significantly larger free spectral range compared to the EOM approach.
In the following, we first introduce the concept and theory of the reconstruction with a synthetic frequency lattice in Sec.~\rsect{concept}. Then, we describe the experimental setup and demonstrate the reconstruction of signals with two and four spectral lines in Sec.~\rsect{experiment}. Finally, we present conclusions and outlook in Sec.~\rsect{conclusions}.
\section{Concept of reconstruction with synthetic frequency lattice}
\lsect{concept}
We propose to use a synthetic frequency lattice to transform input signals into a form where a single-shot spectral intensity measurement at the output would allow full reconstruction of the input spectral phase and coherence. In a synthetic lattice, discrete frequency components effectively become lattice sites, where light can couple or hop to nearest-neighbor or farther frequencies corresponding to the short- and long-range couplings on a lattice.
Such frequency lattices allow complex spectral manipulation of light within a single spatial mode of an optical fiber or waveguide~\cite{Yuan:2016-741:OL, Bell:2017-1433:OPT, Bersch:2009-2372:OL}.
We build on the recently developed approaches based on spatial lattices~\cite{Minardi:2012-3030:OL, Minardi:2015-13804:PRA, Titchener:2016-4079:OL, Oren:2017-993:OPT, Titchener:2018-19:NPJQI, Wang:2019-41:OPT}, and reveal that light propagation and reshaping
in nonlinearly induced synthetic lattices can be utilized to allow full reconstruction of an input signal from a simple spectral intensity measurement at the output. Specifically, the spectral amplitude, phase and relative coherence of the signal can be recovered.
The system is all-optical, with
the synthetic lattice induced through optical nonlinearity and controlled via a pump, which
can be altered quickly to provide different functionalities. Furthermore, manipulation of light within the spectral dimensions, without resorting to spatial multiplexing, makes such devices very compact.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Fig_1_v2}
\caption{Concept of signal reconstruction.
An input signal composed of a number of different frequency channels undergoes a transformation in a synthetic lattice with first and second order coupling between frequency channels. The observed output spectra can then uniquely determine the input spectral coherence function of the signal for a specially tailored form of the synthetic lattice transformation. }
\label{fig:1}
\end{figure*}
We consider an input signal which spectrum consists of a number of discrete frequency components separated by a constant spectral spacing $\Omega$, as shown in Fig.~\ref{fig:1}.
After propagating through the synthetic lattice, which introduces coupling between the frequency lines, the input signal is spread across an increased number of synthetic lattice sites.
In the regime of coherent frequency conversion, the output spectral intensity pattern is dependent on the relative phases and coherence of the input signal. Importantly, we find that, for certain parameters of the synthetic lattice,
it is sufficient to measure just the output spectral intensities in order to perform robust reconstruction of the input spectrum and phase.
The propagation of a fully coherent signal through the synthetic lattice, which can be induced experimentally in an optical fiber as we discuss in the following Sec.~\rsect{experiment}, can be described by coupled-mode equations~\cite{Bell:2017-1433:OPT} for a set of complex amplitudes $a_n$ at frequency components $n \Omega$,
\begin{equation}\label{eq:hamiltonian}
\frac{\mathrm{d}a_n}{\mathrm{d} z}=i \sum_{j=1}^{+\infty} \left[ C_j a_{n+j}+C^\ast_j a_{n-j} \right],
\end{equation}
where $C_j = 2\gamma\sum_{m} A_m(0) A_{m-j}^\ast(0)$ is the $j$-th order coupling coefficient, $\gamma$ is the effective nonlinearity, and $A_m(z)$ are the complex amplitudes of the envelope of pump spectral components.
We determine the lattice dispersion in the framework of Eq.~(\ref{eq:hamiltonian}) by considering the plane-wave solutions in the form $a_n = a_0 \exp[i k n + i \beta(k) z]$, where $k$ is the wavenumber, and find the propagation constant as:
\begin{equation}\label{eq:beta}
\beta(k) = 2 \sum_{j=1}^{+\infty} {\rm Re}(C_j e^{i k j}) .
\end{equation}
The output state after propagation through the fiber length $L$ is
related to the input state via a linear transformation which can be expressed through the transfer matrix ${\bf T}$,
\begin{equation}\label{eq:output}
a_n(z=L) = \sum_i {\bf T}_{n,n'}(L) a_{n'}(z=0).
\end{equation}
We obtain the transfer matrix elements by performing a Fourier transform of Eq.~\reqt{hamiltonian} as
\begin{equation}\label{eq:Tn}
{\bf T}_{n,n'}(z) = \frac{1}{2 \pi} \int_{-\pi}^{\pi} dk \exp\left[ i k (n-n') + i \beta(k) z \right] .
\end{equation}
In general, noise and fluctuations may affect the coherence of comb spectrum. Such partially coherent signals can be characterized by the mutual coherence or complex visibility~\cite{Mandel:1995:OpticalCoherence} between different pairs of spectral lines,
\begin{equation}\label{eq:visibility}
V_{n,m}(z) = \langle \tilde{a}_n(z) \tilde{a}_m^\ast(z) \rangle ,
\end{equation}
where $\tilde{a}_n(z)$ are the fluctuating complex field amplitudes of the spectral comb lines and angled brackets denote averaging. It is established~\cite{Mandel:1995:OpticalCoherence} that any partially coherent field can be represented as an incoherent mixture of coherent fields $a_m^{(p)}$ indexed by $p$, such that
\begin{equation}\label{eq:visibilityMixture}
V_{n,m}(z) = \sum_p a_n^{(p)}(z) [a_m^{(p)}(z)]^\ast .
\end{equation}
We substitute Eq.~\reqt{output} into Eq.~\reqt{visibilityMixture} and obtain
\begin{equation}\label{eq:visibilityTransfer}
V_{n,m}(z) = \sum_p \sum_{n',m'} {\bf T}_{n,n'}(z) a_{n'}^{(p)}(0) \left[{\bf T}_{m,m'}(z) a_{m'}^{(p)}(0)\right]^\ast = \sum_{n',m'} {\bf T}_{n,n'}(z) {\bf T}_{m,m'}^\ast(z) V_{n',m'}(0).
\end{equation}
The output comb intensities, which can be directly measured by a spectrometer, are then found as
\begin{equation}\label{eq:forwards_eq}
I_{n}(z=L)
\equiv V_{n,n}(L)
=\sum_{n',m'}
T_{n,n'}(L) T_{n,m'}^\ast(L) \, V_{n',m'}(z=0) .
\end{equation}
Therefore the input complex visibility can be retrieved
by inverting Eq.~(\ref{eq:forwards_eq}) to find $V_{n',m'}(z=0)$ in terms of the output spectral intensities $I_{n}(z=L)$. Considering the input signal to consist of $N_{\rm in}$ spectral lines, and the output spread out to $N_{\rm out}$ lines, we require that the number of measurements is equal to or larger than the number of unknowns, $N_{\rm out} \ge N_{\rm in}^2$.
The robustness of the inversion process can be characterized by the ``condition number'', $\kappa=||\mathbf{M}||\; ||\mathbf{M}^{-1}||$, where $\mathbf{M}$ is the instrument matrix that directly links the observable ($I_n(z=L)$) and the reconstructed quantities ($V_{n',m'}(z=0)$) and $||\cdot||$ denotes the norm of a matrix.
The condition number quantifies the amplification of the measurement errors and noise of the output intensities in the process of reconstruction of the input
complex visibility function~\cite{Press:2007:NumericalRecipes, Minardi:2015-13804:PRA, Titchener:2018-19:NPJQI, Wang:2018-1104:SCI}. A high condition number implies that small errors in the measurement data would be highly amplified after the inversion, making the result of reconstruction unreliable. Thus for an accurate reconstruction, we require a low condition number.
As we discuss in the following, robust reconstruction, associated with a small condition number close to a theoretical optimum, requires specially tailored simultaneous short- and long-range couplings in the synthetic frequency lattice. We note that the reconstruction procedure is also possible in the presence of absorption, which effect can be incorporated in the coupled-mode Eq.~(\ref{eq:hamiltonian}), while the general approach formulated in Eqs.~(\ref{eq:output}),(\ref{eq:visibility})-(\ref{eq:forwards_eq}) remains the same.
\section{Experimental comb reconstruction via the synthetic lattice in an optical fiber} \lsect{experiment}
We realize the synthetic frequency lattice through the process of four-wave mixing Bragg scattering (FWM-BS) in a $\chi^{(3)}$ nonlinear fiber \cite{Bell:2017-1433:OPT}, see Fig.~\ref{fig:2}(a). Here, the lattice sites are represented by discrete signal frequency channels, which are coupled together by nonlinear frequency conversion. In this system, controllable coupling is made possible by shaping the spectrum of the optical pump. A broad phase-matched bandwidth for FWM-BS is achieved by placing the signal and pump spectra on either side of the zero-dispersion wavelength. One of the unique features of this approach is that it allows long-range coupling, which enables efficient reconstruction of complex input signal states by measuring the output spectra.
\begin{figure*}
\centering
\includegraphics[width=.8\textwidth]{Fig_2_v3}
\caption{
(a)~Experimental setup. A mode locked fiber laser is spectrally shaped to create a set of pump (red) and signal (blue) spectral lines.
(b)~Example pump and signal spectra after being shaped by the pulse shapers and before coupling into the nonlinear fiber in (a).
(c)~The dispersion of the synthetic lattice shown with the red line. The dashed lines show the contributions of the first- and second-order coupling to the dispersion, for $C_1 = 2 C_2$.
(d)~The condition number of the transformation as a function of first and second order coupling rates. The red dashed line shows the subset of the parameter space with $C_1 = 2 C_2$ which was investigated experimentally.
(e)~Experimentally measured output spectra for a frequency space quantum walk for input state $\ket{0}+i\ket{1}$.
(f)~(left hand axis) Experimental condition number of the quantum walk transformation shown in~(e). Increasing the pump power increases the coupling rates, effectively moving along the red line in Fig.~\ref{fig:2}(d).
(right hand axis) Fidelity of the reconstruction of the state $\ket{0}+i\ket{1}$ as a function of pump power.}
\label{fig:2}
\end{figure*}
We experimentally demonstrate the reconstruction of a signal consisting of up to four frequency channels, but in principle the method is applicable to more complicated signals.
We choose a pump spectrum consisting of $3$ different frequency components around $1540$~nm wavelength, separated by $\Omega = 2\pi \times 50$\,GHz [Fig.~\ref{fig:2}(b)]. By having three equally spaced pump spectral lines with $A_1=A_2=A_3$, the signal can be up- or down-shifted by $\Omega$ or $2\Omega$ via the FWM-BS. This realizes a synthetic lattice with first and second order coupling coefficients $C_1 = 2 C_2 = 4 \gamma |A_1|^2$. The corresponding lattice dispersion $\beta(k)$ is presented in Fig.~\ref{fig:2}(c), with the contributions of the first and second order coupling shown individually as dashed lines.
We first numerically
investigate the inversion problem for the case where the input signal is limited to two input synthetic lattice sites. We study the effect of varying the coupling rates $C_1$ and $C_2$, while assuming that other couplings are zero ($C_{j>2}\equiv 0$),
and the normalized fiber length is held constant at $L=1$. For each set of values of $C_1$ and $C_2$ we calculated the transfer function according to Eq.~(\ref{eq:Tn}) and the corresponding condition number.
The plot of the condition number vs. the coupling rates is shown in Fig.~\ref{fig:2}(d), indicating that inversion will be practical given $C_1$ and $C_2$ are chosen in an optimal region. Importantly, both short- and long-range couplings need to be present simultaneously, with $C_{1,2} > 0$.
Next, we experimentally measure the transfer function of the nonlinear fiber system for a range of different coupling rates. This measurement fully incorporates the effect of losses both on the pump and signal, which enables reconstruction under realistic experimental conditions.
We vary the
power of the three equal-amplitude pump spectral lines $A_1=A_2=A_3$ to simultaneously control the first- and second-order coupling rates with $C_1=2 C_2$, as indicated by the dashed line in Fig.~\ref{fig:2}(d).
The transfer function of the system for a given set of pump powers was determined by performing measurements for a set of four different input signals. We used the basis set $\ket{\omega_0}$, $\ket{\omega_1}$, $\ket{\omega_0}+\ket{\omega_1}$ and $\ket{\omega_0}+i\ket{\omega_1}$, where $\omega_0$ and $\omega_1$ correspond to the wavelengths of $1561.7$~nm and $1562.1$~nm, respectively. The output states resulting from the input basis state $\ket{\omega_0}+i\ket{\omega_1}$ are shown in Fig.~\ref{fig:2}(e), where the x-axis indicates the pump power.
We use the experimentally determined transfer function to calculate the condition number of the inverse problem as a function of pump power, see Fig.~\ref{fig:2}(f). We also solved the inverse problem using the data from the basis state $\ket{\omega_0}+i\ket{\omega_1}$ as the input, then calculated the fidelity of the reconstructed state with the input. This fidelity is shown by the right hand y-axis on Fig.~\ref{fig:2}(f). The decrease in fidelity with increasing pump power indicates that the assumption that the fiber system preforms linear transformation on the input signal breaks down, as other effects begin to emerge at higher pump powers. We use Fig.~\ref{fig:2}(f) to determine the pump power which minimizes the condition number and maximizes the fidelity in order to allow the most accurate reconstruction of arbitrary input states.
A pump power of $135\:\mu W$ was chosen to enable the optimal reconstruction accuracy of input signals. We then tested our reconstruction method experimentally by coupling a sequence of random input signals into the fiber. The general form of the input signals was $a_n = \cos(\theta/2)\ket{\omega_0}+e^{i\phi}\sin(\theta/2)\ket{\omega_1}$, where $\theta$ and $\phi$ are randomly chosen, and implemented using a waveshaper. Then the output spectra at the end of the fiber are measured and used to reconstruct the input signal. An example of the real and imaginary parts of a reconstructed input complex visibility function is shown in Figs.~\ref{fig:3}(a) and~\ref{fig:3}(b), respectively. The reconstructions were carried out for many random input signals parametrized by $\theta$ and $\phi$ as shown in Fig.~\ref{fig:3}(c). We observe that the reconstructed signals' fidelity to the input that the waveshaper was configured to produce is typically above $96\%$.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Fig_3_v2}
\caption{
(a)~Real and (b)~imaginary parts of an example reconstructed two-channel spectral coherence function. Fidelity to the target state is $97.8\%$. (c)~Reconstruction fidelity shown by color of random states in the form $\cos(\theta/2)\ket{\omega_0}+e^{i\phi}\sin(\theta/2)\ket{\omega_1}$ vs. the parameters $\phi$ and $\theta$ as indicated by labels.
The red arrow points to a state from (a) and (b).}
\label{fig:3}
\end{figure*}
We also demonstrate the reconstruction of more complex input signals. As an example, we consider an input consisting of four lattice sites centered at frequencies $1561.6$~nm, $1562.0$~nm, $1562.4$~nm, and $1562.8$~nm, which we denote by the kets $\ket{\omega_1}$, $\ket{\omega_2}$, $\ket{\omega_3}$, and $\ket{\omega_4}$, respectively. First the transfer function of the fiber and pump system was determined by measuring the output spectra of a set of 16 basis states. Then this was used to reconstruct a number of random input states. The output spectra as a function of pump power of two representative input states are shown in Figs.~\ref{fig:4}(a) and~\ref{fig:4}(d). The pump power providing optimum condition number of 24.7 for inversion was determined to be $360\:\mu W$, as indicated by the dashed vertical lines in Figs.~\ref{fig:4}(a) and~\ref{fig:4}(d).
The real and imaginary parts of the reconstructed complex visibility functions are shown in Figs.~\ref{fig:4}(b,c) and \ref{fig:4}(e,f). We obtain a high fidelity of the reconstructed states to the input at $93.9\%$ and $96.7\%$, respectively.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Fig_4_v4}
\caption{
(a,d)~Output spectra after a quantum walk in the synthetic lattice as a function of the pump power for the input state (a)~$\ket{\omega_1}+i\ket{\omega_2}-\ket{\omega_3}-i\ket{\omega_4}$ and (d)~$i\ket{\omega_1}+\ket{\omega_2}+i\ket{\omega_3}+\ket{\omega_4}$.
(b,e)~Real and (c,f)~imaginary parts of the spectral coherence function, reconstructed from the the output spectra measured at $360\:\mu W$ pump power [red dotted line in (a,d)] as shown on the corresponding plots on the left.
Fidelity of the reconstruction in (b,c)~is $93.9\%$ and in (e,f)~$96.7\%$.
}
\label{fig:4}
\end{figure*}
\section{Conclusion and outlook} \lsect{conclusions}
In conclusion, we have experimentally investigated the use of synthetic lattices for single-shot reconstruction of frequency combs.
We have shown that the full complex visibility of input spectral comb signals can be reconstructed just by measuring its output spectra after propagating through a synthetic lattice with specially optimized short- and long-range couplings.
This technique is naturally suited to characterizing input signals consisting of a number of evenly spaced narrow frequency components. Furthermore, there appears an interesting potential to extend the approach for characterizing non-equidistant spectra by using more complex pump profiles to induce couplings with incommensurate frequency spacings.
The developed single-shot nonlinear-optical frequency comb characterisation is also well-suited for weak non-classical light, since the nonlinear process of FWM-BS does not introduce quantum noise. Recently, there has been significant progress in quantum optics in the spectral domain~\cite{Reimer:2016-1176:SCI, Kues:2017-622:NAT}. Frequency domain Hong-Ou-Mandel interference has proven stable and practical to implement~\cite{Kobayashi:2016-441:NPHOT, Imany:2018-2760:OL}, and frequency-encoded photonic states are now poised to enable scalable quantum information processing~\cite{Lukens:2017-8:OPT, Lu:2018-30502:PRL, Lu:2018-1455:OPT}. Having access to fast and simple means to perform a full tomographic characterization of a quantum state in the frequency domain would be indispensable. We anticipate a potential in applying our approach based on synthetic frequency lattices in the quantum regime, where the measurement of output multi-photon correlations between different combinations of the spectral lines can enable full reconstruction of the input quantum density matrix based on the mathematical formalism developed in the spatial domain~\cite{Titchener:2016-4079:OL, Oren:2017-993:OPT, Titchener:2018-19:NPJQI, Wang:2019-41:OPT}, bringing the benefits of simplicity and speed of all-optical realization.
\section*{ACKNOWLEDGMENTS}
We acknowledge support by the Australian Research Council (ARC): Discovery Project (DP160100619, DE180100070, and DP190100277); Centre of Excellence CUDOS (CE110001018); Laureate Fellowship (FL120100029).
|
1,108,101,563,228 | arxiv |
\section{Benchmark Characteristics and Debloating Settings} \label{sec:chiselSettings}
Table~\ref{tab:benchmark1} lists the programs in Benchmark\_1 and the supplied inputs for debloating, and mentions various size metrics about the original programs. Table~\ref{tab:chiselbenchinput} provides the list of supplied inputs that we used for debloating the programs in Benchmark\_2. This list of inputs are obtained from~\cite{RAZOR}.
\begin{table}[!h]
\centering
\small
\caption{Characteristics of the benchmarks in \textit{Benchmark\_1}.}
\scalebox{0.9}{
\begin{tabular}{|p{1.3cm}|p{1.55cm}|p{0.8cm}|p{0.65cm}|p{1.1cm}|p{1cm}|}
\hline
\multirow{2}{*}{Program} & \multirow{2}{*}{\shortstack{Supplied\\ Inputs}} & \multicolumn{4}{c|}{Original} \\\cline{3-6}
& & \shortstack{\# IR \\ Inst.} & \shortstack{\#\\ Func.} & \shortstack{\# Basic\\ Blocks} & \shortstack{Binary \\ Size $^a$} \\ \hline
basename & \-\-suffix=.txt & 4,083 & 96 & 790 & 26,672 \\ \hline
basenc & base64 & 8,398 & 156 & 1,461 & 44,583 \\ \hline
comm & -12 & 5,403 & 110 & 972 & 32,714 \\ \hline
date & -R & 29,534 & 166 & 6,104 & 89,489 \\ \hline
du & -h & 5,0727 & 466 & 8,378 & 180,365 \\ \hline
echo & -E & 4,095 & 89 & 811 & 27,181 \\ \hline
fmt & -c & 5,732 & 115 & 1,095 & 79,676 \\ \hline
fold & -w30 & 4,623 & 100 & 893 & 29,669 \\ \hline
head & -n3 & 6,412 & 119 & 1,175 & 37,429 \\ \hline
id & -G & 5,939 & 125 & 1,172 & 36,985 \\ \hline
kill& -9 & 4,539 & 96 & 898 & 31,649 \\ \hline
realpath & -P & 8,092 & 155 & 1,419 & 41,946 \\ \hline
sort & -u & 25,574 & 329 & 3,821 & 116,119 \\ \hline
uniq & -d & 5,634 & 115 & 1,092 & 37,159 \\ \hline
wc & -l & 7,076 & 130 & 1,219 & 41,077 \\ \hline
\end{tabular}}
\label{tab:benchmark1}
\footnotesize{$^a$Total binary size obtained via the GNU \texttt{size} utility.}
\end{table}
\begin{table}[!h]
\centering
\caption{Input settings for the programs in \textit{Benchmark\_2} (obtained from~\cite{RAZOR}).}
\begin{tabular}{|c|c|}
\hline
Program & Supplied Inputs \\ \hline
chown & -h, -R \\ \hline
date & -d, –rfc-3339, -utc \\ \hline
gzip & -c\\ \hline
rm & -f, -r \\ \hline
sort & -r, -s, -u, -z \\ \hline
uniq & -c, -d, -f, -i, -s, -u, -w \\ \hline
\end{tabular}
\label{tab:chiselbenchinput}
\end{table}
\section{Neck Miner Evaluation}\label{apndx:neckMinerEvaluation}
Table~\ref{tab:neckMinerResults} describes the neck miner evaluation results. The second column indicates whether multiple neck locations are matching the control-flow properties. The third column describes if the location of the selected neck location is inside the main function.
Table~\ref{tab:capability} shows \mbox{\textsc{LMCAS}}\xspace performed debloating based on various debloating settings but using the same neck locations that has been identified in each program. This experiment shows the neck location is independent of the input arguments.
\begin{table}[!htb]
\centering
\caption{Neck Miner Results. The second column indicates if there are multiple neck locations to select from. The third column indicates whether the identified neck location is inside the \texttt{main} function}
\scalebox{0.9}{
\begin{tabular}{|c|c|c|}
\hline
\textbf{Program} & \textbf{Multiple Neck Locations} & \textbf{Inside \texttt{main}} \\ \hline
basename 8.32 & \xmark & \cmark\\ \hline
basenc 8.32 & \xmark & \cmark\\ \hline
comm 8.32 & \xmark & \cmark\\ \hline
date 8.32 & \cmark & \cmark\\ \hline
du 8.32 & \xmark & \cmark\\ \hline
echo 8.32 & \xmark & \cmark\\ \hline
fmt 8.32 & \xmark & \cmark\\ \hline
fold 8.32 & \xmark & \cmark\\ \hline
head 8.32 & \xmark & \cmark\\ \hline
id 8.32 & \cmark & \cmark\\ \hline
kill 8.32 & \xmark & \cmark\\ \hline
realpath 8.32 & \xmark & \cmark\\ \hline
sort 8.32 & \xmark & \cmark\\ \hline
uniq 8.32 & \xmark & \cmark\\ \hline
wc 8.32 & \cmark & \cmark\\ \hline
chown-8.2 & \xmark & \cmark\\ \hline
date-8.21 & \cmark & \cmark\\ \hline
rm-8.4 & \xmark & \cmark\\ \hline
sort-8.16 & \xmark & \cmark\\ \hline
uniq-8.16 & \xmark & \cmark\\ \hline
gzip-1.2.4 & \cmark & \xmark\\ \hline
tcpdump-4.10.0 & \xmark & \cmark\\ \hline
objdump-2.33 & \xmark & \cmark\\ \hline
readelf-2.33 & \cmark & \xmark \\ \hline
diff-2.8 & \xmark & \cmark\\ \hline
Nginx-1.19.0 & \cmark & \xmark\\ \hline
\end{tabular}
}
\label{tab:neckMinerResults}
\end{table}
\begin{table}[!htb]
\centering
\caption{\rev{Debloating subset apps from Benchmark\_1 based on various input arguments using the same neck location identified in each program}}
\small
\scalebox{0.89}{
\begin{tabular}{|p{0.5cm}|p{1.3cm}|p{2.4cm}|p{0.9cm}|p{0.8cm}|p{0.95cm}|}
\hline
\multirow{2}{*}{App} & \multirow{2}{*}{\shortstack{Supplied\\ Inputs}} & \multirow{2}{*}{\shortstack{Required\\ Functionality}} & \multicolumn{3}{l|}{Reduction After Debloating} \\\cline{4-6}
& & & \#Func. & \shortstack{Binary \\ Size} & \shortstack{Total\\ Gadgets} \\ \hline
\multirow{2}{*}{du} & -b & shows number of bytes & 23\% & 15\% & 46\% \\ \cline{2-6}
& -b --time & shows the time of the last modification and number of bytes & 22\% & 14\% & 45\% \\ \hline
\multirow{3}{*}{sort} & -c & check if the file given is already sorted or not & 34\% & 28\% & 54\% \\ \cline{2-6}
& -n & sort a file numerically & 31\% & 25\% & 51\% \\ \cline{2-6}
& -un & sort a file numerically and remove duplicate & 31\% & 25\% & 51\% \\ \hline
\multirow{4}{*}{wc} & -c & character count & 42\% & 21\% & 41\% \\ \cline{2-6}
& -w & word count & 42\% & 21\% & 41\% \\\cline{2-6}
& -lc & line and character count & 43\% & 22\% & 42\% \\ \cline{2-6}
& -wc & word and character count & 42\% & 21\% & 42\% \\\hline
\end{tabular}}
\label{tab:capability}
\end{table}
\section{Code Reduction Comparing with other tools}\label{sec:additionalResultsComparison}
This section provides a detailed code reduction comparison with two more debloating approaches.
\begin{itemize}
\item Debugger-guided Manual debloating. We developed a simple but systematic protocol to perform debloating manually, which we state as Algorithm~\ref{alg:manualAlg}. The goal of this manual approach is to create an approximation for the maximum level of reduction that can be achieved by an average developer.
\item Nibbler~\cite{Nibbler2}. state-of-the-art tool for debloating binary code. It does not generate specialized apps, it rather focuses only on reducing the size of shared libraries.
\end{itemize}
\begin{algorithm}[!htb]
\caption{Debugger-guided Manual Debloating Protocol}
\scriptsize
\label{alg:manualAlg}
\DontPrintSemicolon
\KwIn{App $A$, Input $I$}
\KwOut{App $A\textprime$ }
$varsToRemove \leftarrow \emptyset$\;\label{alg:l:varsToRemove}
$funcToRemove \leftarrow \emptyset$\;\label{alg:l:funcToRemove}
$Executed \leftarrow$ the set of statements that GDB reports are executed, given input $I$ \; \label{alg:l:gdb}
\SetKwProg{Def}{def}{:}{}
$A\textprime \leftarrow A$ \;
\Repeat{no more removals of statements}{
\For{$stmt \in A\textprime$}{ \label{alg:l:loop1-b}
\uIf{$stmt \not\in Executed$}{ \label{alg:l:execLst}
remove stmt from $A\textprime$ \;
$varsToRemove \leftarrow varsToRemove \cup \{\textit{stmt(vars)}\}$ \;
\uIf{$stmt$ is a call site}{
$funcToRemove \leftarrow funcToRemove \cup \{\textit{func}\}$ \; \label{alg:l:loop1-e}
}
}
\While{$varsToRemove$ $\neq \emptyset$ $ \land $ $funcToRemove$ $\neq \emptyset$}{ \label{alg:l:loop2-b}
\For{$func \in funcToRemove$}{
Remove $func$ from funcToRemove \;
\uIf{no occurrence of $func$ exists}{\label{alg:l:occrFunc}
Remove func from A\textprime \;
}
}
\For{$var \in varsToRemove$}{\label{alg:l:occrVar}
Remove $var$ from varToRemove \;
\uIf{no occurrence of $var$ exists}{
Remove var from A\textprime \;\label{alg:l:loop2-e}
}
}
}
\uIf{$A\textprime$ does not build correctly}{\label{alg:l:build-b}
put back $stmt$ \;
undo $var$ and $func$ removal from A\textprime\; \label{alg:l:undo}
}\label{alg:l:build-e}
}
}
\end{algorithm}
We used \textit{Benchmark\_1} to compare the performance of \mbox{\textsc{LMCAS}}\xspace against manual debloating, baseline, Nibbler, and OCCAM.
Figure~\ref{fig:avgReduction} shows the comparison results based on the reduction in the binary size that each tool achieved for each app in \textit{Benchmark\_1}. For computing the binary-size metric, we compiled all debloated apps with \texttt{gcc -O2}, and ran \texttt{size}.
\begin{figure}[!htb]
\centering
\includegraphics[width=1\columnwidth]{images/AvgReductionPerApp.pdf}
\caption{Binary size reduction achieved through manual debloating, baseline, Nibbler, OCCAM, and \mbox{\textsc{LMCAS}}\xspace. (Higher numbers are better.)}
\label{fig:avgReduction}
\end{figure}
\section{\mbox{\textsc{LMCAS}}\xspace Running Time}\label{apndx:runTimeLMCAS}
We measured the running time of \mbox{\textsc{LMCAS}}\xspace. Figure~\ref{fig:analysisTime} shows the breakdown of running time, for \textit{Benchmark\_1}, between (i) partial interpretation, and (ii) Constant Conversion (CS) plus Multi-stage Simplification (MS).
The average total running time is $3.65$ seconds;
the maximum total running time is $13.08$ seconds for analyzing \texttt{sort};
and the lowest total analysis time is $1.19$ seconds for analyzing \texttt{basename}.
Notably, the time for Constant Conversion and Multi-stage Simplification is low:
on average, the time for constant conversion and multi-stage simplification is $0.4$ seconds, while the average time for Partial Interpretation is $3.25$ seconds.
\begin{figure}[!htb]
\centering
\includegraphics[width=1\columnwidth]{images/TimeAnalysis.pdf}
\caption{The time required for partial interpretation (PI) and the partial-evaluation steps (Constant Conversion and Multi-stage Simplifications) for \textit{Benchmark\_1}.
\Omit{\twr{Please change the label on the y-axis to "Running Time", and change the
key from "CC \& MS" to "Partial Evaluation" to be consistent with the caption. We need \underline{all} labeling and terminology to be consistent \underline{everywhere}.}}}
\label{fig:analysisTime}
\end{figure}
\rev{
}
\section{CFI Experiment} \label{apndx:cfi}
\begin{table}[!htb]
\centering
\caption{Total unique gadgets count for original binaries, debloated binaries using \mbox{\textsc{LMCAS}}\xspace}
\small
\scalebox{0.92}{
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{2}{*}{\textbf{Program}} & \multicolumn{4}{c|}{\textbf{Total ROP Count}} \\ \cline{2-5}
& \multicolumn{1}{c|}{\textbf{Original}} & \textbf{LLVM-CFI} & \textbf{LMCAS} & \textbf{\shortstack{LMCAS\\+LLVM-CFI}}\\ \hline
basename & 1794 & 964 & 841 & 261 \\
basenc & 3063 & 1805 & 1309 & 793 \\
comm & 2095 & 1145 & 964 & 794 \\
date & 12119 & 3654 & 3381 & 1592 \\
du & 15094 & 7874 & 8503 & 5873 \\
echo & 1835 & 446 & 876 & 442 \\
fmt & 2496 & 1403 & 1230 & 1158 \\
fold & 2094 & 1168 & 1015 & 769 \\
head & 2671 & 1398 & 1366 & 932 \\
id & 2514 & 1214 & 1183 & 801 \\
kill & 1924 & 1147 & 919 & 1054 \\
realpath & 3073 & 1610 & 1664 & 1658 \\
sort & 7558 & 4804 & 4185 & 3699 \\
uniq & 2516 & 1280 & 1121 & 776 \\
wc & 2225 & 1611 & 1320 & 973 \\
objdump & 115587 & 103241 & 107985 & 80156 \\
readelf & 58186 & 50512 & 56519 & 45320 \\
tcpdump & 82682 & 53205 & 67417 & 50809 \\
Chown & 2890 & 2280 & 2529 & 1998 \\
rm & 3068 & 2316 & 2579 & 2083 \\ \hline
\end{tabular}}
\label{tab:cfiAnalysis}
\end{table}
\section{\mbox{\textsc{LMCAS}}\xspace Framework}
\label{sec:approach}
This section introduces \mbox{\textsc{LMCAS}}\xspace, a lightweight debloating framework that uses sound analysis techniques to generate specialized programs.
Figure~\ref{fig:approach} illustrates the architecture of \mbox{\textsc{LMCAS}}\xspace.
The debloating pipeline of \mbox{\textsc{LMCAS}}\xspace receives as input the
whole-program LLVM bitcode of program $P$, and\Omit{\twr{wording changes}}
performs the following major steps to generate a specialized program $P\textprime$ as bitcode, which is ultimately converted into a binary executable.
\Omit{\twr{Is the overall transformation bitcode-to-bitcode? source-to-bitcode? source-to-binary? bitcode-to-binary? Somewhere we used to say that neck identification was performed on the source code---I may have removed that phrase; that seems like an inessential detail: it may be how it is done in our implementation, but I would expect that, in principle, there is nothing that precludes it from being carried out on the bitcode.} \ma{If we consider that in principle, the neck can be specified on top of the bitcode, then our overall transformation is bitcode-to-bitcode. If this makes sense, I'll do the required changes}}
\begin{itemize}
\item \textbf{Neck Identification Requirements (Section~\ref{subsec:neck}).} As mentioned in Section~\ref{subsec:challenges}, the \textit{neck} is used to partition the code into the part on which symbolic execution is applied, and the part on which a number of optimizing transformations are applied.\Omit{\twr{Reworded:}}
In our current system, neck identification is performed manually.
Section~\ref{subsec:neck} discusses neck identification, and formalizes the problem for which we plan to develop automatic support.
\Omit{\vr{This seems wrong. We do not have an algorithm but only a formalization of the problem we need to solve.} \ma{ture}}
\item \textbf{Symbolic Execution (Section~\ref{subsec:SE}).}
Symbolic execution executes the original application up to the neck, based on a set of supplied inputs that control which functionality should be supported by the specialized application.
The output of the symbolic-execution phase provides (i) a set of global and stack variables and their corresponding values, and
\Omit{\twr{Clarify: presumably this refers to visited functions and basic blocks \emph{up to the neck},} \ma{Yes, thanks for the clarification}}
(ii) a record of the visited functions and their basic blocks up to the neck.
\item \textbf{Constant Conversion (Section~\ref{subsec:cc}).}
This phase converts the variables and their corresponding values captured after symbolic execution (i.e., $V_s$ and $C_s$) to settings of constants at the neck.
This phase also takes advantage of the opportunity to boost the degree of subsequent optimization steps by (i) supporting the conversion of multiple kinds of data types to constants, and (ii) performing constant-conversion in both the configuration logic and the main logic of the program.
\item \textbf{Multi-Stage Simplification (Section~\ref{subsec:msp}).}
This phase applies selected standard and customized LLVM passes for optimizing the program and removing unnecessary functionality.
These optimization steps are arranged and tailored to take advantage of the contribution provided by the constant-conversion phase.
\end{itemize}
\begin{figure}[tb]
\centering
\includegraphics[width=9cm, height=3cm]{images/debloating_v2.png}
\caption{Overview of \mbox{\textsc{LMCAS}}\xspace.}
\label{fig:approach}
\end{figure}
\subsection{Neck Specification} \label{subsec:neck}
\input{neck}
\begin{comment}
The neck defines a splitting point between configuration and main logic in a program $P$ as illustrated in Figure~\ref{fig:neck}. Identifying this location accurately is of paramount importance to optimize and conduct sound analysis. The neck has two major functionalities: (1) terminating the execution of the symbolic execution after reaching the neck. Hence, our neck identification algorithm should be able in an airtight manner to distilling the variables influenced by the supplied inputs. 2) providing boundaries where constant propagation should be performed. All variables captured in the symbolic execution that their uses are located before the neck, they can be safely converted to constants.
In order to automatically identify the location where the symbolic execution should be interrupted, we define the neck identification criteria as a tuple L= <I, V, C, F>, where:
\begin{itemize}
\item \textbf{I} is a set of supplied inputs that activate the intended functionality. These inputs can be in the form of command-line arguments.
\item \textbf{V} is a set of variables that will be affected by the identified inputs \textit{I} and contribute to the activated functionality. This set comprises local and global variables.
\item \textbf{C} is a set of constant values that correspond to the set of variables \textit{V}.
\item \textbf{L} is the location of the neck in the original application and based on it we identify the relevant variables \textit{V}.
\item \textbf{F} is a functionality that will be supported by the specialized application based on the given set of inputs \textit{I}. This functionality should be provided after running the specialized application.
\end{itemize}
To verify if the selected neck location \textit{(L)} is the optimal, the functionality \textit{(F)} should be satisfied after generating the specialized app P\textprime. Otherwise, we keep iterating new nick locations until the functionality \textit{(F)} is satisfied. This process can be guided by relying on heuristic analysis. For example, GNU Core Utilities apply the same approach for parsing the inputs \textit{(I)}.
\end{comment}
\subsection{Symbolic Execution (SE)} \label{subsec:SE}
The symbolic-execution phase is a supporting phase whose goal is to identify---at the neck---the set of variables (and their values) that contribute to the desired functionality.
Symbolic execution is performed only up to the neck;
all functions and basic blocks visited during symbolic execution are recorded.
After symbolic execution terminates, the current symbolic state is saved, and the values of all variables are extracted.
Different types of variables are extracted, including base types (e.g., \texttt{int}, \texttt{char}) and constructed/compound types (e.g., \texttt{enum}, pointer, array, and \texttt{struct}).
\Omit{\twr{Omit starting with
``Our approach supports capturing these various types of data types to perform a sound analysis.'' I think it is something you've written to defend/distinguish your work w.r.t. CHISEL, OCCAM, and TRIMMER, but it is only intrusive in the presentation of \emph{your} work.} \ma{I agree}}
\Omit{\twr{Does the following sentence capture what you intended to say?} \ma{ Yes}}
Consider a network-monitoring tool, such as \texttt{tcpdump}, that supports multiple network interfaces, and takes as input a parameter that specifies which network interface to use.
A specialization scenario might require monitoring a specific interface (e.g., Ethernet) and capturing a specific number of packets: for \texttt{tcpdump}, the command would be ``\texttt{tcpdump -i ens160 -c 100}'' (see the inputs listed in Table~\ref{tab:largeApps}, Section~\ref{sec:evaluation}).
The first argument identifies the Ethernet interface $\texttt{ens160}$, while the second argument specifies that the number of packets to be captured is $100$.
The former argument is a string (treated as an array of characters in C) and the latter is an \texttt{int}.
\Omit{\twr{Omit the following sentence: I don't think it adds anything:}}
Returning to the example from \S\ref{sec:motivation} (Listing~\ref{wc}), Figure~\ref{fig:neck} illustrates the location of the neck.
Suppose that the desired functionality is to count the number of lines (i.e., \texttt{wc -l}).
Table~\ref{tab:vars} shows a subset of the variables and their corresponding values that will be captured and stored in \mbox{\textsc{LMCAS}}\xspace's database after symbolic execution finishes.
\begin{table}[!htb]
\centering
\caption{The set of captured variables $V_s$ and the corresponding values obtained after symbolic execution.}
\begin{tabular}{lllc}
\hline
Variable & Type & Scope & Value \\ \hline
\texttt{total_lines} & \multirow{2}{*}{\texttt{int}} & \multirow{2}{*}{Global} & 0 \\
\texttt{total_chars} & & & 0 \\ \hline
\texttt{flag->count_lines} & \texttt{int} & \multirow{2}{*}{Local} & 1 \\
\texttt{flag->count_chars} & \texttt{char} & & 0 \\ \hline
\end{tabular}
\label{tab:vars}
\end{table}
\subsection{Constant Conversion (CC)} \label{subsec:cc}
\Omit{\twr{I don't understand why you have to do constant conversion in the main logic.
After constant conversion in the pre-neck, why don't all the post-neck simplifications fall out for free via the constant propagation and \texttt{simplifycfg}? \ma{standard constant propagation LLVM pass perform mainly constant folding see (https://releases.llvm.org/6.0.0/docs/Passes.html\#constprop-simple-constant-propagation) i.e., x = 3 + 4 becomes x = 7 (I tested x1= 1, x2= 2, x= x1+x2, but the constprop pass hasn't done constant propagation for x1 and x2 by making x=20), are limited to global variables, and don't handle data types required for our analysis. I also tried the other LLVM pass ipconstprop on the previous example but doesn't propagate constant in the previous example. Furthermore, constant propagation passes don't account for aliasing, therefore we need to perform the post-neck CC. \texttt{simplifycfg} does folding only for control-flow instructions for simplifying and merging BBs. I'll add this explanation at the beginning of Main-logic CC, as we discussed}}
\vr{I agree with Tom here. When you consider LLVM optimizations, you have to consider all of them together, not pick out just one and test it alone. I tested your example and LLVM handles it perfectly.}}
\Omit{\twr{Wording changes OK?}\ma{Yes, thanks}}
We perform constant conversion for variables not only before the neck but also after the neck. These conversions enable further simplifications in later steps.
Constant conversion is performed in two steps: \textit{Configuration Logic Constant-Conversion} and \textit{Main Logic Constant-Conversion}. These two steps are discussed below.
\noindent\textbf{Configuration Logic Constant-Conversion.}
This step aims to propagate constants in the configuration logic to achieve further optimizations. The constant conversion is performed upstream of the neck, thus the variables in the configuration logic are converted to constants in a backward direction based on the captured vales at the neck.
\twr{If I were a reviewer, the following sentences would raise a red flag with me. They sound like nonsense: are you replacing occurrences of variables in the pre-neck with their values at the neck? That can't be right: an occurrence $vo$ of variable $v$ can only be replaced with a constant value $c$ if (i) $c$ was the only value that $vo$ took on whenever that point in the program was reached, and (ii) the replacement value is $c$.
But what if the value of $v$ at the neck is $c'$ and not $c$. Will you replace $vo$ with $c$ or $c'$? When I read this paragraph, it leads me to think that $vo$ is replaced by $c'$, because I don't see anything that makes me think that you have captured the earlier value $c$.}
The def-use chain of global and stack variables is utilized to replace all occurrences of the variables with their corresponding constant values in the program code before the neck.
\twr{The following sentence is problematic.
Yes, some of the statements will become dead, but not all of them do, and you still have to perform a sound transformation of the pre-neck.}
Because some of the program statements will become dead after constant-conversion, the replacement is performed for all occurrences of the set of variables obtained after symbolic execution.
Algorithm~\ref{alg:cc} describes the transformations that are performed during this step. The algorithm receives as input the bitcode of the whole program $P$ generated using WLLVM\footnote{https://github.com/SRI-CSL/whole-program-llvm}, as well as a dictionary (similar to Table~\ref{tab:vars}) that maps the set of variables in $V_s$ captured after symbolic execution and their constant values $C_s$. The set $V_s$ involves global and stack variables (base-type, struct, and pointer variables).
Algorithm~\ref{alg:cc} iterates over the IR instructions to identify
the locations where the variables are accessed, which is indicated by load instructions at Line~\ref{alg:l:fndLoadInst}. Then, we replace the loaded value with the corresponding constant value.
This approach works for global variables (Lines~\ref{alg:l:globals-b}-\ref{alg:l:globals-e}) and stack variables with base types (Lines~\ref{alg:l:base-b}-\ref{alg:l:base-e}).
But for pointers to base variables, Algorithm~\ref{alg:cc} identifies locations
where the pointer is modifying the base variable by looking for store
instructions whose destination operand belongs to the list of pointer variables to base type at Line~\ref{alg:l:PtrBaseVars}.\Omit{\vr{grammar seems off} \ma{where?}}\Omit{\vr{see if
the rewording of the previous sentence is ok. You have not defined what PtrBaseVars is.}\ma{Fixed the PtrBaseVars}} We then replace the source operand of the store operation to use the constant value corresponding to the actual base variable pointed to by the pointer (Lines \ref{alg:l:PtrToBase-replc1}-\ref{alg:l:PtrToBase-replc2}).
\Omit{\vr{How do you handle pointer to pointer} \ma{both pointers point to the same memory location. Therefore, the constant values obtained from KLEE are based on the last modification and thus we don't need to apply alias/pointer analysis}\vr{This is important and is still not addressed. You would need a more general solution using a points-to analysis like you do for main logic.}}
Stack variables that are Structs and pointers to Structs are handled at Lines~\ref{alg:l:strctPtrToStrct-b}-\ref{alg:l:strctPtrToStrct-e}, we first need to identify the memory address that is pointed to by these variables using \Omit{\twr{What exactly is \texttt{GEPInstr}?}}
\texttt{GEPInstr} at Line~\ref{alg:l:strctPtrToStrct-b}.
\texttt{GEPInstr} facilitates tracing back to finding the corresponding struct and pointer-to-struct variables at Line~\ref{alg:l:strctPtrToStrct-traceBack}.
The algorithm then at Line~\ref{alg:l:strctPtrToStrct-findStor} iterates the occurrences of the \texttt{GEPInstr} by leveraging the LLVM method \texttt{users()} to identify store operations that modify the variable.
Finally, it converts the source operand of the store operations to become constant, thus it reinforces the new constant value as described at Line~\ref{alg:l:strctPtrToStrct-e}.
In addition to the tracing back, we also use the recorded element index in symbolic execution to identify which struct element should be converted.
For instance, in the motivating example, the struct pointed to by \texttt{flag} consists of two elements (\texttt{count\_lines} and \texttt{count\_chars}), so we maintain their indices correctly to perform an accurate constant conversion.
For string variables, Alogrithm~\ref{alg:cc} identifies the instructions that represent string variables
at Line~\ref{alg:l:strgVar}. It then creates an array and assigns the string to the created array, which will be converted to a global variable (Lines~\ref{alg:l:crtAry}-\ref{alg:l:crtGbl}) because LLVM treats strings as global variables. Finally, the algorithm identifies store instructions that use the string variable as its destination operand, and overrides source operand of the store instruction to use the constant string value assigned to the created global variable (Lines~\ref{alg:l:findstrStore}-\ref{alg:l:replaceStoreStr}).
According to the motivating example (see Listing~\ref{wc}), no replacement is performed for the global variables \texttt{total\_lines} and \texttt{total\_chars} before the neck, because of the absence of any occurrences. But the replacement is performed for the elements of the pointer to the struct \texttt{flag}. Therefore, the occurrences of $\texttt{flag->count\_chars}$, $\texttt{flag->count\_lines}$ at lines 13 and 14 are replaced with the their corresponding constant values listed in Table~\ref{tab:vars}.
\Omit{\twr{In Line 3, the iteration is over \emph{every} instruction of $P$. However, I thought constant-conversion was only performed on the post-neck portion of $P$. I thought some different algorithm was employed to simplify the pre-neck portion. \ma{I got your point about the iteration, I'll modify the algorithm to stop instruction iteration at the neck. For example if there is a condition located before the neck `if (x)`, and SE provided the value of x is 0, this means the condition is always false, so I need to convert x here to 0 though it's before the neck, therefore the simplifyCFG can recognize this branch will not be taken and thus simplifies it. Otherwise, simplifyCFG can't remove the branch} In any case, it does not seem sound to use the values from the neck in the pre-neck portion: the values are the result of a computation performed by the code up to the neck.}\ma{I need to do the pre-neck CC, because this will facilitate the simplification steps that will be performed subsequently.}}
The constant conversion operates using updated and accurate values, because symbolic execution phase handles situations where a pointer indirectly updates the value of a memory location.
\Omit{The set of global variables may include a plethora of variables (not just the variables in Table~\ref{tab:vars}).
However, not all global variables in $V_s$ contribute to the functionality that the \mbox{\textsc{LMCAS}}\xspace user desires to enable in the specialized program. Therefore, the constant-conversion step at Line $2$ removes global variables in $V_s$ that are irrelevant.
Removing irrelevant variables 1) reduces the overhead of the specialization process, and 2) supports generating accurate specialized applications by avoiding side effects (i.e., \Omit{\twr{How can missing a variable cause a side effect?} \ma{crashing, run forever, wrong output}} missing a variable, or converting at a wrong location)\Omit{\twr{``resulted after the constant conversion'' is not grammatical; do you mean ``caused by constant conversion''?} \ma{ yes}}
caused by constant conversion
\Omit{\vr{This paragraph is very confusing to me. Do we have some examples of
variables that are irrelevant? \ma{e.g., argc} I too do not see why this should be required.
Moreover, Line 2 is a blackbox. Can you explain the algorithm for
FilterGlobalVars()? \ma{explained in the next paragraph "irrelevant global variables will be removed by comparing"}}}}
\Omit{\twr{You say ``all irrelevant variables'' whereas I would say just ``irrelevant variables.'' It is a red flag to say ``all'' because the problem of identifying irrelevant variables is undoubtedly undecidable. Now it occurs to me that the problem statement for this phase must be ``Identify a subset of all definitely relevant variables,'' which means that you are over-approximating the set of (all) irrelevant variables.} \ma{Thanks for explaining that. I agree.}}
\Omit{\twr{Please start the paragraph by saying what the purpose of this filtering step is.
In hidden notes, you seemed to have agreed that the specification of the step is to identify a "subset of all definitely relevant variables".
You seem to be achieving that by removing an overapproximation of irrelevant variables.
}
\Omit{
\twr{I don't understand what Line 2 does: if SE gives you a set of global variables, and LLVM gives you the set of all global variables, what are you checking for? At best you can check that the former is contained in the latter.} \ma{yes, this is exactly what I did, check intersection between LLVM list and KLEE list of global variables}
\twr{If SE has supplied a set of \emph{global variables} in parameter \textit{GlobalVars}, you already know that they are global variables, and hence it is unnecessary to check that they are global variables w.r.t. the LLVM list. Your implementation may perform such a check as a way to catch bugs, but there is no reason to include such a check in the pseudo-code---it is just an irrelevant detail.}\ma{Agreed, it's not a major step. I will remove it from the algorithm and from the text.}}
\noindent\textbf{Main Logic Constant-Conversion.}
\Omit{\vr{incomplete sentence.}\ma{thanks. I fixed it}}This step is applied only for the set of variables that are not modified in the main logic.\Omit{\twr{Not understanding ``by iterating the usage of the variable''. It sounds like you are doing reachability in some kind of graph (built as part of the LLVM IR?) Can you describe it that way?} \ma{ I added a description in the 1st place when I mentioned the API users()}}
\Omit{\twr{wording changes OK -- to end of paragraph?}\ma{Yes, thanks}}We identify variables with this property using the def-use chain at each variable occurrence of interest.
The absence of a store instruction indicates that no write operation is performed on the variable;
in this case, the value of the variable is not changed after the neck, and it is safe to perform the constant-conversion. \Omit{\twr{Something is definitely incorrect here: This is the section about \emph{main-logic} constant-conversion, but Algorithm~\ref{alg:cc} is for \emph{configuration logic}}\ma{I'll fix it} \vr{looking forward to seeing the fix/}\ma{I need to revisit. I'm not quite sure how to put preneck and postneck in one algorithm}
Although LLVM provides a set of standard constant propagation passes, these passes do not handle data types considered in our analysis, mainly perform constant folding, and lack of handling aliasing. Therefore, we also adapt our Algorithm~\ref{alg:cc} to address the challenges presented by aliasing~\cite{PGL-014}, as described at Lines~\ref{alg:l:alias}-\ref{alg:l:alias-e}:
constant-conversion should be performed for
a variable occurrence only if it can be shown that it is not written to after the neck, either directly or via an alias.
Generally, our analysis needs only to be worried about aliasing for a small set of variables maintained by the dictionary $Map$ (mapping of $V_s$ to $C_s$ in Algorithm~\ref{alg:cc}).
To this end, we use LLVM alias-analysis APIs to obtain the set of all variables that might write to the memory locations in which we are interested. \Omit{\vr{Can we add a bit more detail? As a reviewer, I will be interested in words like ``interprocedural'' and ``may-alias.''}\ma{I explained mayAlias, not quite sure where are you referring to by interprocedural}}\Omit{\ma{I don't want to put low-level details, but in case you've any concerns, LLVM provides an API to perform AliasAnalysis, so my implementation leverage these APIs to provide me the set of aliases (mayAlias not mustAlias, over-approximate, but to avoid wrong constant conversion) of a certain memory location, then I check if none of them is modified after the neck to proceed with the CC. Just to be clear, I haven't implemented the AliasAnalysis, it's provided by LLVM}
Finally, the constant conversion is performed after all these variables are examined to verify they do not make any modification after the neck.
\Omit{\vr{slightly reworded this para}\ma{thanks}}
In the motivating example (see Listing~\ref{wc}), the global variables \texttt{total\_lines} and \texttt{total\_chars} are not converted to constants because their values are modified after the neck at Lines 17 and 18. However, the struct elements $\texttt{flag->count\_chars}$, $\texttt{flag->count\_lines}$ used in the conditional statements at Lines 17-20 are replaced with the their corresponding constant values listed in Table~\ref{tab:vars}. This makes the conditions at Lines 17 and 19 always false, but the conditions at Lines 18 and 20 are always true. These post-neck constant-conversions facilitate the simplifications performed in the next phase Multi-Stage Simplification.
\Omit{\vr{The language needs improvement, I am not able to understand much of the
stuff here. Do we have a pointer analysis or not? \ma{No, that's why I didn't mention that. I basically, was benefiting from def-use chain API provided by LLVM. A single storeinstr after the neck indicates the variable's value is modified and I was doing this for all variables in case of pointer to variable} Note that we can work with a
conservative analysis that provides soundness guarantees. For example, if the
address of a variable is never taken, we can make it constant without worry. \ma{I don't get your point here "Note that we can work with a conservative analysis ... without worry", because I think what you described here is a summary of the previous paragraph.}}}
\begin{comment}
\begin{algorithm}[!htb]
\small
\caption{\mbox{\textsc{LMCAS}}\xspace Constant-Conversion
\label{alg:cc}
\DontPrintSemicolon
\KwIn{program P, Map <$V_s, C_s$>
L=$NeckLoc$}
\KwOut{$P_{cc}$}
\SetKwProg{Def}{def}{:}{}
$P_{cc} \longleftarrow P$ \;
\Repeat{$Instr \neq L$}{
\For{$Instr \in P_{cc}$}{
\If{$Instr$ is $LoadInstr$}{
\textcolor{blue}{\tcc{Handle Global Variables}}
\If{$Instr(oprnd) \in Globals$}{
$replaceWithConst(oprnd)$\;
}
\textcolor{blue}{\tcc{Handle Base Variables}}
\If{$Instr(oprnd) \in AllocInstr$ $\land$ $oprnd \in BaseVars$}{
$replaceWithConst(oprnd)$\;
}
}
\textcolor{blue}{\tcc{Handle Pointer to Base Variables}}
\If{$Instr$ is $StoreInstr$}{
\If{$Instr(oprnd2) \in AllocInstr$ $\land$ $Instr(oprnd) \in PtrBaseVars$}{ \label{alg:l:PtrBaseVars}
\If{oprnd before $L$}{
new $\longleftarrow$ createStoreInstr(oprnd.val)\;
ReplaceInstWithInst(Instr, new)\;
}
}
}
\textcolor{blue}{\tcc{Handle Struct \& Pointer to struct Variables}}
\If{$Instr$ is $GEPInstr$}{
\If{$Instr(oprnd)$ is $AllocInstr$ $\land$ $Instr(oprnd) \in \{StructVars, PtrStrctVars\}$}{
\For{$us \in use(Instr)$}{
\If{$us$ is $StoreIntr$}{
new $\longleftarrow$ createStoreInstr(oprnd.val)\;
ReplaceInstWithInst(us, new)\;
}
}
}
}
\textcolor{blue}{\tcc{Handle String Variables}}
\If{$Instr$ is $AllocInstr$ $\land$ $Instr \in StringVars$}{
ary $\longleftarrow$ CreateArray(Instr.str) \;
gbl $\longleftarrow$ CreateGlobalVar(ary) \;
\For{$us \in use(Instr)$}{
\If{$us$ is $StoreIntr$}{
new $\longleftarrow$ createStoreInstr(gbl) \;
ReplaceInstWithInst(us, new)\;
}
}
}
}
}
\end{algorithm}
\end{comment}
\begin{algorithm}[!htb]
\scriptsize
\caption{\mbox{\textsc{LMCAS}}\xspace Constant-Conversion
\label{alg:cc}
\DontPrintSemicolon
\KwIn{program P, Map <$V_s, C_s$>, NeckLocation=$L$}
\KwOut{$P_{cc}$}
\SetKwProg{Def}{def}{:}{}
$P_{cc} \leftarrow P$ \;
\textcolor{blue}{\tcc{Configuration Logic CC}}
ConstantConversion(PreNeck($P_{cc}$), Map) \;
\textcolor{blue}{\tcc{Main Logic CC}}
\For{$var \in Map.keys$}{
identify aliases of $var$\; \label{alg:l:alias}
\If{none of the aliases modify the address $var$ after $L$ }{\label{alg:l:alias-b}
add all $var$'s aliases to $Map$\; \label{alg:l:alias-e}
} \Else{
remove $var$ and its value from $Map$ \;
}
}
ConstantConversion(PostNeck($P_{cc}$), Map) \;
\Def{ConstantConversion(ProgamPart, Map)}{
\For{$Instr \in ProgamPart$}{
\If{$Instr$ is $LoadInstr$}{\label{alg:l:fndLoadInst}
\textcolor{blue}{\tcc{Handle Global Variables}}
\If{oprnd is Global Variable}{\label{alg:l:globals-b}
$replaceInstrWithConst(Map[oprnd])$\;
}\label{alg:l:globals-e}
\textcolor{blue}{\tcc{Handle Base Variables}}
\If{oprnd is a Base Variable}{\label{alg:l:base-b}
$replaceInstrWithConst(Map[oprnd])$\;
}\label{alg:l:base-e}
}
\textcolor{blue}{\tcc{Handle Pointer to Base Variables}}
\If{$Instr$ is $StoreInstr$}{ \label{alg:l:PtrToBase-b}
\If{destination operand (destOprd) is a pointer to Base Variable}{ \label{alg:l:PtrBaseVars}
new $\leftarrow$ createStoreInstr(Map[destOprd],destOprd)\;\label{alg:l:PtrToBase-replc1}
ReplaceInstWithInst(Instr, new)\;\label{alg:l:PtrToBase-replc2}
}
}\label{alg:l:PtrToBase-e}
\textcolor{blue}{\tcc{Handle Struct \& Pointer to struct Variables}}
\If{$Instr$ is $GEPInstr$}{ \label{alg:l:strctPtrToStrct-b}
\If{oprnd is a struct/pointer to struct}{ \label{alg:l:strctPtrToStrct-traceBack}
\For{Find $StoreIntr$ uses $GEPInstr$}{\label{alg:l:strctPtrToStrct-findStor}
new $\leftarrow$ createStoreInstr(Map[oprnd],oprnd)\;
ReplaceInstWithInst($StoreIntr$, new)\;
}\label{alg:l:strctPtrToStrct-e}
}
}
\textcolor{blue}{\tcc{Handle String Variables}}
\If{$Instr$ is $AllocInstr$ and $Instr$ String Variable}{ \label{alg:l:strgVar}
ary $\leftarrow$ CreateArray(Map[$Instr$]) \;\label{alg:l:crtAry}
gbl $\leftarrow$ CreateGlobalVar(ary) \;\label{alg:l:crtGbl}
\For{Find $StoreIntr$ uses $AllocInstr$}{\label{alg:l:findstrStore}
new $\leftarrow$ createStoreInstr(gbl, $AllocInstr$) \;
ReplaceInstWithInst($StoreIntr$, new)\;
}\label{alg:l:replaceStoreStr}
}
}
}
\end{algorithm}
\begin{algorithm}[!htb]
\scriptsize
\caption{\mbox{\textsc{LMCAS}}\xspace Clean up}
\label{alg:cp}
\DontPrintSemicolon
\KwIn{$P_{cc}$, visitedFunc}
\KwOut{$P$\textprime}
\SetKwProg{Def}{def}{:}{}
$P\textprime\leftarrow P_{cc}$ \;
\textcolor{blue}{\tcc{Remove unused functions}}
CG $\leftarrow$ constructCallGraph($P_{cc}$) \;
\For{$func \in CG$}{
\If{$func\notin visitedFunc~\land$ $func$ is not operand of other instructions}
remove(func) \;
}
}
\For{$func \in CG$}{
\If{$func$ is not operand of other instructions}
removeNodeAndItsDescendant(func) \;
}
}
\textcolor{blue}{\tcc{Remove unused Global Variables}}
\For{$var \in getGlobalList(P_{cc})$}{
\If{$var$ is not operand of other instructions}
remove(var) \;
}
}
\textcolor{blue}{\tcc{Remove unused Stack Variables}}
\For{$func \in CG$}{
\For{$inst \in func$}{
\If{$inst$ is $AllocInst$}{
\If{$inst$ is not operand of other instructions}{
remove(inst) \;
}\uElseIf{$inst$ is a destination operand of only one $storeInst$}{
remove($storeInst$) \;
remove($inst$) \;
}
}
}
}
\end{algorithm}
\subsection{Multi-Stage Simplification (MS)} \label{subsec:msp}
\Omit{\vr{It is surprising you do not mention any of the dead code elimination passes.
LLVM also has a standard sequence of passes based on optimization level. Some of
these passes may be repeated. An informed reader will expect that you either use
the standard sequence or have repetitions of passes to get better optimization.} \ma{ Good point. But I mentioned in the clan up step that LLVM has passes to remove unused code and explained our motivation about not to use the standard dead code elimination passes}}
\Omit{\vr{Apparently LTO can remove unused functions.\\
\url{https://stackoverflow.com/questions/6687630/how-to-remove-unused-c-c-symbols-with-gcc-and-ld}\\
\url{http://lists.llvm.org/pipermail/llvm-dev/2020-January/138660.html} \ma{I didn't come across LTO, I relied on this LLVM documentation to find standard LLVM passes. https://releases.llvm.org/6.0.0/docs/Passes.html}. There is a post (scored 19) on the link that you shared says the generated app was buggy}}
\Omit{\vr{Also, StoreInst not being removed is surprising at least. Maybe give more reason
why that would happen. Just looking very briefly around, it seems StoreInst
removal is supported in dead code elimination.\\
\url{https://github.com/llvm-mirror/llvm/blob/master/lib/Transforms/Scalar/DeadStoreElimination.cpp\#L782}
\ma{Good point. I mentioned how and why this can happen "variables that are initialized but ended up unused"}}}
After variables have been converted to constants, the next step is to perform \Omit{\twr{whole-program}}whole-program optimization to simplify the code and remove unnecessary code.\Omit{\vr{slightly reworded next sentence.}\ma{thanks}}
In this phase, we used existing LLVM passes as well as wrote our own pass to perform powerful simplifications at different levels of granularity in the programs.
In particular, \mbox{\textsc{LMCAS}}\xspace uses the standard LLVM pass for constant propagation to perform constant folding;
it then uses another standard LLVM pass to simplify the control-flow.
\Omit{\twr{Is the following pass the one you wrote?
There is a difference in nuance between "we apply an LLVM pass customized"---which implies that you modified an existing pass---versus "we wrote our own pass"---which means that the pass is your own. Above you said "wrote our own pass", but the language used here does not match.}\ma{The MS phase uses 2 standard LLVM passes and 1 LLVM pass wrote by me, as mentioned at the beginning of this paragraph. I agree the description thereafter was vague. I fixed that}}
Finally, it applies \Omit{\twr{a custom LLVM pass}}our LLVM pass implemented to handle the removal of unnecessary code.
\Omit{
\twr{I think it is a mistake to state "cheap optimization" as the chief criterion. \ma{I'll rewrite it} What is missing is : "The next step is to perform interprocedural constant folding, plus removal of useless and unreachable code." What I understood you to say was that there was no single LLVM pass that performs that work, so you designed your own using a collection of standard LLVM passes (most of which work intraprocedurally) together with passes of your own design\ma{yes, correct summary}. Section 3.4 should tell that story. I suspect it is not relevant to even mention "cheap optimization".}\ma{got your point, I modified the beginning of Section 3.4}}
\begin{itemize}
\item \textbf{Constant Propagation}.
This optimization step folds variables that hold known values by invoking the standard LLVM constant-propagation pass.
\Omit{\twr{Unnecessary: "by passing \texttt{constprop} as an argument of the LLVM command \texttt{opt}." That is some minor detail of your implemenation.} \ma{ got it}}Constant folding allows instructions to be removed.
It also opens up the opportunity for additional simplifications, such as
``Simplifying CFG'' below.
\item \textbf{Simplifying the CFG}.
\mbox{\textsc{LMCAS}}\xspace benefits from the folded constants in the previous step to make further simplifications to the control-flow graph.
\mbox{\textsc{LMCAS}}\xspace leverages a standard LLVM pass, called \texttt{simplifycfg}, to obtain these simplifications.
This pass determines whether the conditions of branch instructions are always true or always false:
unreachable basic blocks are removed, and basic blocks with a single predecessor are merged.
The simplifications in this pass are achieved at the basic-block level.
\item \textbf{Clean Up}.\Omit{\twr{I think you need to say more about what was lacking in LLVM.} \ma{ I'll do}}
In the simplification pass, \mbox{\textsc{LMCAS}}\xspace removes useless code (i.e., no operation uses the result~\cite{cooper2011engineering}) and unreachable code, including dead stack and global variables and uncalled functions.
Although LLVM provides passes to perform aggressive optimization, we wrote a targeted LLVM pass that gives us more control in prioritizing the cleaning up of unneeded code, as described in Algorithm~\ref{alg:cp}.
\hspace*{1.5ex}
The first priority is to remove unused functions.
\Omit{
\twr{Once again, start off by stating the goal. What functions are you trying to remove? To characterize them, two categories might be (i) called only from call-sites before the neck, but not called during SE, and (ii) never called from the set of functions transitively reachable from \texttt{main} (including indirect call-sites).} \ma{thanks for the pointers. I modified the description.}}
The goal is to remove two categories of functions:
(i) those that are called only from call-sites before the neck, but not called during symbolic execution (this step is performed in Algorithm~\ref{alg:cp} Lines 4-6), and (ii) those that are never called from the set of functions transitively reachable from \texttt{main} (including indirect call-sites), as described in Algorithm~\ref{alg:cp} at Lines 7-9.
Function removal is performed after constructing the call graph at Line 3.\Omit{\ma{I added the following for discussing indirect call sites}} Algorithm~\ref{alg:cp} also checks the number of uses of a function at Lines $5$ and $8$ before removing the node to handle indirect call sites. This check prevents the removal of a function used via a function pointer.
\Omit{\twr{What is the difference in information content between "not in ... visited" and "number of uses is zero".} \ma{ I'll clarify that}}
\hspace*{1.5ex}
The focus then shifts to simplifying the remaining functions.
For removing global variables (Lines 10-12), we iterate over the list of global variables obtained through the LLVM API \texttt{getGlobalList} and remove unused variables.
Finally, we remove stack variables (Lines 10-13), including initialized but unused variables by iterating over the remaining functions and erasing unused allocation instructions.
(In general, standard LLVM simplifications do not remove a stack variable that is initialized but is not otherwise used because the function contains a store operation that uses the variable.
Our clean-up pass removes an initialized-but-unused variable by deleting the store instruction, and then deleting the allocation instruction.)
\end{itemize}
\Omit{\twr{Wording changes OK?} \ma{Yes, thanks}}
This phase performs simplifications at both the function level and the instruction level. Consider again the motivating example to create a specialized version of the word-count program (cf., Listing~\ref{wc}) that only counts lines (cf., Listing~\ref{wc-cl}).
After performing the \Omit{\twr{constant-conversion}}constant-conversion phase, both the \texttt{count\_chars} and \texttt{count\_lines} fields of the struct pointed to by stack variable \texttt{flag} are replaced by the constant values $0$ and $1$, respectively (See Table~\ref{tab:vars}).
The simplification steps remove the tests at lines $18$ and $20$ because the values of the conditions are always true.
Because the values of the conditions in the tests at lines $17$ and $19$ are always false, control-flow simplification removes both the tests and the basic blocks in the true-branches.
Furthermore, the removal of these basic blocks removes all uses of the global variable \texttt{total\_chars}, and thus the cleanup step removes it as an unused variable.
\section{\mbox{\textsc{LMCAS}}\xspace Framework}
\label{sec:approach}
\begin{figure}[tb]
\centering
\includegraphics[width=9cm, height=2.5cm]{images/LMCAS_Usenix_V3.pdf}
\caption{\revUN{\mbox{\textsc{LMCAS}}\xspace Workflow.}}
\label{fig:approach}
\end{figure}
This section introduces \mbox{\textsc{LMCAS}}\xspace, a lightweight debloating framework that uses sound analysis techniques to generate specialized programs.
Figure~\ref{fig:approach} illustrates the architecture of \mbox{\textsc{LMCAS}}\xspace.
The debloating pipeline of \mbox{\textsc{LMCAS}}\xspace receives as input the
whole-program LLVM bitcode of program $P$, and\Omit{\twr{wording changes}}
performs the following major phases to generate a specialized program $P\textprime$ as bitcode, which is ultimately converted into a binary executable.
\begin{itemize}
\item \textbf{Neck Miner (Section~\ref{subsec:neckSpec}).}
\revUN{Receives the program to be specialized and modifies it by adding a special function call that marks the neck. Section~\ref{subsec:neckSpec} describes our approach to identifying the neck based on heuristic and structural analysis.}
\item \textbf{Partial Interpretation (Section~\ref{subsec:SE}).}
Interprets the program
up to the neck \revUN{(i.e., terminates after executing the special function call inserted by the neck miner)}, based on the supplied inputs that control which functionality should be supported by the specialized application.
The output of this phase provides a precise partial state of the program at the neck.\revUN{ This partial state comprises the variables that have been initialized during the partial interpretation and their corresponding values.}
\item \textbf{Constant Conversion (Section~\ref{subsec:cc}).}
Incorporates into the program the partial state captured by the partial interpretation.
It converts the variables and their corresponding values captured in the partial state (i.e., $V_s$ and $C_s$) to settings of constants at the neck. This phase also provides the opportunity to boost the degree of subsequent optimization steps by supporting the conversion of multiple kinds of variables to constants.
\item \textbf{Multi-Stage Simplification (Section~\ref{subsec:msp}).}
Applies selected standard and customized LLVM passes for optimizing the program and removing unnecessary functionality.
These optimization steps are arranged and tailored to take advantage of the values introduced by the constant-conversion phase.
\end{itemize}
\subsection{Neck Miner} \label{subsec:neckSpec}
\input{neck}
\begin{comment}
The neck defines a splitting point between configuration and main logic in a program $P$ as illustrated in Figure~\ref{fig:neck}. Identifying this location accurately is of paramount importance to optimize and conduct sound analysis. The neck has two major functionalities: (1) terminating the execution of the symbolic execution after reaching the neck. Hence, our neck identification algorithm should be able in an airtight manner to distilling the variables influenced by the supplied inputs. 2) providing boundaries where constant propagation should be performed. All variables captured in the symbolic execution that their uses are located before the neck, they can be safely converted to constants.
In order to automatically identify the location where the symbolic execution should be interrupted, we define the neck identification criteria as a tuple L= <I, V, C, F>, where:
\begin{itemize}
\item \textbf{I} is a set of supplied inputs that activate the intended functionality. These inputs can be in the form of command-line arguments.
\item \textbf{V} is a set of variables that will be affected by the identified inputs \textit{I} and contribute to the activated functionality. This set comprises local and global variables.
\item \textbf{C} is a set of constant values that correspond to the set of variables \textit{V}.
\item \textbf{L} is the location of the neck in the original application and based on it we identify the relevant variables \textit{V}.
\item \textbf{F} is a functionality that will be supported by the specialized application based on the given set of inputs \textit{I}. This functionality should be provided after running the specialized application.
\end{itemize}
To verify if the selected neck location \textit{(L)} is the optimal, the functionality \textit{(F)} should be satisfied after generating the specialized app P\textprime. Otherwise, we keep iterating new nick locations until the functionality \textit{(F)} is satisfied. This process can be guided by relying on heuristic analysis. For example, GNU Core Utilities apply the same approach for parsing the inputs \textit{(I)}.
\end{comment}
\subsection{Partial Interpretation (PI)} \label{subsec:SE}
Partial interpretation is a supporting phase whose goal is to identify---at the neck---the set of variables (and their values) that contribute to the desired functionality.
Partial interpretation is performed by running a symbolic-execution engine, starting at program entry, and executing the program (using partial program states, starting with the supplied concrete values) along a single path to the neck.
After partial interpretation terminates, the partial state is saved, and the values of all variables are extracted.
Different types of variables are extracted, including base types (e.g., \texttt{int}, \texttt{char}) and constructed/compound types (e.g., \texttt{enum}, pointer, array, and \texttt{struct}).
\Omit{\twr{Omit starting with
``Our approach supports capturing these various types of data types to perform a sound analysis.'' I think it is something you've written to defend/distinguish your work w.r.t. CHISEL, OCCAM, and TRIMMER, but it is only intrusive in the presentation of \emph{your} work.} \ma{I agree}}
\Omit{\twr{Does the following sentence capture what you intended to say?} \ma{ Yes}}
Consider a network-monitoring tool, such as \texttt{tcpdump}, that supports multiple network interfaces, and takes as input a parameter that specifies which network interface to use.
A specialization scenario might require monitoring a specific interface (e.g., Ethernet) and capturing a specific number of packets: for \texttt{tcpdump}, the command would be ``\texttt{tcpdump -i ens160 -c 100}'' (see the inputs listed in Table~\ref{tab:largeApps}, Section~\ref{sec:evaluation}).
The first argument identifies the Ethernet interface $\texttt{ens160}$, while the second argument specifies that $100$ packets should be captured.
The former argument is a string (treated as an array of characters in C); the latter is an \texttt{int}.
\Omit{\twr{Omit the following sentence: I don't think it adds anything:}}
Returning to the example from \S\ref{sec:motivation} (Listing~\ref{wc}), Figure~\ref{fig:neck} illustrates the location of the neck.
Suppose that the desired functionality is to count the number of lines (i.e., \texttt{wc -l}).
Table~\ref{tab:vars} shows a subset of the variables and their corresponding values that will be captured and stored in \mbox{\textsc{LMCAS}}\xspace's database after partial interpretation finishes.
\begin{table}[!htb]
\centering
\caption{Partial program state that contains the set of captured variables $V_s$ and the corresponding values obtained at the neck after partial interpretation of the program in Listing~\ref{wc}.}
\scalebox{0.9}{
\begin{tabular}{lllc}
\hline
Variable & Type & Scope & Value \\ \hline
\texttt{total_lines} & \multirow{2}{*}{\texttt{int}} & \multirow{2}{*}{Global} & 0 \\
\texttt{total_chars} & & & 0 \\ \hline
\texttt{flag->count_lines} & \texttt{int} & \multirow{2}{*}{Local} & 1 \\
\texttt{flag->count_chars} & \texttt{char} & & 0 \\ \hline
\end{tabular}
}
\label{tab:vars}
\end{table}
\subsection{Constant Conversion (CC)} \label{subsec:cc}
This phase aims to propagate constants in the configuration logic to enable further optimizations.
For instance, this phase contributes to removing input arguments that were not enabled during symbolic execution, and thus allows tests that check those inputs to be eliminated.
Constant conversion is a non-standard optimizing transformation because optimization is
performed \emph{upstream} of the neck:
uses of variables in the configuration logic---which comes \emph{before} the neck---are converted to constants, based on values captured at the neck.
The transformations performed during this phase enforce that the state at the neck in the debloated program is consistent with the partial state of constants at the neck that was captured at the end of partial interpretation.
Standard dataflow analyses (e.g. Def-Use)~\cite{Storm} for global and stack variables is used to replace all occurrences of the variables with their corresponding constant values in the program code before the neck.
Because some of the program statements become dead after constant conversion, the replacement is performed for all occurrences (i.e., accesses) of the variables obtained after partial interpretation.
The CC phase receives as input the bitcode of the whole program $P$ generated using WLLVM,\footnote{https://github.com/SRI-CSL/whole-program-llvm} as well as a dictionary (similar to Table~\ref{tab:vars}) that maps the set of variables in $V_s$ captured after symbolic execution to their constant values $C_s$. The set $V_s$ involves global and stack variables (base-type, struct, and pointer variables).
The CC phase then iterates over the IR instructions to identify the locations where the variables are accessed, which is indicated by load instructions.
Then, it replaces the loaded value with the corresponding constant value.
This approach works for global variables and stack variables with base types.
However, for pointers to base variables, it is necessary to identify locations where the pointer is modifying a base variable (by looking for store instructions whose destination-operand type is a pointer to a base type).
The source operand of the store operation is modified to use the constant value corresponding to the actual base variable pointed to by the pointer.
\rev{
For stack variables that are Structs and pointers to Structs, we first need to identify the memory address that is pointed to by these variables, which facilitates tracing back to finding the corresponding struct and pointer-to-struct variables.
We then iterate over the use of the identified memory addresses to determine store operations that modify the variable (corresponding to the memory addresses).
}
Finally, we convert the source operand of the store operations to the appropriate constant.
We also use the element index recorded during symbolic execution to identify which struct element should be converted.
For string variables, we identify the instructions that represent string variables, create an array, and assign the string to the created array. Finally, we identify store instructions that use the string variable as its destination operand, and override the store instruction's source operand to use the constant string value.
In \texttt{wc} (Listing~\ref{wc}), no replacements are performed for global variables \texttt{total\_lines} and \texttt{total\_chars} before the neck: there are no such occurrences.
Replacements are performed for referents of the pointer-to-struct \texttt{flag}:
the occurrences of $\texttt{flag->count\_chars}$ and $\texttt{flag->count\_lines}$ at lines 13 and 14 are replaced with the corresponding values listed in Table~\ref{tab:vars}.
\subsection{Multi-Stage Simplification (MS)} \label{subsec:msp}
This phase begins with the result of constant conversion, and performs whole-program optimization to simplify and remove unnecessary code.
In this phase, we used existing LLVM passes, as well as one pass that we wrote ourselves.
In particular, \mbox{\textsc{LMCAS}}\xspace uses the standard LLVM pass for constant propagation to perform constant folding;
it then uses another standard LLVM pass to simplify the control flow.
Finally, it applies an LLVM pass we implemented to handle the removal of unnecessary code.
\noindent \textbf{Constant Propagation}.
This optimization step folds variables that hold known values by invoking the standard LLVM constant-propagation pass. Constant folding allows instructions to be removed.
\Omit{
It also opens up the opportunity for additional simplifications, such as ``Simplifying CFG'' below.}
\noindent \textbf{Simplifying the CFG}.
\mbox{\textsc{LMCAS}}\xspace benefits from the previous step of constant propagation to make further simplifications by invoking a standard LLVM pass, called \texttt{simplifycfg}.
This pass determines whether the conditions of branch instructions are always true or always false:
unreachable basic blocks are removed, and basic blocks with a single predecessor are merged.
\Omit{
The simplifications in this pass are achieved at the basic-block level. }
\begin{algorithm}[!htb]
\scriptsize
\caption{\mbox{\textsc{LMCAS}}\xspace Clean up}
\label{alg:cp}
\DontPrintSemicolon
\KwIn{$P_{cc}$, visitedFunc}
\KwOut{$P$\textprime}
\SetKwProg{Def}{def}{:}{}
$P\textprime\leftarrow P_{cc}$ \;
\textcolor{blue}{\tcc{Remove unused functions}}
CG $\leftarrow$ constructCallGraph($P'$) \;
\For{$\textit{func} \in CG$}{\label{alg:l:funcloop1-b}
\If{$\textit{func}\notin visitedFunc~\land$ $\textit{func}$ is not an operand of other instructions}{
remove $\textit{func}$ from $P'$ and CG\;\label{alg:l:funcloop1-e}
}
}
\For{$\textit{func} \in CG$}{\label{alg:l:funcloop2-b}
\If{$\textit{func}$ is not an operand of other instructions}
remove $\textit{func}$ from $P'$ and CG \;
remove $\textit{func}$'s descendent nodes from $P'$ and CG if they are not reachable from $\textit{main}$ \;\label{alg:l:funcloop2-e}
}
}
\textcolor{blue}{\tcc{Remove unused Global Variables}}
\For{$var \in getGlobalList(P_{cc})$}{\label{alg:l:gbls-b}
\If{$var$ is not an operand of other instructions}
remove var from $P'$ \;\label{alg:l:gbls-e}
}
}
\textcolor{blue}{\tcc{Remove unused Stack Variables}}
\For{$\textit{func} \in CG$}{\label{alg:l:stack-b}
\For{$inst \in \textit{func}$}{
\If{$inst$ is $AllocInst$}{
\If{$inst$ is not an operand of other instructions}{
remove inst from $P'$ \;
}\uElseIf{$inst$ is a destination operand of only one $storeInst$}{
remove $storeInst$ from $P'$ \;
remove $inst$ from $P'$ \;\label{alg:l:stack-e}
}
}
}
}
\end{algorithm}
\noindent \textbf{Clean Up}.
In the simplification pass, \mbox{\textsc{LMCAS}}\xspace removes useless code (i.e., no operation uses the result~\cite{cooper2011engineering}) and unreachable code, including dead stack and global variables and uncalled functions.
Although LLVM provides passes to perform aggressive optimization, we wrote a targeted LLVM pass that gives us more control in prioritizing the cleaning up of unneeded code, as described in Algorithm~\ref{alg:cp}, which receives the modified program $P_{cc}$ after the CC phase and the list of functions visited during the Partial Interpretation phase (\texttt{visitedFunc}).
The first priority is to remove unused functions.
The goal is to remove two categories of functions:
(i) those that are called only from call-sites before the neck, but not called during symbolic execution (Lines \ref{alg:l:funcloop1-b}-\ref{alg:l:funcloop1-e}), and (ii) those that are never called from the set of functions transitively reachable from \texttt{main}, including indirect call-sites (Lines \ref{alg:l:funcloop2-b}-\ref{alg:l:funcloop2-e}).
Function removal is performed after constructing the call graph at Line 3.\Omit{\ma{I added the following for discussing indirect call sites}}
To handle indirect-call sites, Algorithm~\ref{alg:cp} also checks the number of uses of a function, at Lines $5$ and $8$, before removing the function.
This check prevents the removal of a function invoked via a function pointer.
The focus then shifts to simplifying the remaining functions.
For removing global variables (Lines \ref{alg:l:gbls-b}-\ref{alg:l:gbls-e}), we iterate over the list of global variables
and remove unused variables.
Finally, we remove stack variables (Lines \ref{alg:l:stack-b}-\ref{alg:l:stack-e}), including initialized but unused variables by iterating over the remaining functions and erasing unused allocation instructions.
(In general, standard LLVM simplifications do not remove a stack variable that is initialized but not otherwise used because the function contains a store operation that uses the variable.
Our clean-up pass removes an initialized-but-unused variable by deleting the store instruction, and then the allocation instruction.)
In \texttt{wc} (Listing~\ref{wc}),
after the CC phase both the \texttt{count\_chars} and \texttt{count\_lines} fields of the struct pointed to by stack variable \texttt{flag} are replaced by the constants $0$ and $1$, respectively (see Table~\ref{tab:vars}).
The simplification steps remove the tests at lines $18$ and $20$ because the values of the conditions are always true.
Because the values of the conditions in the tests at lines $17$ and $19$ are always false, control-flow simplification removes both the tests and the basic blocks in the true-branches.
Furthermore, the removal of these basic blocks removes all uses of the global variable \texttt{total\_chars}, and thus the cleanup step removes it as an unused variable.
\section{Background} \label{sec:background}
In this section, we provide the necessary background on program analysis and software debloating that will be used in the remainder of the paper.
\noindent\textbf{Symbolic execution~\cite{SymbolicExecution} and partial evaluation~\cite{partialEvaluation}:} are generalizations of standard interpretation of programs, but there are key differences between them. Specifically, symbolic execution permits interpretation of a program with symbolic values, while partial evaluation precomputes program expressions in terms of the known static input. Therefore, partial evaluation specializes a program with respect to a subset of its input, which is know as \emph{supplied inputs}. In effect, partial evaluation performs optimizations such as loop unrolling, constant propagation and folding, function in-lining, etc. to produce the residual program~\cite{WIPER}. Hence, in contrast to symbolic execution the result of a partial evaluator (i.e. program specializer) is not the value of output variables, but another program (know as residual program or specialized program). Both techniques have been employed together to accomplish various tasks (i.e. software verification~\cite{Interleaving}).
\rev{
\noindent\textbf{KLEE}
\noindent\textbf{LLVM}
}
\noindent\textbf{Stack Variables:} temporary variables created by each function. For instance, the variables \emph{count\_chars} and \emph{count\_lines} are stack variables. Every time a function declares a new variable, it is "pushed" onto the stack. Then every time a function exits, all of the variables pushed onto the stack by that function are deleted. Once a stack variable is freed, that region of memory becomes available for other stack variables.
\section{Conclusion}
\label{sec:Conclusion}
In this paper, we present \mbox{\textsc{LMCAS}}\xspace, a practical and lightweight debloating approach for generating specialized applications.
To speed up our analysis, \mbox{\textsc{LMCAS}}\xspace introduces the neck concept, a splitting point where ``configuration logic'' in a program hands off control to the ``main logic'' of the program. We develop a neck miner for alleviating the amount of manual effort required to identify the neck. \mbox{\textsc{LMCAS}}\xspace only applies partial interpretation up to the neck. The main logic is then optimized according to the values obtained from analyzing the configuration logic. Therefore, \mbox{\textsc{LMCAS}}\xspace eliminates the overhead of demanding techniques and boosts the safety of debloating process. \mbox{\textsc{LMCAS}}\xspace achieves a substantial reduction in the size of programs, and also reduces the attack surface.
\section{Discussion and Limitation} \label{sec:discussion}
\begin{comment}
\begin{table*}[!htb]
\centering
\begin{threeparttable}
\caption{\twr{You might squeeze this table into one column by abbreviating OCCAM, TRIMMER, etc. and shrinking with resizebox.}
Comparison summary with state of the art debloating techniques}
\begin{tabular}{|p{4.2cm}|p{2cm}|p{2.43cm}|p{2cm}|p{1.5cm}|}
\hline
Metric & OCCAM~\cite{OCCAM} & TRIMMER~\cite{TRIMMER} & CHISEL~\cite{Chisel} & \textbf{\mbox{\textsc{LMCAS}}\xspace}\\\hline
Average Analysis Time (sec) & 2.69 & 7.2 & 4167.64 & \textbf{2.5} \\\hline
Size Reduction Rate & $22\%$ & $21\%$ & $91\%$ & $\textbf{25\%}$ \\\hline
ROP gadgets Reduction & 21.5\% & 28.5\% & 66.2\% & \textbf{51.7\%} \\\hline
\# of Evaluated Apps & 3\tnote{1} & 13 & 10 & \textbf{23}\\\hline
Open Source & $\cmark$ & $\xmark$ & $\cmark$ & \textbf{$\cmark$}\\\hline
\end{tabular}
\label{table:comparison}
\begin{tablenotes}
\item[1] \small The number of evaluated apps in the paper is $3$, but the number of the apps in OCCAM GitHub repository is more
\end{tablenotes}
\end{threeparttable}
\end{table*}
\end{comment}
\mbox{\textsc{LMCAS}}\xspace's main goal is to facilitate the process of debloating applications using a lightweight approach while maintaining reliability and soundness. We developed \mbox{\textsc{LMCAS}}\xspace to satisfy the five debloating requirements introduced in~\cite{Chisel}.
Table~\ref{table:comparison} presents a comparison with prior work.
\mbox{\textsc{LMCAS}}\xspace outperforms partial-evaluation-based approaches (OCCAM and TRIMMER) for application specialization.
Although it was infeasible to compare with TRIMMER experimentally because neither TRIMMER nor the specialized programs that it produced were available, \mbox{\textsc{LMCAS}}\xspace achieves better reduction rates compared to the results reported for TRIMMER.
\mbox{\textsc{LMCAS}}\xspace reduces the size of the binary by $25\%$, on average (see Figure~\ref{fig:avgReduction}), while TRIMMER reduces the size by $21\%$~\cite{TRIMMER}.
\begin{table}[!htb]
\centering
\caption{\label{table:comparison}Comparison summary with state-of-the-art debloating techniques.}
\scalebox{0.7}{
\begin{threeparttable}
\begin{tabular}{|p{2.8cm}|p{1.9cm}|p{2.2cm}|p{1.8cm}|p{1.2cm}|}
\hline
Metric & OCCAM~\cite{OCCAM} & TRIMMER~\cite{TRIMMER} & CHISEL~\cite{Chisel} & \textbf{\mbox{\textsc{LMCAS}}\xspace}\\\hline
Average Analysis Time (sec) & 2.69 & 7.2 & 4167.64 & \textbf{2.5} \\\hline
Size Reduction Rate & $22\%$ & $21\%$ & $91\%$ & $\textbf{25\%}$ \\\hline
ROP gadgets Reduction & 21.5\% & 28.5\% & 66.2\% & \textbf{51.7\%} \\\hline
\# of Evaluated Apps & 3\tnote{1} & 13 & 10 & \textbf{23}\\\hline
Open Source & $\cmark$ & $\xmark$ & $\cmark$ & \textbf{$\cmark$}\\\hline
\end{tabular}
\begin{tablenotes}
\item[1] \small The number of evaluated apps in the paper is $3$; the number of apps in OCCAM's GitHub repository is greater.
\end{tablenotes}
\end{threeparttable}}
\end{table}
While reinforcement-learning approaches like CHISEL can achieve a high reduction rate, the majority of the debloated programs that CHISEL produces either crash or do not work properly in some other way.
CHISEL's analysis is conducted on top of the source code without including libraries.
In contrast, \mbox{\textsc{LMCAS}}\xspace performs its analysis over the LLVM IR using bitcode that includes the code for all libraries.
Furthermore, CHISEL suffers from overhead and usability issues because the developer potentially needs to create many test cases to cover even a single debloating scenario.
\Omit{\twr{The ideal here is to say what the threat to validity are, and what we did to mitigate the threats.}}
In the following, we outline threats to the validity of our approach along with the applied mitigations:
\begin{itemize}
\item \textbf{Incorrect neck identification}. Misidentifying the neck may
lead to incorrect debloating. For example on of our requirements constrains the neck candidates by the property that the neck is executed only once.
As a corollary, the neck cannot appear in a loop.
\Omit{\twr{Restatement OK? There used to be something about dominance, but it didn't make sense.} \ma{thank, Tom. the rewording clarifies the mitigation better now}}
\Omit{\vr{I rewrote this. Feel free to discard if you don't like.}}
\Omit{\vr{While correct, needs rewording. \#2 probably subsumes \#1. Also, the
dominance is guaranteed only under a specific input assignment.} \ma{How about this? if still not satisfying. Can you please help to improve the wording.}}
\hspace{1.5ex}
\Omit{\twr{Moved here from Sect.~\ref{subsec:neck}, as it represents a threat to validity.}}
There can be situations where the some approaches to neck identification discussed in Sect.~\ref{subsec:neck} fails;
in particular, the criterion of finding a dominator in the CFG may not be the right condition.
For instance, suppose that the application is essentially the union of two separate applications $A_1$ and $A_2$ that share some common code $C$;
suppose that they were combined by introducing a command-line parameter that causes a dispatch to one of two different places (corresponding to $A_1$ and $A_2$), at which point subsequent app-specific command-line processing is performed, and then $A_1$ or $A_2$ is run (and both ultimately call the common code $C$).
Note that $C$ is not dominated by either the end of command-line processing for $A_1$ (say, $a_1$) or the end of command-line processing for $A_2$ ($a_2$).
In this case, one would want the neck to be a cut-set: $\{ a_1, a_2 \}$.
Although this situation is possible, in our somewhat limited experience to date, we have not encountered it.
Moreover, it is essentially trivial to change from working with a single-point neck to working with a cut-set neck (i.e. in this case neck is a set of statements).
\item \textbf{Precision of Constant Conversion}.
\Omit{\vr{Feels verbose for PLDI. Maybe just say: LMCAS performs a
conservative but sound conversion of variables into constants.}}\Omit{\twr{Wording of this part OK?}\ma{yes, thanks}}\mbox{\textsc{LMCAS}}\xspace relies on converting a subset of the variables into constants.
If one of the variables could have been converted to a constant, but was not identified by \mbox{\textsc{LMCAS}}\xspace as being convertible (and therefore no occurrences of the variable are changed), no harm is done: that situation just means that some simplification opportunities may be missed.
On the other hand, if some variable occurrences are converted to constants unsoundly, the debloated program may not work correctly.
\hspace{1.5ex}
We mitigate this issue by: (1) avoiding the conversion of some variables to constants (e.g., \texttt{argc}) because the symbolic-execution input is different than the one required by the specialized program: for instance, with \texttt{wc} the file name is not supplied during symbolic execution, while the file name is supplied to the debloated program;
(2) leveraging existing LLVM APIs (i.e., \texttt{getUser}) that track the uses of variables to capture the final constant values at the end of the symbolic execution.
This approach overcomes situations where a pointer indirectly updates the value of a location, and ensures that the \Omit{\twr{OK?}\ma{Yes, thanks}} pre-neck constant-conversion step operates using updated and accurate constant values.
(3) Limiting the constant-conversion after the neck to unmodified variables (where "unmodified" accounts for aliasing).
\Omit{\vr{The content deserves better explanation. The only thing I understand
from this right now is that you have a sound (but possibly conservative)
alias analysis.} \ma{yes, that's true.}}
\Omit{\twr{I don't understand the next sentence.} \ma{I restate what we did in the Main Logic Constant-Conversion. Basically, the variable won't be converted to constant until we check that none of the alias variables is modified}}
\Omit{\twr{What are you trying to say in the next three sentences? I read it as saying that \mbox{\textsc{LMCAS}}\xspace needs to perform dataflow analysis, but you did not implement dataflow analysis. However, my understanding is that you used existing LLVM dataflow passes, as well as at least one of your own design. If that is the case, then what is the threat to validity?} \ma{I our approach partially managed to overcome some of the challenges. For example, we don't perform inter-procedural constant propagation.}}
\Omit{\item \textbf{Dynamically-linked Libraries}.
Our KLEE-based symbolic-execution phase works on a complete program with all libraries already linked statically.
\twr{I don't get the next sentence (which is awkwardly worded. If I understand correctly, you are proposing a generalized tool in which the resulting debloated program will make use of dynamically linked libraries. What is the threat to validity?}
Therefore, we are planning to port the results of the KLEE analysis and apply them to an LLVM module that can be compiled to a dynamically-linked binary.}
\end{itemize}
\section{Discussion and Limitations} \label{sec:discussion}
\begin{comment}
\begin{table*}[!htb]
\centering
\begin{threeparttable}
\caption{\twr{You might squeeze this table into one column by abbreviating OCCAM, TRIMMER, etc. and shrinking with resizebox.}
Comparison summary with state of the art debloating techniques}
\begin{tabular}{|p{4.2cm}|p{2cm}|p{2.43cm}|p{2cm}|p{1.5cm}|}
\hline
Metric & OCCAM~\cite{OCCAM} & TRIMMER~\cite{TRIMMER} & CHISEL~\cite{Chisel} & \textbf{\mbox{\textsc{LMCAS}}\xspace}\\\hline
Average Analysis Time (sec) & 2.69 & 7.2 & 4167.64 & \textbf{2.5} \\\hline
Size Reduction Rate & $22\%$ & $21\%$ & $91\%$ & $\textbf{25\%}$ \\\hline
ROP gadgets Reduction & 21.5\% & 28.5\% & 66.2\% & \textbf{51.7\%} \\\hline
\# of Evaluated Apps & 3\tnote{1} & 13 & 10 & \textbf{23}\\\hline
Open Source & $\cmark$ & $\xmark$ & $\cmark$ & \textbf{$\cmark$}\\\hline
\end{tabular}
\label{table:comparison}
\begin{tablenotes}
\item[1] \small The number of evaluated apps in the paper is $3$, but the number of the apps in OCCAM GitHub repository is more
\end{tablenotes}
\end{threeparttable}
\end{table*}
\end{comment}
\begin{comment}
\mbox{\textsc{LMCAS}}\xspace's main goal is to facilitate the process of debloating applications using a lightweight approach, while preserving program semantics and maintaining reliability.
\mbox{\textsc{LMCAS}}\xspace supports converting various types of variables to constants, which enables standard LLVM compiler-optimization techniques to achieve a high degree of simplification in the process of generating specialized programs.
\mbox{\textsc{LMCAS}}\xspace outperforms partial-evaluation-based approaches (OCCAM and TRIMMER) for application specialization.
Unfortunately, it was not possible to make a head-to-head comparison with TRIMMER because neither TRIMMER nor the specialized programs that it produced were available. However, \mbox{\textsc{LMCAS}}\xspace achieves better reduction rates compared to the results reported for TRIMMER.
\mbox{\textsc{LMCAS}}\xspace reduces the size of the binary by $25\%$, on average (see Figure~\ref{fig:avgReduction}), while TRIMMER reduces the size by $21\%$~\cite{TRIMMER}. \rev{Also, according to the TRIMMER paper~\cite{TRIMMER}, the average running time is $7.2$ seconds, which would make \mbox{\textsc{LMCAS}}\xspace $2.75$x faster than TRIMMER.}
\end{comment}
\begin{comment}
\begin{table}[!tb]
\centering
\caption{\label{table:comparison}Comparison summary with state-of-the-art debloating techniques.}
\scalebox{0.7}{
\begin{threeparttable}
\begin{tabular}{|p{2.7cm}|p{1.6cm}|p{1.6cm}|p{1.6cm}|p{2cm}|}
\hline
Metric & OCCAM & RAZOR & CHISEL & \mbox{\textsc{LMCAS}}\xspace\\\hline
Average Analysis Time (sec) & $2.69$ & $11.8$ & $4167.64$ & $2.5$ \\\hline
Size Reduction Rate & $22\%$ & $47.08\%$ & $83.4\%$ & $25\%$ \\\hline
ROP gadgets Reduction & $21.5\%$ & $25.1\%$ & $66.2\%$ & $51.7\%$ \\\hline
\end{tabular}
\begin{tablenotes}
\item[1] \small The number of evaluated apps in the paper is $3$; the number of apps in OCCAM's GitHub repository is greater.
\end{tablenotes}
\end{threeparttable}}
\end{table}
While aggressive debloating approaches like CHISEL can achieve a high reduction rate, the majority of the debloated programs that CHISEL produces either crash or do not work properly in some other way.
CHISEL's analysis is conducted on top of the source code without including libraries.
\end{comment}
\begin{comment}
In contrast, \mbox{\textsc{LMCAS}}\xspace performs its analysis over the LLVM IR using bitcode that includes the code for all libraries.
Furthermore, CHISEL suffers from overhead and usability issues because the developer potentially needs to create many test cases to cover even a single debloating scenario.
\Omit{\twr{The ideal here is to say what the threat to validity are, and what we did to mitigate the threats.}}
\end{comment}
\begin{comment}
The CC phase of \mbox{\textsc{LMCAS}}\xspace aims to enforce the concrete state captured at the neck.
The constant-conversion method sketched in Section~\ref{subsec:cc} can make some substitutions of constants that lead to incorrect debloating results (e.g., when the variables converted in the configuration logic are initialized based on the enabled argument, and later reassigned to).
In this work, we did not encounter this situation in any of the datasets that we used (\texttt{Benchmark\_1-3}), and thus
the performed constant conversion was sound.
\end{comment}
\begin{comment}
However, our plan in the future is to incorporate data-flow analysis to handle situations where the influenced variables are modified in the configuration logic after being initialized.
Tools such as SVF~\cite{svfPaper} and PhASAR~\cite{PhASAR} can be leveraged to identify precisely the variables that can be converted to constants.
\end{comment}
\begin{comment}
\rev{
\noindent\textbf{Automating Neck Identification:} the neck is identified manually in this work. However, in Section~\ref{subsec:neckSpec}, we formalized the neck-identification process and described control-flow (CFG) properties that we applied during the manual identification.
We envision a process of identifying the neck automatically comprising the following steps:
\begin{itemize}
\item Identifying articulation points in the CFG, starting from the main function. This will yield several statements as articulation points, which represent candidates for the neck.
\item Eliminating irrelevant articulation points by testing the candidates for properties like reachability, dominance, and not inside loops.
\end{itemize}
Performing these steps relies on analyzing various graphs, such as the CFG and its dominator tree. We also foresee the need to leverage some data-flow analysis like taint analysis to support the neck-identification process.
}
\end{comment}
\begin{comment}
\begin{figure}[tb]
\centering
\includegraphics[width=0.5\columnwidth]{images/cfg.pdf}
\caption{\rev{Simplified CFG of the main function in Listing~\ref{wc}. The configuration-logic and main-logic nodes are grouped.}}
\label{fig:cfg}
\end{figure}
\end{comment}
\rev{
\noindent\textbf{Generality of the neck concept.}
This concept applies to various types of programs. Our evaluation involves $26$ command-line programs, in which the neck can be easily identified, without discarding any program because the neck identification was not possible. We also inspected \texttt{Nginx} and observed that ALL directives in the configuration file are read before reaching the main logic (as in command-line programs). But this partitioning approach into configuration and main logic cannot be applied to Event-driven programs, which require constant interaction with the user to handle the functionalities required by the user.
Therefore, the configuration of program features is performed at various locations. However, we foresee our partitioning approach applies to Event-driven programs that apply server architecture, whose life-cycle is divided into initialization and serving phases~\cite{Temporal}.
Our future work will consider evaluating such programs.
}
\noindent\revUN{\textbf{Incorrect neck identification}.
Misidentifying the neck may lead to incorrect debloating. But the neck miner incorporates a set of heuristics and structural features to recommend accurate neck locations. The heuristic analysis techniques aid the neck identification by pinpointing the starting point where the neck miner establishes its analysis. For instance, the GNU Coreutils programs use a particular idiom for parsing command-line parameters; in principle, a special-purpose algorithm could be designed for identifying the neck for programs that use that idiom. Then the neck miner applies a set of structural requirements to constrain the neck candidates. For example, the properties that the neck is executed only once and the neck is an articulation point. These structural properties reflect the nature of the neck definition.}
\noindent\textbf{Precision of Constant Conversion}.
\mbox{\textsc{LMCAS}}\xspace relies on converting a subset of the variables in the captured concrete state into constants.
If one of the variables could have been converted to a constant, but was not identified by \mbox{\textsc{LMCAS}}\xspace as being convertible (and therefore no occurrences of the variable are changed), no harm is done: that situation just means that some simplification opportunities may be missed.
On the other hand, if some variable occurrences are converted to constants unsoundly, the debloated program may not work correctly.
We mitigate this issue by: (1) avoiding the conversion of some variables to constants (e.g., \texttt{argv}) because it carries out the rest of inputs (i.e., delayed inputs) different than the one required by the specialized program: for instance, with \texttt{wc} the file name is not supplied during partial interpretation, while the file name is supplied to the debloated program;
(2) leveraging existing LLVM APIs (i.e., \texttt{getUser}) that track the uses of variables to capture the final constant values at the end of the partial interpretation.
This approach overcomes situations where a pointer indirectly updates the value of a location, and ensures that the pre-neck constant-conversion step operates using updated and accurate constant values.
\noindent\revUN{\textbf{Reducing the Attack Surface}.
\mbox{\textsc{LMCAS}}\xspace reduces the attack surface in various ways by removing some known CVEs and eliminating some code-reuse attacks.
\Omit{\twr{The next sentence doesn't mean anything: remove. At best, one can say that the capabilities of \mbox{\textsc{LMCAS}}\xspace and LLVM-CRI are incomparable.} \ma{agreed}}
Our future work will enforce stronger security properties: specializing control-flow by restricting the set of allowed indirect control-flow transfers in the remained code by incorporating CFI techniques~\cite{Ancile,enforce_CFI} \Omit{\twr{What does it mean to tighten CFI checks (beyond the constraints that CFI already enforces)?}} and disabling security-critical system calls that are unused in the main logic~\cite{Temporal}.\Omit{\twr{What does it mean to "remove ... calls that are unused"? Instead of "remove" do you mean "disable"? Or will the disabling be done inside the OS kernel, in which case what is \mbox{\textsc{LMCAS}}\xspace contributing (maybe just information about which system calls to disable)?} \ma{thanks for the correction, I meant disabling. Yes, this would be the contribution, but we already do this when we check the use of functions before removing them, therefore, what we need to do know, just capturing precisely the system calls.}}}
\section{Evaluation}
\label{sec:evaluation}
This section presents our experimental evaluation of \mbox{\textsc{LMCAS}}\xspace.
We address the following research questions:
\begin{itemize}
\item \textbf{Efficiency:}
What is the debloating performance of \mbox{\textsc{LMCAS}}\xspace with respect to \mbox{\textsc{LMCAS}}\xspace analysis time, and the amount that apps are reduced in size? (\ref{subsec:Efficiency})
\item \textbf{Security:}
Can \mbox{\textsc{LMCAS}}\xspace reduce the attack surface? (\ref{subsec:security})
\item \textbf{Scalability:}
How scalable is \mbox{\textsc{LMCAS}}\xspace in debloating large apps? (\ref{subsec:Scalability})
\item \textbf{Robustness:}
How robust are the debloated programs produced by \mbox{\textsc{LMCAS}}\xspace? (\ref{subsec:robustness})
\end{itemize}
\begin{comment}
\noindent\textit{\textbf{Implementation}}.\Omit{\twr{MISSING: we do not say how neck analysis was performed.} \ma{ manually, but based on the neck identification algorithm, by satisfying the requirement that the neck should be executed only once and it dominates the main logic}}
In the current implementation of \mbox{\textsc{LMCAS}}\xspace, neck identification is performed manually, and all other steps are performed automatically.
The \mbox{\textsc{LMCAS}}\xspace user marks the neck location by adding a particular function call to the program being analyzed.
The user has the responsibility of identifying an appropriate neck location that satisfies the constraints defined in Section~\ref{subsec:neckSpec} (i.e., the neck should be executed only once, and should dominate the main logic).
For the $15$ GNU CoreUtils programs in \texttt{Benchmark\_1}, the total time to perform the manual analysis was $4$ hours.
After examining three programs, we noticed these programs use a particular idiom for parsing command-line parameters;
thereafter, neck identification for the rest of the programs was much quicker (around 10 minutes/program).
We modified KLEE 2.1~\cite{KLEE} to perform the Partial Interpretation phase and to stop and capture the set of affected variables and their corresponding values after the neck is reached.
\rev{
In essence, KLEE is being used as an execution platform that supports “partial concrete states.” For LMCAS, none of the inputs are symbolic, but only part of the full input is supplied. In the case of word-count, the input is “wc -c”, with no file-name supplied. Only a single path to the neck is followed, at which point KLEE returns its state (which is a partial concrete state).
The second column in Tables \ref{tab:benchApps} \& \ref{tab:largeApps} describes the inputs supplied to KLEE for the different examples.
}
We also developed two LLVM passes using LLVM 6.0 to perform constant-conversion (CC) and the clean-up step of the MS phase.
\end{comment}
\noindent\textit{\textbf{Experimental Setup}}. Our evaluation relies on three datasets, as shown in Table~\ref{tab:dataset}.
\textit{Benchmark\_1} contains $15$ programs from GNU Coreutils v8.32 (see Table~\ref{tab:benchApps}).
\textit{Benchmark\_2} contains five programs obtained from ChiselBench (see Table~\ref{table:cve}).\footnote{https://github.com/aspire-project/chisel-bench}
\textit{Benchmark\_3} consists of three programs (see Table~\ref{tab:largeApps}).
\rev{
The selection of programs in Benchmark_1 was motivated by their use in prior papers on software debloating. We used Benchmark_2 because it provides us a list of CVE vulnerabilities and corresponding apps; considering this dataset facilitates our evaluation of the removal of CVEs, and allows us to compare against the CVE-removal performance of Chisel. While Benchmark_1 shows the size-reduction performance and analysis time of our approach under simple settings.
}
\begin{comment}
Our evaluation relies on three datasets, as shown in Table~\ref{tab:dataset}.
For RQ1, we used \textit{Benchmark\_1}, which contains $15$ programs from GNU Coreutils v8.32 (see Table~\ref{tab:benchApps}), to evaluate the reduction rate produced by \mbox{\textsc{LMCAS}}\xspace and the time for analysis.
\textit{Benchmark\_1} is also used in the security evaluation (RQ3).
\textit{Benchmark\_2} contains five programs (see Table~\ref{table:cve}) obtained from ChiselBench;\footnote{https://github.com/aspire-project/chisel-bench}
it is used in RQ1 and RQ3 to compare with the state of the art.
\textit{Benchmark\_2} is also used in the robustness experiment in RQ3.
The programs in this dataset are listed in Table~\ref{table:cve}.
Finally, in RQ2, we used \textit{Benchmark\_3}, which consists of three programs (listed in Table~\ref{tab:largeApps}), to evaluate scalability.
\end{comment}
\Omit{\twr{MISSING: say what test suite you used to test debloated apps}\ma{I used one test case, and compared the output before and after debloating. If the output is the same, this indicates debloating is performed correctly}.
\twr{Not sure what to say: that is not a very comprehensive approach.
At least you ran AFL on \textit{Benchmark\_3} for RQ2.
Could you still run AFL on the debloated programs in \textit{Benchmark\_1} and \textit{Benchmark\_2}?}\ma{I'll do that, but due to time limitation, I'll run AFL for 2-3 hrs/app}}
All experiments were conducted on an Ubuntu $18.04$ machine with a 2.3GHz Quad-Core Intel i7 processor and 16GB RAM, except the fuzzing experiment (RQ2), for which we used an Ubuntu $18.04$ machine with a 3.8GHz Intel(R) Core(TM) i7-9700T CPU and 32GB RAM.
\begin{table}[!tb]
\centering
\caption{Benchmark sets used in the evaluation.}
\scalebox{0.9}{
\begin{tabular}{llcl}
\hline
Source & Label & \# of apps & Scope \\ \hline
\thead{GNU Coreutils 8.32} & Benchmark\_1 & 15 & RQ1, 2, 4 \\ \hline
\thead{CHISEL Benchmark} & Benchmark\_2 & 5 & RQ1, 2, 4\\ \hline
\thead{Tcpdump \& GNU Binutils} & Benchmark\_3 & 3 & RQ3 \\ \hline
\end{tabular}
}
\label{tab:dataset}
\end{table}
\noindent\textit{\textbf{Compared tools and approaches}}. To evaluate the effectiveness of \mbox{\textsc{LMCAS}}\xspace, we compared with the following tools and approaches:
\begin{itemize}
\begin{comment}
\item
\textit{Manual debloating}.
We developed a simple but systematic protocol to perform debloating manually, which we state as Algorithm~\ref{alg:manualAlg}. The goal of this manual approach is to create a baseline for the maximum level of reduction that can be achieved by an average developer.\Omit{\twr{Comments on Algorithm~\ref{alg:manualAlg}.
The second removal loop is still not satisfactory.
Unintelligible to the reader: lines 16, 17, 20, 21; you are assuming they understand certain operations that you never explain.\ma{at these lines, I just iterate the worklists varsToRemove and funcToRemove, and I remove each element from the list, so the while loop terminates.}
Worse, I don't see how lines 15-18 do the correct thing: for a given function $f$ in funcToRemove, you need to check whether the program still has any call-sites that call $f$: if there are no call-sites, then you can remove $f$. But you have to keep iterating: you could have already considered $g$, which was kept because there was a call on $g$ in $f$. Now you want to remove $g$ as well, and so on . . .\ma{If g calls f, but there is no call for g, then the definition of func g is removed, and in the next iteration the definition of f will be removed. I was using grep to search for the occurrences of functions/variables}
Finally, there is the issue of indirect function calls. The simplest approach is that you need to know the set of functions whose address has been taken (say $f$, $g$, and $h$); then all indirect call-sites are considered as if they are calls on $f$, $g$, and $h$. (If you have done points-to analysis, then the set can be narrowed to the set of functions that the points-to analysis tells you.)
Line 16 is still completely mysterious---callSiteSet is never used.\ma{I fixed it}}\ma{A general note on the manual analysis, I didn't perform any program analysis, if a variable/function is accidentally removed, then the app won't be compiled successfully or the app will crash or generate wrong output. So I undo the last removal.So it's kind of try-error approach. The goal of this approach is constituting a baseline about the level of reduction that can be manually achieved by an average developer}}
We first create work-lists of variables and functions that are candidates for removal (Lines~\ref{alg:l:varsToRemove}-\ref{alg:l:funcToRemove}). We then execute app $A$ under the control of GDB\footnote{https://www.gnu.org/software/gdb/}
to identify the set of statements that are executed based on input $I$, as described in Algorithm~\ref{alg:manualAlg} at Line~\ref{alg:l:gdb}. The manual debloating is performed over two main loops. In the first loop at Lines~\ref{alg:l:loop1-b}-\ref{alg:l:loop1-e}, we iterate over all statements in the source code starting from the main function.
If the statement does not belong to the list of executed statements ($Executed$), then we remove the statement and record its variables in the list of candidate variables to be removed ($varsToRemove$), but if the statement is a callsite, the function name is recorded in $funcToRemove$.
In the second loop at Lines~\ref{alg:l:loop2-b}-\ref{alg:l:loop2-e}, one iterates over the lists of variables and functions that are candidates for removal: the goal is to identify functions never invoked and variables never used by performing text-based search to determine the occurrences of the variable/functions (Lines~\ref{alg:l:occrFunc} and~\ref{alg:l:occrVar}), so that they can be removed, if they appeared only once in the remained source code. Finally, we compile the app to verify that it builds correctly; otherwise, we put back the removed statements, including removed variables and functions in the iteration (Lines~\ref{alg:l:build-b}-\ref{alg:l:build-e}).
\hspace{1.5ex}
The resulting application is compiled and tested to see that it yields correct results.
This approach does not require prior knowledge of the application or advanced experience in program analysis---and thus average developers should be able to apply it to obtain only desired functionality.
\end{comment}
\item Baseline. We establish the baseline by compiling each app's LLVM bitcode at the -O2 level of optimization. This baseline approach was used in prior work~\cite{TRIMMER}.
\item Manual debloating. We developed a simple but systematic protocol to perform debloating manually, which we state as Algorithm $2$ (in the supplementary materials~\cite{materials}). The goal of this manual approach is to create a baseline for the maximum level of reduction that can be achieved by an average developer.
\item OCCAM~\cite{OCCAM}. The system most comparable to the approach used by \mbox{\textsc{LMCAS}}\xspace. However, OCCAM does not perform constant propagation, and thus omits a major component of partial evaluation.
\item CHISEL~\cite{Chisel}. It requires the user to identify wanted and unwanted functionalities and uses reinforcement learning to perform the reduction process.
\item RAZOR~\cite{RAZOR}. similar to CHISEL, RAZOR relies on test cases to drive the debloating, but also incorporates heuristic analysis to improve soundness.
\item Nibbler~\cite{Nibbler2}. state-of-the-art tool for debloating binary code. It does not generate specialized apps, it rather focuses only on reducing the size of shared libraries.
\begin{comment}
\rev{
We avoided running Chisel on Benchmark\_1, not only because its analysis takes a long time, but also because it requires transforming the programs into a single meta-file, generating test cases, and writing a script according to a certain template to identify the required features. These steps are a time-consuming process.
}
\end{comment}
\end{itemize}
\begin{comment}
\begin{algorithm}[!htb]
\caption{Best-Effort Manual Debloating Protocol}
\label{alg:manualAlg}
\DontPrintSemicolon
\KwIn{App $A$, Input $I_s$}
\KwOut{App $A\textprime$ }
$relevantVars \longleftarrow \emptyset$\;
$varsToRemove \longleftarrow \emptyset$\;
$funcToRemove \longleftarrow \emptyset$\;
$notExecuted \longleftarrow$ the set of statements that will not be executed, given input I \;
\SetKwProg{Def}{def}{:}{}
\Def{Debloat($A, I_s$)}{
$A\textprime \longleftarrow A$ \;
\Repeat{no more stmt removal}{
\For{$stmt \in A\textprime$}{
\uIf{$stmt$ is $callSite$ \&\& $notExecuted$}{
$A\textprime \longleftarrow remove(stmt)$\;
$funcToRemove \longleftarrow add(func)$\;
\uIf{$stmt(vars) \in I$}{
relevantVars $\longleftarrow$ relevantVars $\cup$ \{var\} \;
$var \longleftarrow initVal(var)$\;
}\Else{
$varsToRemove \longleftarrow add(var)$ \;
$A\textprime \longleftarrow remove(stmt)$\;
}
}
\While{$varsToRemove$ $\neq \emptyset$ $ \land $ $funcToRemove$ $\neq \emptyset$}{
\For{$func \in funcToRemove$}{
$callSiteSet \longleftarrow remove(func)$\;
\uIf{func is not called}{
Remove func from A\textprime \;
}
}
\For{$var \in varsToRemove$}{
$varsToRemove \longleftarrow remove(var)$\;
\uIf{$var$ not used}{
Remove var from A\textprime \;
}
}
}
}
}
\end{algorithm}
\end{comment}
\begin{comment}
\begin{algorithm}[!htb]
\caption{Manual Debloating Protocol}
\scriptsize
\label{alg:manualAlg}
\DontPrintSemicolon
\KwIn{App $A$, Input $I$}
\KwOut{App $A\textprime$ }
$varsToRemove \leftarrow \emptyset$\;\label{alg:l:varsToRemove}
$funcToRemove \leftarrow \emptyset$\;\label{alg:l:funcToRemove}
$Executed \leftarrow$ the set of statements that GDB reports are executed, given input $I$ \; \label{alg:l:gdb}
\SetKwProg{Def}{def}{:}{}
$A\textprime \leftarrow A$ \;
\Repeat{no more removals of statements}{
\For{$stmt \in A\textprime$}{ \label{alg:l:loop1-b}
\uIf{$stmt \not\in Executed$}{ \label{alg:l:execLst}
remove stmt from $A\textprime$ \;
$varsToRemove \leftarrow varsToRemove \cup \{\textit{stmt(vars)}\}$ \;
\uIf{$stmt$ is a call site}{
$funcToRemove \leftarrow funcToRemove \cup \{\textit{func}\}$ \; \label{alg:l:loop1-e}
}
}
\While{$varsToRemove$ $\neq \emptyset$ $ \land $ $funcToRemove$ $\neq \emptyset$}{ \label{alg:l:loop2-b}
\For{$func \in funcToRemove$}{
Remove $func$ from funcToRemove \;
\uIf{no occurrence of $func$ exists}{\label{alg:l:occrFunc}
Remove func from A\textprime \;
}
}
\For{$var \in varsToRemove$}{\label{alg:l:occrVar}
Remove $var$ from varToRemove \;
\uIf{no occurrence of $var$ exists}{
Remove var from A\textprime \;\label{alg:l:loop2-e}
}
}
}
\uIf{$A\textprime$ does not build correctly}{\label{alg:l:build-b}
put back $stmt$ \;
undo $var$ and $func$ removal from A\textprime\; \label{alg:l:undo}
}\label{alg:l:build-e}
}
}
\end{algorithm}
\end{comment}
\begin{comment}
\begin{figure*}[h!]
\centering
\subfloat[Average reduction in size achieved through manual debloating, OCCAM, and \mbox{\textsc{LMCAS}}\xspace, using four different size metrics. (Higher numbers are better.)]{
\begin{minipage}[b][0.33\width]{
0.3\textwidth}
\centering
\includegraphics[width=1\textwidth]{images/AvgReductionPerApp.pdf}
\label{fig:avgReduction}
\end{minipage}}
\hfill
\subfloat[Running times of \mbox{\textsc{LMCAS}}\xspace, CHISEL, and OCCAM based on \textit{Benchmark\_2} (ChiselBench).]{
\begin{minipage}[b][1\width]{
0.3\textwidth}
\centering
\includegraphics[width=1\textwidth]{images/OCCAM_CHISEL_LMCAS_V2.png}
\label{fig:analysisTimeComparison}
\end{minipage}}
\hfill
\subfloat[Reduction in the total number of gadget occurrences (SYS, ROP and JOP) between \mbox{\textsc{LMCAS}}\xspace and OCCAM for \textit{Benchmark\_1}. (Higher numbers are better.)]{
\begin{minipage}[b][1\width]{
0.3\textwidth}
\centering
\includegraphics[width=1\textwidth]{images/rop.png}
\label{fig:rop}
\end{minipage}}
\caption{Evaluation results of \mbox{\textsc{LMCAS}}\xspace}
\end{figure*}
\end{comment}
\subsection{Efficiency} \label{subsec:Efficiency}
We conducted two experiments to evaluate the efficiency of \mbox{\textsc{LMCAS}}\xspace.
First, we measured the reduction rate that \mbox{\textsc{LMCAS}}\xspace achieves, and compared it with two state-of-the-art tools.
Second, we measured the time required for performing partial interpretation and code simplification.
\begin{table}[!tb]
\centering
\small
\caption{Characteristics of the benchmarks in \textit{Benchmark\_1}.}
\scalebox{0.9}{
\begin{tabular}{|p{1.3cm}|p{1.55cm}|p{0.8cm}|p{0.65cm}|p{1.1cm}|p{1cm}|}
\hline
\multirow{2}{*}{App} & \multirow{2}{*}{\shortstack{Enabled\\ Arguments}} & \multicolumn{4}{c|}{Original} \\\cline{3-6}
& & \shortstack{\# IR \\ Inst.} & \shortstack{\#\\ Func.} & \shortstack{\# Basic\\ Blocks} & \shortstack{Binary \\ Size $^a$} \\ \hline
basename & \-\-suffix=.txt & 4,083 & 96 & 790 & 26,672 \\ \hline
basenc & base64 & 8,398 & 156 & 1,461 & 44,583 \\ \hline
comm & -12 & 5,403 & 110 & 972 & 32,714 \\ \hline
date & -R & 29,534 & 166 & 6,104 & 89,489 \\ \hline
du & -h & 5,0727 & 466 & 8,378 & 180,365 \\ \hline
echo & -E & 4,095 & 89 & 811 & 27,181 \\ \hline
fmt & -c & 5,732 & 115 & 1,095 & 79,676 \\ \hline
fold & -w30 & 4,623 & 100 & 893 & 29,669 \\ \hline
head & -n3 & 6,412 & 119 & 1,175 & 37,429 \\ \hline
id & -G & 5,939 & 125 & 1,172 & 36,985 \\ \hline
kill& -9 & 4,539 & 96 & 898 & 31,649 \\ \hline
realpath & -P & 8,092 & 155 & 1,419 & 41,946 \\ \hline
sort & -u & 25,574 & 329 & 3,821 & 116,119 \\ \hline
uniq & -d & 5,634 & 115 & 1,092 & 37,159 \\ \hline
wc & -l & 7,076 & 130 & 1,219 & 41,077 \\ \hline
\end{tabular}}
\label{tab:benchApps}
\footnotesize{$^a$Total binary size obtained via the GNU \texttt{size} utility.}
\end{table}
\noindent\textbf{Reduction Rate}.
We used \textit{Benchmark\_1} to compare the performance of \mbox{\textsc{LMCAS}}\xspace against manual debloating, baseline, Nibbler, and OCCAM.
Figure~\ref{fig:avgReduction} shows the comparison results based on the reduction in the binary size that each tool achieved for each app in \textit{Benchmark\_1}. For computing the binary-size metric, we compiled all debloated apps with \texttt{gcc -O2}, and ran \texttt{size}.
We report the sum of the sizes of all sections in the binary file (text + data + bss) because this quantity reflects the outcome of our simplifications across all sections. \mbox{\textsc{LMCAS}}\xspace achieved significant higher reduction rate (i.e., around the double) in comparison with manual debloating, baseline, and Nibbler.
This result is due to the fact that the clean-up step of \mbox{\textsc{LMCAS}}\xspace can remove nodes in the call-graph that correspond to functions in binary libraries that are not used.
Although the reduction rates of \mbox{\textsc{LMCAS}}\xspace and OCCAM are close (geometric mean binary size reduction is $25\%$ and $22\%$, respectively), some of the specialized programs generated by OCCAM are not reliable.
In particular, of the applications in Table~\ref{tab:benchApps}, five of the OCCAM-debloated applications ($33\%$) crash or generate inaccurate results. In contrast, all of the \mbox{\textsc{LMCAS}}\xspace-debloated applications run correctly (i.e., \mbox{\textsc{LMCAS}}\xspace preserves apps' behaviour).
\begin{figure}[!htb]
\centering
\includegraphics[width=1\columnwidth]{images/AvgReductionPerApp.pdf}
\caption{Binary size reduction achieved through manual debloating, baseline, Nibbler, OCCAM, and \mbox{\textsc{LMCAS}}\xspace. (Higher numbers are better.)}
\label{fig:avgReduction}
\end{figure}
\begin{comment}
\begin{algorithm}[!htb]
\caption{Best-Effort Manual Debloating Protocol}
\label{alg:manualAlg}
\DontPrintSemicolon
\KwIn{App A, Input I}
\KwOut{App A\textprime \Omit{\twr{The procedure never uses A\textprime} \ma{ because it's the final output of the procedure}} }
$irrelevantVars \longleftarrow \emptyset$\;
$callSiteSet \longleftarrow \emptyset$\;
\SetKwProg{Def}{def}{:}{}
\Def{Debloat($A, I$)}{
Go to the main function\;
$update(relevantVar)$\;
$update(irrelevantVars)$\;
\twr{A function is different than a call-site; which do you mean? \ma{ initially, the callSiteSet contains a list of all functions in the program} Also, why is the next statement an assignment but the later update of callSites not an assignment}\;
$update(callSiteSet) \longleftarrow functions(A)$\;
\While{$irrelevantVars$ $!=\emptyset$ $\&$ $callSiteSet$ $!=\emptyset$}{
\For{$func \in callSiteSet$}{
$callSiteSet \longleftarrow remove(func)$\;
\uIf{func is not called}{
$A \longleftarrow remove(func)$\;
$update(callSites)$\;
$update(irrelevantVars)$\;
}
}
\For{$var \in irrelevantVars$}{
$irrelevantVars \longleftarrow remove(var)$\;
\uIf{$var$ not used}{
$A \longleftarrow remove(var)$\;
}\Else{
$replace(var) \longleftarrow initital(var)$\;
$A \longleftarrow remove(var)$\;
}
}
}
$A \longleftarrow remove(irrelevant IncludeStmts)$\;
}
\end{algorithm}
\end{comment}
\noindent\textbf{Running Time.}
We measured the analysis time of \mbox{\textsc{LMCAS}}\xspace for \textit{Benchmark\_1}. The average total analysis time is $3.65$ seconds;
the maximum total analysis time is $13.08$ seconds for analyzing \texttt{sort};
and the minimum total analysis time is $1.19$ seconds for analyzing \texttt{basename}.
Notably, the time for constant conversion and multi-stage simplification is low:
on average, the time for constant conversion and multi-stage simplification is $0.4$ seconds, while the average time for Partial Interpretation is $3.25$ seconds.
\begin{comment}
We measured the running time of \mbox{\textsc{LMCAS}}\xspace. Figure~\ref{fig:analysisTime} shows the breakdown of running time, for \textit{Benchmark\_1}, between (i) symbolic execution, and (ii) Constant Conversion (CS) plus Multi-stage Simplification (MS).
The average total running time is $3.65$ seconds;
the maximum total running time is $13.08$ seconds for analyzing \texttt{sort};
and the lowest total analysis time is $1.19$ seconds for analyzing \texttt{basename}.
Notably, the time for Constant Conversion and Multi-stage Simplification is low:
on average, the time for constant conversion and multi-stage simplification is $0.4$ seconds, while the average time for Partial Interpretation is $3.25$ seconds.
\begin{figure}[!htb]
\centering
\includegraphics[width=1\columnwidth]{images/TimeAnalysis.png}
\caption{The time required for symbolic execution and the partial-evaluation steps (Constant Conversion and Multi-stage Simplifications) for \textit{Benchmark\_1}.
\Omit{\twr{Please change the label on the y-axis to "Running Time", and change the
key from "CC \& MS" to "Partial Evaluation" to be consistent with the caption. We need \underline{all} labeling and terminology to be consistent \underline{everywhere}.}}}
\label{fig:analysisTime}
\end{figure}
\end{comment}
We also compared the running time of \mbox{\textsc{LMCAS}}\xspace against those of CHISEL, RAZOR, and OCCAM on \textit{Benchmark\_2}.
The results are shown in Figure~\ref{fig:analysisTimeComparison}.
The running times for \mbox{\textsc{LMCAS}}\xspace, RAZOR, and OCCAM are significantly lower than the time for CHISEL.
These results show that our debloating approach is more efficient than previous ones.
\mbox{\textsc{LMCAS}}\xspace substantially outperforms CHISEL and slightly outperforms OCCAM and RAZOR because it carries out a relatively lightweight debloating approach.
As a result, \mbox{\textsc{LMCAS}}\xspace runs up to $1500$x, $3.9$x, and $1.2$x faster on average than CHISEL, RAZOR, and OCCAM, respectively.
\begin{figure}[!htb]
\centering
\includegraphics[width=1\columnwidth]{images/OCCAM_CHISEL_LMCAS_V3.pdf}
\caption{\Omit{\twr{The y-axis says "Analysis Time''. Make it consistent with the text, which I changed to ``running time''.}}
Running times of \mbox{\textsc{LMCAS}}\xspace, CHISEL, RAZOR, and OCCAM based on \textit{Benchmark\_2} (ChiselBench).}
\label{fig:analysisTimeComparison}
\end{figure}
\subsection{Security} \label{subsec:security}
We evaluated the capabilities of \mbox{\textsc{LMCAS}}\xspace to reduce code-reuse attacks and to remove known vulnerabilities.
We evaluated these security aspects from two angles: (i) we attempt to reproduce (by executing the app) the vulnerability after debloating the apps to see if the vulnerabilities were eliminated;
(ii) we measured the reduction in the number of code-reuse attacks\Omit{attack surface} by counting the number of\Omit{return-oriented programming (ROP)} eliminated gadgets in compiled versions of the original and reduced programs using ROPGadget~\cite{ROPGadget}, an analysis tool for enumerating different types of gadgets in a binary.
\noindent\textbf{Vulnerability Removal}.
To test the ability to mitigate vulnerabilities, we used the five programs in \textit{Benchmark\_2}.
Table~\ref{table:cve} presents a comparison between \mbox{\textsc{LMCAS}}\xspace, RAZOR, OCCAM, and CHISEL.
\mbox{\textsc{LMCAS}}\xspace and OCCAM removed CVEs from 4 out of the 5 programs, but the debloated \textit{sort-8.16} using OCCAM was not behaving correctly at run-time. While CHISEL and RAZOR removed CVEs from 3 programs.
\mbox{\textsc{LMCAS}}\xspace could not remove the vulnerability in \textit{date-8.21} because the bug is located in the core functionality of this program.
When undesired functionality is too intertwined with the core functionality---e.g., multiple quantities are computed by the same loop---then \mbox{\textsc{LMCAS}}\xspace may not be able to remove some undesired functionality because---to the analysis phases of \mbox{\textsc{LMCAS}}\xspace---it does not appear to be useless code.
In such cases, \mbox{\textsc{LMCAS}}\xspace retains the desired functionality.
In contrast, CHISEL tries to remove all undesired functionality.
\begin{table}[!tb]
\centering
\caption{Vulnerabilities after debloating by \mbox{\textsc{LMCAS}}\xspace and CHISEL. $\cmark$ means the CVE vulnerability is eliminated; \Omit{\twr{Rewording OK?}} $\xmark$ means that it was not removed.}
\scalebox{0.75}{
\begin{threeparttable}
\begin{tabular}{llcccc}
\hline
App & CVE ID & RAZOR & CHISEL & OCCAM & \mbox{\textsc{LMCAS}}\xspace \\ \hline \hline
chown-8.2 & CVE-2017-18018 & $\cmark$ & $\cmark$ & $\cmark$ & $\cmark$ \\ \hline
date-8.21 & CVE-2014-9471 & $\cmark$ & $\xmark$ & $\xmark$ & $\xmark$ \\ \hline
rm-8.4 & CVE-2015-1865 & $\xmark$ & $\xmark$ & $\xmark$ & $\cmark$ \\ \hline
sort-8.16 & CVE-2013-0221 & $\cmark$ & $\cmark$ & $\cmark$ & $\cmark$ \\ \hline
uniq-8.16 & CVE-2013-0222 & $\xmark$ & $\cmark$ & $\cmark$ & $\cmark$\\ \hline
\end{tabular}
\begin{tablenotes}
\item[1] \small \revUN{CHISEL and RAZOR CVE removal is obtained from the corresponding publication.
\item[2] \small Although OCCAM removed the CVE in rm-8.4, but the debloated version gets stuck at run-time.}
\end{tablenotes}
\end{threeparttable}
}
\label{table:cve}
\end{table}
\noindent\textbf{Gadget Elimination}.
Code-reuse attacks leverage existing code snippets in the executable, called gadgets, as the attack payload~\cite{CSET19}.
Those gadgets are often categorized---based on their last instruction---into Return-Oriented Programming (ROP), Jump-Oriented Programming (JOP), and syscall (SYS)~\cite{Piece-Wise,CARVE}.
Code-reuse attacks can be mitigated using software diversification, which can be achieved through application specialization.
Software debloating is a specialization technique that can both reduce the number of exploitable gadgets and change the locations of gadgets, thus diversifying the binary.
In this experiment, we used \textit{Benchmark\_1} and observed noticeable reductions in the total gadget count \Omit{\twr{What are SYS and JOP? Are you saying that you have two other figures like Figure~\ref{fig:rop} for two other kinds of gadgets? If so, break that out into a separate paragraph (after this one), and report an aggregate number like the geometric means for the SYS and JOP measurements. Have I explained the subtleties of computing geometric mean of percentages to you? You cannot just compute the geometric means of the raw percentage numbers.} \ma{I apologies for the incorrect description, but Fig 6 reports the 3 types of gadgets SYS, JOP and ROP. I fixed that by making sure everything is consistent that we report total gadgets}}(occurrences of ROP, SYS, and JOP gadgets), as illustrated in Figure~\ref{fig:rop}.
\Omit{\twr{How is average computed? Say which kind of average you are reporting: arithmetic mean or geometric mean.} \ma{Arithmetic}}The average reduction (arithmetic mean) in the total number of gadgets is $51.7\%$ and the maximum reduction is $72.1\%$ (for \texttt{date}), while OCCAM reduces the total number of gadgets by $21.5\%$, on average, with a maximum reduction of $80.3\%$ (for \texttt{echo}).
For one program, \texttt{sort}, OCCAM increases the total number gadgets by $6.2\%$.
With \mbox{\textsc{LMCAS}}\xspace, the number of SYS gadgets is reduced to $1$ for $14$ out of the $15$ applications.
\mbox{\textsc{LMCAS}}\xspace caused an increase in the number of SYS gadgets in one application (\texttt{sort}), but still produced an overall decrease when considering ROP and JOP. A similar increase is also observed with TRIMMER in three applications~\cite{TRIMMER}.
\begin{figure}[htb]
\centering
\includegraphics[width=\columnwidth]{images/rop.png}
\caption{Reduction in the total number of gadget occurrences (SYS, ROP, and JOP) between \mbox{\textsc{LMCAS}}\xspace and OCCAM for \textit{Benchmark\_1}. (Higher numbers are better.)}
\label{fig:rop}
\end{figure}
\subsection{Scalability} \label{subsec:Scalability}
We evaluated the capability of \mbox{\textsc{LMCAS}}\xspace to handle \textit{large-scale programs} that maintain complex input formats, such as object files, binaries, and network packets.
We used \textit{Benchmark\_3} in this experiment. The programs considered in \textit{Benchmark\_3} have been used in prior work to evaluate scalability~\cite{OCCAM,SymCC}, including the scalability of KLEE~\cite{moklee}.
Accordingly, we use the following applications\footnote{
Because deciding which lines of code belong to an individual application is difficult, lines of code (LOC) are for the whole application suites from which our benchmarks are taken.
We used \texttt{scc} (https://github.com/boyter/scc) to report LOC.
}
to show the scalability of \mbox{\textsc{LMCAS}}\xspace:
\begin{itemize}
\item \texttt{tcpdump}~\cite{tcpdump} (version 4.10.0; 77.5k LOC) analyzes network packets.
We link against
its accompanying libpcap library (version 1.10.0; 44.6k LOC).
\item \texttt{readelf} and \texttt{objdump} from GNU Binutils~\cite{Binutils} (version 2.33; 78.3k LOC and 964.4k of library code\footnote{
We only consider lines in the binutils folder and the following dependencies: libbfd, libctf, libiberty, and libopcodes.
}).
\texttt{readelf} displays information about ELF files, while \texttt{objdump} displays information about object files.
\end{itemize}
We debloated these programs using different inputs to illustrate the capability of \mbox{\textsc{LMCAS}}\xspace to handle and debloat programs based on single and multiple inputs.
For example, we debloated \texttt{readelf} based on one argument \texttt{(-S)} and nine arguments \texttt{(-h -l -S -s -r -d -V -A -I)}. Table~\ref{tab:largeApps} breaks down the analysis time in terms of Partial Interpretation (third column) and the combination of Constant Conversion and Multistage Simplification (fourth column).
For these programs, the time for symbolic execution is lower than that for the LLVM-based simplifications.
This situation is expected because these programs contain a large number of functions, so a longer time is needed for the LLVM simplifications steps.
The inclusion of third-party libraries diminishes the reduction rate in the binary size, which is clearly illustrated in the achieved reduction rate for \texttt{tcpdump}.
\begin{table}[!tb]
\centering
\caption{Scalability analysis of large applications, based on various inputs.
\Omit{\twr{This table is now inconsistent with Figure \ref{fig:analysisTimeComparison}: one refers to CC \& MS; the other refers to "partial-evaluation steps". I prefer CC \& MS because it is more specific.} \ma{I fixed that, everything now is based on CC \& MS}}
}
\small
\scalebox{0.9}{
\begin{tabular}{|p{1.2cm}|p{1.8cm}|p{0.8cm}|p{1.2cm}|p{1.9cm}|}
\hline
App & Inputs & PI (sec) & CC \& MS (sec)& Binary Size\newline Reduction Rate\\ \hline
\multirow{2}{*}{tcpdump} & -i ens160 & 48.1 & 173.1 & 2\%\\ \cline{2-5}
& -i ens160 -c 5 & 48.2 & 201.7 &2\% \\ \hline
\multirow{2}{*}{readelf} & -S & 10.6 & 41.7 & 4.8\%\\ \cline{2-5}
& -h -l -S -s -r -d -V -A -I & 20.15 & 72.4 & 4.71\%\\ \hline
\multirow{2}{*}{objdump} & -x & 40.84 & 246.17 & 5.65\%\\ \cline{2-5}
& -h -f -p & 48.07 & 320.11 & 5.71\%\\ \hline
\end{tabular}
}
\label{tab:largeApps}
\end{table}
\subsection{Robustness} \label{subsec:robustness}
We assessed the robustness of the debloated programs using fuzzing.
The aim was to test whether a debloated app functioned correctly and did not crash.
We used AFL (version 2.56b), a state-of-the-art fuzzing tool~\cite{afl}, to perform this experiment.
AFL has been used to improve coverage~\cite{SymCC} and to verify robustness~\cite{Chisel}.
Because our analysis is performed on LLVM bitcode, we could not use AFL to instrument the source code (i.e., gray-box mode);
instead, we used AFL's black-box mode, which requires using the QEMU emulator (https://www.qemu.org/).
We compiled debloated programs with \texttt{gcc -O2} to generate the binary file for each program.
Because black-box mode is 2-5x slower than gray-box mode, we ran AFL on the debloated programs created from \textit{Benchmark\_1} for 2 hours/program;
however, we ran AFL on the debloated programs created from \textit{Benchmark\_2} for six days.
AFL did not bring out any failures or crashes of the debloated programs in either dataset.
This experiment provides additional confidence in the correctness of the debloated programs created using \mbox{\textsc{LMCAS}}\xspace.
\begin{comment}
\rev{
\subsection{RQ5 (Debloating Capabilities)}
Table~\ref{tab:benchApps} presents \mbox{\textsc{LMCAS}}\xspace debloating settings for a subset of benchmark apps from Benchmark\_1 based on certain arguments (the second column) to conduct performance evaluation. However, \mbox{\textsc{LMCAS}}\xspace offers debloating based on various combination of input arguments and is not limited to the enabled arguments in the second column in Table~\ref{tab:benchApps}. Therefore, this section shows \mbox{\textsc{LMCAS}}\xspace's capability to perform debloating based on various input arguments for a subset apps from Benchmark\_1. For example, in Table~\ref{table:capability}, \mbox{\textsc{LMCAS}}\xspace supports different arguments and/or combination of arguments to specialize apps according to the required functionalities. The last three columns provide the reduction rate w.r.t number of functions, binary size, and total gadgets in the generated specialized programs. We also ran AFL on the debloated programs and it did not report any failures or crashes.
}
\begin{table}[!htb]
\centering
\caption{\rev{Debloating subset apps from Benchmark\_1 based on various input arguments}}
\small
\scalebox{0.89}{
\begin{tabular}{|p{0.5cm}|p{1.45cm}|p{2.4cm}|p{1.4cm}|p{0.8cm}|p{0.95cm}|}
\hline
\multirow{2}{*}{App} & \multirow{2}{*}{\shortstack{Enabled\\ Arguments}} & \multirow{2}{*}{\shortstack{Required\\ Functionality}} & \multicolumn{3}{l|}{Reduction After Debloating} \\\cline{4-6}
& & & \#Functions & \shortstack{Binary \\ Size} & \shortstack{Total\\ Gadgets} \\ \hline
\multirow{2}{*}{du} & -b & shows number of bytes & 23\% & 15\% & 46\% \\ \cline{2-6}
& -b --time & shows the time of the last modification and number of bytes & 22\% & 14\% & 45\% \\ \hline
\multirow{3}{*}{sort} & -c & check if the file given is already sorted or not & 34\% & 28\% & 54\% \\ \cline{2-6}
& -n & sort a file numerically & 31\% & 25\% & 51\% \\ \cline{2-6}
& -un & sort a file numerically and remove duplicate & 31\% & 25\% & 51\% \\ \hline
\multirow{4}{*}{wc} & -c & character count & 42\% & 21\% & 41\% \\ \cline{2-6}
& -w & word count & 42\% & 21\% & 41\% \\\cline{2-6}
& -lc & line and character count & 43\% & 22\% & 42\% \\ \cline{2-6}
& -wc & word and character count & 42\% & 21\% & 42\% \\\hline
\end{tabular}}
\label{table:capability}
\end{table}
\end{comment}
\section{Evaluation}
\label{sec:evaluation}
This section presents our experimental evaluation of \mbox{\textsc{LMCAS}}\xspace. We address the following research questions:
\begin{itemize}
\item \textbf{Effectiveness of Neck Miner:}
How \textit{accurate} is the neck miner in identifying the neck location? (\ref{subsec:neckMiner})
\item \textbf{Optimizing Debloating Process:}
Does \mbox{\textsc{LMCAS}}\xspace \textit{speed up} the debloating process w.r.t. running time? (\ref{subsec:Optimizing})
\item \textbf{Functionality Preserving and Robustness:} Does \mbox{\textsc{LMCAS}}\xspace produce \textit{functional} programs? and how \textit{robust} are the debloated programs produced by \mbox{\textsc{LMCAS}}\xspace?
(\ref{subsec:Preserving})
\item \textbf{Code Reduction:}
What is the debloating performance of \mbox{\textsc{LMCAS}}\xspace w.r.t. the amount that programs are \textit{reduced in size}? (\ref{subsec:Reduction})
\item \textbf{Security:}
Can \mbox{\textsc{LMCAS}}\xspace reduce the \textit{attack surface}? (\ref{subsec:security})
\item \textbf{Scalability:}
How \textit{scalable} is \mbox{\textsc{LMCAS}}\xspace in debloating large apps? (\ref{subsec:Scalability})
\end{itemize}
\noindent\textit{\textbf{Experimental Setup}}. Our evaluation relies on three datasets, as shown in Table~\ref{tab:dataset}.
\textit{Benchmark\_1} contains $15$ programs from GNU Coreutils v8.32 (see Table~\ref{tab:benchmark1}).
\textit{Benchmark\_2} contains six programs obtained from ChiselBench (see Table~\ref{table:cve}).\footnote{https://github.com/aspire-project/chisel-bench}
\textit{Benchmark\_3} consists of three programs (see Table~\ref{tab:largeApps}).
\rev{
The selection of programs in Benchmark_1 was motivated by their use in prior papers on software debloating. We used Benchmark_2 because it provides us a list of CVE vulnerabilities and corresponding apps; considering this dataset facilitates our evaluation of the removal of CVEs, and allows us to compare against the CVE-removal performance of Chisel and RAZOR.
}
All experiments were conducted on an Ubuntu $18.04$ machine with a 2.3GHz Quad-Core Intel i7 processor and 16GB RAM, except the fuzzing experiment, for which we used an Ubuntu $18.04$ machine with a 3.8GHz Intel(R) Core(TM) i7-9700T CPU and 32GB RAM.
\begin{table}[!tb]
\centering
\caption{Benchmark sets used in the evaluation.}
\scalebox{0.9}{
\begin{tabular}{llcl}
\hline
Source & Label & \# of apps \\ \hline
\thead{GNU Coreutils 8.32} & Benchmark\_1 & 15 \\ \hline
\thead{CHISEL Benchmark} & Benchmark\_2 & 6 \\ \hline
\thead{Tcpdump \& GNU Binutils} & Benchmark\_3 & 3 \\ \hline
\end{tabular}
}
\label{tab:dataset}
\end{table}
\noindent\textit{\textbf{Compared tools and approaches}}. To evaluate the effectiveness of \mbox{\textsc{LMCAS}}\xspace, we compared with the following tools and approaches:
\begin{itemize}
\item Baseline. We establish the baseline by compiling each app's LLVM bitcode at the -O2 level of optimization. This baseline approach was used in prior work~\cite{TRIMMER}.
\item OCCAM~\cite{OCCAM}. The system most comparable to the approach used by \mbox{\textsc{LMCAS}}\xspace. However, OCCAM does not perform constant propagation, and thus omits a major component of partial evaluation.
\item CHISEL~\cite{Chisel}. It requires the user to identify wanted and unwanted functionalities and uses reinforcement learning to perform the reduction process.
\item RAZOR~\cite{RAZOR}. Similar to CHISEL, RAZOR relies on test cases to drive the debloating but incorporates heuristic analysis to improve soundness. RAZOR performs debloating for binary code, while the others operate on top of the LLVM IR code.
\end{itemize}
\revUN{We considered CHISEL and RAZOR as they represent state-of-the-art tools that are applying aggressive debloating techniques. While we selected OCCAM because it is a state-of-the-art partial evaluation tool and thus is the closest to \mbox{\textsc{LMCAS}}\xspace. Comparing with these various tools facilitates the verify the capabilities and effectiveness of \mbox{\textsc{LMCAS}}\xspace.}
\subsection{Effectiveness of the Neck Miner} \label{subsec:neckMiner}
\revUN{In this experiment, we measured the effectiveness of the neck miner in facilitating neck identification.
Our evaluation involved the $26$ programs specified in Table~\ref{tab:dataset}.
These programs belong to various projects: Coreutils, Binutils, Diffutils, Nginx and Tcpdump. For all $26$ programs, neck mining was successful, and the identified neck location was used to perform debloating.}
\revUN{For some programs, such as GNU \texttt{wc} and \texttt{date}, there were multiple candidate neck locations before the shortest-distance criterion was applied at Line~\ref{alg:nm-closestStmt} of Algorithm~\ref{alg:neckMiner}.
(Table~\ref{tab:neckMinerResults} in the Appendix~\ref{apndx:neckMinerEvaluation} contains the full set of results.)
}
\revUN{The neck location is inside the \texttt{main} function for the majority of the programs, except \texttt{readelf} and \texttt{Nginx}.
With the help of the neck miner, it took only a few minutes for the one manual step (Line~\ref{alg:nm-matchStmtProp} of Algorithm~\ref{alg:neckMiner}) needed to identify the neck locations.
More specifically, for each program the analysis time for the heuristic analysis was $2$ seconds on average, and it took $5-10$ minutes to perform the manual part of structural analysis.
This amount of time is acceptable, given that neck identification is performed only once per program.
}
\begin{comment}
\revUN{The heuristic analysis identified correctly the scope of the neck location where the structural analysis can be conducted. But the performance of the heuristic analysis can be improved by considering additional heuristics. For example, in the $15$ GNU Coreutils programs in \texttt{Benchmark\_1}, we noticed these programs use a particular idiom for parsing command-line parameters. Incorporating such idioms in the heuristic analysis would be useful for projects (i.e., BusyBox~\cite{BusyBox}) that require tiny versions \twr{What is the connection between the idioms and "requir[ing] tiny versions"? I don't see those to things as being connected, so the sentence seems like a non sequitur.}of many common UNIX utilities or any other group of programs.}
\end{comment}
\revUN{As mentioned in Section~\ref{subsec:neckSpec}, the neck is identified only once for each program: the same neck can be used, regardless of what arguments are supplied.
To verify this aspect, we debloated various programs based on different supplied inputs.
For example, we debloated \texttt{sort} and \texttt{wc} based on $4$ and $5$ input settings, respectively, and (for each program) the same neck location was used with all debloating settings.
Similarly, a single neck location is used for multiple debloatings for each of the programs listed in Tables~\ref{tab:largeApps} and \ref{tab:capability} (Appendix~\ref{apndx:neckMinerEvaluation}).
}
\subsection{Optimizing Debloating Process} \label{subsec:Optimizing}
\revUN{We compared the running time of \mbox{\textsc{LMCAS}}\xspace against those of CHISEL, RAZOR, and OCCAM on \textit{Benchmark\_2}. We used this benchmark because it was used by both CHISEL and RAZOR, thus the used test cases are available; otherwise, we need to come up with a set of test cases, which is not trivial. Debloating settings in this experiment are listed in Table~\ref{tab:chiselbenchinput}.
As depicted in Figure~\ref{fig:analysisTimeComparison}, the running times for \mbox{\textsc{LMCAS}}\xspace and OCCAM are significantly lower than the time for aggressive debloating techniques CHISEL and RAZOR.
As a result, \mbox{\textsc{LMCAS}}\xspace runs up to $1500$x, $4.6$x, and $1.2$x faster on average than CHISEL, RAZOR, and OCCAM, respectively. This result illustrates \mbox{\textsc{LMCAS}}\xspace substantially speeds up the debloating process in contrast to aggressive debloating tools, but also slightly outperforms partial evaluation debloating techniques.}
\begin{figure}[!htb]
\centering
\includegraphics[width=1\columnwidth]{images/OCCAM_CHISEL_LMCAS_V4.pdf}
\caption{\Omit{\twr{The y-axis says "Analysis Time''. Make it consistent with the text, which I changed to ``running time''.}}
Running times of \mbox{\textsc{LMCAS}}\xspace, CHISEL, RAZOR, and OCCAM based on \textit{Benchmark\_2} (ChiselBench).}
\label{fig:analysisTimeComparison}
\end{figure}
\subsection{Functionality Preserving and Robustness} \label{subsec:Preserving}
\revUN{In this experiment, we ran the binaries before and after debloating against given test cases to understand their robustness.
The majority of the programs debloated in \textit{Benchmark\_2} using RAZOR and CHISEL are suffering from run-time issues. The issues include crashing, infinite loop, or performing unexpected operations. These issues are reported and discussed in~\cite{RAZOR}. In our experiment, we found that all the debloated programs by CHISEL contain these issues. Among the programs in \textit{Benchmark\_2}, all but one of the OCCAM-debloated programs work correctly, and all of the \mbox{\textsc{LMCAS}}\xspace-debloated applications run correctly.}
\revUN{Since \mbox{\textsc{LMCAS}}\xspace and OCCAM have comparable results over \textit{Benchmark\_2}, we extended our evaluation by debloating programs in \textit{Benchmark\_1} according to the settings in Table~\ref{tab:benchmark1}.
Five of the OCCAM-debloated programs ($33\%$) crash (i.e., segmentation fault) or generate inaccurate results, as reported in Table~\ref{tab:funcPreserving}. In contrast, all of the \mbox{\textsc{LMCAS}}\xspace-debloated programs run correctly (i.e., \mbox{\textsc{LMCAS}}\xspace preserves programs' behavior).}
We further assessed the robustness aspect of the debloated programs using fuzzing. It has been used to verify the robustness of debloated programs in~\cite{Chisel}. The aim was to test whether a debloated programs by \mbox{\textsc{LMCAS}}\xspace functioned correctly and did not crash. We used AFL (version 2.56b), a state-of-the-art fuzzing tool~\cite{afl}, to perform this experiment. We used AFL's black-box mode because our analysis is performed on LLVM bitcode; therefore, we could not use AFL to instrument the source code.
We ran AFL on the debloated programs created from \textit{Benchmark\_1} and \textit{Benchmark\_2} for six days. AFL did not bring out any failures or crashes of the debloated programs in either dataset. This experiment provides additional confidence in the correctness of the debloated programs created using \mbox{\textsc{LMCAS}}\xspace.
\begin{table}[]
\centering
\caption{Evaluation of functionality preserving after debloating by \mbox{\textsc{LMCAS}}\xspace and OCCAM for programs in Benchmark\_1. $\cmark$ means functionality is correctly preserved; otherwise: crashing (C), Infinite Loop(L), or Wrong operation (W).}
\scalebox{0.9}{
\begin{tabular}{|c|c|c|}
\hline
\multicolumn{1}{|l|}{\textbf{Program}} & \multicolumn{1}{l|}{\textbf{OCCAM}} & \multicolumn{1}{l|}{\textbf{LMCAS}} \\ \hline
basename & $\cmark$ & $\cmark$ \\ \hline
basenc & $\cmark$ & $\cmark$ \\ \hline
comm & $\cmark$ & $\cmark$ \\ \hline
date & $\cmark$ & $\cmark$ \\ \hline
du & W & $\cmark$ \\ \hline
echo & $\cmark$ & $\cmark$ \\ \hline
fmt & $\cmark$ & $\cmark$ \\ \hline
fold & $\cmark$ & $\cmark$ \\ \hline
head & L & $\cmark$ \\ \hline
id & $\cmark$ & $\cmark$ \\ \hline
kill & W & $\cmark$ \\ \hline
realpath & $\cmark$ & $\cmark$ \\ \hline
sort & L & $\cmark$ \\ \hline
uniq & $\cmark$ & $\cmark$ \\ \hline
wc & C & $\cmark$ \\ \hline
\end{tabular}
}
\label{tab:funcPreserving}
\end{table}
\subsection{Code Reduction} \label{subsec:Reduction}
We used Benchmark\_1 to compare the performance of LMCAS against baseline and OCCAM. Figure~\ref{fig:FineGrainedReductionO2_OCCAM_LMCAS} shows the average reduction in size, using four different size metrics, achieved by baseline, OCCAM, and \mbox{\textsc{LMCAS}}\xspace.
All size measures, except binary size, are taken from the LLVM Intermediate Representation (IR).
For computing the binary-size metric, we compiled all debloated apps with \texttt{gcc}, and ran \texttt{size}.
We report the sum of the sizes of all sections in the binary file (text + data + bss) because this quantity reflects the outcome of our simplifications across all sections. \mbox{\textsc{LMCAS}}\xspace achieved significant higher reduction rate (i.e., around the double) in comparison with baseline and OCCAM.
This result is due to the fact that the clean-up step of \mbox{\textsc{LMCAS}}\xspace can remove nodes in the call-graph that correspond to functions in binary libraries that are not used.
Although the reduction rates of \mbox{\textsc{LMCAS}}\xspace and OCCAM are close (geometric mean binary size reduction is $25\%$ and $22\%$, respectively), some of the specialized programs generated by OCCAM are not reliable (as discussed in Section~\ref{subsec:Preserving}).
\begin{figure}[!htb]
\centering
\includegraphics[width=1\columnwidth]{images/FineGrainedReductionO2_OCCAM_LMCAS.pdf}
\caption{Average reduction in size achieved through baseline, OCCAM, and \mbox{\textsc{LMCAS}}\xspace, using four different size metrics. (Higher numbers are better.)}
\label{fig:FineGrainedReductionO2_OCCAM_LMCAS}
\end{figure}
\revUN{Although the baseline shows a higher average reduction rate at instruction and basic block levels, its average reduction at binary size is the worse. Indeed, it increases the binary size for two programs (i.e., basenc and kill), as depicted in Figure~\ref{fig:binaryReductionO2_LMCAS_OCCAM} (Appendix~\ref{sec:additionalResultsComparison} presents extended results), which shows the comparison results based on the reduction in the binary size that each tool achieved for each app in \textit{Benchmark\_1}.}
\begin{figure}[!htb]
\centering
\includegraphics[width=1\columnwidth]{images/BinaryReduction.pdf}
\caption{Binary size reduction achieved through baseline OCCAM, and \mbox{\textsc{LMCAS}}\xspace. (Higher numbers are better.)}
\label{fig:binaryReductionO2_LMCAS_OCCAM}
\end{figure}
\subsection{Security benefits of \mbox{\textsc{LMCAS}}\xspace} \label{subsec:security}
We evaluated the capabilities of \mbox{\textsc{LMCAS}}\xspace to reduce code-reuse attacks and to remove known vulnerabilities. Therefore, we conducted the following three experiments: (i) we attempt to reproduce (by executing the app) the vulnerability after debloating the apps to see if the vulnerabilities were eliminated;
(ii) we measured the reduction in the number of code-reuse attacks\Omit{attack surface} by counting the number of\Omit{return-oriented programming (ROP)} eliminated gadgets in compiled versions of the original and reduced programs
(iii) we compared the degree of gadget reduction achieved by \mbox{\textsc{LMCAS}}\xspace and LLVM-CFI.
\noindent\textbf{Vulnerability Removal}.
To test the ability to mitigate vulnerabilities, we used six programs in \textit{Benchmark\_2} because this benchmark contains a set of known CVEs.
Table~\ref{table:cve} presents a comparison between \mbox{\textsc{LMCAS}}\xspace, RAZOR, OCCAM, and CHISEL.
\mbox{\textsc{LMCAS}}\xspace and OCCAM removed CVEs from $5$ out of the $6$ programs, but the debloated \texttt{sort-8.16} using OCCAM was not behaving correctly at run-time. While CHISEL and RAZOR removed CVEs from $4$ programs. Although OCCAM removed CVEs from $3$ programs including \texttt{rm-8.4}, but the debloated \texttt{rm-8.4} shows unexpected infinite-loop behaviour. We suspect that OCCAM may remove loop condition checks.
\mbox{\textsc{LMCAS}}\xspace could not remove the vulnerability in \texttt{date-8.21} because the bug is located in the core functionality of this program.
When undesired functionality is too intertwined with the core functionality---e.g., multiple quantities are computed by the same loop---then \mbox{\textsc{LMCAS}}\xspace may not be able to remove some undesired functionality because---to the analysis phases of \mbox{\textsc{LMCAS}}\xspace---it does not appear to be useless code.
In such cases, \mbox{\textsc{LMCAS}}\xspace retains the desired functionality.
In contrast, CHISEL tries to remove all undesired functionality.
\begin{table}[!tb]
\centering
\caption{Vulnerabilities after debloating by \mbox{\textsc{LMCAS}}\xspace and CHISEL. $\cmark$ means the CVE vulnerability is eliminated; $\xmark$ means that it was not removed.}
\scalebox{0.75}{
\begin{threeparttable}
\begin{tabular}{llcccc}
\hline
App & CVE ID & RAZOR & CHISEL & OCCAM & \mbox{\textsc{LMCAS}}\xspace \\ \hline \hline
chown-8.2 & CVE-2017-18018 & $\cmark$ & $\cmark$ & $\cmark$ & $\cmark$ \\ \hline
date-8.21 & CVE-2014-9471 & $\cmark$ & $\xmark$ & $\xmark$ & $\xmark$ \\ \hline
gzip-1.2.4 & CVE-2015-1345 & $\cmark$ & $\cmark$ & $\xmark$ & $\cmark$ \\ \hline
rm-8.4 & CVE-2015-1865 & $\xmark$ & $\xmark$ & $\xmark$ & $\cmark$ \\ \hline
sort-8.16 & CVE-2013-0221 & $\cmark$ & $\cmark$ & $\cmark$ & $\cmark$ \\ \hline
uniq-8.16 & CVE-2013-0222 & $\xmark$ & $\cmark$ & $\cmark$ & $\cmark$\\ \hline
\end{tabular}
\begin{tablenotes}
\item[1] \small \revUN{CHISEL and RAZOR CVE removal is obtained from the corresponding publication.
\item[2] \small Although OCCAM removed the CVE in rm-8.4, but the debloated version suffers from infinite loop at run-time.}
\end{tablenotes}
\end{threeparttable}
}
\label{table:cve}
\end{table}
\noindent\textbf{Gadget Elimination}.
Code-reuse attacks leverage existing code snippets in the executable, called gadgets, as the attack payload~\cite{CSET19}.
Those gadgets are often categorized---based on their last instruction---into Return-Oriented Programming (ROP), Jump-Oriented Programming (JOP), and syscall (SYS)~\cite{Piece-Wise,CARVE}.
Code-reuse attacks can be mitigated using software diversification, which can be achieved through application specialization.
Software debloating is a specialization technique that can both reduce the number of exploitable gadgets and change the locations of gadgets, thus diversifying the binary.
In this experiment, we used \textit{Benchmark\_1} and observed noticeable reductions in the total gadget count \Omit{\twr{What are SYS and JOP? Are you saying that you have two other figures like Figure~\ref{fig:rop} for two other kinds of gadgets? If so, break that out into a separate paragraph (after this one), and report an aggregate number like the geometric means for the SYS and JOP measurements. Have I explained the subtleties of computing geometric mean of percentages to you? You cannot just compute the geometric means of the raw percentage numbers.} \ma{I apologies for the incorrect description, but Fig 6 reports the 3 types of gadgets SYS, JOP and ROP. I fixed that by making sure everything is consistent that we report total gadgets}}(occurrences of ROP, SYS, and JOP gadgets), as illustrated in Figure~\ref{fig:rop}.
We used ROPgadget~\cite{ROPGadget} for counting the number of gadgets.
\Omit{\twr{I think we need to clarify what is meant by the phrase ``total number of gadgets.''
From hidden comments, you say you mean SYS + JOP + ROP gadgets.
However, there is still ambiguity about whether you are counting
gadget \emph{kinds} or gadget \emph{instances}.
That is if the program contains 5 instances of gadget g1, 3 of gadget g2,
and 12 of gadget g3, is that ``three total gadgets'' or ``twenty total gadgets''?
} \ma{20 if all gadgets are unique}}
\Omit{\twr{How is average computed? Say which kind of average you are reporting: arithmetic mean or geometric mean.} \ma{Arithmetic}}The average reduction (arithmetic mean) in the total number of unique gadgets is $51.7\%$ and the maximum reduction is $72.1\%$ (for \texttt{date}), while OCCAM reduces the total number of unique gadgets by $21.5\%$, on average, with a maximum reduction of $80.3\%$ (for \texttt{echo}).
For one program, \texttt{sort}, OCCAM increases the total number gadgets by $6.2\%$.
With \mbox{\textsc{LMCAS}}\xspace, the number of SYS gadgets is reduced to $1$ for $14$ out of the $15$ applications.
\mbox{\textsc{LMCAS}}\xspace caused an increase in the number of SYS gadgets in one application (\texttt{sort}), but still produced an overall decrease when considering ROP and JOP. A similar increase is also observed with TRIMMER in three applications~\cite{TRIMMER}.
\begin{figure}[htb]
\centering
\includegraphics[width=\columnwidth]{images/rop.png}
\caption{Reduction in the total number of unique gadget occurrences (SYS, ROP, and JOP) between \mbox{\textsc{LMCAS}}\xspace and OCCAM for \textit{Benchmark\_1}. (Higher numbers are better.)}
\label{fig:rop}
\end{figure}
\noindent
\revUN{\textbf{Control-Flow Integrity (CFI).}
CFI is a prominent mechanism for reducing a program's attack surface by preventing control-flow hijacking attacks:
CFI confines the program to some specific set of CFG paths, and prevents the kinds of irregular transfers of control that take place when a program is attacked.
Although CFI does not specifically aim to reduce the number of gadgets, others have observed empirically that CFI reduces the number of unique gadgets in programs \cite{enforce_CFI,Fine-CFI,Ancile}.
Thus, we compared the degree of gadget reduction achieved by \mbox{\textsc{LMCAS}}\xspace and LLVM-CFI (a state-of-the-art static CFI mechanism)~\cite{Ancile}.
We compiled the LLVM bitcode of our suite of programs using \texttt{clang}, with the flags \texttt{-fsanitize=cfi -fvisibility=default}.
Among the $20$ programs analyzed, \mbox{\textsc{LMCAS}}\xspace outperformed LLVM-CFI on $12$ programs (60\%) by creating a program with a smaller total number of unique gadgets. (Table~\ref{tab:cfiAnalysis} in the Appendix~\ref{apndx:cfi} contains the full set of results.)
The last column in Table~\ref{tab:cfiAnalysis} shows that a significant reduction in unique gadgets---beyond what either \mbox{\textsc{LMCAS}}\xspace or LLFM-CFI is capable of alone---is obtained by first applying \mbox{\textsc{LMCAS}}\xspace and then LLVM-CFI.}
\subsection{Scalability} \label{subsec:Scalability}
We evaluated the capability of \mbox{\textsc{LMCAS}}\xspace to handle \textit{large-scale programs} that maintain complex input formats, such as object files, binaries, and network packets.
We used \textit{Benchmark\_3} in this experiment. The programs considered in \textit{Benchmark\_3} have been used in prior work to evaluate scalability~\cite{OCCAM,SymCC}, including the scalability of KLEE~\cite{moklee}.
Accordingly, we use the following applications\footnote{
Because deciding which lines of code belong to an individual application is difficult, lines of code (LOC) are for the whole application suites from which our benchmarks are taken.
We used \texttt{scc} (https://github.com/boyter/scc) to report LOC.
}
to show the scalability of \mbox{\textsc{LMCAS}}\xspace:
\begin{itemize}
\item \texttt{tcpdump}~\cite{tcpdump} (version 4.10.0; 77.5k LOC) analyzes network packets.
We link against
its accompanying libpcap library (version 1.10.0; 44.6k LOC).
\item \texttt{readelf} and \texttt{objdump} from GNU Binutils~\cite{Binutils} (version 2.33; 78.3k LOC and 964.4k of library code\footnote{
We only consider lines in the binutils folder and the following dependencies: libbfd, libctf, libiberty, and libopcodes.
}).
\texttt{readelf} displays information about ELF files, while \texttt{objdump} displays information about object files.
\end{itemize}
We debloated these programs using different inputs to illustrate the capability of \mbox{\textsc{LMCAS}}\xspace to handle and debloat programs based on single and multiple inputs.
For example, we debloated \texttt{readelf} based on one argument \texttt{(-S)} and nine arguments \texttt{(-h -l -S -s -r -d -V -A -I)}. Table~\ref{tab:largeApps} breaks down the analysis time in terms of Partial Interpretation (third column) and the combination of Constant Conversion and Multistage Simplification (fourth column).
For these programs, the time for symbolic execution is lower than that for the LLVM-based simplifications.
This situation is expected because these programs contain a large number of functions, so a longer time is needed for the LLVM simplifications steps.
The inclusion of third-party libraries diminishes the reduction rate in the binary size, which is clearly illustrated in the achieved reduction rate for \texttt{tcpdump}.
\begin{table}[!tb]
\centering
\caption{Scalability analysis of large applications, based on various inputs.
\Omit{\twr{This table is now inconsistent with Figure \ref{fig:analysisTimeComparison}: one refers to CC \& MS; the other refers to "partial-evaluation steps". I prefer CC \& MS because it is more specific.} \ma{I fixed that, everything now is based on CC \& MS}}
}
\small
\scalebox{0.9}{
\begin{tabular}{|p{1.2cm}|p{1.8cm}|p{0.8cm}|p{1.2cm}|p{1.9cm}|}
\hline
Program & Supplied Inputs & PI (sec) & CC \& MS (sec)& Binary Size\newline Reduction Rate\\ \hline
\multirow{2}{*}{tcpdump} & -i ens160 & 48.1 & 173.1 & 2\%\\ \cline{2-5}
& -i ens160 -c 5 & 48.2 & 201.7 &2\% \\ \hline
\multirow{2}{*}{readelf} & -S & 10.6 & 41.7 & 4.8\%\\ \cline{2-5}
& -h -l -S -s -r -d -V -A -I & 20.15 & 72.4 & 4.71\%\\ \hline
\multirow{2}{*}{objdump} & -x & 40.84 & 246.17 & 5.65\%\\ \cline{2-5}
& -h -f -p & 48.07 & 320.11 & 5.71\%\\ \hline
\end{tabular}
}
\label{tab:largeApps}
\end{table}
\begin{comment}
\rev{
\subsection{RQ5 (Debloating Capabilities)}
Table~\ref{tab:benchApps} presents \mbox{\textsc{LMCAS}}\xspace debloating settings for a subset of benchmark apps from Benchmark\_1 based on certain arguments (the second column) to conduct performance evaluation. However, \mbox{\textsc{LMCAS}}\xspace offers debloating based on various combination of input arguments and is not limited to the enabled arguments in the second column in Table~\ref{tab:benchApps}. Therefore, this section shows \mbox{\textsc{LMCAS}}\xspace's capability to perform debloating based on various input arguments for a subset apps from Benchmark\_1. For example, in Table~\ref{table:capability}, \mbox{\textsc{LMCAS}}\xspace supports different arguments and/or combination of arguments to specialize apps according to the required functionalities. The last three columns provide the reduction rate w.r.t number of functions, binary size, and total gadgets in the generated specialized programs. We also ran AFL on the debloated programs and it did not report any failures or crashes.
}
\begin{table}[!htb]
\centering
\caption{\rev{Debloating subset apps from Benchmark\_1 based on various input arguments}}
\small
\scalebox{0.89}{
\begin{tabular}{|p{0.5cm}|p{1.45cm}|p{2.4cm}|p{1.4cm}|p{0.8cm}|p{0.95cm}|}
\hline
\multirow{2}{*}{App} & \multirow{2}{*}{\shortstack{Enabled\\ Arguments}} & \multirow{2}{*}{\shortstack{Required\\ Functionality}} & \multicolumn{3}{l|}{Reduction After Debloating} \\\cline{4-6}
& & & \#Functions & \shortstack{Binary \\ Size} & \shortstack{Total\\ Gadgets} \\ \hline
\multirow{2}{*}{du} & -b & shows number of bytes & 23\% & 15\% & 46\% \\ \cline{2-6}
& -b --time & shows the time of the last modification and number of bytes & 22\% & 14\% & 45\% \\ \hline
\multirow{3}{*}{sort} & -c & check if the file given is already sorted or not & 34\% & 28\% & 54\% \\ \cline{2-6}
& -n & sort a file numerically & 31\% & 25\% & 51\% \\ \cline{2-6}
& -un & sort a file numerically and remove duplicate & 31\% & 25\% & 51\% \\ \hline
\multirow{4}{*}{wc} & -c & character count & 42\% & 21\% & 41\% \\ \cline{2-6}
& -w & word count & 42\% & 21\% & 41\% \\\cline{2-6}
& -lc & line and character count & 43\% & 22\% & 42\% \\ \cline{2-6}
& -wc & word and character count & 42\% & 21\% & 42\% \\\hline
\end{tabular}}
\label{table:capability}
\end{table}
\end{comment}
\section{Implementation}
\noindent\textit{\textbf{Neck Miner}}.
\revUN{This component is implemented as an LLVM analysis pass.
In command-line programs, we use LLVM's def/use API to track the use of argv.
For configuration-file programs, we iterate over the LLVM IR code to identify call-sites for the pre-identified file-parsing APIs.
The developer has the responsibility of identifying the program locations that satisfy the structural properties from \S\ref{Se:StructuralAnalysis}.
(i.e., locations that are executed only once, and dominate the main logic).
This task is relatively easy because the developer can rely on existing LLVM analysis passes to compute the dominance tree and verify the structural properties.
We argue that such efforts are manageable. More importantly, they are one-time efforts.
(Such a semi-automated approach has also been used in prior work~\cite{mpi} and completely manual~\cite{Temporal}.)
Finally, the neck location is marked by adding a special function call to the program being analyzed.}
\noindent\textit{\textbf{Partial Interpretation}}.
\revUN{Our implementation uses KLEE~\cite{KLEE} to perform the partial interpretation because it (1) models memory with bit-level accuracy, and (2) can handle interactions with the outside environment---e.g., with data read from the file system or over the network---by providing models designed to explore various possible interactions with the outside world.}
We modified KLEE 2.1 to stop and capture the set of affected variables and their corresponding values after the neck is reached.
\rev{In essence, KLEE is being used as an execution platform that supports “partial concrete states.”
For LMCAS, none of the inputs are symbolic, but only part of the full input is supplied.
In the case of word-count, the input is “wc -l”, with no file-name supplied.
Only a single path to the neck is followed, at which point KLEE returns its state (which is a partial concrete state).
The second column in Tables \ref{tab:largeApps} and \ref{tab:benchmark1} describes the inputs supplied to KLEE for the different examples.
}
\noindent\textit{\textbf{Multi-Stage Simplification.}}
\revUN{We also developed two LLVM passes using LLVM 6.0 to perform constant-conversion (CC) and the clean-up step of the MS phase.
We implemented these passes because of the absence of existing LLVM passes that perform such functionalities.
We tried existing LLVM passes like global dead-code elimination (DCE) to remove unused code.
However, global DCE is limited to handle only global variables (and even some global variables cannot be removed).
We also noticed that not all stack variables are removed, so in our clean-up pass we employ def-use information to identify stack variables that are loaded but not used.
Also, the removal of indirect calls is not provided by LLVM.
To prevent the removal of functions invoked via a function pointer, our clean-up pass checks that the number of uses of a function is zero before removing the function.
}
\section{Introduction} \label{sec:introduction}
The software stack is becoming increasingly bloated. This software growth decreases performance and increases security vulnerabilities.
\emph{Software debloating} is a mitigation approach that downsizes programs while retaining certain desired functionality.
Although static program debloating can thwart unknown possibilities for attack by reducing the attack surface~\cite{attackSurface},
prior work has generally not been effectiv
, due to the
overapproximation of static program analysis:
a lot of bloated code remains because these tools statically determine the set of functions to be removed using a combination of analysis techniques, such as unreachable-function analysis and global constant propagation~\cite{BlankIt}.
More aggressive debloating approaches (e.g., RAZOR~\cite{RAZOR} and Chisel~\cite{Chisel}) can achieve more reduction;
however, \revUN{they involve demanding techniques:} the user needs to define a comprehensive set of test cases to cover the desired functionalities, \revUN{generate many traces of the program, and perform extensive instrumentation}.
\revUN{The computational expense of these steps leads to a high-overhead debloating process.
The work on RAZOR acknowledges the challenge of generating test cases to cover all code, and incorporates heuristics to address this challenge.} Furthermore, these aggressive approaches often break soundness, which can lead the debloated programs to crash or execute incorrectly.
These issues make such approaches unsafe and impractical~\cite{BlankIt}.
Partial evaluation is a promising program-specialization technique.
It is applied in prior work~\cite{OCCAM,TRIMMER}, but prior work has suffered from the overapproximation inherent in static program analysis.
In particular, existing implementations rely only on the command-line arguments to drive the propagation of constants.
However, they do not capture precisely the set of variables that are affected by the supplied inputs.
Constant propagation is performed only for global variables that have one of the base types (\texttt{int}, \texttt{char}), and no attempt is made to handle compound datatypes (\texttt{struct}). Therefore,
this approach leaves a substantial amount of unwanted code in the ``debloated program'', which
reduces the security benefits and does not preserve the behaviour of the program after debloating due to unsound transformations (i.e., incorrect constant folding~\cite{TRIMMER}). (our evaluation in~\S\ref{sec:evaluation} shows multiple programs debloated by OCCAM are crashing or behave unexpectedly.)
\Omit{\vr{Only one point, so why (i)?\ma{thanks}}}
\Omit{\twr{I don't understand the second point, which seems backwards: it will \emph{include} the features that are controlled by the variables that partial evaluation ignores.\Omit{\ma{to address John's concern about soundness mentioned in the next statement. So PE in prior work can also exclude features, I had this case when OCCAM returns empty output}}}
\twr{You are thinking about a point that is far to complicated for the introduction. If you are worried about the word "soundness", then remove the word, remove the label "(i)" above, remove all of point "(ii)" below, and end the previous sentence at "reduces the security benefits".}}\Omit{\ma{(ii) can exclude required features that are enabled by the ignored variables.}}
\Omit{\vr{Incomplete sentence. Also, we do not have an evaluation of performance.}\ma{thanks}}
In this paper, we present Lightweight Multi-Stage Compiler-Assisted Application Specialization (\mbox{\textsc{LMCAS}}\xspace), a new software-debloating framework.
\mbox{\textsc{LMCAS}}\xspace relies on the observation that, in general, programs consist of two components:
(a) \textit{configuration logic}, in which the inputs are parsed, and
(b) \textit{main logic}, which implements the set of functionalities provided by the program.
We call the boundary between the two divisions the \textit{neck}.
\revUN{\mbox{\textsc{LMCAS}}\xspace captures a partial state of the program by interpreting the configuration logic based on the supplied inputs. The partial state comprises concrete values of the variables that are influenced by the supplied inputs,
\mbox{\textsc{LMCAS}}\xspace then applies partial-evaluation optimizations to generate the specialized program. These optimizations involve: converting the influenced variables at the neck into constants, applying constant propagation, and performing multiple stages of standard and customized compiler optimizations.
}
\revUN{\mbox{\textsc{LMCAS}}\xspace makes significant and novel extensions to make debloating much safer in a modern context.
The extensions involve optimizing the debloating process and improving its soundness.
Specifically, we optimize the debloating process by introducing the \textit{neck} concept, which eliminates demanding techniques that require: (1) executing the whole program, (2) generating many traces, (3) performing extensive instrumentation, and (4) obtaining a large set of tests.
We demonstrate the soundness of our approach by validating the functionality of $23$ programs after debloating and under various settings.
The achieved soundness is driven by: capturing a precise partial state of the program, supporting various data types, and performing guided constant conversion and clean-up.
}
Our evaluation demonstrates that \mbox{\textsc{LMCAS}}\xspace is quite effective: on average, \mbox{\textsc{LMCAS}}\xspace achieves $25\%$ reduction in the binary size and $40\%$ reduction in the number of functions in the specialized applications.
\mbox{\textsc{LMCAS}}\xspace reduces the attack surface of code-reuse attacks by removing $51.7\%$ of the total gadgets, and eliminates $83\%$ of known CVE vulnerabilities.
On average, \mbox{\textsc{LMCAS}}\xspace runs $1500$x, $4.6$x, and $1.2$x faster than the state-of-the-art debloating tools CHISEL, RAZOR, and OCCAM, respectively.
\revUN{Hence, LMCAS strikes a favorable trade-off between functionality, performance, and security.
}
\begin{comment}
We demonstrate the scalability and robustness of \mbox{\textsc{LMCAS}}\xspace on three large applications: \texttt{tcpdump}, \texttt{readelf}, and \texttt{objdump}.
\end{comment}
The contributions of our work are as follows:
\begin{enumerate}
\item \revUN{We propose the novel idea of dividing programs into configuration logic and main logic to reduce the overhead of debloating process
}
\item \revUN{We develop a neck miner to enable the identification of such boundary with high accuracy, substantially alleviating the amount of manual effort required to identify the neck.}
\Omit{\twr{The previous sentence is way off the mark.
It makes it sound like the paper is about dynamic analysis, and that partitioning is a technique that you have introduced to improve a dynamic-analysis tool.}}
\item \revUN
We apply the principles of partial evaluation to generate specialized programs based on supplied inputs. A partial program state is captured by conducting partial interpretation for the program by executing the configuration logic according to the supplied inputs. The program state is enforced by applying compiler optimizations for the main logic to generate the specialized program.}
\item We develop the \mbox{\textsc{LMCAS}}\xspace prototype based on LLVM. \mbox{\textsc{LMCAS}}\xspace harnesses symbolic execution to perform partial interpretation and applies a set of LLVM passes to perform the compiler optimizations. We also carried out an extensive evaluation based on real-world applications to demonstrate the low overhead, size reduction, security, and practicality of \mbox{\textsc{LMCAS}}\xspace.
\Omit{\twr{I would split this point in two: the ``principles'' and the prototype.
The principles are the core idea found in the paper, and should go either first or second in the list of contributions (probably second, after the idea that many programs can be split into configuration and main logic. The prototype can be folded into the bullet point about evaluation.}}
\item We will make \mbox{\textsc{LMCAS}}\xspace implementation
and its artifacts available to the community.
\end{enumerate}
\section{Motivation and Background} \label{sec:motivation}
In this section, we present a motivating example, and review necessary background material on program analysis and software debloating used in the remainder of the paper.
Finally, we discuss debloating challenges illustrated by the motivating example, and describe our solutions for addressing these challenges
\subsection{Motivating Example}
Listing~\ref{wc} presents a scaled-down version of the UNIX word-count utility \texttt{wc}.
It reads a line from the specified stream (i.e., \texttt{stdin}), counts the number of lines and/or characters in the processed stream, and prints the results.
Thus, this program implements the same action that would be obtained by invoking the UNIX word-count utility with the \texttt{-lc} flag (i.e., \texttt{wc -lc}). Although Listing~\ref{wc} merely supports two counting options, it is still bloated if the user is only interested in enabling the functionality that counts the number of lines.
\input{listings/toy_wc_struct}
Put another way, Listing~\ref{wc} goes
against the \textit{principle of least privilege}~\cite{plp}~\cite{Cimplifier}:
code that implements unnecessary functionality can contain security vulnerabilities that an attacker can use to obtain control or deny service---bloated code may represent an opportunity for privilege escalation.
For instance, the character-count functionality ``\texttt{wc -c}'' processes and decodes the provided stream of characters via the function \texttt{decodeChar} (Line 17 of Listing~\ref{wc}).
An attacker might force it to process special characters that \texttt{decodeChar} cannot handle~\cite{wc-bug}.
More broadly, attackers can supply malicious code as specially crafted text that injects shellcode and thereby bypasses input restrictions~\cite{EnglishShellcode}.
\Omit{\vr{Since shellcode here is English text, the characters will be easily
(correctly) processed by wc, I do not see how supplying a file containing
shellcode will suddenly break wc.} \ma{I missed to mention, this is not normal English text, it's a crafted text of the malicious code. So this text can bypass input restrictions}}
Downsizing the program is a way to reduce its attack surface because the ``\texttt{wc -l}'' functionality does not require the character-processing code used in ``\texttt{wc -c}'':
the call to the function \texttt{decodeChar} is completely absent in the specialized version for ``\texttt{wc -l}''
~\revUN{To achieve this goal, debloating should be performed safely without applying high-demand techniques. The next section describes our approach for handling these challenges.}
\begin{comment}
\begin{figure*}
\centering
\begin{minipage}{0.3\textwidth}
\lstset{style=CStyle}
\lstinputlisting[caption={A scaled-down version of the UNIX word-count utility},label={wc}]{listings/toy_wc_v3.c}
\end{minipage}\qquad
\begin{minipage}{0.3\textwidth}
\lstset{style=CStyle}
\lstinputlisting[caption={A specialized word-count equivalent to $wc -l$},label={wc-l}]{listings/toy_wc_1.c}
\end{minipage}\qquad
\begin{minipage}{0.3\textwidth}
\lstset{style=CStyle}
\lstinputlisting[caption={A specialized word-count equivalent to $wc -c$},label={wc},label={wc-c}]{listings/toy_wc_c.c}
\end{minipage}
\end{figure*}
\end{comment}
\begin{comment}
\begin{figure*}
\centering
\begin{minipage}{0.45\textwidth}
\lstset{style=CStyle}
\lstinputlisting[caption={A scaled-down version of the UNIX word-count utility},label={wc}]{listings/toy_wc_struct.c}
\end{minipage}\qquad
\begin{minipage}{0.45\textwidth}
\lstset{style=CStyle}
\lstinputlisting[caption={Specialized word-count versions equivalent to ``wc -c'' (Lines 2-12) and ``wc -l'' (Lines 15-25) \ma{I'm thinking about removing the wc -c in Listing 2 because it doesn't provide any information }},label={wc},label={wc-cl}]{listings/toy_wc_struct-c.c}
\end{minipage}
\end{figure*}
\end{comment}
\subsection{Background}
\noindent Partial evaluation~\cite{partialEvaluation,slicing,tutorialPE} is an optimization and specialization technique that precomputes program expressions in terms of the known static input ---i.e., a subset of the input that is supplied ahead of the remainder of the input.
The result is another program, known as the \emph{residual program}, which is a specialization of the original program. To create the residual program, a partial evaluator performs optimizations such as loop unrolling, constant propagation and folding, function inlining, etc.~\cite{partialEvaluation}.
Partial evaluation and symbolic execution~\cite{SymbolicExecution} both generalize standard interpretation of programs, but there are key differences between them. Specifically, symbolic execution interprets a program using symbolic input values, while partial evaluation precomputes program expressions and simplifies code based on the supplied inputs.
Unlike partial evaluation, one result of symbolic execution is a set of expressions \Omit{\twr{"Expressions" is correct; formulas are a different kind of result that SE gives, for characterizing what input condition is required to follow a particular path.}}for the program's output variables. Because their capabilities are complementary, the two techniques have been employed together to improve the capabilities of a software-verification tool~\cite{Interleaving}.
\revUN{{LLVM~\cite{LLVM:CGO04}} provides robust compiler infrastructures for popular programming languages, including C and C++, and supports a wealth of compiler analyses and optimizations that make it suitable for developing new compiler transforms.
LLVM operates on its own low-level code representation known as the LLVM intermediate representation (LLVM IR).
LLVM is widely used in both academia and industry.
}
\begin{comment}
\noindent\textbf{Stack Variables~\cite{LLVM:CGO04}} are temporary variables
created by each invocation of a function. For instance, the pointer-to-struct
variable \texttt{flag} in Listing~\ref{wc} is a stack variable. For each local
variable declared in a function, a slot for it is reserved in the function's
stack frame. Every time a function exits, all of the variables pushed onto the
stack by that function are deleted, and that region of memory becomes available
for other purposes (e.g., for the stack variables of new stack frames from other
function calls). \Omit{In the LLVM IR, stack variables are created at the beginning of each function using Allocate Instruction (\texttt{AllocaInst}).}
\Omit{\vr{The previous sentence is very implementation-specific. I suggest moving it further down the paper or introducing a 1-paragraph background on LLVM IR.} \ma{agreed.}}
\end{comment}
\subsection{Challenges and Solutions} \label{subsec:challenges}
In this section, we formalize the program-specialization problem illustrated in Listing~\ref{wc}.
In general, there is (i) a program P that provides a set of functionalities $F$,
and (ii) an input space $I$ that contains a set of values that enable certain functionalities in $F$.
Typically, one or more functionalities are enabled based on a set of supplied inputs $I_s$, which are provided as part of a command-line argument or configuration file.
Generating a specialized program $P\textprime$ based on the set of supplied inputs $I_s$ requires identifying a set of variables $V_s=\{v_0, v_1, .., v_n\}$ that are influenced by the supplied inputs, and a corresponding set of constant values $C_s=\{c_0, c_1, .., c_n\}$. The relationship between $V_s$ and $C_s$ is bijective and $I_s \subset I$.
For generating a specialized program $P\textprime$ that (i) retains the required functionalities based on the supplied inputs $I_s$, and (ii) removes irrelevant functionalities, we need to address the challenges discussed below:
\Omit{\twr{You seemn to have adopted the term ``dynamic analysis,'' which seems completely wrong to me: you are just executing the program along a single path, so in what sense is any ``analysis'' going on?
I thought we were calling it ``partial execution.''}
\twr{
Also, you also keep referring to ``the overhead of dynamic analysis'' in other work. I don't think that that comparison helps in this section.
Let's stick to describing our work, and make comparisons only in the Introduction and Related Work.}}
\textbf{Challenge 1:}
\revUN{how to \emph{optimize} the debloating process and avoid high-demand techniques}?
\textbf{Solution.} \revUN{To address this challenge, we propose to interpret the program partially up to a certain point, instead of executing the whole program. We can achieve this partial interpretation by relying on the observation that, in general, programs consist of two components: (a) \textit{configuration logic}, in which the inputs (from the input space $I$) are parsed, and (b) \textit{main logic}, which implements the set of functionalities $F$. We call the boundary point the \textit{neck}.
The \textit{partial interpreter} needs only part of the program state to be available by executing the program upto the neck.
By this means, we optimize the debloating process, yet obtain a precise characterization of the set of variables $V_s$ that are influenced by the supplied arguments.
We then convert these variables to constants, based on the constant values $C_s$ identified by the \textit{partial interpreter}.
These values are then propagated to other parts of the program via partial evaluation.
Consider Listing~\ref{wc} again.
The program $\texttt{wc}$ provides two functionalities, $F = \{\textit{counting\_lines}, \textit{counting\_characters}\}$ and these functionalities can be activated through the two inputs $I = \{l, c\}$. For generating the specialized program $P\textprime$ that retains the \emph{counting\_lines} functionality (i.e., ``\texttt{wc -l}'') based on the supplied input $I_s = \{l\}$, we interpret program $P$ upto the neck (i.e., Line $15$) to identify the set of influenced variables $V_s = \{\texttt{flag->count\_chars}$, $\texttt{flag->count\_lines}$, $\texttt{total\_lines}$, $\texttt{total\_chars}\}$ and the corresponding constant values $C_s = \{0,1,0,0\}$ (the partial state of $P$). We supply this information to the partial evaluator to generate the specialized program.}
\textbf{Challenge 2:}
\revUN{how to simplify the program \emph{sufficiently}, while ensuring that it operates \emph{correctly}, and preserve its functionality and soundness?}
\textbf{Solution.} \revUN{The combination of \textit{partial interpretation} followed by partial-evaluation optimizations holds the promise of achieving significant debloating.
To achieve this promise and preserve the program semantics, it is necessary to handle various data types and complex data structures (i.e., string, pointers, and structs).
By using a precise model of the programming language's semantics, more information about variables and their values is made available, which in turn enables more optimizations to be carried out during program specialization.
Therefore, we need to capture a broad spectrum of variables. For instance, the scaled-down word-count in Listing~\ref{wc} contains a stack variable (\texttt{flag}) and two global variables (\texttt{total\_lines} and \texttt{total\_chars}). Various data types need to be supported as well: the global variables are integers, whereas the stack variable \texttt{flag} is a pointer to a struct that consists of two fields (\texttt{count\_line} and \texttt{count\_chars}). Supporting these various types of variables provides \mbox{\textsc{LMCAS}}\xspace the capability to perform safe debloating and maintain soundness.}
\subsubsection{Heuristic Analysis}
\revUN{This step
guides the structural analysis. It identifies a single location from which the structural analysis can be conducted, and relies on a set of patterns that apply to two categories of programs.
These patterns are described as follows:}
\revUN{\textbf{Command-Line-Program Patterns (Algorithm~\ref{alg:neckMiner} Lines \ref{alg:cmdLineBegin}--\ref{alg:cmdLineEnd}):}
the inputs are provided to this category of programs via command-line arguments.
In C/C++ programs, command-line arguments are passed to \texttt{main()} via the parameters \texttt{argc} and \texttt{argv}:
\texttt{argc} holds the number of command-line arguments, and \texttt{argv[]} is a pointer array whose elements point to the different arguments passed to the program.
Consequently, this analysis step tracks the use of the argument \texttt{argv} (Line~\ref{alg:nm-trackargv}).
Specifically, because \texttt{argv} is a pointer array that, in general, points to multiple arguments, the analysis identifies the uses of \texttt{argv} that are inside of a loop (Line~\ref{alg:nm-insideLoop}).
}
\revUN{\textbf{Configuration-File-Program Patterns (Algorithm~\ref{alg:neckMiner} Lines \ref{alg:ConfigFileBegin}--\ref{alg:ConfigFileEnd}):} this category of programs relies on configuration files to identify the required functionalities.
As a representative example, consider how the neck is identified in Nginx, a web-server program that supports $724$ different configuration options~\cite{EuroSec19}.
Listing~\ref{lst:nginx} presents a simple Nginx configuration file.
The gzip directive at line $8$ is associated with the libz.so library.
In some cases, multiple directives have to be defined to enable a certain capability, such as the SSL-related directives in lines $7$, $9$, and $10$ of Listing~\ref{lst:nginx}.
The heuristic analysis specifies the first location where the configuration file is parsed by certain APIs.
Identifying such APIs is simple because programs use system-call APIs to read files.
For instance, nginx uses the Linux system call \texttt{pread}\footnote{https://man7.org/linux/man-pages/man2/pwrite.2.html} to read the configuration file.}
Finally, after identifying the set of statements that match the various patterns, the heuristic analysis returns the statement that is closest to the CFG's entry point (Line~\ref{alg:nm-closest}), and ties are broken arbitrarily. In the motivating example (Listing~\ref{wc}), the heuristic analysis obtains the statement at Line~$13$ because it is the closest location to the entry point that matches the command-line patterns.
\lstinputlisting[numbers=left,tabsize=2,
basicstyle=\footnotesize,keywordstyle=\color{blue},
numberstyle=\tiny,breaklines=true,numbersep=3pt, frame=tb,
caption=Nginx configuration file,captionpos=b
,keepspaces=true,xleftmargin=.03\textwidth, keywordstyle=\color{blue}\bfseries,
morekeywords={gzip,keepalive_timeout,server, access\_log, error\_log,worker_processes,charset,default_type,include,events,http,listen,ssl\_certificate,ssl\_certificate\_key},label=lst:nginx
]{listings/nginx.config.txt}
\subsubsection{Structural Analysis}
\label{Se:StructuralAnalysis}
\revUN{This step identifies the neck location by analyzing the program's statements, starting from the location specified by the heuristic analysis (Lines~\ref{alg:nm-strtStructAnalysis}-~\ref{alg:nm-strtStructAnalysisFromScope} in Algorithm~\ref{alg:neckMiner}).
It identifies the statements that satisfy a certain set of control-flow properties that are discussed below (Line~\ref{alg:nm-matchStmtProp}).
Because it is possible to have several matching statements, the closest statement to the entry point is selected (Line~\ref{alg:nm-closestStmt}).
(Ties are broken arbitrarily.)
The closest statement is determined by computing the shortest distances in the CFG from the entry point to the neck candidates (Line~\ref{alg:nm-compDistNeck}).
The remainder of this section formalizes the aforementioned control-flow properties.
}
A program $P$ is a 4-tuple $(V,\textit{stmts},\textit{entry},\textit{exit})$, where $V$ is the set
of variables, $\textit{stmts}$ is the set of statements, $\textit{entry} \in \textit{stmts}$ is
the entry point of the program, and $\textit{exit} \in \textit{stmts}$ is the exit of
the program.
As defined in Section~\ref{subsec:challenges}, we assume that there is a set $V_s \subseteq V$, which we call the set of {\it influenced variables} (e.g., command-line parameters of a utility).
Note that $V - V_s$ is the set of ``internal'' or non-influenced variables.
The location of a statement $s \in \textit{stmt}$ is denoted by $\textit{loc}(s)$. For simplicity, we assume that $\textit{Val}$ is the set of values that vars in $V$ can take.
Let $A: V_s \rightarrow Val \cup \{ \star \}$ be a partial assignment to the set of influenced variables (we assume that if $A(v) = \star$, then it has not been assigned a value). An assignment $A': V_s \rightarrow Val$ is {\it consistent} with partial assignment $A$ iff for all $v \in V_s$, if $A(v) \not= \star$, then $A'(v) = A(v)$. A statement $s \in \textit{stmt}$ is a {\it neck} for a program $P=(V,\textit{stmts},\textit{entry},\textit{exit})$ and a partial assignment $A$ (denoted by $\textit{nk}(P,A)$) if the following conditions hold:
\begin{itemize}
\item Given any assignment $A'$ consistent with $A$, $P$ always reaches the neck $\textit{nk}(P,A)$, and the statement corresponding to \textbf{the neck is executed exactly
once}.
This condition rules out the following possibilities:
Given $A'$, the execution of $P$ (i) might never reach the neck (the intuition here is that we do not want to miss some program statements, which would cause debloating to be unsound), or
(ii) the statement corresponding to the neck is inside a loop.
\item
Let $R(nk(P,A))$ be all statements defined as follows: $s \in R(nk(P,A))$ iff $s$ appears after $nk(P,A)$, then $nk(P,A)$ could be \textbf{identified as articulation point} of the CFG of $P$ and one of the connected components
would over-approximate $R(nk(P,A))$. Another structural condition could be defined as follows: $R(nk(P,A))$ is \textbf{the set of all statements that are dominated by the neck}.
\Omit{
\twr{No, "post-dominated" is not correct.
I think you meant "dominated by the neck": we are talking about code downstream from the neck, so the neck is reached first.
Separate from the problem definition: does the implemented algorithm impose the neck-domination condition?}\Omit{\ma{ Yes, as shown in the dom tree of the motivating example that I shared previously}}
}
\Omit{\ma{should I put the dom tree that I generated for our motivating example to prove that.}
\twr{Possibly, but I think Figure 1 is sufficient. I would put my point above (about sometimes needing a cut-set rather than a single cut-point) in Section 5.}}
\end{itemize}
\begin{comment}
Neck identification tries to simultaneously minimize two quantities: $\mu(\textit{nk}(P,A))$
and $\gamma (\textit{Dbl}(R(\textit{nk}(P,A)),A_c))$.
Function $\mu : \textit{stmt} \; \rightarrow \; \Re^+$ is a metric that measures how far the neck is from the entry point.
Function $\gamma: \mathcal{P}(stmt) \; \rightarrow \; \Re^+$ is a metric for the size
of the programs.
There are several possibilities for metrics $\mu$ and $\gamma$.
One is for $\mu (\textit{nk}(P,A))$ to be the length of the longest path in the control-flow graph (CFG) from $\textit{entry}$ to $\textit{nk}(P,A)$.
A simple possibility for $\gamma(\textit{Dbl}(R(\textit{nk}(P,A)),A_c))$ is the number of statements in the program.
\end{comment}
\begin{comment}
\begin{figure}[tb]
\centering
\includegraphics[width=9cm, height=9cm]{images/cfg.main.pdf}
\caption{\rev{Simplified CFG of the main function in Listing~\ref{wc}. Each rectangle represents a basic block and the colored rectangle contains the neck.}}
\label{fig:cfg}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=9cm, height=6cm]{images/dom.main.pdf}
\caption{\rev{Simplified dominance-tree of the main function in Listing~\ref{wc}. Each rectangle represents a basic block and the colored rectangle contains the neck.}}
\label{fig:domtree}
\end{figure}
\end{comment}
\Omit{\twr{I don't understand the next sentence. The reader will wonder why the formalization allows there to be multiple possibilities. Also, the what does it mean "the developer can determine [select?] the neck location" when you go on to say in the very next sentence that the neck location should be the closest to the entry point?} \ma{I rephrase the paragraph to clarify the exact manual step in the neck miner. To make it simple, all steps in the neck miner algorithm are automated except the step at line 19.}}
\revUN{The neck miner is fully automated, except the step at Line~\ref{alg:nm-matchStmtProp} (in Algorithim~\ref{alg:neckMiner}), which currently requires manual intervention.
We argue that such effort is manageable;
moreover, it is a one-time effort for each program.
Once the developer identifies the statements that satisfy the control-flow properties, they are fed to the neck miner to identify the statement that is closest to the entry point in the CFG.
Finally, a special function call (which serves to label the neck location) is inserted before the identified neck location.
}
\Omit{\twr{This paragraph is inconsistent: you talk about "the developer iterating over the code", and in the next sentence "the miner selects".
The reader will wonder "Is the miner the developer or an algorithm?"}\ma{I rephrased the paragraph}}
\revUN{Consider Listing~\ref{wc} again.
The developer iterates over the program code, starting from Line $13$ (specified by the heuristic analysis) to identify the locations that satisfy the control-flow properties.
The developer ignores Lines $13$ and $14$ because they violate the control-flow properties:
they are not articulation points and are inside a loop, so not executed only once.
Line $15$ satisfies the control-flow properties because the statement at this location is executed only once, is an articulation point, and dominates all subsequent statements.
}
\section{Related Work} \label{sec:rw}
A variety of software-debloating techniques have been developed in the research community, mainly over the last three years~\cite{EuroSec19,pldi10,stochasticOptimization,Temporal,RAZOR,BinaryTrimming,BinRec,CARVE,wedDebloat,RedDroid,Nibbler,ESORICS2019,Cozart,Unikernel}. In this section, we discuss various lines of research on software debloating and partial evaluation that are related to our work.
\noindent\revUN{\textbf{Program Partitioning.}
Our work is peripherally related to prior work on program partitioning.
MPI~\cite{mpi} reduces the overhead of provenience tracking and audit logging by portioning programs' execution based on annotated data structures.
Privtrans~\cite{Privtrans} applies program partitioning to integrate privilege separation. Programs are divided into two components: the monitor and slave. These components run as separate processes, but cooperate to perform the same function as the original program.
Glamdring~\cite{Glamdring} partitions applications at the source-code level, minimizing the amount of code placed inside an enclave. Our use of partitioning is for a different purpose: we create exactly two partitions, separated by the neck, and the partition before the neck is used to identify constant values that are then used to specialize the partition that comes after the neck.
\Omit{\twr{I'd like to hear Somesh's opinion of the above discussion.
To me it seems wrong to say that our motivation came from previous work on program partitioning.
That work primarily exploits multiple processes to introduce a degree of protection, which is not what we are doing at all.
One might be able to say that the \emph{analyses} that we do are inspired by program partitioning, but I'm not even sure that that is a correct characterization of our analyses.
Overall, I don't see the point of making such a connection, unless you think that it is easy to make a contrast between work in that area and our work.
But the current text is not bringing out points of contrast.}}
}
\revUN{In the debloating domain, Ghavamnia et al.~\cite{Temporal} propose a debloating approach to reduce the attack surface in server applications. This approach partitions the execution life-cycle of server programs into initialization and execution phases. It then reduces the number of system calls in the execution phase. However, this approach requires manual intervention from the developer to identify the boundary between the two phases, but without providing certain specifications to guide the identification process. In contrast, \mbox{\textsc{LMCAS}}\xspace performs specialization for the main logic and incorporates a neck miner to suggest a possible neck location. The neck miner provides semi-automatic support for the partitioning process and identified the neck correctly in $26$ programs.
}
\Omit{\twr{It seems imperative that we have a well-written comparision with Ghavamnia et al.
Does the work by Ghavamnia et al.\ do \emph{specialization} of the execution-phase code, or does it merely disable certain system calls.
If the latter, then the points of comparision are (i) \mbox{\textsc{LMCAS}}\xspace performs specialization, and (ii) \mbox{\textsc{LMCAS}}\xspace incorporates a neck miner to suggest a set of possible neck locations.
However, this comparison raises some questions that I don't think we address in Section \ref{sec:evaluation}:
(1) How well does the neck miner work---e.g., (a) for what percentage of the examples was the neck we used in the set returned by the neck miner?
(b) What is the average size of the set returned by the neck miner?
Also, ``avoid manual efforts'' is a weak phrasing;
it would be better to replace it with something like ``\mbox{\textsc{LMCAS}}\xspace provides semi-automatic support for the partitioning process by providing a neck miner that ... <some quick summary of success statistics for the neck miner>''
Finally, there is another issue that we don't have time to explore: does \mbox{\textsc{LMCAS}}\xspace have the same notion of ``initialization phase'' as Ghavamnia et al.?
That is, could our neck miner be applied to the examples used by Ghavamnia et al.\ to find the partitioning points that they used?
}}
\noindent\textbf{Partial Evaluation} has been used in numerous domains, including software debloating~\cite{TRIMMER}, software verification~\cite{Interleaving}, and to generate test cases~\cite{Albert2010PETAP}. Bubel et al.~\cite{Interleaving} use a combination of partial evaluation and symbolic execution. Because their capabilities are complementary, the two techniques have been employed together to improve the capabilities of a software-verification tool~\cite{Interleaving}. However, the goals and modes of interaction are different:
in the work of Bubel et al., partial evaluation is used to speed up repeated execution of code by a symbolic-execution engine; in our work, symbolic execution is in service to partial evaluation by finding values that hold at the neck.
\noindent\textbf{Application Specialization}.
For debloating Java programs, JShrink~\cite{JShrink} applies static and dynamic analysis. Storm is a general framework for reducing probabilistic programs~\cite{Storm}. For debloating C/C++ programs TRIMMER~\cite{TRIMMER} and OCCAM~\cite{OCCAM} use partial evaluation.
While TRIMMER overcomes the limitations of OCCAM by including loop unrolling and constant propagation. But in both tools, constant propagation is only performed for global variables, and thus TRIMMER and OCCAM miss specialization opportunities that involve local variables, which makes the debloating process unsafe.
\mbox{\textsc{LMCAS}}\xspace can accurately convert the elements of struct variables into constants.
Furthermore, our analysis considers pointers, both to base types and struct types, which boosts the reliability of \mbox{\textsc{LMCAS}}\xspace.
\revUN{Aggressive debloating tools like CHISEL~\cite{Chisel} and RAZOR~\cite{RAZOR} can achieve a significantly higher reduction rate in the size of specialized applications; however, these tools are prone to run-time issues (e.g., crashing, infinite loops). Furthermore, the debloating process takes a long time because these tools apply burdensome techniques, based on extensive program instrumentation and requiring users to provide a comprehensive set of test cases. RAZOR uses a best-effort heuristic approach to overcome the challenge of generating test cases to cover all code. While \mbox{\textsc{LMCAS}}\xspace applies lightweight techniques using program partitioning. Also, the specialized programs generated by \mbox{\textsc{LMCAS}}\xspace do not suffer from run-time issues.}
\noindent\textbf{Function Specialization}.
Saffire~\cite{Saffire} specializes call-sites of sensitive methods to handle certain parameters based on the calling context.
\Omit{\twr{What does "Piecewise approach" mean?} \ma{it's the name of the tool. My wording was wrong}}Quach et al.~\cite{PieceWise} proposes a a tool called Piecewise for debloating libraries. Piecewise constructs accurate control-flow-graph information at compilation and linking time according to an applications’ usage.
\begin{comment}
\textbf{OS Specialization}.
Partial evaluation have been also used for specializing the network stack in OS kernels~\cite{Bhatia}.
\Omit{\twr{Elaborate the next sentence. What does "invariant parts" mean?} \ma{it means static/constant variables}}
The user needs annotate the constant variables of the program. Unikraft~\cite{Unikraft1,Unikraft2,Unikernels} is an automated system for building specialized POSIX-compliant Operating Systems.
Unikraft decomposes operating systems and libraries into elementary modules called libraries (e.g., schedulers, memory allocators, drivers, filesystems, network stacks, etc.), and then allows users to pick and choose modules to support the desired application.
\mbox{\textsc{LMCAS}}\xspace is orthogonal to Unikraft in the sense that \mbox{\textsc{LMCAS}}\xspace could be used to obtain further specialization of Unikraft modules: \mbox{\textsc{LMCAS}}\xspace trims out unused functions, and performs some degree of optimization of the remaining functions.
\end{comment}
\begin{comment}
\textbf{Container Debloating}.
Rastogi et al.~\cite{Cimplifier} employed dynamic analysis to automatically partition containers to provide better privilege separation across applications. Guo et al.~\cite{CDE} identify the necessary resources to generate a specialized container based on the generated traces of the monitored program.
Our approach is orthogonal to these techniques:
\mbox{\textsc{LMCAS}}\xspace can be used to debloat the applications after the containers are partitioned.
\end{comment} |
1,108,101,563,229 | arxiv | \section{Introduction}
Precise atomic and molecular spectroscopy provides information on the nuclear electromagnetic moments
important for testing theories of nuclear interactions \cite{Lev:99, Gilman:02, Ericson:83, Epelbaum:09, Filin:20b} or even for searching for new physics \cite{Safronowa:18}. This, however, requires a thorough understanding
of the variety of interparticle interactions in atoms and molecules. For instance, the proton mean square
charge radius has been extracted with unprecedented accuracy from the muonic hydrogen Lamb shift \cite{Pohl:10}
only after careful analysis of all the important quantum electrodynamical effects. A similar determination
has recently been performed for the alpha particle from the corresponding measurement in muonic helium
\cite{Pohl:20}. Regarding nuclear magnetic moments, the direct measurement in a Penning trap was performed
only for the proton \cite{Sturm:14}, while magnetic moments of heavier stable nuclei have been determined
by the nuclear magnetic resonance or the atomic hyperfine splitting measurements.
Accordingly, the currently most accurate magnetic moments of deuteron and triton were determined by combining
the nuclear magnetic resonance measurements with precise calculations of the molecular shielding factor \cite{PKP:15}.
Concerning the determination of nuclear magnetic moments from the hyperfine splitting, their accurate calculation
is particularly difficult due to a large contribution from the not-well-known spin-dependent nuclear structure.
The widely accepted Bohr-Weisskopf correction only partially accounts for the nuclear effects \cite{Bohr:50}.
A clear indication of this problem is a strong and still unexplained discrepancy for the Zemach radius
of $^{6}$Li between the nuclear model value \cite{Yerokhin:08} and the result based on the spectroscopic
data of the lithium atom \cite{Puchalski:13}, which has recently been confirmed by independent measurements
and calculations in the Li$^+$ ion \cite{Guan:20, Qi:20}.
In this work we investigate the electric quadrupole moment $Q_d$ of the deuteron on the basis
of the hyperfine splitting in HD and D$_2$ molecules. The total electron spin of such a two-electron
system is zero, and the strengths of all couplings among nuclear spins and the rotational angular
momentum are of the same order of magnitude. Therefore, $Q_d$ can be extracted from the molecular
hyperfine splitting with an accuracy that is limited only by the measurement uncertainty provided
that sufficiently accurate theoretical calculations with all significant contributions are available.
The recent determinations of $Q_d = 0.285\,98(3)$ fm$^2$ \cite{Jozwiak:20} and
$Q_d = 0.285\,783(30)$ fm$^2$ \cite{Pavanello:10} considered to be the most accurate value to date
\cite{Pyykko:18}, neglect or underestimate nonadiabatic effects, i.e. the effects beyond
the commonly employed Born-Oppenheimer (BO) approximation. Indeed, these results disagree with
the recommended value reported in this work (see Eq.~(\ref{EQdrecom})) obtained in the nonadiabatic approach, i.e. without
separation of nuclear and electronic motions.
In the following we describe shortly the theory of the molecular hyperfine splitting,
its accurate calculations with nonadiabatic wave functions, and the determination
of $Q_d$ from the measurements by Ramsey and coworkers.
Henceforth, we use the original notation by Ramsey \cite{Ramsey:56,Ramsey:57}.
\header{Hyperfine Hamiltonian}
There are three angular momenta in the ground electronic state of the heteronuclear HD molecule,
which all couple to each other---the proton spin $\vec{I}_p$, that of the deuteron $\vec{I}_d$,
and the rotational angular momentum $\vec{J}$. The effective Hamiltonian describing these
interactions reads
\begin{align}
H_{\rm hfs} =&\ -c_p\,\vec I_p\cdot\vec J - c_d\,\vec I_d\cdot\vec J \nonumber \\ &
+ \frac{5\,d_1}{(2\,J-1)(2\,J+3)}\,\biggl[
\frac{3}{2}\,(\vec I_p\cdot\vec J)\, (\vec I_d\cdot\vec J) \nonumber \\ &
\hspace{1.5cm} +\frac{3}{2}\,(\vec I_d\cdot\vec J)\, (\vec I_p\cdot\vec J)
-(\vec I_p\cdot\vec I_d)\,\vec{J}^{\;2}\biggr]
\nonumber \\ &
+ \frac{5\,d_2}{(2\,J-1)(2\,J+3)}\,\biggl[
3\,(\vec I_d\cdot\vec J)^2 \nonumber \\ &
\hspace{3cm} +\frac{3}{2}\,(\vec I_d\cdot\vec J)
-\vec I_d^{\;2}\,\vec J^{\;2}\biggr]\,.
\label{01}
\end{align}
The above coefficients $c_p$, $c_d$, $d_1$, and $d_2$ are related, respectively,
to the interactions between the following: the proton spin and molecular rotation,
the deuteron spin and rotation, the proton and deuteron spins, and the electric
quadrupole moment of the deuteron with the electric field gradient \cite{brown_carrington_2003}.
For homonuclear molecules with odd $\vec J$ the effective Hamiltonian takes a simplified form
\begin{align}
H_{\rm hfs} =&\ -c\,\vec I\cdot\vec J
+ \frac{5\,d}{(2\,J-1)(2\,J+3)} \nonumber \\ & \times\biggl[
3\,(\vec I\cdot\vec J)^2
+\frac{3}{2}\,(\vec I\cdot\vec J)
-\vec I^{\;2}\,\vec J^{\;2}\biggr]\,,
\label{02}
\end{align}
where the total nuclear spin $\vec{I}=\vec I_A+\vec I_B$ is equal to 1, and
$c=c_p$, $d = d_1/2$ for H$_2$, and $c=c_d$, $d=d_1-d_2$ for D$_2$.
We will consider all these constants for the first rotational level $J=1$ and $v=0$
and present a short derivation of corresponding formulas followed by numerical calculations
using explicitly correlated wave functions. Both the derivation and the calculations
are performed in the nonadiabatic regime.
The general spin-orbit Hamiltonian is of the form
\begin{align}
\delta H =& \sum_{\alpha,\beta}\frac{e_\alpha\,e_\beta}{4\,\pi}\,\frac{1}{2\,r_{\alpha\beta}^3}
\biggl[\frac{g_\alpha}{m_\alpha\,m_\beta}\,\vec I_\alpha\cdot\vec r_{\alpha\beta}\times\vec p_\beta
\nonumber \\ &\
-\frac{(g_\alpha-1)}{m_\alpha^2}\,\vec I_\alpha\cdot\vec r_{\alpha\beta}\times \vec p_\alpha \biggr],
\label{03}
\end{align}
where the indices $\alpha$ and $\beta$ are for electrons and nuclei.
The gyromagnetic factors
\begin{align}
g_p &= \frac{\mu_p}{\mu_N\,I_p} =5.585\,695\ldots \label{04}\\
g_d &= \frac{\mu_d}{\mu_N\,I_d}\,\frac{m_d}{m_p} = 1.714\,025\ldots\label{05}
\end{align}
are related to the magnetic moment of the proton $\mu_p = 2.792\,847\,344\,63(82)\,\mu_N$
and the deuteron $\mu_d = 0.857\,438\,2338(22)\,\mu_N$, respectively \cite{CODATA:18}.
The $g$ factor is a dimensionless quantity convenient for use in the magnetic moment formulas.
In particular, the coupling of the nuclear spin $\vec I_A$
to the molecular rotation, using Eq. (\ref{03}), is
\begin{align}
\delta_A H &= \vec I_A\cdot\vec Q_A\label{06}\\
\vec Q_A &= -\sum_{b}\frac{\alpha}{2\,r_{A b}^3}\!
\biggl[\frac{g_A}{m_A\,m_e}\,\vec r_{A b}\times\vec p_b-
\frac{(g_A-1)}{m_A^2}\,\vec r_{A b}\times \vec p_A \biggr]
\nonumber \\ &\quad
+\frac{\alpha}{2\,r_{A B}^3}
\biggl[\frac{g_A}{m_A\,m_B}\,\vec r_{A B}\times\vec p_B-
\frac{(g_A-1)}{m_A^2}\,\vec r_{A B}\times \vec p_A \biggr] \label{07}
\end{align}
and the spin-rotation coefficient is thus
\begin{align}
c_A = \frac{1}{2}\,\mathrm{i}\,\epsilon^{ijk}\,\langle\phi^i|Q_A^j|\phi^k\rangle\,, \label{08}
\end{align}
where $\phi^i$ is the wave function for the first rotational state
with normalization $\langle\phi^i|\phi^i\rangle = 1$.
The nuclear spin-spin direct interaction can be effectively written as
\begin{align}
\delta H =&\ I_A^i\,I_B^j\,Q_{AB}^{ij} \label{09}\\
Q_{AB}^{ij} =&\ \frac{g_A\,g_B}{4\,m_A\,m_B}\,\frac{\alpha}{r_{AB}^3}\,\biggl(\delta^{ij}-3\,\frac{r_{AB}^i\,r_{AB}^j}{r_{AB}^2}\biggr)
\label{10}
\end{align}
and the corresponding $d_1$ coefficient is
\begin{align}
d_1 = -\frac{1}{5}\,\langle\phi^i| Q_{AB}^{ij} |\phi^j\rangle \label{Ed1}
\end{align}
The omitted part of the spin-spin interaction, proportional to $\delta^3(r_{AB})$,
is negligibly small.
The interaction of a particle with the charge $e$, possessing the electric quadrupole moment $Q^{ij}$ with
the gradient of the electric field, is given by
\begin{align}
\delta H =&\ -\frac{e}{6}\,Q^{ij}\,\partial_jE^i\,.\label{12}
\end{align}
For a particle with a definite spin $I\geq 1$, the $Q^{ij}$, as a traceless and symmetric tensor,
can be expressed in terms of a single scalar electric quadrupole moment $Q$ defined by
\begin{align}
Q^{ij} =&\ \frac{Q}{I\,(2I-1)}\,\biggl(\frac{3}{2}\,I^i\,I^j + \frac{3}{2} I^j\,I^i -\delta^{ij}\,\vec I^2\biggr).\label{13}
\end{align}
Referring to Eq.~(\ref{01}), the Ramsey's constant $d_2$ becomes (in atomic units)
\begin{align}
d_2&=-\alpha^2\,\frac{Q\,q}{10\,\lambdabar^2},\label{Ed2}
\intertext{where}
q&=\left\langle\phi^i\left|\frac{\partial^2 V}{\partial r_A^i\,\partial r_A^j}\,
-\frac{\delta^{ij}}{3}\frac{\partial^2 V}{\partial r_A^k\,\partial r_A^k}\right|\phi^j\right\rangle\label{Eq}
\end{align}
is the electric field gradient at the nucleus $A$, $V$ is the Coulomb potential of Eq.~(\ref{EV}),
and $\lambdabar$ is the reduced Compton wavelength of the electron.
\header{Numerical calculations}
The nonadiabatic wave function $\Psi$ is obtained from the variational principle
with the following nonrelativistic Hamiltonian for the hydrogen molecule
\begin{equation} \label{16}
H = T + V\,,
\end{equation}
where (in atomic units)
\begin{eqnarray}
T &=& \frac{ \vec p_A^{\,2}}{2\,m_A} + \frac{ \vec p_B^{\,2}}{2\,m_B} + \frac{\vec p_1^{\,2}}{2} + \frac{\vec p_2^{\,2}}{2} \,, \label{17}\\
V &=& \frac{1}{r_{AB}} -\frac{1}{r_{A1}} - \frac{1}{r_{A2}}-\frac{1}{r_{B1}} - \frac{1}{r_{B2}} +\frac{1}{r_{12}} \,.\label{EV}
\end{eqnarray}
Here, indices $A$, $B$ and $1$, $2$ denote nuclei and electrons, respectively.
The nuclear masses are those currently recommended by CODATA \citep{CODATA:18}.
The wave function $\Psi$ depends on four particle coordinates
$\Psi = \Psi(\vec r_A, \vec r_B, \vec r_1, \vec r_2)$.
In the center of mass frame the total momentum vanishes
$\vec{p}_A + \vec p_B + \vec p_1 + \vec p_2 = 0 $,
and thus we may assume that the wave function $\Psi$ depends only on the coordinate differences.
In the variational approach the wave function is represented as a linear combination
\begin{equation}
\Psi = \sum_k^N c_k\, \psi_k(\vec r_A, \vec r_B, \vec r_1, \vec r_2)\,,\label{19}
\end{equation}
of properly symmetrized basis functions
\begin{equation}
\psi_k = (1 \pm P_{A\leftrightarrow B})\,(1+P_{1\leftrightarrow 2})\,\phi_k (\vec r_A, \vec r_B, \vec r_1, \vec r_2), \label{Epsik}
\end{equation}
%
where $P_{i\leftrightarrow j}$ is the particle exchange operator. In the $J=1$ state
of H$_2$ and D$_2$ the wave function is antisymmetric with respect to the exchange
of nuclear spatial variables and symmetric in electronic spatial variables,
whereas in the heteronuclear HD molecule,
only electronic symmetry is imposed, and both (nuclear) symmetric and antisymmetric
basis functions are employed. For $J=1$ the functions $\phi_k$ in Eq. (\ref{Epsik})
are the nonadiabatic explicitly correlated Gaussians (naECG) of the form
\begin{align}
\phi^i_k =&\ r^i\,r_{AB}^n \label{21} \\ \times &
e^{-a_{k,1}\, r^2_{AB}-a_{k,2}\, r^2_{A1}-a_{k,3}\, r^2_{A2}-a_{k,4}\, r^2_{B1}-a_{k,5}\, r^2_{B2}-a_{k,6}\, r^2_{12}}, \nonumber
\end{align}
where $\vec{r}$ (or $r^i$) is the factor representing the angular momentum $J=1$, and is
either $\vec{r}_{AB}$, $\vec{r}_{A1}$, $\vec{r}_{A2}$, $\vec{r}_{B1}$, $\vec{r}_{B2}$ or $\vec{r}_{12}$.
The nonlinear $a_k$ parameters are optimized individually for each basis function $\phi_k$.
The powers $n$ of the internuclear coordinate $r_{AB}$, needed to represent accurately
the vibrational part of the wave function, are restricted to even integers
and are generated randomly for each basis function from the log-normal distribution
within the limited 0-80 range. Moreover, the number of basis functions with the particular prefactor
$\vec {r}$\,$r_{AB}^n$ is subject to additional discrete optimization. As a result, the nonrelativistic
energy reaches an accuracy of about $10^{-11}$ (see Table~\ref{Tconv}).
\renewcommand{\arraystretch}{1.0}
\begin{table*}[!htb]
\caption{Convergence of the nonrelativistic energy $E$ (in a.u.), hyperfine splitting
parameters (in kHz), and the electric field gradient $q$ (in a.u.) calculated using naECG
wave functions for the first rotational level $(v,J) = (0,1)$.
The extrapolated nonadiabatic results (NA) are compared to the Born-Oppenheimer
(BO) values \cite{Komasa:20} and with the results of measurements by Ramsey
{\em et al.{}}~\cite{Harrick:53,Quinn:58,Code:71}.
The NA values do not include here the uncertainties due to omitted relativistic
and QED effects, which can be estimated by the relative factor of $\alpha^2\approx5\times10^{-5}$,
where $\alpha$ is the fine structure constant.}
\label{Tconv}
\begin{ruledtabular}
\begin{tabular}{cx{3.18}@{\extracolsep{\fill}}*{2}{x{3.6}}*{2}{x{5.10}}}
\multicolumn{6}{c}{H$_2$} \\
Basis &\cent{E}& \cent{c_p} & & \cent{d=d_1/2} & \\[1ex]
\hline
512 & -1.163\,485\,167\,695 & 112.393\,16 & & 57.643\,938\,904\,75 \\
1024 & -1.163\,485\,172\,061 & 113.889\,80 & & 57.643\,937\,929\,23 \\
1536 & -1.163\,485\,172\,209 & 113.904\,33 & & 57.643\,937\,899\,30 \\
2048 & -1.163\,485\,172\,287 & 113.911\,88 & & 57.643\,937\,895\,69 \\
NA & -1.163\,485\,172\,314\,0(1)^a & 113.920(8) & & 57.643\,937\,891(6) \\[2ex]
BO, \cite{Komasa:20} & & 114.00(12) & & 57.69(6) \\
NA $-$ BO& & -0.08(12) & & -0.05(6) \\
Measured, \cite{Harrick:53} & & 113.904(30) & & 57.671(24) \\[2ex]
\multicolumn{6}{c}{HD} \\
Basis &\cent{E}& \cent{c_p} & \cent{c_d} & \cent{d_1} & \cent{q} \\[1ex]
\hline
512 &-1.165\,065\,367\,519 & 83.740\,0 & 13.281\,98 & 17.761\,872\,179\,98 & 0.334\,510\,892 \\
1024 &-1.165\,065\,376\,045 & 85.446\,7 & 13.146\,42 & 17.761\,872\,391\,72 & 0.334\,493\,539 \\
1536 &-1.165\,065\,376\,735 & 85.550\,6 & 13.135\,55 & 17.761\,872\,423\,13 & 0.334\,492\,630 \\
2048 &-1.165\,065\,376\,858 & 85.598\,6 & 13.126\,23 & 17.761\,872\,410\,33 & 0.334\,491\,813 \\
NA &-1.165\,065\,376\,941\,65(3)^a& 85.63(4) & 13.117(9) & 17.761\,872\,414(13) & 0.334\,491\,0(8)\\[2ex]
BO, \cite{Komasa:20} & & 85.675(60) & 13.132(9) & 17.773(12) & 0.334\,7(3) \\
NA $-$ BO& & -0.04(7) & -0.014(14) & -0.012(12) &-0.000\,2(3) \\
Measured, \cite{Quinn:58} & & 85.600(18) & 13.122(11) & 17.761(12) & \\[2ex]
\multicolumn{6}{c}{D$_2$} \\
Basis &\cent{E}& & \cent{c_d} & \cent{d_1} & \cent{q} \\[1ex]
\hline
512 &-1.166\,896\,428\,705 & & 8.723\,72 & 2.737\,626\,131\,22 & 0.335\,240\,662 \\
1024 &-1.166\,896\,432\,071 & & 8.763\,77 & 2.737\,626\,043\,01 & 0.335\,233\,684 \\
1536 &-1.166\,896\,432\,230 & & 8.765\,41 & 2.737\,626\,038\,04 & 0.335\,232\,171 \\
2048 &-1.166\,896\,432\,323 & & 8.766\,20 & 2.737\,626\,037\,75 & 0.335\,231\,377 \\
NA &-1.166\,896\,432\,359\,76(4)^a & & 8.767\,4(10) & 2.737\,626\,037\,4(12) & 0.335\,230\,7(7)\\[2ex]
BO, \cite{Komasa:20} & & & 8.770(5) & 2.739(2) & 0.335\,35(18) \\
NA $-$ BO& & &-0.003(5) &-0.002(2) &-0.000\,12(18) \\
Measured, \cite{Code:71} & & & 8.768(3) & & \\
\end{tabular}
\end{ruledtabular}
\flushleft
$^a$ This is a reference energy obtained from explicitly correlated exponential functions
\cite{PK:18a,PK:18b,PK:19}
\end{table*}
\header{Hyperfine parameters}
The hyperfine parameters for the hydrogen molecule isotopologues obtained with the above wave function
are presented in Table~\ref{Tconv}. The numerical convergence for the spin-orbit couplings
$c_p$ and $c_d$ is relatively slow, and the resulting numerical uncertainties are not negligible.
Most importantly, their difference from BO values fits within the uncertainties,
which indicates that the estimation of the magnitude of nonadiabatic effects by the ratio
of the electron mass to the nuclear reduced mass is correct.
Moreover, the nonadiabatic and BO values are, within uncertainties, in agreement with
the Ramsey measurements.
In contrast, the numerical convergence of $d_1$ and $q$ parameters is very fast,
and the corresponding inaccuracy is negligible compared to the uncertainty due to
unknown higher order relativistic and QED effects. Again, the difference
with BO values is consistent with the estimate of nonadiabatic effects, represented
as an inaccuracy of the BO values. We also note that the nonadiabatic $d_1$ for HD
agrees with Ramsey's measurements up to its uncertainty, while for H$_2$ it fits within
$1.2\,\sigma$. Regarding the $q$ parameter, the fast numerical convergence of the gradient of
the electric field enables six significant digits to be quoted.
Our recommended nonadiabatic value of
\begin{equation}
q=0.335\,230\,7(7) \text{ a.u. }
\end{equation}
obtained for D$_2$
will be used in the next paragraph for the determination of the deuteron quadrupole moment $Q_d$.
We note that this value differs by $0.000\,6$ a.u. from the $q=0.334\,66$ a.u obtained in
the pioneering nonadiabatic calculations by Bishop and Cheung \cite{Bishop:79}.
This difference is relatively large and shows that the former results in \cite{Bishop:79} are not
accurate enough to draw definite conclusions about the magnitude of the nonadiabatic effects.
\begin{figure}[b]
\includegraphics[scale=0.21]{QdFig}
\caption{Comparison of literature data for $Q_d$.}
\label{FQdFig}
\end{figure}
\header{The quadrupole moment of deuteron}
$Q_d$ can be determined most accurately from the coupling
constant $d$ measured by the molecular-beam magnetic resonance method of Code and Ramsey \cite{Code:71}
for D$_2$ in the first rotational level. We obtain $d_2$ from the difference $d_2=d_1-d$
of calculated $d_1$ and measured $d$. Finally, we evaluate the quadrupole moment from
\begin{align}\label{EQd}
Q_d &= -\frac{d_2}{2\,Ry\,c}\,\frac{10\,\lambdabar^2}{\alpha^2\,q}
\intertext{obtaining the recommended value of}
Q_d&=0.285\,699(15)(18) \text{ fm$^2$. } \label{EQdrecom}
\end{align}
The results of such calculations are summarized in Table~\ref{TQd}.
A comparison with literature data reported within the last fifty years is presented in Table~\ref{TQd}
and Fig~\ref{FQdFig}. The first uncertainty assigned to our $Q_d$ is due to unknown higher order relativistic and QED corrections to $q$, which are estimated by a relative factor $\alpha^2$. The second uncertainty comes from the measurement of $d$, while numerical uncertainties are negligible.
We should mention that the second order magnetic dipole interaction,
being also a kind of relativistic correction, leads to the pseudoquadrupole effect
estimated by Ramsey to be of about $10^{-5}$ \cite{Ramsey:53b}
and this contribution is included in our $\alpha^2$ uncertainty.
This nonadiabatic $Q_d$ is in agreement with
our BO value \cite{Komasa:20}, the relative uncertainty of which was estimated
by the ratio of the electron mass to the nuclear reduced mass. It is in significant
disagreement, though, with the recent most accurate determination by Pavanello {\em et al.{}}~\cite{Pavanello:10}
and J\'o\'zwiak {\em et al.{}}~\cite{Jozwiak:20} (see Table~\ref{TQd} and Fig~\ref{FQdFig}).
It is in agreement, however, with the revised result by Bishop, quoted in 1983 by Ericson
{\em et al.{}}~\cite{Ericson:83},
which served for a long time as a reference value for $Q_d$. A comparison of our result with recent
literature data indicates the significance of nonadiabatic effects and also draws attention
to the need for the correct use of the BO potential, as the above mentioned results
\cite{Pavanello:10,Jozwiak:20,Komasa:20}
differ from each other due to different potentials used to average the $q$ parameter.
To verify our result, we used the obtained value of $Q_d$ to evaluate
the $d_2$ parameter for the HD molecule in a $J=1$ state, and we achieved a perfect agreement with $d_2$
as measured by Ramsey {\em et al.{}}~\cite{Quinn:58} (see Table~\ref{TQd}).
\renewcommand{\arraystretch}{1.0}
\begin{table}[!bh]
\caption{Determination of the deuteron quadrupole moment (in fm$^2$) from D$_2$ hyperfine parameters
and its verification for HD.}
\label{TQd}
\begin{ruledtabular}
\begin{tabular*}{0.65\textwidth}{c@{\extracolsep{\fill}}x{3.18}l}
Quantity & & Source \\
& \centt{D$_2$} & \\
\hline
$d$ & 25.241\,4(14)\text{ kHz} & Code \& Ramsey, 1971 \cite{Code:71} \\[1ex]
$d_1$ & 2.737\,626\,037\,4(12)\text{ kHz}& This work, Eq.~(\ref{Ed1}) \\
$d_2$ &-22.503\,8(14)\text{ kHz} & $d_2 = d_1-d$ \\
$q$ & 0.335\,230\,7(7)\text{ a.u.} & This work, Eq.~(\ref{Eq}) \\[2ex]
$Q_d$ & 0.285\,699(15)(18)^a& This work, Eq.~(\ref{EQd})\\
& 0.285\,4_{-17}^{+38\ b} & Filin {\em et al.{}}, 2020 \cite{Filin:20a} \\
& 0.285\,98(3) & J\'o\'zwiak {\em et al.{}}, 2020 \cite{Jozwiak:20} \\
& 0.285\,6(2) & Komasa {\em et al.{}}, 2020 \cite{Komasa:20} \\
& 0.285\,783(30) & Pavanello {\em et al.{}}, 2010 \cite{Pavanello:10} \\
& 0.285\,90(30) & Bishop revisited, 1983, cited in \cite{Ericson:83} \\
& 0.286\,0(15) & Bishop \& Cheung, 1979 \cite{Bishop:79} \\
& 0.286\,0(15) & Reid \& Vaida, 1972/5 \cite{Reid:72,Reid:75Errata} \\[1ex]
& \centt{HD} & \\ \hline
$q$ & 0.334\,491\,0(8)\text{ a.u.} & This work, Eq.~(\ref{Eq}) \\
$d_2$ &-22.454\,2(14)^c\text{ kHz}& This work, Eq.~(\ref{Ed2}) \\
$d_2$ &-22.454(6)\text{ kHz} & Ramsey {\em et al.{}}, 1958 \cite{Quinn:58} \\
Diff. & -0.000(6)\text{ kHz} \\
\end{tabular*}
\end{ruledtabular}
\flushleft
$^a$ The first uncertainty accounts for missing relativistic correction, the second one is
due to the experimental
uncertainty of $d$.\\
$^b$ Calculated in the framework of $\chi$EFT. \\
$^c$ This uncertainty comes from the experimental uncertainty in $Q_d$.
\end{table}
\header{Summary}
The deuteron quadrupole moment $Q_d$ is determined with the highest accuracy among all nuclei
in the periodic table \cite{Pyykko:18}. This accurate result can be used not only in precise
atomic and molecular
structure calculations but also can serve as a benchmark for the nuclear structure theory.
Indeed, for a long time, $Q_d$ could not be reproduced by any modern potential model
(see e.g.~\cite{Ericson:83} and references therein). It was not until very recently
very recently Filin {\em et al.{}}~\cite{Filin:20a} reported $Q_d=0.285\,4_{-17}^{+38}$ fm$^2$
obtained from chiral effective field theory ($\chi$EFT), which is in very good agreement with our result.
This agreement opens the possibility of better understanding the spin-dependent
nuclear structure effects in atomic spectra, particularly in muonic deuterium hyperfine splitting,
where significant discrepancies with measurement \cite{Pohl:16} have been reported
\cite{Kalinowski:18}.
Apart from the deuteron quadrupole moment, by accounting for nonadiabatic effects,
we obtained all the other hyperfine constants in very good agreement with Ramsey's
molecular-beam magnetic resonance measurements (see Table I).
However, current theory includes only the leading relativistic effects.
Because the inclusion of higher order relativistic and QED corrections is certainly within reach,
more accurate measurements are desirable. This may open new windows
for high-precision tests of fundamental interactions on the molecular scale.
This research was supported by National Science Center (Poland) Grant No. 2016/23/B/ST4/01821
as well as by a computing grant from Pozna\'n Supercomputing and Networking Center and by PL-Grid Infrastructure.
|
1,108,101,563,230 | arxiv | \section{Introduction}
Recent experiments in spin qubit systems demonstrated full single-qubit control via electrical ac driving at microwave frequencies \cite{Medford_PRL13,Shulman_NatCom14, Laucht_SciAdv15, Veldhorst_Nat15, Kim_NPJ15}. Advantages of such an approach include the ability to operate the qubit while staying near a sweet spot where the charge noise is considerably suppressed \cite{Medford_PRL13, Kim_NPJ15} while simultaneously filtering out low-frequency noise off-resonance with the driving field. However, designing an ac control pulse is not necessarily a trivial matter. There are a variety of approaches for designing pulses with some set of desirable qualities such as speed, finite bandwidth, low fluence, robustness against pulse length errors, etc.
The standard approach in spin qubit experiments is simply to use the rotating wave approximation (RWA). RWA is rather ubiquitous in a variety of contexts, arising, for example, when a laser is used to couple two atomic states. When the driving field is near resonance with the transition frequency between the states, RWA gives a straightforward prescription for full control of the two-level system by manipulating the phase and the duration of the ac control pulse. With the exception of extremely intense lasers, the amplitude of the ac control field is typically small when compared to the resonant frequency, and is extensively employed in atomic and molecular physics. However, semiconductor qubits have a much smaller two-level splitting than typical atomic systems and can be operated in a regime where RWA is no longer valid, i.e., the driving ac field strength is not necessarily much smaller than the qubit energy spacing. In practice, semiconductor qubits are often operated with weak driving in order to take advantage of the simplicity of the RWA control design. However, there is no reason stronger driving could not be used in conjunction with state-of-the-art pulse-shaping or some other control method to perform qubit operations. In fact, strong driving is attractive for semiconductor qubit operations because it leads to both faster gate operations and less exposure to background noise. The latter is particularly important at this stage, as large-scale quantum computation requires errors per gate to be below a given threshold for fault-tolerant error correction (FTEC) \cite{Knill_PRSA98, Aharonov_SIAM08,Gottesman_thesis97}, so one needs gate times substantially shorter than the $T_2$ coherence time. This threshold is often estimated to be $\sim 10^{-4}$ \cite{Preskill_PRSA98, DiVincenzo_Fortschritte2000, Steane_PRA03, Knill_Nat10, Brown_PRA11}, and even for more recent surface codes with thresholds of $\sim 0.7\%$ \cite{Fowler_PRA12} the massive overhead is greatly reduced if one can reduce errors down to near the $10^{-4}$ level \cite{Martinis_15}. This fact also provides motivation to design gates accurately -- even in the absence of any environmental noise, using control pulses based on an RWA analysis in a strong driving regime would obviously introduce completely avoidable deterministic errors (which would have to be corrected introducing avoidable overhead in the quantum computing operations)! The objective of the first part of this work is to quantify the error due to a RWA pulse design by considering realistic experimental situations involving different qubit architectures. Such a quantification in the specific context of spin qubits is very important as a practical guide, and here we provide the necessary analysis.
\begin{table*
\caption{\label{tab:exper_parameters}
Estimated infidelity due to RWA for different semiconductor qubit experiments. Single-spin qubit parameters are listed for donor electrons (e), nuclear spins (n), and gate-confined quantum dots in silicon (Si).
}
\renewcommand{\arraystretch}{1.3}
\tabcolsep=0.9 cm
\begin{tabular}{ccccc}%
\hline\hline
& $\omega_z/2\pi$ & $\omega_x/2\pi$ & $1-\bar{\mathcal{F}}^{\rm Max}_{\rm rwa}$ & Reference \\
\hline
RX qubit & 210-370MHz & 41-112MHz & 1.0-3.8$\times 10^{-3}$ & Ref.~\cite{Medford_PRL13}
\\
ST qubit & 60MHz & 20MHz & $4.6\times 10^{-3}$& Ref.~\cite{Shulman_NatCom14}
\\
Hybrid & 11.5GHz & 110MHz & $3.8\times 10^{-6}$& Ref.~\cite{Kim_NPJ15}
\\
\multirow{3}{*}{\renewcommand{\arraystretch}{1.0}$\begin{array}{c}\textrm{single}\\ \textrm{spin}\\ \textrm{qubit}\end{array}$}
&(e) 43GHz $\quad$ & 48kHz& $5.2\times 10^{-14}$ & \multirow{2}{*}{ Ref.~\cite{Laucht_SciAdv15}} \\
&(n) 97MHz $\quad$ & 10kHz& $4.6\times 10^{-10}$\\
&(Si) 39GHz & 730kHz& $1.5\times 10^{-11}$ & Ref.~\cite{Veldhorst_Nat15}
\\
\hline\hline
\end{tabular}
\end{table*}
It turns out that to achieve faster gating for certain oscillatory driven quantum-dot systems such as the RX \cite{Medford_PRL13} or ST qubit \cite{Shulman_NatCom14}, one can no longer use RWA as a guide for constructing the control pulse. In the second part of this paper we carry out a numerical analysis of a specific form of control pulse and present the key pulse control parameters for experiments to achieve certain important gates with much higher speed.
We do not search for the shortest possible solution, as in optimal control theory, but instead we catalog all realizations within a simple, highly constrained space. Some elegant optimal control approaches apply to two \cite{Khaneja_PRA01} and three qubits \cite{Khaneja_PRA02}, but we consider only single-qubit operations. Other approaches focus on accounting for the RWA resonance shift \cite{Ashhab_PRA07, Lu_PRA12, Yan_PRA15, Romhanyi_PRB15}, but we wish to keep the entire dynamics, including the rapid counter-oscillating terms. There are a wealth of pulse-shaping approaches, such as analytical reverse-solving \cite{Barnes_PRL12, Barnes_SciRep15}, multi-parameter optimization schemes such as CRAB \cite{Doria_PRL11, Scheuer_NJP14}, and other iterative updating algorithms for pulse sequences such as GRAPE \cite{Khaneja_JMR05} and Krotov's approach \cite{Krotov_book95, Maximov_JCP08} based on optimal control theory. In comparison, our approach focuses on simple sinusoidal pulses with four parameters (the durations and phases of the two pulses), and our theoretical results will serve to guide a straightforward experimental calibration of the control pulses. Optimal control theory or pulse-shaping techniques may well be more desirable for a variety of reasons in a given case. What we show in the second part of our work, though, is that one can already consider driving in the strongly non-RWA regime with a relatively simple, two-pulse, numerical construction. We believe that the specificity and the concreteness of our results as well as the simplicity of the proposed pulses make our work of immediate usefulness in the context of faster gate control in various semiconductor qubit experiments going on worldwide.
An overview of this paper is as follows. The first part of this paper quantifies the deterministic error induced using RWA pulse design in ST, RX, as well as spin-charge hybrid \cite{Kim_NPJ15} and different single-spin qubit \cite{Laucht_SciAdv15, Veldhorst_Nat15} systems. Both the nominally resonant (when field frequency equals the two-level splitting) and nonzero detuning cases are covered. We find that for the RX qubit \cite{Medford_PRL13, Taylor_PRL13} and the resonantly-driven ST qubit \cite{Shulman_NatCom14,Levy_PRL02, Klauser_PRB06}, the deterministic RWA-induced errors are already larger than the fault tolerance threshold with the experimentally reported parameters, which grow even larger if the gates are implemented faster in an attempt to reduce random environment-induced errors. Therefore special attention will be paid to these two cases. However, the analysis applies generally to any strongly driven two-level system, and should be applicable to other qubit systems where fast gate operations are desirable.
Following the numerical analysis, we apply higher-order perturbation theory (two orders beyond the leading order, which is RWA) on the converted Floquet Hamiltonian \cite{Shirley_thesis63,Shirley_PRB65} and obtain an analytical expression for the state-averaged gate infidelity induced by using only an RWA (i.e. the zeroth-order) analysis to design gate pulses. We show that there is an intermediate parameter regime where incorporating higher-order corrections in the gate suppresses the deterministic error by orders of magnitude. This is quite relevant to the currently on-going ST and RX experiments, and could be useful for pulse designs to also reduce random errors, as previously done with piecewise constant pulses \cite{Wang_NatCom12, Kestner_PRL13}.
In the second part of this work, we go beyond perturbative approaches such as RWA, and solve numerically exactly for pulse control parameters that realize important single-qubit gates for a large range of driving strengths. This both eliminates deterministic error and increases gate speed, thus indirectly also reducing random error. Our solutions increase the gate speed by more than two orders of magnitude, making our results important for future experiments involving quantum error corrections in semiconductor qubits. We use a simple pulse sequence with only four parameters. Experimental implementation in the nonideal solid-state environment would proceed by calibrating the pulses in a small region of parameter space near our numerical solution.
The structure for the rest of the paper is as follows (and we also provide a guidance for the key results). Section~\ref{subsec:RWA_sinu_numer} gives a systematic numerical quantification of the averaged RWA infidelity, with the underlying leading-order effect worked out analytically [Eq.~(\ref{eq:infidelity_rwa_ext})] in Sec.~\ref{subsec:analytic_extension}. The discussion is tailored to various key semiconductor qubit systems summarized in Table~\ref{tab:exper_parameters}, with each system's position in parameter space clearly mapped out in Fig.~\ref{fig:4}. Section~\ref{sec:fast_gate} works out the exact pulse control parameters to realize Hadamard and $\frac{\pi}{8}$ gates beyond the RWA regime, with key results summarized in Figs.~\ref{fig:Sinu_Hada} and \ref{fig:Sinu_PiOver8}. An extension to shifted oscillatory coupling is demonstrated in Appendix~\ref{app:shifted_sinu}. Section~\ref{sec:conclusion} concludes this work, where a comparison of our gate speed to the quantum speed limit is also given.
\section{Analysis of RWA errors in semiconductor qubit systems}\label{sec:Error_analysis}
We start with the Hamiltonian for a quantum two-level system,
\begin{eqnarray}
H_{\rm sin}= \frac{J_z}{2}\sigma_z + \frac{J_x(t)}{2} \sigma_x, \label{eq:H_exact}
\end{eqnarray}
where the bases of the Pauli matrices $\sigma_{x,z}$ are the qubit states $|0\rangle$ and $|1\rangle$, $J_z= \hbar \omega_z$ is the two-level splitting in the absence of driving, and the subscript ``$\rm sin$'' indicates that the off-diagonal control field $J_x(t)$ has the sinusoidal form
\begin{eqnarray}\label{eq:J_x_t}
J_x(t)= \hbar\omega_x \cos (\omega t+\Phi).
\end{eqnarray}
with a driving angular frequency $\omega$ and phase $\Phi$. When $\omega$ is tuned close enough to $\omega_z$, and the coupling strength is weak ($|J_x/ J_z|\ll 1$), one may employ the well-known rotating wave approximation by ignoring the so-called counter-rotating part of $J_x$ in Eq.~(\ref{eq:J_x_t}). The approximated Hamiltonian is therefore \cite{Rabi_PR37, Bloch_PR40, Vandersypen_PMP04}
\begin{eqnarray}\label{eq:H_rwa}
H_{\rm rwa} &=& \frac{J_z}{2}\sigma_z + \frac{\hbar\omega_x}{4} \left[ e^{-i(\omega t+\Phi)}\frac{\sigma_+}{2} +e^{i(\omega t +\Phi)}\frac{\sigma_-}{2}\right]
\nonumber\\
&=& \hbar \left[\frac{\omega_z}{2}\sigma_z + \frac{\omega_x}{4} \hat{\mathbf{b}}(t)\cdot\bm\sigma \right]
\end{eqnarray}
in the same laboratory frame of Eq.~\eqref{eq:H_exact}, where the conventional notations $\sigma_\pm\equiv\sigma_x\pm i\sigma_y$, $\bm\sigma\equiv\sigma_x \hat{\mathbf{x}} + \sigma_y \hat{\mathbf{y}} + \sigma_z \hat{\mathbf{z}} $, and $\hat{\mathbf{b}}(t)=\cos(\omega t+\Phi)\hat{\mathbf{x}}+\sin(\omega t+\Phi)\hat{\mathbf{y}}$. Obviously, Eq.~(\ref{eq:H_rwa}) is much easier to use [than Eq.~(\ref{eq:H_exact})] in interpreting and understanding experiments, and therefore, in many applications, Eqs.~(\ref{eq:H_exact}) and (\ref{eq:H_rwa}) are considered equivalent (often uncritically), and this is referred to as RWA in the literature.
In this section we examine how good this approximation is for the range of parameters $\omega_z$, $\omega_x$ and $\omega$ actually used in various semiconductor qubit systems. Note that the Rabi frequency becomes $\omega_x/2$ upon dropping the counter-rotating part of driving terms. We therefore use the corresponding nominal Rabi period $T_R\equiv 4\pi/\omega_x$ to compare resonant and off-resonant cases where applicable.
We focus on the semiconductor qubit platforms where oscillatory control fields may be or have already been utilized. Most attention is given to the RX (Sec.~\ref{subsec:RWA_sinu_numer}) and various ST qubit systems (Sec.~\ref{subsec:RWA_sinu_numer} and more in Sec.~\ref{sec:fast_gate}), which can be operated in the strong coupling regime where the RWA may break down. In comparison, we also study other systems such as spin-charge hybrid qubits and different kinds of single-spin qubits, for which we reconfirm that the RWA is safe for the control parameters concerned (even from the stringent quantum error correction constraint considerations). However, we will show quantitatively how efforts to speed up gate operations for scalable quantum computation ($T_R<\{T_2/10^3, T^*_2\}$, $T^*_2$ being the ensemble coherence time \cite{Hanson_RMP07}) would enlarge the error due to using RWA, and we give a detailed numerical study in Sec.~\ref{subsec:RWA_sinu_numer}. Focusing on the leading orders of the errors, we present analytical results that can be used to greatly reduce errors as one leaves the strong coupling regime in Sec.~\ref{subsec:analytic_extension}.
\subsection{Systematic numerical study of RWA infidelity}\label{subsec:RWA_sinu_numer}
As an $SU(2)$ unitary matrix, the evolution operators corresponding to the Hamiltonian in Eqs.~\eqref{eq:H_exact} and \eqref{eq:H_rwa} may be written as
\begin{eqnarray}\label{eq:U_axial_way}
U_\lambda(t) &=& \exp\left[-i \hat{\mathbf{n}}_\lambda(t)\cdot \bm{\sigma} \frac{\phi_\lambda(t)}{2}\right]
\nonumber\\
&=& \cos\frac{\phi_\lambda(t)}{2} -i \hat{\mathbf{n}}_\lambda(t) \cdot \bm{\sigma} \sin\frac{\phi_\lambda(t)}{2},
\end{eqnarray}
where $\hat{\mathbf{n}}$ and $\phi$ can be physically interpreted as the rotation axis and angle, respectively, and $\lambda$ is used to denote ``$\rm sin$'' or ``$\rm rwa$''. The deviation due to RWA from the desired rotation can then be expressed as
\begin{eqnarray}\label{eq:delta_U_rwa}
\delta U_{\rm rwa} = U^\dag_{\rm rwa} U_{\rm sin} =\exp\left[-i \delta\hat{\mathbf{n}}\cdot \bm{\sigma} \frac{\delta\phi}{2}\right],
\end{eqnarray}
where
\begin{eqnarray}
\cos\frac{\delta\phi(t)}{2} &=& \cos\frac{\phi_{\rm rwa}}{2} \cos\frac{\phi_{\rm sin}}{2} \nonumber\\
&&+ \hat{\mathbf{n}}_{\rm rwa}\cdot \hat{\mathbf{n}} _{\rm sin} \sin\frac{\phi_{\rm rwa}}{2} \sin\frac{\phi_{\rm sin}}{2},
\end{eqnarray}
and a similar relation follows for $\delta\mathbf{n}(t)$ \cite{footnote_delta_n}.
In this work we evaluate this error using the state-averaged infidelity $1-\bar{\mathcal{F}}_{\rm rwa}$, straightforwardly defined as \cite{Bruss_PRA98,Bowdrey_PLA02}
\begin{eqnarray}\label{eq:benchmark_2}
1\!-\!\bar{\mathcal{F}}_{\rm rwa}\!\!&=&\! 1\!-\! \frac{1}{4\pi} \int \!d\Omega {\rm Tr}\left[\left(\rho_\Omega \delta U_{\rm rwa}\right)\left(\rho_\Omega \delta U_{\rm rwa}^\dag\right) \right]
\nonumber\\
&=&\frac{1}{2}\!-\!\frac{1}{3}\!\! \sum_{i=x,y,z}\! {\rm Tr}\left[\!\left(\frac{\sigma_i}{2}\delta U_{\rm rwa}\right) \!\left(\frac{\sigma_i}{2} \delta U_{\rm rwa}^\dag\right) \right]
\end{eqnarray}
where $\rho_\Omega$ is the density matrix for a spinor with the solid angle $\Omega$. Using Eq.~(\ref{eq:delta_U_rwa}) the above equation can be expressed as
\begin{eqnarray}\label{eq:infidelity_in_delta_phi}
1-\bar{\mathcal{F}}_{\rm rwa}= \frac{2}{3} \sin^2\frac{\delta\phi(t)}{2}.
\end{eqnarray}
It is clear that the maximum infidelity is 2/3 from Eq.~(\ref{eq:infidelity_in_delta_phi}). In the following, we study individually different qubit systems for the error induced by using RWA to design the operation.
\begin{figure}[!htbp]
\centering
\includegraphics[width=8.5cm]{fig1_new.eps}
\caption{Averaged infidelity due to RWA, $1-\bar{\mathcal{F}}_{\rm rwa}$, as a function of time for the zero detuning case ($\omega=\omega_z$). Results for various driving strengths $\omega_x/\omega_z$ are shown: (a) 1, (b) 0.3, (c) 0.1, and (d) 0.01.} \label{fig:1}
\end{figure}
\textit{Resonant exchange (RX) qubit.} The resonant-driven version of the exchange-only triple-dot qubits \cite{DiVincenzo_Nat00, Laird_PRB10, Gaudreau_NatPhys12, Medford_NatNano13, Braakman_NatNano13} has been developed \cite{Medford_PRL13, Taylor_PRL13} with the intention to filter out low-frequency charge noises. In the experiment of Ref.~\cite{Medford_PRL13}, the control fields can be expressed phenomenologically as $J_z \approx p^2/\epsilon_0$ and $J_x\approx \sqrt{3} p^2 \epsilon/\epsilon_0^2$ where $p$ is the tunnel coupling between the neighboring quantum dots, $\epsilon_0$ measures the voltage shift of the middle gate, and the oscillatory $\epsilon(t)$ is the relative detuning with respect to the center of the (111) charge region. In the RWA regime the key quantity $\omega_x/\omega_z=J_x/J_z\approx \sqrt{3} \epsilon/\epsilon_0$. From the parameters provided in Ref.~\cite{Medford_PRL13} we have found that $\omega_x/\omega_z$ is around 15\%-30\% (see Table~\ref{tab:exper_parameters} for more details), corresponding to an RWA error of approximately $0.4\%$ [cf. Fig.~\ref{fig:1}(b)]. While this error is small compared to those caused by nuclear and charge noises, it still exceeds the typical target threshold of $10^{-4}$ for quantum error correction, meaning that even after the environmental noises have been eliminated (e.g. through dynamical decoupling or other techniques), the RWA remains a hurdle to fault tolerant quantum computing. Unlike environmental noise, the RWA error is systematic and must be treated in a different way than standard techniques such as dynamical decoupling which are typically used to mitigate environmental decoherence. This is precisely what motivates our work.
We further discuss the ranges of the coupling strengths in the RX qubit and their consequences on RWA. Reference~\cite{Medford_PRL13} has demonstrated that by increasing the middle gate voltage, both the qubit splitting $J_z$ and the resonant coupling $J_x$ increase. The latter however increases much faster: While $\omega_z/2\pi$ is increased from 0.355 to 1.98 GHz (a factor of $\sim5$), $d(\omega_x/4\pi) /dV_l$ is increased from 0.07 GHz/mV to 5 GHz/mV (a factor of $\sim70$). This indicates that while enhancing the control fields is appealing due to potentially faster gate operation and smaller influence from noise, it will actually lead to even larger RWA error. The corresponding infidelity is a few percent [cf. Fig.~\ref{fig:1}(a)] or larger, which can be comparable to those contributed from environmental noises.
On the other hand, if $\omega_x$ is kept small, the gate speed becomes an issue. Typically one requires that each quantum gate operation is at least $10^3$ times faster then the coherence time, which sets a lower bound on $\omega_x$. This is estimated below. When $\omega_z/2\pi=0.2$ GHz, quantum coherence time $T_2$ reaches 20 $\mu$s as measured experimentally \cite{Medford_PRL13}, which imposes $\omega_x/4\pi \geq 10^3/T_2 = 0.05$ GHz. This ratio $\omega_x/\omega_z = 1/2$ gives a fairly good estimate on how slow the Rabi oscillation is allowed to go for quantum computation, corresponding to an RWA infidelity of about 1\% which is unacceptably high. Ways to correct the problem include prolonging the coherence time and enlarging $\omega_z$ while keeping $\omega_x$ small by fine tuning the gate voltages, although other noises may play a role in the latter approach.
\textit{Singlet-Triplet (ST) qubit.} The ST qubit \cite{Petta_Science05} can similarly benefit from the oscillatory driving to filter out the quasi-static charge noise. In order to conform with the Hamiltonian Eq.~\eqref{eq:H_exact}, we take $|\uparrow\downarrow\rangle$ and $|\downarrow\uparrow\rangle$ as the qubit bases so that $J_z=g^* \mu_B \Delta B_z$ corresponds to the magnetic field gradient and $J_x$ is the exchange interaction between the singlet and triplet states which can be made oscillatory. Such oscillating control has been demonstrated in Ref.~\cite{Shulman_NatCom14}, although the experiments are done for the different purpose of dynamical Hamiltonian estimation\cite{Klauser_PRB06}. In this experiment, $\omega_x/\omega_z$ is estimated to be approximately $1/3$ which means that the RWA error is large [cf. Table~\ref{tab:exper_parameters} and Fig.~\ref{fig:1}(b)]. One contributing factor to the large ratio of $\omega_x/\omega_z$ is the small $\Delta B_z$ one is able to maintain in this experiment. This factor has since improved to reach about a few GHz for $\omega_z$\cite{Yacoby_footnote} meaning that the RWA error can be greatly reduced.
The ST qubits may also be implemented across different platforms including GaAs and Si. While many of these qubits are being operated using piecewise constant pulses \cite{Foletti_NatPhys09, Petersen_PRL13, Wu_PNAS14}, migration to oscillating pulses is straightforward thanks to the efficient electrical control over $J_x$ using gate voltages. In order to gain insight into the RWA error if these qubits were to be operated using oscillating pulses, we summarize the range of parameters $\omega_x$ and $\omega_z$ in Table~\ref{tab:parameter_ST}. We have included cases with different methods to generate the magnetic field gradient, including dynamical nuclear spin polarization\cite{Foletti_NatPhys09,Shulman_NatCom14}, micromagnets in GaAs\cite{Petersen_PRL13}, micromagnets in Si \cite{Wu_PNAS14}, and the gradient produced by the donor hyperfine fields\cite{Kalra_PRX14}. From the table we can see that one has considerable freedom to tune the ratio $\omega_x/\omega_z$ in a wide range. In order to keep the RWA error below $10^{-3}$ we must have $\omega_x/\omega_z \lesssim 0.1$, which however implies slow gates. In Sec.~\ref{sec:fast_gate}, we shall present a method to avoid using RWA, allowing access to large ratios of $\omega_x/\omega_z$ without introducing errors and implementing precisely-designed fast gates.
\begin{table
\caption{\label{tab:parameter_ST} Representative parameters $\omega_z/2\pi=g^* \mu_B \Delta B_z/2\pi\hbar$ and the gate controlled exchange coupling amplitude $\omega_x/2\pi$ from the major recent ST systems with static control field. The shortcut notations are, (D)NP=(dynamical) nuclear polarization and m.magnet=micromagnet.}
\renewcommand{\arraystretch}{2.0}
\tabcolsep=0.15 cm
\begin{tabular}{cccc}%
\hline\hline
& $\omega_z/2\pi$ & $\omega_x/2\pi$ & Reference \\
\hline
DNP in GaAs & 1 GHz & $\leq 25$ \!GHz & Ref.~\cite{Foletti_NatPhys09}
\\
\renewcommand{\arraystretch}{1.0}$\begin{array}{c}\textrm{NP by m.magnet}\\ \textrm{ in GaAs}\end{array}$
& 3 GHz & $\leq 25$ \!GHz & Ref.~\cite{Petersen_PRL13}
\\
m.magnet in Si/SiGe & 0.014 GHz & $\leq 25$ \!GHz & Ref.~\cite{Wu_PNAS14}
\\
\renewcommand{\arraystretch}{1.0}$\begin{array}{c}\textrm{P donor hyperfine}\\ \textrm{in Si(proposal)}\end{array}$ & 0.12 GHz & $\leq 1.2$ \!GHz & Ref.~\cite{Kalra_PRX14}
\\
\hline\hline
\end{tabular}
\end{table}
\begin{figure}[!htbp]
\centering
\includegraphics[width=8.5cm]{fig2_new.eps}
\caption{Averaged infidelity due to RWA, $1-\bar{\mathcal{F}}_{\rm rwa}$, as a function of time, for different relative detunings ($\delta \omega/\omega_z$ where $\delta \omega=\omega-\omega_z$) and coupling strengths ($\omega_x/\omega_z$), which are, respectively, (a) 0.1 and 0.01, (b) 0.01 and 0.001, (c) 0.1 and 0.1, and (d) 0.01 and 0.01. The insets plot $1-\bar{\mathcal{F}}_{\rm rwa}$'s behavior in the long time limit where another time period emerges.} \label{fig:2}
\end{figure}
\textit{Spin-charge hybrid qubit.} Recently the microwave-driven gate operation \cite{Kim_NPJ15} of the hybrid qubit \cite{Kim_Nat14} has been experimentally realized in Si nanostructures. As in the RX qubit, ac control of the hybrid qubit provides straightforward two-axis manipulation while suppressing the charge noise. In the experiment of Ref.~\cite{Kim_NPJ15} a $55\times 2\pi$ MHz Rabi oscillation is performed with the resonant frequency of 11.52 GHz. This $\omega_x/\omega_z$ ratio coincides with Fig.~\ref{fig:1}(d) which shows a negligible RWA error ($1-\bar{\mathcal{F}}_{\rm rwa}<5\times 10^{-6}$). However, this Rabi frequency is still relatively slow given the short $T_2$ time of 150 ns \cite{Kim_NPJ15}. To push for faster operations, possibly by dynamically modulating tunnel coupling \cite{Kim_NPJ15}, we expect that the RWA error increases to the case shown in Fig.~\ref{fig:1}(b). Again, the method we shall present in Sec.~\ref{subsec:analytic_extension} and Sec.~\ref{sec:fast_gate} will shed light on how one may reconcile the need of fast gate operation and low RWA error.
\textit{Single-spin qubit}. Single spin qubit manipulation utilizing Kane's proposal \cite{Kane_Nat98} has been realized in silicon with phosphorus ($^{31}\!$P) donors \cite{Laucht_SciAdv15}. Proximity gate voltage is demonstrated to modify the resonance frequencies of both the electron and nuclear spin under a static magnetic field $B_0$ via the Stark shift\cite{Laucht_SciAdv15}, as the change of localized electron distribution affects the effective $g$ factor and the hyperfine coupling. In this experiment, $B_0$ and the effective field from hyperfine interaction are equal or greater than $10^4$ of the oscillatory field amplitude $B_1$, with effective $\omega_z$ and $\omega_x$ shown in Table~\ref{tab:exper_parameters}. In this regime, RWA is clearly safe to use due to its negligible error. Moreover, the demonstrated Rabi frequency is already close to $2\pi\times 10^3/T_2$ with $T_2>10 (100)$ ms for electron (nuclear) spins \cite{Laucht_SciAdv15}, which have met the requirement of FTEC.
In another recent work using resonant-driven qubit control in non-purified silicon\cite{Veldhorst_Nat15}, a similar static field $B_0=1.4$ T has been used and the Rabi period of about 2.4 $\mu$s have been demonstrated. The $T_2$ time can reach $28$ ms, and together with a relatively stronger $J_z$ one may satisfy the FTEC requirement while keeping the RWA error negligible.
Having discussed several experimental platforms with the assumption that the driving frequency is on resonance with the qubit level splitting, we turn our attention to the detuning of driving frequency from exact resonance ($\delta \omega=\omega-\omega_z$), which is also an important RWA parameter and could affect the error. $\delta \omega$ is relevant in various qubit operations and calibrations. Ramsey fringes \cite{Ramsey_PR50} have been used in several of the aforementioned experiments \cite{Shulman_NatCom14, Kim_NPJ15, Laucht_SciAdv15, Veldhorst_Nat15} to obtain ensemble coherence $T^*_2$ time with $\delta\omega$ reaching up to $0.1 \omega_z$ \cite{Shulman_NatCom14}, and to calibrate the resonance frequency $\omega_z$ \cite{Ramsey_PR50}. Moreover, in the single-donor spin qubits \cite{Laucht_SciAdv15}, the detuning can be controlled by tuning $\omega_z$, as opposed to $\omega$, by proximity to the top gate. From Fig.~\ref{fig:2}, we see that when $\delta\omega/\omega_x$ reaches about 1/10, it starts to worsen the RWA appreciably. However, when $\delta \omega$ further increases such that $\delta\omega/\omega_x\gtrsim1$, its contribution to the infidelity stops growing: this is the far off-resonance regime where the qubit is not rotated. Another interesting feature is that the error due to detuning grows over time, which is absent in the zero-detuning case (cf. Fig.~\ref{fig:1}). This is relevant for the wide `non-interacting' region in the Ramsey experiment, as its mechanism assumes that the driving pulse keeps oscillating in the background to accumulate phase difference with Larmor precession in the RWA limit \cite{Ramsey_PR50}. We also observe that the shape of $1-\bar{\mathcal{F}}_{\rm rwa}$ versus $t/T_R$ is nearly the same when $\delta\omega/\omega_x$ is similar. The insets in Fig.~\ref{fig:2} show the long time behavior of $1-\bar{\mathcal{F}}_{\rm rwa}$ where a much slower frequency emerges. There the upper bound of $1-\bar{\mathcal{F}}_{\rm rwa}=2/3$ [Eq.~(\ref{eq:infidelity_in_delta_phi})] is eventually reached. These and other features can be explained by the leading-order analytical results we carry out, as presented in the next subsection.
\subsection{Error within a perturbative extension to RWA}\label{subsec:analytic_extension}
Analytical expressions of the qubit evolution are useful in quantum gate operations for many reasons. In particular, they are simple to use and avoid the numerical difficulty of integrating oscillating functions. Here we apply an extended higher-order analytical approximation \cite{Shirley_PRB65,Shirley_thesis63} and look deeper into the error caused by RWA.
We follow the theory developed by Shirley and others\cite{Shirley_PRB65,Shirley_thesis63} for going beyond RWA in a perturbative expansion. The periodic time-dependent Hamiltonian is transformed into a time-independent Floquet Hamiltonian, which is then solved by successive applications of quasi-degenerate perturbation theory in terms of the small parameters involved in this problem: $\omega_x/ \omega_z$ and $\delta\omega/\omega_z$. To do this, we start with an exact transformation
\begin{eqnarray}
\tilde{H}_{\rm sin}=U_\omega H_{\rm sin} U_\omega^\dag + i\hbar \frac{\partial U_\omega}{\partial t} U_\omega^\dag
\end{eqnarray}
where $U_\omega=e^{i\omega\sigma_z t/2}$ converts the reference frame to the rotating one. In this frame, the eigenstate becomes quasi-degenerate with $\omega\approx \omega_z$ in the absence of the driving term, and
\begin{eqnarray}\label{eq:H_sinu_rotating}
\tilde{H}_{\rm sin}= \frac{\hbar}{2}\left(\begin{array}{cc}
\omega_z-\omega & \frac{1}{2} \omega_x(1+ e^{2i \omega t}) \\
\frac{1}{2} \omega_x(1+ e^{-2i \omega t}) & -\omega_z+\omega
\end{array}\right).
\end{eqnarray}
Dropping the fast oscillating terms $e^{\pm 2i\omega t}$ reduces the Hamiltonian to the RWA one. The exact equation for the evolution operator in this frame should follow
\begin{eqnarray}\label{eq:exact_HU_rot}
i\hbar \frac{\partial \tilde{U}_{\rm sin}(t)}{\partial t} &=& \tilde{H}_{\rm sin} \tilde{U}_{\rm sin}(t).
\end{eqnarray}
Using the unitarity of $U$, $U_{22}=U^*_{11}$ and $U_{12}=-U^*_{21}$, where $U_{11}$ and $U_{12}$ are the Cayley-Klein parameters, the matrix equation Eq.~(\ref{eq:exact_HU_rot}) reduces to one for a vector state $[\tilde{U}_{\rm sinu,11}(t), \tilde{U}_{\rm sinu,21}(t)]^T$ with the initial conditions $U_{\rm sinu,11}(0)=1$ and $U_{\rm sinu,21}(0)=0$.
By transforming Eq.~(\ref{eq:exact_HU_rot}) into a corresponding Floquet equation with basis states incorporating the $e^{2i n\omega t}$ factor \cite{Shirley_PRB65,Shirley_thesis63}, a second-order perturbation analysis gives two general solutions, $(\tilde{U}_{\rm ext,11}(t), \tilde{U}_{\rm ext,21}(t))^T$,
\begin{eqnarray}\label{eq:nu_1}
\nu_1\!=\!\left(\!\!\begin{array}{c}
\sqrt{\Delta\omega'-\delta \omega'}- \frac{\omega_x}{8\omega} \sqrt{\Delta\omega'+\delta \omega'} e^{2i \omega t}
\\
\sqrt{\Delta\omega'+\delta \omega'}+ \frac{\omega_x}{8\omega} \sqrt{\Delta\omega'-\delta \omega'} e^{-2i \omega t}
\end{array} \!\!\right) \!\!\!\frac{e^{-i\frac{\Delta \omega''t}{2}}}{\sqrt{2\Delta\omega'}}
\end{eqnarray}
and
\begin{eqnarray}\label{eq:nu_2}
\nu_2\!=\!\left(\!\!\begin{array}{c}
\sqrt{\Delta\omega'+\delta \omega'}+ \frac{\omega_x}{8\omega} \sqrt{\Delta\omega'-\delta \omega'} e^{2i \omega t}
\\
-\sqrt{\Delta\omega'-\delta \omega'}+ \frac{\omega_x}{8\omega} \sqrt{\Delta\omega'+\delta \omega'} e^{-2i \omega t}
\end{array} \!\!\!\!\right) \!\!\!\!\frac{e^{i\frac{\Delta \omega''t}{2}}}{\sqrt{2\Delta\omega'}}
\end{eqnarray}
where
\begin{eqnarray}
&\delta \omega' = \delta\omega \left(1- \frac{\omega_x^2}{64 \omega^2} \right)-\frac{\omega_x^2}{16 \omega},&
\nonumber\\
&\Delta\omega' = \sqrt{\frac{\omega_x^2}{4}+\delta \omega'^2}, \quad \Delta \omega''= \left(1-\frac{\omega_x^2}{64\omega^2}\right) \Delta\omega' .&
\end{eqnarray}
The well known first-order correction to the physical resonance frequency, the Bloch-Siegert shift $\omega_x^2/16 \omega$ \cite{Bloch_PR40}, is the last term of $\delta \omega'$. Considering the initial conditions for $\tilde{U}_{\rm sin}$, we have
\begin{eqnarray}\label{eq:U_solution_2nd}
\left(\begin{array}{c}
\tilde{U}_{\rm ext,11}(t) \\ \tilde{U}_{\rm ext,21}(t) \end{array} \right)
= C_1 \nu_1+ C_2 \nu_2,
\end{eqnarray}
with constant coefficients
\begin{eqnarray}
C_1 \!\!&=&\!\! \left(1+ \frac{\omega_x^2}{64\omega^2}\right)^{-1} \!\!\frac{\sqrt{\Delta\omega'-\delta \omega'}- \frac{\omega_x}{8\omega} \sqrt{\Delta\omega'+\delta \omega'} }{\sqrt{2\Delta\omega'}},\qquad\label{eq:coef_C1}
\\
C_2\! \!&=& \!\!\left(1+ \frac{\omega_x^2}{64\omega^2}\right)^{-1} \!\! \frac{\sqrt{\Delta\omega'+\delta \omega'}+ \frac{\omega_x}{8\omega} \sqrt{\Delta\omega'-\delta \omega'} }{\sqrt{2\Delta\omega'}}.\qquad\label{eq:coef_C2}
\end{eqnarray}
Taking the $\{\delta\omega, \omega_x\}/\omega_z\rightarrow 0$ limit recovers the results from RWA. Apart from the higher-order correction to the resonance shift, we see from Eqs.~(\ref{eq:nu_1}) and (\ref{eq:nu_2}) that the oscillating terms $e^{\pm 2i \omega t}$ also play a major role, the contribution of which can be readily identified from the oscillating nature of curves shown in Figs.~\ref{fig:1} and \ref{fig:2}.
Converting the solution $U_{\rm ext}$ implied from Eq.~(\ref{eq:U_solution_2nd}) into the form of Eq.~(\ref{eq:U_axial_way}), we can express the infidelity corresponding to $U_{\rm ext}$ in terms of the deviation angle $\delta\phi$ as Eq.~\eqref{eq:infidelity_in_delta_phi}. In the RWA limit, $\tilde{\phi}_{\rm rwa}=\Delta\omega t$ and $\hat{\tilde{\mathbf{n}}}_{\rm rwa}=(\omega_x,0,-2\delta\omega)$ [$\Phi$ in Eq.~(\ref{eq:J_x_t}) has been set to be 0].
\begin{figure}[!htbp]
\centering
\includegraphics[width=8.5cm]{fig3_new.eps}
\caption{Averaged infidelity $1-\bar{\mathcal{F}}_{\rm ext}$ from Eq.~(\ref{eq:U_solution_2nd}). The parameters in panels (a)-(c) are the same as in Figs.~\ref{fig:1}(a) and \ref{fig:1}(b) and Fig.~\ref{fig:2}(c), respectively. In (d) we plot the difference in the return probability between the exact solution and both the RWA and the extended analytical results for an initial $|0\rangle$ state and $\omega_x/\omega_z=0.3$ and $\delta \omega/\omega_z=0.1$. } \label{fig:3}
\end{figure}
The infidelity of $U_{\rm ext}$ compared to the exact evolution $U_{\rm sin}$, $1-\bar{\mathcal{F}}_{\rm ext}$, is much smaller than that corresponding to RWA $1-\bar{\mathcal{F}}_{\rm rwa}$, as expected. This can also be seen from Fig.~\ref{fig:3}. The infidelities $1-\bar{\mathcal{F}}_{\rm ext}$ as shown in Figs.~\ref{fig:3}(a) and \ref{fig:3}(c) are orders of magnitude smaller than what is shown in Figs.~\ref{fig:1} and \ref{fig:2}, which are well within the quantum error correction threshold. Even for very large $\omega_x/\omega_z$ ratio [cf. Fig.~\ref{fig:3}(a)], the second-order perturbation results show a small error $\sim 8\times 10^{-4}$. In Fig.~\ref{fig:3}(d), we also present the improved accuracy using the return probability for the initial $|0\rangle$ state, $P_0$. Again we see a substantially smaller error for $U_{\rm ext}$ than for $U_{\rm rwa}$. Thus, the analytical perturbative results should be applicable in many practical situations already giving substantially more accurate results than RWA.
As the analytical solutions presented above satisfactorily approximate the exact sinusoidal result, we shall use them to investigate the RWA error across a wider parameter range. Tracing the RWA error versus time, three different frequency components emerge and play important roles. As mentioned, a highly oscillatory component is always present with the counter-rotating frequency $2\omega$ in the rotating frame, and is the origin for the approximation in the RWA. This high frequency component is modulated by a combination of two lower frequencies, emergent from the interference of Floquet eigen-frequencies in the zeroth (RWA) and second-order perturbation. They correspond to $\Delta\omega''/2$ in the last factor of the two general solutions [Eqs.~(\ref{eq:nu_1}) and (\ref{eq:nu_2})] and its limiting value in RWA $\Delta\omega=\sqrt{\omega_x^2/4+\delta \omega^2}$. They produce a slow and a relatively fast component,
\begin{multline}
\Delta\omega'' - \Delta\omega
=- \frac{\omega^2_x}{16\Delta\omega \omega_z} \!\Bigg[ \delta \omega +\frac{ \delta\omega^2\!+\! \Delta\omega^2}{4 \omega_z}
-\frac{\omega^4_x}{128 \omega_z \Delta\omega^2}\Bigg] \\
+\mathcal{O} \left[\left\{\frac{\delta \omega}{\omega_z}, \frac{\omega_x}{\omega_z}\right\}^5\right], \label{eq:small_freq}
\end{multline}
\begin{equation} \label{eq:mid_freq}
\frac{1}{2}(\Delta\omega'' + \Delta\omega)
\approx \Delta\omega.
\end{equation}
These two frequencies explain the envelope modulations in Figs.~\ref{fig:1} and \ref{fig:2} (the faster one) and in the inset of Fig.~\ref{fig:2} (the slower one). Taking $\delta \omega/\omega_z=0.1$ and $\omega_x/\omega_z=0.01$ for example, the leading contribution from Eq.~(\ref{eq:small_freq}) makes $\sim \omega_x/800$, matching $T\sim 400 T_R$ in the inset of Fig.\ref{fig:2} (a). Contrasting $\tilde{U}_{\rm rwa}$ and $\tilde{U}_{\rm ext}$ in the form of Eq.~(\ref{eq:benchmark_2}), we are able to obtain all leading-order terms for $1-\bar{\mathcal{F}}_{\rm rwa}$ in parameters $\frac{\delta \omega}{\omega_z}$ and $\frac{\omega_x}{\omega_z}$ after some algebra,
\begin{widetext}
\begin{eqnarray}\label{eq:infidelity_rwa_ext}
1\!-\!\bar{\mathcal{F}}_{\rm rwa}(t)\!&\!=\!&\! \frac{\omega^2_x}{96\Delta\omega^4 \omega_z^2} \left[
\frac{3\omega_x^4 \!+\!24\omega_x^2 \delta\omega^2 \!+\!64\delta\omega^4 }{32}\!+ \!\frac{1}{16} \omega_x^2 \Delta \omega^2 \delta \omega^2\;t^2
\!-\! \frac{\omega_x^2 \Delta\omega^2}{4} (\cos 2\omega t\! +\!\delta \omega \; t\; \sin2\omega t)
\right.\nonumber\\
&&\left. \qquad\quad
+(\delta\omega^2 \!+\!\Delta\omega^2)\! \left(\frac{1}{8}\omega^2_x\cos\Delta\omega t\! -\!\Delta\omega^2 \cos2\omega t \cos\Delta\omega t\! -\!\delta\omega \Delta \omega \sin2\omega t \sin \Delta\omega t \! \right)\!
\right]\!+\!\mathcal{O} \left[\left\{\frac{\delta \omega}{\omega_z},\! \frac{\omega_x}{\omega_z}\right\}^3\right].\qquad
\end{eqnarray}
\end{widetext}
From the equation above, we estimate the magnitude of error during one Rabi cycle as a function of $\frac{\delta \omega}{\omega_z}$ and $\frac{\omega_x}{\omega_z}$, which can be written as
\begin{widetext}
\begin{multline}\label{eq:infidelity_Max}
[1-\bar{\mathcal{F}}_{\rm rwa}]^{\rm Max} \approx \frac{\omega^2_x}{96\Delta\omega^4 \omega_z^2} \left[
\frac{3\omega_x^4 +24\delta\omega^2 \omega_x^2 +64\delta\omega^4 }{32}
+ \frac{1}{16}\delta \omega^2 \omega_x^2 \Delta \omega T_R^2
+ \frac{\omega_x^2 \Delta\omega^2}{4} \sqrt{1 +(\delta \omega \; T_R)^2}\right.
\\\left. +(\delta\omega^2 +\Delta\omega^2)\sqrt{ \left(\frac{\omega^2_x}{8} +\Delta\omega^2\right)^2 +(\delta\omega \Delta \omega )^2}
\right].
\end{multline}
\end{widetext}
The numerical evaluation of Eq.~\eqref{eq:infidelity_Max} is summarized in Fig.~\ref{fig:4}, which gives an estimate of the upper bound of the RWA error for a wide range of parameters. Three lines are drawn where $[1-\bar{\mathcal{F}}_{\rm rwa}]^{\rm Max}$ coincides with the threshold values $10^{-4}$, $10^{-3}$ and $10^{-2}$. Several points in the parameter space relevant to experiments are also marked.
\begin{figure}[!htbp]
\centering
\includegraphics[width=8.5cm]{fig_Max_Error.eps}
\caption{Contour plot for the infidelity amplitude in the first Rabi cycle, $[1-\bar{\mathcal{F}}_{\rm rwa}]^{\rm Max}$ in Eq.~(\ref{eq:infidelity_Max}), as a function of relative detuning ($\delta \omega/\omega_z$) and driving strength ($\omega_x/\omega_z$). We mark the $10^{-2}$, $10^{-3}$ and $10^{-4}$ contour lines. Three red crosses mark experimental operation points in the RX experiment (Figs. 2-4 in \cite{Medford_PRL13}), the light blue diamond marks the resonant ST experiment \cite{Shulman_NatCom14}, the red solid circle marks the hybrid qubit \cite{Kim_NPJ15}, and the light blue square marks the weakly driven single-spin experiments \cite{Laucht_SciAdv15,Veldhorst_Nat15} (see Table~\ref{tab:exper_parameters}).} \label{fig:4}
\end{figure}
\section{Fast gate operations with arbitrary oscillatory driving amplitude}\label{sec:fast_gate}
\begin{figure}[h]
\centering
\includegraphics[width=8.5cm]{RWA_Hada.eps}
\caption{Two-pulse solutions for the Hadamard gate in the resonant RWA limit. $\Phi_{1,2}$ and $t_{1,2}$ are the phases and durations of the two pulses. The solution is plotted parametrically, with (a) $\{\Phi_1,\Phi_2\}$, and (b) $\{t_1, t_2\}$ versus the total gate time, $t_{Tot}=t_1+t_2$. Times are scaled by the driving frequency, $\omega_x$. Note that all four curves are closed or periodic. The usual RWA solution, $\{\Phi_1=\frac{\pi}{2},\Phi_2=\pi, \frac{\omega_x}{2}t_1=\frac{\pi}{2}, \frac{\omega_x}{2}t_2=\pi\}$ in our notation, is marked by solid dots. These dots also give a sense of how to associate points on the multivalued curves at a given total time.
}\label{fig:gate_RWA}
\end{figure}
We now consider the construction of gate operations with strong driving outside the range of validity of the RWA (and its perturbative generalization). Two oscillatory pulses of the form $\omega_x \cos\left(\omega t_i + \Phi_i\right)$ are sufficient to achieve any single-qubit logic gate, which contains 3 degrees of freedom: two specifying the axis and one the rotation angle. Even though we have taken fixed driving frequency and amplitude, there are still 4 degrees of freedom in the two pulse sequence, i.e., the phase and duration of each segment. We expect, then, that there is an infinite number of two-pulse realizations for a given logical gate, with the solution vector forming a curve in the 4D space $\{\Phi_1,\Phi_2, t_1,t_2\}$. We show below that this is indeed true, and select an optimum solution with the minimum total time. This allows us to leave the weak coupling limit and increase the gate speed while still having simple pulse controls, as in RWA, as opposed to a complex pulse-shaping technique. This scheme should be useful for the current semiconductor qubit experiments.
The essential set of single-qubit gates necessary for universal quantum computation can be the Hadamard (H) gate and the `$\pi/8$' (T) gate, i.e., a $\pi$ rotation around $\hat{\mathbf{x}}+\hat{\mathbf{z}}$ direction and a $\pi/4$ rotation around $z$-axis, respectively. In the form compatible with $U_{22}=U^*_{11}$ and $U_{12}=-U^*_{21}$, one has
\begin{eqnarray}
U_{H} &=& \frac{1}{\sqrt{2}}\left(\begin{array} {cc} i& i \\ i & -i \end{array}\right), \label{eq:U_H}\\
U_{T} &=& \left(\begin{array} {cc} e^{-i\pi/8}& 0 \\ 0 & e^{i\pi/8} \end{array}\right),\label{eq:U_T}
\end{eqnarray}
up to an inconsequential global sign. Forming the H or T gate by two sinusoidal pulses amounts to searching for solutions to the differential Eq. \eqref{eq:exact_HU_rot} such that
\begin{eqnarray}\label{eq:U_2pulses}
\tilde{U}_{\rm sin}(t_2) \tilde{U}_{\rm sin}(t_1) = U_{H/T},
\end{eqnarray}
that is, to a root-searching problem,
\begin{eqnarray}
|\tilde{\mathbf{U}}_{I\!I}- \mathbf{U}_{H/T}|=0,
\end{eqnarray}
where the two vectors are
\begin{eqnarray}
\tilde{\mathbf{U}}_{\!I\!I}\!\!=\!\left( \!\begin{array}{c}
\!\tilde{U}_{\!{\rm sin},\!1\!1}(t_2) \tilde{U}_{\!{\rm sin},\!1\!1}(t_1\!) - \tilde{U}^*_{\!{\rm sin},\!2\!1}(t_2) \tilde{U}_{\!{\rm sin},\!2\!1}(t_1\!) \!\\
\!\tilde{U}_{\!{\rm sin},\!2\!1}(t_2) \tilde{U}_{\!{\rm sin},\!1\!1}(t_1\!) + \tilde{U}^*_{\!{\rm sin},\!1\!1}(t_2) \tilde{U}_{\!{\rm sin},\!2\!1}(t_1\!)\! \end{array}\! \right),
\end{eqnarray}
and $\mathbf{U}_{H(T)}=(U_{H(T),11},U_{H(T),21})^T$. Here the logical gates are realized in the rotating frame as they are in the usual RWA limit, and we take the resonant case $\omega= \omega_z$, though our approach is not specific to that case. We numerically carry out the (rather computationally demanding) search over the pulse durations $t_1$ and $t_2$, and phases $\Phi_{1}$ and $\Phi_2$.
Displaying the solutions is somewhat difficult, but Fig.~\ref{fig:gate_RWA} gives a visualization of all possible two-pulse solutions for the Hadamard gate in the resonant RWA limit. We show it here as a point of comparison for the strong driving case below. Furthermore, one can see that the most obvious solution, $\{\Phi_1,\Phi_2, \frac{\omega_x}{2}t_1, \frac{\omega_x}{2}t_2\}=\{\frac{\pi}{2},\pi,\frac{\pi}{2}, \pi\}$, is not the solution with the minimal total time. Rather, there is a solution with $t_1=t_2$ that is the fastest. The set of solution vectors may be thought of as a closed string in the 4D parameter space. This space is actually a hypertorus, as the RWA Hamiltonian for the two pulses is periodic in both $\Phi_{1,2}$ and $t_{1,2}$. We can also note in Fig.~\ref{fig:gate_RWA} a symmetry between a pair of solutions at the same $t_{Tot}$: $t_{1,2}\leftrightarrow t_{2,1}$ \& $\Phi_{1,2}\leftrightarrow 2\pi-\Phi_{2,1}$ \cite{footnote_symmetry}. For the $\frac{\pi}{8}$ gate in this RWA limit, the plots are even simpler since the solutions are of the form $\{\Phi_2= \Phi_1 +\frac{\pi}{8}, \frac{\omega_x}{2}t_2= \frac{\omega_x}{2} t_1=\pi \}$. This forms a straight line perpendicular to the $\{t_1, t_2\}$ plane bent in the hypertorus.
\begin{figure*}[!htbp]
\centering
\includegraphics[width=18cm]{Fig_Hada_Sinu.eps}
\caption{Two-pulse solutions for the resonant Hadamard gate with $\omega_x/\omega_z\leq 4$. Times and frequencies are given in units such that $\omega_z=1$. (a) Minimal total time (scaled by driving strength) vs driving strength. The inset shows the same time in terms of the fixed unit. (b) Pulse segment times (scaled by driving strength) vs driving strength. Red solid and blue dashed curves represent the first and second pulses respectively, throughout. (c) Pulse segment phases vs driving strength. (d) Parametric plot of pulse segment times $t_1$ and $t_2$ vs total time for $\omega_x = \omega_z$. (e) Parametric plot of pulse segment phases $\Phi_1$ and $\Phi_2$ vs total time for $\omega_x = \omega_z$. (Note the $2\pi$ periodicity in $\Phi_{1,2}$.) The arrows are a guide to the eyes, indicating which values on the multivalued curves belong to the same solution. The minimal $t_{Tot}$ solution is enclosed by the dotted cyan box.
(f) Parametric plot of pulse segment times $t_1$ and $t_2$ vs total time for $\omega_x = 0.1\omega_z$. (g) Parametric plot of pulse segment phases $\Phi_1$ and $\Phi_2$ vs total time for $\omega_x = 0.1\omega_z$.
(h)-(j) The evolution operators corresponding to the minimal $t_{Tot}$ solutions for $\omega_x/\omega_z= 0.1$ and 1 vs time. The axial vector $\tilde{\bm{\omega}}$ describes the cumulative effect of the evolution up to time $t$ in terms of an effective axis of rotation, $\{\tilde{\omega}_{\theta}, \tilde{\omega}_{\phi}\}$ and an angle, $|\tilde{\bm{\omega}}|$.
}\label{fig:Sinu_Hada}
\end{figure*}
With this visual representation in mind, we now show the solutions outside the RWA limit. First we consider the Hadamard gate for example. The shape of the string (or strings) of solutions in the 4D variable space now depends on the driving strength, $\omega_x$. Figures~\ref{fig:Sinu_Hada} (f) and (g) explicitly show the whole solutions for $\omega_x=0.1\omega_z$. As this is fairly weak driving, the solutions are very similar to those in the RWA limit shown in Fig.~\ref{fig:gate_RWA}, with small, rapidly oscillating deviations due to the counter-rotating terms. The solutions for stronger driving of $\omega_x=\omega_z$ are shown in Figs.~\ref{fig:Sinu_Hada}(d) and \ref{fig:Sinu_Hada}(e). There the distortion compared to the RWA case of Fig.~\ref{fig:gate_RWA} is dramatic.
To give some idea of the dynamics of these highly nontrivial solutions, we use the solution parameters with minimal total gate time and plot their detailed gate evolution in Figs.~\ref{fig:Sinu_Hada}(h)-(j). The evolution matrix after the two pulses, $\tilde{U}_{I\!I}(t_1+t_2)=\tilde{U}_{\rm sin}(t_1)\tilde{U}_{\rm sin}(t_2)$, has to end up equal to $U_H$. We show how this operator evolves in time to the desired one, representing the operator in terms of a rotation axis and angle. In the RWA limit, the evolution operator has a fixed rotation axis and an angle that increases linearly with time in the rotating frame. Figures~\ref{fig:Sinu_Hada}(h)-(j) depict the total axial vector $\tilde{\bm{\omega}}(t)$'s direction $\{\tilde{\omega}_{\theta}, \tilde{\omega}_{\phi}\}$ and magnitude $|\tilde{\bm{\omega}}|$ during these two pulses, which indeed accomplish $\tilde{\omega}_{\theta}=\pi/4$, $\tilde{\omega}_{\phi}=0$ and $|\tilde{\bm{\omega}}|=-\pi$ of $U_H$. (Note that this indeed occurs much faster for $\omega_x=\omega_z$ than for $\omega_x=0.1\omega_z$.)
\begin{figure*}[!htbp]
\centering
\includegraphics[width=18cm]{Fig_PiOver8_Sinu.eps}
\caption{Two-pulse solutions for the resonant $\pi/8$ phase gate with $\omega_x/\omega_z\leq 12$. Times and frequencies are given in units such that $\omega_z=1$. (a)-(j) follow the same structure as in Fig.~\ref{fig:Sinu_Hada} for the Hadamard gate. Most features are within $\omega_x\leq4 \omega_z$, but the additional insets in (b) and (c) show the extended range $0<\omega_x\leq12 \omega_z$. (e) and (g) only show $\Phi_{1,2}$ over an interval of $\pi$ as that is the periodicity in this case (see Appendix~\ref{app:shifted_sinu} for details).
}\label{fig:Sinu_PiOver8}
\end{figure*}
Finally, we select the minimal total time solution and map out the parameters of this optimal solution for a wide range of driving amplitude in Figs.~\ref{fig:Sinu_Hada} (b) and (c). We show the minimal total time, scaled by $\omega_x$, as a function of $\omega_x$ in Fig.~\ref{fig:Sinu_Hada} (a). Note that $\omega_x t_{Tot}$ does not vary much as $\omega_x$ increases from zero to the order of $\omega_z$, indicating that the time there is roughly inversely proportional to driving amplitude in that regime. Around $\omega_x \sim 2\omega_z$, $t_{Tot}$ begins to saturate at around $2\omega_z^{-1}$, as can be seen in the set of Fig.~\ref{fig:Sinu_Hada} (a). As a result, the curve increases linearly, indicating that the time saturates at some constant value regardless of how much more strongly the qubit is driven. This is not unexpected: In the RWA regime, gate speed scales as $1/\omega_x$, and evidently this scaling roughly holds even up to $\omega_x\sim\omega_z$. In the other extreme, when $\omega_x\gg \omega_z$, the counter-rotating coupling is nearly stationary within time $1/\omega_x$, and one is in the static coupling limit where the gate speed is limited by $\omega_z$ independent of the value of $\omega_x$. The transition point between these two regimes evidently depends on the specific logic gate, as we will show below. The main practical point though is that the duration of a Hadamard gate in the saturated, strongly driven regime is at least two orders of magnitude shorter than in the weakly driven regime where RWA applies. This should motivate using the fast gate constructions shown in this work. We believe that the optimal fast gate construction outlined here going beyond RWA is the simplest method for achieving FTEC in semiconductor qubit operations.
Figure~\ref{fig:Sinu_PiOver8} presents the same results for the $\frac{\pi}{8}$ phase gate. We again see in Fig.~\ref{fig:Sinu_PiOver8}(a) the similar trend of gate operation speed-up until saturation (as in Fig.~\ref{fig:Sinu_Hada}), though here for the $\pi/8$ gate the saturation occurs at somewhat stronger driving than for the Hadamard gate. The solution string shapes for $\omega_x=0.1\omega_z$ and $\omega_x=\omega_z$ are very similar to each other, although differences become visible when $\omega_x$ further increases (not shown).
We have also found equivalent solutions for an important extension of the standard driving model where the oscillations no longer need be centered at zero, $J_x=\hbar[\omega_{\rm ave}+\omega_x\cos(\omega t+\Phi)]$.
This is relevant for a larger category of qubit systems, and in particular, the ST qubit whose exchange coupling between $|\!\uparrow \downarrow\rangle$ and $|\!\downarrow\uparrow\rangle$ cannot dynamically change sign. The stationary term $\hbar\omega_{\rm ave}$ is more effective than the counter-rotating part to cause distortion of the Rabi oscillation when $\omega_{\rm ave}$ is not much smaller than $\omega_z$. Notwithstanding, the gate speed-up with increasing $\omega_x$ remains robust. The detailed results and discussions are given in Appendix~\ref{app:shifted_sinu}. This generic extension also serves to demonstrate the flexibility of our procedure for handling different oscillatory drivings. We can readily modify the coupling model to any experimentally relevant situation and calculate the pulse control parameters for desired logic gates, allowing easy experimental access to fast operations. Our work can also be easily generalized to other qubits where fast operations are desirable (e.g. superconducting qubits).
\section{Conclusion}\label{sec:conclusion}
If desired, semiconductor spin qubits could be operated for faster gate operations in a strong driving regime, far beyond the threshold where RWA gives an accurate description of the dynamics. The strong driving is indeed very desirable to speed up operations, but the design of gate operations in that regime requires appropriate protocols different from the RWA scheme. This is particularly true given the stringent precision requirements of fault-tolerant quantum computing. In this work we have quantified the well-known limitations of RWA for specific spin qubit platforms by carefully analyzing the experimental situations, and also determined the extent to which a perturbative approach can extend the applicability of an analytic approach to gate design. The analytical approach we developed, which may work when the driving is not too strong (and hence the gate operations are not very fast), is simple and practical to use. We find that for the resonant-exchange and ac-driven singlet-triplet qubits in particular, a completely new approach is required to circumvent the limitations of RWA.
While one could focus on the aspect of minimizing the gate time through any of the standard pulse-shaping or optimal control techniques, in this work we have instead considered the solutions that exist within a simple, highly constrained space. This has inherent heuristic value, and the ability to map out the full set of solutions is unusual among numerical optimal control approaches, and may prove useful in informing local searches in the experimental parameter space for non-RWA gating pulses. Our desired gate operation is built from only two applications of a simple sinusoidal pulse. Our complete numerical calculations of the pulse control parameters for the Hadamard and $\frac{\pi}{8}$ logic gates have been plotted as a visual look-up table to synthesizing gates over a wide range of driving amplitudes. While this means of presentation is necessarily imprecise, any real experimental implementation will have imperfections not included in the theoretical model that will perturb the solutions we have presented anyway. The true power of our results is that they provide a guide to the on-chip calibration of fast gating operations.
For convenience, we provide a quantitative comparison with the ultimate quantum speed limit. It is emphasized that the optimization of quantum gates or evolution operators [on $SU(2)$, or diffeomorphic to the $S^3$ space] \cite{Khaneja_PRA01, DAlessandro_IEEE01, Wu_PLA02, Boozer_PRA12, Garon_PRA13, Avinadav_PRB14} are different than that of the state transfer which is only parametrized by two real numbers on the Bloch sphere \cite{Boscain_JMP06, Hegerfeldt_PRL13}. In the weak-driving RWA regime, the minimal times to achieve the Hadamard and $\frac{\pi}{8}$ gates are $3.938/\frac{\Omega}{2}$ and $\frac{\sqrt{15}}{4}\pi/\frac{\Omega}{2}$ ($\hbar\Omega$ is the bound of the control field), respectively, by resorting to Pontryagin maximum principle \cite{Pontryagin_book74, Garon_PRA13} and minimizing the action \cite{Boozer_PRA12}. The conventional RWA implementation of the Hadamard and $\frac{\pi}{8}$ gates spend longer times, $\frac{3\pi}{2}/\frac{\omega_x}{2}$ and $2\pi/\frac{\omega_x}{2}$, as limited by the fixed control field axis (to be $\hat{\mathbf{x}}$). In the single axis strong-driving regime, the absolute minimal time is $\frac{\pi}{2\omega_z}$ or $\frac{\pi}{4\omega_z}$, for the Hadamard or $\frac{\pi}{8}$ gate, respectively \cite{Khaneja_PRA01}, whereas our strong-driving $t_{Tot}$ in Figs.~\ref{fig:Sinu_Hada}(a) and \ref{fig:Sinu_PiOver8}(a) are within a factor of 2. Thus, this comparison also serves to demonstrate the overall high performance of our scheme despite of its simple design.
This simple yet highly constrained control pulse is not intended to compete with those using optimal control theory \cite{Krotov_book95, Khaneja_JMR05, Maximov_JCP08}, such as the recent single-qubit implementation by Scheuer \textit{et al} \cite{Scheuer_NJP14}. As a concrete comparison, the method of Ref.~\cite{Scheuer_NJP14} produces high-fidelity state rotations whose speed is less than the relevant quantum speed limit by a mere 3\% by optimizing over 15 parameters and incorporating realistic technical constraints of the pulse generator, while our method produces exact gates whose speed is less than the relevant quantum speed limit by 50\% or less by optimizing over only four parameters in an idealized theoretical scenario. We hasten to add, however, that in many ways this is an apples-to-oranges comparison, not least because the quantum speed limits for rotations from a specific input state to a specific output state, as in Ref.~\cite{Scheuer_NJP14}, are different from the speed limits for implementing a state-independent gate operation (in fact, the latter are not even fully worked out yet \cite{Avinadav_PRB14}). Nonetheless, this illustrates the point that for a specific goal or cost functional (i.e., some combination involving fidelity, time, bandwidth, etc.), techniques such as that of Scheuer \textit{et al} \cite{Scheuer_NJP14} may offer better performance. However, the attraction of the pulses we have presented is their theoretically rudimentary form and the ability to map out the entire solution set within the comparatively small parameter space in order to aid calibration.
As environmental dephasing errors increase with operation time, the two orders of magnitude speed-up permitted by non-RWA pulses is a promising tool for enabling fault-tolerant quantum computation with spin qubits. For a given experiment with specified constraints and objectives, a more sophisticated approach may even allow further speed-up or incorporate additional robustness. Going beyond RWA will eventually be essential for resonant exchange and singlet-triplet qubits in order to reach the error correction thresholds for quantum computation.
This work is supported by LPS-MPO-CMTC. XW acknowledges support from City University of Hong Kong (Projects No. 9610335 and No. 7200456).
|
1,108,101,563,231 | arxiv | \section{Introduction}
Frustrated quantum spin systems have been
the subjects of considerable interest.
One of the intriguing systems is the $S=1/2$ quantum spin Kitaev model
on the honeycomb lattice~\cite{Kitaev2006}, where
strong frustration emerges due to direction-dependent Ising interactions.
It is known that this model is exactly solvable and
its ground state is a quantum spin liquid state, where
spin degrees of freedom are decoupled by
itinerant Majorana fermions and
$Z_2$ fluxes~\cite{Kitaev2006,Feng2007,Chen2007,Chen2008}.
Two energy scales for distinct degrees of freedom
yield interesting finite temperature properties such as
double peak structure in the specific heat, spin dynamics,
and thermal Hall coefficients
at low temperatures~\cite{Nasu2015,Yamaji2016,Yoshitake2016,Nasu2017,suzuki2018pre,yamaji2018pre}.
Since magnetic properties of certain Mott insulators
with strong spin-orbit coupling should be described by the Kitaev model and its
extentions~\cite{Jackeli2009},
a lot of theoretical studies for these models~\cite{Chaloupka2013,Yamaji2014,Suzuki2015,PhysRevB.96.144414,Nakauchi,SuzukiYamaji} and
experimental studies
on candidates materials such as
$A_2$IrO$_3$($A$=Na, Li)~\cite{Singh2010,Singh2012,Comin2012,Choi2012},
$\alpha$-RuCl$_3$~\cite{Plumb2014,Kubota2015,Sears2015,Majumder2015}
and $\rm H_3LiIr_2O_6$~\cite{Bette2017}
have been done so far.
The key of these features characteristic of the Kitaev model
is the existence of the local conserved quantity
defined at each plaquette in the honeycomb lattice,
which is responsible for the absence of the long-range spin correlations.
From a fundamental viewpoint of quantum spin systems,
a question arises whether or not such interesting properties
appear in generalized Kitaev models with arbitrary spins.
It is known that
in the general spin $S$ Kitaev model~\cite{PhysRevB.78.115116},
there exist local $Z_2$ conserved quantities and
the ground state is nonmagnetic.
In spite of the presence of the local invariants, details of its excitations
and finite temperature properties remain unclear.
Very recently thermodynamic properties in the Kitaev model
with spin $S\le 5/2$
has been examined for an 8-site cluster~\cite{SuzukiYamaji}.
It has been found that,
in the specific heat, double peak structure appears
only for the $S=1/2$ and $S=1$ models, while a single peak appears
in the $S>1$ model.
However, the system size may not be large enough
to discuss thermodynamic properties for
the Kitaev models on the honeycomb lattice.
In addition, the role of the local conserved quantities was still unclear.
Therefore, it is necessary to clarify
how the $Z_2$ degrees of freedom inherent
in the Kitaev model affects
ground state and finite temperature properties.
In this Letter, we mainly focus on
the ground state and finite temperature properties of the $S=1$ Kitaev model.
First, we show the existence of the global parity symmetry
in addition to the $Z_2$ symmetry in each plaquette
in the system~\cite{PhysRevB.78.115116}.
Using these conserved quantities,
we examine ground state and finite-temperature properties in the large clusters up to 24 sites.
We clarify that ground state has no degeneracy and
belongs to the subspace with zero-fluxes, which is consistent with
the semiclassical results~\cite{PhysRevB.78.115116}.
We find that a lowest excited state belongs to
the same subspace as the ground state and the gap monotonically decreases
with increasing system size, which suggests that
the ground state of the $S=1$ Kitaev model is gapless.
Furthermore, using the thermal pure quantum (TPQ) state methods,
we demonstrate that the multiple peak structure appears
in the specific heat, similar to the $S=1/2$ Kitaev model~\cite{Nasu2015}.
Thermodynamic properties in the generic spin-$S$ Kitaev models with $S\le 2$
are also addressed.
We consider the $S=1$ Kitaev model given by
\begin{align}
H=-J\sum_{\langle i,j \rangle_x}S_i^xS_j^x
-J\sum_{\langle i,j \rangle_y}S_i^yS_j^y
-J\sum_{\langle i,j \rangle_z}S_i^zS_j^z,\label{eq:H}
\end{align}
where $S_i^\alpha$ is an $\alpha(=x,y,z)$ component of an $S=1$ spin operator
at the $i$th site in the honeycomb lattice.
The ferromagnetic interaction $J(>0)$ is defined
on three different types of the nearest
neighbor bonds, $x$ (red), $y$ (blue), and $z$ bonds (green),
respectively [see Fig. \ref{fig:model}(a)].
\begin{figure}[htb]
\centering
\includegraphics[width=6.5cm]{model.pdf}
\caption{
(a) Kitaev model on the honeycomb lattice.
Red, blue, and green lines denote $x, y$, and $z$ bonds, respectively.
Clusters used in the exact diagonalization and TPQ methods are represented
by the bold lines.
(b) Plaquette with sites marked $1-6$ is shown for
the corresponding operator $W_p$ defined in Eq.~(\ref{eq:Wp}).
}
\label{fig:model}
\end{figure}
In this $S=1$ model, the basis sets $|\alpha\rangle\; (\alpha=x,y,z)$
are convenient~\cite{Tomishige}, which is explicitly given as
\begin{align}
|x\rangle &= -\frac{1}{\sqrt{2}}\left(|1\rangle -|-1\rangle \right),\\
|y\rangle &= \frac{i}{\sqrt{2}}\left(|1\rangle +|-1\rangle \right),\\
|z\rangle &= |0\rangle,
\end{align}
where $|m\rangle\; (m=-1, 0, 1)$ is an eigenstate of
the $z$-component of the spin operator $S^z$ with an eigenvalue $m$.
In this representation, we obtain the following relations as
\begin{align}
S^\alpha|\beta\rangle = i \epsilon_{\alpha\beta\gamma}|\gamma\rangle.
\label{eq:S}
\end{align}
Now, we consider the local Hamiltonian for the $x$ bond
between $i$ and $j$ sites, $H_{ij}^x=JS_i^xS_j^x$.
When a certain state $|\psi\rangle$ includes an $x$ state
in $i$th and/or $j$th sites,
$H_{ij}^x|\psi\rangle=0$ due to eq.~(\ref{eq:S}).
Then, nonzero components in the Hamiltonian $H_{ij}^x$ are explicitly given as
\begin{align}
H_{ij}^x|y_i z_j\rangle&=J|z_i y_j\rangle,\\
H_{ij}^x|z_i y_j\rangle&=J|y_i z_j\rangle,\\
H_{ij}^x|y_i y_j\rangle&=-J|z_i z_j\rangle,\\
H_{ij}^x|z_i z_j\rangle&=-J|y_i y_j\rangle.
\end{align}
We here note that the numbers of $y$ and $z$ states are not conserved,
but their parity is conserved.
When one considers all Kitaev interactions
in the system,
there exist global parity symmetries
for the number of $\alpha(=x,y,z)$ states.
In other word, the Kitaev Hamiltonian eq.~(\ref{eq:H}) commutes
the parity operators for $\alpha$ states as
\begin{align}
P_\alpha&=e^{i\pi n_\alpha}
=\exp\left[i\pi \sum_i \left(S_i^\alpha\right)^2\right]
\end{align}
where $n_\alpha(=\sum_i [1-(S_i^\alpha)^2 ])$ is the number operator
for $\alpha$ states, and
$P_x, P_y$, and $P_z$ commute with each other.
In addition, there exists a local conserved quantity
defined at each plaquette $p$ composed of the sites labeled as
$1, 2, \cdots, 6$ [see Fig.~\ref{fig:model}(b)]~\cite{PhysRevB.78.115116}:
\begin{align}
W_p&=\exp\Big[i\pi \left(S_1^x+S_2^y+S_3^z+S_4^x+S_5^y+S_6^z\right)\Big].
\label{eq:Wp}
\end{align}
This operator commutes with not only $W_{p'}$ on another plaquette $p'$ and
the Hamiltonian eq.~(\ref{eq:H}), but also the parity operators $P_\alpha$.
Then, the Hilbert space of the Hamiltonian can be classified into
each subspace specified by ${\cal S}=[\{p_\alpha\}; \{w_p\}]$,
where $p_\alpha(=\pm 1)$ and $w_p(=\pm 1)$ are eigenvalues of the operators
$P_\alpha$ and $W_p$.
This enables us to perform the numerical calculations for large clusters
in the Lanczos and TPQ state methods.
For example, the Hilbert space in the cluster with $N=24$ is divided into
2,048 subspaces, in which the maximum rank of
the block-diagonalized Hamiltonian is 140,279,025.
Applying the Lanczos and TPQ state methods to the systems
with periodic boundary conditions, we discuss ground state and
finite temperature properties.
In our study, we deal with the $S=1$ Kitaev model
on the clusters with $N=12, 18$, and 24,
which are shown in Fig.~\ref{fig:model}(a).
Calculating the lowest energy for each cluster,
we find that the ground state has no degeneracy and
always belongs to the subspace ${\cal S}_G=[\{p_\alpha=+1\}, \{w_p=+1\}]$,
corresponding to the zero-flux sector.
This is consistent with the results obtained from
the semiclassical approach~\cite{PhysRevB.78.115116}.
The ground state energy for each cluster is shown in Fig.~\ref{fig:ene}(a).
\begin{figure}[htb]
\includegraphics[width=9cm]{ene.pdf}
\caption{
(a) Ground state energy as a function of $1/N$.
(b) Excitation gap as a function of $1/N$.
Circles, triangles, and squares represent the gap between the ground state
and lowest excited state in the subspaces ${\cal S}_G, {\cal S}_P$, and ${\cal S}_O$,
respectively.
For convenience, we have plotted the results for $N=18a$ and $N=18b$
with tiny offsets.
}
\label{fig:ene}
\end{figure}
In the thermodynamic limit, the ground state energy appears to be given by
$E/N\sim -0.65J$
although the systematic finite size scaling
is necessary to be deduced quantitatively.
We also discuss the elementary excitations in the $S=1$ Kitaev model,
examining all subspaces for the clusters.
Figure~\ref{fig:ene}(b) shows lowest excitation energies in the subspaces
${\cal S}_G, {\cal S}_P=[\{p_\alpha=+1\}]-{\cal S}_G$
and ${\cal S}_O=[\{p_\alpha=+1\}]^c$,
where $[\{p_\alpha\}]$ is the subspace with eigenvalues ${p_\alpha}$,
and ${\cal S}^c$ means the complement space of ${\cal S}$.
The excited states strongly depend on the size and/or shape of clusters,
as shown in Fig.~\ref{fig:ene}(b).
Therefore, we believe that the largest cluster $N=24$ with isotropic geometry
should capture the nature of the excitations in the thermodynamic limit.
In the $N=24$ system, the lowest excited state
belongs to the subspace ${\cal S}_G$,
which is the same subspace for the ground state.
Furthermore, we find that
the excitation gap in the subspace ${\cal S}_G$
monotonically decreases with increasing the system size
and is much smaller than the exchange constant $J$.
These suggest that in the thermodynamic limit,
the $S=1$ Kitaev model has gapless excitations in the same subspace
${\cal S}_G$.
This is similar to the case with the $S=1/2$ Kitaev model,
where the gapless dispersion for Majorana fermions is obtained
in the subspace corresponding to the zero-flux sector~\cite{Kitaev2006}.
Next, we examine thermodynamic properties in the $S=1$ Kitaev model.
We calculate thermodynamic quantities for a small cluster with $N=12$,
obtaining all eigenenergies by means of the exact diagonalization method.
Furthermore, we make use of the TPQ states for larger clusters.
As for two kinds of clusters with $N=18$,
we evaluate thermodynamic quantities and expectation values
by means of hundred independent TPQ states.
By contrast, the numerical cost is expensive for the $N=24$ site cluster,
and only two TPQ states are treated.
The obtained results are shown in Fig.~\ref{fig:tpq}.
\begin{figure}[htb]
\includegraphics[width=9cm]{tpq.pdf}
\caption{
(a) Specific heat, (b) entropy, (c) the expectation values of
local conserved quantity and nearest-neighbor spin correlations
as a function of the temperature for the systems.
Shaded areas
are estimated
by using standard deviation of the results obtained from TPQ states.
}
\label{fig:tpq}
\end{figure}
We clearly find in Fig.~\ref{fig:tpq}(a),
the multiple peak structure in the specific heat $C$,
which is consistent with the previous work~\cite{SuzukiYamaji}.
Now, we focus on the peak structure around $T=T_H\sim 0.8J$.
Since the specific heat around $T_H$ is almost independent of the cluster size,
this peak structure appears in the thermodynamic limit.
On the other hand, low temperature structure is sensitive to the cluster size,
which originates from the fact that
low-energy excitations depend on the size and/or shape of clusters
[see Fig.~\ref{fig:ene}(b)].
Nevertheless, the peak appears around $T_L/J\sim 0.02$
except for the smallest cluster with $N=12$,
which suggests that the low temperature peak indeed appears
in the thermodynamic limit.
In addition to these features, the shoulder or peak structure appears
in lower energy region
$T/J\sim 0.009$, as shown in Fig.~\ref{fig:tpq}(a).
Since it strongly depends on the clusters,
further detailed analysis with larger clusters is necessary
to clarify whether or not such low-temperature behavior appears in the thermodynamic limit.
We also calculate the entropy $S$, and
the expectation values of the local conserved quantity
$W$, and spin-spin correlations $C_s$, which are defined as
\begin{align}
W&=\frac{1}{N_p}\sum_p \langle W_p \rangle,\\
C_s&=\frac{1}{N_B}\sum_{\langle i,j\rangle_\alpha} \langle S_i^\alpha S_j^\alpha\rangle,
\end{align}
where $N_p(=N/2)$ is the number of plaquettes, and
$N_B(=3N/2)$ is the number of bonds.
Note that $C_s$ is proportional to the energy as $C_s=-E/(JN_B)$.
The results are shown in Figs.~\ref{fig:tpq}(b) and (c).
It is found that,
decreasing temperatures,
nearest-neighbor spin-spin correlations develop around $T=T_H$ and
the plateau behavior appears in the curve of the entropy
at $S/N\sim 1/2\log 3$ around $T/J\sim 0.1$.
On the other hand, the expectation value $W$ changes rapidly
at lower temperature $T_L/J\sim 0.02$.
Therefore, the peaks in the specific heat at $T=T_H$ and $T_L$
should be related to spin-spin correlations and $Z_2$ local conserved quantity.
From the above discussions, we have confirmed that, in the $S=1$ Kitaev model,
at least, double peak structure appears in the specific heat.
This originates from two distinct temperature dependences
for nearest-neighbor spin-spin correlations and the local conserved quantity,
which are essentially the same as those
in the $S=1/2$ Kitaev model~\cite{Nasu2015}.
Then, similar finite-temperature behavior is naively expected
even in larger spin $S$ models
since they have the local conserved quantity~\cite{PhysRevB.78.115116}.
To clarify this, we also examine finite-temperature properties for
the Kitaev models with $S=3/2$ and $S=2$,
using the TPQ states for the clusters with $N=16$ and $N=12$, respectively.
First, we have compared the results for the $N=12$ and $N=16$ clusters
in the spin $S=3/2$ system, and have confirmed that
the higher temperature peak is not changed in the specific heat.
Therefore, we believe that
the $N=12$ site calculations can capture higher temperature peak
in the thermodynamic limit.
Meanwhile, the characteristic temperature for low temperature peak
in the specific heat
depends on the system size,
which is similar to the $S=1$ case (see Fig.~\ref{fig:tpq}).
\begin{figure}[htb]
\includegraphics[width=8cm]{raw.pdf}
\caption{
(a) Specific heat and (b) normalized entropy
as a function of the temperature for the Kitaev model with spin $S\le 2$,
where $S_\infty=N\log (2S+1)$.
Shaded areas are estimated by using standard deviation of the results
obtained from initial random states.
Data with error bars for $S=1/2$ Kitaev model has been obtained
by the Monte Carlo simulations with $N=800$ sites~\cite{Nasu2015,Nasu2017}.
Dashed lines represent the high-temperature behavior $C= (T^*/T)^2$,
where $T^*=JS(S+1)/\sqrt{6}$.
Inset of the figure (a) shows the spin dependence of
the characteristic temperatures $T_H$ and $T^*$
and the square of its specific heat $C$.
Inset of the figure (b) shows the entropy as a function of $T/(JS)$.
}
\label{fig:tpqS}
\end{figure}
Figure~\ref{fig:tpqS} shows the specific heat and entropy
in the Kitaev models with $S\le 2$.
Namely, the $S=1/2$ results are obtained by the Monte Carlo method
for $N=800$ sites~\cite{Nasu2015,Nasu2017}.
The results for $S=1, 3/2$ and $2$ are obtained by the TPQ state methods
for $N=24$, $16$ and $12$ clusters, respectively.
We find multiple peak structures in the specific heat
for the spin-$S$ Kitaev model in Fig.~\ref{fig:tpqS}(a).
This is contrast to the previous work~\cite{SuzukiYamaji},
where a single peak is observed in the specific heat
for the $S>1$ cases in an 8-site cluster.
This should originate from the fact that
the system size treated in previous works is too small
to discuss finite temperature properties in the thermodynamic limit.
Now, we focus on the higher-temperature peak in the specific heat,
which can be described quantitatively in our methods.
It is found that the characteristic temperature and the height of the peak
monotonically increase with increasing the amplitude of spin $S$,
as shown in Fig.~\ref{fig:tpqS}(a).
To clarify the spin dependence,
we also plot its characteristic temperature $T_H$ and
the value of the specific heat at $T_H$
in the inset of Fig.~\ref{fig:tpqS}(a).
We numerically find that these quantities are almost scaled as
$T_H/J \propto S$ and $C(T_H)/N\propto \sqrt{S}$.
These are in contrast to the spin dependence of
the characteristic temperature $T^*$ derived from
the high-temperature expansion:
the specific heat is expanded as $C/N = (T^*/T)^2 + O(T^{-3})$
with $T^*=JS(S+1)/\sqrt{6}$.
The difference in the characteristic temperatures
implies that the fractionalization of spin occurs
in the spin-$S$ Kitaev models
as well as the $S=1/2$ Kitaev model~\cite{Kitaev2006}.
To confirm this, we calculate the temperature dependence of
the entropy, as shown in Fig.~\ref{fig:tpqS}(b).
We find that the plateau appears in the curve of the entropy at
$S/S_\infty\sim 1/2$,
suggesting that spin with the amplitude $S$ is fractionalized
into two halves with distinct energy scales.
This is corroborated by the fact that the temperature of normalized entropy
$S/S_\infty$ is well scaled by $JS$,
as shown in the inset of Fig.~\ref{fig:tpqS}(b).
We naively expect that the decrease of temperatures induces
another peak in the specific heat, where the residual entropy is released.
Unfortunately, the low-temperature peak could not be discussed quantitatively
since the clusters treated in the present study are not enough large
to describe low energy excitations correctly.
Therefore, the systematic analysis with larger clusters are needed
to clarify whether peak and/or shoulder structures appears or not
in the lower temperature region,
which is left for a future work.
In summary, we have studied an $S=1$ Kitaev model
to discuss ground-state and finite temperature properties.
We have demonstrated the existence of parity symmetry as well as
the local conserved quantity, which enables us
to perform the calculations in large clusters up to twenty-four sites.
We have found that the ground state is singlet and
its energy is estimated as $E/N\sim -0.65$.
In addition, a lowest excited state belongs to
the same subspace as the ground state and the gap monotonically decreases
with increasing system size, which suggests that
the ground state of the $S=1$ Kitaev model is gapless.
By means of the TPQ states,
we have examined finite temperature properties and clarified that
the multiple peak structure in the specific heat appears in the Kitaev models
with spin $S\le 2$.
This should be a common feature of the general $S$ Kitaev model,
which might originate from the presence of a local conserved quantity
on each plaquette in the lattice.
This work is supported by Grant-in-Aid for Scientific Research from
JSPS, KAKENHI Grant Number JP17K05536, JP16H01066 (A.K.) and
JP16K17747, JP16H02206, JP16H00987 (J.N.).
Parts of the numerical calculations were performed
in the supercomputing systems in ISSP, the University of Tokyo.
\bibliographystyle{jpsj}
|
1,108,101,563,232 | arxiv | \section{Introduction}
Ultrathin films are essential for many applications in nanoscale science and technology. Considerable scientific interest is devoted to thin oxide films at metal surfaces \cite{KrScNaShKoVa05,RoReSc07,FrRiSc07}, for which applications are found in catalysis, semiconductor devices as well as mechanical wear and corrosion protection. Correctly assessing the thermodynamic stability of such films is an important issue \cite{Ca06}. New thin film surface alloys have been discovered and the stability of nano-sized patterns in the ultrathin films investigated with respect to strain and chemical effects \cite{ThOzScBaAsHoChHw01,OzAsHo02}. Formation of ultrathin films is also observed in metal-ceramic materials after sintering of nanoparticles into a bulk nanocrystalline material \cite{LaLoDo04}.
In this Letter, we consider an interface between two dissimilar materials, a metal-ceramic interface, and the propensity to form, at the interface, an ultrathin film of a secondary phase, whose corresponding bulk phase is thermodynamically unstable. We develop a model for the thin film formation and stability as function of temperature and show that it can be stabilized by interfacial effects.
For modeling of interfaces with complex bonding such as metal-ceramic interfaces, first-principles calculations in the framework of density functional theory (DFT) has proved to be a very useful tool \cite{Fi96,DuHaLu00,ChWaAlLa05}. However, phase stabilities at high temperatures are cumbersome to predict with sufficient accuracy using DFT \cite{TuAbBuFrGrKaKoMaOhPiScZh07} and therefore we combine DFT calculations with driving forces of nucleation from well-developed thermodynamic modeling techniques \cite{SuJaAn85}. These modeling techniques describe bulk phases but lack information on interfacial effects.
We apply the developed methodology to WC-Co cemented carbides. These materials are produced by means of powder metallurgy, where powders of carbides and metal Co are sintered together into a hard and dense material with excellent mechanical properties. The materials are used in cutting and wear resistant tools and are of considerable industrial importance. Much effort has been invested in developing nanostructured cemented carbides by reducing the mean WC grain size in the finished product, because a fine and homogeneous microstructure has superior mechanical properties \cite{FaWa08}. To control the microstructure and retain a fine grain size throughout the sintering, dopants in the form of VC, $\text{Cr}_3\text{C}_2$, TiC, TaC or NbC powders are added, of which VC has proved to be the most effective inhibitor in ultrafine alloys \cite{ScBoLu95}. The grain growth inhibiting effect of the dopants is well established, whereas the inhibiting mechanism on the atomic scale is not understood. Experimental high-resolution electron microscopy (HREM) studies \cite{JaYaIkSaTaOkTa98,YaIkSa00,YaIkWaSaTaOkTa01,LaHaThLa02,LaThHaTh03} have shown the existence of V- and Cr-rich films of cubic carbide structure at WC/Co interfaces in VC- and $\text{Cr}_3\text{C}_2$-doped materials, respectively, at low temperatures, after liquid phase sintering and cooling. In order to predict the possible growth inhibiting effect of such films, it is decisive to determine whether these can exist at high temperature liquid phase sintering conditions, where most of the grain growth occurs and where HREM cannot be applied.
\section{Model}
\subsection{General description}
The interface energy $\gamma$ can be expressed as the excess Gibbs energy per unit interface area due to the presence of an interface \cite{SuBa96},
\begin{equation}
\label{eq:interfaceEnergy}
\gamma = \frac{1}{A} \left( G - \sum_i n_i \mu_i \right).
\end{equation}
$G$ is the Gibbs energy for the entire considered system, which contains an interface of area $A$. $n_i$ is the number of atoms in the system of component $i$ with a corresponding chemical potential $\mu_i$ determined by the reservoirs with which the system is in equilibrium. In the present case, the reservoirs are stoichiometric WC and a binder phase. The latter consists of mainly Co (and will be denoted ``Co'') but also of dissolved W, C, and M atoms, where M = V, Cr, Ti, etc. depending on the doping.
We consider the propensity to form a thin film consisting of $N$ layers of carbide phase, here denoted MC, at the interface between WC and the binder phase. For sufficiently thick films, the interface can be regarded as split into a WC/MC interface and a MC/Co interface separated by a slab of MC. The energy for such a film-covered interface can then be decomposed as
\begin{equation}
\label{eq:gamma_film}
\gamma_\text{film} = \gamma_\text{WC/MC} + \gamma_\text{MC/Co} + N \left( \Delta \bar{g}_\text{MC} + \bar{e}_\text{MC} \right),
\end{equation}
where $\gamma_\text{WC/MC}$ ($\gamma_\text{MC/Co}$) is the interface energy between WC and MC (MC and Co). The last term of Eq.~(\ref{eq:gamma_film}) represents the energetic cost per layer of building the MC phase and, thus, increases linearly with the number of layers $N$ of the film. For the corresponding energy per formula unit we use the notation $\Delta g_\text{MC} = S \Delta \bar{g}_\text{MC}$, where $S$ is the interface area of one formula unit. The energy $\Delta g_\text{MC}$ is the negative driving force of nucleation. Assuming a stoichiometric carbide phase, it equals
\begin{equation}
\Delta g_\text{MC} = g_\text{MC} - \mu_\text{M} - \mu_\text{C},
\end{equation}
where $g_\text{MC}$ is the Gibbs energy per formula unit of MC and $\mu_\text{M}$ ($\mu_\text{C}$) is the chemical potential of M (C). A positive value of $\Delta g_\text{MC}$ thus implies that the MC phase is unstable. We consider coherently strained thin films and the last quantity of Eq.~(\ref{eq:gamma_film}), $\bar{e}_\text{MC}$, is the strain energy per layer needed to bring the film into epitaxy with the underlying WC.
The energy $\gamma_\text{film}$ should be compared with the energy $\gamma_\text{WC/Co}$ of a WC/Co interface not covered by any MC film. The film is stable if $\gamma_\text{film} < \gamma_\text{WC/Co}$ and, hence, a positive value of
\begin{equation}
\label{eq:deltaGamma}
\Delta \gamma_\text{MC} = \gamma_\text{WC/Co} - \left( \gamma_\text{WC/MC} + \gamma_\text{MC/Co} \right)
\end{equation}
indicates that film formation is possible. An upper limit of the film thickness $N_\text{limit}$ is found where the energy of the film-covered interface matches the energy of the uncovered interface, $\gamma_\text{film}=\gamma_\text{WC/Co}$, which yields
\begin{equation}
\label{eq:N_limit}
N_\text{limit} = \frac{\Delta \gamma_\text{MC}}{\Delta \bar{g}_\text{MC} + \bar{e}_\text{MC}}.
\end{equation}
If $N_\text{limit}$ is larger than a few atomic layers, the interface may contain a thin film.
The decomposition of $\gamma_\text{film}$ in Eq.~(\ref{eq:gamma_film}) is valid only if the film is sufficiently thick, so that the interior of the film is bulk-like and the interface energies are well-defined. For thin films, $\gamma_\text{film}$ will deviate from the linear dependence on $N$, which implies that $\gamma_\text{film}$ can have a minimum as function of $N$ for a finite film thickness. In equilibrium, this optimal film thickness $N_\text{eq}$ will be attained. A schematic illustration of $\gamma_\text{film}$ is given in Figure~\ref{fig:gamma_film_sch}.
\begin{figure}
\begin{center}
\includegraphics{fig1}
\end{center}
\caption{\label{fig:gamma_film_sch}A schematic illustration of $\gamma_\text{film}$ as function of film thickness $N$. The circles denote the energy in an atomistic treatment and the solid line follows the decomposition in Eq.~(\ref{eq:gamma_film}).}
\end{figure}
\subsection{Specific model and method}
In the present paper we concentrate on VC-doped WC-Co. Results for other MC dopants will be presented elsewhere \cite{JoWa}. WC has a simple hexagonal lattice structure. After liquid-phase sintering in Co, the WC grains are predominantly delimited by their basal $\left( 0001 \right)$ and prismatic $\left(1 0 \bar{1} 0 \right)$ surfaces \cite{Ex79}. Here, we focus on the basal surface, which can be either W- or C-terminated. We choose the WC surface and the cubic VC phase to be oriented according to
\begin{equation}
\label{eq:orientation}
\left( 0001 \right)_\text{WC} \parallel \left( 111 \right)_\text{VC}, \left[ \bar{1} 2 \bar{1} 0 \right]_\text{WC} \parallel \left[ \bar{1} 1 0 \right]_\text{VC}.
\end{equation}
This orientation relationship between WC and MC has been experimentally established for both V- and Cr-doped WC-Co \cite{LaHaThLa02,DeBaPaLaAl04}.
Interface calculations are performed using DFT and the current computational methodology has been developed in a series of papers \cite{ChDuWa02,ChWa03,ChWa04}. Computational results have been compared with experimental data \cite{ChDuWa02,ChWa03} and the use of the generalized gradient approximation (GGA) for the exchange-correlation effects within DFT in interface calculations has been motivated \cite{ChWa04}. Further computational details can be found in \cite{onlinesupp09}.
To model the interfaces, we use standard supercell slab geometries, where the interfaces are made coherent by straining the VC and Co phases to match the WC basal surface. The associated strain energies are minimized by relaxing the interplanar distances to account for the Poisson effect. In the calculation of interface energies, the strain energies are canceled, so that $\gamma_\text{WC/VC}$, $\gamma_\text{VC/Co}$, and $\gamma_\text{WC/Co}$ represent the chemical contribution to the interface energy and do not depend on system size.
To define an absolute value of the interface energy, a definite value of the carbon chemical potential $\mu_\text{C}$ is required. Within the two-phase region (WC+Co) of the W-C-Co system, $\mu_\text{C}$ can be varied in a small interval limited by the formation of either graphite or a metal-carbide phase $\text{(W,Co)}_6\text{C}$ \cite{FrMa08}. We assume carbon to be in equilibrium with graphite and use the corresponding computed value for $\mu_\text{C}$ \cite{ChWaAlLa05}.
\section{Results and discussion}
\subsection{Interface energies}
In the $\left \langle 0001 \right \rangle_\text{WC}$ and $\left \langle 111 \right \rangle_\text{VC}$ directions, the WC and VC phases consist of alternating layers of metal and carbon atoms. Assigning the positions $A$ to $(0,0)$, $B$ to $(\frac{1}{3}, \frac{2}{3})$ and $C$ to $(\frac{2}{3}, \frac{1}{3})$ with respect to hexagonal lattice vectors in the $(0001)_\text{WC}$ plane, the stacking sequence for WC is $\ldots \text{W}_A \text{C}_B \text{W}_A \text{C}_B \ldots$ and for VC $\ldots \text{V}_A \text{C}_C \text{V}_B \text{C}_A \text{V}_C \text{C}_B \ldots$ along $\left [ 0001 \right ]_\text{WC}$ and $\left [ 111 \right ]_\text{VC}$, respectively. We have calculated interface energies for all possible stacking sequences involving the $A$, $B$ and $C$ positions and varying atomic terminations of the phases. Only stoichiometric VC compounds are considered. To assign a value of $\gamma_\text{WC/VC}$, we pick the lowest energy among the different stackings and terminations. The resulting interface energy is found to be slightly negative, $\gamma_\text{WC/VC} = -0.03 \ \text{J/m}^2$ \cite{onlinesupp09}.
The predicted WC/VC interface structure can be compared with experimental findings. By comparing experimental and simulated HREM images of a WC(0001)/VC(111) interface in a VC-doped WC-Co, a particular interfacial stacking has been suggested \cite{LaHaThLo04}. We find that the lowest energy is obtained for $\ldots \text{C}_B \text{W}_A \text{C}_B \text{V}_A \text{C}_C \text{V}_B \text{C}_A \ldots$ \cite{onlinesupp09} which is in complete agreement with the stacking suggested in Ref.~\cite{LaHaThLo04}.
The WC/Co and VC/Co interfaces need to be treated with more concern, since at relevant sintering temperatures, the Co binder is in its liquid phase. A solid-liquid interface energy is usually considered as stemming from two separate parts (see, e.g. \cite{Wa80}): a chemical contribution due to differing atomic bonding at the interface, and a structural contribution due to disturbance of the liquid's structure caused by the interface. One should notice, that only the difference between the two interface energies $\gamma_\text{WC/Co}$ and $\gamma_\text{VC/Co}$ enters into the expression (\ref{eq:deltaGamma}) for the key quantity $\Delta \gamma_\text{VC}$. Therefore, the geometrical similarity of the WC(0001) and VC(111) surfaces implies that structural contributions to $\Delta \gamma_\text{VC}$ will to large extent cancel.
We choose to model the WC/Co and VC/Co interfaces in the same geometry as the WC/VC interface, which means that we are effectively modeling WC(0001)/Co(111) and VC(111)/Co(111) interfaces with different Co stackings. Following the preceding argument, we do not expect the particular choice of Co orientation to influence $\Delta \gamma_\text{VC}$ in any decisive way, although existing experimental observations of this orientation between WC and solid Co \cite{MoSt96_1,BoLaLoMi08} do motivate its use as model geometry. At high temperatures, the motion of liquid Co atoms will constantly change the atomic structure at the interface. Therefore, we use a mean value of energies for different stackings of Co in the interface to approximate $\gamma_\text{WC/Co}$ and $\gamma_\text{VC/Co}$, respectively. In Figure~\ref{fig:interface} we depict the six different atomic arrangements of Co used in calculating the mean energy of the carbide/Co interface. The resulting value for the interface energy can be regarded as a good approximation for an incoherent interface \cite{ChWaLaAl07}. The termination of the WC or VC slabs (metallic or C) is consistently chosen to yield the lowest mean interface energy.
\begin{figure}
\begin{center}
\includegraphics{fig2}
\end{center}
\caption{\label{fig:interface}The optimal atomic structure of a WC/VC/Co interface containing two layers of stoichiometric VC. Top: Atomic positions projected onto a plane with normal $[\bar{1} 2 \bar{1} 0]_\text{WC}$. Bottom: The positions of the V, C and Co atoms closest to the interface projected onto the interface plane. The six depicted atomic configurations of Co are used to calculate a mean value of the carbide/Co interface energy for the thin film configuration. The same set of relative translations and rotations of Co with respect to the topmost carbide layer is also used in calculating WC/Co and VC/Co interface energies.}
\end{figure}
We find that for WC/Co, the interface is C-terminated with an energy $\gamma_\text{WC/Co} = 1.01 \ \text{J/m}^2$ \cite{onlinesupp09}. For VC/Co, the interface is V-terminated with $\gamma_\text{VC/Co} = 0.00 \ \text{J/m}^2$ \cite{onlinesupp09}. The resulting difference in interface energy is $\Delta \gamma_\text{VC} = 1.03 \ \text{J/m}^2$, which we henceforth will assume to be temperature independent.
The final result of $\Delta \gamma_\text{VC}$ does not strongly depend on the choice of averaging of the various Co stackings in the interfaces. For instance, by following the above procedure but disregarding the two most high-energetic configurations in each interface, $\Delta \gamma_\text{VC}$ increases by $0.14 \ \text{J/m}^2$. The impact on $\Delta \gamma_\text{VC}$ due to ordering of the liquid Co atoms in the vicinity of the WC or VC interfaces is therefore expected to be maximally of this size. Temperature-dependent deviations from our estimation of $\Delta \gamma_\text{VC}$ may arise from differences in the vibrational spectrum of V, W, and C atoms at the interface as compared with their respective reference states. Following the procedure of Reuter and Scheffler \cite{ReSc01}, we estimate this effect to be maximally $\pm 0.3 \ \text{J/m}^2$. Considering both these effects, we expect $\Delta \gamma_\text{VC}$ to be determined within an accuracy of $\pm 0.35 \ \text{J/m}^2$. By accounting for a mixed carbide layer instead of the stoichiometric VC compounds considered so far, further entropic effects are introduced (see below).
\subsection{Thin film energy and composition}
In addition to $\Delta \gamma_\text{VC}$, the energetic costs, $\Delta \bar{g}_\text{VC}$ and $\bar{e}_\text{VC}$, to build the VC thin films are needed to predict stability. To compress VC from its calculated equilibrium lattice parameter in the interface, $a_\text{VC}/\sqrt{2} = 2.944 \ \text{\r{A}}$, to the calculated WC lattice parameter, $a_\text{WC} = 2.919 \ \text{\r{A}}$, an energy of only $\bar{e}_\text{VC} = 0.011 \ \text{J/m}^2$ ($0.005 \ \text{eV}$ per VC formula unit) is required.
The driving force of nucleation, $- \Delta g_\text{VC}$, depends sensitively on the temperature and composition of the system. The most stable cubic carbide is not stoichiometric VC, but a mixed $\text{(V,W,Co)C}_x$ phase. In its at $1593 \ \text{K}$ experimentally measured composition, V constitute 75\% and W 23\% of the metallic atoms and the C vacancy concentration is 11\% \cite{FrMa08}. To determine the driving force of nucleation we have used the Thermocalc software \cite{SuJaAn85} with a database for Cr and V additions in the W-C-Co system \cite{FrMa08}. The database has been assessed from experimental observations using the Calphad method and models temperature dependent Gibbs energies of the phases occurring in the W-C-Co system.
In Figure~\ref{fig:VC_Deltag}, the negative driving force of nucleation $\Delta g_{\text{(V,W,Co)C}_x}$ for a $\text{(V,W,Co)C}_x$ phase of optimal composition with respect to Gibbs energy in a W-C-Co-V system in equilibrium with WC and graphite is given as a function of temperature for varying V/Co atomic ratio $y_i$. For low temperatures, $\Delta g_{\text{(V,W,Co)C}_x} = 0$ for all considered doping levels, meaning that stable $\text{(V,W,Co)C}_x$ precipitates exist at these chemical conditions. The sharp increase of $\Delta g_{\text{(V,W,Co)C}_x}$ at temperatures around 1550~K is due to the melting of the binder phase, which significantly increases its solubility of V. At the doping levels $y_2$ and $y_3$ of Figure~\ref{fig:VC_Deltag}, the thermodynamic calculation (which lacks any model for interface stabilization) predicts that all $\text{(V,W,Co)C}_x$ phase would dissolve as the binder melts. A discontinuous change in the slope of $\Delta g_{\text{(V,W,Co)C}_x}$ is also found near $1300 \ \text{K}$ for $y_1$. This change is associated with the transition of the solid Co binder from its ferro- to paramagnetic phase. We conclude that for relevant doping concentrations, $\Delta g_{\text{(V,W,Co)C}_x}$ varies between 0 and 0.1~eV (corresponding to 0 and $0.22 \ \text{J/m}^2$ per layer of $\text{(V,W,Co)C}_x$).
\begin{figure}
\begin{center}
\includegraphics{fig3}
\end{center}
\caption{\label{fig:VC_Deltag}$\Delta g_{\text{(V,W,Co)C}_x}$ as function of temperature $T$ in a W-C-Co-V system in equilibrium with WC and graphite. $y_i$ denotes the V/Co atomic ratio and $y_1 = 6.4 \cdot 10^{-3}$, $y_2 = 2.6 \cdot 10^{-2}$, $y_3 = 5.1 \cdot 10^{-2}$ and $y_4 = 7.7 \cdot 10^{-2}$.}
\end{figure}
We can now summarize our results for the propensity of thin film formation. In Figure~\ref{fig:VC_stab}, the solid line corresponds to the result from Eq.~(\ref{eq:gamma_film}) for VC. Here, we set $\Delta g_\text{VC} = 0.06 \ \text{eV}$, which is a reasonable choice based on the results from our thermodynamic modeling at relevant sintering temperatures. With this value of $\Delta g_\text{VC}$, Eq.~(\ref{eq:N_limit}) suggests that stable films of thickness less than $N_\text{limit} = 7$ may exist.
In the interface calculations, we have approximated the film carbide phase with a stoichiometric VC phase. The real film may well have a composition similar to the bulk $\text{(V,W,Co)C}_x$ phase, though finding the optimal composition would require numerous large interface calculations. To assess the effect of composition, we have performed a calculation for a mixed $(\text{V}_{0.75},\text{W}_{0.25}) \text{C}$ phase, whose composition is close to the experimentally measured one of bulk $\text{(V,W,Co)C}_x$ \cite{FrMa08}. We obtain $\gamma_{(\text{V}_{0.75},\text{W}_{0.25})\text{C}/\text{Co}} = -0.15 \ \text{J/m}^2 < \gamma_\text{VC/Co}$ \cite{onlinesupp09} and, therefore, we expect that for the true composition, the interfacial effect will further stabilize the thin film.
\subsection{Explicit atomistic layer-by-layer treatment}
The expression for $\gamma_\text{film}$ in Eq.~(\ref{eq:gamma_film}) is only valid if the film is sufficiently thick. For thinner films, the details of the atomic layer-by-layer structure has to be taken into account. We have therefore explicitly modeled the atomic layer-by-layer structure of thin films of cubic VC at the WC/Co interface. The number of possible stacking sequences of VC in the WC/VC/Co interface is large, but can be decreased by omitting obviously high-energetic configurations. In total, we have considered 76 different VC stacking sequences with six corresponding Co stackings for each of these \cite{onlinesupp09}.
In Figure~\ref{fig:VC_conv}, the energy of a WC/Co interface containing a VC film is given as a function of the number of metallic V layers in the film for different VC stacking sequences. The results for the stackings corresponding to the minimal energy in the WC/VC interface described earlier are circled. In the plot, we show the result without adding any contribution from the energetic cost per layer to build the thin film ($\Delta \bar{g}_\text{VC} + \bar{e}_\text{VC} = 0$). The result should then converge to $\gamma_\text{WC/VC} + \gamma_\text{VC/Co}$ for sufficiently thick films. We find that the result is close to the asymptotic value already for two V layers, while the result for one V layer deviates both energetically and structurally with $\ldots \text{C}_B \text{W}_A \text{C}_B \text{V}_C \text{Co} \ldots$ being the optimal stacking in this case. The optimal structure in the case of two V layers is depicted in Figure~\ref{fig:interface}.
\begin{figure}
\begin{center}
\includegraphics{fig4}
\end{center}
\caption{\label{fig:VC_conv}Interface energy $\gamma_\text{film}$ as a function of the number of V layers $N$ in a cubic VC film in the WC/Co interface for different stackings of VC. + (x) denotes a C-terminated (metal-terminated) carbide/Co interface. The stacking sequence that corresponds to the minimal WC/VC interface energy is circled. No energetic cost per layer to build the thin film is added ($\Delta \bar{g}_\text{VC} + \bar{e}_\text{VC} = 0$).}
\end{figure}
Our final results are presented as circles in Figure~\ref{fig:VC_stab}, where we again set $\Delta g_\text{VC} = 0.06 \ \text{eV}$. To calculate $\gamma_\text{film}(N)$, we take for each number of V layers $N$ the minimum energy among the stackings and terminations presented in Figure~\ref{fig:VC_conv} and add $N \left( \Delta \bar{g}_\text{VC} + \bar{e}_\text{VC} \right)$. The fast convergence of the interface energies toward the asymptotic value of Eq.~(\ref{eq:gamma_film}) implies that a thin film with two V layers will be the most stable configuration for a rather wide interval of $\Delta g_\text{VC}$, $0.01 \ \text{eV} \lesssim \Delta g_\text{VC} \lesssim 0.2 \ \text{eV}$. For no value of $\Delta g_\text{VC}$ will one V layer be the most stable configuration. With the values used in Figure~\ref{fig:VC_stab}, we find that the interface with a VC film is favored by $0.68 \ \text{J/m}^2$ compared to the interface without segregated VC. This value lies well outside our estimated error bound of $\pm 0.35 \ \text{J/m}^2$ indicating that in equilibrium, the WC(0001) surface is indeed covered by VC.
\begin{figure}
\begin{center}
\includegraphics{fig5}
\end{center}
\caption{\label{fig:VC_stab}Interface energy $\gamma_\text{film}$ as a function of the number of V layers $N$ in a cubic VC film in the WC/Co interface. $\Delta g_\text{VC}$ is set to $0.06 \ \text{eV}$ per formula unit, which is an appropriate value for relevant doping conditions and sintering temperatures. A thin film of two V layers will be the most stable configuration for a rather wide interval of $\Delta g_\text{VC}$, $0.01 \ \text{eV} \lesssim \Delta g_\text{VC} \lesssim 0.2 \ \text{eV}$.}
\end{figure}
Our prediction of a few stable V layers agrees very well with HREM imaging \cite{LaHaThLa02,LaThHaTh03,LaHaThLo04,LaLoDo04}, where thin films on WC(0001) in VC-doped material are seen to have a thickness of two and sometimes a few additional metallic layers. An interpretation of the additional layers is that they precipitate as $\Delta g_\text{VC} \rightarrow 0$ during cooling. The VC-covered surface is also a likely nucleation site for further growth of the VC phase, which sometimes builds up as a precipitate on WC(0001) during cooling \cite{LaHaThLa02,LaHaThLo04}.
\subsection{Grain growth inhibiting effect}
It is generally believed that the rate limiting step of grain growth in WC-Co systems is the dissolution and/or reprecipitation reactions occuring at WC/Co interfaces. Several authors have therefore attributed the growth inhibiting effect of V to a proposed interference with these reactions either through blocking active growth centers at the WC surface \cite{BoScLu92,KaTeHa04} or through the presence of segregated V at the WC surface \cite{JaYaIkSaTaOkTa98,LaThHaTh03}, which would hinder W diffusion into or out of the WC grain. Our prediction that ultrathin VC films may be present and stable also at liquid-phase sintering temperatures is supportive of the latter explanation.
\section{Conclusions}
We have developed an atomistic model for understanding thin film formation at interfaces in bulk composite materials. We combine extensive density functional theory (DFT) calculations for interfaces and thin films with thermodynamic modeling techniques for multicomponent systems. The latter technique is used to obtain relevant chemical potentials for the various components as function of temperature and doping conditions.
The theory is applied to thin film formation in VC-doped WC-Co cemented carbides. It is predicted that under realistic doping conditions, ultrathin films may exist at high temperature liquid phase sintering conditions where most of the grain growth occurs.
By explicit DFT modeling of the thin films as function of the number of atomic layers, we find that the most stable configuration consists of only two V layers over a large temperature range and we show how the atomistic picture approaches the thick film limit with increasing number of layers.
The films, whose corresponding bulk phase is thermodynamically unstable, are stabilized by interfacial effects. Compared with the original interface in the absence of a thin film, there is an energy gain associated with the creation of the two sub-interfaces involved in the thin film formation, which more than compensates for the energetic cost of the thin film itself.
In conclusion, our results explain the formation of VC films in WC/Co interfaces. The presented method can be applied to theoretically assess the stability of other carbide films in WC/Co interfaces.
\section*{Acknowledgments}
Financial support from the Swedish Research Council, Sandvik and Seco Tools is gratefully acknowledged. Computations have been performed on SNIC resources. We thank Susanne Norgren, Sandvik Tooling, for providing thermodynamic calculations.
\bibliographystyle{tPHL}
|
1,108,101,563,233 | arxiv | \section{The Afterglow Light Curve}
The afterglow of GRB 110918A was imaged for over 40 days after the trigger with GROND in the $g'r'i'z'JHK_{S}$ bands (outlined in Sect.~\ref{sec:Observations:subsec:Afterglow:subsubsec:GROND}). Utilising the deep observations of the host, the underlying contribution from the host galaxy was subtracted using
the High Order Transform of PSF and Template Subtraction package, HOTPANTS\footnote{\url{http://www.astro.washington.edu/users/becker/hotpants.html}} \texttt{v5.1.10b}. The resulting afterglow light curve can be seen in Fig.~\ref{fig:lc_grb}, the raw data can be found in Tables~\ref{tab:stars_grboptphot} and~\ref{tab:stars_grbnirphot} and the host subtracted data can be found in Tables~\ref{tab:stars_grbsuboptphot} and~\ref{tab:stars_grbsubnirphot}. The standard stars used in $g'r'i'z'$ for relative calibration can be found in Table~\ref{tab:stars_optref}.
\begin{figure}[h!]
\includegraphics[width=9.5cm]{imgs/sub_lc.pdf}
\caption{The afterglow light curve of GRB 110918A obtained with the 7 channel imager GROND (host subtracted).}
\label{fig:lc_grb}
\end{figure}
\begin{table*}
\centering
\caption{Optical reference stars.}
\begin{tabular}{l l l l l l}
R.A. & Dec. & $g'$ & $r'$ & $i'$ & $z'$ \\ \hline
(J2000) & (J2000) & $\mathrm{mag_{AB}}$ & $\mathrm{mag_{AB}}$ & $\mathrm{mag_{AB}}$ & $\mathrm{mag_{AB}}$ \\ \hline
02:10:16.65 & -27:06:22.7 & $19.85\pm0.03$ & $19.64\pm0.03$ & $19.53\pm0.04$ & $19.54\pm0.04$ \\
02:10:11.56 & -27:04:53.9 & $20.28\pm0.03$ & $19.86\pm0.03$ & $19.72\pm0.04$ & $19.72\pm0.05$ \\
02:10:13.25 & -27:07:04.0 & $19.82\pm0.03$ & $19.48\pm0.03$ & $19.30\pm0.03$ & $19.27\pm0.04$ \\
02:10:12.62 & -27:08:12.2 & $20.83\pm0.04$ & $19.37\pm0.03$ & $17.96\pm0.03$ & $17.27\pm0.03$ \\
\hline\hline
\end{tabular}
\label{tab:stars_optref}
\end{table*}
\begin{table*}
\centering
\caption{GROND photometric data $g'r'i'z'$.}
\begin{tabular}{l l l l l l}
$T_{mid}-T_{0}$ & Exposure & $g'$ & $r'$ & $i'$ & $z'$ \\ \hline
s & s & $\mathrm{mag_{AB}}$ & $\mathrm{mag_{AB}}$ & $\mathrm{mag_{AB}}$ & $\mathrm{mag_{AB}}$ \\ \hline
193892 & 691 & $20.43\pm0.04$ & $20.15\pm0.02$ & $19.92\pm0.04$ & $19.80\pm0.07$ \\
194311 & 1526 & $20.39\pm0.04$ & $20.16\pm0.03$ & $19.93\pm0.03$ & $19.73\pm0.08$ \\
194724 & 699 & $20.42\pm0.03$ & $20.18\pm0.02$ & $19.92\pm0.04$ & $19.80\pm0.06$ \\
204203 & 683 & $20.48\pm0.03$ & $20.25\pm0.02$ & $20.05\pm0.03$ & $19.87\pm0.06$ \\
204615 & 1508 & $20.48\pm0.03$ & $20.25\pm0.02$ & $20.02\pm0.03$ & $19.84\pm0.06$ \\
205025 & 689 & $20.51\pm0.03$ & $20.28\pm0.02$ & $20.02\pm0.03$ & $19.89\pm0.06$ \\
214526 & 693 & $20.56\pm0.03$ & $20.33\pm0.02$ & $20.09\pm0.04$ & $19.98\pm0.06$ \\
215017 & 1674 & $20.56\pm0.04$ & $20.33\pm0.02$ & $20.11\pm0.03$ & $19.90\pm0.07$ \\
215504 & 700 & $20.56\pm0.05$ & $20.32\pm0.03$ & $20.12\pm0.04$ & $19.91\pm0.07$ \\
290361 & 691 & $21.09\pm0.02$ & $20.81\pm0.02$ & $20.66\pm0.03$ & $20.40\pm0.06$ \\
290775 & 1519 & $21.11\pm0.03$ & $20.88\pm0.03$ & $20.65\pm0.04$ & $20.44\pm0.06$ \\
291188 & 693 & $21.09\pm0.03$ & $20.83\pm0.02$ & $20.62\pm0.03$ & $20.44\pm0.05$ \\
381456 & 1727 & $21.60\pm0.02$ & $21.30\pm0.02$ & $21.06\pm0.03$ & $20.96\pm0.05$ \\
553350 & 1732 & $22.19\pm0.04$ & $21.96\pm0.03$ & $21.59\pm0.06$ & $21.30\pm0.07$ \\
725709 & 1727 & $22.62\pm0.05$ & $22.46\pm0.05$ & $22.02\pm0.09$ & $21.54\pm0.10$ \\
981853 & 3455 & $22.89\pm0.06$ & $22.76\pm0.07$ & $22.02\pm0.09$ & $21.60\pm0.09$ \\
1507300 & 5327 & $23.09\pm0.07$ & $22.84\pm0.08$ & $22.05\pm0.07$ & $21.75\pm0.09$ \\
\hline\hline
\end{tabular}
\tablefoot{
All magnitudes have been corrected for Galactic foreground reddening. No correction has been made to subtract the flux contribution from the underlying host galaxy.
}
\label{tab:stars_grboptphot}
\end{table*}
\begin{table*}
\centering
\caption{GROND photometric data $JHK_{s}$.}
\begin{tabular}{l l l l l}
$T_{mid}-T_{0}$ & Exposure & $J$ & $H$ & $K_{s}$ \\ \hline
s & s & $\mathrm{mag_{AB}}$ & $\mathrm{mag_{AB}}$ & $\mathrm{mag_{AB}}$ \\ \hline
126393 & 82 & $18.93\pm0.08$ & $18.76\pm0.08$ & .... \\
194337 & 1579 & $19.40\pm0.09$ & $19.18\pm0.12$ & $18.84\pm0.15$ \\
204643 & 1560 & $19.50\pm0.08$ & $19.19\pm0.12$ & $18.90\pm0.16$ \\
215045 & 1726 & $19.58\pm0.10$ & $19.35\pm0.12$ & $18.84\pm0.15$ \\
290803 & 1571 & $19.78\pm0.09$ & $19.64\pm0.12$ & $18.90\pm0.16$ \\
381482 & 1773 & $20.21\pm0.10$ & $19.73\pm0.16$ & $19.03\pm0.16$ \\
553375 & 1779 & $20.53\pm0.14$ & $20.17\pm0.17$ & $19.25\pm0.19$ \\
725734 & 1774 & $20.82\pm0.16$ & $20.43\pm0.21$ & $19.64\pm0.19$ \\
981879 & 3455 & $21.08\pm0.15$ & $20.59\pm0.18$ & $19.56\pm0.28$ \\
1507330 & 5379 & $20.77\pm0.12$ & $20.65\pm0.16$ & $19.50\pm0.27$ \\
\hline\hline
\end{tabular}
\tablefoot{
All magnitudes have been corrected for Galactic foreground reddening. No correction has been made to subtract the flux contribution from the underlying host galaxy.
}
\label{tab:stars_grbnirphot}
\end{table*}
\begin{table*}
\centering
\caption{GROND host subtracted photometric data $g'r'i'z'$.}
\begin{tabular}{l l l l l l}
$T_{mid}-T_{0}$ & Exposure & $g'$ & $r'$ & $i'$ & $z'$ \\ \hline
193892 & 691 & $20.48\pm0.04$ & $20.14\pm0.02$ & $19.93\pm0.02$ & $19.87\pm0.03$ \\
194724 & 699 & $20.52\pm0.03$ & $20.08\pm0.03$ & $19.93\pm0.03$ & $19.86\pm0.04$ \\
204203 & 683 & $20.46\pm0.04$ & $20.16\pm0.02$ & $20.02\pm0.02$ & $19.88\pm0.05$ \\
205025 & 689 & $20.51\pm0.04$ & $20.26\pm0.02$ & $20.03\pm0.03$ & $19.91\pm0.04$ \\
214527 & 693 & $20.65\pm0.03$ & $20.28\pm0.02$ & $20.13\pm0.03$ & $20.05\pm0.03$ \\
215504 & 700 & $20.63\pm0.04$ & $20.30\pm0.02$ & $20.18\pm0.03$ & $20.14\pm0.04$ \\
290776 & 1519 & $20.99\pm0.04$ & $20.70\pm0.03$ & $20.52\pm0.03$ & $20.43\pm0.04$ \\
381457 & 1727 & $21.67\pm0.03$ & $21.31\pm0.02$ & $21.14\pm0.03$ & $21.05\pm0.04$ \\
553350 & 1732 & $22.43\pm0.04$ & $22.04\pm0.03$ & $21.89\pm0.04$ & $21.75\pm0.05$ \\
725709 & 1727 & $23.09\pm0.05$ & $22.65\pm0.04$ & $22.53\pm0.06$ & $22.51\pm0.06$ \\
981853 & 3455 & $23.78\pm0.06$ & $23.30\pm0.06$ & $23.31\pm0.10$ & $23.03\pm0.28$ \\
1507300 & 5327 & $>24.80$ & $>24.15$ & $>22.84$ & $>22.01$ \\
\hline\hline
\end{tabular}
\tablefoot{
All magnitudes have been corrected for Galactic foreground reddening.
}
\label{tab:stars_grbsuboptphot}
\end{table*}
\begin{table*}
\centering
\caption{GROND host subtracted photometric data $JHK_{s}$.}
\begin{tabular}{l l l l l}
$T_{mid}-T_{0}$ & Exposure & $J$ & $H$ & $K_{s}$ \\ \hline
s & s & $\mathrm{mag_{AB}}$ & $\mathrm{mag_{AB}}$ & $\mathrm{mag_{AB}}$ \\ \hline
126444 & 393 & $18.74\pm0.05$ & $18.47\pm0.07$ & .... \\
193921 & 745 & $19.42\pm0.06$ & $19.04\pm0.05$ & .... \\
194338 & 790 & $....$ & $....$ & $18.92\pm0.08$ \\
194751 & 752 & $19.54\pm0.06$ & $19.25\pm0.06$ & .... \\
204231 & 736 & $19.35\pm0.06$ & $19.39\pm0.06$ & .... \\
204643 & 780 & $....$ & $....$ & $19.05\pm0.07$ \\
205052 & 742 & $19.48\pm0.06$ & $19.29\pm0.06$ & .... \\
214555 & 746 & $19.59\pm0.06$ & $19.45\pm0.08$ & .... \\
215045 & 863 & $....$ & $....$ & $19.17\pm0.07$ \\
215531 & 753 & $19.49\pm0.06$ & $19.30\pm0.06$ & .... \\
290803 & 1571 & $20.12\pm0.08$ & $19.80\pm0.07$ & $>18.93$ \\
381482 & 1773 & $20.67\pm0.10$ & $20.22\pm0.08$ & $>19.34$ \\
553375 & 1779 & $>20.48$ & $>20.91$ & $>18.44$ \\
725734 & 1774 & $>20.71$ & $>20.55$ & $>18.91$ \\
981879 & 3455 & $>20.80$ & $>20.80$ & $>18.44$ \\
1507329 & 5379 & $>20.77$ & $>20.49$ & $>19.09$ \\
\hline\hline
\end{tabular}
\tablefoot{
All magnitudes have been corrected for Galactic foreground reddening.
}
\label{tab:stars_grbsubnirphot}
\end{table*}
\section{The Afterglow's Sight-Line Spectrum}
The spectra obtained with GMOS and OSIRIS (see Sect.~\ref{sec:Observations:subsec:HostGalaxy:subsubsec:Osiris}) reveal many absorption lines of gas along the line of sight toward the afterglow, specifically of the following species: \ion{Fe}{II} (2344,2374,2382,2586,2600), \ion{Mg}{II} (2803,2796), \ion{Mg}{I} (2853) and \ion{Ca}{II} (3935,3970). The equivalent widths of the metals are listed in Table~\ref{tab:spectra_equivalentwidths}.
\begin{table*}
\center
\caption{Equivalent widths measured for the absorption lines of the afterglow.}
\begin{tabular}{l l l l l l l} \\ \hline
$\lambda_{\mathrm{obs}}$ & Feature & Contaminants & $\mathrm{EW}_{\mathrm{obs}}$ & $\mathrm{EW}_{\mathrm{rest}}$ & & $z$\\ \hline
\AA & & & & \\ \hline
$4650.4$ & \ion{Fe}{II2344.2} & \ion{Fe}{$\mathrm{II^{*}}$2345.0} & $4.3\pm0.4$ & $2.2\pm0.2$ & $0.9838$ \\
\hline
$4713.4$ & \ion{Fe}{II2374.5}\tablefootmark{\dag} & $....$ & $....$ & $....$ & $0.9850$ \\
$4720.5$ & $....$ & \ion{Fe}{$\mathrm{II^{*}}$2381.5} & $8.4\pm0.6$ & $4.2\pm0.3$ & $....$ \\
$4727.2$ & \ion{Fe}{II2382.8}\tablefootmark{\dag} & $....$ & $....$ & $....$ & $0.9389$ \\
\hline
$5132.7$ & \ion{Fe}{II2586.7} & \ion{Mn}{II2594.5} & $3.7\pm0.4$ & $1.9\pm0.2$ & $0.9843$ \\
$5157.7$ & \ion{Fe}{II2600.2} & \ion{Mn}{II2696.7}, \ion{Fe}{$\mathrm{II^{*}}$2586.7} & $6.3\pm0.5$ & $3.2\pm0.2$ & $0.9836$ \\
\hline
$5547.3$ & \ion{Mg}{II2796.4}\tablefootmark{\dag} & $....$ & $....$ & $....$ & 0.9837 \\
$5540.8$ & $....$ & $....$ & $11.9\pm0.4$ & $6.0\pm0.2$ & $....$ \\
$5560.1$ & \ion{Mg}{II2803.5}\tablefootmark{\dag} & $....$ & $....$ & $....$ & 0.9833 \\
\hline
$5660.2$ & \ion{Mg}{I2853.0} & $....$ & $4.3\pm0.4$ & $2.2\pm0.2$ & $0.9840$ \\
$7806.1$ & \ion{Ca}{II3934.8} & $....$ & $4.0\pm0.5$ & $2.0\pm0.2$ & $0.9839$ \\
$7877.6$ & \ion{Ca}{II3969.6} & $....$ & $3.3\pm0.6$ & $1.7\pm0.3$ & $0.9845$ \\
\hline\hline
\end{tabular}
\tablefoot{The redshift was determined for those lines with no contaminants.
\tablefoottext{\dag}{Blended lines.}
}
\label{tab:spectra_equivalentwidths}
\end{table*}
\section{The Host's Emission Lines}
Two spectra of the host galaxy were obtained with OSIRIS and X-shooter (see Sect.~\ref{sec:Results:subsec:Host:subsubsec:OpticalSpectrum}), showing the following emission lines: H$\alpha$~and H$\beta$~transitions from the Balmer series and also forbidden transitions of [\ion{O}{II}]~and [\ion{N}{II}]~(only [\ion{O}{II}]~emission was detected with OSIRIS, and so for consistency only the X-shooter emission lines are shown). All of the 2D spectral images and 1D Gaussian fits can be seen in Fig~\ref{fig:lines}.
\begin{figure*}
\subfloat[]{\label{sub:1}\includegraphics[width=8cm]{imgs/halpha.pdf}}\quad\quad
\subfloat[]{\label{sub:2}\includegraphics[width=8cm]{imgs/hbeta.pdf}}
\subfloat[]{\label{sub:3}\includegraphics[width=8cm]{imgs/nii.pdf}}\quad\quad
\subfloat[]{\label{sub:4}\includegraphics[width=8cm]{imgs/oii.pdf}}
\caption{The 2D spectra of the host of GRB 110918A, depicting four different emissions ([\ion{N}{II}], [\ion{O}{II}], H$\alpha$, H$\beta$). Overplotted is our Gaussian fit where areas that overlay telluric lines are shown in white and excluded from the fit. All the values presented are raw values and do not include slit-loss or extinction corrections. Each image has been smoothed in both pixel directions for presentation purposes. {\bf (a)}: The Balmer series transition H$\alpha$. {\bf (b)}: The Balmer series transition H$\beta$. {\bf (c)}: The forbidden transition [\ion{N}{II}]. {\bf (d)}: The forbidden transition [\ion{O}{II}]. }
\label{fig:lines}
\end{figure*}
\end{appendix}
\section{Conclusion}
\label{sec:Conclusion}
We observed the afterglow of GRB 110918A and its associated host galaxy and obtained photometry and spectroscopy of both. The extensive follow-up campaign has allowed us to measure the afterglow sight-line extinction as well as the attenuation of the galaxy's stellar and gas-phase component. We further derive the host's integrated SFR, stellar mass and gas-phase metallicity. In summary, this burst has revealed the following properties with respect to the long GRB population:
\begin{enumerate}
\item The SED determined stellar mass of $\log_{10}\left(M_{*}\right/\mathrm{M_{\sun}})=10.68\pm0.16$ makes the host of GRB 110918A one of the most massive galaxies selected by a GRB at $z\sim1$.
\item GRB 110918A is the first relatively unobscured afterglow ($A^{\mathrm{GRB}}_{V}=0.16\,\mathrm{mag}$) that has been detected in a very massive host galaxy, suggesting that the geometry of dust is more clumpy than homogeneous or local dust has been destroyed by the progenitor.
\item The optical/NIR spectrum reveals a solar metallicity environment ($0.9-1.7\,\mathrm{Z_{\sun}}$, depending on the chosen diagnostic), making it one of the most metal-rich long GRB host galaxies found yet.
\item Using the fundamental metallicity relation and the measured SFR, stellar mass and metallicity, we show that the host of GRB 110918A is no different to star forming galaxies selected through their own stellar light.
\item The large energy output from the $\gamma$-ray emission of GRB 110918A and the large metallicity content of the host galaxy, is in strong contradiction with there being an anti-correlation between energy output of the GRB and environmental metallicity.
\item Finally, the solar abundance of metals contradicts a cut-off for host galaxies of $Z<0.5\,\mathrm{Z_{\sun}}$, even if a chemical dispersion of $\sim0.3\,\mathrm{dex}$ existed.
\end{enumerate}
\section{Observations and Data Reduction}
\label{sec:Observations}
\subsection{Swift -XRT Spectra}
\label{sec:Observations:subsec:Afterglow:subsubsec:XRT}
At the time of the IPN trigger {\it Swift}~\citep{Gehrels04a} was both in the South Atlantic Anomaly and Earth-occulted, and so no trigger~\citep{Krimm11a} was initiated in the Burst Alert Telescope~\citep[][BAT]{Barthelmy05a}. However, the {\it Swift} X-ray telescope~\citep[][XRT]{Burrows05a} began observing the field of GRB 110918A (see Fig.~\ref{fig:fits_grb}) at $T_{0}+107.4\,\mathrm{ks}$ until $\sim40\,\mathrm{d}$ later. The XRT spectrum shows no signs of spectral evolution, remaining with a constant hardness ratio of $\sim0.85$ for its entire emission. We extract a spectrum at the time interval of $T_{0}+140\, \rm ks$ to $T_{0}+250\, \rm ks$ to coincide with our optical/NIR wavelength observations (see Fig.~\ref{fig:sed_grb_broadband}). The XRT spectral data were obtained from the public {\it Swift} archive and were regrouped to ensure at least 20 counts per bin in the standard manner, using the \emph{grappha} task from the HEAsoft package with response matrices from CALDB \texttt{v20120209}. We assume a Galactic hydrogen column of $\mathrm{N^{Gal}_{H,X}}=1.68\times10^{20}\,\mathrm{cm^{-2}}$~\citep{Kalberla05a} in the direction of the burst.
\subsection{GROND Optical/NIR Photometry}
\label{sec:Observations:subsec:Afterglow:subsubsec:GROND}
\begin{figure}
\includegraphics[trim=0 0 0 0, clip=True, width=9.0cm]{imgs/fits_host.pdf}
\caption{A GROND 60 minute stacked $i'$ band image of the GRB 110918A host galaxy. The slit arrangement used in the acquisition image is shown for both the OSIRIS afterglow spectrum (red line) and host spectrum (blue line).}
\label{fig:fits_grb}
\end{figure}
The Gamma-Ray Burst Optical Near-infrared Detector~\citep[GROND;][]{Greiner08a} mounted at the MPG/ESO $2.2~\mathrm{m}$ telescope at La Silla, Chile, began its follow-up campaign of GRB 110918A $29.2~\mathrm{hrs}$ after the trigger simultaneously in the $g^{\prime}r^{\prime}i^{\prime}z^{\prime}JHK_{s}$ filters \citep{Elliott11a}. A mosaic of 5 pointings was carried out to cover the full IPN error box ($\sim20\arcmin \mathrm\times20\arcmin$) and the GRB optical afterglow candidate was detected at the location R.A. (J2000) = $02^{h}10^{m}09.34^{s}$, Dec. (J2000) = $-27\degr06\arcmin19.7\arcsec$, in GROND's NIR chips, located just outside the IPN error box, consistent with the X-ray \citep{Mangano11a} and optical \citep{Tanvir11a} position with an uncertainty of $0.2\arcsec$. GROND observations continued for over 1 month after the GRB trigger and an underlying host was discovered \citep[see also][]{Oksanen11b}. Deep images of 3600 s in the NIR and 4500 s in the optical were obtained with GROND of the host galaxy at $T_{0}+36.37\,\mathrm{d}$.
Image reduction and photometry of the GROND observations were carried out using standard IRAF tasks~\citep{Tody93a} in the way outlined in~\citet{Kruehler08a} and~\citet{Yoldas08a}. In brief, a point-spread function (PSF) was obtained from the bright stars within the field and applied to the afterglow photometry. The absolute calibration of the optical photometry was achieved by observing a Sloan Digital Sky Survey (SDSS) field~\citep{2011ApJS..193...29A} at R.A. (J2000) = $01^{h}40^{m}00.0^{s}$, Dec. (J2000) = $-18\degr03\arcmin00.0\arcsec$ and the GRB field consecutively. The NIR absolute calibration was obtained from the Two Micron Sky Survey (2MASS) stars~\citep{Skrutskie06a} within the field of the GRB. As a result of the extension of the galaxy (see Fig.~\ref{fig:fits_grb}) an aperture size of $3.2\arcsec$ was used on the deep host galaxy images, within which the aperture flux flattened in a curve-of-growth analysis and the zero points were correspondingly corrected.
The scope of this paper does not involve a full analysis of the afterglow emission. However, afterglow flux measurements are required at certain time intervals to implement slit loss corrections to the optical spectra and to determine the local extinction of the afterglow. Therefore, we only present the required data and direct the reader to future work for a full analysis of the afterglow\footnote{For the interested reader the afterglow light curves are found in the Appendix.}. For the slit loss correction of the afterglow, we obtain the following brightnesses in the AB system at a mid time of $T_{0}+2.2\,\mathrm{d}$: $g'=20.50\pm0.05\,\mathrm{mag}$, $r'=20.20\pm0.04\,\mathrm{mag}$, $i'=19.96\pm0.04\,\mathrm{mag}$, $z'=19.83\pm0.07\,\mathrm{mag}$, $J=19.46\pm0.09\,\mathrm{mag}$, $H=19.04\pm0.10\,\mathrm{mag}$, $K_{s}=18.66\pm0.19\,\mathrm{mag}$. The magnitudes are uncorrected for a Galactic dust reddening of $E(B-V)^{Gal}=0.020\,\mathrm{mag}$ corresponding to an extinction of $A^{\mathrm{Gal}}_{V}=0.062\,\mathrm{mag}$ for $R_{V}=3.1$~\citep{Schlegel98a}.
\subsection{WFI Optical Photometry}
\label{sec:Observations:subsec:Afterglow:subsubsec:WFI}
Further deep observations of the host galaxy were made 392 d after the trigger with the Wide Field Imager~\citep[][WFI]{Baade99a}, also mounted on the MPG/ESO 2.2m telescope, in the standard broadband filters $\mathrm{BB}\#B\mathrm{/123\_ESO878}$ ($B$) and $\mathrm{BB}\#U\mathrm{/150\_ESO878}$ ($U$). Two sets of images were taken, the first on 25 October 2012 consisting of 1800 s in $U$ and 600 s in $B$ and the second set was on 26 October 2012, consisting of 150 s in $U$ and 75 s in $B$. A calibration field was obtained on 26 October 2012 in both the $U$ and $B$ filters and the standard field SA113+158\footnote{\url{www.eso.org/sci/observing/tools/standards/Landolt.html}} was used as a primary calibrator. The photometry was carried out in the same way as the GROND images, and the magnitudes converted into the AB system using the ESO magnitude converter\footnote{\url{http://archive.eso.org/mag2flux}}.
\subsection{WISE IR Photometry}
\label{sec:Observations:subsec:Afterglow:subsubsec:WISE}
The Wide-field Infrared Survey Explorer~\citep[][WISE]{Wright10a} All-Sky Source Catalogue\footnote{\url{http://irsa.ipac.caltech.edu}} reveals a source at the position of the host galaxy of GRB 110918A, with an $11\sigma$ and $3\sigma$ detection in the $W1$ and $W2$ bands, centred at 3.4$\mu$m and 4.6$\mu$m respectively. The \emph{wmpro} magnitudes were used, which are the magnitudes retrieved from profile-fitting photometry (or the magnitude of the 95\% confidence brightness) and converted into the AB system using the WISE conversion factors\footnote{\url{http://wise2.ipac.caltech.edu/docs/release/allsky/expsup/sec4_4h.html}}. Galactic reddening corrections were made using the $A^{\mathrm{Gal}}_{V}$ conversions determined by~\citet{Jarrett12a}.
\subsection{GMOS Optical Spectroscopy}
\label{sec:Observations:subsec:Afterglow:subsubsec:Gmos}
The first spectrum of the afterglow was taken with the Gemini Multi-Object Spectrographs~\citep[][GMOS]{Hook04a} on the Gemini North telescope (Mauna Kea) starting at 12:52 UT on 20 September 2011, 1.6 d after the GRB trigger. Four exposures of 500 s each were obtained using the R400 grism and a 1\arcsec ($\sim8.0\,\mathrm{kpc}$ projected at $z=0.984$) slit width. Two of the spectra were obtained with a central wavelength of $6000\,$\AA~and the other two with $6050\,$\AA~to cover the detector gaps. In addition, a spatial dither was used to cover the amplifier boundaries. The resulting spectrum covers the range $3930 - 8170\,$\AA. We reduced the data with tasks within the {\it Gemini.GMOS} package and {\it IRAF}, \texttt{v1.11}, using flat field and arc lamp frames taken directly before and after the science data.
\subsection{OSIRIS Optical Spectroscopy}
\label{sec:Observations:subsec:HostGalaxy:subsubsec:Osiris}
The second spectrum of the afterglow was obtained using the Optical System for Imaging and low Resolution Integrated Spectroscopy~\citep[][OSIRIS]{Cepa00a} mounted on the $10.4\,\mathrm{m}$ Gran Telescopio Canarias (Roque de los Muchachos) starting at 13:00 UT on 21 September 2011, 2.2 d after the GRB trigger. Three exposures of 900 seconds each were taken using the R500B grism and a 1\arcsec slit width obtained at the parallactic angle. The resulting spectrum covers the range $4400 - 8700\,$\AA. Data were reduced and calibrated using standard procedures in IRAF. The spectrum was flux calibrated using G157-34 as a standard star and the 1D spectrum was scaled to the GROND afterglow photometry to correct for slit losses (correction factor of $\sim1.9$).
A spectrum of the host was obtained on 11 November 2011, $T_{0}+54.1$ d after the GRB trigger. A sequence of three 1200 s exposures were obtained with OSIRIS using the R1000R grism and a $1.0\arcsec$~slit width, covering the wavelength range of $5100\,$\AA~to $10000\,$\AA. The spectrum was flux calibrated using the standard star G191-B2B and corrected for slit losses by scaling the 1D spectrum to the photometry of the host galaxy obtained using GROND (correction factor of $\sim3.5$).
\subsection{X-Shooter Optical/NIR Spectroscopy}
\label{sec:Observations:subsec:HostGalaxy:subsubsec:Xshooter}
We further observed the host of GRB~110918A with the cross-dispersed echelle spectrograph X-shooter \citep{2011arXiv1110.1944V} on the Very Large Telescope Kueyen (UT2). X-shooter has three individual arms taking spectra simultaneously in the range of $3000\,\mathrm{\AA}$ to $5600\,\mathrm{\AA}$ (UVB arm), $5600\,\mathrm{\AA}$ to $10\,200\,\mathrm{\AA}$ (VIS arm), and $10\,200\,\mathrm{\AA}$ to $24\,800\,\mathrm{\AA}$ (NIR arm). Three different sets of observations were carried out on 17 December 2012, 07 January 2013, and 16 January 2013, respectively, with a position angle of $59^{\circ}$ East of North. They consisted of a pair of nodded frames with exposure times of $1200$~s in each of the UVB/VIS arm and $2\times600$~s in the NIR arm. The slit width was 1\farc{6}, 1\farc{5} and 0\farc{9} yielding a resolution measured on arc-lamp frames of $R \sim \lambda/\Delta\lambda=3200$, 4900, and 5300 in the UVB, VIS and NIR arm, respectively. The NIR slit includes a blocking filter for the $K$-band limiting our effective wavelength coverage to $<20\,500\,\mathrm{\AA}$, but providing lower background levels in the $J$ and $H$-band.
Each of the individual observations were reduced and wavelength- and flux-calibrated separately using standard procedures within the X-shooter pipeline \texttt{v2.0.0} \citep{2006SPIE.6269E..80G} supplied by ESO. The individual two-dimensional frames were then stacked using variance weighting in a heliocentric reference frame, and the one-dimensional spectra were extracted using an optimal extraction method. Given the extent of the target, slit-losses are substantial. Similar to the OSIRIS and GMOS spectroscopy, we scaled the well-detected continuum of the X-shooter data to the available photometric host SED. The consistency between matching factors derived from different photometric data in the individual arms\footnote{We derive factors for the $r'$-, $i'$- and $z'$ -band data in the VIS arm of $3.1 \pm 0.3$, $2.8 \pm 0.3$, $2.8 \pm 0.3$, and $3.5 \pm 0.4$, $3.3 \pm 0.3$ for the $J$ and $H$-band in the NIR arm. The offset between the VIS and NIR arm is readily explained by the smaller slit width in the NIR arm.}, provides confidence that the absolute flux-calibration in the final X-shooter spectrum is accurate to better than $\sim 20$\% over the full wavelength range of interest.
\section{Discussion}
\label{sec:Discussion}
\subsection{Host Galaxy Identification}
\label{sec:Discussion:subsec:hostidentification}
We have used
absorption lines from metal ions in the afterglow spectrum \citep[see also][for an extensive sample]{Fynbo09a}, and forbidden/recombination lines from the host galaxy to determine the redshift of the GRB \citep[see also][]{Kruehler12b}, but it is in principle possible that the GRB lies at a higher redshift.
To investigate if the host galaxy of GRB 110918A has been misidentified we calculate the commonly used p-value, $p\left(m\right)=1-\exp\left(-\pi r_{i}^{2}\sigma\left(\le m \right) \right)$, which is the probability of finding a galaxy of magnitude $m$ (or brighter) overlapping the GRB within an effective radius $r_{\rm{i}}$, assuming that galaxies are Poisson distributed throughout the sky~\citep{Bloom02a}. This neglects any type of galaxy clustering, however, recent work indicates that GRB locations do not preferentially lie in areas of strong galaxy overabundances~\citep[][]{Cucchiara12a,Sudilovsky13a}. The number of galaxies brighter than $m$ per square arcsecond is given by $\sigma$, taken from~\citet{Bloom02a} and calculated from the work of~\citet{Hogg97a}.
The burst location of GRB 110918A is seen to be offset from the bright centroid of the host by $12\,\mathrm{kpc}$, however, in comparison to the half-light radius of the host galaxy, $R_{\frac{1}{2}}=10.6\,\mathrm{kpc}$, the offset is consistent with the long GRB population, which has a median offset of $R_{\mathrm{offset}}/R_{\frac{1}{2}}\sim1$~\citep{Bloom02a}. We follow~\citet{Bloom02a} and set $r_{i}=2\times R_{\frac{1}{2}}=2.66\arcsec$. Synthetic $R_{C}$-band \citep{Bessel79a} photometry of the GRB 110918A host galaxy using the best fit galaxy template taken from Sect.~\ref{sec:Results:subsec:HostSED:subsubsec:PhotometricSED}, and the conversions given in~\citet{Rossi12a}, results in $R_{C}=21.7\,\mathrm{mag_{Vega}}$. This yields a probability of chance association of $p=0.01$, making this galaxy highly likely the host of GRB 110918A.
The non-detection of the Lyman forest above $\sim$4500 {\AA} implies a strong upper limit of $z<2.7$. Therefore,
using our knowledge of the strength of spectral features in GRB environments and their distribution ~\citep{deUgartePostigo12a}, we can estimate the likelihood of the GRB having occurred between redshift 1.0 and 2.7 and yet not having detectable absorption lines at the redshift of the host in its spectrum.
We calculate the detection limits for \ion{Mg}{ii} and \ion{C}{iv} doublets as described by~\citet{deUgartePostigo12a} and find that the lines would have to be weaker than
99.7\% of a normal long GRB sample
to have happened at a redshift between 1 and 2.7.
Furthermore, the properties of the absorber (strong \ion{Mg}{II} absorption and vigorous star formation, see Sect.~\ref{sec:Results:subsec:afterglowspectra}), are very common in other afterglow observations, and do not indicate a different physical nature. Combining the arguments presented above, we consider the redshift of the GRB, and accordingly the physical association between GRB and galaxy, robust.
\subsection{Host Environment in the Context of the GRB-Host Population}
\label{sec:Discussion:subsec:HostGRBEnvironment}
The mass-metallicity relation of field galaxies \citep[e.g.,][]{Tremonti04a} has been studied in depth to high redshift~\citep{Savaglio05a,Erb06a,Yabe12a}. Similarly, the dust content of a given galaxy is also well known to correlate with stellar mass~\citep[e.g.,][]{Garn10a,Zahid13a}. To illustrate the behaviour for GRB hosts, we show the average host galaxy extinction versus the stellar mass of the host galaxy alongside the correlation determined by~\citet{Garn10a} in Fig.~\ref{fig:hostav}. The GRB hosts are taken from~\citet[][SG09]{Savaglio09a},~\citet[][MN11]{Mannucci11a},~\citet[][KR11]{Kruehler11a} and~\citet[][PL13]{Perley13a} and converted to a Chabrier IMF if need be. The correlation of~\citet{Garn10a} has been determined from SDSS galaxies with $z<0.7$ and so we limited our GRB sample to galaxies with $z<1.0$.
GRB hosts follow the distribution obtained from field galaxies well, with a possible excess of dusty systems at stellar masses of $10^{9-10}\,M_{\sun}$. Given the inherent systematic difficulties of determining the dust reddening in galaxies, and the heterogeneous selection of targets (in particular the KR11 and PL13 samples were initially selected to contain a lot of dust), this trend should not be over-interpreted. What seems clear is that the host of GRB~110918A is at the high end of the distribution of stellar masses for GRB hosts, and there is no strong discrepancy between GRB-selected galaxies and field galaxies in the relation between their dust content and stellar mass.
\begin{figure}
\includegraphics[trim=0cm 0cm 0cm 0cm, clip=True, width=9.5cm]{imgs/hostav.pdf}
\caption{The average line of sight extinction of the host galaxy vs. the stellar mass. Values from SG09 and MN11 have been converted from $E(B-V)^{\mathrm{gas}}$ to $E(B-V)^{\mathrm{stars}}$ using the relation from~\citet{Calzetti00a}. The dotted-black line is the polynomial fit determined by~\citet{Garn10a} and the grey region denotes the uncertainty of $0.3\,\mathrm{dex}$.}
\label{fig:hostav}
\end{figure}
Secondly we plot the host's stellar mass vs. the GRB's line of sight extinction in Fig.~\ref{fig:grbav}. \citet[][]{Perley13a} have highlighted that throughout the covered galaxy-mass scale, there is a very tight correlation between stellar-mass and sight-line extinction probed by the GRBs. Quite surprisingly, this correlation between afterglow dust and galaxy mass is found to be stronger than for any other physical property of the galaxy (PL13). From Fig.~\ref{fig:grbav} it can be seen that hosts selected due to high afterglow extinction (green, KR11; brown PL13) have systematically more massive and dust extinguished sight lines than the optically selected hosts (blue, SG09). Outliers to this trend such as GRB 061222A or GRB 100621A have already been noted~\citep[e.g.,][]{Kruehler11a,Perley09a}, that were within blue, low-mass galaxies that were locally strongly extinguished along the line of sight. GRB 110918A is the first example of a dust-poor line of sight with a galaxy mass at the high-end of the distribution (i.e., $\log_{10}\left(\frac{M_{*}}{\rm M_{\sun}}\right)>10.5$). While in principle, cases like GRB 110918A would be easy to identify (bright afterglow, easy localization, bright host), no comparable example has been reported in the literature to date.
The host of GRB 110918A shows similar host-integrated extinction ($A^{\mathrm{stars}}_{V}=0.90\,\mathrm{mag}$) to galaxies of a similar mass range (e.g., $M_{*}>10^{10}\,\mathrm{M_{\sun}}$ in Fig.~\ref{fig:hostav} and $M_{*}>4\times10^{9}\,\mathrm{M_{\sun}}$ in Fig.~15 of~\citet{Perley13a} have an $A^{\mathrm{stars}}_{V}\gtrsim1.0\,\mathrm{mag}$). However, in comparison to the systems of similar mass, GRB 110918A exhibits at least 10 times less extinction along the GRB line of sight. Therefore, it is possible that: (i) the geometry of dust within the host of GRB110918A is more patchy than homogeneous in comparison to the rest of the massive GRB host population, in agreement with the example of GRB 100621A and 061222A, whereby clumpy dust was one explanation for having a highly extinguished afterglow within an unobscured galaxy~\citep{Kruehler11a,Perley13a}, or (ii) the progenitor had enough time to destroy local dust from its UV emission~\citep[see][and referencest therein]{Perley13a}.
\begin{figure}
\includegraphics[trim=0cm 0cm 0cm 0cm, clip=True, width=9.5cm]{imgs/grbav.pdf}
\caption{The GRB line of sight $A^{\mathrm{GRB}}_{V}$ plotted against the stellar mass of the host galaxy. The extinction values have been obtained from~\citet{Kann06a},~\citet{Kann10a},~\cite{Greiner11a},~\citet{Schady12a} and~\citet{Perley13a}.}
\label{fig:grbav}
\end{figure}
\subsection{Fundamental Metallicity Relation}
\label{sec:Discussion:subsec:FMR}
The difference between galaxies of long GRBs and that of normal star forming field galaxies is still an on going debate. We have derived estimates for the mass, metallicity and SFR of the host of GRB 110918A, which facilitates comparing this galaxy with respect to normal star forming galaxies through the fundamental metallicity relation~\citep[FMR;][]{Mannucci11a}. The plane of the FMR was derived from star forming SDSS galaxies in the mass range $9.2<\log_{10}\left(\frac{M_{*}}{\mathrm{M_{\sun}}}\right)<11.4$ and is described by:
\begin{equation}
12+\log(O/H) = 8.90 + 0.47\times(\mu_{0.32}-10),
\end{equation}
\noindent where $\mu_{0.32}=\log_{10}\left(M_{*}/\mathrm{M_{\sun}}\right)-0.32\times\log\left(\mathrm{SFR/M_{\sun}yr^{-1}}\right)$. Using the SED-determined mass and the \ion{H}{$\alpha$}-determined SFR, the metallicity from the FMR is $12+\log (O/H)=8.98\pm0.08$, in agreement with the metallicity from the \ion{N}{II}/\ion{H}{$\alpha$} line ratio of $12+\log (O/H)=8.93\pm0.13$. The method used in our metallicity estimate is the same as the one used by MN11 to construct the FMR, in order to ensure a direct comparison. The estimated errors are purely based on the uncertainties of the mass and SFR, and any systematic uncertainties from the method used to fit the stellar mass have been ignored.
The agreement in the characteristic properties of the host galaxy of GRB 110918A with the FMR (see Fig.~\ref{fig:fmr}) shows that the host galaxy has no deficit of metals in comparison to normal field galaxies, in line with the conclusions of~SG09, MN11, KR11, and ~\citet{Michalowski12a}. This illustrates that the mass and SFR of a GRB-selected galaxy, at least for this one event, can be used as a fair proxy for the metallicity even in the solar, or super-solar regime.
\begin{figure}
\includegraphics[width=9.5cm]{imgs/fmr.pdf}
\caption{The metallicity determined from the fundamental metallicity relation (taken from MN11) vs. the parametric quantity $\mu_{0.32}$, plotted in grey for a range of star formation rates ($\rm SFR=0-100\,M_{\sun}yr^{-1}$). Real quantities are plotted for MN11 (magenta squares), GRB 080605~\citep[cyan upward-pointing triangle]{Kruehler12a}, and GRB 110918A (red). The host of GRB 110918A is well described by the SDSS determined FMR.}
\label{fig:fmr}
\end{figure}
\subsection{Metallicity and Long GRB Progenitors}
\label{sec:Discussion:subsec:metallicityandprogenitors}
Many authors have attributed the fact that most long GRB host galaxies exhibit low metallicities as the result of an environmental preference, rather than the effect of the FMR~\citep[e.g.,][]{Modjaz08a,Graham12a,Perley13a}. This dependence on metallicity has also led to the prediction that the lower the progenitor metallicity, the larger the angular momentum, and thus the higher the energy output ($E_{\gamma,\mathrm{iso}}$) of the GRB~\citep{MacFadyen99a}. Initial studies indeed showed an anti-correlation between these two quantities~\citep{Stanek06a}, together with a cut-off metallicity above which long GRBs (for $z<0.2$) are no longer created, i.e., $Z<0.15\,\mathrm{Z_{\sun}}$. More recent studies, however, which include long GRBs at cosmological redshifts and exclude sub-luminous GRBs~\citep{Wolf07a,Levesque10a} indicate that there is no clear anti-correlation between metallicity and the GRB's energy output, as shown in Fig.~\ref{fig:eiso_met}. The prompt emission of GRB 110918A yielded an energy output of $E_{\gamma,\mathrm{iso}}=1.9\times10^{54}\,\mathrm{erg}$~\citep{Frederiks11a} within the top $2\%$ of the GRB population~\citep[][Frederiks et al. 2013]{Amati08a}. This makes GRB 110918A one of the most energetic long GRBs yet observed and its host one of the most metal rich galaxies, in contradiction to the idea of a correlation between $E_{\gamma,\mathrm{iso}}$ and metallicity.
\begin{figure}
\includegraphics[width=9.5cm]{imgs/eiso_met.pdf}
\caption{The isotropic-equivalent energy release in $\gamma$-rays of GRBs plotted against the gas-phase metallicity of the host galaxy. Blue data are taken from~\citet{Levesque10a}.}
\label{fig:eiso_met}
\end{figure}
Recently, \citet{Perley13a} performed an extensive photometric study of host galaxies selected from a sample of dark bursts, limiting the selection biases present in previous works. However, while the inclusion of dark GRB hosts increases the consistency of GRB hosts with the star formation weighted sample of field galaxies, there is still a clear lack of high-mass galaxies at $z \lesssim 1.5$. Associating the galaxy mass with metallicity, this provides indirect evidence for a metallicity effect in GRB hosts. A similar conclusion was reached based on a comparison of long GRB hosts with supernovae hosts~\citep{Graham12a}, namely that long GRB hosts show a strong preference for lower metallicity environments relative to other populations of star forming galaxies, with a metallicity cut-off of $Z<0.5\,\mathrm{Z_{\sun}}$. This cut-off is not consistent with the host galaxy of GRB 110918A, even if metallicity dispersions of $\sim0.3\,\rm dex$ are considered~\citep{Niino11a}.
\section{Introduction}
\label{sec:Introduction}
During their prompt emission, long gamma-ray bursts (GRBs) are the brightest objects in the Universe, easily reaching isotropic-equivalent luminosities as high as $\sim10^{54}\,\mathrm{erg\, s^{-1}}$. Their observed association to supernovae events~\citep[e.g.,][]{Galama98a,Hjorth03a, Stanek03a,Matheson03a,DellaValle11a,Hjorth12a} has tightly linked them to the death of massive stars. The GRB itself is then believed to result from accretion of matter onto the newly formed rapidly rotating black hole or compact object in the collapsar model~\citep{Woosley93a, Paczynski98a, MacFadyen99a}. The lack of hydrogren and helium in the spectra of GRB-supernovae classify them as type Ic, supporting the notion that GRB progenitors are likely Wolf-Rayet like stars~\citep[for a review of supernova classifications see, e.g.,][]{Filippenko97a}. Given that these type of stars undergo vigorous mass loss from stellar winds, metallicity constraints ($Z<0.3\,\mathrm{Z_{\sun}}$) on the progenitor are postulated to ensure that an accretion disk is still formed around the black hole~\citep{Hirschi05a, Yoon05a, Woosley06a}.
The possible association of long GRBs with massive stars supported the idea that they could be used as complementary and independent tracers of star formation, especially at high redshifts ($z\gtrsim4$), due to their very high luminosities~\citep[see e.g.,][]{Daigne06a,Li08a,Kistler09a,Ishida11a}. However, to have full confidence in these studies the intrinsic evolutionary effects in long GRB production must be understood and the galactic environments preferred by the progenitor need to be quantified~\citep[e.g.,][]{Butler10a,Wang11a,Salvaterra12a,Robertson12a,Elliott12a}. Of particular interest is the relation between the galaxies selected by GRBs and the star formation weighted population of field galaxies. To be direct and unbiased tracers of star formation, the relative rates of GRBs in galaxies of various physical properties should be the same in galaxies taken from samples that trace the global star formation density at a given redshift. Studies based on these galaxy samples are most commonly performed at $z \lesssim 1.5$, where the star formation of field galaxies is largely recovered by state-of-the-art deep-field surveys.
Initial work showed that many long GRB host galaxies had a low mass, low metallicity, as well as blue colours and were actively star forming~\citep[see e.g.,][]{Fruchter99a, LeFloch03a, Berger03a, Christensen04a, Tanvir04a}. This seemed directly in line with the requirements of the collapsar model. Further work carried out with larger samples~\citep[e.g.][]{Savaglio09a}, showed again similar characteristics. However, at a given mass and star formation rate, long GRB hosts were also found to be no different to the normal population of star forming galaxies at the same redshift \citep{Mannucci11a}. However, these initial studies neglected the contribution from galaxies hosting dust-extinguished afterglows, often termed dark bursts \citep[e.g.,][]{Perley09a, Greiner11a}. Galaxies hosting dark bursts are systematically more massive and have a higher dust-content than the previously-established population localized with optically bright afterglows~\citep{Kruehler11a, Hjorth12a, Rossi12a, Perley13a, Christensen11a, deUgartePostigo12a}.
Despite the inclusion of this more evolved galaxy population, there are still less GRBs in massive galaxies than expected based on their contribution to the overall star formation rate (SFR), at least at $z < 1.5$ \citep{Perley13a}, indicative of a GRB explosion mechanism dependent on metallicity. It is however important to note that these above conclusions are inferred indirectly through stellar mass as a metallicity proxy, and while the photometric samples of GRB hosts have reached integrated number statistics of 100 and above \citep[e.g.,][]{Hjorth12a, Perley13a}, the most crucial measurement of gas-phase metallicity has only been performed in a hand-full cases at $z \gtrsim 1$ \citep{Levesque10d, Kruehler12a}.
Only a few host galaxies with substantial gas-phase metallicities around or above solar~\citep[e.g.,][]{Levesque10b} that directly violate the proposed cut-off in galaxy metallicity have been observed to date. There is thus still lively debate in the literature about the nature of GRB hosts, and their relation to the star formation weighted galaxy population as a whole \citep[e.g.,][]{Niino11a, Mannucci11a, Kocevski11a, Graham12a}. GRB hosts with high stellar mass and high global metallicity are hence of primary interest for GRB host studies, as they directly probe this allegedly forbidden parameter space. A robust understanding of the galactic environments in which GRBs form would then add confidence in their use as cosmological probes, beyond the limits of deep survey studies \citep[e.g.,][]{Tanvir12a, Basa12a}.
Here, we present spectroscopy and photometry of the host galaxy and afterglow of the luminous GRB 110918A, detected on the 18th of September 2011 at $T_{0}$=21:26:57 UT \citep{Hurley11a}. This burst had one of the highest fluences of any GRB observed over the last 20 years~\citep[together with GRB 021206;][]{Wigger08a} and had the highest peak flux ever detected by {\it Konus-Wind} \citep[][Frederiks et al. 2013 in prep.]{Golenetskii11a,Frederiks11a}, located at a redshift of $z=0.98$. The massive, metal rich host galaxy and unobscured afterglow of GRB 110918A challenges the current view of the connection between local and global environments and allow us to investigate the preferred conditions for the formation of a long GRB.
The paper is arranged as follows: first we describe the observations carried out by both ground and space based instruments and their corresponding reduction in Sect.~\ref{sec:Observations}. Second, the resulting properties ascertained from the SEDs and spectra of the GRB and its host are described in Sect.~\ref{sec:Results}. Finally, we discuss our findings and their implications for the population of long GRBs in Sect.~\ref{sec:Discussion} and conclude in Sect.~\ref{sec:Conclusion}. We adopt the convention that the GRB flux density is described by $F_{\nu}\left(t\right)\propto t^{-\alpha}\nu^{-\beta}$, reported errors are at the $1\sigma$ confidence level and we assume a $\Lambda$CDM cosmology: $H_{0} = 71\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$, $\Omega_{M} = 0.27$ and $\Omega_{\Lambda}=0.73$. We use a \citet{Chabrier03a} initial mass function and abundances throughout the text.
\section{Results}
\label{sec:Results}
\subsection{The Afterglow Sight-Line: Dust, Star Formation Rate and Gas}
\label{sec:Results:subsec:broadbandsed}
\label{sec:Results:subsec:afterglowspectra}
A broadband SED was constructed from the optical/NIR GROND photometry (see Sect.~\ref{sec:Observations:subsec:Afterglow:subsubsec:GROND}) at a mid time of $T_{0}+194\,\mathrm{ks}$ and X-ray data between $T_{0}+140\,\rm ks$ and $T_{0}+250\,\rm ks$. The SED was fit in a standard manner \citep[e.g.,][]{Filgas11a}, assuming that the afterglow emission is well described by the standard synchrotron mechanism.
The best fit single power-law ($\chi^{2}/\mathrm{d.o.f.}=85/73$) is shown in Fig.~\ref{fig:sed_grb_broadband} with a spectral slope of $\beta=0.70\pm0.02$, a hydrogen column density of $\rm N^{GRB}_{H,X}=1.56^{+0.52}_{-0.46}\times10^{21}\,cm^{-2}$ and a line of sight extinction of $A^{\mathrm{GRB}}_V=0.16\pm0.06\,\mathrm{mag}$, assuming a Small Magellanic Cloud (SMC) like dust with $R_{V}=2.93$ in the parametrization of \citet{1992ApJ...395..130P}. Despite yielding an improved fit, a broken power-law model is not warranted as the number of free parameters increases ($\chi^{2}/\mathrm{d.o.f.}=83/71$), implying the improvement is not statistically significant. Nevertheless, the resulting parameters of the best fit broken power-law, as well as from different dust models, are consistent with the uncertainties of the power-law values below the break at $\sim 0.6$~keV and do not alter our conclusions.
\begin{figure}
\center
\includegraphics[trim=0cm 0cm 0cm 0cm, clip=True, width=9cm]{imgs/sed_grb_broadband.pdf}
\caption{Broadband SED of GRB 110918A, including optical, NIR and X-ray data. The SED was constructed using GROND data at a mid time of $T_{0}+194\,\mathrm{ks}$ and X-ray data between $T_{0}+140\,\rm ks$ and $T_{0}+250\,\rm ks$. The best-fit parameters for a power-law ($\chi^{2}/\mathrm{d.o.f.}=85/73$) are: a spectral slope of $\beta=0.70\pm0.02$, a hydrogen column density of $\rm N^{GRB}_{H,X}=1.56^{+0.52}_{-0.46}\times10^{21}\,cm^{-2}$ and a line of sight extinction of $A^{\mathrm{GRB}}_V=0.16\pm0.06\,\mathrm{mag}$, assuming SMC like dust.}
\label{fig:sed_grb_broadband}
\end{figure}
Using the procedure outlined by~\citet{deUgartePostigo12a} with the two afterglow spectra obtained with GMOS ($T_{0}+1.6\mathrm{d}$) and OSIRIS ($T_{0}+2.2\mathrm{d}$), we detect the transition of several metal ions including \ion{Fe}{II}, \ion{Mg}{II} and \ion{Mg}{I} at a common redshift of $z=0.984\pm0.001$ (see Tables \ref{tab:phot_par} and \ref{tab:spectra_equivalentwidths}), consistent with galactic winds or star bursting periods~\citep{Fynbo09a, Nestor11a, Christensen11a, RodriguezHidalgo12a}. In comparison with a long GRB sample~\citep{deUgartePostigo12a} we find that it has stronger absorption features than 80\% of the sample.
\subsection{The Host's Stellar Component: Dust Attenuation, Star Formation Rate and Stellar Mass}
\label{sec:Results:subsec:HostSED:subsubsec:PhotometricSED}
The host of GRB 110918A was detected in 11 different filters ranging from the ultra-violet to $4.5\mu$m, yielding a well-sampled photometric SED (see Fig.~\ref{fig:_sedhost} and Table~\ref{tab:phot_host}). To estimate the global properties of the host galaxy we employed standard techniques that use stellar population synthesis to estimate stellar masses, as outlined thoroughly in~\citet{Ilbert09a}. We constructed a grid of galaxy templates based on the models taken from~\citet{Bruzual03} over a wide parameter space consisting of: a range of ages ($0-1.35\times10^{9}\mathrm{yr}$), star formation histories ($\propto e^{\tau},\tau=0.1,0.3,1,2,3,5,10,15,30$), reddening values ($E(B-V)=0-0.4\,\mathrm{mag}$), a single attenuation law~\citep[starburst;][]{Calzetti00a} and metallicities ($Z=0.004,0.008,0.02$). Emission lines were also included, whereby the emission lines were estimated from the predicted UV luminosity and converted to a star formation rate using \citet{Kennicutt98a}. For each template a SED was constructed for the filters required and a $\chi^{2}$ was calculated using the Photometric Analysis for Redshift Estimate routines, LePHARE~\footnote{\url{www.cfht.hawaii.edu/~arnouts/LEPHARE/lephare.html}} \texttt{v2.2}~\citep{Arnouts99a, Ilbert06a}. The best fit template was the one that gave the minimum $\chi^{2}$ and the corresponding uncertainties for each parameter were obtained from the grid of $\chi^{2}$ values. Systematic uncertainties of up to an average of $0.2-0.3\,\mathrm{dex}$ are expected in the stellar mass value due to the adopted stellar population models and extinction laws~\citep[see e.g.,][and references therein]{Kruehler11a}.
The filter response curves for the $W1$ and $W2$ bands were obtained from~\citet{Wright10a} and for the $U$ and $B$ bands from the ESO web pages\footnote{\url{www.eso.org/sci/facilities/lasilla/instruments/wfi/inst/filters}}.
The results of the best fit template, which had a $\chi^{2}/\mathrm{\#\,Bands}=5.4/11$, had the following parameters: a mass of $\log_{10}\left(\mathrm{\frac{M_{*}}{M_{\sun}}}\right)=10.68\pm0.16$, a star formation rate $\mathrm{SFR_{SED}}=66^{+50}_{-30}\,\mathrm{M_{\odot}}$, a reddening of $E(B-V)^{\rm{stars}}=0.26\pm0.15\,\mathrm{mag}$, and a starburst age of $\tau=0.7^{+1.4}_{-0.4}\,\mathrm{Gyr}$.
\begin{figure}
\centering
\includegraphics[trim=0.5cm 0cm 0cm 0cm, clip=True, width=9.0cm]{imgs/sed_host.pdf}
\caption{The SED of the host of GRB 110918A obtained using GROND, WFI and WISE data, amounting to 11 filters: $UBg'r'i'z'JHK_{s}W1W2$ from left to right. The best-fit spectrum is depicted in black.}
\label{fig:_sedhost}
\end{figure}
\begin{table}
\caption{Host galaxy magnitudes.}
\begin{center}
\begin{tabular}{c c c c}
Filter & Instrument & Magnitude & Uncertainty \\
& & $(\mathrm{mag_{AB}})$ & $(\mathrm{mag_{AB}})$ \\
\hline\hline
$U$ & WFI & 22.98 & 0.25 \\
$B$ & WFI & 22.45 & 0.16 \\
$g'$ & GROND & 22.49 & 0.15 \\
$r'$ & GROND & 22.07 & 0.05 \\
$i'$ & GROND & 21.26 & 0.06 \\
$z$ & GROND & 20.78 & 0.06 \\
$J$ & GROND & 19.92 & 0.11 \\
$H$ & GROND & 19.73 & 0.17 \\
$K_s$ & GROND & 19.61 & 0.22 \\
$W1$ & WISE & 19.56 & 0.10 \\
$W2$ & WISE & 20.28 & 0.36 \\
\hline\hline
\end{tabular}
\tablefoot{Corrected for Galactic foreground reddening. The observations in $g'r'i'z'JHK_{s}$ were obtained $36.37\,\mathrm{d}$ after the burst. The $UB$ observations were obtained 392 d after the burst. The $W1$ and $W2$ photometry were obtained prior to the burst by the WISE Survey.}
\label{tab:phot_host}
\end{center}
\end{table}
\subsection{The Host's Gas-Phase Component: Dust Extinction, Star Formation Rate and Metallicity}
\label{sec:Results:subsec:Host:subsubsec:OpticalSpectrum}
The first host spectrum of GRB 110918A was obtained with OSIRIS/GTC in the optical wavelength range, $\sim50$ days after the trigger, and the one second was obtained with X-shooter/VLT more than $460$ days post trigger. The X-shooter spectrum extends our spectral coverage to the NIR and thus to the wavelength range where important tracers of star formation rate and metallicity are located. In summary, we clearly detect the H$\alpha$ and H$\beta$ transition from the Balmer series, as well as the forbidden transitions of [\ion{O}{II}]($\lambda\lambda$3726, 3729) and [\ion{N}{II}]($\lambda$6584). The emission lines corresponding to [\ion{O}{III}]($\lambda\lambda$4959,5007) are cosmologically redshifted to regions of low sensitivity for both OSIRIS and X-shooter, and are thus not detected.
The velocity profile of the emission lines is clearly resolved by our X-shooter data and spans approximately $500\,\rm{km\,s^{-1}}$ in velocity space (see Fig.~\ref{fig:lines}). It displays a conspicuous two-humped profile, with the two peaks of the emission lines separated by 200~$\rm{km\,s^{-1}}$. However, we do not observe a spatial tilt in the line-shape as would have been expected from a largely rotationally-supported galaxy (unless it is face on), and both peaks appear at the same spatial position in the two-dimensional spectrum. Line fluxes were measured by numerically integrating the available data, and cross-checked with fitting Gaussians. Both procedures return consistent values, and from the X-shooter spectrum, we measure global\footnote{Here and in the following, the error includes both the statistical error of the measurement as well as the error in slit-loss correction.} emission-lines fluxes of $f_{[\ion{O}{II}]} = (19.0 \pm 3.1)\times$$\,10^{-17}\mathrm{erg\,s^{-1}\,cm^{-2}}$, $f_{\rm{H}\beta} = (9.5 \pm 1.9)\times$$\,10^{-17}\mathrm{erg\,s^{-1}\,cm^{-2}}$, $f_{\rm{H}\alpha}= (47.8 \pm 4.9)\times$$\,10^{-17}\mathrm{erg\,s^{-1}\,cm^{-2}}$~and $f_{\ion{[N}{II]}}= (15.3 \pm 3.3)\times$$\,10^{-17}\mathrm{erg\,s^{-1}\,cm^{-2}}$. The OSIRIS spectrum yields $f_{[\ion{O}{II]}]} = (20.0 \pm 2.8)\times$$\,10^{-17}\mathrm{erg\,s^{-1}\,cm^{-2}}$, fully consistent with the X-shooter value.
Assuming case B recombination~\citep{Osterbrock89a} and using the standard values for electron density ($10^{2}\,\mathrm{cm^{-3}} \lesssim n_{\rm e} \lesssim 10^{4}\,\mathrm{cm^{-3}}$) and temperature ($T_{\rm e}\sim10^{4}\,\rm K$), the Balmer ratio of H$\alpha$/H$\beta$~implies an $E(B-V)^{\rm{gas}}=0.57^{+0.24}_{-0.22}$ or visual extinction $A_V^{\rm{gas}}=1.8^{+0.8}_{-0.7}$~mag towards the star forming regions assuming a Milky-Way like extinction law\footnote{Different local extinction laws yield comparable results, as there is little difference in the wavelength range of the H$\alpha$~and H$\beta$~Balmer lines.}. It is worth mentioning, that this is the luminosity-weighted reddening/extinction of the gas-phase. This value is typically found to be a factor of around two larger than the stellar $E(B-V)^{\rm{stars}}$ from the photometric SED model \citep[e.g.,][]{Calzetti00a}, consistent with our measurements for GRB~110918A. The H$\alpha$~line flux implies a SFR of $\rm SFR_{H\alpha}=41^{+28}_{-16}\,$$\rm{M_{\sun}\,yr^{-1}}$, following \citet[][]{Kennicutt98a} with a \citet{Chabrier03a} IMF.
Using the different emission line ratios, we can measure the gas-phase metallicity of the galaxy hosting GRB~110918A \citep[see e.g.,][ for an extensive summary on those techniques]{2008ApJ...681.1183K}.
At $z\sim1$ we
we are limited to the diagnostic ratios based on H$\alpha$, [\ion{O}{II}]~and [\ion{N}{II}]. When using the ratio of [\ion{N}{II}]~and H$\alpha$~as a metallicity tracer, uncertainties in the reddening or chromatic slit-losses are not going to affect the overall results, because the respective lines are located very close in wavelength space. However, [\ion{N}{II}]/H$\alpha$~saturates at high metallicities ([\ion{N}{II}]/H$\alpha$ $\sim 0.3$), while [\ion{N}{II}]/[\ion{O}{II}]~does not. The final uncertainties are thus comparable in both indicators, and we measure 12 + log(O/H)$_{\rm{N2H\alpha}} = 8.93 \pm 0.13$ and 12 + log(O/H)$_{\rm{N2O2}} = 8.85_{-0.18}^{+0.14}$ using the formulation of \citet{2006A&A...459...85N}. Different calibrations of the strong-line diagnostics yield 12 + log(O/H)$_{\rm{N2H\alpha}} = 8.63\pm0.08$ \citep{2004MNRAS.348L..59P} or 12 + log(O/H)$_{\rm{N2O2}} = 8.86_{-0.14}^{+0.10}$ \citep{2002ApJS..142...35K}.
The inherent systematic uncertainty
of typically 0.1-0.2 dex \citep[e.g.,][]{2006A&A...459...85N, 2008ApJ...681.1183K} has not been included. These measurement imply metallicities between 0.9 and 1.7 times solar. All physical parameters of the galaxy hosting GRB~110918A are summarized in Table~\ref{tab:phot_par}.
\begin{table}
\caption{Physical parameters of the galaxy hosting GRB~110918A}
\begin{center}
\begin{tabular}{c c c }
\hline
Quantity & Unit/Method & Value \\
\hline
\hline
$E(B-V)^{\rm{GRB}}$ & mag & $0.05\pm0.02$ \\
$N_{\rm{X,H}}^{\rm{GRB}}$ & $10^{21}\,\rm{cm}^{-2}$ & $1.56^{+0.52}_{-0.46}$ \\
EW$_{\rm rest}^{\rm{GRB}}$ & (\AA) \ion{Mg}{II}(2796, 2803) & 6.0 \\
$\rm SFR^{GRB}_{[OII]}$ & $\rm{M_{\sun}\,yr^{-1}}$ & $2.3\pm0.7$ \\
\hline
Stellar Mass & $\log(M_{*}/M_{\sun})$ & $10.69\pm0.16$ \\
Half-light radius & kpc & $10$ \\
$E(B-V)^{\rm{stars}}$ & mag &$0.26\pm0.15$ \\
$\rm SFR_{SED}$ & $\rm{M_{\sun}\,yr^{-1}}$ & $66^{+50}_{-30}$ \\
$\tau$ & Gyr & $0.7^{+1.4}_{-0.4}$ \\
\hline
$SFR_{H\alpha}$ & $\rm{M_{\sun}\,yr^{-1}}$ & $41^{+28}_{-16}$ \\
$E(B-V)^{\rm{gas}}$ & mag & $0.57^{+0.24}_{-0.22}$ \\
12+$\log(\rm{O/H})$ & [\ion{N}{II}]/H$\alpha$$^{(a)}$ & $8.93 \pm 0.13$ \\
12+$\log(\rm{O/H})$ & [\ion{N}{II}]/H$\alpha$$^{(b)}$ & $8.63 \pm 0.08$ \\
12+$\log(\rm{O/H})$ & [\ion{N}{II}]/[\ion{O}{II}]$^{(a)}$ & $8.86^{+0.10}_{-0.14}$ \\
12+$\log(\rm{O/H})$ & [\ion{N}{II}]/[\ion{O}{II}]$^{(c)}$ & $8.85^{+0.14}_{-0.18}$ \\
\hline
\end{tabular}
\tablefoot{All values use a Chabrier IMF, and take into account the statistical uncertainty of the measurements, as well as the uncertainty in the slit-loss and dust-correction factor if applicable.
$^{(a)}$ Following \citet{2006A&A...459...85N}.
$^{(b)}$ Following \citet{2004MNRAS.348L..59P}.
$^{(c)}$ Following \citet{2002ApJS..142...35K}.
}
\label{tab:phot_par}
\end{center}
\end{table}
|
1,108,101,563,234 | arxiv | \section{Introduction}
The conventional approach to supersymmetry breaking in models of supergravity (SUGRA)
is to assume some form of spontaneous breaking in a hidden sector, mediated
to the visible sector with contains the MSSM via gravitational interactions \cite{Chamseddine:1982jx}.
The overall mass scale is then set by the gravitino mass $m_{3/2}$, and all other masses
for squarks, sleptons, gauginos and higgsinos come out roughly proportional to it,
by demanding the cancellation of the vacuum energy.
Low energy supersymmetry, as required by a natural explanation of the Higgs potential,
fixes $m_{3/2}$ to the electro-weak scale. Now, recently it was argued \cite{Arkani-Hamed:2004fb} that
the fine tuning problem of the Higgs mass may be insignificant compared to the
fine tuning of the cosmological constant, and that an anthropic selection mechanism
(see e.g.\ \cite{landscape})
may then involve actual fine tuning of MSSM parameters.
The mass pattern that was proposed under the name of
split supersymmetry \cite{Giudice:2004tc} has
all the MSSM fermions at the electro-weak
scale, whereas all scalars, except for the one fine tuned Higgs doublet,
get ultra-heavy at a high mass scale. This scenario has attracted some
attention recently \cite{recentwork}. More concretely, the challenge for model building is to
keep the gauginos and higgsinos light, while letting the scalars become very heavy.
The motivation for this originates from supersymmetric grand unification, even
without low energy supersymmetry in the usual sense, and the model is designed to keep
the merits of gauge coupling unification as in the MSSM.
Here, we pursue the perspectives of such patterns in the mass spectrum in the context of
SUGRA and string theory models, based on the paradigm of spontaneous breaking
in a hidden sector.
\\
Given the above mentioned relations that govern gravity mediated supersymmetry breaking,
it seems very hard to achieve hierarchically split mass scales. If all masses are proportional
to $m_{3/2}$, there is no room for flexibility. There is however a loophole to the argument,
which has not so far been explored in the conventional approach in any depth, probably because
it quickly leads to large masses, which were thought unacceptable. It consists of
assuming not only auxiliary F-terms but also D-terms to be generated. In global
supersymmetry, this is well known under the label of supersymmetry breaking mediated
by an anomalous $U(1)$, where large masses can be
avoided \cite{Dvali:1996rj,Binetruy:1996uv}, as will be seen later.
The mechanism basically adds a Fayet-Iliopoulos (FI) term for an extra
anomalous $U(1)$ gauge symmetry, independent of the F-terms which may also
be present. In local supergravity, the two contribute both to the vacuum energy, and thus get tied
together at roughly the same scale. Still, the contribution to scalar masses can
be very different, since the F-terms are mediated via gravity, while the D-terms are
mediated via gauge interactions, which opens up the possibility to have
hierarchically split mass scales, splitting scalars charged under the relevant $U(1)$,
from all other fields, i.e.\ gauginos, higgsinos as well as scalars not charged under
the relevant $U(1)$'s.
This is not quite along the lines of high scale supersymmetry breaking, as advocated in
split supersymmetry, where the gravitino mass itself was assumed at
the high scale, and the main difficulty lies in keeping the gauginos and higgsinos
lighter than $m_{3/2}$.\footnote{To achieve this, it is usually assumed that gravity
mediation of gaugino masses can be avoided first of all. Furthermore, one has to find ways to suppress
contributions from anomaly mediation \cite{Randall:1998uk}. Since the latter is not fully understood within
string theory (see \cite{Antoniadis:2004qn}) and we include the effects of gravity mediation anyway, we will
not consider anomaly mediation in the following.}
Instead, we propose extra contributions to the masses of charged scalars,
which make them heavier than $m_{3/2}$, while the fermions including the gravitino remain light.
\\
The purpose of this paper is to study the confluence of this combined approach with
F- and D-terms in supergravity and string theory models. By this we mean that we
assume that some hidden sector dynamics generates F- and D-terms at some scale
of supersymmetry breaking, but we do not present a full dynamical model, how this happens.
Instead, we analyze the various scenarios that can emerge in the visible sector, and in particular
identify classes of models which generically lead to hierarchically split mass scales.
\subsection{FI-terms in global supersymmetry}
In many models of grand unification, compactification of higher dimensional supergravity
or string theory, the minimal gauge symmetries that can be achieved at low energies
involve various extra abelian $U(1)$ gauge factors beyond the Standard Model gauge group,
i.e.\ the total gauge symmetry is $SU(3)_C\times SU(2)_L\times U(1)^n$, where
among the $U(1)$ there is also the hypercharge. In string theory it often happens
that some of the extra factors are actually anomalous, the anomaly being canceled by a
(generalized) Green-Schwarz (GS) mechanism, in which the gauge boson develops a
Stueckelberg mass and decouples (see e.g.\ \cite{Klein:1999im}). In any case, it is then permissible to add
FI-terms $\xi_a D_a$ to the supersymmetric Lagrangian, one
for each $U(1)_a$. The D-term potential in global supersymmetry is
\begin{eqnarray}
{\cal V}_D ~=~ \sum_a \frac{g_a^2}{2} D_a^2 = \sum_a \frac{g_a^2}{2}
\Big( \sum_i Q_a^i |\tilde f_i|^2 + \xi_a \Big)^2
\end{eqnarray}
and thus $\xi_a>0$ leads to the formation of a condensate for at least some
field $\tilde f_i$ of negative charge $Q_a^i<0$. This breaks the gauge symmetry spontaneously, but
supersymmetry can be restored at the minimum if $\langle D_a \rangle =0$.\footnote{In string theory,
the FI-parameter is usually a function of the moduli, $\xi_a=\xi_a(T_I, \bar T_{\bar I})$.
Therefore, turning on the FI-term can correspond to a flat direction $|\tilde f_i|^2 = \xi_a$
in the total potential, for $Q_a^i=-1$.}
In the MSSM it is usually assumed that the FI-term of the hypercharge is
absent or very small, and does not play a role in the Higgs potential. \\
Whenever the auxiliary field obtains a non-vanishing expectation value $\langle D_a \rangle \not=0$,
supersymmetry is broken, and mass terms are generated for all the charged scalars,
\begin{eqnarray}
m_i^2 ~=~ \sum_a g_a^2 Q_a^i \langle D_a \rangle \ ,
\end{eqnarray}
where it is now assumed that the charges are positive for the MSSM fields, to avoid
breaking of the Standard Model gauge symmetries. This scenario can be achieved in a global
supersymmetric model with a single extra $U(1)_X$
by adding two scalars $\phi^\pm$ to the MSSM, singlets under
$SU(3)_c\times SU(2)_L\times U(1)_Y$, but with charges $\pm 1$ under
$U(1)_X$ \cite{Dvali:1996rj}. The crucial ingredient is an interaction in the superpotential
of the form
\begin{eqnarray} \label{supo}
W_\pm = m \phi^+\phi^-\ .
\end{eqnarray}
Minimizing the full potential
\begin{eqnarray}
{\cal V} = m^2 \left( |\phi^+|^2 + |\phi^-|^2 \right) +
\frac{g_X^2}{2} \Big( \sum_i Q_X^i |\tilde f_i|^2 +|\phi^+|^2-|\phi^-|^2 +\xi_X \Big)^2
\end{eqnarray}
drives the fields to
\begin{eqnarray}
\langle \phi^+ \rangle =0\ , \quad
\langle \phi^- \rangle^2 = \xi_X- \frac{m^2}{g_X^2} \ ,
\end{eqnarray}
and
\begin{eqnarray} \label{F-terms1}
\langle D_X \rangle ~=~ \frac{m^2}{g_X^2}\ , \quad
\langle F_{\phi^+} \rangle ~=~ m \sqrt{\xi_X} +\, \cdots \ .
\end{eqnarray}
Gaugino masses may originate from higher dimensional operators, and
are suppressed by powers of
$M_{\rm Pl}$\footnote{The Planck mass $M_{\rm Pl}$
is defined so that $M_{\rm Pl}=\kappa^{-1}=(8\pi G)^{1/2}=2.4\times 10^{18}$ GeV.},
\begin{eqnarray} \label{gaugino}
m_\lambda ~\sim~ \frac{1}{M_{\rm Pl}^2} \langle F_{\phi^+}\phi^- + F_{\phi^-}\phi^+ \rangle
~\sim~ m \frac{\xi_X}{M_{\rm Pl}^2}\ .
\end{eqnarray}
Assuming $m \sim {\cal O}({\rm TeV})$ and $\xi \sim {\cal O}(M_{\rm Pl}^2)$
on gets masses at the electro-weak scale.
Depending on the precise scale and the
charges of the MSSM scalars under the extra $U(1)_X$, these contributions to
their masses can be very important in the soft breaking Lagrangian.
A central point to notice here is the fact that the masses that follow from
the FI-terms are directly proportional to the expectation values of the auxiliary $D_a$ fields,
they are mediated by the anomalous $U(1)_X$, whereas the masses induced via the F-terms are
suppressed by $M_{\rm Pl}$ through their mediation by gravity. The function of
the extra fields $\phi^\pm$ lies in absorbing the potentially large FI parameter
$\xi_a$, such that $\langle D_a \rangle^{1/2} \sim {\cal O}({\rm TeV})$, consistent with the
standard scenario of superpartner masses at the electro-weak scale.
\\
We will discuss this type of model and its modifications in the frame work
of supergravity and string theory. However, before getting into the details of the
extended model, we discuss how such FI-terms arise in string theory.
\subsection{A single anomalous $U(1)$ and the heterotic string}
The four-dimensional GS mechanism consists of the cancellation
of the anomalies of the usual one-loop triangle diagrams with tree-level exchange of an axionic
scalar $\sigma$. This refers to the mixed abelian-gravitational
and abelian-non-abelian anomalies at the same time.
The relevant terms in the action are usually written
in terms of the Hodge-dual 2-form $B_{\mu\nu}$, related to $\sigma$ by
$\partial_\mu \sigma \sim \epsilon_{\mu\nu\rho\kappa} \partial^\nu B^{\rho\kappa}$, as
\begin{eqnarray} \label{gscpl}
c_A \partial_\mu B_{\nu\rho} \omega_3^{\mu\nu\rho} + m_X \epsilon^{\mu\nu\rho\kappa} B_{\mu\nu} F^X_{\rho\kappa}
\end{eqnarray}
where $c_A$ and $m_X$ are two coupling constants, $\omega_3$ the Chern-Simons
(CS) 3-form, and $F^X$ the gauge field strength of the relevant $U(1)_X$.
Now, $\sigma$ is the imaginary part of some complex scalar in a chiral multiplet.
For the heterotic string, the only such scalar that
participates in the GS mechanism is the dilaton-axion field $S|_{\theta=\bar\theta=0}=s + i\sigma$ (see
\cite{Lalak:1999bk} for an overview).
Its action is described by the K\"ahler potential
${\cal K} (S,\bar S)= -\ln(S+\bar S)$. Since Eq.(\ref{gscpl}) implies a non-linear gauge transformation
$\delta_X S \sim m_X\epsilon_X$ under the $U(1)_X$, the gauge invariance demands a redefinition of the
K\"ahler potential $\ln(S+\bar S) \rightarrow \ln(S+\bar S - m_X V_X)$, where
$V_X$ is the vector multiplet superfield. The Lagrangian then involves a Stueckelberg mass term
with mass proportional to $m_X$ for the gauge boson of this $U(1)_X$,
which absorbs the scalar $\sigma$ as its longitudinal component.
In addition, an FI-term $\xi_X D_X$ is present, with $\kappa^2 \xi_X \sim m_X/s$.
For the heterotic string, this FI-term is generated at one-loop
and the coefficient reads \cite{Dine:1987xk}
\begin{eqnarray}
\kappa^2 \xi_X ~\sim~ \frac{m_X}{s} ~\sim~ \frac{g_X^2 {\rm tr}(Q_X)}{192\pi^2} \ .
\end{eqnarray}
In the presence of an interaction (\ref{supo})
the scalar fields $\tilde f_i$ charged under the $U(1)_X$ acquire masses given by
\begin{eqnarray}
m_{i}^2 ~=~ Q_X^i m^2 ~\sim~ {\cal O}({\rm TeV})\ .
\label{mass1}
\end{eqnarray}
The remarkable feature of Eq.(\ref{mass1}) is that it is independent of
the FI-parameter. Thus, the vector boson gets a mass of the order of $m_X$, which is close to
the Planck scale, whereas the charged sfermions and gauginos
remain massless at the high scale, and get masses of the order of the electro-weak scale.
This is the standard scenario of supersymmetry breaking via an anomalous $U(1)$ with GS
mechanism.
\\
\subsection{Multiple Anomalous $U(1)$ Symmetries and D-branes}
Orientifold string compactifications \cite{Angelantonj:2002ct} usually involve more than one
anomalous $U(1)$ factor. While in the heterotic string it is only the
axionic partner of the dilaton
that participates in the GS anomaly cancellation, now
all the axionic scalars that follow from the reduction of
the RR forms from ten dimensions can do so \cite{Ibanez:1998qp}.
In orientifold compactifications of type IIB strings, the
relevant RR scalars originate from the twisted sectors.
The FI-parameters $\xi_a$ are then functions of
the expectation values of the real parts of these twisted scalars, instead of the dilaton $s$.
For a special example of this class of models, in a toroidal
orbifold $\mathbb T^6/\mathbb Z_3$, it was
shown, that no FI-term was generated at one-loop,
consistent with the fact that the twisted scalars vanish at the
orbifold point \cite{Poppitz:1998dj}.
As another class of models, orientifolds with intersecting
(type IIA) or magnetized (type IIB) D-branes have been studied extensively
in the recent past, most prominently for their very attractive features to produce
Standard Model or MSSM like gauge groups and spectra. For these,
untwisted RR scalars participate in the GS mechanism. Again the FI-term at
tree-level (i.e.\ from a disc diagram, or the
dimensional reduction of the Born-Infeld action) is
proportional to the modulus that combines with the
axionic scalar from the GS mechanism into a complex scalar.
The GS couplings analogous to Eq.(\ref{gscpl}) now involve many scalars
\begin{eqnarray} \label{gs2}
\sum_{I,A} c^I_A \partial_\mu B^I_{\nu\rho} \omega_3^{\mu\nu\rho}
+ \sum_{a,I} m_a^I \epsilon^{\mu\nu\rho\kappa} B^I_{\mu\nu} F^a_{\rho\kappa} \ ,
\end{eqnarray}
where $I$ labels the scalars $\sigma_I$, given by
$\partial_\mu \sigma_I \sim \epsilon_{\mu\nu\rho\kappa} \partial^\nu B^{I\rho\kappa}$, and
$a$ the anomalous $U(1)_a$ factors with field strengths $F^a$ for the superfield $V_a$.
The constants $c_A^I$ are labeled by $A$ for the different anomalies, i.e.\ the different CS
forms that can appear. We let
$T_I|_{\theta=\bar\theta=0} = t_I + i\sigma_I$ be the complex scalars.
Again Eq.(\ref{gs2}) implies that the $T_I$ transform under $U(1)_a$ whenever the coupling
coefficient $m_a^I\not=0$. Then the K\"ahler coordinate $T_I$ is replaced in the
following way
\begin{eqnarray}
{\cal K}(T_I+\bar T_I) ~\rightarrow~ {\cal K}(T_I+\bar T_I - \sum_a m_a^I V_a ) \ .
\end{eqnarray}
Depending on the precise form of the K\"ahler potential, a FI-term will be generated
from this expression, that will depend on the vacuum expectation value of $t_I$.
The simplest expression would be
\begin{eqnarray}
\kappa^2 \xi_a ~\sim~ \sum_I m_a^I t_I \ .
\label{13}
\end{eqnarray}
It was stressed in \cite{Ibanez:1998qp} that the $\xi_a$ given by Eq.(\ref{13})
can in principle be of any size, as opposed to the result for the heterotic string case.
Another important observation is to note, that the FI-terms are not necessarily tied to
anomalous gauge symmetries, but only the non-vanishing Stueckelberg coupling $m_a^I\not=0$
has to exist.\footnote{As mentioned earlier, we shall here not attempt to model the dynamics of the
hidden sector in any detail, and therefore also do not try to answer, how
the FI-parameters are generated dynamically.
Since they are moduli-dependent functions, a meaningful answer would
have to address the moduli-stabilization at the same time.
}
\section{FI-terms in supergravity and string theory}
We now examine the patterns of soft supersymmetry breaking that arise
from an effective string theory Lagrangian with one or more FI-terms, motivated by
the appearance of multiple $U(1)$ factors in orientifold models, that can develop FI-terms.
\subsection{The vacuum energy}
The degrees of freedom of the model are assumed to be given by the fields $f_i$ of the MSSM,
the extra gauge vector multiplets for the $U(1)_a$, the moduli $T_I$ of the gravitational sector, plus
the axion-dilaton $S$, which includes the fields that participate in the GS mechanism, producing
Stueckelberg masses for the gauge bosons and FI-terms. Furthermore, we can add extra fields like the
$\phi^\pm$ of the globally supersymmetric model,
with charges $\pm \alpha_a$ under $U(1)_a$. The effective scalar potential
is given by the ${\cal N}=1$ supergravity formula \cite{Cremmer:1982en}
\begin{eqnarray}
{\cal V} ~=~ - \kappa^{-4}e^{-G} [G^{M\bar N}G_MG_{\bar N} +3] + {\cal V}_D \ ,
\end{eqnarray}
with
\begin{eqnarray}
G ~=~ - \kappa^2 {\cal K}- \ln(\kappa^6 WW^{\dagger}) \ ,
\end{eqnarray}
where indices $M,N$ run over all fields.
We define the dilaton and moduli fractions of the vacuum energy by
\begin{eqnarray}
|\gamma_S|^2=-\frac{1}{3} G^{S\bar S}G_SG_{\bar S}\ , \quad
|\gamma_I|^2=-\frac{1}{3}
G^{I\bar I}G_IG_{\bar I}\ ,
\end{eqnarray}
and $|\gamma_{\pm}|^2$ in a similar fashion.
It also turns out to be useful to introduce the following combinations
\begin{eqnarray}
&&
m_{3/2} = \kappa^{-1} e^{-G/2}\ , \quad
m_\pm = e^{\kappa^2 {\cal K}/2} m \ , \quad
x = \kappa \langle \phi^+ \rangle \ , \quad
y = \kappa \langle \phi^-\rangle \ . \quad
\end{eqnarray}
Similarly, all other fields are made dimensionless.
Imposing the restriction on the model
that the vacuum energy vanishes (through fine tuning) one has
\begin{eqnarray}
|\gamma_S|^2 + \sum_I |\gamma_I|^2 + |\gamma_+|^2+ |\gamma_-|^2
+\frac{1}{3m_{{3}/{2}}^2M_{\rm Pl}^2}
\sum_a \frac{g_a^2}{2} D_a^2 =1\ ,
\end{eqnarray}
where $D_a = \alpha_a |\phi^+|^2-\alpha_a |\phi^-|^2 + \xi_a$.
This implies an immediate bound on the expectation values of the auxiliary fields
$F_I = D_I W= \partial_I W + \kappa^2(\partial_I {\cal K}) W$ and $D_a$,
\begin{eqnarray}
\langle F_I \rangle ~\lesssim~ m_{{3}/{2}} M_{\rm Pl}\ , \quad
\langle D_a \rangle ~\lesssim~
m_{{3}/{2}} M_{\rm Pl}\ ,
\label{19}
\end{eqnarray}
where we ignore prefactors involving the K\"ahler potential.
As long as $m_{{3}/{2}}\sim{\cal O}({\rm TeV})$, Eq.(\ref{19}) implies
roughly $\langle D_a \rangle^{1/2} \lesssim 10^{10-13}\, {\rm GeV}$,
which is the usual intermediate supersymmetry breaking scale in SUGRA models.
The masses that are generated by the F-terms are given by
$F_I/M_{\rm Pl}\sim{\cal O}({\rm TeV})$, whereas the D-terms would be
able to produce much larger mass terms proportional to $D_a^{1/2}$.
This means that the mass parameter in the superpotential (\ref{supo}), which
had to be fine tuned to the electro-weak scale, can now also be assumed as large as the
intermediate scale, $m\lesssim \sqrt{m_{{3}/{2}}M_{\rm Pl}}$.
Further, we note that the scenario with a Planck
scale sized FI-parameter, as is unavoidable for the
heterotic string in the presence of an anomalous $U(1)$,
is only consistent with a Planck scale sized gravitino mass.
In orientifold D-brane models, as mentioned earlier, the FI-parameter can in principle
be of any value, and the problem does not occur.
\\
In scenarios with split supersymmetry, the gravitino mass itself is not restricted to a small value. However,
gravity mediation generically leads to a contribution to gaugino and higgsino masses which
is proportional to the gravitino mass, and therefore $m_{{3}/{2}}\sim{\cal O}({\rm TeV})$
is unavoidable in the present context. This then really puts an upper bound $10^{10-13}\, {\rm GeV}$ on
the high mass scale allowed for the sleptons and squarks.
\\
Before going into the various scenarios, let us first assemble a few general definitions and formulas
for the supergravity version of the model of supersymmetry breaking mediated by one or many anomalous $U(1)$.
For the K\"ahler potential ${\cal K}$ we write
\begin{eqnarray}
{\cal K} &=& {\cal K}_{\rm hid}(T_I,\bar T_{\bar I}) +
{\cal K}_{i\bar\imath}(T_I,\bar T_{\bar I}) f_{i}f_{\bar\imath}^{\dagger}
+ {\cal K}_+(T_I,\bar T_{\bar I}) \phi^+\phi^{+\dagger}
+ {\cal K}_-(T_I,\bar T_{\bar I}) \phi^-\phi^{-\dagger} \ ,
\nonumber
\end{eqnarray}
where we have now included $S$ among the $T_I$.
The gauge kinetic functions are moduli-dependent,
\begin{eqnarray}
f_a ~=~ f_a(T_I)\ , \quad \frac{1}{g_a^2}~=~ \Re(f_a) \ ,
\end{eqnarray}
but independent of $\phi^\pm$.
For the superpotential we assume the following factorized form
\begin{eqnarray}
W ~=~ W_{\rm MSSM}(f_i) + W_{\rm hid}(T_I) + W_\pm(\phi^+,\phi^-) ~=~ m\phi^+\phi^- + W_0
\end{eqnarray}
where $W_{\rm MSSM}$ contains the quark, lepton and Higgs fields, $W_{\rm hid}$
contains the fields of the hidden sector which break supersymmetry spontaneously
by generating auxiliary field components for
$F_I = D_IW_{\rm hid}$, while $W_\pm$ is still given by
Eq. (\ref{supo}).\footnote{This implies an assumption on the absence of any coupling among MSSM fields
and $\phi^\pm$ in the superpotential, which may be problematic in the context of a concrete model.
Furthermore, we also ignored any cross-coupling in the K\"ahler potential, where in principle the
moduli-dependence of the various coefficients could also involve $\phi^\pm$.}
With this, the total D-term potential ${\cal V}_D$ is given by
\begin{eqnarray}
{\cal V}_D = {\cal V}_D^{\rm MSSM} + \sum_a\frac{g_a^2}{2} \Big(\sum_i Q_a^i {\cal K}_{i\bar\imath} |\tilde f_i|^2
+\alpha_a {\cal K}_+ |\phi^+|^2-\alpha_a {\cal K}_- |\phi^-|^2 +\xi_a\Big)^2\ .
\label{d1}
\end{eqnarray}
Here ${\cal V}_D^{\rm MSSM}$ is the D-term arising from
the $SU(2)_L\times U(1)_Y$ sector, which will not be important.
The standard expressions for the soft breaking terms that originate from the F-terms only,
are \cite{Kaplunovsky:1993rd}
\begin{eqnarray}
m_\lambda ~=~ \frac{1}{2\Re(f_a)} F^I \partial_I f_a\ ,
\end{eqnarray}
for gaugino masses, and
\begin{eqnarray}
m_{{\rm gr,}i}^2 ~=~ m_{3/2}^2 - F^I \bar F^{\bar J} \partial_I \partial_{\bar J} \ln( {\cal K}_{i\bar\imath} )
\end{eqnarray}
for scalar masses. For $F_I \sim m_{3/2}M_{\rm Pl}$, both masses are of the order of $m_{3/2}$.
\subsection{The simplest model}
The simplest model that already displays the effects of the FI-terms is given
by assuming one or more FI-terms being generated by extra $U(1)_a$ gauge factors, and only
including the MSSM fields with arbitrary positive charges, but leaving out the
extra fields $\phi^\pm$.\footnote{It may sound very restrictive to allow only positive charges here,
and it really would be in any reasonable model derived from a GUT or string theory. However,
there are well known cases in string theory compactifications, where higher order corrections
in the derivative expansion of the effective action (such as the Born-Infeld Lagrangian)
lead to a lifting of tachyonic negative masses in the presence of FI-terms, even if some fields have
negative charge. We will come to explain this in some more detail later.}
In that case, supersymmetry is broken, and the D-terms are trivially given by
\begin{eqnarray}
\frac{g_a^2}{2} \langle D_a^2 \rangle ~=~ \frac{g_a^2}{2} \xi_a^2\ .
\end{eqnarray}
Together with potential F-terms, they generate masses
\begin{eqnarray}
m_i^2 ~=~ m_{{\rm gr,}i}^2 + \sum_a g_a^2 Q_a^i {\cal K}_{i\bar\imath} \xi_a
\end{eqnarray}
for all charged scalars.
On the other hand, the masses of gauginos (and higgsinos) are unaffected by the D-term, at least at leading
order, and would be dictated by the F-terms to be of order $m_{3/2}$.
In a scenario with $m_{3/2}\sim {\cal O}({\rm TeV})$, and $D_a \sim 10^{10-13}\, {\rm GeV}$ for at least one
FI-term, this provides a hierarchical split of energy scales, however, the high scale cannot move
up all the way to the Planck scale. The most simple charge
assignment would give positive charges of order one to all MSSM fields, except the Higgs,
and thus the sfermion sector
of the MSSM would become very heavy and undetectable at LHC.\footnote{This charge assignment would
indeed render the $U(1)$ anomalous.}
There may of course also be other interesting
patterns to consider, such as different charges for the three generations, different charges for
different $SU(5)$ multiplets, which would leave the gauge coupling unification intact,
or different charges for left- and right-handed fields, which could be more easily realized in
certain D-brane models from string theory.
\subsection{The full potential with a single $U(1)_X$}
The essence of the model of \cite{Dvali:1996rj},
where an anomalous $U(1)$ with its FI-term is responsible for
supersymmetry breaking, is the scalar condensate for $\phi^-$ that cancels the contribution
of the FI-parameter to the D-term, up to small electro-weak scale sized contribution, given the
interaction (\ref{supo}) in the superpotential. Since such a condensate breaks the gauge symmetry,
one has to add extra chiral multiplets $\phi^\pm$ to the MSSM, neutral under the MSSM gauge
symmetries. The model of \cite{Dvali:1996rj} is within the framework of global supersymmetry, and it
is of obvious interest to study the embedding of this original model into supergravity.
So, we focus first on the case when there is just one extra $U(1)_X$,
which develops a FI-term.
The potential can be written as
\begin{eqnarray}
\kappa^4 {\cal V} &=& \kappa^4 {\cal V}_{\rm hid} +
e^{\kappa^2 {\cal K}} \Big[ \kappa^2 m^2 \left( |x|^2 + |y|^2 \right)
+ \kappa^6 |W|^2 \left( |{\cal K}_+ x|^2 + |{\cal K}_- y|^2 -3 \right)
\\
&&
\hspace{.5cm}
+~ \kappa^4 m \left( {\cal K}_+ + {\cal K}_- \right) \left( xy \bar W+\bar x\bar y W\right) \Big]
+ \frac{g_X^2}{2} \Big [ {\cal K}_+|x|^2 - {\cal K}_-|y|^2 +\xi_X \Big]^2 \ ,
\nonumber
\end{eqnarray}
where we also have replaced $\xi_X \rightarrow \kappa^{-2} \xi_X$, and defined
\begin{eqnarray}
{\cal V}_{\rm hid} ~=~ e^{\kappa^2 {\cal K}} {\cal K}^{I\bar J} D_I W D_{\bar J}\bar W
\end{eqnarray}
for the contribution of the hidden sector. It is essential for the fine tuning of the vacuum energy.
The two minimization conditions are $\partial_x V = \partial_y V = 0$. To keep things simple,
we now also set all relevant phases to zero, i.e.\ we treat $x,y,W$ as real.
Then one gets the following two equations
\begin{eqnarray}
&& {\cal K}_+ x [ \kappa^2 g_X^2 D_X + \kappa^4 {\cal V}_F] + \frac{m_\pm^2}{M_{\rm Pl}^2} [x+ xy^2 ({\cal K}_+ + {\cal K}_-)]
\\
&& \hspace{3cm} + \frac{m_{3/2}^2}{M_{\rm Pl}^2} {\cal K}_+^2 x
+ \frac{m_\pm m_{3/2}}{M_{\rm Pl}^2} y [({\cal K}_+ x)^2+({\cal K}_- y)^2 +2{\cal K}_- -3] ~=~0\ ,
\nonumber \\
&&
{\cal K}_- y [- \kappa^2 g_X^2 D_X + \kappa^4 {\cal V}_F] + \frac{m_\pm^2}{M_{\rm Pl}^2} [y+ x^2y ({\cal K}_+ + {\cal K}_-)]
\nonumber \\
&& \hspace{3cm} + \frac{m_{3/2}^2}{M_{\rm Pl}^2} {\cal K}_-^2 y
+ \frac{m_\pm m_{3/2}}{M_{\rm Pl}^2} x [({\cal K}_+ x)^2+({\cal K}_- y)^2 +2{\cal K}_+ -3] ~=~0\ .
\nonumber
\end{eqnarray}
where ${\cal V}_F$ and $D_X$ are defined so that ${\cal V} = {\cal V}_F + \frac12 g_X^2 D_X^2$,
where ${\cal V}_F$ and ${\cal V}_{\rm hid}$ are related by
\begin{eqnarray}
{\cal V}_F &=& {\cal V}_{\rm hid} +
\Big[ m_{\pm}^2 M_{\rm Pl}^2 \left( |x|^2 + |y|^2 \right)
+ |m_{3/2}|^2 M_{\rm Pl}^2\left( |{\cal K}_+ x|^2 + |{\cal K}_- y|^2 -3 \right)
\\
&&
\hspace{1cm}
+~ m_{\pm}M_{\rm Pl}^2 \left( {\cal K}_+ + {\cal K}_- \right) \left( xy \bar m_{3/2}+\bar x\bar y m_{3/2}\right) \Big],
\nonumber
\end{eqnarray}
Note that the redefined mass parameters are field-dependent.
In the following, we also restrict to canonically normalized fields, and set
${\cal K}_+={\cal K}_-=1$.
We have illustrated the full potential in figure 1, using values
\begin{eqnarray} \
\kappa^4 {\cal V}_{\rm hid} ~=~ 3\ , ~~
\kappa^6 e^{\kappa^2{\cal K}}|W_0|^2 =1\ , ~~
\kappa^2 e^{\kappa^2{\cal K}} m^2=1 \ , ~~
g_X=\sqrt{200} \ , ~~
\kappa^2\xi_X=10^{-1}\ .
\nonumber
\end{eqnarray}
This corresponds to setting some parameters equal, dividing the total potential
by $M_{\rm Pl}^2 m_{3/2}^2$, and rescaling the Planck mass by many order of magnitude,
so that $M_{\rm Pl}/m_{3/2}=10$, just
to suppress the very steep part of the potential. It turns out, that for the relevant
range of parameters $x\neq 0 \neq y$ at the minimum.
\begin{figure}[h]
\begin{center}
\resizebox{6cm}{!}{\psfig{figure=pot1.eps,width=4cm}}
\caption{Rescaled potential ${\cal V}(x,y)$} \label{pot1}
\end{center}
\end{figure}
Since the minimization of the F-term potential basically consists of balancing
terms that scale like $m_\pm^2 |\phi^\pm|^2$, $m_{3/2}^2 |\phi^\pm|^2$
or with inverse powers of $M_{\rm Pl}$, with the negative term
$m_{3/2}^2 M_{\rm Pl}^2$, it is intuitively clear that $x$ and $y$ will
be roughly bounded through the most dangerous term by
$m_{3/2} m^{-1}_\pm$ or ${\cal O}(1)$. This is of course only
valid as long as $|\phi^\pm|^2 \gg \xi_X$, otherwise $|\phi^-|^2$ gets tied to $\xi_X$.
In any case one has that $F_{\phi^\pm}/M_{\rm Pl}\lesssim m_{3/2}$, which is sufficient for light gauginos.
\subsection{Limiting cases: $m_{3/2}=0$ or $m_\pm=0$}
We consider now some limiting cases for the above. To see how the
model of \cite{Dvali:1996rj} emerges, we
take the flat limit by setting
\begin{eqnarray}
m_{3/2}~=~ 0\ .
\end{eqnarray}
The two minimization conditions reduce to equations homogeneous in $x,y$ respectively.
One can convince oneself that $x=0$ is stable. The
solution then reduces to the known case of global supersymmetry, where
\begin{eqnarray}\label{sol1}
x~=~0\ , \quad y^2 ~=~ \xi_X + 1 \pm \sqrt{ 1+\frac{2m_\pm^2}{g_X^2M_{\rm Pl}^2}}
~\sim~ \xi_X - \frac{m_\pm^2}{g_X^2M_{\rm Pl}^2} \ .
\label{m32zero}
\end{eqnarray}
The auxiliary fields are also identical to (\ref{F-terms1}),
since $D_I W =\partial_I W$. Thus, in this limit, all masses are of the order of $m_\pm\sim m$.
\\
On the other hand, it is interesting to study the supergravity corrections to the case that
allowed to restore supersymmetry in global supersymmetry, when the
mass term in the superpotential is absent,
\begin{eqnarray}
m_\pm ~=~ 0\ .
\end{eqnarray}
In this case the solution is given by
\begin{eqnarray}
x^2 + y^2 ~=~ 2 - \frac{{\cal V}_{\rm hid}}{m_{3/2}^2 M_{\rm Pl}^2} \ , \quad
x^2 - y^2 ~=~ - \xi_X \ .
\label{mzero}
\end{eqnarray}
Obviously, there is no supersymmetry breaking by D-terms, and $\langle D_X\rangle =0$.
Actually, in this limit the vacuum energy
cannot be fine tuned without extra contributions to the potential, irrespective of ${\cal V}_{\rm hid}$,
since it is always given by $- m_{3/2}^2M_{\rm Pl}^2$. This is however an artifact of the limit $m_\pm=0$, and
for non-vanishing $m_\pm$, the value at the minimum
can be shifted by varying ${\cal V}_{\rm hid}$. Finally as a special case of
Eq.(\ref{mzero}), one may consider the case
\begin{eqnarray}
{\cal V}_{\rm hid}~=~ (2-\xi_X)m_{3/2}^2M_{\rm Pl}^2
\end{eqnarray}
which gives
\begin{eqnarray}
x~=~0\ , \quad y^2~=~\xi_X\ .
\label{mzero1}
\end{eqnarray}
The values of Eq.(\ref{mzero1}) give the minimum of the potential that is in some sense
analogous to the case Eq.(\ref{m32zero}).
\subsection{Multiple $U(1)$ gauge symmetries}
With multiple FI-terms in the potential, it is clear that supersymmetry breaking
can occur more generically. We have seen above, that FI-terms that cannot be canceled by
scalar condensates lead to large scale masses, whereas a scalar condensate with
a single FI-term was able to lower the masses to the electro-weak
scale $m_{3/2}$, similar to \cite{Dvali:1996rj}.
If multiple FI-terms now come accompanied by the same number of extra charged fields to
develop condensates, then the Lagrangian would just be a sum of identical copies of the one
of the previous sections, and nothing new is found. An interesting case arises, when
there is a mismatch, and not all the FI-parameters can be absorbed by fields like $\phi^-$,
and thus some large masses can be generated.
We now analyze the situation, when there still is only a single set of extra charged
fields $\phi^\pm$, that takes a non-vanishing expectation value, but there are multiple $U(1)_a$
gauge symmetries, under which it is charged.
The potential is only modified by summing over D-terms,
\begin{eqnarray}
\kappa^4 {\cal V}_D &=&
\sum_a \frac{g_a^2}{2} \Big [ \alpha_a {\cal K}_+ |x|^2 - \alpha_a{\cal K}_- |y|^2 +\xi_a \Big]^2 \ ,
\nonumber
\end{eqnarray}
again making $\xi_a$ dimensionless by $\xi_a\rightarrow\kappa^{-2} \xi_a$.
Regarding the vacuum energy cancellation, the new minimization conditions read
\begin{eqnarray}
&& {\cal K}_+ x \Big[ \kappa^2 \sum_a \alpha_a g_a^2 D_a + \kappa^4 {\cal V}_F \Big]
+ \frac{m_\pm^2}{M_{\rm Pl}^2} [x+ xy^2 ({\cal K}_+ + {\cal K}_-)]
\\
&& \hspace{3cm} + \frac{m_{3/2}^2}{M_{\rm Pl}^2} {\cal K}_+^2 x
+ \frac{m_\pm m_{3/2}}{M_{\rm Pl}^2} y [({\cal K}_+ x)^2+({\cal K}_- y)^2 +2{\cal K}_- -3] ~=~0\ ,
\nonumber \\
&&
{\cal K}_- y \Big[ -\kappa^2 \sum_a \alpha_a g_a^2 D_a + \kappa^4 {\cal V}_F \Big]
+ \frac{m_\pm^2}{M_{\rm Pl}^2} [y+ x^2y ({\cal K}_+ + {\cal K}_-)]
\nonumber \\
&& \hspace{3cm} + \frac{m_{3/2}^2}{M_{\rm Pl}^2} {\cal K}_-^2 y
+ \frac{m_\pm m_{3/2}}{M_{\rm Pl}^2} x [({\cal K}_+ x)^2+({\cal K}_- y)^2 +2{\cal K}_+ -3] ~=~0\ .
\nonumber
\end{eqnarray}
It is not necessary to study the general solution here, since the multiple FI-terms already
break supersymmetry without the superpotential (\ref{supo}), and one can therefore set $m_\pm=0$.
With ${\cal K}_+={\cal K}_-=1$ one can easily solve for $x,y$, finding
\begin{eqnarray}
x^2 + y^2 ~=~ 2 - \frac{{\cal V}_{\rm hid}}{m_{3/2}^2 M_{\rm Pl}^2} \ , \quad
x^2 - y^2 ~=~ - \frac{\sum_a g_a^2 \xi_a }{M_{\rm Pl}^2 \sum_a g_a^2}\ .
\end{eqnarray}
Thus, in a generic situation, where not all $\xi_a$ are equal, supersymmetry will be
broken, and $\langle D_a\rangle \sim \xi_a \sim m_{3/2} M_{\rm Pl}$.
In this case, the vacuum energy is given by the negative F-term contribution plus D-terms, and can
be fine tuned even without ${\cal V}_{\rm hid}$.
The F-terms are given by $F_{\phi^\pm}=\kappa^2 \phi^\pm W_0 \lesssim m_{3/2}M_{\rm Pl}$.
The contributions to the scalar masses now look like (with dimensionful $\xi_a$)
\begin{eqnarray}
m_i^2 ~=~ m_{{\rm gr},i}^2 + \sum_a g_a^2 Q_a^i D_a ~\sim~ \sum_a g_a^2 Q_a^i \xi_a +\, \cdots
\end{eqnarray}
where the terms in parentheses can be of the same order of magnitude, but the
gravity-mediated contributions are negligible. This realizes the split supersymmetry scenario,
if $\xi_a\sim 10^{10-13}\, {\rm GeV}$, for at least two FI-parameters, but with
a low gravitino mass $m_{3/2}\sim {\cal O}({\rm TeV})$.
\\
It may also be interesting to note that at the same time all other soft breaking
parameters will get modified in the presence of FI-terms and scalar field condensates.
This happens through the prefactor $e^{\kappa^2{\cal K}}$ in the total potential.
For instance, the bi-linear couplings $B$ and the tri-linear couplings $A$ are (see e.g.\ \cite{Nath:2002nb})
\begin{eqnarray}
A_{\alpha\beta\gamma}^0\ , ~~B_{\alpha\beta}^0 ~\propto~
m_{3/2} e^{\kappa^2{\cal K}/2} ~=~ m_{3/2} (S+\bar S)^{1/2} e^{\frac12 (|x|^2 + |y|^2) + \, \cdots} \ ,
\end{eqnarray}
which may lead to extra suppression factors, depending on the model. Further, we note that the
Higgs mixing parameter $\mu$ that enters in the superpotential in the form $\mu H_1H_2$,
where $H_1$ and $H_2$ are the two Higgs doublets of MSSM, remains essentially unaffected.
This is so because
in string/supergravity scenarios one expects the $\mu$ term to arise from the K\"ahler
potential using a
K\"ahler tranformation after the spontaneous breaking of supersymmetry has taken place. The K\"ahler
transformation is sensitive to the F-term breaking and not the D-term. Thus, one expects
the same mechanism that produces a $\mu$ term of electro-weak size for supergravity models to
hold in this case as well.
\subsection{Scenarios with partial mass hierarchies}
To summarize, there are several scenarios that are possible,
which would lead to different patterns in the mass spectrum:
$(i)$ There is only one FI-term, and an extra scalar field $\phi^-$ beyond the
MSSM fields which forms a vacuum expectation value. Here all the soft scalar masses will be of
electro-weak size. $(ii)$ In the case of two or more $U(1)_a$ with non-vanishing $\xi_a$,
and the charges $Q^i_a$ non-zero and sufficiently generic, the scalar masses are
of the order of $\xi_a$, while the gauginos stay light. One can thus get
a hierarchical splitting of scalar and fermion masses.
$(iii)$ One may also achieve hybrid
scenarios, when only some of the charges $Q_a^i$ are non-vanishing, or
such that the high mass terms just cancel out. Below we consider two
specific scenarios with partial splitting of scalars.
The implications of this partially split scenario will be very
different from the usual SUGRA scenario and also from the high scale
supersymmetry scenario of split supersymmetry.
\subsubsection{Model I: $2+1$ generations}
As the first model we consider the case with
non-vanishing charges for the first and second
generation of squarks and sleptons, but vanishing charges for the
third generation. In this circumstance
the former will
develop super heavy masses and will not be accessible at the LHC,
as opposed to the third generation.
The above implies that the dangerous flavor changing neutral currents will be automatically
suppressed. Some of the signals of this scenario will be very unique, such as
the decay of the gluino ($\tilde g$). In SUGRA its
decay modes are as follows
\begin{eqnarray}
\tilde g ~\rightarrow~ \bar u_i\tilde u_i\, ,\ \bar d_i \tilde d_i
~\rightarrow~
\left\{ { u_i \bar u_i \tilde \chi_j^0\, ,\ d_i\bar d_i \tilde \chi_j^0
\atop u_i \bar d_i \tilde \chi_k^-\, ,\ d_i\bar u_i \tilde \chi_k^+ } \right. \ .
\end{eqnarray}
where $i=1,2,3$ are the generational indices, $j=1,2,3,4$,
and $k=1,2$ for neutralinos and charginos.
However, for the case of the model under consideration
the decay through the first two generations is highly suppressed, and the gluino
decay can only occur through third generation squarks
via the modes $\tilde g\rightarrow b\bar b \tilde \chi_j^0$ with
admixtures of $\tilde g\rightarrow b\bar t \tilde\chi_k^+$,
$\tilde g\rightarrow t\bar b \tilde\chi_k^-$, and $\tilde g\rightarrow t\bar t \tilde\chi_j^0$,
depending on the gluino mass.
There will be no contribution to the anomalous magnetic moment of the muon at the one-loop level.
The following set of phases can arise: $\theta_{\mu}$, $\xi_i$,
and $\alpha_A$, where $\theta_{\mu}$ is the phase of the Higgs mixing parameter $\mu$,
$\xi_i$ ($i=1,2,3$) are the phases of the $SU(3)_c$, $SU(2)_L$ and $U(1)_Y$ gaugino masses,
and $\alpha_A$ is the phase of the common trilinear coupling for the third generation scalars
(However, it should be kept in mind that not all the phases are independent and only
certain combinations enter in physical processes).
Because of the super heavy nature of the first two generations, the one-loop
contributions to the electric dipole mement (edm) of the electron and of the neutron are
highly suppressed. However, there can be higher loop corrections to the edms. Specifically,
the neutron edm can get a contribution from the CP violating dimension six operator.
Unification of gauge coupling constants at the one-loop level will
remain unchanged, although at the two-loop level there will be effects
from the splittings. In this model the staus can be light and thus
co-annihilation of the LSP neutralinos with the staus can occur, allowing
for the possibility that the neutralino relic density could be consistent with the current data.
Finally, proton decay from dimension five operators
would not arise via dressings from the first and second generation
squarks and sleptons, but can arise from the dressings of the
third generation sfermions.
\subsubsection{Model II: ${\bf 5}+{\bf 10}$ split spectrum}
As another illustrative example, we consider a model where the squarks and sleptons belonging to
a {\bf 10} of $SU(5)$ have non-vanishing charges and high scale masses, while the
squarks and sleptons belonging to a {\bf 5} have vanishing
charges. In this case, the only light scalars aside from Higgs
bosons will be $\tilde d^C_i, \tilde e_{Li}, \tilde \nu_i$ ($i=1,2,3$).
In this model, unlike the case of model I, there is a one-loop supersymmetric
contribution to the muon anomalous magnetic moment $g_{\mu}-2$.
The CP phases in this model consist of $\theta_{\mu}$ and $\xi_i$ ($i=1,2,3$).
There also is a one-loop supersymmetric contribution to the edms of
the electron and of the neutron.
Further, the decays of the gluino, the chargino and the neutralino can
occur only via a smaller subset of states and thus their decay widths
will be relatively smaller, though they will still decay within the
detection chamber. Finally proton decay through dimension five operators
will be highly suppressed in this model and the dominant decay will
occur through the usual vector boson interactions.
\section{Hierarchical Breaking in D-brane models}
We turn now to the question of how the models discussed so far fit into
string theory, in particular in the class of
intersecting or magnetized D-brane models \cite{Blumenhagen:2000wh,Ibanez:2001nd}.
These are models within orientifold string compactifications of type II strings with
D-branes that wrap parts of the internal compactification space, and either with
magnetic field backgrounds on the brane world volume (in type IIB) or with the branes intersecting
non-trivially on the internal space (in type IIA) \cite{Bachas:1995ik}.
For these models a great deal about the Lagrangian of their low energy field theory
description is known, e.g.\ in \cite{Cremades:2002te,Kors:2003wf},
and the soft supersymmetry breaking terms in the conventional
setting with F-terms generated in the hidden sector, have been determined.
Furthermore, it is known how FI-terms can
appear \cite{Douglas:1996sw}.\footnote{These models have also been recently discussed in
the context of split supersymmetry in \cite{Antoniadis:2004dt}.}
\\
The gauge group for any single stack of $N_a$ D-branes is $U(N_a)=SU(N_a)\times U(1)_a$ (only
sometimes an $Sp(N_a)$ or $SO(2N_a)$ subgroup thereof), and thus involves
extra abelian $U(1)_a$ factors, when the Standard Model is engineered.
In the first models that were constructed to reproduce the
non-supersymmetric Standard Model \cite{Ibanez:2001nd},
there are four extra $U(1)_a$, including the anomaly-free
hyper charge and gauged $B-L$ quantum numbers, but also two extra $U(1)$ factors which
are anomalous. The relevant anomaly is actually a mixed abelian-non-abelian anomaly, i.e.\
there are triangle diagrams of the type $SU(N_b)^2-U(1)_a$ non-vanishing for either
$SU(2)$ or $SU(3)$. In particular, ${\rm tr}(Q_a)=0$, and there is no gravitational anomaly.
While the first models were non-supersymmetric from the beginning, the structure of charge
assignments, given in table 1 of \cite{Ibanez:2001nd},
and the number of $U(1)_a$ can serve as a representative example.
From this it is clear that these models at least contain two candidate $U(1)_a$, which may
develop FI-terms. In generality, it is known that, when D-branes violate supersymmetry, this
is reflected by FI-terms in the effective theory \cite{Douglas:1996sw}.
The violation of supersymmetry translates into a
violation of the $\kappa$-symmetry on their world volume, and is geometrically captured by a violation of the
so-called calibration conditions. For intersecting D-branes models, this has a very simple
geometric interpretation. Any single brane wraps a three-dimensional internal space. In
the case of an orbifold compactification it is characterized by three angles $\varphi_i^a$, $i=1,2,3$,
one for each $\mathbb T^2$ in $\mathbb T^6=(\mathbb T^2)^3$, measuring the relative
angle of the D-brane with respect to some reference orientifold plane. The supersymmetry condition
reads $\varphi_1^a\pm \varphi_2^a\pm \varphi_3^a=0\, {\rm mod}\, 2\pi$, with some choice of signs.
The FI-parameter at leading order is proportional to the deviation,
\begin{eqnarray}
\alpha' \xi_a ~\sim~ \varphi_1^a\pm\varphi_2^a\pm\varphi_3^a\ {\rm mod}\ 2\pi\ .
\end{eqnarray}
The angles are moduli-dependent quantities, and thus the question if a FI-term is generated
cannot be finally answered without solving the moduli stabilization problem for
the relevant moduli. In the mirror symmetric
type IIB description with magnetized branes this is manifest, and
the relation reads
\begin{eqnarray}
\alpha' \xi_a ~\sim~ \frac{f_a^1}{T_1+\bar T_{\bar 1}}\pm\frac{f_a^2}{T_2+\bar T_{\bar 2}}
\pm\frac{f_a^3}{T_3+\bar T_{\bar 3}} \ ,
\end{eqnarray}
where $f_a^i/(T_i+\bar T_{\bar \imath}) = \tan(\varphi_i^a)$, and the $T_i$ are the three
(dimensionless) moduli, whose real parts measure the sizes of the three $\mathbb T^2$, while the
$f_a^i$ are rational numbers.
The natural scale for the FI-parameter is the string
scale $M_s=(\alpha')^{-1/2}$, and a suppression means that the right-hand-side is small
numerically.
\\
Another very important property of the string theoretic embedding of D-terms is the fact
that tachyonic masses (negative mass squared) can be lifted to positive values.
Inspecting the charge assignments e.g.\ in \cite{Ibanez:2001nd},
one realizes that it is not feasible to have
positive charges under the $U(1)_a$ for all the fields of the MSSM. This would naively mean
that some $m_i^2 \sim g_a^2 Q_a^i D_a$ are negative, which would lead to a breakdown of
gauge symmetry. However, in the particular case
of orbifold models, the exact string quantization can be performed, and the mass spectrum
computed for small FI-parameters, without using effective field theory.
It turns out that for small angles ($\varphi_i^{ab}=\varphi_i^a-\varphi_i^b<\pi/2$),
the mass of the lowest excitation of two intersecting branes $a$ and $b$ is given by
\begin{eqnarray}
\alpha' m_i^2 = \frac12 \sum_{i=1}^3 |\varphi_i^{ab}| - {{\rm max}_{i}}\{ |\varphi_i^{ab}| \} \ ,
\end{eqnarray}
which, for a proper choice of signs,
vanishes precisely if $\xi_a=0$, consistent with the effective description.
However, depending on the angles, there is a region in parameter space, where the
deviation from $\xi_a=0$ only induces positive squared masses, and
no tachyons (see e.g.\ \cite{Rabadan:2001mt} in
this context). This comes as a surprise from the low energy point of view, and is explained by
the presence of higher order corrections in the derivative expansion of the Born-Infeld effective
action, which become important for strings stretching between intersecting branes \cite{Hashimoto:2003pu}.
The conclusion is, that the effective mass that follows from the D-term potential
is corrected to positive values, and we can tolerate
tachyons in the field theoretic models.
\\
Taking these observations together, it seems first of all possible that D-brane models
of the type discussed have mass spectra with important contributions from FI-terms. Since multiple
extra $U(1)_a$ factors exist and turn anomalous, a scenario, where the $D_a$ fields cannot be relaxed to the
electro-weak scale, appears plausible. This would imply a mass hierarchy between charged scalars
and fermions. To solve the moduli stabilization problem, and show convincingly how
the D- and F-terms of the desired magnitude are generated dynamically, of course, remains an open challenge.
One may also want to turn the argument around, to conclude that within the conventional
approach with low energy supersymmetry, the potential presence of many FI-terms is a great danger
for these types of models, and one has to find ways to suppress them dynamically.
\\
Finally, we like to mention a caveat which makes the classes of D-branes models that we
discussed rather not so good candidates to realize split supersymmetry. It is well known
that in D-brane models in general the unification of gauge interactions really only happens at the
string scale, and not in the field theory regime. This means that the original motivation
to keep the gauginos and higgsinos light, while giving up on the scalars, is upset.
The gauge kinetic functions
are moduli-dependent $f_a=f_a(T_I)$, and a unification of couplings requires some
relations among moduli to hold \cite{Blumenhagen:2003jy,Kors:2003wf,Antoniadis:2004dt},
which so far can appear accidentally in various models
in the literature, but do not seem to have any independent justification.
This means, that an essential motivation for split supersymmetry, gauge unification, is only
accidentally realized.
\section{Hierarchical D-term inflation}
In hierarchical supersymmetry breaking of the type discussed here
it seems an intriguing suggestion to relate the
mass scale of the heavy scalars to cosmology. In our model, the potential energy is generated at
the conventional supersymmetry breaking scale $(m_{3/2} M_{\rm Pl})^2$, and with standard values
comes out about $(10^{10-13}\, {\rm GeV})^4$. This is the scale of the individual contributions
of the F- and D-terms to the full potential, and only the fine tuning of the cosmological constant
leads to a cancellation. It is now very tempting to identify the vacuum energy of these individual
components with the vacuum energy that drives inflation, by undoing the fine tuning.
A possible scenario is very easily illustrated
along the lines of D-term inflation \cite{Binetruy:1996xj}.
Roughly speaking, the only required modification of the model
we used so far, with the MSSM extended by one or many extra $U(1)_a$ gauge factors, plus a pair of charged fields
$\phi^\pm$, is to promote the mass term (\ref{supo}) in the superspotential to a dynamical
field $\varphi$, neutral under $U(1)_a$, which plays the role of the inflaton. We now write
\begin{eqnarray}
W ~=~ \varphi \phi^+\phi^- + W_0 \ ,
\end{eqnarray}
and use a canonically normalized K\"ahler potential,
\begin{eqnarray}
{\cal K} ~=~ |\phi^+|^2 + |\phi^-|^2 + |\varphi|^2 + {\cal K}_0\ .
\end{eqnarray}
With this one finds the scalar potential of the form
\begin{eqnarray}
{\cal V} &=& {\cal V}_{\rm hid} + e^{\kappa^2{\cal K}} \Big[ |\varphi\phi^+|^2 + |\varphi\phi^-|^2 + |\phi^+\phi^-|^2 +
\kappa^4 \big( |\varphi|^2 + |\phi^+|^2 + |\phi^-|^2 \big) |W|^2 \nonumber \\
&& \hspace{1cm} + 3 \kappa^2 |\varphi\phi^+\phi^-|^2
- 3\kappa^2 |W_0|^2 \Big] +
\sum_a \frac{g_a^2}{2} \Big[ |\phi^+|^2 - |\phi^-|^2 + \xi_a \Big]^2 \ ,
\end{eqnarray}
where the only negative contribution comes from $-3 \kappa^2 e^{\kappa^2{\cal K}} |W_0|^2$.
Above some threshold value for $\varphi$, the two charged fields
$\phi^\pm$ are stabilized at the origin $\phi^\pm=0$. Their masses at zero are
\begin{eqnarray}
m^2_{\phi^\pm}(\phi^\pm=0) ~=~ \kappa^2 {\cal V}_{\rm hid} + e^{\kappa^2{\cal K}_0}
[(1+\kappa^6|W_0|^2) |\varphi|^2 - 2\kappa^4 |W_0|^2] \pm \sum_a g_a^2 \xi_a \ ,
\end{eqnarray}
which turns positive, when $\langle\varphi\rangle$ is large enough
(but still well below the Planck scale).
The potential simplifies to
\begin{eqnarray}
{\cal V} (\phi^\pm=0) ~=~ {\cal V}_{\rm hid} + \kappa^2 e^{\kappa^2{\cal K}_0} |W_0|^2 [ \kappa^2 |\varphi|^2-3 ]
+ \sum_a \frac{g_a^2}{2} \xi_a^2 \ .
\end{eqnarray}
The inflationary slow-roll condition for the second derivative of the potential is \cite{book}
\begin{eqnarray}
|\eta| ~=~ \left| \frac{\partial_\varphi^2 {\cal V} }{\kappa^2{\cal V}} \right| ~\ll~ 1\ ,
\end{eqnarray}
where $|\eta|\ll 1$ numerically means $|\eta|\sim 0.01$ is acceptable.
This implies that
\begin{eqnarray}
\frac{M_{\rm Pl}^2 \sum_a g_a^2 \xi_a^2}{e^{\kappa^2{\cal K}} |W_0|^2} ~\gtrsim~ 100\ .
\end{eqnarray}
The first derivative is then automatically also small (with $\varphi \sim \sqrt{\xi_a}$),
and inflation can be successful.
This means that the very same D-term vacuum energy that is responsible for the large scalar masses
can drive inflation, if either the D-terms are enhanced or the F-term vacuum energy is
sufficiently suppressed during that period.
Thus, during the de Sitter phase the relation between the Hubble parameter $H$ and
the energy density $\rho$, i.e. the relation $3M_{\rm Pl}^2 H^2 =\rho =\frac{1}{2} \dot\phi^2 +{\cal V}$,
shows that the Hubble expansion in this phase is dominated by the FI-term
\begin{eqnarray}
3M_{\rm Pl}^2H^2 ~\sim~ \sum_a \frac{g_a^2}{2} \xi^2_a \ .
\end{eqnarray}
In summary, if in the phase of large $\varphi$ the vacuum energy of the D-terms is
larger than that of the F-terms by a factor of about 100 or more, the D-term energy
can drive inflation at a scale up to $10^{12-15}\, {\rm GeV}$, slightly above
the mass scale of the heavy scalars. After the end of the slow roll period, $\varphi$ will
eventually fall below the threshold value, and $\phi^\pm$ will form
condensates themselves. This can then lead to a partial relaxation
of the D-term vacuum energy, but a more elaborate model of the hidden sector would be needed to
describe this phase transition properly. In the minimum, one may expect $\varphi$ to settle down to
$\langle \varphi\rangle\sim \sqrt \xi\sim \sqrt{m_{3/2}M_{\rm Pl}}$ on dimensional grounds, which is compatible with
small gaugino masses.\footnote{As
mentioned earlier, we so far ignore here the problem of moduli stabilization, which is even more
severe in the context of inflation with D-brane degrees of freedom, where already the correct choice
of supersymmetric coordinates is a very subtle question \cite{Kachru:2003sx}. The FI-parameters
depend on the moduli fields, and it will be necessary to stabilize the relevant fields
(as for instance along the lines of \cite{Kachru:2003aw}) at a scale larger
that the scale of inflation, or they would have to be considered as dynamical (see also
the last reference of \cite{Binetruy:1996xj}).}
\section{Conclusion}
We have presented a model of supersymmetry breaking in the context of
string and supergravity scenarios by inclusion of both F- and Fayet-Iliopoulos D-terms, arising from
extra $U(1)$ factors in the gauge group. Such extra $U(1)$
gauge symmetries arise quite naturally in
string based models. It was shown that scalars charged under the extra $U(1)$ gain
large masses from the FI-terms, proportional to the charges of the respective scalar
fields under the extra $U(1)$. This leads generically to non-universal masses for
the heavy scalars. The cancellation of the vacuum energy puts an upper bound
of $\sqrt{m_{3/2}M_{\rm Pl}}$ on the scalar masses, and thus also puts a bound on the
FI-term $\xi_X$. The bound on $\xi_X$ can be met in
heterotic string models only for $m_{3/2}$ close to $M_{\rm Pl}$, since there $\xi_X$
is scaled by $M_{\rm Pl}^2$. Thus, heterotic string
scenarios are not preferred from the vacuum energy constraint, when
$m_{3/2}={\cal O}({\rm TeV})$. However, $m_{3/2}={\cal O}({\rm TeV})$ and $\xi_X \sim m_{3/2}M_{\rm Pl}$,
i.e. of size $10^{10-13}\,{\rm GeV}$, could arise in orientifolds
models which allow $\xi_X$ of a variable moduli-dependent size.
The fact that the D-term contributions to the scalar masses depend on their $U(1)$ charges
opens up the possibility of building a new class of models with some scalars (with vanishing
$U(1)$ charges) light and others (with non-vanishing $U(1)$ charges) heavy,
while the gauginos and higgsinos gain masses only of electro-weak size. Further,
the $\mu$ term is essentially unaffected by the FI contribution, and
can be of electro-weak size. We investigated two illustrative examples of models
with light and heavy scalars (Model I and Model II in Sec.~2) and showed that they lead to
significantly different phenomenologies which could be tested at colliders and in
non-accelerator experiment. The class of models we have discussed here are
different from the high scale supersymmetry models of Ref.~\cite{Arkani-Hamed:2004fb},
where all scalars and the gravitino are super heavy. However, which scalars
are heavy and which are light is now a model-dependent question. A further interesting feature
is the possibility that the vacuum energy responsible for generating heavy scalars
may also drive inflation. An analysis of how this can come about was discussed
in Sec.~4. It would be interesting
to investigate more explicit D-brane constructions to build models of the type advocated here.
\section*{Acknowledgements}
B.~K.~would like to thank Angel Uranga for helpful advice.
The work of B.~K.~was supported by the German Science Foundation (DFG) and in part by
funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement
$\#$DF-FC02-94ER40818. The work of P.~N.~was supported in part by
the U.S. National Science Foundation under the grant NSF-PHY-0139967.
|
1,108,101,563,235 | arxiv |
\section{Introduction}
The widespread use of Convolutional Neural Networks (CNNs), as a go-to solution for a wide range of AI-related problems, combined with the high computational load typically associated with their execution, has led researchers to extensively work on the development of hardware accelerators for CNN inference \cite{inmem}\cite{eyeriss}\cite{hyperdrive}\cite{CNN-P}. Availability of this kind of hardware is key in embedded use-cases involving near-sensor processing of data, according to the edge computing paradigm \cite{vestias2019survey}. Among the different solutions available in literature, an important role is played by FPGA-based architectures, and, in more detail, by solutions exploiting the cooperation between general-purpose processors and FPGAs available in modern All-programmable SoCs, e.g. Zynq and Zynq Ultrascale+ devices by Xilinx, that take profit from the efficient implementation of Multiply-And-Accumulate (MAC) operations on the large amount of DSP Slices available \cite{survey}.
CNNs have been employed for computer vision applications, such as image recognition \cite{img_rec_He_Zhang,imagenet_Krizhevsky,deepImage}, object detection \cite{deepFace} or video frame classification \cite{videoClass}. However, convolutional networks have also been extended to deal with time sequences. These CNN variations are usually indicated as Temporal Convolutional Networks (TCNs) \cite{lea2017temporal}. Multiple TCN architectures have been proposed, reaching impressive performance on tasks such as sentence classification \cite{sentClass}, speech recognition \cite{deepSpeech}, text understanding \cite{textUnde}, Natural Language Processing tasks \cite{NLP} and, more recently, machine translation \cite{machineTras}, audio synthesis \cite{wavenet}, language modeling \cite{lang_model} or signal sequence analysis in the healthcare domain, such as action detection \cite{3D_skeleton} or ECG classification \cite{ecg}.
Research work of Bai et al. \cite{bai2018empirical} demonstrates that the exploitation of a Temporal Convolutional Network for typical sequence modelling tasks often outperforms older and better-known Recurrent Neural Networks \cite{lstm} \cite{gru}. \newline
Thus, the rapidly increasing interest in TCNs pushes for investigating on acceleration for these networks. Especially in the embedded domain, where (near) real-time analysis of sequences of data samples acquired by sensors is a common case, accelerating this kind of workload on reconfigurable devices is a very appealing approach.
In this work we explore the capabilities of a state-of-the-art CNN inference accelerator \cite{neu2017} specifically enriched to provide the flexibility needed in TCNs, to support freely selectable \emph{kernel sizes} and \emph{dilated} convolutions, with freely selectable \emph{dilation rates}. We focus on an implementation on low-cost and low-power all-programmable SoCs, more suitable for the integration of edge-computing and IoT processing nodes, considering two widely accessible devices in the Zynq and the Zynq Ultrascale+ families.
Performance is evaluated over various benchmark TCNs, reporting on absolute execution time as well as on the efficiency of the execution with respect to the peak performance imposed by the device resources.
We propose an optimization method relying on \emph{batch processing} to improve efficiency in cases where operational intensity is critical. We also assess the efficiency of the FPGA-based acceleration, comparing with software execution and with state of the art accelerators on bi-dimensional CNNs.
The reminder of this paper is organized as follows: Section \ref{sec:related} discusses the related work. Section \ref{sec:tcn_gen} analyses the TCN model. Section \ref{sec:ref_arch} gives an overview of the accelerator architecture taken as reference for this work. Section \ref{sec:CE} describes implementation details and strategies. The experimental results are presented in Section \ref{sec:hw_impl_eval} and Section \ref{sec:exp_res}. Conclusions are exposed in Section \ref{sec:concl}.
\section{Related Work}
\label{sec:related}
The landscape of FPGA-based accelerators for CNN is crowded and multifaceted \cite{guo2019dl}. Several approaches have been proposed in recent years, focusing both on the embedded domain and on architectures aimed to speed-up execution on cloud servers.
However, work on FPGA acceleration has mainly focused on classic CNN networks.
Yu et al. \cite{yu2019data} developed an FPGA acceleration platform that leverages a unified framework architecture for general-purpose CNN inference acceleration at a data center achieving a throughput comparable with the state-of-the-art GPU in this field, with less latency. This work exploits on-chip DSPs, on a Kintex KU115, arranged as supertile units (SUs), to overcome the computational bound and, together with dispaching-assembling model and broadcast caches, to deal with the memory bound.
Zhang et. al. \cite{Caffeine} proposed Caffeine, a hardware/software library to efficiently accelerate CNNs on FPGAs, leveraging a uniformed convolutional matrix multiplication representation targeting both computation-intensive convolutional layers and communication-intensive fully connected layers of CNN which maximizes the underlying FPGA computing and bandwidth resources utilization. Ma et. al. \cite{RTLCompiler} presented an RTL-level CNN compiler that generates automatically customized FPGA hardware for the inference tasks of CNNs from software to FPGA. The approach proposed by \cite{RTLCompiler} relies on a template accelerator architecture described in Verilog including all the main functions employed by CNNs such as convolutions, pooling, etc, which are automatically customized at design time to match the requirements of the target CNN model.
The proposed methodology is demonstrated with end-to-end FPGA implementations of complex CNN models such as NiN, VGG-16, ResNet-50, and ResNet-152 on two standalone Intel FPGAs, Stratix V and Arria 10, providing average performance up to 720 GOps.
These two frameworks provide huge performance gains when compared to state-of-the-art accelerators and general-purpose CPUs and GPUs. However, both leverage large FPGA devices such as Virtex7 and Arria 10, they mainly target server applications exploiting batching to improve memory access performance and bandwidth utilization.
This approach is less suitable for embedded applications where cheap and compact SoCs integrating embedded processors and FPGAs are desirable, and images have to be processed in real-time.
In this domain, Venieris et. al. \cite{LatencyDriven} presented a latency-driven design methodology for mapping CNNs on FPGAs. As opposed to previously presented approaches mainly intended for bandwidth-driven applications, this work targets real-time applications where the batch size is constrained to one, relying on Xilinx high-level synthesis tools for mapping (i.e. Vivado HLS), demonstrated on relatively simple CNN such as AlexNet, and on a very regular one such as VGG16 featuring only 3$\times$3 kernels, providing a peak performance of 123 GOps. Other work focuses on a template-based approach based on programmable or customizable RTL accelerators proposed in architectures \cite{RTLCompiler}\cite{Snowflake}\cite{GoingDeeper}, more similar to the one that is used in this paper.
SnowFlake \cite{Snowflake} exploits a hierarchical design composed of multiple compute clusters. Each cluster is composed of four vectorial compute units including a vectorial MAC, vectorial max, a maps buffer, weights buffers and trace decoders. SnowFlake provides a computational efficiency of 91\%, and an operating frequency of 250 MHz (best-in-class for CNN accelerators on Xilinx Zynq Z-7045 SoC). However, although the vector processor-like nature of the accelerator is very flexible, delivering significant performance also for 1$\times$1 kernels, it prevents to fully exploit of spatial computation typical of application-specific accelerators, which leads to overheads due to load/store operations necessary to fetch weights and maps from the buffers. This is highlighted by the low utilization of the DSP slices available on the FPGA (i.e. only 256 over 900), and by the performance when executing end-to-end convolutional neural networks, which is lower than that of other architectures including the proposed one even though the operating frequency of the CNN engine is significantly higher.
Several approaches tackling FPGA architectures for image-processing CNN, have explored the reduction of the precision of arithmetic operands to improve energy efficiency. Although most of the architectures available in literature feature a precision of 16-bit (fixed-point)\cite{LatencyDriven, Snowflake, RTLCompiler} numerous reduced-precision implementations have been proposed recently, relying on 8-bit, 4-bit accuracy for both maps and weights, exploiting the resiliency of CNNs to quantization and approximation \cite{GoingDeeper}.
Qiu et. al. \cite{GoingDeeper} proposed a CNN accelerator implemented on a Xilinx Zynq platform exploiting specific hardware to support 8/4 bit dynamic precision quantization, at the cost of 0.4\% loss of classification accuracy.
Other extreme approaches to quantization exploit ternary \cite{Ternary} or binary \cite{FINN} neural-networks accelerators for FPGA. This approach significantly improves the computational efficiency of FPGA Accelerators, allowing to achieve performance level as big as 8 TOPS \cite{Ternary}.
Recent work by Rasoulinezhad et al. \cite{pir-dsp}, starting from the Xilinx DSP slices, proposed an optimized DSP block called PIR-DSP to efficient map 9, 4 and 2 bits data precision MAC operations. It is implemented as a parameterized module generator targeting both FPGAs and ASICs reaching an estimate run time energy decrease up to 31\% for a MobileNet-v2 implementation compared with a standard DSP mode.
Other works, like Wang et al. \cite{wang2020lutnet}, leverage FPGA LUT blocks as inference operators for Binary Neural Network (BNN) achieving up to twice area efficiency compared to state-of-the-art binarized NN implementation and against several standard networks models.
While small networks like MNIST, CIFAR10, SVHN, GTSRB can reach good classification accuracy, the training is still a big challenge for larger networks such as VGG or ResNet \cite{Courbariaux2015a}. The usability of extreme quantization is also not demonstrated for TCN-related tasks, sequence classification and modelling.
Probably the most powerful currently available FPGA-based acceleration engine is the proprietary one offered by Xilinx, which provides an integrated framework, called VitisAI \cite{VitisAI}, that helps designers in mapping CNNs on a templated soft IP called Deep Learning Processing Unit (DPU) \cite{DPU}. The DPU provides impressive performance on CNNs, using quantization and high clock frequency in DSP slices. Quantization is required and can be applied automatically using a dedicated tool included in VitisAI. DSP slices are clocked at very high frequency, using a \emph{DSP Double Data Rate (DDR) technique} \cite{DPU_guide}, which uses a 2x frequency domain to increase peak performance. However, the support for TCN is missing, since the DPU does not support arbitrary dilation, stride and kernel sizes and the the VitisAI quantization process does not support 1D convolutions.
To the best of our knowledge, there are no published FPGA-based accelerators tuned to speed-up inference for generic Temporal Convolutional Networks.
As main novel contributions of this work, we propose:
\begin{itemize}
\item an enriched architectural template supporting efficient TCN inference on FPGA;
\item a methodology for the optimal execution/scheduling of data-transfers exploiting the specific sequence-based structure of data in TCNs
\item a methodology for improving efficiency based on \emph{sample batching} (sequence buffering)
\item the first (to the best of our knowledge) experimental evaluation of the usability of FPGA-based acceleration for TCNs, based on different end-to-end benchmarks and two APSoC devices
\end{itemize}
\section{TCN model generalities}
\label{sec:tcn_gen}
A dilated convolution operation $F$ on element $s$ of a sequence \cite{bai2018empirical} can be defined as:
\begin{equation}
F(s) = (x\ast_d f)(s) = \sum_{i=0}^{k-1} \ f(i)\cdot x_{s-d\cdot i}
\label{eq:tcn_eq}
\end{equation}
where $x\in \mathbf{R}^n $ is a 1-D input sequence, $f:\{0,...,k-1\} \in \mathbf{R}$ is a kernel of size $k$ and $d$ is the $dilation\ rate$.
The sequence of samples that constitute the input of a TCN can be processed both off-line or in a real-time streaming. The second case is very useful in application cases requiring continuous analysis of the input sequence, e.g. aimed at the identification of specific events and/or at promptly closing the loop on data-triggered actuations. This implies that the sequence must be analyzed at every time step, after being updated with a new sample. It is possible to identify the minimal sequence
size to produce a valuable output sample as the \emph{receptive field} that depends on convolutional layer parameters such as the \emph{kernel\_size} and the \emph{dilation rate}:
\begin{equation}
receptive field = 1+ \sum_{l=1}^L \ [k(l) - 1]\times d(l)
\label{eq:rec_field}
\end{equation}
where $l \in {1,2 ... L}$ is a layer of the network.
This can be thought of as a sliding processing window for the input sequence.
\begin{figure}[h]
\centering
\includegraphics[width = \columnwidth]{img/tcn_rec_field_2.pdf}
\caption{TCN execution graph. For this example, see (\ref{eq:rec_field}), $receptivefield=1+(2-1)\times 1+(3-1)\times 2+(4-1)\times3=15$}
%
\label{fig:tcn_rec_field}
\end{figure}
Figure \ref{fig:tcn_rec_field} highlights these concepts.
It is worth noticing that increasing the dilation parameter leads to an increase of the network's memory without affecting its depth.
\section{Reference Accelerator Architecture}
\label{sec:ref_arch}
\begin{figure}[h]
\centering
\includegraphics[width = \columnwidth]{img/NEU_arch_simpler.pdf}
\caption{NEURAghe architecture}
\label{fig:neu_arch}
\end{figure}
Our TCN accelerator is an extension of NEURAghe \cite{neu2017}, a CNN inference accelerator architecture that exploits the cooperation between the ARM Cortex-A9 Processing System (PS) and the Programmable Logic (PL) in Xilinx Zynq devices. Communication at the PS-PL interface is allowed by the PS high-performance general purpose ports. In NEURAghe, as can be seen in Figure \ref{fig:neu_arch}, the Programmable Logic hosts a Convolution Specific Processor (CSP), while the processing system acts as a General Purpose Processor (GPP) dealing with tasks hard to accelerate by parallelization on programmable logic, such as data marshalling or non-Conv layers.
NEURAghe integrates a RISC-V lightweight processor inside the CSP, dedicated to the execution of a firmware that schedules data transfers and convolutions without PS intervention, leaving the latter available for actual computation workload.
In order to support TCN execution, we have enhanced the CPS IP, described in System Verilog and available as open source \footnote{\urlstyle{tt}\url{https://github.com/neuraghe/NEURAghe}}. We have added new features to improve flexibility, requiring substantial modification of the main computational core dedicated to Multiply and Accumulate (MAC) operations execution, called Convolution Engine, and an improvement of the circuitry and procedures managing transfers to/from the DDR memory.
In particular, the Convolution Engine (Figure \ref{fig:ce}) is composed by a MAC Matrix of $N_{cols}$ columns by $N_{rows}$ rows of Sum of Product (SoP) units in charge to calculate the contribution of $N_{cols}$ input features to $N_{rows}$ output features. $N_{rows}$ Shift Adder modules sum together partial result from SoPs in each row with data values resulted from the $N_{cols}$ previously computed input feature partial results read from on-chip memory, enabling successive accumulation over multiple CE runs.
\begin{figure}[h]
\centering
\includegraphics[width = \columnwidth]{img/CE.pdf}
\caption{Convolution Engine. $N_{rows}\times N_{cols}$ MAC Matrix}
\label{fig:ce}
\end{figure}
\section{TCN supporting hardware features}\label{sec:CE}
Considering the typical features of TCNs, we designed the CE according to several design principles. The accelerator:
\begin{itemize}
\item has to be $kernel\_size$ agnostic;
\item must execute convolutions with multiple stride values without performance overhead;
\item must support freely selectable dilation values.
\end{itemize}
\subsection{Freely selectable kernel sizes}
To support arbitrary kernel sizes, we have chosen to dedicate one single DSP cell to compute an entire convolution kernel, reusing it over a number of cycles depending on the kernel size. A new sample of the output feature under production is thus produced after $kernel\_size$ cycles and is ready to be sent to the Shift Adder module. Using one single DSP cell per kernel can easily require the instantiation of a very high number of SoPs, to the aim of exploiting as many resources as possible among those available on the target device.
To keep the MAC Matrix growth feasible, we designed SoPs to be composed by $4$ Xilinx DSP48E primitives, performing $4\ MACs/cycle$, operating in parallel on $4$ different 16-bit samples, as visualized in Figure \ref{fig:sop_elab}, from $4$ neighbour convolution windows in an input feature.
\begin{figure}[h]
\centering
\includegraphics[width = \columnwidth]{img/SoP_elaboration.pdf}
\caption{SoP elaboration scheme}
\label{fig:sop_elab}
\end{figure}
Figure \ref{fig:sop} represents the organization of a SoP.
Considering the template proposed in Figure \ref{fig:ce}, the MAC Matrix implementation requires a deterministic number of DSP48E primitives, as indicated by Equation \ref{DSPscount}.
\begin{equation}\label{DSPscount}
N_{DSPs}= N_{rows}\times N_{cols}\times 4
\end{equation}
\begin{figure}[h]
\centering
\includegraphics[width = \columnwidth]{img/sop_dsp.pdf}
\caption{Sum of Product Unit}
\label{fig:sop}
\end{figure}
\subsection{Flexible activations and weights fetching}
\begin{figure*}[]
\centering
\includegraphics[width = \textwidth]{img/act_n_wei_fetch.pdf}
\caption{Memory transfers}
\label{fig:act_fetch}
\end{figure*}
To enable arbitrary stride and dilation values, the fetching of input samples from the internal memory has been designed to be very flexible. Each CE port dedicated to input samples is endowed by a programmable Activation Source module, while weight kernels are fetched from Weight Memory banks by means of Weights Source modules. Such source modules can be programmed at start-up according to stride, dilation, aspect ratio and size of activations and weight kernels. Fetching of bi-dimensional memory sections is enabled, to support generic CNN execution. \par\noindent
The memory subsystem has been designed to enable conflict-less loading of neighbour convolution windows. The Activation Source module controls $N_{cols}$ ports of the convolution engine. Each one loads samples from a dedicated BRAM module. Considering that four samples, belonging to different windows, can be loaded in the same cycle, to support stride values up to 3 samples (which is sufficient to support most of the TCN use-cases available in literature), each module of the activation memory has to be composed of at least 8 different independently accessible RAMB18 modules. Figure \ref{fig:conflict_bram} represents an example where a different configuration of RAMB18 modules can determine a conflict.\par\noindent
\begin{figure}[H]
\centering
\includegraphics[width = \columnwidth]{img/conflict_bram.pdf}
\caption{BRAM read conflict example. Samples are stored in BRAM modules using interleaving. Consecutive samples are stored in adjacent RAMB18 banks. 4 DSP slices in a SoP units, with stride 2, in the first cycle of the convolution, load respectively samples 0,2,4,6. With four RAMB18 banks, samples 0-4 and 2-6 are on the same bank, creating a conflict.}
\label{fig:conflict_bram}
\end{figure}
Moreover, the Weight source module controls one port per each SoP in the matrix, which has to be implemented by at least one RAMB18 module. \par\noindent
Finally, the CE has a set of ports that are used to write results and to load previously computed partial results, when a convolution requires to accumulate over several accelerator operations. These ports are controlled by a Partial Result Source module and by an Output Sink module. Each of these modules controls $N_{rows}$ ports, each one writing/reading four samples simultaneously. Thus BRAM modules in the corresponding memory region are composed by at least 8 RAMB18 modules each. \par\noindent Samples and weights, in the experiments presented in this paper, are all using a 16-bit data format, thus RAMB18 modules are configured to expose two 16-bit addressable ports and can be 1024 words deep. \par\noindent
Considering the described organization, a given architectural configuration requires a number of RAMB18 primitives that can be deterministically estimated as indicated in Equation \ref{BRAMScount}.
\begin{equation}\label{BRAMScount}
N_{BRAMs}= N_{rows}\times N_{cols} \\
+ N_{cols} \times 8 \\
+ (N_{rows}\times 2)\times 8 + \\
32
\end{equation}
The first component corresponds to weight memory, the second to the modules storing input activations, the third to output and partial results memories. 32 blocks are used to implement the RISC-V scheduler instruction memory and private memory.
\subsection{TCN support in firmware}\label{sec:SCHED}
The programming model used in NEURAghe envisions the ARM-based processing system in the Zynq platforms to execute a C program implementing the neural network inference. When the accelerator implemented in the programmable logic has to be used. The program in the PS sends commands describing the layer to be executed. The RISC-V soft-core in the accelerator decodes the command and decomposes the layer in sub-operations, namely partial convolutions in the accelerator and data transfers from/to the off-chip memory, executing an optimized firmware which is also coded in C. The firmware uses a double-buffering technique, to allow the accelerator to overlap transfers phases with convolutions, as represented in Figure \ref{fig:scheduling}, reducing as much as possible idle times in the CE to exploit the processing capabilities of DSP slices with maximum efficiency.
\begin{figure}[H]
\centering
\includegraphics[width = \columnwidth]{img/scheduling.pdf}
\caption{Scheduling scheme}
\label{fig:scheduling}
\end{figure}
When considering TCN executions, the previously described paradigm has to be applied to the characteristics of the algorithm. For every Convolutional Layer in a TCN, given its \emph{kernel\_size} and a \emph{dilation}, it is possible to consider a local \emph{receptive field}, indicated as $RF_{local}$ in Equation \ref{loc_rec_field}, that is the minimum amount of layer's input samples per channel needed to produce a valuable output sample.
\begin{equation}\label{loc_rec_field}
RF_{local}= 1 + (k - 1)\times d
\end{equation}
\begin{figure}[h]
\subfloat[\label{fig:B=1}]{\includegraphics[width = 0.5\columnwidth]{img/hwce_feed_B=1.pdf}}
\hfill
\subfloat[\label{fig:B=B}]{\includegraphics[width = 0.5\columnwidth]{img/hwce_feed_B=B.pdf}}
\caption{TCN execution on NEURAghe}
\label{fig:tcn_exec}
\end{figure}
Figure \ref{fig:B=1} shows how input and output transfers are implemented for TCNs.
At every new time step, input to a layer is updated by adding one new sample to input features. In order to execute the layer, the firmware triggers DMA transfers to load to the activation input memory $RF_{local}$ samples per each input feature. After the execution, one output sample is produced per output feature. Output samples are sent to DDR using an output DMA transfer, to be stored until the next time step.
\subsection{Improving through batch processing} \label{batch proc}
Although the previous approach provides the minimum classification/recognition latency, executing the network every time a new sample is available to update the input sliding window, it can determine performance to be bandwidth limited.
This is because all of network parameters/weights must be loaded for every layer. So, despite the double-buffered scheduling strategy, transfer and computation phases hardly overlap, affecting the \emph{operational intensity} of the application.
\begin{figure}[h]
\centering
\includegraphics[width = \columnwidth]{img/roofline_model_ECG_3.pdf}
\caption{Use case benchmark's Roofline Model for the Convolution Specific Processor (12x4 MAC Matrix in Z-7020 SoC) with respect to different batch sizes}
\label{fig:roofline}
\end{figure}
The roofline model in Figure \ref{fig:roofline} shows performance trend starting from the sample-by-sample processing (leftmost red cross), on a use-case that will be presented in the following.
If the latency constraint is not extremely tight, it is possible to pre-buffer input samples in order to process longer sample sequences (batch) and produce more output with every execution.
Figure \ref{fig:B=B} shows the transfer scheduling when \emph{batch size} is increased.
As the \emph{batch size} increases the architecture gets closer and closer to the computational limit (rightmost red cross on the roofline plot of Figure \ref{fig:roofline}) because of the growing number of operations performed. In this way, it is possible to increase the utilization of computing resources and to gain efficiency.
As may be noticed in Figure \ref{fig:roofline}, when sample batch size is small, besides being on the badwidth-limited region of the plot, points in the roofline model are also distant from the theoretical achievable performance. This is due to overheads related with CE programming/warm-up, and to initial and final input/output transfers, which cannot overlap with convolutions. The impact of this overhead is limited when the operational intensity increases, reducing the distance between points and the theoretical roofline model.
\section{Hardware Implementation Evaluation}
\label{sec:hw_impl_eval}
\subsection{Design Space Exploration}
A designer willing to use the NEURAghe template, on a given target SoC, has multiple possible architectural configurations available. In order to perform a careful selection, it is possible to perform a simple design space exploration and to choose a near-optimal setup from the performance point of view. To select the architectures presented in this paper, we have used a simple grid search that evaluates multiple configurations, featuring different values of $N_{rows}$ and $N_{cols}$, to maximize the number of SoPs, while keeping the number of used DSPs and BRAMs in the range of availability imposed by the target SoC.
Table \ref{tab:board_utiliz} shows utilization numbers estimated using the Equations \ref{DSPscount} and \ref{BRAMScount}. Light-gray coloured cells in the table indicate which configurations are not implementable in a Z7020 SoC, due to excessive DSP or BRAM utilization. Darker-coloured cells indicate configurations that are not feasible in an Ultrascale+ ZU3EG device.
\begin{table}[h]
\caption{RAMB18 and DSP utilization in NEURAghe architecture with respect to the MAC Matrix shape}
\begin{center}
\setlength{\arrayrulewidth}{0.3mm}
\renewcommand{1} \begin{tabular}{*{1}{c}|*{1}{c}*{1}{c|}*{1}{c}*{1}{c}*{1}{c}*{1}{c}}{1.2}
\begin{tabular}{*{1}{|c}*{1}{|c}|*{9}{c|}}
\cline{1-11}
& \multicolumn{9}{c|}{$N_{cols}$} & \\
\cline{1-11}
& \tikzmark{startbram} 144 \tikzmark{endbram} & 156 & 168 & 180 & 192 & 204 & 216 & 228 & 240 & \multirow{2}{0.8em}{4}\\
& \tikzmark{startdsp} 64 \tikzmark{enddsp} & 80 & 96 & 112 & 128 & 144 & 160 & 176 & \tikzmark{startone} 192 \tikzmark{endone} & \\
\cline{2-11}
& 164 & 177 & 190 & 203 & 216 & 229 & 242 & 255 & 268 & \multirow{2}{0.8em}{5}\\
& 80 & 100 & 120 & 140 & 160 & 180 & 200 & \tikzmark{starttwo} 220 \tikzmark{endtwo} & \cellcolor{lightgray}240 & \\
\cline{2-11}
& 184 & 198 & 212 & 226 & 240 & 254 & 268 & \cellcolor{lightgray}282 & \cellcolor{lightgray}296 & \multirow{2}{0.8em}{6}\\
\multirow{9}{0.8em}{\rotatebox{90}{$N_{rows}$}} & 96 & 120 & 144 & 168 & 192 & 216 & \cellcolor{lightgray}240 & \cellcolor{lightgray}264 & \cellcolor{lightgray}288 & \\
\cline{2-11}
& 204 & 219 & 234 & 249 & 264 & 279 & \cellcolor{lightgray}294 & \cellcolor{lightgray}309 & \cellcolor{lightgray}324 & \multirow{2}{0.8em}{7}\\
& 112 & 140 & 168 & 196 & \cellcolor{lightgray}224 & \cellcolor{lightgray}252 & \cellcolor{lightgray}280 & \cellcolor{lightgray}308 & \cellcolor{lightgray}336 & \\
\cline{2-11}
& 224 & 240 & 256 & 272 & \cellcolor{lightgray}288 & \cellcolor{lightgray}304 & \cellcolor{lightgray}320 & \cellcolor{lightgray}336 & \cellcolor{lightgray}352 & \multirow{2}{0.8em}{8}\\
& 128 & 160 & 192 & \cellcolor{lightgray}224 & \cellcolor{lightgray}256 & \cellcolor{lightgray}288 & \cellcolor{lightgray}320 & \cellcolor{lightgray}352 & \cellcolor{gray}384 & \\
\cline{2-11}
& 244 & 261 & 278 & \cellcolor{lightgray}295 & \cellcolor{lightgray}312 & \cellcolor{lightgray}329 & \cellcolor{lightgray}346 & \cellcolor{lightgray}363 & \cellcolor{lightgray}380 & \multirow{2}{0.8em}{9}\\
& 144 & 180 & 216 & \cellcolor{lightgray}252 & \cellcolor{lightgray}288 & \cellcolor{lightgray}324 & \tikzmark{startthree} \cellcolor{lightgray}360 \tikzmark{endthree} & \cellcolor{gray}396 & \cellcolor{gray}432 & \\
\cline{2-11}
& 264 & \cellcolor{lightgray}282 & \cellcolor{lightgray}300 & \cellcolor{lightgray}318 & \cellcolor{lightgray}336 & \cellcolor{lightgray}354 & \cellcolor{lightgray}372 & \cellcolor{lightgray}390 & \cellcolor{lightgray}408 & \multirow{2}{0.8em}{10}\\
& 160 & 200 & \cellcolor{lightgray}240 & \cellcolor{lightgray}280 & \cellcolor{lightgray}320 & \cellcolor{lightgray}360 & \cellcolor{gray}400 & \cellcolor{gray}440 & \cellcolor{gray}480 & \\
\cline{2-11}
& \cellcolor{lightgray}284 & \cellcolor{lightgray}303 & \cellcolor{lightgray}322 & \cellcolor{lightgray}341 & \cellcolor{lightgray}360 & \cellcolor{lightgray}379 & \cellcolor{lightgray}398 & \cellcolor{lightgray}417 & \cellcolor{gray}436 & \multirow{2}{0.8em}{11}\\
& 176 & 220 & \cellcolor{lightgray}264 & \cellcolor{lightgray}308 & \cellcolor{lightgray}352 & \cellcolor{gray}396 & \cellcolor{gray}440 & \cellcolor{gray}484 & \cellcolor{gray}528 & \\
\cline{2-11}
& \cellcolor{lightgray}304 & \cellcolor{lightgray}324 & \cellcolor{lightgray}344 & \cellcolor{lightgray}364 & \cellcolor{lightgray}384 & \cellcolor{lightgray}404 & \cellcolor{lightgray}424 & \cellcolor{gray}444 & \cellcolor{gray}464 & \multirow{2}{0.8em}{12}\\
& 192 & \cellcolor{lightgray}240 & \cellcolor{lightgray}288 & \cellcolor{lightgray}336 & \cellcolor{gray}384 & \cellcolor{gray}432 & \cellcolor{gray}480 & \cellcolor{gray}528 & \cellcolor{gray}576 & \\
\hline
& 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & \\
\hline
\end{tabular}
\end{center}
\label{tab:board_utiliz}
\begin{center}
\begin{tabular}{*{1}{c}*{4}{|c}}
& avail. & avail. & & out of \\
& RAMB18 & DSP & selected & resources \\
\hline
Z7020 & 280 & 220 & \tikzmark{startlone} \tikzmark{endlone} & \cellcolor{lightgray} \\
\hline
ZU3EG & 432 & 360 & \tikzmark{startltwo} \tikzmark{endltwo} & \cellcolor{gray} \\
\end{tabular}
\end{center}
\begin{tikzpicture}[remember picture,overlay]
\draw[rounded corners,black,thick]
([shift={(-0.5\tabcolsep,-0.5ex)}]pic cs:startbram)
rectangle
([shift={(0.5\tabcolsep,2ex)}]pic cs:endbram);
\draw [ -stealth ]
([shift={(0\tabcolsep,0)}]pic cs:startbram)
--
([shift={(-11\tabcolsep,0.1)}]pic cs:startbram) node [above] {RAMB};
\draw[rounded corners,black,thick]
([shift={(-0.5\tabcolsep,-0.5ex)}]pic cs:startdsp)
rectangle
([shift={(0.5\tabcolsep,2ex)}]pic cs:enddsp);
\draw [ -stealth ]
([shift={(0\tabcolsep,0)}]pic cs:startdsp)
--
([shift={(-8\tabcolsep,-0.1)}]pic cs:startdsp) node [left] {DSP};
\foreach \Val in {one,two}
{
\draw[rounded corners,green,thick]
([shift={(-0.5\tabcolsep,-0.5ex)}]pic cs:start\Val)
rectangle
([shift={(0.5\tabcolsep,4.8ex)}]pic cs:end\Val);
}
\draw[rounded corners,red,thick]
([shift={(-0.5\tabcolsep,-0.5ex)}]pic cs:startthree)
rectangle
([shift={(0.5\tabcolsep,4.8ex)}]pic cs:endthree);
\draw[rounded corners,green,thick]
([shift={(-0.5\tabcolsep,-0.5ex)}]pic cs:startlone)
rectangle
([shift={(1\tabcolsep,1ex)}]pic cs:endlone);
\draw[rounded corners,red,thick]
([shift={(-0.5\tabcolsep,-0.5ex)}]pic cs:startltwo)
rectangle
([shift={(1\tabcolsep,1ex)}]pic cs:endltwo);
\end{tikzpicture}
\end{table}
\subsection{Implementation on different SoCs}
We have implemented the NEURAghe configurations with different Convolution Engine's MAC Matrix shapes, selected after the DSE process, on two target platforms: a Xilinx Zynq Z-7020 and a Xilinx Zynq Ultrascale+ ZU3EG. For the Z-7020 we have selected two configurations, featuring a similar number of DSP slices and similar clock frequencies. Table \ref{tab:occup_12x4} shows resource occupation on the reconfigurable logic of a first configuration implemented on the Z-7020, featuring a 12$\times$4 MAC matrix of SoP units, that can be clocked up to 120 MHz, providing peak performance of 46 GOPS/s. Table \ref{tab:occup_11x5} shows results related with a similar configuration that integrates an 11$\times$5 MAC matrix, using slightly more DSPs but clockable up to 110 MHz, with a peak performance of 48.4 GOPS/s. Finally, Table \ref{tab:occup_ultra} describes the results of the implementation of a configuration, featuring a 9$\times$10 MAC matrix of SoP modules, on the ZU3EG. This design uses all the DSP slices on the chip and can be clocked at 180 MHz, providing a peak of 129.6 GOPS/s.
\begin{table}[h!]
\caption{Resource occupation on a Xilinx Zynq Z-7020 (12x4 MAC matrix)}
\centering
\begin{tabular}{*{1}{c}||*{5}{c}}
& DSP & BRAM & LUTs & LUTs & Regs \\
& & & (logic) & SR & \\
\hline
\hline
Used & 192 & 120 & 47230 & 259 & 26942 \\
\hline
Available & 220 & 140 & 53200 & 53200 & 106400 \\
\hline
\% & 87.27 & 85.71 & 88.78 & 1.49 & 25.32 \\
\multicolumn{6}{c}{} \\
\end{tabular}
\label{tab:occup_12x4}
\end{table}
\begin{table}[h!]
\caption{Resource occupation on a Xilinx Zynq Z-7020 (11x5 MAC matrix)}
\centering
\begin{tabular}{*{1}{c}||*{5}{c}}
& DSP & BRAM & LUTs & LUTs & Regs \\
& & & (logic) & SR & \\
\hline
\hline
Used & 220 & 128 & 42964 & 283 & 26962 \\
\hline
Available & 220 & 140 & 53200 & 53200 & 106400 \\
\hline
\% & 100 & 91.4 & 80.76 & 1.63 & 25.34 \\
\multicolumn{6}{c}{} \\
\end{tabular}
\label{tab:occup_11x5}
\end{table}
\begin{table}[h!]
\caption{Resource occupation on a Xilinx Zynq UltraScale+ ZU3EG (9x10 MAC matrix)}
\centering
\begin{tabular}{*{1}{c}||*{5}{c}}
& DSP & BRAM & LUTs & LUTs & Regs \\
& & & (logic) & SR & \\
\hline
\hline
Used & 360 & 354 & 47857 & 159 & 26463 \\
\hline
Available & 360 & 432 & 70560 & 70560 & 141120 \\
\hline
\% & 100 & 81.94 & 67.82 & 0.4 & 18.75 \\
\multicolumn{6}{c}{} \\
\end{tabular}
\label{tab:occup_ultra}
\end{table}
\section{Experimental Results}
\label{sec:exp_res}
We tested the actual level of performance that can be achieved using the accelerator on three different benchmarks, trying to cover as much as possible the landscape of TCNs presented in literature:
\begin{itemize}
\item a \emph{plain} TCN network for ECG monitoring and classification (Goodfellow et al. \cite{ecg}), that performs classification over single lead ECG waveforms and reaches around $90\%$ average accuracy. We call this benchmark \emph{ECG} in the following. This benchmark exposes real-time constraints and can be used to assess the usability of the accelerator in this kind of context.
\item A network, called hereafter Res-TCN, based on residual units, with a structure taken from the Resnet family presented in \cite{resnet}. The TCN is presented in (\cite{3D_skeleton}) authors started from a 3D human action recognition dataset with 3D full skeleton annotations to extract a 1D feature representation per frame resulting in a 150-dimensional vector.
\item A more complex network, based on WaveNet (Van DenOord et al. [14]), for Polyphonic Note Transcription of Time-Domain Audio Signal ([19]). We indicate this use-case as WN-PNT.
\end{itemize}
We have considered three system implementations using NEURAghe, two of them, 11$\times$5 and 12$\times$4, implemented on a Zedboard development board, with the MAC matrix respectively clocked at 70 and 80 MHz, and one implemented on the Ultra96 development board, with MAC Matrix clocked at 180 MHz.
Table \ref{tab:mem_foot} shows the comparison between on-chip memory capabilities for each system implementation and the memory footprint of each use-case network that regards the overall occupation of weight kernels and input features through all layers, considering the sample batch size values that will be used in the following. As may be noticed, all the networks must be adequately managed with the scheduling strategy presented in section \ref{sec:SCHED}, since their activation and weight memory footprint exceeds the memory available on the considered low-cost hardware platforms.
\begin{table}[h!]
\caption{Implementation related on-chip memory capabilities and use-case networks memory footprint}
\centering
\begin{tabular}{*{1}{c}|*{3}{c}*{1}{c}*{1}{c}}
& \multicolumn{3}{c}{input features [kB]} & weight kernels [kB] \\
\hline
\multirow{2}{8em}{12x4 on Z-7020 on-chip memory} & \multicolumn{3}{c}{\multirow{2}{1em}{192}} & \multirow{2}{1em}{96} \\
& & \\
\hline
\multirow{2}{8em}{11x5 on Z-7020 on-chip memory} & \multicolumn{3}{c}{\multirow{2}{1em}{176}} & \multirow{2}{1em}{110} \\
& & \\
\hline
\multirow{2}{8em}{9x10 on ZU3EG on-chip memory} & \multicolumn{3}{c}{\multirow{2}{1em}{144}} & \multirow{2}{1em}{180} \\
& & \\
\hline
ECG: & \textbf{B=1} & \textbf{B=8} & \textbf{B=348} & \\
memory footprint & 216,8 & 250,8 & 1696,5 & 11495,6 \\
\hline
Res-TCN: & \textbf{B=1} & \multicolumn{2}{c}{\textbf{B=144}} & \\
memory footprint & 37,3 & \multicolumn{2}{c}{511,5} & 5424,8 \\
\hline
WN-PTN: & \textbf{B=1} & \multicolumn{2}{c}{\textbf{B=504}} & \\
memory footprint & 553,5 & \multicolumn{2}{c}{5846,49} & 3328,8 \\
\end{tabular}
\label{tab:mem_foot}
\end{table}
\subsection{ECG classification use-case}
The network consists of several computational blocks mostly made of a 1D Convolutional layer, a batch normalization layer, a ReLU and a dropout stage, operating on streams of 16-bit samples, acquired at 300 Hz frequency.
In this case, we have assessed three different operating modes:
\begin{itemize}
\item \textit{minimum latency mode},
\item \textit{maximum throughput mode},
\item \textit{real-time execution}.
\end{itemize}
The first one provides classification in output after minimum latency. The network is executed as soon as possible per every input sample.
In the second operating mode, sample batching is used extensively, to maximize throughput without considering latency as an optimization objective. In the third mode, sample batching is exploited just enough to reach a throughput that allows respecting the real-time constraint posed by the input sampling frequency (300 Hz).
Figure \ref{fig:effic_compar_zed} shows the performance achieved on this benchmark by configurations implemented on the Zedboard, while Figure \ref{fig:time_compar_zed} shows execution times.
As expected, without sample batching, performance is significantly bandwidth-limited. In minimum latency mode ($B=1$ in the Figure), efficiency is very low for every layer, and consequently on the whole network. DSPs have very long idle times and actual performance is very far from the peak. Execution time is around 17 ms with the 12$\times$4 matrix, and a bit higher with the 11$\times$5.
To design the maximum throughput mode, we have iteratively tested different batch size values, to identify which value saturates the benefits achievable with sample batching. For this benchmark, such value is B=348, corresponding to a latency of around 1.2$s$. In this case, both configurations can operate very near to the peak performance, with an average efficiency of around 0.9. Some layers, especially Type 1, are still less efficient, however, their contribution to the overall execution time is limited. In maximum throughput mode, each network execution takes around 140 ms, producing in output classification of 348 sample consecutive sample sequences, corresponding to one sample every 0.4$ms$.
Finally, the real-time constraint requires to process one sequence in less than 3.3 $ms$. To design the real-time operating mode, by exploring the batch size, we identified $B=8$, to be the lowest value enabling that respects such requirement. Execution time is around 17 $ms$, corresponding to around 2 $ms$ per sample.
Since 12$\times$4 is more efficient than 11$\times$5 in all modes, in this case higher frequency is more important than the number of MACs, due to better utilization with the specific layer characteristics (the matrix is partially unused in the final operations of a convolution when the number of output features is not a multiple of 5).
\begin{table}[h!]
\caption{Convolutional Layer characteristics for Network ECG \cite{ecg}}
\centering
\scriptsize
{\def1} \begin{tabular}{*{1}{c}|*{1}{c}*{1}{c|}*{1}{c}*{1}{c}*{1}{c}*{1}{c}{1} \begin{tabular}{*{1}{c}||*{1}{c}|*{1}{c}|*{1}{c}|*{1}{c}|*{1}{c}|*{1}{c}|*{1}{c}|*{1}{c}}
& type & type & type & type & type & type & type & type \\
& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
$input features$ & 1 & 320 & 256 & 256 & 128 & 128 & 128 & 64 \\
$output features$ & 320 & 256 & 256 & 128 & 128 & 128 & 64 & 64 \\
$Kernel\_size$ & 24 & 16 & 16 & 8 & 8 & 8 & 8 & 8 \\
$dilation\ rate$ & 1 & 2 & 4 & 4 & 6 & 8 & 8 & 8 \\
\end{tabular}}
\label{tab:CL_charact}
\end{table}
\begin{figure}[h]
\centering
\includegraphics[width = \columnwidth]{img/effic_on_goodfellow_compar.pdf}
\caption{Efficiency comparison on ECG \cite{ecg} for NEURAghe 12x4 and 11x5 MAC matrix configurations}
\label{fig:effic_compar_zed}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width = \columnwidth]{img/time_on_goodfellow_compar.pdf}
\caption{Execution Time comparison on ECG \cite{ecg} for NEURAghe 12x4 and 11x5 MAC matrix configurations}
\label{fig:time_compar_zed}
\end{figure}
The results are similar on the Ultra96 board, with increased performance, obviously, due to the higher clock frequency and the higher number of SoP modules. Efficiency and execution time are shown respectively in Figure \ref{fig:effic_compar_ultra} and in \ref{fig:time_compar_ultra}. Using minimum latency, efficiency is very low due to the bandwidth limits. Using $B=348$ we have around 0.9 efficiency corresponding to around 0.1 $ms$ per sample. To respect real-time constraint with minimum latency we can use $B=4$.
\begin{figure}[h]
\centering
\includegraphics[width = \columnwidth]{img/effic_on_goodfellow_ultra.pdf}
\caption{Efficiency on ECG \cite{ecg} for NEURAghe 9x10 MAC matrix configuration in ZU3EG}
\label{fig:effic_compar_ultra}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width = \columnwidth]{img/time_on_goodfellow_ultra.pdf}
\caption{Execution Time on ECG \cite{ecg} for NEURAghe 9x10 MAC matrix configuration in ZU3EG}
\label{fig:time_compar_ultra}
\end{figure}
\subsection{Res-TCN use-case}
This use case considers execution of the network presented in \cite{3D_skeleton}, made of stacked residual units, which perform 1D convolutions.
Table \ref{tab:skeleton_char_perf} shows the characteristics for the different types of Convolutional Layer in the network.
\begin{table}[]
\caption{Convolutional Layer characteristics for Res-TCN \cite{3D_skeleton}}
\centering
\scriptsize
{\def1} \begin{tabular}{*{1}{c}|*{1}{c}*{1}{c|}*{1}{c}*{1}{c}*{1}{c}*{1}{c}{1} \begin{tabular}{*{1}{c}||*{1}{c}|*{1}{c}|*{1}{c}|*{1}{c}|*{1}{c}|*{1}{c}}
& type & type & type & type & type & type \\
& 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
$ input features $ & 150 & 64 & 64 & 128 & 128 & 256 \\
$ output features $ & 64 & 64 & 128 & 128 & 256 & 256 \\
$ Kernel\_size $ & 8 & 8 & 8 & 8 & 8 & 8 \\
$ stride $ & 1 & 1 & 2 & 1 & 2 & 1 \\
\end{tabular}}
\label{tab:skeleton_char_perf}
\end{table}
Figure \ref{fig:effic_time_3D_hum} shows efficiency and execution time on this benchmark, highlighting contributions of the single layers.
Behaviour is similar to the ECG use case on both boards. Executing the processing after each sample ($B=1$), the efficiency is very limited for all the layers. Increasing $B$, the accelerator provides much better throughput. The layers executed less efficiently are the early layers, especially the first type, that pays for a significant underutilization of the MAC Matrix, due to the low number of input features. However, its contribution to the overall efficiency is marginal. Layers that show longer execution time are placed at the end of the TCN graph. These layers account for the highest contribution to the overall workload and, as shown in Figure \ref{fig:time_3D_hum} and \ref{fig:time_3D_hum_ultra}, are executed quite efficiently when sample batching is used ($B=144$). Sample batching is more effective on the Zedboard, that integrates a MAC Matrix with less input and output ports, thus requires longer runs to complete convolution, which reduces the impact of data-transfer and accelerator warm-up overheads.
\begin{figure}[h]
\subfloat[\label{fig:effic_3D_hum}]{\includegraphics[width = \columnwidth, height = 0.55\columnwidth]{img/effic_3D_human.pdf}}
\hfill
\subfloat[\label{fig:time_3D_hum}]{\includegraphics[width = \columnwidth, height = 0.55\columnwidth]{img/time_3D_human.pdf}}
\caption{Efficiency trend and Execution Time on Res-TCN \cite{3D_skeleton} for different batch sizes for NEURAghe 12x4 and 11x5 matrix configuration in Z-7020 SoC}
\label{fig:effic_time_3D_hum}
\end{figure}
\begin{figure}[h]
\subfloat[\label{fig:effic_3D_hum_ultra}]{\includegraphics[width = \columnwidth, height = 0.5\columnwidth]{img/effic_3D_human_ultra.pdf}}
\hfill
\subfloat[\label{fig:time_3D_hum_ultra}]{\includegraphics[width = \columnwidth, height = 0.5\columnwidth]{img/time_3D_human_ultra.pdf}}
\caption{Efficiency trend and Execution Time on Res-TCN \cite{3D_skeleton} for different batch sizes for NEURAghe 9x10 matrix configuration in ZU3EG SoC}
\label{fig:effic_time_3D_hum_ultra}
\end{figure}
\subsection{WN-PNT use-case}
This case considers a network derived from WaveNet (Van Den Oord et al. \cite{wavenet}) for Polyphonic Note Transcription of Time-Domain Audio Signal (\cite{polyWaveNet}). It is made of $20$ stacked residual blocks composed by a $1\times 1$ skip connection and a Dilated Convolutional block of $128$ channels each. \emph{Filter Size} is $2$ and the \emph{dilation factor} grows like $2^{k}$ where the residual block index $k \in \{0,1,2 ... 9,0,1,2 ... 9\}$.
\begin{figure}[h]
\subfloat[\label{fig:effic_poliph_ultra}]{\includegraphics[width = \columnwidth, height = 0.5\columnwidth]{img/effic_poliph_ultra.pdf}}
\hfill
\subfloat[\label{fig:time_poliph_ultra}]{\includegraphics[width = \columnwidth, height = 0.5\columnwidth]{img/time_poliph_ultra.pdf}}
\caption{Efficiency trend and Execution Time on WN-TCN \cite{polyWaveNet} for different batch sizes for NEURAghe 9x10 matrix configuration in ZU3EG}
\label{fig:effic_time_poliph}
\end{figure}
This case is especially challenging, due to the small size of kernels ($kernel\_size=2$ in most layers) and because the sample acquisition frequency is $16KHz$, posing hard constraints about real-time execution. Thus we focus on the Ultra-96 board, relying on its higher clock frequency to achieve the required throughput.
As may be noticed in Figure \ref{fig:effic_poliph_ultra}, in general, this use case is executed less efficiently on the platform, none of the layers reaches more than 0.56 efficiency. This is because the execution of the actual convolution kernels takes only two cycles and thus hardly overlaps with input data transfers. Batch size can be used to increase the reuse of weights, once they are loaded to the weight memory region, but, at the same time, has an impact on the duration of input and output transfers. To compensate this issue, considering that the total size of the local receptive fields in the network allows for the continuous storage on the on-chip memory for almost all layers, (unless for types 8, 9 and 10), we have slightly modified the scheduling described in Figure \ref{fig:scheduling}. In this use-case, we transfer to/from DDR only new output/input samples, while keeping the rest of the local receptive fields in the on-chip memory. As shown in Figure \ref{fig:time_poliph_ultra}, using $B=504$ we can process 504 samples in around $19 ms$, thus respecting the real-time constraint posed by the use-case.
\subsection{Assessment of hardware vs. software speed-up}
In order to evaluate the benefits obtained by on-FPGA acceleration, we have compared the execution of the convolution-related load of the three benchmarks on NEURAghe, with the execution on an A53 quad-core ARM processor, using the ARM Compute Library optimized functions \footnote{Since ARM-CL does not support dilated 1D convolution, for the sake of comparison, we have evaluated execution time on the A53 on analogous convolution layers, without dilation. Considering that in dilated convolutions input samples are not adjacent in memory, the exploitation of the vector processing in the ARM-CL could be suboptimal, thus the execution time reported in Figure \ref{fig:neu_a53_time} for the A53 could be underestimated. Our own software implementation (unoptimized) of a dilated convolution performs one order of magnitude slower than what reported for ARM-CL.}. As may be noticed, NEURAghe provides up to 10.8x, 10.7x and 5.7x speed-up on the three reference benchmarks, and considering the power consumption of the two platforms (around $0.9\ W$ for the A53 cores and around $3.3\ W$ for NEURAghe), PL-based acceleration provides improvement, in terms of power efficiency, corresponding to 33,8 GOPS/s/W , 26,3 GOPS/s/W and 15 GOPS/s/W respectively.
\begin{figure}[h]
\subfloat[\label{fig:neu_a53_time}]{\includegraphics[width = \columnwidth, height = 0.5\columnwidth]{img/NEUvsA53_time.pdf}}
\hfill
\subfloat[\label{fig:neu_a53_watt}]{\includegraphics[width = \columnwidth, height = 0.5\columnwidth]{img/NEUvsA53_gopss_watt.pdf}}
\caption{Execution time and power efficiency comparison between software execution on a Cortex-A53 quad core and NEURAghe (ultra-96).}
\label{fig:neu_a53}
\end{figure}
\subsection{Comparison to other APSoC based accelerators}
As mentioned, literature is lacking TCN evaluations on FPGA-based accelerators. The only attempt available is \cite{waveFPGA}, which is strictly customized for a autoregressive TCN and is implemented on a high-end board non usable in the embedded domain. However, when designing the TCN-supporting version of NEURAghe, we have considered compatibility with 2D convolutions for classic CNN acceleration. Moreover, the improved flexibity of the architecture, in terms of kernel size and stride, allows for some improvements with respect to other approaches in the literature targeting low-to-mid All-programmable SoCs.
In Table \ref{tab:comp_previous}
we report comparative results with state-of-the-art on two well known networks for image classification, ResNet-18 and VGG-16. We compare to other accelerator architectures, \cite{neuESL}, \cite{sharma2016high} and \cite{venieris2017latency}, that are implemented on the same kind of hardware and use the same 16-bit data precision.
On VGG-16, that exposes quite regular kernel sizes and stride values, our work shows comparable performance with respect to the alternatives. It executes convolutions slightly faster than \cite{neuESL} and \cite{sharma2016high} and around 13\% slower than \cite{venieris2017latency}. On ResNet-18, which exposes more variable $kernel\ sizes$ and $strides$, we can reduce a lot of overheads that must be payed by more \emph{static} architectures, executing the whole convolution workload 40\% faster than \cite{neuESL}.
\begin{table}[h]
\caption{Comparison between this work, previous version (\cite{neuESL}) and other works on ResNet-18 and VGG-16. Xilinx Zynq Z-7020}
\centering
\scriptsize
{\def1} \begin{tabular}{*{1}{c}|*{1}{c}*{1}{c|}*{1}{c}*{1}{c}*{1}{c}*{1}{c}{1} \begin{tabular}{*{1}{c}|*{1}{c}*{1}{c|}*{1}{c}*{1}{c}*{1}{c}*{1}{c}}
& \textbf{This} & & \textbf{This} & & & \\
& \textbf{Work} & \cite{neuESL} & \textbf{Work} & \cite{neuESL} & \cite{venieris2017latency} & \cite{sharma2016high} \\
\hline
& \multicolumn{2}{c|}{ResNet-18} & \multicolumn{4}{c}{VGG-16} \\
\hline
Xilinx Zynq SoC & \textbf{Z-7020} & Z-7020 & \textbf{Z-7020} & Z-7020 & Z-7020 & Z-7020 \\
Freq. [MHz] & \textbf{120} & 120 & \textbf{120} & 120 & 125 & 150 \\
GOPS/s & \textbf{26,5} & 16,1 & \textbf{42,62} & 42.48 & 48.53 & 31.38 \\
\end{tabular}}
\label{tab:comp_previous}
\end{table}
\section{Conclusions and future work}\label{sec:concl}
In this work, we have presented an accelerator architecture supporting Temporal Convolutional Networks, implemented on FPGA-based SoCs. We have presented the accelerator features serving this specific computing pattern, such as the capability of supporting arbitrary \emph{kernel\_size}, \emph{dilation rate} and \emph{stride values} without overhead.
We have also shown how data transfers from-to off-chip memory can be managed in TCNs and how performances improve by changing the computational paradigm, going from a \emph{latency constrained} approach to a \emph{batched} approach, that trades latency for throughput.
We applied this method to three notable TCNs and using two different SoCs as a target, analyzing throughput, latency and efficiency figures, and the capabilities of the system to respect real-time constraints.
Results show that using sample batching, we can achieve efficiency up to 0.96, 0.86 and 0.57 on the three use-cases. In the two use-cases with real-time requirements, an adequate sample batching can be used to achieve sufficient throughput to timely process all input samples.
We also compared the execution of the use-cases on a Cortex A53 quad-core, to evaluate the achievable speedup, noticing up to 10x execution time reduction, with an improvement in power efficiency that ranges from 2x to 3.5x depending on the benchmark.
Finally, we validated the compatibility of proposed architectural solutions with state-of-the-art CNNs used in the image processing domain, by direct comparison with previous work on APSoC-based accelerator, showing similar performance with respect to CNN-targeting alternatives when regular network topologies are targeted and 40\% improvement when targeting more irregular patterns.
\balance
\bibliographystyle{IEEEtran}
|
1,108,101,563,236 | arxiv | \section{Introduction}
The aim of this note is to present a two-dimensional differential calculus on the quantum disc algebra, which has no counterpart in the classical limit, but admits a well-defined (albeit different from the one in \cite{BegMaj:spe}) integral, and restricts properly to the quantum cone algebras. In this way the results of \cite{Brz:dif} are extended to other classes of non-commutative surfaces and to higher forms. The presented calculus is associated to an orthogonal pair of skew-derivations, which arise as a particular example of skew-derivations on generalized Weyl algebras constructed recently in \cite{AlmBrz:ske}. It is also a fundamental ingredient in the construction of the Dirac operator on the quantum cone \cite{BrzDab:spe} that admits a twisted real structure in the sense of \cite{BrzCic:twi}.
The reader unfamiliar with non-commutative differential geometry notions is referred to \cite{Brz:abs}.
\section{A differential calculus on the quantum disc}\label{sec.diff}
Let $0<q<1$. The coordinate algebra of the quantum disc, or the quantum disc algebra $\mathcal{O}(D_q)$ \cite{KliLes:two} is a complex $*$-algebra generated by $z$ subject to
\begin{equation}\label{disc}
z^*z - q^2zz^* = 1-q^2.
\end{equation}
To describe the algebraic contents of $\mathcal{O}(D_q)$ it is convenient to introduce a self-adjoint element $x = 1-zz^*$, which $q^2$-commutes with the generator of $\mathcal{O}(D_q)$, $xz = q^2zx$. A linear basis of $\mathcal{O}(D_q)$ is given by monomials
$x^kz^l, x^k{z^*}^{l}$.
We
view $\mathcal{O}(D_q)$ as a ${\mathbb Z}$-graded algebra, setting $\deg(z)=1$, $\deg(z^*)=-1$. Associated with this grading is the degree-counting automorphism
$\sigma:\mathcal{O}(D_q) \to \mathcal{O}(D_q)$, defined on homogeneous $a\in \mathcal{O}(D_q)$ by $\sigma(a) = q^{2\deg(a)} a$. As explained in \cite{AlmBrz:ske} there is an orthogonal pair of skew-derivations $\partial,\bar\partial: \mathcal{O}(D_q)\to \mathcal{O}(D_q)$ twisted by $\sigma$ and given on the generators of $\mathcal{O}(D_q)$ by
\begin{equation}\label{partial}
\partial(z) = z^*, \quad \partial(z^*) = 0, \qquad \bar\partial(z) = 0, \quad \bar\partial(z^*) = q^2z,
\end{equation}
and extended to the whole of $\mathcal{O}(D_q)$ by the (right) $\sigma$-twisted Leibniz rule. Therefore, there is also a corresponding first-order differential calculus $\Omega^1 (D_q)$ on $\mathcal{O}(D_q)$, defined as follows.
As a left $\mathcal{O}(D_q)$-module, $\Omega^1 (D_q)$ is freely generated by one forms $\omega, \bar\omega$. The right $\mathcal{O}(D_q)$-module structure and the differential $d :\mathcal{O}(D_q) \to \Omega^1 (D_q)$ are defined by
\begin{equation}\label{diff}
\omega a = \sigma(a) \omega, \quad \bar\omega a = \sigma(a) \bar\omega, \qquad d(a) = \partial(a)\omega + \bar\partial(a)\bar\omega.
\end{equation}
In particular,
\begin{equation}\label{dz}
dz = z^*\omega = q^2\omega z^*, \qquad dz^*= q^2z \bar\omega = \bar\omega z,
\end{equation}
and so, by the commutation rules \eqref{diff},
\begin{equation}\label{omegadz}
\omega = \frac{q^{-2}}{1-q^2}\left(dz z - q^4zdz\right), \qquad \bar\omega = \frac{q^{-2}}{1-q^2}\left( z^*dz^* - q^2dz^*z^*\right).
\end{equation}
Hence $\Omega^1 (D_q) = \{\sum_i a_i db_i\; |\; a_i,b_i\in \mathcal{O}(D_q)\}$, i.e.\ $(\Omega^1 (D_q), d)$ is truly a first-order differential calculus not just a degree-one part of a differential graded algebra. The appearance of $q^2-1$ in the denominators in \eqref{omegadz} indicates that this calculus has no classical (i.e.\ $q=1$) counterpart.
The first-order calculus $(\Omega^1 (D_q), d)$ is a $*$-calculus in the sense that the $*$-structure extends to the bimodule $\Omega^1 (D_q)$ so that $(a \nu b)^* = b^*\nu^* a^*$ and $(da)^* =d(a^*)$, for all $a,b\in \mathcal{O}(D_q)$ and $\nu \in \Omega^1 (D_q)$, provided $\omega^* = \bar\omega$ (this choice of the $*$-structure justifies the appearance of $q^2$ in the definition of $\bar\partial$ in equation \eqref{partial}). From now on we view $(\Omega^1 (D_q), d)$ as a $*$-calculus, which allows us to reduce by half the number of necessary checks.
Next we aim to show that the module of 2-forms $\Omega^2 (D_q)$ obtained by the universal extension of $\Omega^1 (D_q)$ is generated by the anti-self-adjoint 2-form\footnote{One should remember that the $*$-conjugation takes into account the parity of the forms; see \cite{Wor:dif}.}
\begin{equation}\label{volume}
\mathsf{v} = \frac{q^{-6}}{q^2-1}(\omega^*\omega + q^8\omega\omega^*), \qquad \mathsf{v}^* = - \mathsf{v}
\end{equation}
and to describe the structure of $\Omega^2 (D_q)$. By \eqref{diff}, for all $a\in \mathcal{O}(D_q)$,
\begin{equation}\label{va}
\mathsf{v} a = \sigma^2(a) \mathsf{v}.
\end{equation}
Combining commutation rules \eqref{diff} with the relations \eqref{dz} we obtain
\begin{equation}\label{zdz}
z^* dz = q^2dzz^*, \qquad dzz - q^4 zdz = q^2(1-q^2)\omega,
\end{equation}
and their $*$-conjugates. The differentiation of the first of equations \eqref{zdz} together with \eqref{diff} and \eqref{disc} yield
\begin{equation}\label{omom}
\omega\omega^* =(1-x)\mathsf{v}, \qquad \omega^*\omega = q^6(q^2x -1)\mathsf{v},
\end{equation}
which means that $\omega\omega^*$ and $\omega^*\omega$ are in the module generated by $\mathsf{v}$. Next, by differentiating $\omega z^* = q^{-2}z^*\omega$ and $\omega z = q^2z\omega$ and using \eqref{dz} and \eqref{diff} one obtains
\begin{equation}\label{domegaz}
d\omega z^* = q^{-2} z^*d\omega + z(\omega^*\omega+q^4\omega\omega^*), \quad d\omega z = q^2zd\omega + (q^2+q^{-2})z^*\omega^2.
\end{equation}
The differentiation of $dz = z^* \omega$ yields
\begin{equation}\label{zdom}
z^*d\omega = -q^2z\omega^*\omega.
\end{equation}
Multiplying this relation by $z$ from left and right, and using commutation rules \eqref{disc} and \eqref{diff} one finds that
$
(1-x) d\omega = q^{-4} z^*d\omega z.
$
Developing the right hand side of this equality with the help of the second of equations \eqref{domegaz} we find
\begin{equation}\label{domega}
d\omega = \frac{1+q^{-4}}{q^2-1} {z^*}^2 \omega^2.
\end{equation}
Combining \eqref{domegaz} with \eqref{domega} we can derive
\begin{equation}\label{zomega}
{z^*}^3\omega^2 = -z\frac{q^8}{q^4 +1}\left(\omega^*\omega +q^4\omega\omega^*\right) .
\end{equation}
The multiplication of \eqref{zomega} by $z^3$ from the left and right and the usage of \eqref{disc}, \eqref{diff} give
\begin{subequations}\label{omegas}
\begin{equation}\label{omega1}
(1-x)(1-q^{-2}x)(1-q^{-4}x) \omega^2 = - \frac{q^8}{q^4 +1}z^4\left(\omega^*\omega +q^4\omega\omega^*\right) ,
\end{equation}
\begin{equation}\label{omega2}
(1-q^2x)(1-q^{4}x)(1-q^{6}x) \omega^2 = - \frac{q^8}{q^4 +1}z^4\left(\omega^*\omega +q^4\omega\omega^*\right) .
\end{equation}
\end{subequations}
Comparing the left hand sides of equations \eqref{omegas}, we conclude that
\begin{equation}\label{xomegas}
x\omega^2 = 0 = \omega^2 x \quad \mbox{and, by $*$-conjugation,} \quad x\omega^{*2} = 0 = \omega^{*2} x,
\end{equation}
and hence in view of either of \eqref{omegas}
\begin{equation}\label{omega.sq}
\omega^2 = - \frac{q^8}{q^4 +1}z^4\left(\omega^*\omega +q^4\omega\omega^*\right).
\end{equation}
By \eqref{omom}, the right hand side of \eqref{omega.sq} is in the module generated by $\mathsf{v}$, and so is $\omega^2$ and its adjoint $\omega^{*2}$. Thus, the module $\Omega^2 (D_q)$ spanned by all products of pairs of one-forms is indeed generated by $\mathsf{v}$.
Multiplying \eqref{domega} and \eqref{zdom} by $x$ and using relations \eqref{xomegas} we obtain
\begin{equation}\label{xzomegas1}
xz\omega^*\omega = 0 = \omega^*\omega xz.
\end{equation}
Following the same steps but now starting with the differentiation of
$dz^*\!=\!q^2z \omega^*$ (see \eqref{dz}), we obtain the complementary relation
\begin{equation}\label{xzomegas2}
xz\omega\omega^* = 0 = \omega\omega^* xz.
\end{equation}
In view of the definition of $\mathsf{v}$, \eqref{xzomegas1} and \eqref{xzomegas2} yield
$
xz\mathsf{v} = 0 = \mathsf{v}xz.
$
Next, the multiplication of, say, the first of these equations from the left and right by $z^*$ and the use of \eqref{disc} yield
$
x(1-x)\mathsf{v} =0$ and $x(1-q^2x)\mathsf{v} =0$.
The subtraction of one of these equations from the suitable scalar multiple of the other produces the necessary relation
\begin{equation}\label{xv}
x\mathsf{v} = 0 = \mathsf{v} x,
\end{equation}
which fully characterises the structure of $\Omega^2 (D_q)$ as an $\mathcal{O}(D_q)$-module generated by $\mathsf{v}$. In the light of \eqref{xv}, the ${\mathbb C}$-basis of $\Omega^2 (D_q)$ consists of elements $\mathsf{v}z^n$, $ \mathsf{v}z^{*m}$, and hence, for all $w\in \Omega^2 (D_q)$, $wx = xw =0$, i.e., $\Omega^2 (D_q)$ is a torsion (as a left and right $\mathcal{O}(D_q)$-module). Since $\mathcal{O}(D_q)$ is a domain and $\Omega^2 (D_q)$ is a torsion, the dual of $\Omega^2 (D_q)$ is the zero module, hence, in particular $\Omega^2 (D_q)$ is not projective. Again by \eqref{xv}, the annihilator of $\Omega^2 (D_q)$,
$$
\mathrm{Ann}(\Omega^2 (D_q)) := \{ a\in \mathcal{O}(D_q) \; |\; \forall w\in \Omega^2 (D_q), \, aw=wa=0\},
$$
is the ideal of $\mathcal{O}(D_q)$ generated by $x$. The quotient $\mathcal{O}(D_q)/ \mathrm{Ann}(\Omega^2 (D_q))$ is the Laurent polynomial ring in one variable, i.e.\ the algebra $\mathcal{O}(S^1)$ of coordinate functions on the circle. When viewed as a module over $\mathcal{O}(S^1)$, $\Omega^2 (D_q)$ is free of rank one, generated by $\mathsf{v}$. Thus, although the module of 2-forms over $\mathcal{O}(D_q)$ is neither free nor projective, it can be identified with sections of a trivial line bundle once pulled back to the (classical) boundary of the quantum disc.
With \eqref{xv} at hand, equations \eqref{omom}, \eqref{omega.sq}, \eqref{domega} and their $*$-conjugates give the following relations in $\Omega^2 (D_q)$
\begin{subequations}\label{full}
\begin{equation}
d\omega = q^8 z^2\mathsf{v}, \quad d\omega^* = -z^{*2}\mathsf{v}, \quad \omega\omega^* = \mathsf{v}, \quad \omega^*\omega = -q^6\mathsf{v},
\end{equation}
\begin{equation}
\omega^2 = q^{12}\frac{q^2 -1}{q^4+1}z^4\mathsf{v}, \qquad \omega^{*2} = q^{-4}\frac{q^2 -1}{q^4+1}z^{*4}\mathsf{v}.
\end{equation}
\end{subequations}
One can easily check that \eqref{full}, \eqref{xv} and \eqref{va} are consistent with \eqref{diff} with no further restrictions on $\mathsf{v}$.
Setting $\Omega^n (D_q) =0$, for all $n>2$, we thus obtain a 2-dimensional calculus on the quantum disc.
\section{Differential calculus on the quantum cone}
The quantum cone algebra $\mathcal{O}(C^N_q)$ is a subalgebra of $\mathcal{O}(D_q)$ consisting of all elements of the ${\mathbb Z}$-degree congruent to 0 modulo a positive natural number $N$. Obviously $\mathcal{O}(C^1_q)= \mathcal{O}(D_q)$, the case we dealt with in the preceding section, so we may assume $N>1$. $\mathcal{O}(C^N_q)$ is a $*$-algebra generated by the self-adjoint $x=1-zz^*$ and by $y=z^N$, which satisfy the following commutation rules
\begin{equation}\label{cone}
xy = q^{2N} yx, \qquad yy^* = \prod_{l=0}^{N-1}\left(1-q^{-2l}x\right), \qquad y^*y = \prod_{l=1}^{N}\left(1-q^{2l}x\right).\end{equation}
The calculus $\Omega(C^N_q)$ on $\mathcal{O}(C^N_q)$ is obtained by restricting of the calculus $\Omega(D_q)$, i.e.\ $\Omega^n(C^N_q) = \{ \sum_{i}a_0^i d(a_1^i)\cdots d(a_n^i)a_{n+1}^i\; |\; a_k^i\in \mathcal{O}(C^N_q)\}$. Since $d$ is a degree-zero map $\Omega(C^N_q)$ contains only these forms in $\Omega(D_q)$, whose ${\mathbb Z}$-degree is a multiple of $N$. We will show that all such forms are in $\Omega(C^N_q)$. Since $\deg(\omega)= 2$, $\deg(\omega^*) =-2$ and $\deg(\mathsf{v})=0$,
this is equivalent to
$$
\Omega^1(C^N_q) = \mathcal{O}(D_q)_{\overline{-2}}\, \omega \oplus \mathcal{O}(D_q)_{\overline{2}}\, \omega^*, \qquad \Omega^2(C^N_q)= \mathcal{O}(C^N_q) \mathsf{v},
$$
where $\mathcal{O}(D_q)_{\overline{s}} = \{a\in \mathcal{O}(D_q)\; |\; \deg(a) \equiv s \!\! \mathbf{Mod}\! N\}$.
As an $\mathcal{O}(C^N_q)$-module, $\mathcal{O}(D_q)_{\overline{-2}}$ is generated by $z^{N-2}$ and ${z^*}^2$, hence to show that $\mathcal{O}(D_q)_{\overline{-2}}\, \omega\subseteq \Omega^1(C^N_q)$ suffices it to prove that $z^{N-2}\omega, {z^*}^2\omega \in \Omega^1(C^N_q)$. Using the Leibniz rule one easily finds that
$$
d y = \left(\qn{N}{q^2} -q^{-2N+4} \qn{N}{q^4}x\right) z^{N-2}\omega,
$$
where
$\qn{n}{s} := \frac{ s^n -1}{s-1}$. Hence, in view of \eqref{disc} and \eqref{diff},
\begin{subequations}\label{ydy}
\begin{equation}\label{y*dy}
y^* dy = \qn{N}{q^2}\left( 1 - q^4\frac{\qn{N}{q^4}}{\qn{N}{q^2}} x\right) \prod_{l=3}^N\left(1-q^{2l}x\right) {z^*}^2\omega,
\end{equation}
\begin{equation}\label{dyy*}
dy y^* = q^{-2N}\qn{N}{q^2}\left( 1 - q^{-2N+4}\frac{\qn{N}{q^4}}{\qn{N}{q^2}} x\right) \prod_{l=0}^{N-3}\left(1-q^{-2l}x\right) {z^*}^2\omega.
\end{equation}
\end{subequations}
The polynomial in $x$ on the right hand side of \eqref{y*dy} has roots in common with the polynomial on the right hand side of \eqref{dyy*} if and only if there exists an integer
$k\in \left[-2N+2, -N-1\right] \cup \left[2, N-1\right]$
such that
\begin{equation}\label{crit}
q^{2k}(q^{2N} +1) = {q^2}+1.
\end{equation}
Equation \eqref{crit} is equivalent to
$
{q^2} \qn{N+k-1}{q^2} + \qn{k}{q^2} =0,
$
with the left hand side strictly positive if $k>0$ and strictly negative if $k\leq -N$. So, there are no solutions within the required range of values of $k$.
Hence the polynomials \eqref{y*dy}, \eqref{dyy*} are coprime, and so there exists a polynomial (in $x$) combination of the left had sides of equations \eqref{ydy} that gives ${z^*}^2\omega$. This combination is an element of $\Omega^1(C^N_q)$ and so is ${z^*}^2\omega$.
Next,
$$
{z^*}^2\omega \, y = q^{2N}(1-{q^2}x)(1-q^4 x)z^{N-2}\omega ,
$$
$$
y {z^*}^2\omega = (1-q^{-2N+4}x)(1-q^{-2N+2}x)z^{N-2}\omega,
$$
so again there is an $x$-polynomial combination of the left hand sides (which are already in $\Omega^1(C^N_q)$) giving $z^{N-2}\omega$. Therefore, $\mathcal{O}(D_q)_{\overline{-2}}\, \omega\subseteq \Omega^1(C^N_q)$. The case of $\mathcal{O}(D_q)_{\overline{2}}$ follows by the $*$-conjugation.
Since $z^2\omega^*$, ${z^*}^2\omega$
are elements of $\Omega^1(C^N_q)$,
\begin{equation}\label{o*o1}
\Omega^2(C^N_q) \ni z^2\omega^*{z^*}^2\omega = q^{-4}(1-x)(1-q^{-2}x) \omega^*\omega = -q^2 \mathsf{v},
\end{equation}
by the quantum disc relations and \eqref{full} and \eqref{xv}.
Consequently, $\mathsf{v} \in \Omega^2(C^N_q)$. Therefore, $\Omega(C^N_q)$ can be identified with the subspace of $\Omega(D_q)$, of all the elements whose ${\mathbb Z}$-degree is a multiple of $N$.
\section{The integral}\label{sec.int}
Here we construct an algebraic integral associated to the calculus constructed in Section~\ref{sec.diff}.
We start by observing that since $\sigma$ preserves the ${\mathbb Z}$-degrees of elements of $\mathcal{O}(D_q)$ and $\partial$ and $\bar\partial$ satisfy the $\sigma$-twisted Leibniz rules, the definition \eqref{partial} implies that $\partial$ lowers while $\bar\partial$ raises degrees by $2$. Hence, one can equip $\Omega^1 (D_q)$ with the ${\mathbb Z}$-grading so that $d$ is the degree zero map, provided $\deg(\omega) = 2$, $\deg(\omega^*) = -2$. Furthermore, in view of the definition of $\sigma$, one easily finds that
\begin{equation}\label{q-deriv}
\sigma^{-1}\circ\partial \circ\sigma = q^4 \partial, \qquad \sigma^{-1}\circ\bar\partial \circ\sigma = q^{-4} \bar \partial,
\end{equation}
i.e.\ $\partial$ is a $q^4$-derivation and $\bar\partial$ is a $q^{-4}$-derivation. Therefore, by \cite{BrzElK:int}, $\Omega (D_q)$ admits a divergence, for all right $\mathcal{O}(D_q)$-linear maps $f:\Omega^1 (D_q)\to \mathcal{O}(D_q)$, given by
\begin{equation}\label{div}
\nabla_0(f) = q^4\partial\left(f\left(\omega\right)\right) + q^{-4}\bar\partial\left(f\left(\omega^*\right)\right) .
\end{equation}
Since the $\mathcal{O}(D_q)$-module $\Omega^2 (D_q)$ has a trivial dual, $\nabla_0$ is flat. Recall that by the {\em integral} associated to $\nabla_0$ we understand the cokernel map of $\nabla_0$.
\begin{thm}\label{thm.int}
The integral associated to the divergence \eqref{div} is a map $\Lambda: \mathcal{O}(D_q) \to {\mathbb C}$, given by
\begin{equation}\label{int}
\Lambda(x^kz^l) = \lambda \frac{\qn{k+1}{q^2}}{\qn{k+1}{q^4}} \delta_{l,0}, \qquad \mbox{for all $k\in {\mathbb N},\, l\in {\mathbb Z}$},
\end{equation}
where, for $l<0$, $z^l$ means ${z^*}^{-l}$ and $\lambda \in {\mathbb C}$.
\end{thm}
\begin{proof}
First we need to calculate the image of $\nabla_0$. Using the twisted Leibniz rule and the quantum disc algebra commutation rules \eqref{disc}, one obtains
\begin{equation}\label{part1}
\partial(x^k) = -q^{-2}\qn{k}{q^4} x^{k-1}{z^*}^2.
\end{equation}
Since $\partial(z^*) =0$, \eqref{part1} means that all monomials $x^k {z^*}^{l+2}$ are in the image of $\partial$ hence in the image of $\nabla_0$. Using the $*$-conjugation we conclude the $x^k {z}^{l+2}$ are in the image of $\bar\partial$ hence in the image of $\nabla_0$. So $\Lambda$ vanishes on (linear combinations of) all such polynomials.
Next note that
\begin{equation}\label{part2}
\partial(z^2) = ({q^2}+1) - (q^4+1)x,
\end{equation}
hence
$$
\partial(z^*z^2 - q^4z^2z^*) = (1-q^4)z^*, \quad \partial(z^*z^2 - {q^2}z^2z^*) =(1-{q^2})(1+q^4)xz^*.
$$
This means that $z^*$ and $xz^*$ are in the image of $\partial$, hence of $\nabla_0$. In fact, all the $x^kz^*$ are in this image which can be shown inductively. Assume $x^kz^* \in \mathrm{Im}(\partial)$, for all $k\leq n$. Then using the twisted Leibniz rule, \eqref{part1} and \eqref{part2} one finds
\begin{equation}\label{part3}
\partial(x^nz^2) = -{q^2}\qn{N}{q^4} x^{n-1} + ({q^2}+1)\qn{n+1}{q^4} x^n - \qn{n+2}{q^4}x^{n+1}.
\end{equation}
Since $\partial(z^*) =0$, equation \eqref{part3} implies that $\partial(z^nz^2z^*)$ is a linear combination of monomials $x^{n-1}z^*$, $x^nz^*$ and $x^{n+1}z^*$. Since the first two are in the image of $\partial$ by the inductive assumption, so is the third one. Therefore, all linear combinations of $x^kz^*$ and $x^kz$ (by the $*$-conjugation) are in the image of $\nabla_0$.
Put together all this means that $\Lambda$ vanishes on all the polynomials
$\sum_{k,l=1}^n(c_{kl}x^kz^l + c'_{kl}x^k{z^*}^l).
$
The rest of the formula \eqref{int} can be proven by induction. Set $\lambda = \Lambda(1)$. Since $\Lambda$ vanishes on all elements in the image of $\nabla_0$, hence also in the image of $\partial$, the application of $\Lambda$ to the right hand side of \eqref{part1} confirms \eqref{int} for $k=1$. Now assume that \eqref{int} is true for all $k\leq n$. Then the application $\Lambda$ to the right hand side of \eqref{part3} followed by the use of the inductive assumption yields
\begin{eqnarray*}
\qn{n+2}{q^4}\Lambda\left(x^{n+1}\right) &=& {q^2}\qn{N}{q^4} \Lambda\left(x^{n-1}\right) -({q^2}+1)\qn{n+1}{q^4} \Lambda\left(x^n\right) \\
&=& \lambda\left( ({q^2}+1)\qn{n+1}{q^2}-{q^2}\qn n{q^2} \right) = \lambda\qn{n+2}{q^2}.
\end{eqnarray*}
Therefore, the formula \eqref{int} is true also for $n+1$, as required.
\end{proof}
The restriction of $\Lambda$ to the elements of $\mathcal{O}(D_q)$, whose ${\mathbb Z}$-degree is a multiple of $N$ gives an integral on the quantum cone $\mathcal{O}(C^N_q)$.
\subsection*{Acknowledgment}
The work on this project began during the first author's visit to SISSA, supported by INdAM-GNFM. He would like to thank the members of SISSA for hospitality. The second author was supported in part by the Simons Foundation grant 346300 and the Polish Government MNiSW 2015-2019 matching fund.
|
1,108,101,563,237 | arxiv | \section{Introduction}
\subsection{The parameterized halting problem}
The complexity of the following parameterized halting problem is still wide
open.
\npprob{\ensuremath{p\text{-}\textsc{Halt}}}{$n\in \mathbb N$ in unary and a nondeterministic Turing
machine~$\mathbb M$} {$|\mathbb M|$, the size of $\mathbb M$}{Does $\mathbb
M$ accept the empty input in at most $n$ steps?}
The importance of \ensuremath{p\text{-}\textsc{Halt}}\ is derived from its close connections to central
problems in proof complexity and descriptive complexity
theory~\cite{cheflu12,nasremvia05}. Among others, there is a logic for
$\PTIME$ if $\ensuremath{p\text{-}\textsc{Halt}}$ can be decided by an algorithm in time $n^{f(|\mathbb
M|)}$ for some function $f: \mathbb N\to \mathbb N$. Sofar, however, such
algorithms have been ruled out only under a certain very strong
\emph{non-standard} complexity-theoretic hypothesis and only for computable
$f$~\cite{cheflu10,cheflu12}.
Thus, lower bounds on $\ensuremath{p\text{-}\textsc{Halt}}$ are poorly understood and of fundamental
interest. A seemingly modest and natural starting point is
\begin{conj}\label{conj:phaltAC}
$\ensuremath{p\text{-}\textsc{Halt}}\notin \mathsf{para}\text{-}\mathsf{AC}^0$.
\end{conj}
Here, $\mathsf{para}\text{-}\mathsf{AC}^0$ is the analogue of (uniform) $\mathsf{AC}^0$ in the parameterized
world. One easily sees that $\ensuremath{p\text{-}\textsc{Halt}}$ is in a \emph{nonuniform} version of
$\mathsf{para}\text{-}\mathsf{AC}^0$: for fixed $k\in\mathbb{N}$, let $\mathbb M_{k,0}, \ldots, \mathbb
M_{k,\ell_k-1}$ list all nondeterministic Turing machines of size $k$ and let
$n_{k,i}$ be the minimal $n$ such that $\mathbb M_{k,i}$ accepts the empty
input in $n$ steps; if there is no such $n$, let $n_{k,i}:=\infty$. Then, on
instances $(1^n,\mathbb M)$ with parameter $|\mathbb M|=k$, $\ensuremath{p\text{-}\textsc{Halt}}$ is
decided by the following family of simple Boolean functions:
\begin{align*}
F_{n,k}(& x_0\ldots x_{n-1}, y_0\ldots y_{k-1})
= \bigvee_{\substack{\text{$i<\ell_k$ such} \\
\text{that $n_{k,i}\le n$}}}
\big(x_0\ldots x_{n-1}= 1^n \wedge y_0\ldots y_{k-1}= \mathbb M_{k,i}\big).
\end{align*}
Observe that $F_{n,k}$ can be understood as a circuit of depth $2$ and size
$O(k\cdot \ell_k\cdot n)$.
Conjecture~\ref{conj:phaltAC} is highly plausible and might appear to be
within reach because $\mathsf{AC}^0$ is well-understood and,
in particular, \cite{cflowerac} establishes (unconditional) $\mathsf{para}\text{-}\mathsf{AC}^0$ lower
bounds for many well-studied parameterized problems. It deserves some genuine
interest because its failure implies that $\mathsf{AC}^0$, or equivalently,
\emph{$(+, \times)$-invariant $\mathsf{FO}$} is captured by some logic. However, we
failed to prove the conjecture after years of attempts and only now
understand why: it implies that nondeterministic exponential time $\mathsf{NE}$ is
distinct from the linear time hierarchy $\mathsf{LINH}$. This connection can be
further tightened by considering the following variant of $\ensuremath{p\text{-}\textsc{Halt}}$:
\npprob{$\ensuremath{p\text{-}\textsc{Halt}}_=$}{$n\in \mathbb N$ in unary and a nondeterministic Turing
machine $\mathbb M$}{$|\mathbb M|$}{Does $\mathbb M$ accept the empty input
in \emph{exactly} $n$ steps.}
While the classical problems underlying $\ensuremath{p\text{-}\textsc{Halt}}_=$ and $\ensuremath{p\text{-}\textsc{Halt}}$ are easily
seen to be equivalent, we shall see that their parameterized versions behave
quite differently. In fact, $\ensuremath{p\text{-}\textsc{Halt}}_=$ appears to be
harder than $\ensuremath{p\text{-}\textsc{Halt}}$, e.g., a simple Boolean function family like $F_{n,k}$
for $\ensuremath{p\text{-}\textsc{Halt}}$ is not known to exist for $\ensuremath{p\text{-}\textsc{Halt}}_=$. We refer to
Section~\ref{sec:concl} for some discussion.
We show:
\begin{theo}\label{thm:phaltequal}\
\begin{enumerate}\itemsep=0pt
\item[(i)] $\ensuremath{p\text{-}\textsc{Halt}}_=\in \mathsf{para}\text{-}\mathsf{AC}^0$ if and only if $\mathsf{NE}\subseteq \mathsf{LINH}$.
\item[(ii)] $\ensuremath{p\text{-}\textsc{Halt}}_=\in \mathsf{para}\text{-}\mathsf{AC}^0$ implies $\ensuremath{p\text{-}\textsc{Halt}}\in \mathsf{para}\text{-}\mathsf{AC}^0$.
\end{enumerate}
\end{theo}
\subsection{The \textup{MRDP}\ theorem}
Thus, to settle Conjecture \ref{conj:phaltAC} one might try to first separate
$\mathsf{NE}$ from $\mathsf{LINH}$. We tie this question to the provability of the
Matiyasevich-Robinson-Davis-Putnam (\textup{MRDP}) theorem~\cite{mrdp} in bounded
arithmetic. This theorem states that $\Sigma_1$-definable sets are
\emph{Diophantine} and it is a long standing open problem whether it is
provable in $I\Delta_0$, i.e., Peano arithmetic with induction restricted to
$\Delta_0$-formulas.
Wilkie observed~\cite{wil80} that a positive answer would imply the collapse
of $\mathsf{LINH}$ to $\mathsf{NLIN}$ (nondeterministic linear time), and therefore
$\NP= \co\NP$ and $\mathsf{NE}\not\subseteq \mathsf{LINH}$. We derive the latter consequence
from an apparently much weaker provability assumption:
\begin{theo}\label{thm:mrdp}
If $I\Delta_0$ proves \textup{MRDP}\ for small numbers, then $\mathsf{NE}\not\subseteq
\mathsf{LINH}$.
\end{theo}
Roughly, that \emph{$I\Delta_0$ proves \textup{MRDP}\ for small numbers} means that
the equivalence of any $\Delta_0$-formula $\varphi(\bar x)$ to some
Diophantine formula is proved in $I\Delta_0$ for all $\bar x$ of logarithmic
order. Model-theoretically, the equivalence holds in any $I\Delta_0$-model
for all $\bar x$ from the initial segment of numbers $x$ such that $2^x$
exists, while proof-theoretically, we allow an
$I\Delta_0$-proof to use exponentiation, but only once.
Such limited use of exponentiation has been studied in bounded
arithmetic~\cite{kra}. An unlimited use of exponentiation is sufficient:
Gaifman and Dimitracopoulos~\cite{gaifdim82} showed that $I\Delta_0+\forall
x\exists y\ 2^x{=}y$ does prove \textup{MRDP}. Kaye~\cite{kaye} proved \textup{MRDP}\ using
only induction for bounded existential formulas plus an axiom stating the
totality of a suitable function of exponential growth. It is asked
in~\cite[p.188]{gaifdim82} whether $I\Delta_0$ plus the totality of $x^{\log
x}$, or of $x^{\log\log x}$ etc. proves \textup{MRDP}, and it is easy to see that if
the answer to the first (or any) of these questions is positive, then
$I\Delta_0$ proves \textup{MRDP}\ for small numbers. Intuitively, this provability is
much weaker than $I\Delta_0$-provability.
\subsection{$\Delta_0$ truth}
Theorem~\ref{thm:mrdp} follows from our main result concerning the complexity
to decide the truth of $\Delta_0$-sentences:
\npprob[10.5]{\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}}{$n\in \mathbb N$ in unary and a $\Delta_0$-formula
$\varphi(x)$}{$|\varphi|$, the size of $\varphi$}{$\mathbb{N}\models\varphi(n)$\ ?}
Intuitively, taking $|\varphi|$ as the parameter means shifting attention to
inputs where $n$ is much larger than $|\varphi|$. This is a natural focus.
Classical work of Paris and Dimitracopolous~\cite{paris} took $n$ to be
nonstandard and related the complexity of truth definitions for
$\Delta_0$-formulas to the complexity-theoretic hypotheses that $\mathsf{LINH}$ or
$\PH$ does not collapse.
Wilkie proved a weak version of the former hypothesis by showing that
$\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$ restricted to quantifier-free formula inputs can be decided in
space $f(k)+O(\log n)$ where $k:= |\varphi|$ is the parameter and $f:\mathbb{N}\to
\mathbb{N}$ a computable function~\cite[Proof of Lemma 3.1]{wil80}.
The straightforward algorithm decides $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$ in space $f(k)\cdot \log n$.
Can $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$ be decided in space $f(k)+ O(\log n)$? Maybe with
nondeterminism? Can it be decided in time $f(k)\cdot n^{O(1)}$? Maybe with
nondeterminism, i.e., is $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\in
\para\NP$?
At present all these questions are wide open. Our main result
(Theorem~\ref{thm:truthupper}) shows that such \emph{upper bounds} on the
parameterized complexity of $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$ imply \emph{lower bounds} in classical
complexity theory. Notably,
\begin{theo}\label{thm:truthNP}
If $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\in \para\NP$, then $\mathsf{NE}\not\subseteq \mathsf{LINH}$.
\end{theo}
Theorem~\ref{thm:truthNP} follows from our analysis of $\ensuremath{p\text{-}\textsc{Halt}}_=$ and the
following unconditional lower bound:
\begin{theo}\label{thm:truthAC}
$\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\not\in\mathsf{para}\text{-}\mathsf{AC}^0$.%
\end{theo}
The proof is based on diagonalization or, more specifically, the
undefinability of truth. Furthermore, it relies on the classical
result~\cite{BIS} of descriptive complexity theory that, roughly speaking,
equates $\mathsf{AC}^0$ and first-order logic with built-in arithmetic.
\subsection{$\mathsf{AC}^0$-bi-immunity}
Could Conjecture~\ref{conj:phaltAC} be false? We give further evidence for
its truth by establishing a connection to the existence of $\mathsf{AC}^0$-bi-immune
sets in $\NP$. Recall, a problem $Q$ is \emph{$\mathsf{AC}^0$-bi-immune} if neither
$Q$ nor its complement contain an infinite subset in $\mathsf{AC}^0$.
\begin{theo}\label{thm:immu}
If $\NP$ contains an $\mathsf{AC}^0$-bi-immune problem, then $\ensuremath{p\text{-}\textsc{Halt}}\not\in\mathsf{para}\text{-}\mathsf{AC}^0$.
\end{theo}
It is a standard hypothesis that $\NP$ contains even $\P$-bi-immune problems
and this follows from the measure hypothesis~\cite{mayor}. Whether $\NP$
contains at least $\mathsf{AC}^0$-bi-immune problems has been asked once it was
realized~\cite{ghs,abhh} that deterministic time hierarchy theorems hold with
bi-immunity (or, equivalently~\cite{schoening}, almost everywhere) while this
is open for nondeterministic time~\cite{abhh,forsan}.
While Zimand~\cite{zimand} obtained some partial positive answers, Allender
and Gore~\cite{allgor90} showed that this has different answers relative to
different oracles.\footnote{\cite{allgor90} studies $\mathsf{AC}^0$-immunity but
their oracle constructions can be adapted to $\mathsf{AC}^0$-bi-immunity.} This
indicates that also refuting Conjecture~\ref{conj:phaltAC} might be
non-trivial.
\subsection{Outline}
Much of the technical work consists in connecting the dots between results of
various subareas of logic and complexity, namely classical, parameterized and
descriptive complexity theory and formal arithmetic. Section~\ref{sec:prelim}
reviews the results we need and fixes our notation. The technicalities are
somewhat subtle, in particular, the move from $\ensuremath{p\text{-}\textsc{Halt}}$ to $\ensuremath{p\text{-}\textsc{Halt}}_=$ is
crucial. Section~\ref{sec:halt} proves Theorem~\ref{thm:phaltequal} and
various variants of it via an analysis of $\ensuremath{p\text{-}\textsc{Halt}}_=$: it is in a strong sense
the hardest among what we call \emph{almost tally} problems in $\para\NP$.
Such problems have instances consisting of a natural number in unary notation
plus a short binary string where short means having a length effectively
bounded in terms of the parameter. Section~\ref{sec:truth} proves
Theorem~\ref{thm:truthAC}. This together with the results in
Section~\ref{sec:halt} implies Theorem~\ref{thm:truthNP} and various
variants. Section~\ref{sec:mrdp} derives (a strengthening of)
Theorem~\ref{thm:mrdp} from Theorem~\ref{thm:truthNP}. Section~\ref{sec:immu}
proves Theorem~\ref{thm:immu}. The final section discusses the role of
uniformity, and exhibits the different behaviours of our parameterized
problems $\ensuremath{p\text{-}\textsc{Halt}}$, $\ensuremath{p\text{-}\textsc{Halt}}_=$ and $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$.
\section{Preliminaries}\label{sec:prelim}
Standard monographs are~\cite{papad,arora} for classical complexity theory,
\cite{df,fg,df2} for parameterized complexity theory, \cite{hp,kayebook} for
formal arithmetic, and~\cite{immer,ebbflu} for descriptive complexity theory.
\subsection{Classical complexity}
A \emph{(classical) problem} is a subset of $\{0,1\}^*$, the set of finite
binary strings. The length of a binary string $x\in\{0,1\}^*$ is denoted
$|x|$. For $n\in \mathbb N$ we let $1^n$ denote the binary string consisting
of $n$ many~$1$'s. We use multitape Turing machines with alphabet $\{0,1\}$
as our basic model of computation. When considering \emph{dlogtime} Turing
machines, i.e.\ deterministic machines running in time $O(\log n)$, it is
understood that they access their input via an address tape
(see~e.g.~\cite{BIS}). As usual, $\P$ and $\NP$ denote deterministic and
nondeterministic polynomial time~$n^{O(1)}$, and $\mathsf{E}$ and $\mathsf{NE}$ denote
deterministic and nondeterministic exponential time with linear exponent
$2^{O(n)}$. The \emph{linear time hierarchy} $\mathsf{LINH}$ is the set of problems
acceptable by alternating Turing machines in linear time $O(n)$ with $O(1)$
alternations. $\LINSP$ and $\NLINSP$ denote deterministic and
nondeterministic linear space $O(n)$. Clearly,
\[
\mathsf{LINH}\subseteq \LINSP\subseteq \NLINSP\subseteq \mathsf{E}\subseteq \mathsf{NE}.
\]
Following~\cite{BIS} we define (dlogtime uniform) $\mathsf{AC}^0$ as the set of
problems decided by \emph{$\mathsf{AC}^0$-circuit families} $\big(\mathsf C_n\big)_{n\in
\mathbb N}$:
\begin{enumerate}\itemsep=0pt
\item[--] $\mathsf C_n$ is a circuit (with $\wedge, \vee, \neg$ gates and
unbounded fan-in) with $n$ variables, size $\le n^c$ and depth $\le d$,
where $c,d\in \mathbb N$ are two constants independent of $n$;
\item[--] there is a \textup{dlogtime}\ Turing machine which given $\angle{1^n, i,
b}$ where $n,i\in \mathbb N$ and $b\in \{0,1\}$ decides whether the
$i$-th bit of the binary encoding of $\mathsf C_n$ is $b$.
\end{enumerate}
Here, for binary strings $x= x_0\cdots x_{|x|-1}$ and $y= y_0\cdots
y_{|y|-1}$ we use the standard pairing
\begin{equation}\label{eq:pairing}
\angle{x,y}:=
x_0x_0\cdots x_{|x|-1}x_{|x|-1}01y_0y_0\cdots y_{|y|-1}y_{|y|-1},
\end{equation}
and similarly for more arguments. The above definition is somewhat sensitive
to the choice of the binary encoding of a circuit. An appropriate choice
would be to encode~$C_n$ by the list of strings in the \emph{direct
connection language} corresponding to $n$; we refer to~\cite{BIS} for
details.
For $n\in\mathbb{N}$ we let $\mathit{bin}(n)\in\{0,1\}^*$ denote the binary expansion of $n$;
it has length $\ceil{\log(n+1)}$. For $x\in \{0,1\}^*$ let $\mathit{num}(x)$ be the
natural number with binary expansion $1x$, i.e., $\mathit{bin}(\mathit{num}(x))= 1x$. For a
problem $Q$ let
\[
\mathit{un}(Q):= \left\{1^{\mathit{num}(x)}\mid x\in Q\right\}.
\]
The last statement of the following is~\cite[Proposition~5]{allgor90}, and
the first two are trivial:
\begin{prop}[\cite{allgor90}]\label{prop:ag}
Let $Q$ be a problem. Then:
\begin{enumerate}\itemsep=0pt
\item[(i)] $Q\in \mathsf{NE}$ if and only if $\mathit{un}(Q)\in \NP$.
\item[(ii)] $Q\in \mathsf{E}$ if and only if $\mathit{un}(Q)\in \P$.
\item[(iii)] $Q\in \mathsf{LINH}$ if and only if $\mathit{un}(Q)\in \mathsf{AC}^0$.
\end{enumerate}
\end{prop}
\subsection{Parameterized complexity}
A \emph{parameterized problem} is a pair $(Q, \kappa)$ of an
\emph{underlying} classical problem $Q\subseteq \{0,1\}^*$ and a polynomial
time computable \emph{parameterization} $\kappa:\{0,1\}^*\to \mathbb N$
mapping an instance $x\in \{0,1\}^*$ to its \emph{parameter} $\kappa(x)\in
\mathbb N$.
E.g., $\ensuremath{p\text{-}\textsc{Halt}}$ has underlying classical problem $\big\{\angle{1^n, \mathbb
M}\bigmid$ the nondeterministic Turing machine $\mathbb M$ accepts the empty
input in at most $n$ steps$\big\}$ and a parameterization $\kappa$ that maps
strings of the form $\angle{1^n, \mathbb M}$ to $|\mathbb M|$ and other
strings to, say,~0.
The para-operator~\cite{flugro03} turns a classical complexity class into a
parameterized one (the most important intractable parameterized classes are
not of this form, however). The class $\para\P= \FPT$ contains the
parameterized problems $(Q,\kappa)$ that are {\em fixed-parameter tractable},
i.e., decidable in deterministic time $f(\kappa(x))\cdot |x|^{O(1)}$ for some
computable $f:\mathbb{N}\to\mathbb{N}$. Similarly, $\para\NP$ denotes nondeterministic time
$f(\kappa(x))\cdot |x|^{O(1)}$ (for any computable $f$), $\para\L$ denotes
deterministic space $f(\kappa(x))+O(\log|x|)$, and $\para\NL$ denotes
nondeterministic such space. Clearly,
$$
\para\L\subseteq\para\NL\subseteq\FPT\subseteq\para\NP.
$$
The central parameterized
class in this paper is $\mathsf{para}\text{-}\mathsf{AC}^0$. It is characterized as follows:
\begin{prop}[\cite{cflowerac}]\label{prop:paraAC}
Let $(Q, \kappa)$ be a parameterized problem such that $Q$ is decidable and
$\kappa$ is computable by an $\mathsf{AC}^0$-circuit family. Then the following are
equivalent.
\begin{enumerate}\itemsep=0pt
\item[(i)] $(Q,\kappa)\in \mathsf{para}\text{-}\mathsf{AC}^0$.
\item[(ii)] There is a family $(\mathsf C_{n,k})_{n,k\in \mathbb N}$ of circuits
such that
\begin{enumerate}\itemsep=0pt
\item[--] there are a computable $f:\mathbb N\to \mathbb N$ and
$c,d\in \mathbb N$ such that for all $n,k\in \mathbb N$ the
circuit $\mathsf C_{n,k}$ has $n$ inputs, size at most $ f(k)\cdot n^c$,
and depth at most $d$;
\item[--] for all $x\in \{0,1\}^*$ we have
\[
x\in Q\iff \mathsf C_{|x|, \kappa(x)}(x)=1;
\]
\item[--] there are a computable $g:\mathbb N\to \mathbb N$ and a
deterministic Turing machine which given as input $\angle{1^n,
1^k, i, b}$ where $n,k,i\in \mathbb N$ and $b\in \{0,1\}$ decides
in time $g(k)+ O(\log n)$ whether the $i$-th bit of the binary
encoding of $\mathsf C_{n,k}$ is $b$.
\end{enumerate}
\item[(iii)] There are a computable $h: \mathbb N\to \mathbb N$ and an
$\mathsf{AC}^0$-circuit family $(\mathsf C_n)_{n\in \mathbb N}$ such that for all
$x\in \{0,1\}^*$ with $|x|\ge h(\kappa(x))$ we have
\[
x\in Q\iff \mathsf C_{|x|}(x)=1.
\]
\end{enumerate}
\end{prop}
According to the terminology of~\cite{flugro03}, (iii) states that
$(Q,\kappa)$ is \emph{eventually in $\mathsf{AC}^0$}.
\subsection{Formal arithmetic}
We let $L_\textup{ar}:=\{+,\times,0,1,<\}$ be the language of arithmetic with
binary function symbols $+,\times$, constants $0,1$ and a binary relation
symbol $<$. The {\em standard $L_\textup{ar}$-structure}, denoted $\mathbb{N}$, has
universe $\mathbb{N}$ and interprets the symbols in the obvious way. Every
$L_\textup{ar}$-term $p$ computes a polynomial with coefficients in $\mathbb{N}$ and of
total degree at most $|p|$. We do not distinguish terms $p$ or formulas
$\varphi$ from their binary encodings, so $|p|$ and $|\varphi|$ denote the
lengths of these encodings. Writing $\varphi(\bar x)$ for a formula $\varphi$
means that \emph{all} free variables of~$\varphi$ are among~$\bar x$. A {\em
sentence} is a formula without free variables.
A \emph{$\Delta_0$-formula} is an $L_\textup{ar}$-formula obtained from atomic
formulas, Boolean connectives, and bounded quantifiers $\exists x{<}p$,
$\forall x{<}p$ where $p$ is an $L_\textup{ar}$-term not involving $x$; e.g.,
$\exists x{<}p\; \varphi$ stands for $\exists x (x{<}p\wedge \varphi)$.
$\Sigma_1$- and $\Pi_1$-{\em formulas} are obtained from $\Delta_0$-formulas
by existential and universal quantification, respectively.
\begin{theo}[\textup{MRDP}]
For every $\Delta_0$-formula $\varphi(\bar x)$ there are $L_\textup{ar}$-terms
$p(\bar x,\bar y),q(\bar x,\bar y)$ such that
\[
\mathbb{N}\models \forall\bar x\big(\varphi(\bar x)\leftrightarrow
\exists\bar y\; p(\bar x,\bar y){=}q(\bar x,\bar y)\big).
\]
\end{theo}
G\"odel showed that computable functions are $\Sigma_1$-definable. The \textup{MRDP}\
theorem improves this to an existential definition:
\begin{cor}\label{cor:fxy}
For every computable $f:\mathbb N\to \mathbb N$ there is a quantifier-free
$L_\textup{ar}$-formula $\varphi_f(x, y, \bar z)$ such that for every $n,m\in
\mathbb N$
\begin{eqnarray*}
f(n)= m & \iff &
\mathbb{N} \models \exists \bar z\; \varphi_f(n,m,\bar z).
\end{eqnarray*}
\end{cor}
We are mainly concerned with finite arithmetical structures with universe
\[
[n]:=\{0,\ldots, n-1\}
\]
for some $n\in\mathbb{N}$ with $n\ge 2$, and
therefore consider the relational version
\[
L_\textup{ar}^\textup{r}
\]
of $L_\textup{ar}$ where $+$, $\times$ are ternary relation symbols. The standard
$L_\textup{ar}^\textup{r}$-structure with universe $\mathbb{N}$, also denoted $\mathbb{N}$, interprets
$+$, $\times$ by the graphs of addition and multiplication, respectively. For
$n\in \mathbb{N}$ with $n\ge 2$, the standard $L^\textup{r}_\textup{ar}$-structure with universe
$[n]$, simply denoted $n$, is the substructure of $\mathbb{N}$ with universe $[n]$,
i.e., it interprets the symbols in $L^\textup{r}_\textup{ar}$ by $+^n:=\{(k,\ell,m)
\in[n]^3\mid k+\ell=m\}$, $\times^n:= \{(k,\ell,m)\in [n]^3\mid k\cdot \ell=
m\}$, $0^n:=0$, $1^n:=1$ and $<^n:=\{(k,\ell)\in[n]^2\mid k<\ell\}$.
For every $L^\textup{r}_\textup{ar}$-formula $\varphi(\bar x)$ and $\bar n, n\in \mathbb{N}$ with
$1, \bar n<n$ we have
\begin{eqnarray*}
\mathbb{N}\models \varphi^{<n}(\bar n)
& \iff &
n\models \varphi(\bar n),
\end{eqnarray*}
where $\varphi^{<y}$ is obtained from $\varphi$ by replacing all quantifiers
$\exists z$, $\forall z$ by $\exists z{<}y$, $\forall z{<}y$.
\begin{rem}\label{rem:fxy}
Corollary~\ref{cor:fxy} holds for a quantifier-free $L^\textup{r}_\textup{ar}$-formula
$\varphi_f(\bar x, y, \bar z)$. Indeed, it is straightforward to express an
$L_\textup{ar}$-term equality by an existential $L^\textup{r}_\textup{ar}$-formula.
\end{rem}
\subsection{Descriptive complexity}
A binary string $x= x_0\cdots x_{n-1}\in \{0,1\}^*$ of length $n\ge 2$ is
often identified with the \emph{string structure} $\str S(x)$ defined as the
$L^\textup{r}_\textup{ar}\cup\{\mathit{ONE}\}$-expansion of the standard $L^\textup{r}_\textup{ar}$-structure
$n$ that interprets the unary relation symbol $\mathit{ONE}$ by
\[
\mathit{ONE}^x:=\{i\in[n]\mid x_i=1\},
\]
i.e., $\str S(x)= \big([n], +^n, \times^n, 0^n, 1^n, <^n, \mathit{ONE}^x\big)$.
We shall work with the following descriptive characterization of (dlogtime
uniform) $\mathsf{AC}^0$:
\begin{theo}[\cite{BIS}]\label{thm:BIS}
A problem $Q$ is in $\mathsf{AC}^0$ if and only if there is an
$L^\textup{r}_\textup{ar}\cup\{\mathit{ONE}\}$-sentence~$\varphi$ such that for every $x\in
\{0,1\}^*$ with $|x|\ge 2$:
\begin{eqnarray*}
x\in Q & \iff & \str S(x)\models \varphi.
\end{eqnarray*}
\end{theo}
This result and Proposition~\ref{prop:paraAC}~(iii) imply what is to be our
working definition of $\mathsf{para}\text{-}\mathsf{AC}^0$: the parameterized problems that are
\emph{eventually definable}.
\begin{cor}\label{cor:evtfo}
Let $(Q, \kappa)$ be a parameterized problem such that $Q$ is decidable and
$\kappa$ is computable by an $\mathsf{AC}^0$-circuit family.
Then $(Q, \kappa)$ is in $\mathsf{para}\text{-}\mathsf{AC}^0$ if and only if $(Q,
\kappa)$ is \emph{eventually definable}: there are a computable $h: \mathbb{N}\to \mathbb{N}$
and an $L^\textup{r}_\textup{ar}\cup \{\mathit{ONE}\}$-sentence $\varphi$ such that for all $x\in
\{0,1\}^*$ with $|x|\ge h(\kappa(x))$:
\begin{eqnarray*}
x\in Q & \Longleftrightarrow & \str S(x)\models \varphi.
\end{eqnarray*}
\end{cor}
In descriptive complexity the role of reductions is played by
interpretations. Let $L,L'$ be languages consisting of relation symbols and
constants. Let $w\in \mathbb{N}$ with $w\ge 1$. An \emph{interpretation $\inter I$ of
$L'$ in $L$ (of width $w$)} is given by an $L$-formula
$\varphi_{\textup{uni}}(\bar x)$, an $L$-formula $\varphi_R(\bar x_0, \cdots,
\bar x_{r-1})$ for each $r$-ary relation symbol $R\in L'$, and an $L$-formula
$\varphi_c(\bar x)$ for every constant $c\in L'$; here, $\bar x$, $\bar x_i$
are $w$-tuples of variables. Given an $L$-structure $A$ define the
$L'$-structure $A^{\inter I}$ as follows. It has universe $A^{\inter I}:=
\big\{\bar a\in A^w\bigmid A\models \varphi_{\textup{uni}}(\bar a)\big\}$,
interprets an $r$-ary $R\in L'$ by $\big\{(\bar a_0,\ldots,\bar a_{r-1})\in
(A^{\inter I})^r \bigmid A\models \varphi_R(\bar a_0, \ldots, \bar a_{r-1})
\big\}$, and a constant $c\in L'$ by the unique $\bar a\in A^{\inter I}$
satisfying $\varphi_c(\bar x)$ in $A$. If this uniqueness is violated or if
the universe $A^{\inter I}$ is empty, then $A^{\inter I}$ is not defined. If
$B\cong A^I$ for some~$I$, we say $B$ is {\em interpretable} in $A$. The
following is standard.
\begin{lem}\label{lem:interpretation}
Let $\inter I$ an interpretation of $L'$ in $L$ of width $w$ and $I'$ an
interpretation of $L''$ in $L'$ of width $w'$. Further let $A$ be an
$L$-structure such that $A^I$ is defined.
\begin{enumerate}\itemsep=0pt
\item[(i)] For every $L'$-formula $\varphi(x,y,\ldots)$ there is an
$L$-formula $\varphi^I(\bar x,\bar y,\ldots)$ where $\bar x, \bar y,
\ldots$ are $w$-tuples of variables such that for all $\bar a, \bar b,
\ldots \in A^I$:
\begin{eqnarray*}
A^I \models \varphi(\bar a, \bar b, \ldots)
& \iff &
A \models \varphi^I(\bar a, \bar b, \ldots).
\end{eqnarray*}
\item[(ii)] There is an interpretation $I'\circ I$ of $L''$ in $L$ of width
$w\cdot w'$ such that if $(A^I)^{I'}$ is defined, then so is
$A^{I'\circ I}$ and
\[
A^{I'\circ I }\cong (A^I)^{I'}.
\]
\end{enumerate}
\end{lem}
\medskip
The following is folklore, and a proof can be found
in~\cite[Appendix]{sch05}.
\begin{lem}\label{lem:karyinterpretation}
Let $d\in\mathbb{N}$.
\begin{enumerate}\itemsep=0pt
\item[(i)] For every $n\ge 2$ the standard $L^\textup{r}_\textup{ar}$-structure $n^d$
is interpretable in the standard $L^\textup{r}_\textup{ar}$-structure~$n$. In fact,
there is an interpretation $\inter I_d$ of width $d$ such that
$n^d\cong n^I$ for every $n\ge 2$, and the isomorphism maps each
$a<n^d$ to the length $d$ representation of $a$ in base~$n$.
\item[(ii)] There is an $L^\textup{r}_\textup{ar}$-formula $\mathit{BIT}(x, y)$ such that for
every $n\ge 2$ and all $i, j\in [n]$:
\begin{eqnarray*}
n\models \mathit{BIT}(i,j)
& \iff &
\text{the $j$-th bit of $\mathit{bin}(i)$ is $1$}.
\end{eqnarray*}
\end{enumerate}
\end{lem}
\section{$\ensuremath{p\text{-}\textsc{Halt}}$ and $\mathsf{NE}$ versus $\mathsf{LINH}$}\label{sec:halt}
In this section we first introduce a workable notion of reduction that
preserves $\mathsf{para}\text{-}\mathsf{AC}^0$, prove Theorem~\ref{thm:phaltequal}, and then consider
some generalizations and variants that will be instrumental later in
Section~\ref{sec:truth} for the proof of Theorem~\ref{thm:truthNP} and its
variants.
\subsection{Eventually definable reductions}
A \emph{parameterized} reduction from a parameterized problem $(Q,\kappa)$ to
another $(Q',\kappa')$ is a reduction $r:\{0,1\}^*\to\{0,1\}^*$ from $Q$ to
$Q'$ such that $\kappa'\circ r\le f\circ \kappa$ for some computable function
$f: \mathbb{N}\to \mathbb{N}$.
\begin{defn}\label{def:kappaevt}
Let $\kappa$ be a parameterization. A function $r:\{0,1\}^*\to \{0,1\}^*$ is
\emph{$\kappa$-eventually definable} if there are a computable $h:\mathbb{N}\to\mathbb{N}$
and an interpretation~$\inter I$ such that
\begin{eqnarray*}
\str S(x)^{\inter I} & \cong & \str S(r(x))
\end{eqnarray*}
for all $x\in\{0,1\}^*$ with $|x|\ge h(\kappa(x))$.
\end{defn}
\begin{exa}\label{exa:internum}
The function
\[
\angle{1^n, x} \mapsto
1^{\displaystyle \mathit{num}(\langle \mathit{bin}(n),x\rangle)}
\]
where $n\in \mathbb{N}$, $x\in \{0,1\}^*$ is $\kappa$-eventually definable where
$\kappa$ maps $\angle{1^n, x}$ to $|x|$ (both functions map arguments that
are not of the required form to, say, $0$).
\end{exa}
\proof Note $\mathit{num}(\angle{\mathit{bin}(n), x})< 2^{|\angle{\mathit{bin}(n), x}|+ 1}\le
2^{O(\log n+ |x|)}$. Choose a constant $d\in \mathbb{N}$ and a computable $h: \mathbb{N}\to
\mathbb{N}$ such that $\mathit{num}(\angle{\mathit{bin}(n), x})< n^d$ and $\mathit{num}(x)< n$ whenever $n\ge
h(|x|)$. We describe an interpretation of $\str S(1^{\mathit{num}(\angle{\mathit{bin}(n),
x})})$ in $\str S(\angle{1^n, x})$ whenever $n\ge h(|x|)$. It will be clear
that the interpretation does not depend on $n$, $x$.
Let $(n,\mathit{num}(x))$ be the expansion of the standard $L^\textup{r}_\textup{ar}$-structure
$n$ that interprets a new constant by $\mathit{num}(x)\in[n]$. This is interpretable
in $\str S(\angle{1^n, x})$ using $\mathit{BIT}$. By
Lemma~\ref{lem:karyinterpretation}, also $(n^d, \mathit{num}(x))$ is interpretable in
$\str S(\angle{1^n, x})$. But this structure defines ($n$ and)
$\mathit{num}(\angle{\mathit{bin}(n), x})\in [n^d]$ using $\mathit{BIT}$. Thus, $\str
S(1^{\mathit{num}(\angle{\mathit{bin}(n), x})})$ is interpretable in $\str S(\angle{1^n, x})$
as claimed. \proofend
Recall, a function $r:\{0,1\}^*\to\{0,1\}^*$ is \emph{honest} if $|r(x)|\ge
|x|^{\Omega(1)}$.
\begin{lem}\label{lem:transitive}
Assume that $r,r': \{0,1\}^*\to \{0,1\}^*$ are $\kappa$- and
$\kappa'$-eventually definable, respectively, that $\kappa'\circ r\le f\circ
\kappa$ for some computable $f:\mathbb{N}\to \mathbb{N}$, and that $r$ is honest. Then
$r'\circ r$ is $\kappa$-eventually definable.
\end{lem}
\proof Choose $\inter I,h$ for $r$ and $\inter I',h'$ for $r'$ according to
Definition~\ref{def:kappaevt}. We can assume
that~$h'$ is nondecreasing. Choose $n_0, c\in\mathbb{N}$ such that $|r(x)|\ge
|x|^{1/c}$ for all $x\in\{0,1\}^*$ with
$|x|\ge n_0$. Define $g:\mathbb{N}\to\mathbb{N}$ by
\[
g(k):= \max\big\{h'(f(k))^c,h(k),n_0\big\}.
\]
We claim that $I'\circ I$ and $g$ witness that $r'\circ r$ is
$\kappa$-eventually definable. To verify this let $x\in\{0,1\}^*$ satisfy
$|x|\ge g(k)$ where $k:= \kappa(x)$. Then $|r(x)|\ge |x|^{1/c}\ge h'(f(k))\ge
h'(\kappa'(r(x)))$ using that~$h'$ is nondecreasing. Hence $\str
S(r(x))^{I'}\cong \str S(r'(r(x)))$. Then $\str S(x)^{I'\circ I}\cong \str
S(r'(r(x)))$ because $\str S(x)^I\cong\str S(r(x))$, because $|x|\ge h(k)$.
\proofend
\begin{defn}\label{defn:evtdef}
Let $(Q, \kappa)$ and $(Q', \kappa')$ be parameterized problems. An
\emph{eventually definable reduction from $(Q, \kappa)$ to $(Q', \kappa')$}
is a parameterized reduction from $(Q, \kappa)$ to $(Q', \kappa')$ that is
honest and $\kappa$-eventually definable.
\end{defn}
\begin{rem}\label{rem:evtdef}
A parameterized problem is in $\mathsf{para}\text{-}\mathsf{AC}^0$ if and only if there is an eventually
definable reduction from $(Q,\kappa)$ to a trivial problem, say, $(Q_0,
\kappa_0)$ for $Q_0$ the set of strings starting with $0$ and $\kappa_0$ is
constantly $0$.
\end{rem}
It is straightforward to check that this reducibility is transitive and
preserves membership in~$\mathsf{para}\text{-}\mathsf{AC}^0$:
\begin{lem}\label{lem:ACclosed}
Assume there is an eventually definable reduction from $(Q, \kappa)$ to $(Q',
\kappa')$.
\begin{enumerate}\itemsep=0pt
\item[(i)] If there is an eventually definable reduction from $(Q',
\kappa')$ to $(Q'', \kappa'')$, then there is one from $(Q, \kappa)$ to
$(Q'', \kappa'')$.
\item[(ii)]If $(Q', \kappa')\in \mathsf{para}\text{-}\mathsf{AC}^0$, then $(Q, \kappa)\in \mathsf{para}\text{-}\mathsf{AC}^0$.
\end{enumerate}
\end{lem}
\proof
(i) follows from Lemma~\ref{lem:transitive}. (ii) follows from (i) and
Remark~\ref{rem:evtdef}. \proofend
\subsection{The complexity of $\ensuremath{p\text{-}\textsc{Halt}}_=$}
It is known that the question whether $\ensuremath{p\text{-}\textsc{Halt}}_=$ is fixed-parameter tractable
is closely related to the relationship of $\mathsf{E}$ and $\mathsf{NE}$:
\begin{theo}[\cite{aumdom08,cheflu09}]\label{thm:phaltENE}
$\ensuremath{p\text{-}\textsc{Halt}}_=\in \FPT$ if and only if $\mathsf{NE}\subseteq \mathsf{E}$.
\end{theo}
Theorem~\ref{thm:phaltequal}~(i) is a $\mathsf{para}\text{-}\mathsf{AC}^0$-analogue of this result.
\medskip\noindent \emph{Proof of Theorem~\ref{thm:phaltequal}:} (ii) follows easily
from the equivalence that a nondeterministic Turing machine $\mathbb M$
accepts the empty input in at most $n$ steps if and only if $\mathbb M$
accepts the empty input in exactly $n'$ steps for some $n'\le n$.
To prove (i), first assume $\mathsf{NE}\subseteq \mathsf{LINH}$ and let $Q$ be the classical
problem underlying $\ensuremath{p\text{-}\textsc{Halt}}_=$ but with input $n$ encoded in binary:
\nprob{$Q$}{$n \in \mathbb N$ in \emph{binary} and a nondeterministic Turing
machine $\mathbb M$}{Does $\mathbb M$ accept the empty input in exactly $n$
steps?}
Clearly, $Q\in \mathsf{NE}$, so by assumption and Proposition~\ref{prop:ag}~(iii) we
have $\mathit{un}(Q)\in \mathsf{AC}^0$. Recall
\begin{align*}
\mathit{un}(Q)=
\Big\{1^{\mathit{num}(\angle{\mathit{bin}(n), \mathbb M})} \Bigmid
\begin{array}{l}
\text{the nondeterministic Turing machine $\mathbb M$} \\
\text{accepts the empty input
in exactly $n$ steps}
\end{array}
\Big\}.
\end{align*}
By Example~\ref{exa:internum} the map $\angle{1^n, \mathbb M} \mapsto
1^{\mathit{num}(\angle{\mathit{bin}(n), \mathbb M})}$ is eventually definable with respect to
the parameterization of $\ensuremath{p\text{-}\textsc{Halt}}_=$. It is a honest parameterized reduction to
$(\mathit{un}(Q), \kappa)$ where~$\kappa$ maps $1^{\mathit{num}(\angle{\mathit{bin}(n), \mathbb M})}$
to $|\mathbb{M}|$. Since $(\mathit{un}(Q), \kappa)\in \mathsf{para}\text{-}\mathsf{AC}^0$,
Lemma~\ref{lem:ACclosed} implies $\ensuremath{p\text{-}\textsc{Halt}}_=\in \mathsf{para}\text{-}\mathsf{AC}^0$.
\medskip
Conversely, assume $\ensuremath{p\text{-}\textsc{Halt}}_=\in \mathsf{para}\text{-}\mathsf{AC}^0$. Let $Q\subseteq \{0,1\}^*$ be a
problem in $\mathsf{NE}$. To show that $Q\in \mathsf{LINH}$, it suffices to prove $\mathit{un}(Q)\in
\mathsf{AC}^0$ again by Proposition~\ref{prop:ag}~(iii).
As $Q \in \mathsf{NE}$ there is a nondeterministic Turing machine $\mathbb M$ and a
constant $c \in \mathbb N$ such that $\mathbb M$ accepts $Q$ in time at most
$\mathit{num}(x)^c-2|x|-2$. Consider the nondeterministic
Turing machine~$\mathbb M^*$\label{page:Mstar} that started with the empty
input runs as follows:
\begin{center}
\fbox{
\begin{minipage}[t]{15cm}
\begin{algorithm}
\im0 guess $y\in\{0,1\}^*$
\im0 simulate $\mathbb M$ on $y$
\im0 \IF\ $\mathbb M$ rejects, \THEN\ reject
\im0 make dummy steps such that so far the total
running time is $\mathit{num}(y)^c$
\im0 accept.
\end{algorithm}
\end{minipage}
}
\end{center}
Line~1 takes exactly $2|y|+2$ many steps by moving the head forth and back on
some tape,
so the dummy steps in line~4 are possible. Since $\mathit{num}$ is injective, we have
\begin{eqnarray}\label{eq:mes}
x \in Q & \iff & \text{$\mathbb M^*$ accepts the empty input tape
in exactly $ \mathit{num}(x)^c+1$ many steps}.
\end{eqnarray}
Since $\mathbb M^*$ is a fixed machine, $\ensuremath{p\text{-}\textsc{Halt}}_=\in \mathsf{para}\text{-}\mathsf{AC}^0$ implies that
the classical problem
\[
Q':=\Big\{ 1^n\mid \text{$\mathbb M^*$ accepts the empty input tape
in exactly $n+1$ many steps}\Big\}
\]
is in $\mathsf{AC}^0$. Choose a first-order sentence $\varphi$ for $Q'$ according to
Theorem~\ref{thm:BIS}. Lemma~\ref{lem:karyinterpretation} gives an
interpretation $\inter I$ such that $\str S(1^n)^{\inter I}\cong \str
S(1^{n^c})$ for all $n\ge 2$. Then $1^{n^c}\in Q'$ is equivalent to $\str
S(1^n)\models \varphi^{\inter I} $. Thus the r.h.s.\ in~\eqref{eq:mes} is
equivalent to $\str S(1^{\mathit{num}(x)})\models \varphi^{\inter I}$ provided
$\mathit{num}(x)\ge 2$, i.e., $x$ is non-empty. The l.h.s.\ in \eqref{eq:mes} is
equivalent to $1^{\mathit{num}(x)}\in \mathit{un}(Q)$. Thus $\varphi^{\inter I}$ witnesses
that $\mathit{un}(Q)\in \mathsf{AC}^0$ according to Theorem~\ref{thm:BIS}. \proofend
\begin{rem}\label{rem:xac0}
The direction from left to right only required a $\mathsf{AC}^0$-circuit family for
instances of $\ensuremath{p\text{-}\textsc{Halt}}_=$ with the fixed machine $\mathbb M^*$. This implies
that the assertions in Theorem~\ref{thm:phaltequal}~(i) are equivalent to
$\ensuremath{p\text{-}\textsc{Halt}}_=\in \XAC{}$ (see Definition~\ref{df:xac}).
\end{rem}
\subsection{Almost tally problems}
Recall that a classical problem $Q\subseteq \{0,1\}^*$ is \emph{tally} if
$Q\subseteq\{1\}^*$. All parameterized problems mentioned in the introduction
are almost tally in the following sense:
\begin{defn}
A parameterized problem $(Q,\kappa)$ is \emph{almost tally} if
\[
Q\subseteq\big\{\angle{1^n,x} \mid n\in\mathbb{N},x\in\{0,1\}^*\big\}
\]
and there is a computable $f:\mathbb{N}\to\mathbb{N}$ such that for all $n\in\mathbb{N}$,
$x\in\{0,1\}^*$
\[
|x|\le f(\kappa(\angle{1^n, x})).
\]
\end{defn}
Theorem~\ref{thm:phaltequal}~(ii) holds not only for $\ensuremath{p\text{-}\textsc{Halt}}$ but for every
almost tally problem in $\para\NP$. In fact, $\ensuremath{p\text{-}\textsc{Halt}}_=$ is the hardest almost
tally problem in $\para\NP$:
\begin{lem}\label{lem:phaltalmtally}
For every almost tally problem in $\para\NP$ there is an eventually definable
reduction to $\ensuremath{p\text{-}\textsc{Halt}}_=$.
\end{lem}
\proof Let $(Q, \kappa)\in \para\NP$ be almost tally. The identity is a
parameterized reduction from $(Q, \kappa)$ to its re-parameterization $(Q,
\kappa')$ where $\kappa'(\angle{1^n, x}):=|x|$ for all $n\in \mathbb{N}$, $x\in
\{0,1\}^*$. We can therefore assume that $\kappa= \kappa'$.
Let $\mathbb M$ be a nondeterministic Turing machine that accepts $Q$ and on
input $\angle{1^n, x}$ runs in time at most $f(k)\cdot n^{c}$ where $c\in
\mathbb{N}$, $f: \mathbb{N}\to \mathbb{N}$ is a computable function, and $k:=|x|$.
Define $g:\mathbb{N}^2\to\mathbb{N}$ by
\[
g(m,k):= m^{c+1}+ 2m+ 2k+ 2.
\]
For $x\in\{0,1\}^*$ with $k:=|x|$, consider the nondeterministic Turing
machine $\mathbb M_x$\label{page:Mx} that on the empty input runs as follows:
\begin{center}
\fbox{
\begin{minipage}[t]{15cm}
\begin{algorithm}
\im0 nondeterministically write $\angle{1^m,x}$ for some $m\in \mathbb{N}$
\im0 simulate $\mathbb M$ on $\angle{1^m,x}$
\im0 \IF\ $\mathbb M$ does not halt or rejects, \THEN\ reject
\im0 make dummy steps such that so far the total
running time is $g(m,k)$
\im0 accept.
\end{algorithm}
\end{minipage}
}
\end{center}
Step~1 can be implemented to take exactly $2+2m+2+2k$ many steps
(recall~\eqref{eq:pairing}), so the dummy steps in line~4 are possible if
$m>f(k)$. Note that for each $k$, the function $m\mapsto g(m,k)$ is
injective. Thus, if $n>f(k)$, we have
\begin{eqnarray*}
\angle{1^n,x}\in Q
& \Longleftrightarrow &
\angle{1^{g(n,k)+1},\mathbb M_x}\in\ensuremath{p\text{-}\textsc{Halt}}_=.
\end{eqnarray*}
Choose a parameterized honest reduction from $(Q,\kappa)$ to $\ensuremath{p\text{-}\textsc{Halt}}_=$ that
maps $\angle{1^n,x}$ with $n>f(k)$ to $\angle{1^{g(n,k)+1}, \mathbb M_x}$.
We verify that it is eventually definable.
Choose a computable $h:\mathbb{N}\to\mathbb{N}$ and $d\in\mathbb{N}$ such that $n\ge h(k)$ implies
$n> f(k)$ and $n^d> g(n,k)+ 1, \mathit{num}(x), \mathit{num}(\mathbb M_x)$ for all $x\in
\{0,1\}^*$ of length $|x|= k$.
Let $\angle{1^n, x}$ satisfy $n> h(k)$ where $k:= |x|$. By
Lemma~\ref{lem:karyinterpretation}, $\str S(\angle{1^n, x})$ interprets the
expansion $(n^{d}, k, \mathit{num}(x))$ of the standard $L^\textup{r}_\textup{ar}$-structure
$n^d$ by two constants $c_0$ and $c_1$ denoting~$k$ and $\mathit{num}(x)$. We are
left to show that $\big(n^{d}, k, \mathit{num}(x)\big)$ interprets $\str
S(\angle{1^{g(n,k)+1}, \mathbb M_x})$. Using $\mathit{BIT}$, it suffices to show that
both $g(n,k)+1$ and $\mathit{num}(\mathbb M_x)$ are definable in
$\big(n^{d},k,\mathit{num}(x)\big)$.
The former being clear, we show the latter. Consider a computable function
$M: \mathbb{N}\to \mathbb{N}$ that maps $\mathit{num}(x)$ to $\mathit{num}(\mathbb M_x)$. Choose a
quantifier-free $L^\textup{r}_\textup{ar}$-formula $\varphi_M(x, y, \bar z)$ according to
Remark~\ref{rem:fxy}. We can assume that $h$ grows fast enough so that for
all $\ell\in\mathbb{N}$
\[
\mathbb{N}\models \exists \bar z {<} h(\ell)\ \varphi_M(\ell,M(\ell),\bar z).
\]
Then $\exists \bar z\ \varphi_M(c_1,y,\bar z)$ defines $\mathit{num}(\mathbb M_x)$ in
the structure $(n^{d},k,\mathit{num}(x))$. \proofend
It is straightforward to infer from Proposition~\ref{prop:ag} that
$\mathsf{NE}\subseteq\mathsf{LINH}$ if and only if every tally problem in $\NP$ is in
$\mathsf{AC}^0$. We don't know of a similarly easy proof of the following
parameterized variant of this observation. Instead, our proof relies on our
analysis of $\ensuremath{p\text{-}\textsc{Halt}}_=$:
\begin{cor}\label{cor:almtallyNE}
$\mathsf{NE}\subseteq\mathsf{LINH}$ if and only if every almost tally problem in $\para\NP$
is in $\mathsf{para}\text{-}\mathsf{AC}^0$.
\end{cor}
\begin{proof}
The l.h.s.\ is equivalent to $\ensuremath{p\text{-}\textsc{Halt}}_=\in\mathsf{para}\text{-}\mathsf{AC}^0$ by
Theorem~\ref{thm:phaltequal}~(i). And by Lemmas~\ref{lem:phaltalmtally} and
\ref{lem:ACclosed}, $\ensuremath{p\text{-}\textsc{Halt}}_=\in\mathsf{para}\text{-}\mathsf{AC}^0$ is equivalent to the r.h.s..
\proofend
\end{proof}
\subsection{Variants}
For the optimistic reader, Corollary~\ref{cor:almtallyNE} is an approach to
separate $\mathsf{NE}$ from~$\mathsf{LINH}$. From this perspective, it is of interest to ask
whether finding an almost tally problem outside $\mathsf{para}\text{-}\mathsf{AC}^0$ but in a natural
subclass of $\para\NP$ implies stronger separations of natural complexity
classes. We verify the following variants of Corollary~\ref{cor:almtallyNE}:
\begin{lem}\label{lem:almtallyvariants} \
\begin{enumerate}\itemsep=0pt
\item[(i)] $\mathsf{E}\subseteq \mathsf{LINH}$ if and only if every almost tally problem in
$\FPT$ is in $\mathsf{para}\text{-}\mathsf{AC}^0$.
\item[(ii)] $\NLINSP\subseteq \mathsf{LINH}$ if and only if every almost tally
problem in $\para\NL$ is in $\mathsf{para}\text{-}\mathsf{AC}^0$.
\item[(iii)] $\LINSP\subseteq \mathsf{LINH}$ if and only if every almost tally
problem in $\para\L$ is in $\mathsf{para}\text{-}\mathsf{AC}^0$.
\end{enumerate}
\end{lem}
\proof The proof of (i) is analogous to the proof of
Corollary~\ref{cor:almtallyNE} using the subproblem of $\ensuremath{p\text{-}\textsc{Halt}}_=$ where the
input machine $\mathbb M$ is deterministic. Similarly the proof of (iii) is
analogous to the proof of (ii). We show how (ii) is proved by modifying the
proof of Corollary~\ref{cor:almtallyNE}.
Consider the following variant of $\ensuremath{p\text{-}\textsc{Halt}}$:
\npprob{$\ensuremath{p\text{-}\textsc{Halt}}^*_=$}{$n,m\in \mathbb N$ in unary with $n\le m$ and a
nondeterministic Turing machine $\mathbb M$}{$|\mathbb M|$}{Does $\mathbb M$
accept the empty input in \emph{exactly} $n$ steps and space at most
$\floor{\log m}$?}
It is clear that this problem is in $\para\NL$.
\medskip
\noindent \textit{Claim 1.} $\ensuremath{p\text{-}\textsc{Halt}}^*_=\in \mathsf{para}\text{-}\mathsf{AC}^0$ if and only if
$\NLINSP\subseteq \mathsf{LINH}$.
\medskip
\noindent \textit{Proof of the Claim 1.} Assume $\NLINSP\subseteq \mathsf{LINH}$ and
let $Q$ be the classical problem underlying $\ensuremath{p\text{-}\textsc{Halt}}^*_=$ but with the inputs
$n, m$ encoded in binary. Clearly, $Q\in \NLINSP \subseteq \mathsf{LINH}$, so
$\mathit{un}(Q)\in \mathsf{AC}^0$ by Proposition~\ref{prop:ag}~(iii). Similarly as
Example~\ref{exa:internum} one sees that $\angle{1^n, 1^m, \mathbb M}\mapsto
1^{\mathit{num}(\angle{n, m, \mathbb M})}$ is eventually definable. Then $\ensuremath{p\text{-}\textsc{Halt}}^*_=
\in \mathsf{para}\text{-}\mathsf{AC}^0$ follows as before.
Conversely, assume $\ensuremath{p\text{-}\textsc{Halt}}^*_= \in \mathsf{para}\text{-}\mathsf{AC}^0$ and let $Q\in \NLINSP$. Choose a
nondeterministic Turing machine $\mathbb M$ accepting $Q$ that on input $x\in
\{0,1\}^*$ runs in time at most
\[
\frac{\mathit{num}(x)^{c}}{10c(|x|+2)} - 10c(|x|+ 2)- |x|
\]
and uses space at most $c\cdot |x|$; here $c\in \mathbb{N}$ is a suitable constant.
Define $\mathbb M^*$ as in page~\pageref{page:Mstar} but with the following implementation details. For
the simulation in line 2, first initialize a length $c(|y|+ 2)$ binary
counter using exactly $10c(|y|+ 2)$ steps, and increase it using exactly
$10c(|y|+ 2)$ many steps for each simulated step of $\mathbb M$. In line~4
continue increasing the counter in this way until it reaches
$\mathit{num}(y)^c/(10c(|y|+ 2))$. For long enough $y$, the binary representation of
this number can be computed in time at most $\mathit{num}(y)$ and space $O(|y|)$
(where the constant in the O-notation depends on $c$). This computation can
be done in parallel to the simulation in lines 2 and 4. Hence, line~5
completes exactly $\mathit{num}(y)^c+1$ steps, and uses space at most $d\cdot |y|$
for a suitable~$d\ge c$.
Thus, we arrive at the following variant of~\eqref{eq:mes}. For long enough
$x\in\{0,1\}^*$:
\begin{eqnarray*}
x \in Q & \iff & \text{$\mathbb M^*$ accepts the empty input
in exactly $ \mathit{num}(x)^c+1$ many steps}\\
&&\text{and space at most $\floor{\log(\mathit{num}(y)^d)}$}.
\end{eqnarray*}
Our assumption $\ensuremath{p\text{-}\textsc{Halt}}_=^*\in \mathsf{para}\text{-}\mathsf{AC}^0$ implies that the classical problem
\begin{align*}
Q':=\big\{\angle{1^n,1^m}\bigmid\ & n\le m \text{ and $\mathbb M^*$ accepts the empty input
in exactly $n+1$ many steps}\\
&\text{and space at most $\floor{\log m}$}\big\}
\end{align*}
is in $\mathsf{AC}^0$. Now $\mathit{un}(Q)\in\mathsf{AC}^0$ (and hence $Q\in\mathsf{LINH}$) follows as before
using an interpretation~$\inter I$ such that $\str S(1^n)^{\inter I}\cong
\str S(\big\langle 1^{n^c}, 1^{n^d}\big\rangle)$. \hfill$\dashv$
\medskip
\noindent \textit{Claim 2.} For every almost tally problem in $\para\NL$
there is an eventually definable reduction to $\ensuremath{p\text{-}\textsc{Halt}}^*_=$.
\medskip
\noindent \textit{Proof of the Claim 2.} Let $(Q, \kappa)\in \para\NL$ be
almost tally and $\mathbb M$ be a nondeterministic Turing machine that
accepts $Q$ and that on input $\angle{1^n, x}$ runs in time at most
$f(k)\cdot n^{c}$ and space at most $f(k)+ c\cdot\log n$ where $c\in \mathbb{N}$,
$f:\mathbb{N}\to \mathbb{N}$ is a computable function, and $k:= \kappa(\angle{1^n, x})= |x|$.
For $x\in \{0, 1\}^*$ with $k:= |x|$, define the nondeterministic Turing
machine $\mathbb M_x$ as in page~\pageref{page:Mx} but
with a different $g$ (chosen below) and line 1 changed to
nondeterministically write some $m\in \mathbb{N}$ in binary in exactly
$2\ceil{\log(m+ 1)}+ 2$ steps. The simulation in line 2 is done as in the
previous claim maintaining a length $(c+1)\ceil{\log(m+1)}$ binary counter.
It further maintains the position of $\mathbb M$'s head on the input tape
\big(which we can assume to be at most $|\angle{1^m,x}|+1$\big) and uses it
to compute the currently scanned bit. If $k<m$, both the maintenance of the
counter and position can be done in exactly $10c\ceil{\log(m+1)}$ steps. In
line 4 the binary counter is updated until it reaches $m^{c+1}$. Hence line~4
is completed after exactly $g(m,k):=m^{c+1}\cdot 10c\ceil{\log(m+ 1)}+
2\ceil{\log(m+1)}+ 2$ steps. The dummy steps in line~4 are possible if
$m>f(k)$. In this case the computation takes space at most $d\log m$ for
suitable $d\in\mathbb{N}$. Thus, if $n> f(k)$, we have
\begin{eqnarray*}
\angle{1^n, x}\in Q
& \Longleftrightarrow &
\angle{1^{g(n,k)+1},1^{n^d},\mathbb M_x} \in \ensuremath{p\text{-}\textsc{Halt}}^*_=.
\end{eqnarray*}
Similarly as seen in the proof of Lemma~\ref{lem:phaltalmtally}, this implies
the claim. \hfill$\dashv$
\medskip
It now suffices to show that $\ensuremath{p\text{-}\textsc{Halt}}^*_= \in \mathsf{para}\text{-}\mathsf{AC}^0$ if and only if every
almost tally problem in $\para\NL$ is in $\mathsf{para}\text{-}\mathsf{AC}^0$. The forward direction
follows from Claim~2 and Lemma~\ref{lem:ACclosed}. And if $\ensuremath{p\text{-}\textsc{Halt}}^*_= \not\in
\mathsf{para}\text{-}\mathsf{AC}^0$, then we get an almost tally problem in $\para\NL \setminus \mathsf{para}\text{-}\mathsf{AC}^0$
by rewriting inputs $\angle{1^n, 1^m, \mathbb M}$ of $\ensuremath{p\text{-}\textsc{Halt}}_=^*$ to
$\angle{1^{\angle{n, m}}, \mathbb M}$ where $\angle{n, m}$ is a pairing
function on $\mathbb{N}$. \proofend
We find it worthwhile to explicitly point out the following direct corollary
concerning the parameterized halting problem for {\em deterministic} Turing
machines:
\begin{cor}
If $\ensuremath{p\text{-}\textsc{DHalt}}\not\in \mathsf{para}\text{-}\mathsf{AC}^0$, then $\mathsf{E}\not\subseteq \mathsf{LINH}$.
\npprob{$\ensuremath{p\text{-}\textsc{DHalt}}$}{$n\in \mathbb N$ in unary and a deterministic Turing
machine $\mathbb M$}{$|\mathbb M|$}{Does $\mathbb M$ accept the empty input
in at most $n$ steps?}
\end{cor}
\section{On the parameterized complexity of $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$}\label{sec:truth}
This section first observes that $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$ is ``the same'' as a basic
parameterized model-checking problem, uses this to prove the lower bound
$\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\not\in\mathsf{para}\text{-}\mathsf{AC}^0$ (Theorem~\ref{thm:truthAC}), and finally, based on
the previous section, infers consequences from {\em upper bounds} on the
parameterized complexity of $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$, including Theorem~\ref{thm:truthNP}.
\subsection{Model-checking arithmetic}
We observe that $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$ is ``the same'' as the parameterized model-checking
problem for first-order logic over finite standard $L^\textup{r}_\textup{ar}$-structures:
\npprob{$\pMC(L^\rel_\arit)$}{$n\ge 2$ in unary and an $L^\textup{r}_\textup{ar}$-sentence
$\varphi$}{$|\varphi|$}{$n\models \varphi$?}
\begin{lem}\label{lem:fcttorel}
There is a computable function that maps every $\Delta_0$-formula
$\varphi(x)$ to an $L^\textup{r}_\textup{ar}$-sentence~$\psi$ such that for all $n\in \mathbb{N}$
with $n\ge 2$:
\begin{eqnarray}\label{eq:fcttorel}
\mathbb{N}\models \varphi(n)
& \Longleftrightarrow &
n\models \psi.
\end{eqnarray}
Further, there is a computable function that maps every
$L^\textup{r}_\textup{ar}$-sentence $\psi$ to a $\Delta_0$-formula~$\varphi(x)$ such that
\eqref{eq:fcttorel} holds all $n\in\mathbb{N}$ with $n\ge2$.
\end{lem}
\proof
For the second
assertion define $\varphi(x)$ as~$\psi^{<x}$ with atoms rewritten in the functional language $L_\textup{ar}$. The first assertion
is folklore, see~\cite[Proposition~2.2]{gaifdim82}. We give a brief sketch
for completeness. It is routine to compute, given a $\Delta_0$-formula
$\varphi(\bar x)$, a constant $c_\varphi>1$ and an $L^\textup{r}_\textup{ar}$-formula
$\psi_0(\bar x)$ such that
\begin{eqnarray*}
\mathbb{N}\models \varphi(\bar n)&\Longleftrightarrow&\mathbb{N}\models\psi_0^{<m}(\bar n)
\end{eqnarray*}
for all $\bar n,m\in\mathbb{N}$ with $m\ge \max\{\bar n,2\}^{c_\varphi}$. Hence, for
$n\ge 2$, the truth of $\varphi(n)$ is equivalent to $n^{c_\varphi} \models
\psi_0(n)$. Since the number $n$ is definable in the standard
$L^\textup{r}_\textup{ar}$-structure $n^{c_\varphi}$ (as the minimal element whose
$c_\varphi$-th power does not exist), we can replace $\psi_0(n)$ by some
sentence~$\psi_1$. Then set $\psi:=\psi_1^{\inter I_{c_\varphi}}$ for the
interpretation $\inter I_{c_\varphi}$ from
Lemma~\ref{lem:karyinterpretation}. \proofend
\subsection{A lower bound} In this subsection we prove Theorem~\ref{thm:truthAC}. We fix
a proper elementary extension $M$ of the standard $L_\textup{ar}^\textup{r}$-model $\mathbb{N}$,
and a nonstandard element $a\in M\setminus \mathbb{N}$. We let $<^M$ denote the
interpretation of $<$ in $M$. We need a simple lemma:
\begin{lem}\label{lem:Mf}
Let $f:\mathbb N\to \mathbb N$ be a computable function. Then there is an
$L_\textup{ar}^\textup{r}$-formula $\chi_f(x, y)$ such that
for every $k\in \mathbb N$ and every $b<^Ma$:
\begin{eqnarray*}\label{eq:chif}
f(k)=b & \iff & M \models \chi_f^{<a}(k,b).
\end{eqnarray*}
\end{lem}
\proof
Let $\exists z\varphi(x,y,z)$ define $f$ in $\mathbb{N}$ where $\varphi$ is $\Delta_0$. As in the proof of Lemma~\ref{lem:fcttorel} let $\psi(x,y,z)$ be an $L_\textup{ar}^\textup{r}$-formula such that for all $k,\ell,m,n\in\mathbb{N}$ with $k,\ell,m<n$:
\begin{eqnarray}\label{eq:fpsi}
\mathbb{N}\models\varphi(k,\ell,m)\iff\mathbb{N}\models\psi^{<n}(k,\ell,m)
\end{eqnarray}
Here, $\mathbb{N}$ ambiguously denotes the standard model in the respective languages. Set
$$
\chi_f(x,y):=\exists z\psi(x,y,z).
$$
If $f(k)=b$, then $b\in\mathbb{N}$ and there is $m\in\mathbb{N}$ such that $\mathbb{N}\models\psi^{<n}(k,b,m)$ for all $n>k,b,m$, so $M\models\psi^{<a}(k,b,m)$ as $k,b,m<^Ma$, and $M\models \chi^{<a}_f(k,b)$. Conversely, assume $M\models \chi^{<a}_f(k,b)$ for $b<^Ma$ and $f(k)\neq b$; then
$\exists u,y,z(k,y,z<u\wedge \psi^{<u}(k,y,z)\wedge y\neq f(k))$ holds in $M$ and hence in $\mathbb{N}$; by \eqref{eq:fpsi}, $\mathbb{N}\models\varphi(k,\ell,m)$ for some $\ell,m\in\mathbb{N}$ with $\ell\neq f(k)$; this contradicts the fact that $\exists z\varphi(x,y,z)$ defines~$f$ in $\mathbb{N}$.
\proofend
Some notation: for $n\in\mathbb{N}$ define the $L^\textup{r}_\textup{ar}$-formula $\text{``}x{=}n\text{''}$ by $\text{``}x{=}0\text{''}:= x{=}0$ and $\text{``}x{=}(n+1)\text{''}:=\exists y(\text{``}y{=}n\text{''}\wedge +(y,1,x))$. For an $L^\textup{r}_\textup{ar}$-formula $\varphi(y,\bar x)$ set $\varphi(\underline{n},\bar x):=\exists y(\text{``}y{=}n\text{''}\wedge\varphi(y,\bar x))$; we understand $\varphi^{<z}(\underline{n},\bar x)$ as
$\big(\varphi(\underline{n},\bar x)\big)^{<z}$. If $n<m$, then both
$(\text{``}x{=}n\text{''})^{<m}$ and $\text{``}x{=}n\text{''}$ define $n$ in $\mathbb{N}$, so $\varphi^{<m}(\underline{n},\bar x)$ and
$\varphi^{<m}(n,\bar x)$ are equivalent in $\mathbb{N}$. In particular, for
every $n\in\mathbb{N}$:
\begin{eqnarray}\label{eq:subst}
M\models\varphi^{<a}(\underline{n},\bar x)\leftrightarrow\varphi^{<a}(n,\bar x) )
\end{eqnarray}
\medskip
\noindent{\em Proof of Theorem~\ref{thm:truthAC}:} For contradiction, assume
otherwise, so $\pMC(L^\rel_\arit)$ $\in \mathsf{para}\text{-}\mathsf{AC}^0$ by Lemma~\ref{lem:fcttorel}.
By~Corollary~\ref{cor:evtfo}, there is an increasing computable function $h:
\mathbb N\to \mathbb N$ and a sentence $\textit{sat}$ such that for every
$n\in \mathbb N$ and every $L^\textup{r}_\textup{ar}$-sentence $\varphi$ with $n>
h(\mathit{num}(\varphi))$ we have
\begin{eqnarray}\label{eq:sat}
n \models \varphi
& \iff &
\str S\big(\angle{1^n, \varphi}\big) \models \textit{sat}.
\end{eqnarray}
For $k<n$, let $(n,k)$ denote the expansion of the standard
$L^\textup{r}_\textup{ar}$-structure $n$ that interprets a constant $c$ by $k$. It is
clear that there is an interpretation $\inter I$ (independent of $n,
\varphi$) such that $(n,\mathit{num}(\varphi))^{\inter I}\cong\str S(\angle{1^n,
\varphi})$ for all $\varphi$ with $\mathit{num}(\varphi)<n$. Replacing in
$\textit{sat}^{\inter I}$ the constant $c$ by a new variable $x$ gives an
$L^\textup{r}_\textup{ar}$-formula $\textit{true}(x)$ such that for $n>
h(\mathit{num}(\varphi))\ge \mathit{num}(\varphi)$:
\begin{eqnarray}\nonumber
\str S\big(\angle{1^n, \varphi}\big) \models \textit{sat}
& \iff &
n \models \textit{true}\big(\mathit{num}(\varphi)\big) \\\label{eq:true}
& \iff &\mathbb{N}\models \textit{true}^{<n}\big(\mathit{num}(\varphi)\big),
\end{eqnarray}
where $\mathbb{N}$ is the standard $L^\textup{r}_\textup{ar}$-model. Since $h:\mathbb N\to
\mathbb N$ is computable, there is an $L^\textup{r}_\textup{ar}$-formula $\text{``}
h(x)< y\text{''}$ with the obvious meaning. Note the l.h.s.\ of
\eqref{eq:sat} is equivalent to $\mathbb{N}\models\varphi^{<n}$. Combining
\eqref{eq:sat} and \eqref{eq:true} we get
\[
\mathbb{N}\models \text{``}h(\mathit{num}(\varphi))< y\text{''}\to
\big(\varphi^{<y}
\leftrightarrow \textit{true}^{<y}(\mathit{num}(\varphi))\big)
\]
for every $L^\textup{r}_\textup{ar}$-sentence $\varphi$. But $M\models \text{``}
h(\mathit{num}(\varphi))<a\text{''}$, hence
\begin{eqnarray}\label{eq:Mvarphi}
M \models \varphi^{<a}
\leftrightarrow \textit{true}^{<a}(\mathit{num}(\varphi))
\end{eqnarray}
for every $L^\textup{r}_\textup{ar}$-sentence $\varphi$. As stated in \cite[proof of
Proposition 3]{paris} this contradicts Tarski's undefinability of truth. We
include the details as they are omitted in~\cite{paris}.
The function which for every $L^\textup{r}_\textup{ar}$-formula~$\varphi(x)$ maps
$\mathit{num}(\varphi)$ to $\mathit{num}(\varphi(\underline{\mathit{num}(\varphi)}))$ is computable.
So by Lemma~\ref{lem:Mf}, there is a formula $\textit{sub}(x,y)$ such that
for every formula $\varphi(x)$ and every $b\in M$ with $b<^M a$:
\begin{eqnarray}\label{eq:sub}
b= \mathit{num}(\varphi(\underline{\mathit{num}(\varphi)}))
& \iff &
M\models \textit{sub}^{<a}( \mathit{num}(\varphi), b)
\end{eqnarray}
Define $\chi(x) := \forall y\big(\textit{sub}(x,y) \to \neg
\textit{true}(y))$ and $\theta:=\chi(\underline{\mathit{num}(\chi)})$, and note
\begin{equation}\label{eq:numtheta}
\mathit{num}(\theta)= \mathit{num}(\chi(\underline{\mathit{num}(\chi)})).
\end{equation}
We arrive at the desired contradiction:
\[
\begin{array}{lrll}
& M \models \theta^{<a} \iff
& M\models \forall y{<}a \big(\textit{sub}^{<a}(\mathit{num}(\chi),y)\to \neg \textit{true}^{<a}(y)\big)
& \text{by~\eqref{eq:subst}} \\
& \iff &
M \models \textit{sub}^{<a}(\mathit{num}(\chi),b) \to \neg \textit{true}^{<a}(b)
& \text{for all $b<^M a$} \\
& \iff &
M\models \neg \textit{true}^{<a}(\mathit{num}(\theta))
& \text{by \eqref{eq:sub} and \eqref{eq:numtheta}} \\
& \iff & M\not\models \theta^{<a}
& \text{by~\eqref{eq:Mvarphi}}.
\end{array}
\]
\vspace{-.7cm} \proofend
\subsection{Upper bounds}
Based on our analysis of halting problems in Section~\ref{sec:halt}, we now
see that various \emph{upper bounds} on the complexity of $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$ imply
separations of classical complexity classes from $\mathsf{LINH}$. This is our main
result. The first assertion is Theorem~\ref{thm:truthNP}:
\begin{theo}\label{thm:truthupper}\
\begin{enumerate}\itemsep=0pt
\item[(i)] If $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\in \para\NP$, then $\mathsf{NE}\not\subseteq \mathsf{LINH}$.
\item[(ii)] If $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\in \FPT$, then $\mathsf{E}\not\subseteq \mathsf{LINH}$.
\item[(iii)] If $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\in \para\NL$, then $\NLINSP\not\subseteq \mathsf{LINH}$.
\item[(iv)] If $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\in \para\L$, then $\LINSP\not\subseteq \mathsf{LINH}$.
\end{enumerate}
\end{theo}
\proof Since $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$ is an almost tally problem, (i) follows from
Theorem~\ref{thm:truthAC} and Corollary~\ref{cor:almtallyNE}. The other
assertions follow using Lemma~\ref{lem:almtallyvariants}. \proofend
\section{Provability of the \textup{MRDP}\ theorem}\label{sec:mrdp}
In this section we prove Theorem~\ref{thm:mrdp} as a corollary to
Theorem~\ref{thm:truthNP} via Parikh's theorem~\cite{par71}:
\begin{theo}\label{thm:parikh}
Let $T$ be a $\Pi_1$-theory and $\varphi(\bar x, \bar y)$ a
$\Delta_0$-formula. If $T$ proves $\exists \bar y\; \varphi(\bar x, \bar y)$,
then $T$ proves $\exists \bar y{<}p(\bar x)\; \varphi(\bar x, \bar y) $ for
some term $p(\bar x)$.
\end{theo}
Here, a \emph{theory} is a set of sentences, and a {\em $\Pi_1$-theory} is a
set of $\Pi_1$-sentences. For example, $I\Delta_0$ is (equivalent to) a
$\Pi_1$-theory.
\begin{defn}
A theory \emph{$T$ proves \textup{MRDP}} if for every $\Delta_0$-formula
$\varphi(\bar x)$ there are $L_\textup{ar}$-terms $p(\bar x, \bar y)$ and $q(\bar
x, \bar y)$ such that $T$ proves
\begin{align*}
\varphi(\bar x)
\leftrightarrow \exists \bar y\; p(\bar x, \bar y){=} q(\bar x, \bar y).
\end{align*}
\end{defn}
As mentioned in the introduction it is a long standing open problem whether
$I\Delta_0$ proves \textup{MRDP}\ but it is known that adding exponentiation
suffices. Intuitively, the following concept asks whether \textup{MRDP}\ can be
proved using exponentiation only once.
\begin{defn}\label{def:smallnumber}
A theory \emph{$T$ proves \textup{MRDP}\ for small numbers} if for every $k\in
\mathbb N$ and every $\Delta_0$-formula $\varphi(\bar x)= \varphi(x_0,
\ldots, x_{k-1})$ there are $L_\textup{ar}$-terms $p(\bar x, \bar y)$ and $q(\bar
x, \bar y)$ such that $T$ proves
\begin{eqnarray}\label{eq:smallmrdp}
\textstyle\bigwedge_{i<k} 2^{x_i}{\le}z
& \to & \Big(\varphi(\bar x) \leftrightarrow
\exists \bar y\; p(\bar x, \bar y){=} q(\bar x, \bar y)\Big).
\end{eqnarray}
\end{defn}
\noindent Here, $2^x{\le}y$ stands for a well-known
$\Delta_0$-formula~\cite[Section~V.3.(c)]{hp}. The following strengthens
Theorem~\ref{thm:mrdp}:
\begin{theo}
Let $T$ be a true $\Pi_1$-theory. Moreover, assume that $T$ is computably
enumerable. If $T$ proves \textup{MRDP}\ for small numbers, then
$\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\in\para\NP$
and thus $\mathsf{NE}\not\subseteq \mathsf{LINH}$.
\end{theo}
\proof
Assume $T$ proves \eqref{eq:smallmrdp} for $\varphi(x)$, and hence
\[
2^{x}{\le}z
\wedge \varphi(x) \to
\exists \bar y\; p(x, \bar y){=} q(x, \bar y).
\]
By Theorem~\ref{thm:parikh} $\exists\bar y$ can be replaced by $\exists \bar
y{<}r(x,z)$ for some term $r(x,z)$. But since $T$ proves \eqref{eq:smallmrdp}
for $\varphi(x)$, $T$ proves
\[
2^{x}{\le}z
\to\big( \varphi(x) \leftrightarrow
\exists \bar y{<}r(x,z)\; p(x, \bar y){=} q(x, \bar y)\big).
\]
Since $T$ is computably enumerable, such terms $p,q,r$ can be computed from
$\varphi$. Given an instance $\angle{1^n,\varphi}$ of $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$, compute
$p,q,r$ as above, guess $\bar m<r(n,2^n)$ and check $p(n, \bar m){=} q(n,
\bar m)$. Note the length of the guess $\bar m$ is $O(|r|\cdot|\bar y|\cdot
n)$. The check can be done in time $(|p|\cdot |q|\cdot |r|\cdot n)^{O(1)}$.
Thus, $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\in\para\NP$. Now apply Theorem~\ref{thm:truthNP}. \proofend
It would be interesting to find variants of the this result that infer
$\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\in \FPT$ or $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\in \para\NL$ from certain provabilities of
\textup{MRDP}\ or other arithmetical statements. Note this implies stronger
separations of complexity classes by Theorem~\ref{thm:truthupper}.
\section{\ensuremath{p\text{-}\textsc{Halt}}\ and a universal $\mathsf{AC}^0$-easy set in $\NP$}\label{sec:immu}
In this section we prove Theorem~\ref{thm:immu}. We use the following
technical lemma stating, roughly, that every computable function is dominated
by a computable injection which is $\mathsf{AC}^0$-invertible.
\begin{lem}\label{lem:h}
Let $f: \mathbb N\to \mathbb N$ be computable. Then there is an
increasing $h:\mathbb N\to \mathbb N$ with the following properties.
\begin{enumerate}\itemsep=0pt
\item[(i)] $h(n)\ge f(n^2)$ for every $n\in \mathbb N$.
\item[(ii)] $1^n\mapsto 1^{h(n)}$ is computable in time
$h(n)^{O(1)}$.
\item[(iii)] There is an $L^\textup{r}_\textup{ar}$-sentence $\varphi_h$ such that for
every $x\in\{0,1\}^*$ with $|x|\ge 2$:
\begin{eqnarray*}
\str S(x)\models\varphi_h
& \iff &
x= 1^{h(n)} \text{for some $n\in\mathbb{N}$}.
\end{eqnarray*}
\item[(iv)] There is an $L^\textup{r}_\textup{ar}$-formula $\varphi(x)$ that defines
$n$ in $ \str S(1^{h(n)})$ for every $n\ge 2$.
\end{enumerate}
\end{lem}
\proof Given a deterministic Turing machine $\mathbb M$ and $x\in \{0,1\}^*$
we let $y_{\mathbb M, x}\in\{0,1\}^*$ encode the computation of $\mathbb M$
on $x$. This encoding can be chosen so that:
\begin{enumerate}\itemsep=0pt
\item[(a)] $x\mapsto y_{\mathbb M, x}$ is computable in time $|y_{\mathbb
M, x}|^{O(1)}$.
\item[(b)] $\big\{\angle{x, y_{\mathbb M, x}}\bigmid x\in
\{0,1\}^*\big\}\in \mathsf{AC}^0$.
\end{enumerate}
Now let $\mathbb M_f$ be a Turing machine that computes $1^n\mapsto
1^{f(n)}$. Let $\mathbb M$ be the machine that on input~$1^n$ runs $\mathbb
M_f$ on $1^{i^2}$ for every $i\le n$. Define the \emph{increasing} function
$h:\mathbb N\to \mathbb N$ by
\begin{equation}\label{eq:hdefinition}
h(n)= \mathit{num}\big(\angle{1^n, y_{\mathbb M, 1^n}}\big)
\end{equation}
Clearly, the string $y_{\mathbb M_f, 1^{n^2}}$ encoding the computation of
$\mathbb M_f$ on input $1^{n^2}$ has length at least $f(n^2)$. Similarly,
$|y_{\mathbb M, 1^n}|\ge f(n^2)$. Thus $h(n)\ge f(n^2)$ for every $n\in
\mathbb N$, i.e., (i) holds.
(ii) holds by (a). To show (iii), Theorem~\ref{thm:BIS} and (b) imply that
there is an $L^\textup{r}_\textup{ar}$-sentence $\varphi$ that holds precisely in the
string structures of the form $\str S\big(\mathit{bin}(h(n))\big)$ for $n\in\mathbb{N}$.
Using $\mathit{BIT}$, there is an interpretation $\inter I$ such that $\str
S(1^m)^{\inter I}\cong\str S(\mathit{bin}(m))$ for every $m> 2$, so
$\varphi_h:=\varphi^{\inter I}$ holds precisely in the string structures of
the form $\str S(1^{h(n)})$ for $n\in\mathbb{N}$ (we have $h(n)>2$ for all $n\in\mathbb{N}$).
Trivially, $n$ is definable in $\str S(\mathit{bin}(h(n)))$, so (iv) follows using
the interpretation $I$ above. \proofend
Theorem~\ref{thm:immu} is an easy consequence of the following stronger
result, and we view it as good evidence for the truth of
Conjecture~\ref{conj:phaltAC}.
\begin{theo}\label{thm:universal}
Assume $\ensuremath{p\text{-}\textsc{Halt}}\in \mathsf{para}\text{-}\mathsf{AC}^0$. Then there is an infinite tally problem $X$ such
that for every $Q\in\NP$ we have $Q\cap X\in \mathsf{AC}^0$.
\end{theo}
\noindent \textit{Proof of Theorem~\ref{thm:immu} from
Theorem~\ref{thm:universal}:} Assume $\ensuremath{p\text{-}\textsc{Halt}}\in \mathsf{para}\text{-}\mathsf{AC}^0$ and let $Q\in \NP$.
Let $X$ be as stated in Theorem~\ref{thm:universal}. Then either $Q\cap X$ or
$(\{0,1\}^*\setminus Q)\cap X$ is infinite. By Theorem~\ref{thm:universal}
they are both in $\mathsf{AC}^0$. Hence $Q$ is not $\mathsf{AC}^0$-bi-immune. \proofend
\noindent \textit{Proof of Theorem~\ref{thm:universal}:}
By~Corollary~\ref{cor:evtfo} there is a computable increasing
function $f:\mathbb N\to \mathbb N$ and an
$L^\textup{r}_\textup{ar}$-sentence $\varphi$ such that for every $\angle{1^n, \mathbb
M}$ with $n \ge f(|\mathbb M|)$:
\begin{eqnarray} \label{eq:phaltvarphi}
\str S\big(\angle{1^n, \mathbb M}\big)\models \varphi
& \iff &\text{$\mathbb M$ accepts the empty input tape in at most $n$ steps}.
\end{eqnarray}
Now let $h:\mathbb N\to \mathbb N$ be the increasing function as stated in
Lemma~\ref{lem:h}. In particular, there is a deterministic Turing machine
$\mathbb M_h$ and a constant $c\ge 1$ such that on input $1^m$ the machine
$\mathbb M_h$ outputs the string $1^{h(m)}$ in time $h(m)^c$. The desired
set $I$ is defined by
\[
X:= \big\{1^{h(m)} \mid m\ge 2\big\}.
\]
By Lemma~\ref{lem:h}~(iii) the sentence $\varphi_h$ witnesses $X\in\mathsf{AC}^0$
according to Theorem~\ref{thm:BIS}.
Now let $Q\subseteq \{0,1\}^*$ be a problem in $\NP$. In particular, there is
a nondeterministic Turing machine $\mathbb M_Q$ and a constant $d\ge 1$ such
that $\mathbb M_Q$ accepts $x$ in time $|x|^d$.
Define the nondeterministic Turing machine $\mathbb M_{Q,h,m}$ to run
$\mathbb M_h$ on $1^m$ to produce output $1^{h(m)}$ and then run $\mathbb
M_Q$ on $1^{h(m)}$. This machine runs in time
\[
n(m):= h(m)^c + h(m)^{d}.
\]
Choose a constant $e\in\mathbb{N}$ such that $m\ge |\mathbb M_h|+ |\mathbb
M_Q|+e$ implies $m^2\ge |\mathbb M_{Q,h,m}|$. Then
\begin{align*}
n(m) \ge h(m)\ge f(m^2)
\ge f(|\mathbb M_{Q,h,m}|).
\end{align*}
Hence, by \eqref{eq:phaltvarphi}, for $m\ge |\mathbb M_h|+ |\mathbb
M_Q|+e$:
\begin{eqnarray}\notag
1^{h(m)}\in Q & \iff &
\text{$\mathbb M_{Q,h,m}$ accepts the empty input in at most $n(m)$ steps} \\\label{eq:Mm}
& \iff &
\str S(\angle{1^{n(m)},\mathbb M_{Q,h,m}})\models\varphi.
\end{eqnarray}
Lemma~\ref{lem:h}~(iv) implies that there is an interpretation $\inter I$
such that for every $m\in \mathbb N$
\[
\str S(1^{h(m)})^{\inter I}
= \str S\big(\angle{1^{n(m)}, \mathbb M_{Q,h,m}}\big).
\]
By Theorem~\ref{thm:BIS} it suffices to show that for every $x\in \{0,1\}^*$
with $|x|\ge h(|\mathbb M_h|+ |\mathbb M_Q|+ e)$:
\begin{eqnarray*}
x\in Q\cap X
& \iff &
\str S(x)\models \varphi_h\wedge \varphi^{\inter I}.
\end{eqnarray*}
Assume $x\in Q\cap X$. Then $x=1^{h(m)}$ for some $m\ge 2$ and $\str
S(x)\models\varphi_h$. Since $|x|=h(m)\ge h(|\mathbb M_h|+ |\mathbb M_Q|+e)$
and $h$ is increasing, we have $m\ge |\mathbb M_h|+ |\mathbb M_Q|+e$. Thus
$x=1^{h(m)}\in Q$ implies $\str S(\angle{1^{n(m)}, \mathbb M_{Q,h,m}})\models
\varphi$ by~\eqref{eq:Mm}, and $\str S(1^{h(m)})\models \varphi^{\inter I}$
follows.
Conversely, assume $S(x)\models\varphi_h\wedge\varphi^{\inter I}$. By $\str
S(x)\models\varphi_h$, we have $x\in X$, so $x=1^{h(m)}$ for some $m\ge 2$.
By $\str S(1^{h(m)})\models\varphi^{\inter I}$ we have $\str
S(\angle{1^{n(m)}, \mathbb M_{Q,h,m}})\models\varphi$. This implies
$x=1^{h(m)}\in Q$ by~\eqref{eq:Mm} because, as above, $m\ge |\mathbb M_h|+
|\mathbb M_Q|+e$. \proofend
\section{Problem comparison}\label{sec:concl}
\subsection{The role of uniformity}
Our proof of the lower bound $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\notin \mathsf{para}\text{-}\mathsf{AC}^0$
(Theorem~\ref{thm:truthAC}) makes crucial use of the uniformity condition in
the definition of $\mathsf{para}\text{-}\mathsf{AC}^0$. To shed some light on this dependence, we relax
the uniformity condition as follows.
\begin{defn}\label{df:xac} Let $(Q,\kappa)$ be a parameterized problem and $d,k\in\mathbb{N}$. The {\em $k$-th slice of $(Q,\kappa)$} is the classical problem $\{x\in Q\mid\kappa(x)=k\}$.
The class $\XAC{}$ contains $(Q,\kappa)$ if and only if $\mathsf{AC}^0$ contains every slice of $(Q,\kappa)$.
The class $\XAC{d}$ contains $(Q,\kappa)$ if and only if $\CAC d$ contains every slice of $(Q,\kappa)$; here,
$\CAC d$ denotes the class of problems decided by
dlogtime uniform circuit families of polynomial size and depth $d$.
\end{defn}
Clearly,
\begin{equation}\label{eq:xac}
\textstyle
\mathsf{para}\text{-}\mathsf{AC}^0\subseteq \bigcup_{d\in\mathbb{N}}\XAC d\subseteq\XAC{}
\end{equation}
and $\XAC 1\not\subseteq \mathsf{para}\text{-}\mathsf{AC}^0$ as witnessed, e.g., by an undecidable
problem with parameterization~$\mathit{num}$.
The class $\XAC{}$ is important in our context because it is a natural upper bound on $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$:
\begin{prop}\label{prop:truthxac} $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\in\XAC{}$.
\end{prop}
\proof It suffices to show that for every $\Delta_0$-formula $\varphi(x)$
the problem $\{1^n\mid\mathbb{N}\models\varphi(n)\}$
belongs to~$\mathsf{AC}^0$. But this problem is $\mathit{un}(Q)$ for $Q:=\{x\in\{0,1\}^*\mid \mathbb{N}\models\varphi(\mathit{num}(x))\}$. Clearly $Q\in\mathsf{LINH}$, so
$\mathit{un}(Q)\in\mathsf{AC}^0$ follows from Proposition~\ref{prop:ag}.
\proofend
We show that it is likely difficult to improve
Theorem~\ref{thm:truthAC}
to $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}
\not\in\bigcup_{d\in\mathbb{N}}\XAC d$. This somewhat artificial class also exhibits
the different behaviors of the parameterized problems $\ensuremath{p\text{-}\textsc{Halt}}$,
$\ensuremath{p\text{-}\textsc{Halt}}_=$, and $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$.
\begin{theo}\label{thm:xac}\
\begin{enumerate}\itemsep=0pt
\item[(i)] $\ensuremath{p\text{-}\textsc{Halt}}\in \XAC 2$.
\item[(ii)] $\ensuremath{p\text{-}\textsc{Halt}}_=\in \XAC d$ for some $d\in\mathbb{N}$ if and only if
$\mathsf{NE}\subseteq \mathsf{LINH}$.
\item[(iii)] $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\in \XAC d$ for some $d\in\mathbb{N}$ if and only if $\mathsf{LINH}$
collapses.
\end{enumerate}
\end{theo}
\proof (i) has been observed in the introduction and (ii) follows from
Remark~\ref{rem:xac0} and \eqref{eq:xac}. To see (iii), assume $\mathsf{LINH}$ collapses. Paris and
Dimitracopolous~\cite[Proof of Proposition 4]{paris} showed that this implies
the following. There is an $L^\textup{r}_\textup{ar}$-formula $\lambda(x,y)$ such that
for every $\Delta_0$-formula $\varphi(x)$ there are
$c_\varphi,d_\varphi,e_\varphi\in \mathbb{N}$ such that for all $n\ge c_\varphi$
\begin{eqnarray*}
\mathbb{N}\models \varphi(n)
& \iff & n^{d_\varphi}\models\lambda(n,e_\varphi)
\end{eqnarray*}
For each fixed $\varphi$ there is an $\mathsf{AC}^0$-family that given $1^n$ decides
whether $n$ satisfies the the r.h.s.. The size of this family is bounded
$n^{f_\varphi}$ for some $f_\varphi\in\mathbb{N}$ depending on $\varphi$, but the
depth of this family is determined by the quantifier alternation rank of
$\lambda$ and, in particular, does not depend on $\varphi$. This implies
$\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\in \XAC d$ for some $d\in\mathbb{N}$.
Conversely, assume $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\in \XAC d$ and let $Q\in\mathsf{LINH}$. It is well
known (see e.g.\ \cite[Ch.V, Lemma 2.13]{hp}) that
there is a $\Delta_0$-formula that is satisfied by $\mathit{num}(x)$ if and only if
$x\in Q$. Fixing this formula in the input to $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$, the assumption
implies that there is a dlogtime uniform circuit family $(C_{n})_n$ of
polynomial size and depth $d$ such that for all $x\in\{0,1\}^*$:
\begin{eqnarray*}
x\in Q & \iff & C_{\mathit{num}(x)}(1^{\mathit{num}(x)})=1.
\end{eqnarray*}
It suffices to show that, given $x$, the r.h.s. can be checked by an
alternating machine in linear time with $d$ alternations. This is
straightforward by guessing a path through $C_{\mathit{num}(x)}$. E.g., if the output
gate is a $\vee$-gate, the machine existentially guesses an input gate $g_1$
to it, and if it is a $\wedge$-gate it universally guesses $g_1$. Depending
on the type of $g_1$ it either existentially or universally guesses an input
gate $g_2$ to $g_1$, and so on. When reaching (with $g_{d-1}$ or earlier) an
input gate or a negation thereof, the machine checks it is satisfied by the
corresponding bit of $x$. Each guess requires~$O(|x|)$ bits. Checking that
e.g.~$g_2$ is an input to $g_1$ can be done in time logarithmic in the size
of~$C_{\mathit{num}(x)}$, that is, in time~$O(|x|)$. We omit further
details.\proofend
\subsection{Reducibilities}
In this section we draw some corollaries concerning how our problems
$\ensuremath{p\text{-}\textsc{Halt}},\ensuremath{p\text{-}\textsc{Halt}}_=$ and $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$ compare with respect to our notion of
reducibility. Saying that a (parameterized) problem is {\em reducible} to
another means that there is an eventually definable reduction. Two problems
are {\em equivalent} if they are reducible to one another.
The picture is as follows: an arrow indicates reducibility, $\equiv$ means
equivalence.
\[
\begin{array}{ccc}
&\ensuremath{p\text{-}\textsc{Spec}}&\\
\hspace*{7ex}\nearrow&&\hspace{-11ex}\nwarrow\\
\ensuremath{p\text{-}\textsc{Halt}}_=&\not\equiv&\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\\
\uparrow&&\\
\ensuremath{p\text{-}\textsc{Halt}}&&
\end{array}\]
In particular, we show unconditionally that $\ensuremath{p\text{-}\textsc{Halt}}_=$ and $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$ are not
equivalent and both are reducible to yet another almost tally problem of
central importance to mathematical logic, namely the following parameterized
version of the spectrum problem:
\npprob{$\ensuremath{p\text{-}\textsc{Spec}}$}{$n\in \mathbb N$ in unary and a first-order sentence $\varphi$}
{$|\varphi|$}{Does $\varphi$ have a model of size $n$?}
Recall that having a model of size $n$ means that $n$ belongs to the spectrum
of $\varphi$.
\medskip
We start comparing $\ensuremath{p\text{-}\textsc{Halt}}$ and $\ensuremath{p\text{-}\textsc{Halt}}_=$. Clearly, $\ensuremath{p\text{-}\textsc{Halt}}$ is reducible to
$\ensuremath{p\text{-}\textsc{Halt}}_=$. Concerning the converse we note that Theorem~\ref{thm:xac}~(i),
(ii) implies:
\begin{cor}
If $\ensuremath{p\text{-}\textsc{Halt}}_=$ is reducible to $\ensuremath{p\text{-}\textsc{Halt}}$, then $\mathsf{NE}\subseteq\mathsf{LINH}$.
\end{cor}
Adapting a mode of speech from~\cite{cfslice}, call an almost tally problem
$(Q,\kappa)$ {\em slicewise monotone} if $(1^n,x)\in Q$ implies $(1^m,x)\in
Q$ for all $x\in\{0,1\}^*$ and all $n,m\in\mathbb{N}$ with $n<m$.
One can show that $\ensuremath{p\text{-}\textsc{Halt}}$ is the hardest such problem in $\para\NP$. This is
an easy modification of the proof Lemma~\ref{lem:phaltalmtally} and
strengthens \cite[Proposition~11]{cfslice}:
\begin{cor} Every almost tally problem in $\para\NP$ that is slicewise monotone is reducible to~$\ensuremath{p\text{-}\textsc{Halt}}$.
\end{cor}
We turn to $\ensuremath{p\text{-}\textsc{Halt}}_=$ and $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$.
\begin{cor} \
\begin{enumerate}\itemsep=0pt
\item[(i)] If $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$ is reducible to $\ensuremath{p\text{-}\textsc{Halt}}_=$, then $\mathsf{NE}\not\subseteq\mathsf{LINH}$.
\item[(ii)] If $\ensuremath{p\text{-}\textsc{Halt}}_=$ is reducible to $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$, then $\mathsf{NE}\subseteq\mathsf{LINH}$.
\item[(iii)] $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$ and $\ensuremath{p\text{-}\textsc{Halt}}_=$ are not equivalent.
\end{enumerate}
\end{cor}
\proof (iii) follows from (i) and (ii).
For (i), assume $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$ is reducible to $\ensuremath{p\text{-}\textsc{Halt}}_=$. Then $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}\in\para\NP$ and
$\mathsf{NE}\not\subseteq\mathsf{LINH}$ follows by Theorem~\ref{thm:truthNP}.
For (ii), assume $\ensuremath{p\text{-}\textsc{Halt}}_=$ is reducible to $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$. Then $\ensuremath{p\text{-}\textsc{Halt}}_=\in\XAC{}$ by Proposition~\ref{prop:truthxac} and hence $\mathsf{NE}\subseteq\mathsf{LINH}$ by Remark~\ref{rem:xac0}.
\proofend
Finally, we turn to $\ensuremath{p\text{-}\textsc{Spec}}$:
\begin{prop} Both $\ensuremath{p\text{-}\textsc{Halt}}$ and $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$ are reducible to $\ensuremath{p\text{-}\textsc{Spec}}$.
\end{prop}
\proof It is straightforward to compute from a nondeterministic Turing machine $\mathbb M$ a first-order sentence $\varphi_{\mathbb M}$ that has a model of size $n$ if and only if $\mathbb M$ accepts the empty input in exactly $n$ steps.
Concerning $\ensuremath{p\text{-}\Delta_0\text{-}\textsc{Truth}}$, by Lemma~\ref{lem:fcttorel}, it suffices to show that $\pMC(L^\rel_\arit)$ is
reducible to $\ensuremath{p\text{-}\textsc{Spec}}$: map an instance $(1^n,\varphi)$ of $\pMC(L^\rel_\arit)$ to $(1^n,\varphi\wedge\psi)$ where $\psi$ is
an $L^\textup{r}_\textup{ar}$-sentence whose finite models are exactly those isomorphic to some standard finite $L^\textup{r}_\textup{ar}$-structure.
\proofend
Observe $\ensuremath{p\text{-}\textsc{Spec}}$ can be solved in nondeterministic time $n^{f(k)}$ for some computable $f:\mathbb{N}\to\mathbb{N}$ where $k:=|\varphi|$ is the parameter. Can the parameter be moved out of the exponent? We find in worthwhile to explicitly point out the following direct corollary of the previous proposition and Theorem~\ref{thm:truthNP}:
\begin{cor} If $\ensuremath{p\text{-}\textsc{Spec}}\in\para\NP$, then $\mathsf{NE}\not\subseteq\mathsf{LINH}$.
\end{cor}
\bibliographystyle{plain}
|
1,108,101,563,238 | arxiv | \section{Introduction}
Understanding the dynamics of mobile agents is important for robots to operate autonomously in environments with pedestrians, vehicles, animals, or other moving entities.
Autonomous driving, service robot navigation through crowds, and similar applications all rely on trajectory prediction to help avoid dynamic obstacles.
Predicting trajectories is also an important form of \emph{intention inference}, where the intention to move in a certain direction or to a given location is inferred~\cite{best_bayesian_2015,Rhinehart_2019_ICCV,eiffert_probabilistic_2020,kavindie_2021}. Examples involving pedestrian trajectories include goal-directed inference~\cite{best_bayesian_2015,Rhinehart_2019_ICCV}, modelling social interactions~\cite{eiffert_probabilistic_2020}, and industrial applications with cobots~\cite{dinh_cobot_2015,cobot_overview}.
We are interested in leveraging environmental context to predict the abstract motion of an agent.
Obstacles in the environment, for example, place constraints on the trajectories we wish to predict.
The environment, therefore, is a rich source of information for trajectory prediction that can be seen to split the set of all possible trajectories into equivalence classes. These equivalence classes summarise the agent's high-level motion in a sparse identifier, sometimes referred to as an \emph{$h$-signature}, that specifies properties such as whether the trajectory passes above or below a certain obstacle, and how many times it does so.
Having this identifier is powerful, as it gives a general idea of the motion in a sparse representation that can be used as the basis for developing efficient computational frameworks.
For example, $h$-signatures have been used in various path planning contexts~\cite{bhattacharya_search-based_2010, bhattacharya_persistent_2015, bhattacharya_topological_2015,mccammon_topological_2021}.
In trajectory prediction, topological concepts like $h$-signatures allow the extraction of salient, predictive features of trajectories that arise from obstacles or other features of the environment.
Existing trajectory prediction methods, however, typically are based on a geometric representation~(e.g., \cite{zhi_kernel_2020,kavindie_2021}) and can be computationally prohibitive for long-term prediction.
These methods operate on full trajectories and consider all differences between them, including minor differences.
The number of small and insignificant variations, even between topologically equivalent trajectories, can easily become computationally overwhelming.
These geometric algorithms thus trade-off prediction horizon with computational efficiency~\cite{kavindie_2021,Rhinehart_2019_ICCV,eiffert_path_2020,zhi_kernel_2020}.
In this paper, we propose a new topological approach to trajectory prediction by leveraging the salient features of paths summarised as by $h$-signatures. $h$-signatures are both compact in size and provide global information about a trajectory as they only encode how a trajectory moves past an obstacle as opposed to an overall trajectory. The use of $h$-signatures in trajectory prediction can therefore circumvent the aforementioned trade-off that other methods face.
We thus present our method for topology-informed trajectory prediction, which introduces the notion of \emph{partial $h$-signatures}, defined for incomplete trajectories. The concept of partial $h$-signatures decomposes trajectory prediction into high-level and low-level prediction problems.
As initial solutions to these problems, we present a high-level prediction algorithm using VOMP~\cite{ron_power_1993} and a low-level prediction algorithm based on a hierarchical GMM. We demonstrate the behaviour of our method with a simple synthetic dataset, and then extensively evaluate its performance compared to a baseline GMM method without topological knowledge using the ATC shopping mall dataset~\cite{atc_dataset}.
Results show that our method performs as well or better in terms of geometric error, and considerably better in identifying the correct high-level motion.
This indicates that $h$-signatures are a meaningful high-level representation for predicting agent trajectories, thus useful for applications such as target search~\cite{wakulicz_active_2021}.
\begin{figure}[t!]
\centering
\includegraphics[width=0.99\columnwidth]{figs/prob_form.png}
\caption{Illustration of the problem considered in this paper. The task is to predict high-level motions then use them to inform lower level trajectory predictions.}
\vspace{-3.5ex}
\label{fig:prob_form}
\end{figure}
\section{Related Work}
Trajectory prediction is a well-studied problem in robotics. Model-based approaches use an estimated or learned dynamic model of an agent's motion to predict an agent's state forward in time~\cite{zhi_spatiotemporal_2019,wakulicz_active_2021}. Model-free learning based methods can be used where dynamics are cumbersome to model. For example, by treating trajectory prediction as a sequence generation problem, recurring neural networks (RNNs) and long-short term memory networks (LSTMs) have been implemented in various contexts with large success~\cite{altche_lstm_2017, shi_lstm-based_2018, eiffert_path_2020, bi_large-scale_2022}. Other approaches have introduced mixture models such as Gaussian mixture models (GMMs) and the kernel trajectory map~\cite{zhi_kernel_2020} to capture the multi-modal nature of an agent's possible future paths, showing greater prediction accuracy over single-mode approaches~\cite{wiest_probabilistic_2012, Ivanovic_2019_ICCV, zyner_naturalistic_2020}.
Beyond removing the burden of acquiring a dynamic model, the power of model-free learning methods lie in their ability to incorporate contextual information into the model. For example, social interactions between agents have been captured in various architectures to improve pedestrian trajectory prediction in crowded environments~\cite{eiffert_probabilistic_2020, vemula_social_2018, alahi_social_2016, Ivanovic_2019_ICCV}. Where crowds are extremely dense and interactions are complex, bulk properties of crowds have been learned instead~\cite{kiss_probabilistic_2021,kiss_constrained_2022}. Even the intent of an agent may be learned, and has been proposed as a powerful contextual cue for long-term trajectory prediction problems~\cite{best_bayesian_2015,yao_bitrap_2021,eiffert_probabilistic_2020,kavindie_2021}.
Contextual information common and relevant to all trajectory prediction scenarios is the environment within which the agent is travelling. Obstacles, unreachable regions and road structures -- the \textit{topological features} of an environment -- all dictate how an agent can move through an environment. It seems then that topological information should be a fruitful contextual cue for trajectory prediction. Indeed, topology-aware learning methods have been introduced for trajectory prediction, outperforming a GMM-based approach that does not take advantage of this contextual cue~\cite{pokorny_topological_2016,frederico_carvalho_long-term_2019}.
\section{Background and problem formulation}
\subsection{Homotopy Theory in Robotics}\label{sec:homotopy_theory}
Two paths $\tau_1$, $\tau_2$ in a topological space $\mathcal{D}$ with common start and ending points are \textit{homotopic} if there exists a continuous transformation or deformation from one to the other~\cite{hatcher_algebraic_2002}. Sets of paths homotopic to one another are named \textit{homotopy classes}. Non-trivial homotopy classes arise as a result of obstacles in the space, as deforming some paths into others would require moving through an obstacle and breaking the continuity requirement. This notion is depicted in Fig.~\ref{fig:homotopy}. Paths $\tau_1$ and $\tau_2$ are homotopic as they can be continuously deformed into each other, indicated by the dotted paths between them. These paths are however not homotopic to $\tau_3$, as any deformation into $\tau_3$ would require moving through the obstacle $\mathcal{O}$.
In the context of robotics, obstacles in the environment split the space of an agent's possible trajectories between points from a single homotopy class into a countable number of homotopy classes. The obstacles therefore dictate the number of unique ways an agent may travel through the space from one point to another. For example, in Fig.~\ref{fig:homotopy} the obstacle $\mathcal{O}$ splits the space of trajectories into those that move `above' the obstacle, those that move `below', and those that wind around the obstacle any number of times before moving to the end point. Such abstraction of the high-level motions available to an agent is a powerful tool for navigation, prediction and tracking tasks often encountered in robotics~\cite{bhattacharya_search-based_2010, bhattacharya_persistent_2015, pokorny_topological_2016, frederico_carvalho_long-term_2019, mccammon_topological_2021}.
\subsection{$h$-signatures as Homotopy Invariants}\label{sec:background:invariants}
To identify which homotopy class a trajectory $\tau$ belongs to, one must compute a \textit{homotopy invariant} -- a unique identifier $h(\tau)$ of a trajectory's homotopy class such that $h(\tau_1) = h(\tau_2)$ if and only if $\tau_1$ and $\tau_2$ are homotopic. There are many ways to construct a homotopy invariant. The most simple and commonly used is detailed in~\cite{hatcher_algebraic_2002} and coined the \textit{$h$-signature} of a trajectory in~\cite{bhattacharya_topological_2015}. Non-intersecting rays are drawn from within each obstacle to the boundary of the environment. These rays are commonly taken to be parallel and emanate upwards from the centres of obstacles. Then to compute the $h$-signature of path $\tau$, a `word' is constructed by following the path and appending letter `$n$' if $\tau$ crosses the ray corresponding to the $n$-th obstacle from left to right, and the letter `$-n$' if $\tau$ crosses from right to left. The final word may then be \textit{reduced} by cancelling all consecutive appearances of $n$ and $-n$. For example, the $h$-signature $(1,2,-2,3)$ reduces to $(1,3)$.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\columnwidth]{figs/homtopy.png}
\caption{Illustrative example of paths of equal ($\tau_1$, $\tau_2$) and differing ($\tau_3$) homotopy classes.\label{fig:homotopy}}
\vspace{-1.5em}
\end{figure}
\subsection{Problem Formulation}
Consider an agent traversing through a planar environment $\mathcal{D}$ containing $n$ obstacles $O = \{\mathcal{O}_1,\ldots,\mathcal{O}_{n}\}$.
The agents trajectory is denoted $\mathbf{X} = \{ \mathbf{x}_{1}, \ldots \mathbf{x}_{T} \}$.
We assume that the agent begins at a start location on the boundary $\delta\mathcal{D}$ of the environment and travels to an end location on $\delta\mathcal{D}$. We are given a dataset of $K$ historical, fully observed trajectories $\mathcal{X} = \{X_{\text{obs}}^1,\ldots,X_{\text{obs}}^K\}$, with full knowledge of obstacles.
Further, in the online setting, we have a partial noisy measurement of trajectory $\mathbf{Y}_{\text{obs}} = \{\mathbf{y}_1,\ldots,\mathbf{y}_{T_{\mathrm{obs}}}\}$ up to time $T_{\mathrm{obs}}$, with a known sensor model $P(\mathbf{y}_{t} \mid \mathbf{x}_{t})$.
We are then interested in predicting the agent's high-level motion through the environment.
An agent's `high-level motion' can be abstracted from its trajectory in many different ways. For example, one might learn trajectory clusters with a GMM and treat each cluster as a unique high-level motion.
We instead propose that high-level motion is best abstracted by the trajectory's homotopy class and thus that prediction of a trajectory's $h$-signature is the most appropriate solution. Formally, the problem is:
\begin{problem} (High-level prediction)\label{prob:high-level} Given the partial, noisy measurements of the trajectory $\mathbf{Y}_{\mathrm{obs}}$, predict the $h$-signature $\hat{h}$ of the robot's future full trajectory.
\end{problem}
While the solution to Problem~\ref{prob:high-level} can be used as a tool in many robotics problems, here we are interested in how the $h$-signature may be used to produce a topology-informed low-level prediction for a trajectory:
\begin{problem} (Low-level prediction)\label{prob:low-level} Given the predicted $h$-signature $\hat{h}$ associated with an agent's partially observed measurements $\mathbf{Y}_{\mathrm{obs}}$, predict the full trajectory $\mathbf{X}$.
\end{problem}
\section{Topology-informed trajectory prediction}
\subsection{Overview}\label{sec:overview}
Problems~\ref{prob:high-level} and~\ref{prob:low-level} are challenging to solve directly without modification, as illustrated in the factor graphs shown in Fig.~\ref{fig:pgm}.
In a pedantic Bayesian formulation~(Fig.~\ref{fig:pgm}~\subref{fig:pgm:naive}), one would first predict the underlying trajectory for the entire duration, and subsequently predict the corresponding $h$-signature.
This is because the measurements are conditionally independent of the $h$-signature given the trajectory, and the only given relationship between the trajectory and the $h$-signature is the computation process outlined in Sec.~\ref{sec:background:invariants}.
In other words, in this view, low-level trajectory prediction precedes high-level prediction, limiting its effectiveness.
Instead, we propose to circumvent the low-level prediction by introducing the notion of partial $h$-signature (green lines, Fig.~\ref{fig:pgm}~\subref{fig:pgm:ours}).
Unlike the usual $h$-signature, partial $h$-signatures can be obtained from an incomplete trajectory, as we detail in Sec.~\ref{sec:partial}.
Given the partial $h$-signature, we predict the full $h$-signature.
This is achieved with a variable-order Markov process (VOMP) model (red line, Fig.~\ref{fig:pgm}~\subref{fig:pgm:ours}) trained on a dataset (Sec.~\ref{sec:vomp}).
The full $h$-signature can then be used to predict the low-level trajectory, with a model learnt using the historical dataset.
As we readily have access to trajectories and their associated $h$-signature, this can be as simple as learning a mixture of experts for low-level trajectory prediction within each homotopy class.
To this end, we demonstrate the use of a hierarchical Gaussian mixture model (GMM) in Sec.~\ref{sec:gmm}.
\subsection{Partial $h$-signatures}\label{sec:partial}
As described in Sec.~\ref{sec:homotopy_theory}, homotopy classes exist only for paths between two fixed start and end points. In the context of this paper trajectories are assumed to start and end on different boundary points of the environment. Similar to~\cite{mccammon_topological_2021}, we ensure our description of homotopy classes is valid by applying a quotient map, mapping all boundary points to a single quotient point while preserving the topology of the space. Then, the set of all homotopy classes here is over paths between the quotient point and itself.
With this in mind, the notion of a \textit{partial} $h$-signature is ill-defined in a topological sense. Much like a full $h$-signature, it is calculated by constructing a `word' according to a path's ray crossings. However, it is extracted from an incomplete trajectory, i.e. one that has not yet returned to the quotient point. It is thus crucial to note that a partial $h$-signature is not an identifier of a homotopy class but rather a \textit{predictor}.
To predict full $h$-signatures from partial ones, a notion of compatibility between the two is needed. For a given partial $h$-signature $p$, compatible full $h$-signatures are those whose prefix is $p$. In other words, the set of all $h$-signatures compatible with $p$ is $\mathcal{H}(p) = \{h \mid \exists p', h = pp^{\prime} \}$.
\subsection{High-level Prediction using Topological Variable Order Markov Processes}\label{sec:vomp}
\begin{figure}[t!]
\centering
\subfloat[Pedantic model.]{\includegraphics[width=0.73\columnwidth]{figs/factor_graph_original.pdf}\label{fig:pgm:naive}} \\
\subfloat[Proposed model.]{\includegraphics[width=0.73\columnwidth]{figs/factor_graph_ours.pdf}\label{fig:pgm:ours}}
\caption{Probabilistic graphical models depicting the problem. Directed arrows from $A$ to $B$ imply availability of a model of $B$ given $A$. Dashed arrows are ignored during inference.}
\vspace{-1.5em}
\label{fig:pgm}
\end{figure}
As $h$-signatures in an environment with $n$ obstacles are simply `words' constructed from an alphabet $\mathcal{A} = \{1,..,n,-1,...,-n\}$, prediction of a full $h$-signature given partial $h$-signature can be viewed as a sequence completion problem. The sparse nature of the $h$-signature as a representation of high-level motion allows for relatively simple techniques to be used for sequence generation. We propose a VOMP~\cite{ron_power_1993} for this purpose. Like LSTMs, VOMPs can learn dependencies in data that are of varying length. However, VOMPs are capable of producing probabilistic predictions where LSTMs are not. This is crucial for our approach, as a probability distribution over full $h$-signatures is required to produce probabilistic low-level predictions (blue lines, Fig.~\ref{fig:pgm}~\subref{fig:pgm:ours}).
We represent a VOMP with a probabilistic suffix automaton (PSA) as in~\cite{ron_power_1993}.
Here, a PSA state is a partial $h$-signature $p$ of up to some maximum length $L>0$ constructed from alphabet $\mathcal{A}$ as $p = a_{1}\ldots a_{l}$, $0 \leq l \leq L$. Transitions between two states $p$ and $p^{\prime}$ are allowed only if there exists some $a \in \mathcal{A}$ such that $p^{\prime}$ is a \textit{suffix} of $ap$. Allowed transitions have associated with them a probability that the transition will occur. Transition probabilities learned offline can be used to produce probability distributions over future states online. An example PSA is drawn in Fig.~\ref{fig:vmmp}~\subref{fig:vmmp:automata}.
We follow~\cite{ron_power_1993} closely to train a VOMP over $h$-signatures offline, and adapt their online prediction process to better suit the prediction of $h$-signatures. Our algorithms for online prediction and offline learning are detailed below.
\subsubsection{Online Prediction}
The trained VOMP outputs probabilities of arbitrarily long $h$-signatures. To find the probability of any $h$-signature $P(h)$ one simply takes a corresponding walk through the PSA, multiplying transition probabilities from state to state. However, we are interested in finding the conditional distribution $P(h \mid p)$ over possible full $h$-signatures $h$ given a partial $h$-signature $p$ extracted from a partially observed trajectory. Letting $\mathcal{H}(p)$ be the set of all full $h$-signatures compatible with $p$, the conditional probability $P(h \mid p)$ of any $h \in \mathcal{H}(p)$ can be calculated in typical fashion,
\begin{equation}\label{eq:naive_conditioning}
P(h \mid p) = \frac{P(h)}{\sum_{h \in \mathcal{H}(p)} P(h) }.
\end{equation}
However, the set $\mathcal{H}(p)$ of compatible $h$-signatures is of infinite size. Just as agents may walk paths of varying lengths through an environment, the lengths of compatible full $h$-signatures varies. For example, if an agent is observed passing from left to right above the first obstacle in an environment their partial $h$-signature is $(1)$. Later, the agent may or may not pass above any other obstacle. Then, their full $h$-signature may be $(1)$ or $(1,\ldots)$.
To handle these nuances, we assume that the longest possible $h$-signature an agent will take through the environment is the maximum length $h$-signature present in training data. That is, for any $h \in \mathcal{H}(p)$, $P(h) = 0$ if $|h| > \max(|p|)$. Then, we weigh $h$-signature probabilities in Eqn.~\ref{eq:naive_conditioning} by the probability of observing an $h$-signature of that length in data. Thus, the VOMP is queried in order to make the adjusted calculation
\begin{equation}
P(h \mid p) = \frac{P(h) \cdot P(|h|)}{\sum_{h \in \mathcal{H}(p)} P(h) \cdot P(|h|) }.
\end{equation}
\subsubsection{Offline Learning}
\begin{algorithm}[t!]
\caption{PSA offline learning} \label{alg:offline}
\textbf{Inputs:} $\epsilon$, $L$, $\mathcal{A}$, $h$-signature data \\
\textbf{Output:} Trained PSA
\begin{algorithmic}[1]
\State initialise tree $T$
\State $\mathcal{P} \gets \{a \mid a \in \mathcal{A}, P(a) \geq \epsilon\}$
\While{$\mathcal{P}$ not empty}:
\State $p \gets \mathcal{P}$.pop()
\If{$\mathcal{E}(p, \mathrm{suffix}(p)) \geq \epsilon$}:
\State add path to $p$ to $T$
\EndIf
\If{$|p| \leq L$}:
\State $\mathcal{P} \gets \mathcal{P}\cup\{ap \mid a \in \mathcal{A}, P(ap) \geq \epsilon \}$
\EndIf
\EndWhile
\For{all leaves $r$ in $T$}:
\If{longest $\mathrm{prefix}(r)$ not in $T$}:
\State add path to $r$ to $T$
\EndIf
\EndFor
\State PSA $\gets$ leaves of $T$
\end{algorithmic}
\end{algorithm}
To facilitate online prediction of full $h$-signatures given partial $h$-signatures, a prediction suffix tree (PST) (Fig.~\ref{fig:vmmp}~\subref{fig:vmmp:tree}) is built over the alphabet $\mathcal{A}$ admitted by all possible $h$-signatures given the environment. The PST is then converted to a PSA (Fig~\ref{fig:vmmp}~\subref{fig:vmmp:automata}) as per~\cite{ron_power_1993}. A high-level overview of this process is provided in Alg.~\ref{alg:offline}.
To construct the PST, it is first initialised with root node corresponding to the `empty' $h$-signature labelled $()$. Paths to suffixes are then successively added to the tree if the suffix has sufficiently strong predictive power. Specifically, a child node labelled with partial $h$-signature $ap$ is added to parent node $p$ if some measure of statistical difference $\mathcal{E}$ between $P(\cdot \mid p)$ and $P(\cdot \mid ap)$ is above a user-defined threshold $\epsilon$. The metric used is the KL divergence scaled by the probability of observing $ap$,
\begin{equation}
\mathcal{E}(ap, p) = P(ap) D_{KL}(P(\cdot \mid ap) || P(\cdot \mid p)).
\end{equation}
This scaling factor $P(ap)$ serves to avoid the addition of suffixes that have very low probability of occurring, yet give large KL divergence. Suffixes up to length $L$ are tested for their predictive power and added.
To calculate $P(a\mid p)$ and $P(p)$ from training data, Laplace's rule of succession is used. Denoting $\mathcal{F}(p)$ as the frequency with which partial $h$-signature $p$ appears in data, $\mathcal{F}(p^{\mathrm{C}})$ is then used to denote the frequency of the complement event; the frequency of observing any other partial $h$-signature of length $|p|$. Then,
\begin{equation}
P(p) \approx \frac{\mathcal{F}(p) + 1}{\mathcal{F}(p^{\mathrm{C}}) + |\mathcal{A}|}.
\end{equation}
Similarly, denoting the frequency with which letter $a$ follows $p$ in observations by $\mathcal{F}(a|p)$ and the frequency of observing any other letter after $p$ by $\mathcal{F}(a^{\mathrm{C}}\mid p)$,
\begin{equation}\label{eq:trans_prob}
P(a \mid p) \approx \frac{\mathcal{F}(a|p)+1}{\mathcal{F}(a^{\mathrm{C}}\mid p) + |\mathcal{A}|}.
\end{equation}
After construction, taking the leaves of the PST gives the states of the corresponding PSA. However, this simple action may not always admit a valid transition between all states. In this case, leaves must be added to the PST to ensure a complete PSA. Nodes are added to the PST until, for every leaf in the PST, the longest prefix of the leaf exists in the PST. When this condition is true, the leaves of the PST are guaranteed to give a complete PSA~\cite{ron_power_1993}. Transition probabilities between states of the PSA correspond directly to the transition probabilities of the PST found via Eqn.~\ref{eq:trans_prob} and are used to calculate probabilities online.
\begin{figure}[t!]
\centering
\subfloat[Prediction suffix tree.]{\includegraphics[width=0.49\linewidth]{figs/vmmp_tree.png}\label{fig:vmmp:tree}}
\subfloat[Probabilistic suffix automaton.]{\includegraphics[width=0.49\linewidth]{figs/automata.png}\label{fig:vmmp:automata}}
\caption{Equivalent methods of representing a VOMP. Values along edges between states are transition probabilities.}\label{fig:vmmp}
\vspace{-1.5em}
\end{figure}
\subsection{Illustration of $h$-signature Prediction}
Here, an illustration of VOMP's high-level motion prediction process is given. A simplistic toy environment and dataset was created, shown in Fig.~\ref{fig:high_level_vmmmp}~\subref{fig:train_toy}. Trajectory data was created by running Dijkstra's algorithm on a graph over the environment to find shortest-distance paths from randomly selected points on the left boundary to random points on all other boundaries. Thus, in this simple dataset all trajectories move from left to right, and the set of possible homotopy classes are denoted by $h$-signatures $\{(), (1), (1,2)\}$.
Prediction of $h$-signatures over time is shown in the following figures Fig.~\ref{fig:high_level_vmmmp}~\subref{fig:toy_vmmp_1}~-~\subref{fig:toy_vmmp_3}. At time $t=8$, the partial $h$-signature is $()$, and the VOMP predicts that all $h$-signature trajectories are possible in the future, with probabilities $P(()) = 0.28$, $P((1)) = 0.37$, $P((1,2)) = 0.35$. At $t=17$ the $()$ class is predicted with probability $0$ now the observed trajectory has passed above the first obstacle, and $P((1)) = 0.52$, $P((1,2)) = 0.48$. In the final time step the VOMP correctly assesses that the trajectory will have full $h$-signature $(1,2)$ with probability 1 as the trajectory has passed the second obstacle. Coloured regions correspond to regions in which one may expect the agent to be in the future, with probabilities indicated by opacity. These regions demonstrate the predictive power of the VOMP output.
\begin{figure}[t!]
\centering
\subfloat[Data\label{fig:train_toy}]{\includegraphics[width=0.24\columnwidth]{figs/train_set_small.png}}
\subfloat[$t=8$\label{fig:toy_vmmp_1}]{\includegraphics[width=0.24\columnwidth]{figs/rndm_starts_1.png}}
\subfloat[$t=17$]{\includegraphics[width=0.24\columnwidth]{figs/rndm_starts_2.png}}
\subfloat[$t=44$\label{fig:toy_vmmp_3}]{\includegraphics[width=0.24\columnwidth]{figs/rndm_starts_3.png}}
\caption{VOMP $h$-signature prediction for a test trajectory over time. Black trajectory is the observed test trajectory. Coloured regions correspond to straight-line path completions that lie within predicted homotopy classes. Colour coding of homotopy classes is as in the training set. Alpha of the regions reflects the VOMP output probability for that class.}
\label{fig:high_level_vmmmp}
\vspace{-1.5em}
\end{figure}
\subsection{Low-level Prediction using hierarchical Gaussian Mixture Models}\label{sec:gmm}
We present a hierarchical GMM as an example implementation of the low-level prediction algorithm for retrieving a probability distribution $ P(\mathbf{X} \mid h)$ over the trajectory $\mathbf{X}$ given the final $h$-signature $h$.
To this end, we simply cluster the trajectories from the training dataset into their homotopy classes (i.e. having the same $h$-signature $h$), and fit a GMM for each class,
\begin{equation}\label{eq:prior_gmm_hsig}
P(\mathbf{X} \mid h) = \sum_{c} w^{(c, h)} \mathcal{N}( \mathbf{X} \mid \mathbf{M}^{ (c, h)}, \Sigma^{(c, h)}).
\end{equation}
Here, for a given $h$-signature $h$, $w^{(c, h)}$, $\mathbf{M}^{ (c, h)}$ and $\Sigma^{(c, h)}$ are the weight, mean and covariance respectively of a component $c \in [1, N_{C}]$.
$\mathcal{N}(\mathbf{X} \mid \mathbf{M}, \Sigma )$ is the multivariate normal distribution over $\mathbf{X}$ with mean vector $\mathbf{M}$ and covariance $\Sigma$.
In doing so, the low-level patterns in the trajectories can be captured by computing the full covariance matrix $\Sigma$, thereby capturing correlations between positions at particular times.
From this distribution, one can derive a fully probabilistic prediction of the trajectory given position measurements $\mathbf{Y}_{\mathrm{obs}}$ and partial $h$-signature $p$. To simplify computation, we treat the partial $h$-signature $p$ as an additional independent measurement to $\mathbf{Y}_{\mathrm{obs}}$.
Then, the partial $h$-signature is conditionally independent of $\mathbf{X}$ given the full $h$-signature $h$, and we can easily condition on the partial $h$-signature as a weighted sum $P(\mathbf{X} \mid p) = \sum_{h} P(\mathbf{X} \mid h) P(h \mid p)$. This effectively only scales the weights of the GMM by $P(h\mid p)$ given by the VOMP.
Subsequently, we can further condition on the actual measurements $\mathbf{Y}_{\mathrm{obs}}$. Following standard methods,
\begin{equation}
P(\mathbf{X} \mid p, \mathbf{Y}_{\mathrm{obs}}) = \sum_{c, h} \hat{w}_{\mathrm{obs}}^{(c, h)} \mathcal{N}(\mathbf{X} \mid \mathbf{\hat{M}}_{\mathrm{obs}}^{(c, h)}, \mathbf{\hat{\Sigma}}_{\mathrm{obs}}^{(c, h)} ).
\end{equation}
Here, the conditional mean and covariance $\mathbf{\hat{M}}_{\mathrm{obs}}^{(c, h)}$, $\mathbf{\hat{\Sigma}}_{\mathrm{obs}}^{(c, h)}$ are calculated in the same manner as standard conditional Gaussian distribution~\cite[Sec.~8.1.3]{matrixcookbook} given measurements.
The conditional weights are calculated as
\begin{equation}
\hat{w}_{\mathrm{obs}}^{(c, h)} \propto w_{t}^{(c, h)} P(h \mid p) \mathcal{N}( \mathbf{Y}_{\mathrm{obs}} \mid \mathbf{M}^{ (c, h)}, \Sigma^{(c, h)} + \sigma_{\mathbf{Y}}^2 I), \label{eq:conditioning:weights}
\end{equation}
followed by normalisation. The last term is the marginal likelihood of observing $\mathbf{Y}_{t}$ within each mixture component.
\begin{figure*}
\centering
\begin{tabular}{c}
\subfloat[The ATC shopping mall environment and a subset of the data~\cite{atc_dataset}.]{\includegraphics[width=0.285\linewidth]{figs/atc_train.png}\label{fig:atc_data}}
\end{tabular}
\begin{tabular}{c}
\subfloat[VOMP $t=1$]{\includegraphics[width=0.13\textwidth]{figs/atc_data_new_vmmp_gmm_traj300_000.png}\label{fig:gmmp_1}}
\subfloat[VOMP $t=12$]{\includegraphics[width=0.13\textwidth]{figs/atc_data_new_vmmp_gmm_traj300_001.png}\label{fig:gmmp_2}}
\subfloat[VOMP $t=34$]{\includegraphics[width=0.13\textwidth]{figs/atc_data_new_vmmp_gmm_traj300_003.png}\label{fig:gmmp_3}}
\subfloat[VOMP $t=56$]{\includegraphics[width=0.13\textwidth]{figs/atc_data_new_vmmp_gmm_traj300_005.png}\label{fig:gmmp_4}}
\subfloat[VOMP $t=78$]{\includegraphics[width=0.13\textwidth]{figs/atc_data_new_vmmp_gmm_traj300_007.png}\label{fig:gmmp_5}} \\
\subfloat[Naive $t=1$]{\includegraphics[width=0.13\textwidth]{figs/atc_data_new_naive_gmm_traj300_000.png}\label{fig:naive_1}}
\subfloat[Naive $t=12$]{\includegraphics[width=0.13\textwidth]{figs/atc_data_new_naive_gmm_traj300_001.png}\label{fig:naive_2}}
\subfloat[Naive $t=34$]{\includegraphics[width=0.13\textwidth]{figs/atc_data_new_naive_gmm_traj300_003.png}\label{fig:naive_3}}
\subfloat[Naive $t=56$]{\includegraphics[width=0.13\textwidth]{figs/atc_data_new_naive_gmm_traj300_005.png}\label{fig:naive_4}}
\subfloat[Naive $t=78$]{\includegraphics[width=0.13\textwidth]{figs/atc_data_new_naive_gmm_traj300_007.png}\label{fig:naive_5}}
\end{tabular}
\caption{Snapshots of predictions output by a GMM with $h$-signature context given by the VOMP~\protect\subref{fig:gmmp_1}~-~\protect\subref{fig:gmmp_5} and without~\protect\subref{fig:naive_1}~-~\protect\subref{fig:naive_5}. Black dashed trajectory is the ground truth test trajectory. Shaded regions show the variance of around the mean trajectory in solid colour. Transparency is proportional to weight. Colours in~\protect\subref{fig:gmmp_1}~-~\protect\subref{fig:gmmp_5} indicate $h$-signatures, while colours in~\protect\subref{fig:naive_1}~-~\protect\subref{fig:naive_5} indicate mixture components.}
\label{fig:low_level_gmmp}
\vspace{-1em}
\end{figure*}
\section{Experimental Results}
\subsection{Experimental Setup}
We experimentally illustrate the benefits of high-level prediction by comparison against a standard GMM without topological knowledge.
We use the ATC shopping mall dataset~\cite{atc_dataset} shown in Fig.~\ref{fig:low_level_gmmp}~\subref{fig:atc_data}.
We retrieved a portion of the dataset containing 17558 trajectories.
We selected 9230 trajectories among the dataset trajectories contained within the area shown in Fig.~\ref{fig:low_level_gmmp}~\subref{fig:atc_data} that satisfy the border crossing assumption.
Among these trajectories, we randomly selected 5538 trajectories to create a training dataset, and 1000 trajectories for testing.
Trajectories are interpolated over a discrete number of timesteps and used to evaluate the prediction performance of the two pipelines.
Three metrics are used for comparison.
Most immediately, we consider the standard \emph{average displacement error}~(ADE),
which measures the deterministic error in the maximum likelihood prediction.
This is defined as
\begin{equation}
\text{ADE} = \frac{1}{T} \sum_{t} || \mathbf{x}_{t} - \mathbf{\hat{M}_{t, obs}}^{ (c^{*}, h^{*})} ||,
\end{equation}
where $(c^{*}, h^{*})$ denotes the index of the component with the highest weight.
Since the framework is fully probabilistic, we also need to account for the uncertainty estimates produced.
We also consider the \emph{average Mahalanobis distance}~(AMD), which accounts for the weights and covariances in the GMM:
\begin{equation}
\text{AMD} =
\sum_{c, h} \frac{\hat{w}_{\mathrm{obs}}^{ (c, h) }}{T} \sum_{t} \Theta_{t}^{{ (c, h) }^{\mathrm{T}}} (\mathbf{\hat{\Sigma}_{t, obs}}^{ (c, h ) })^{-1} \Theta_{t}^{ (c, h) },
\end{equation}
where $\Theta_{t}^{ (c, h) } = (\mathbf{x}_{t} - \mathbf{\hat{M}_{t, obs}}^{ (c, h) })$.
Lastly, we would like to characterise the performance of the high-level prediction made.
To do so, we compare the \emph{KL divergence} (KLD) between the posterior GMM weights given the full trajectory and a partial trajectory, defined as
\begin{equation}
\text{KLD} = \sum_{c, h} \hat{w}_{T}^{ (c, h) } (\log \hat{w}_{T}^{ (c, h) } - \log \hat{w}_{\mathrm{obs}}^{ (c, h) }).
\end{equation}
For fairness, we use the same number of mixtures for the naive GMM as in the topology-informed GMM. All GMMs used are trained using the implementation in the \textsc{scipy} library, using the default arguments.
\subsection{Results}
Figure~\ref{fig:atc_results} shows the results of comparisons between naive GMM (red) and our topology-informed approach (green) in terms of ADE (top), AMD (middle), and KLD (bottom).
In terms of ADE, our approach outperforms the naive approach after some time, namely showing a $34.8$\% improvement in the median around halfway through the trajectory ($t = 56$), and up to $69.4$\% over time. This increase in performance is because the topology-informed approach gains more information as the target
crosses obstacles, and this topological information improves the predictive power of the VOMP.
Since the naive GMM was given the same number of components as the topology-informed one calculated based on the number of homotopy classes, it is unsurprising that the naive GMM performs relatively well.
In practice, a major challenge in deploying GMMs is selecting the right number of mixture components. In fact, the naive GMM is initialised with a number directly informed by the number of homotopy classes present in the data, providing a small topological cue to the baseline method.
Nonetheless, the AMD metric shows that our approach performs significantly better than or equal to the naive approach. Specifically, we see a $72.3$\% improvement over the naive halfway through the trajectory, with a maximum of $80.3$\% improvement over time. This implies that the uncertainty predictions from the topology-informed approach are more \emph{consistent} with the actual error compared to the naive approach. This is because our approach pre-clusters trajectories that are `similar', improving the fitness of the GMM model within each cluster.
Further, our approach consistently outperforms in terms of KLD by a great margin, with $100$\% improvement achieved by halfway through the trajectory. This means that early trajectory GMM weights of our approach better represented the final GMM weights. Viewing GMM components as an alternative description of high-level motion to $h$-signatures, this implies that $h$-signatures are indeed better descriptors.
Figure~\ref{fig:low_level_gmmp} shows the behaviour of our approach and baseline GMM.
Early on, we see both methods give highest weight to components that reasonably predict the general motion of the trajectory. However, the naive approach begins to produce poor predictions in an area of the environment where many training paths diverge. This `crossroads' is visible in the sample data plotted in Fig.~\ref{fig:low_level_gmmp}~\subref{fig:atc_data}. In comparison, our approach continues to correctly predict the same $h$-signature at this crossroads. This demonstrates the power of topological information in imbuing low-level predictions with robustness to deviations in trajectories.
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{figs/full_plot_new_2.png}
\caption{Experimental results. Red: Naive GMM, Green: Our approach. Lower is better. Markers show the median. Error bars show 25\% and 75\% quantiles.}
\label{fig:atc_results}
\vspace{-1.5em}
\end{figure}
\section{Conclusion}
We presented a framework for trajectory prediction using homotopy classes.
The core of the framework is the notion of partial $h$-signature, which allows prediction of the full trajectory at both abstract and geometric levels.
Under this framework, we presented VOMP and hierarchical GMMs as minimal implementations of the high-level and low-level prediction components, the combination of which was shown to outperform baselines without topological knowledge.
Our framework opens many exciting opportunities for future work.
The high-level and low-level components are modular, and can be replaced with many other algorithms of varying sophistication and efficiency.
Further, we expect the saliency and sparseness of $h$-signatures will be instrumental when directly used in planning problems such as target search in cluttered environments~\cite{wakulicz_active_2021}.
We believe similar topological features can be defined in other application domains such as marine buoys~\cite{argo,floats2012}.
\bibliographystyle{IEEEtran}
|
1,108,101,563,239 | arxiv | \section{Introduction}\label{Introduction}
In recent years, there has been enormous progress with integrable conformal quantum field theories in dimensions higher than two. The most important example is planar $\mathcal{N}=4$ super-Yang-Mills theory (SYM) in four dimensions. Here an exact description exists at strong coupling due to the AdS/CFT correspondence \cite{Maldacena_1999,Gubser_1998,witten1998anti}. Moreover, the model appears to be integrable at arbitrary values of the coupling constant \cite{Beisert_2011}. While an exact proof is still missing, this apparent fact has been used to calculate a large number of quantities at weak, strong, and even finite coupling. What appears to be crucial for integrability is the conformal symmetry of the model in conjunction with a second, {\it dual} conformal symmetry \cite{Drummond_2007}. These two symmetries combine into an infinite-dimensional Yangian symmetry, which leads to integrability \cite{Ferro:2018ygf}. It should be stressed, from the derivation of the AdS/CFT correspondence, that the two copies of conformal symmetry should each be {\it global}, i.e.\ they should also be present for large, finite (ordinary and dual) conformal transformations in Minkowski space. A topical issue is to understand the integrability directly on the level of the Feynman diagrammatic expansion for correlation functions of $\mathcal{N}=4$ SYM. As we will review below, infinitesimal dual conformal symmetry indeed appears even on the level of the simplest radiative correction to four-point functions at one-loop order: The well-known box integral. However, vexingly, global invariance does not strictly hold for all large transformations. Although known, this rather subtle symmetry breaking has never been analysed in detail, and is the main focus of this paper: We relate the branched structure of the known analytic expression for the box integral to equivalence classes of four points in Minkowski-space with respect to conformal transformations at fixed values of the conformal cross-ratios. This required a systematic classification of the latter, which to the knowledge of the authors has not been worked out in full detail before. Furthermore, we studied a novel set of conformal four-point configurations in Minkowski space which cover all of the kinematic space. We describe the simplest of these configurations here, which involve moving two points (rather than the standard one point) to conformal infinity, and demonstrate how these configurations can simplify the calculation of the box integral. The consequences of our findings for the integrability program of $\mathcal{N}=4$ SYM will be studied in future work.
\newline
\newline
The scalar one-loop massless box integral in dual momentum space (figure \ref{figg16}) is written
\begin{align}\label{2t}
I(x_1,x_2,x_3,x_4)=\int \frac{d^4x_5}{i\pi^2}\frac{x_{13}^2x_{24}^2}{(x_{15}^2+i\epsilon)(x_{25}^2+i\epsilon)(x_{35}^2+i\epsilon)(x_{45}^2+i\epsilon)},
\end{align}
where $x_{ij}^2\equiv (x_i-x_j)^2$. $x_i\in \mathbb{R}^{1,3}$ are dual momenta \cite{Drummond_2010}, related to the usual QFT momenta\footnote{The $x_i$ can only be defined from $p_i$ up to an overall translation.} $p_i$ by $p_i=x_i-x_{i+1}$, with summation in the index taken mod 4.
\begin{figure}[h!]
\begin{center}
\begin{adjustbox}{max totalsize={.34\textwidth}{.27\textheight},center}
\begin{tikzpicture}{scale=0.2}
\Large
\draw[dotted] (-2,2)--(-3.2,3.2);
\draw[dotted] (2,2)--(3.2,3.2);
\draw[dotted] (2,-2)--(3.2,-3.2);
\draw[dotted] (-2,-2)--(-3.2,-3.2);
\draw[dotted] (-2,2)--(2,2);
\draw[dotted] (-2,2)--(-2,-2);
\draw[dotted] (-2,-2)--(2,-2);
\draw[dotted] (2,-2)--(2,2);
\draw[thick] (-2.8,0)--(2.8,0);
\draw[thick] (0,-2.8)--(0,2.8);
\filldraw (0,0) circle (3pt);
\filldraw (-2.8,0) circle (3pt);
\filldraw (0,2.8) circle (3pt);
\filldraw (0,-2.8) circle (3pt);
\filldraw (2.8,0) circle (3pt);
\node at (-0.5,2.8){$x_1$};
\node at (-0.5,-2.8){$x_3$}; \node at (-2.8,0.4){$x_4$}; \node at (2.8,0.4){$x_2$};
\normalsize
\end{tikzpicture}
\end{adjustbox}
\end{center}
\caption{One-loop massless box diagram in dual momentum space. Internal points are integrated over. Dotted lines show the diagram in momentum space.}
\label{figg16}
\end{figure}
We take the $x_i$ to be sufficiently generic, so that in particular $x_{ij}^2\neq 0$ and we are away from light cone singularities\footnote{Away from these singularities the integral (\ref{2t}) is finite in four dimensions.}. The $+i\epsilon$ indicates the usual Feynman prescription for momentum-space propagators in Minkowski space in $(+--\hspace{0.09cm}-)$ signature, which we use throughout this work\footnote{In a direct position space interpretation of \eqref{2t} one should replace all four $+i\epsilon$'s by $-i\epsilon$'s.}. The integral (\ref{2t}) appears ubiquitously at next-to-leading order in the calculation of both scattering amplitudes and correlation functions in four-dimensional gauge theories. Concrete examples in planar $\mathcal{N}=4$ super-Yang-Mills (SYM) include the one-loop four gluon amplitude \cite{Bern_2005} and the one-loop correlation function of four protected operators \cite{Eden_1999}. In fact these two results are equal in a certain light cone limit \cite{Eden_2011}. More recently (\ref{2t}) has appeared directly as the first of a series of fishnet Feynman integrals in a strongly twisted version of $\mathcal{N}=4$ SYM \cite{Basso_2017}. It has been shown to possess a remarkable symmetry known as dual conformal invariance \cite{Broadhurst:1993ib,Drummond_2007}, which is best seen in Euclidean space. The Euclidean integral corresponding to (\ref{2t}) is
\begin{align}\label{3t}
I^E(x_1,x_2,x_3,x_4)=\int \frac{d^4 x_5}{\pi^2}\frac{x_{13}^2x_{24}^2}{x_{15}^2x_{25}^2x_{35}^2x_{45}^2},
\end{align}
where $x_1,\hspace{0.05cm} x_2,\hspace{0.05cm} x_3,\hspace{0.05cm} x_4$ are non-coincident points in Euclidean space $\mathbb{R}^4$. The dual conformal invariance of (\ref{3t}) refers to its invariance under conformal transformations in the $x$ variables
\begin{align}
I^E(x_1,x_2,x_3,x_4)=I^E(x'_1,x'_2,x'_3,x'_4),\hspace{1cm} x_i'=Ax_i,
\end{align}
where $A:\mathbb{R}^4\rightarrow \mathbb{R}^4$ is a conformal transformation on $\mathbb{R}^4$, i.e.\ an element of $\text{Conf}(\mathbb{R}^4)\simeq SO(1,5)/\mathbb{Z}_2$. As such the Euclidean integral (\ref{3t}) depends only on a conformal invariant $z$ and its complex conjugate $\bar{z}=z^*$. It was calculated, for example in \cite{Usyukina:1992jd}, to be
\begin{align}\label{4t}
I^E(x_1,x_2,x_3,x_4)=I^E(z)=\frac{2\text{Li}_2(z)-2\text{Li}_2(\bar{z})+\log(z\bar{z})\log(\frac{1-z}{1-\bar{z}})}{z-\bar{z}}.
\end{align}
Because $\bar{z}=z^*$ in Euclidean space (\ref{4t}) is essentially proportional to the Bloch-Wigner function, which has many interesting analytic properties \cite{Zagier:2007knq}. It is because of this conformal invariance that (\ref{3t}) is referred to as the \textit{dual conformal} box integral.
\newline
\newline
The situation in Minkowski space, where $z$ and $\bar{z}$ can either be independent real numbers or complex conjugate pairs, is more subtle. The dual conformal box integral in Minkowski space (\ref{2t}) has been calculated many times, first by 't Hooft and Veltman in \cite{tHooft:1978jhc}. More compact representations were obtained in \cite{Denner:1991qq} and \cite{Duplan_i__2002}. Common features of these calculations are the Feynman parametric and Mellin Barnes representations for the integral (\ref{2t}). The result in \cite{Duplan_i__2002} is the only one which gives any hint of the underlying dual conformal symmetry of the integral. It is given essentially in terms of the conformal invariants $z$ and $\bar{z}$, plus some corrections which depend explicitly on the signs of the kinematics\footnote{$x_{ij}^2$ coincide with the usual four particle kinematics $p_i^2,s,t$ in momentum space.} $x_{ij}^2$. These corrections encode a subtle breaking of dual conformal invariance in Minkowski space, which is caused by the Feynman $i\epsilon$, here serving as an IR regulator, required to properly define the integral.
\newline
\newline
The $i\epsilon$ regulator encodes causality in the propagators, and the breaking of conformal invariance in Minkowski space is due to the fact that the conformal group on Minkowski space $\text{Conf}(\mathbb{R}^{1,3})\simeq SO^+(2,4)/\mathbb{Z}_2\simeq SU(2,2)/\mathbb{Z}_4$ does not preserve causality. In particular spacelike separations can be mapped to timelike separations, and vice versa. To define the action of the conformal group on Minkowski space properly, we must move to the compactification $\mathbb{R}_c^{1,3}$ \cite{book}. This compactification can be expressed in numerous ways, for example in terms of the group $U(2)$ \cite{Uhlmann1963THECO} or as the projective null cone in $\mathbb{R}^{2,4}$ \cite{Dirac:1936fq}. The conformal group then acts naturally on both of these spaces, although causality in the original Minkowski space is still not preserved. In order to preserve a notion of causality, it is necessary to move to the universal cover of compactified Minkowski space $\Tilde{\mathbb{R}}^{1,3}_c$ and consider the universal covering group $\reallywidetilde{\text{Conf}}(\mathbb{R}^{1,3})$ \cite{go1975}. In this work however we will focus on the conformal geometry of compactified Minkowski space, expressed in terms of $U(2)$. The value of the box integral (\ref{2t}) can change under finite special conformal transformations (SCTs) which change the signs of the kinematics $x_{ij}^2$. Geometrically this corresponds to a point `crossing infinity', i.e.\ moving from a point on Penrose's $\mathscr{I}^\pm$ to the antipodal point on $\mathscr{I}^{\mp}$. In the compactification these points are identified and crossing infinity still represents a continuous operation. For infinitesimal SCTs there are no issues and the integral (\ref{2t}) is \textit{locally} conformally invariant.
\newline
\newline
Another way to phrase the calculation of the Minkowskian box integral is ``how can one `analytically continue' the function (\ref{4t}) to Minkowski space?'' The Euclidean integral (\ref{3t}) can be obtained from (\ref{2t}) via a Wick rotation in certain kinematic regions, for example when all of the kinematics are spacelike ($x_{ij}^2<0$). In other regions (\ref{3t}) can only be Wick rotated to (\ref{2t}) up to logarithms of the conformal invariants $z$ and $\bar{z}$, so that the final answer is on another `branch' of the Bloch-Wigner function (\ref{4t}) \cite{hodges2010box}. There has been much work regarding the analytic continuation of Euclidean correlators to Minkowski space, and the subtleties of Wick rotation ~\cite{maldacena2015looking,Hartman_2016,Bautista_2020,qiao2020classification,cornalba2007eikonal}.
\newline
\newline
In this paper we classify all possible kinematic regions for four points in Minkowski space by their conformal structure, and describe how this relates to the branch jumping of the box integral (\ref{2t}). Specifically we study conformally equivalent configurations of four points in compactified Minkowski space. We find that, unlike in Euclidean space, the conformal invariants $z$ and $\bar{z}$ are not enough to determine when two configurations of four points are conformally equivalent. There are two cases, depending on whether $z$ and $\bar{z}$ are complex conjugates or real. If $z$ and $\bar{z}$ are complex conjugates, then all configurations with the same $z$ and $\bar{z}$ are conformally equivalent, as in the Euclidean case. If $z$ and $\bar{z}$ are real, then up to the discrete transformations of the Lorentz group $\mathcal{P},\hspace{0.05cm}\mathcal{T}$ there are two conformal equivalence classes of configurations. These two equivalence classes are exchanged by a reversal of the signs of each of the kinematics $x_{ij}^2$. In each conformal equivalence class we find that there are up to four values of the box integral (\ref{2t}), by explicitly rewriting the result of \cite{Duplan_i__2002}.
\newline
\newline
One new strategy for the calculation of (\ref{2t}) we explored is to exploit the richer structure of `infinity' in Minkowski space. In Euclidean space infinity is a single point $\infty$, whereas in compactified Minkowski space $\mathbb{R}^{1,3}_c$ it is a three-dimensional surface $\mathscr{I}\hspace{0.05cm}\cup\hspace{0.05cm}\iota$ (see section \ref{confcompact}). All configurations of four points in Minkowski space can be conformally mapped to a configuration with two or more points at infinity, such that (importantly) the value of the box integral does not change. Most configurations with real $z,\hspace{0.05cm}\bar{z}$ can be conformally mapped to a special `double infinity' configuration, where the box integral is calculable directly in spacetime using only the residue theorem. The rest of the configurations can be mapped to a configuration with three or four points at infinity, however these are not discussed in this paper as the corresponding integrals are not as easily calculated. These higher infinity configurations, similarly to $(0,1,z,\infty)$ in Euclidean space, furnish a `minimal' set of configurations which fully fix the box integral (and other four-point observables) by pseudo-conformal invariance. They also provide a nice way to generate configurations of four points with a given $z,\bar{z}$, in a given kinematic region.
\newline
\newline
It is worth noting that the box integral has been shown to be Yangian invariant \cite{Chicherin_2017,Chicherin_2018}, and in Euclidean space it has been successfully constrained to the Bloch-Wigner function using the novel `Yangian bootstrap' approach \cite{Loebbert_2020,loebbert2020yangian}. The Minkowskian box integral has also been considered in a momentum-twistor geometry by Hodges \cite{hodges2010box}, who also discussed the subtle breaking of dual conformal invariance and analytic continuation to all kinematic regions of Minkowski space.
\newline
\newline
The structure of this paper is as follows. In section \ref{confcompact} we review the action of the conformal group on the $U(2)$ conformal compactification of Minkowski space. In section \ref{confplane} we study the conformal geometry of four points in compactified Minkowski space, and describe the sets of conformally equivalent configurations. In section \ref{symmetry} we discuss the symmetries of the box integral directly in Minkowski space, and the various representations of the integral which make these symmetries manifest. We give an expression for the box integral (\ref{2t}) in all kinematic regions by specialising the result of \cite{Duplan_i__2002}. In section \ref{doubleinf} we discuss the calculation of the box integral in the special `double infinity' configurations. We found the package \cite{vanOldenborgh:1990yc} useful for numerically testing calculations. We always take logarithms with a branch cut on the negative real axis, so that the corresponding dilogarithms have a cut on $(1,\infty)$.
\section{Conformally Compactified Minkowski Space}\label{confcompact}
There is a slight subtlety in defining the action of the conformal group on Euclidean space $\mathbb{R}^4$. Since the inversion operation $\mathcal{I}:x\rightarrow x/x^2$ is not well-defined on the origin $x=0$, we should add a point `$\infty$' to $\mathbb{R}^4$ which corresponds to the image of the origin under $\mathcal{I}$. The conformal group is well-defined and acts transitively\footnote{A group $G$ acts transitively on a set $X$ if for all $x,y\in X$ there exists $g\in G$ such that $gx=y$.} on the \textit{compactification} $\mathbb{R}_c^4\equiv \mathbb{R}^4\cup\{\infty\}$.
\newline
\newline
The situation is more subtle in Minkowski space $\mathbb{R}^{1,3}$. Defining the light cone of the origin
\begin{align}
L_0\equiv \{x\in\mathbb{R}^{1,3}\hspace{0.05cm}\sp|\hspace{0.05cm}\sp x^2=0\}
\end{align}
we note in this case that $\mathcal{I}$ not well-defined on the entire space $L_0$. We thus are interested in a compactification $\mathbb{R}^{1,3}\rightarrow \mathbb{R}_c^{1,3}$ such that the conformal group is well-defined and acts transitively and continuously on $\mathbb{R}_c^{1,3}$. The points `at infinity' will be identifiable\footnote{Up to an extra `null sphere at infinity' $S^*$, see (\ref{u4}) and figure \ref{figg12}.} with the image $\mathcal{I}(L_0)$, and so will constitute a $3-$dimensional surface.
\newline
\newline
The most common way of compactifying Minkowski space $(\mathbb{R}^{1,3},\eta)$ is the \textit{Penrose compactification} \cite{Penrose:1964ge} and replaces $\mathbb{R}^{1,3}\rightarrow \mathbb{R}^{1,3}_p\equiv \mathbb{R}^{1,3}\cup \partial \mathbb{R}^{1,3}_p$, where $\partial \mathbb{R}^{1,3}_p\equiv\iota^+\cup\iota^-\cup\iota^0\cup\mathscr{I}^+\cup\mathscr{I}^- $. $\iota^0$ is called \textit{spatial} infinity, and in spherical coordinates $(t,r,\theta,\phi)$ corresponds to $r\rightarrow \infty$. $\iota^\pm$ are called \textit{future} and \textit{past} infinity, and correspond to $t\rightarrow \pm\infty$. Both $\iota^0$ and $\iota^\pm$ correspond to individual points on $\partial\mathbb{R}_p^{1,3}$. $\mathscr{I}^\pm$ are called \textit{future null} and \textit{past null} infinity and correspond to $t\pm r\rightarrow \infty$ with $t\mp r$ constant. $\mathscr{I}^{\pm}$ are $3-$dimensional submanifolds of $\partial\mathbb{R}_p^{1,3}$, parametrised by the coordinates $(t\mp r,\theta,\phi)$. Useful coordinates for describing $\mathbb{R}^{1,3}_p$ are
\begin{align}
\chi_+\equiv \arctan(t+r),\hspace{1cm} \chi_-\equiv\arctan(t-r),
\end{align}
and $\tau\equiv \chi_-+\chi_+,\hspace{0.05cm}\sp\hspace{0.05cm} \rho\equiv\chi_+-\chi_-$. The range of the $\tau,\hspace{0.05cm} \rho$ coordinates in $\mathbb{R}^{1,3}_p$ is $\tau\in[-\pi,\pi],\hspace{0.05cm}\rho\in[0,\pi]$. We visualise $\mathbb{R}^{1,3}_p$ in $(\tau,\rho)$ coordinates using an extended Penrose diagram (figure \ref{figg11}).
\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}[thick,scale=0.63, every node/.style={scale=0.785}]
\node[inner sep=0pt] (russell) at (0,0)
{\includegraphics[width=.6\textwidth]{Penrose.pdf}};
\draw[->,thick] (-.015,6.2)--(-0.015,6.22);
\draw[->,thick] (6.2,0.01)--(6.22,0.01);
\node at (6.3,0.35){$\rho$};
\node at (0.35,6.3){$\tau$};
\node at (0.35,5.7){$\iota^+$};
\node at (0.35,-5.6){$\iota^-$};
\node at (5.7,0.35){$\iota^0$};
\node at (3.5,3.2){$\mathscr{I}^+(\theta,\phi)$};
\node at (3.5,-3.2){$\mathscr{I}^-(\theta,\phi)$};
\node at (-3.63,3.2){$\mathcal{A}\mathscr{I}^+(\theta,\phi)$};
\node at (-3.63,-3.2){$\mathcal{A}\mathscr{I}^-(\theta,\phi)$};
\end{tikzpicture}
\end{center}
\caption{Extended Penrose diagram of Minkowski space. Every two-sphere $S^2(\tau,\rho>0)$ is represented by two points, one on the left and one on the right, which are exchanged by the antipodal map $\mathcal{A}$. The dotted lines represent the light cone of the origin \cite{strominger2017lectures}.}
\label{figg11}
\end{figure}
\newline
We want to consider conformal transformations on the Penrose compactification $\mathbb{R}_p^{1,3}$. There are a few issues at $\partial \mathbb{R}_p^{1,3}$. Using infinitesimal SCTs (\ref{u3}) it is possible to move between $\iota^\pm$ and $\iota^0$, and also between $\mathscr{I}^{\pm}(\chi_\mp,\theta,\phi)$ and $\mathscr{I}^{\mp}(\chi_\pm,\mathcal{A}(\theta,\phi))$, where $\mathcal{A}$ is the antipodal map on $S^2$. Therefore in order for the conformal group to act continuously on compactified Minkowski space, we must make the identifications\footnote{The identification $\mathscr{I}^+=\mathcal{A}\mathscr{I}^-$ resembles boundary conditions on fields in asymptotic symmetries \cite{strominger2017lectures}.}
\begin{align}\label{3r}
\iota^+=\iota^-=\iota^0\equiv\iota,\hspace{1cm} \mathscr{I}^+=\mathcal{A}\mathscr{I}^-\equiv \mathscr{I}.
\end{align}
We define $\partial \mathbb{R}^{1,3}_c\equiv \iota\hspace{0.05cm}\cup\hspace{0.05cm}\mathscr{I}$ as $\partial \mathbb{R}_p^{1,3}$ subject to the identifications (\ref{3r}). $\partial \mathbb{R}_c^{1,3}$ can be visualised as a `pinched torus' or croissant (figure \ref{figg12}), and can be parametrised in terms of $\chi\in(-\pi/2,\pi/2]$ and a 2$-$sphere angle $(\theta,\phi)$ for $\chi\neq 0$ \cite{Jadczyk_2011}. We define the conformal compactification $\mathbb{R}^{1,3}_c\equiv \mathbb{R}^{1,3}\cup \partial \mathbb{R}_c^{1,3}$. $\mathbb{R}^{1,3}_c$ can be identified with the group $U(2)$, on which the conformal group acts naturally by $SU(2,2)/\mathbb{Z}_4$.
\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}[thick,scale=1, every node/.style={scale=0.81}]
\draw[thick] (1.33,0) ellipse (1.3 and 1);
\draw[thick] (0,0) arc (180:-180:2);
\draw[blue,thick] (2.62,0) to[out=-50,in=-130] (4,0);
\draw[blue,dotted] (2.62,0) to[out=50,in=130] (4,0);
\draw[red,thick] (2.1,0.8) to[out=0,in=-20] (3.05,1.7);
\draw[red,dotted] (3.05,1.7) to[out=180,in=160] (2.1,0.8);
\draw[red,dotted] (2.1,-0.8) to[out=0,in=20] (3.05,-1.7);
\draw[red,thick] (3.05,-1.7) to[out=180,in=-160] (2.1,-0.8);
\draw[red,thick] (1.3,1) to[out=10,in=-10] (1.2,1.85);
\draw[red,dotted] (1.3,1) to[out=170,in=190] (1.2,1.85);
\draw[red,dotted] (1.3,-1) to[out=10,in=-10] (1.2,-1.85);
\draw[red,thick] (1.3,-1) to[out=170,in=140] (1.2,-1.82);
\filldraw[black] (0.02,0) circle (1pt);
\footnotesize
\node at (-0.6,0){$\iota:\chi=0$};
\node at (4.7,0){$\chi=\pi/2$};
\node at (3.47,1.87){$\chi=\frac{\pi}{3}$};
\node at (3.49,-1.85){$\chi=-\frac{\pi}{3}$};
\node at (1.1,2.1){$\chi=\frac{\pi}{6}$};
\node at (1.2,-2.17){$\chi=-\frac{\pi}{6}$};
\node at (3.35,0.0){\textcolor{blue}{$S^*$}};
\normalsize
\node at (2.25,2.2){$\mathscr{I}$};
\end{tikzpicture}
\end{center}
\caption{Conformal infinity in terms of $\iota$ and $\mathscr{I}$. Surfaces of fixed $\chi$ are $2-$spheres. The parameter $\chi\in(-\pi/2,\pi/2]$ is chosen as $\chi=\pi/2-\chi_+$ mod $\pi$, so $\chi=0$ corresponds to $\iota$.}
\label{figg12}
\end{figure}
To see this, we first note the bijection $H:\mathbb{R}^{1,3}\rightarrow H_{2\times2}$ between Minkowski space and the space of $2\times 2$ Hermitian matrices
\begin{align}
H:x=(x^0,x^1,x^2,x^3)\longrightarrow H(x)\equiv\begin{pmatrix}x^0+x^3&x^1-ix^2\\ x^1+ix^2&x^0-x^3\end{pmatrix}.
\end{align}
In what follows we will label the Hermitian matrices corresponding to points $x,y,z$ in Minkowski space by $X,Y,Z$.
The inverse map is given by
\begin{align}\label{30a}
H^{-1}:X\rightarrow x^\mu=\frac{1}{2}\text{tr}(X\sigma^\mu),
\end{align}
where $\sigma^{\mu}=(\mathbb{I}_2,\sigma^1,\sigma^2,\sigma^3)$. In spherical coordinates $x=(t,r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta)$, the Hermitian matrix corresponding to $x$ is
\begin{align}
X=\begin{pmatrix}t+r\cos\theta& re^{-i\phi}\sin\theta\\ re^{i\phi}\sin\theta&t-r\cos\theta\end{pmatrix}=\Omega(\theta,\phi)^{\dagger}M(t,r)\Omega(\theta,\phi),
\end{align}
where
\begin{align}\label{8r}
\Omega(\theta,\phi)\equiv\begin{pmatrix}\cos(\theta/2)& e^{-i\phi}\sin(\theta/2)\\ -e^{i\phi}\sin(\theta/2)&\cos(\theta/2)\end{pmatrix}\in SU(2),\hspace{1cm} M(t,r)\equiv \begin{pmatrix}t+r&0\\ 0&t-r\end{pmatrix}.
\end{align}
$M(t,r)$ represents the event in Minkowski space at a time $t$, and spatially on the north pole ($\theta=0$) of a sphere of radius $r$. The adjoint action of $\Omega(\theta,\phi)$ rotates this point from the north pole to a general angle $(\theta,\phi)$ on the sphere. In terms of Hermitian matrices we can calculate spacetime intervals via determinants
\begin{align}
(x-y)^2=\det(X-Y).
\end{align}
To compactify, we use the injective Cayley map\footnote{This is analogous to the one-dimensional Cayley map $\mathbb{R}\rightarrow U(1)$, $x\rightarrow \frac{i-x}{i+x}$} $J:H_{2\times 2}\rightarrow U(2)$ defined by
\begin{align}\label{1j}
J:X\longrightarrow J(X)\equiv \frac{i\mathbb{I}_2-X}{i\mathbb{I}_2+X},
\end{align}
which is well-defined because $\det(i\mathbb{I}_2+X)\neq 0$ and $[i\mathbb{I}_2-X,(i\mathbb{I}_2+X)^{-1}]=0$ for all $X\in H_{2\times 2}$. The map (\ref{1j}) is not surjective. The inverse is given
\begin{align}
J^{-1}:U\rightarrow i\frac{\mathbb{I}_2-U}{\mathbb{I}_2+U},
\end{align}
which is not well-defined for those $U\in U(2)$ which satisfy $\det(\mathbb{I}_2+U)=0$. Im$(J)$ can be identified with Minkowski space $\mathbb{R}^{1,3}$ and $U_\infty\equiv U(2)\setminus \text{Im}(J)=\{U\in U(2)\hspace{0.05cm}|\hspace{0.05cm} \det(\mathbb{I}_2+U)=0\}$ can be identified with the conformal boundary $\partial \mathbb{R}_c^{1,3}$ defined above. If $x_1$ and $x_2$ are points in Minkowski space represented by unitary matrices $U_1$ and $U_2$, their spacetime interval can be calculated
\begin{align}\label{25j}
(x_1-x_2)^2=-4\frac{\det(U_1-U_2)}{\det(\mathbb{I}_2+U_1)\det(\mathbb{I}_2+U_2)}.
\end{align}
We can define the spacetime interval between arbitrary unitary matrices $U_1,U_2$ using the formula (\ref{25j}), noting that the expression is of course infinite if either $U_1$ or $U_2$ are on $U_{\infty}$ so that the interval is not well-defined. To define a finite interval between all unitary matrices an unphysical metric $\langle U_1,U_2\rangle\equiv \det(U_1-U_2)$ must be used. It is however possible to define conformally invariant combinations of any four unitary matrices $U_1,U_2,U_3,U_4$ using the physical metric, as seen in section \ref{confplane}.
\newline
\newline
Conformal transformations can be implemented on $\mathbb{R}^{1,3}_c\simeq U(2)$ by $SU(2,2)$ fractional linear transformations \cite{Jadczyk_2011,1976JMP....17...24P}
\begin{align}
U\longrightarrow \mathfrak{C}_GU\equiv (AU+B)(CU+D)^{-1},\hspace{1cm} G=\begin{pmatrix}A&B\\C&D\end{pmatrix}\in SU(2,2),
\end{align}
\begin{align}
SU(2,2)=\{G\in SL(4,\mathbb{C})\hspace{0.05cm}\sp|\hspace{0.05cm}\sp G^{\dagger}KG=K\},\hspace{1cm} K=\begin{pmatrix}\mathbb{I}_2&0\\0&-\mathbb{I}_2\end{pmatrix}.
\end{align}
$\mathfrak{C}$ is the fourfold covering homomorphism $\mathfrak{C}: SU(2,2)\rightarrow\text{Conf}(\mathbb{R}^{1,3})$. $\mathfrak{C}$ acts trivially on the centre $Z(SU(2,2))=\{\pm\mathbb{I}_4,\pm i\mathbb{I}_4\}\simeq \mathbb{Z}_4$ and $\text{Conf}(\mathbb{R}^{1,3})\simeq SU(2,2)/ \mathbb{Z}_4$. The $SU(2,2)$ matrices which correspond to the familiar conformal transformations on Minkowski space are given in appendix \ref{su22}. The set $U_\infty$ can be parametrised explicitly
\begin{align}
U_{\infty}=\left\{\Omega(\theta,\phi)^{\dagger}\begin{pmatrix}-1 & 0\\0&-e^{2i\chi}\end{pmatrix}\Omega(\theta,\phi)\in U(2)\hspace{0.1cm}\Big|\hspace{0.1cm} \chi\in(-\pi/2,\pi/2]\right\},
\end{align}
where $\Omega(\theta,\phi)$ is as defined in (\ref{8r}). With the $U(2)$ formalism it can be checked that $\partial \mathbb{R}_c^{1,3}=U_{\infty}$ and $U_{\infty}=\mathcal{I}(L_0)\cup S^*$. $S^*$ is the \textit{null sphere at infinity} and can be parametrised
\begin{align}\label{u4}
S^*=\left\{\Omega(\theta,\phi)^{\dagger}\begin{pmatrix}-1 & 0\\0&1\end{pmatrix}\Omega(\theta,\phi)\in U(2)\right\}.
\end{align}
$S^*$ is stable under inversions $\mathcal{I}S^*=S^*$. It corresponds to the largest sphere ($\chi=\pi/2$) on the pinched torus (figure \ref{figg12}) and can be reached from the bulk of Minkowski space by taking limits to infinity along the light cone $L_0$. $\iota$ is the image of the origin under $\mathcal{I}$ and corresponds to the $U(2)$ matrix $U=-\mathbb{I}_2$.
\section{Conformal Cross-ratios and Conformal Planes}\label{confplane}
The group of all conformal transformations on compactified Minkowski space $\mathbb{R}^{1,3}_c$ is isomorphic to $O(2,4)/\mathbb{Z}_2$. It is generated by translations, rotations $R\in O(1,3)$, dilatations, and inversions $\mathcal{I}$. The \textit{conformal group} $\text{Conf}(\mathbb{R}^{1,3})$ is defined to be the connected component of the identity of this group, and is isomorphic to $SO^+(2,4)/\mathbb{Z}_2\simeq SU(2,2)/\mathbb{Z}_4$ \cite{book}. We will use the connected component $\text{Conf}(\mathbb{R}^{1,3})$, and explicitly refer to the extra discrete transformations such as $\mathcal{P}$ and $\mathcal{T}$ when needed. For $N\geq 4$ points $U_1, U_2,\dots, U_N\in\mathbb{R}^{1,3}_c$, expressed as unitary matrices, one can form $\frac{N}{2}(N-3)$ independent conformally invariant combinations known as \textit{conformal cross-ratios}. Here we are interested in $N=4$ points, where given $U_1,U_2,U_3,U_4\in\mathbb{R}^{1,3}_c$ the four point cross-ratios $u$ and $v$ can be defined
\begin{align}\label{10j}
u=\frac{\det(U_1-U_2)\det(U_3-U_4)}{\det(U_1-U_3)\det(U_2-U_4)},\hspace{1cm} v=\frac{\det(U_1-U_4)\det(U_2-U_3)}{\det(U_1-U_3)\det(U_2-U_4)}.
\end{align}
If none of the points are on $U_\infty$ then the points can be expressed as usual Minkowski vectors $x_1,x_2,x_3,x_4$ and the cross-ratios can be calculated
\begin{align}\label{9j}
u=\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2},\hspace{1cm} v=\frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}.
\end{align}
If one or more points are on $U_\infty$, then the cross-ratios can be calculated with (\ref{10j}) or by taking appropriate limits of (\ref{9j}). We define $V$ to be the set of $4-$tuples $\boldsymbol{U}=(U_1,U_2,U_3,U_4)$ of points in $\mathbb{R}^{1,3}_c$, such that the cross-ratios are non-singular\footnote{If none of the points are on $U_\infty$ this corresponds to none of the points being lightlike separated.}
\begin{align}\label{11j}
V=\left\{\boldsymbol{U}=(U_1,U_2,U_3,U_4)\hspace{0.1cm}\Big|\hspace{0.1cm} U_i\in\mathbb{R}^{1,3}_c,\hspace{0.1cm} u(\boldsymbol{U}),\hspace{0.05cm} v(\boldsymbol{U})\neq 0,\hspace{0.05cm}\infty\right\} .
\end{align}
Non-singularity of the cross-ratios implies the determinants $\det(U_i-U_j)$ are nonvanishing for $i<j$. Points $\boldsymbol{U}\in V$ will be referred to as \textit{configurations} of $\mathbb{R}^{1,3}_c$. If none of the points in $\boldsymbol{U}\in V$ are on $U_{\infty}$, then we call $\boldsymbol{U}\equiv\boldsymbol{w}$ a \textit{finite} configuration. A finite configuration $\boldsymbol{w}\in V$ can be written in terms of usual Minkowski vectors $\boldsymbol{w}=(x_1,x_2,x_3,x_4)$ with $x_i\in\mathbb{R}^{1,3}$. We denote the set of finite configurations by $V_f\subset V$. The cross-ratios $u$ and $v$ are commonly expressed in terms of other conformally invariant quantities $z$ and $\bar{z}$, defined by
\begin{align}
u=z\bar{z},\hspace{1cm} v=(1-z)(1-\bar{z}).
\end{align}
Note that since we defined $u,v\neq 0$ in (\ref{11j}), for configurations $\boldsymbol{U}\in V$ we have that $z,\bar{z}\neq 0,1$. $z$ and $\bar{z}$ can be extracted from $u$ and $v$ by the formula
\begin{align}\label{2h}
z,\bar{z}=\frac{1+u-v}{2}\pm\sqrt{\left(\frac{1-u-v}{2}\right)^2-uv}.
\end{align}
There are two things to note from (\ref{2h}). The first is that $z$ and $\bar{z}$ can in general be either real numbers or complex conjugate pairs. Secondly since $z$ and $\bar{z}$ appear as the roots of a quadratic equation, without loss of generality we can (and will) always take $\bar{z}\geq z$ when $z,\bar{z}\in\mathbb{R}$ and $\text{Im}(z)>0$ when $z,\bar{z}\in \mathbb{C}\setminus\mathbb{R}$. This is not necessary but we find it useful in practice. In what follows we will refer to $z$ and $\bar{z}$ for a given configuration as the \textit{conformal invariants} for that configuration. Define $V_{z,\bar{z}}$ as the subset of $V$ such that $\boldsymbol{U}=(U_1,U_2,U_3,U_4)\in V$ has conformal invariants $z,\bar{z}$, i.e.\
\begin{align}\label{12j}
V_{z,\bar{z}}=\left\{\boldsymbol{U}\in V\hspace{0.1cm}\spa \Big|\hspace{0.1cm}\spa u(\boldsymbol{U})=z\bar{z},\hspace{0.1cm} v(\boldsymbol{U})=(1-z)(1-\bar{z})\right\}.
\end{align}
Based on the possible values for $z$ and $\bar{z}$, we have the decomposition
\begin{align}\label{3h}
V=\bigcup_{\substack{z,\bar{z}\in \mathbb{R}\setminus \{0,1\}\\z\leq\bar{z}}}V_{z,\bar{z}}\cup\bigcup_{\substack{z\in \mathbb{H}\\\bar{z}=z^*}} V_{z,\bar{z}}\equiv V_{\mathbb{R}}\cup V_{\mathbb{C}},
\end{align}
where $\mathbb{H}\subset \mathbb{C}$ is the upper half-plane. Note that the `boundary' between $V_{\mathbb{R}}$ and $V_{\mathbb{C}}$ occurs when $z=\bar{z}$, and is included in $V_{\mathbb{R}}$. If $A:\mathbb{R}^{1,3}_c\rightarrow \mathbb{R}^{1,3}_c$ is a conformal transformation and $\boldsymbol{U}=(U_1,U_2,U_3,U_4)\in V$, we define $A$ to act on $\boldsymbol{U}$ by
\begin{align}
A:\boldsymbol{U}\rightarrow A\boldsymbol{U}\equiv(AU_1,\hspace{0.05cm} AU_2,\hspace{0.05cm} AU_3,\hspace{0.05cm} AU_4).
\end{align}
There are a number of questions one can ask about the decomposition (\ref{3h}). Ranging over configurations $\boldsymbol{U}\in V$ what are the possible values of $z$ and $\bar{z}$, i.e.\ which of the $V_{z,\bar{z}}$ are non-empty? Furthermore does the conformal group $\text{Conf}(\mathbb{R}^{1,3})$ act transitively on each $V_{z,\bar{z}}$? These questions have been studied in the case of null hexagon configurations in compactified Minkowski space \cite{dorn2012conformal}. We will briefly review these questions for Euclidean space before moving to Minkowski space.
\subsection{Euclidean Space}\label{confplaneeuc}
We consider compactified Euclidean space $\mathbb{R}^{4}_c=\mathbb{R}^4\cup\{\infty\}$. We define $V^E$ to be the set of $4-$tuples of non-coincident points in $\mathbb{R}^{4}_c$, and $V^E_{z,\bar{z}}$ to be the subset of these with conformal invariants $z$ and $\bar{z}$, in analogy with the Minkowskian definitions (\ref{11j}) and (\ref{12j}). In this case it is well-known that for any configuration $\boldsymbol{w}=(x_1,x_2,x_3,x_4)\in V^E$ we have that $\bar{z}=z^*$ and that $\text{Conf}(\mathbb{R}^{4})$ acts transitively on each $V^E_{z,\bar{z}}$.
\newline
\newline
Indeed we can map any $\boldsymbol{w}=(x_1,x_2,x_3,x_4)\in V^{E}$ into a canonical Euclidean configuration $\bar{\boldsymbol{w}}(r,\phi)\in V^{E}$, defined below, as follows. We assume the points are generic so that none of $x_1,x_2,x_3,x_4$ are $\infty$. We make the conformal transformation $A_1\equiv T_{-\mathcal{I}x_1}\hspace{0.05cm}\sp\mathcal{I}\hspace{0.05cm}\sp T_{-x_4}$. Under $A_1$ $\boldsymbol{w}$ transforms
\begin{align}
\boldsymbol{w}\rightarrow \boldsymbol{w}_1\equiv A_1\boldsymbol{w}=(0,y_2,y_3,\infty),
\end{align}
where $y_2\equiv A_1x_2, y_3\equiv A_1x_3$. The stabilising conformal subgroup of the points $0$ and $\infty$ consists of dilatations and rotations $SO(4)\subset SO(1,5)$. Let $|y_3|^2=\rho^2$. Then we can pick $R\in SO(4)$ to rotate $y_3\rightarrow (0,0,0,\rho)$ and then rescale by $1/\rho$ to reach the unit vector $e_4=(0,0,0,1)$. So we let $A_2\equiv D_{1/\rho}R$ and
\begin{align}
\boldsymbol{w}_1\rightarrow \boldsymbol{w}_2\equiv A_2\boldsymbol{w}_1=(0,\hspace{0.05cm}\sp z_2,\hspace{0.05cm}\sp e_4,\hspace{0.05cm}\sp\infty),
\end{align}
where $z_2\equiv A_2y_2$. The stabiliser of $0$, $e_4$, and $\infty$ is $SO(3)\subset SO(4)$ acting on the first 3 coordinates of $\mathbb{R}^4$. We pick $A_3\equiv R'\in SO(3)$ which maps $z_2$ to the 14 plane $z_2\rightarrow (r\sin\phi,0,0,r\cos\phi)$, with $r>0$ and $\phi\in [0,2\pi)$. Defining $A\equiv A_3A_2A_1$ we have shown that given any $\boldsymbol{w}\in V^E$
\begin{align}
\bar{\boldsymbol{w}}(r,\phi)\equiv A\boldsymbol{w}= (0,\hspace{0.05cm}\sp (r\sin\phi,0,0,r\cos\phi),\hspace{0.05cm}\sp e_4,\hspace{0.05cm}\sp\infty)
\end{align}
for some $r>0, \phi\in[0,2\pi)$. We call $\bar{\boldsymbol{w}}(r,\phi)$ the \textit{Euclidean conformal plane configuration} (figure \ref{figg19}).
\begin{figure}[h!]
\begin{center}
\begin{adjustbox}{max totalsize={.34\textwidth}{.27\textheight},center}
\begin{tikzpicture}{scale=0.2}
\draw[->] (-0.75,0)--(5,0);
\draw[->] (0,-0.5)--(0,3);
\node at (-0.2,3){$1$};
\node at (5.2,0){$4$};
\node at (-0.25,0.2){$x_1$};
\filldraw[blue] (0,0) circle (1.75pt);
\node at (1.5,0.25){$x_3$};
\filldraw[blue] (1.5,0) circle (1.75pt);
\filldraw[blue] (1.8,2) circle (1.75pt);
\draw[dotted,thick] (0,0)--(1.8,2);
\node at (1.8,2.25){$x_2$};
\node at (0.77,1.05){$r$};
\filldraw[blue] (4.8,2.75) circle (1.75pt);
\node at (4.8,3){$x_4=\infty$};
\small
\node at (0.33,0.18){$\phi$};
\draw[thick] (0.6,0) arc (0:40:0.69);
\normalsize
\end{tikzpicture}
\end{adjustbox}
\end{center}
\caption{Euclidean conformal plane configuration $\bar{w}(r,\phi)$.}
\label{figg19}
\end{figure}
\newline\newline
The conformal invariants of $\bar{\boldsymbol{w}}(r,\phi)$, and hence of $\boldsymbol{w}$, are calculated to be
\begin{align}
z=re^{i\phi},\hspace{1cm} \bar{z}=re^{-i\phi}.
\end{align}
If $\phi\in (\pi,2\pi)$ we can relabel $\phi\rightarrow 2\pi-\phi$ so that $\text{Im}(z)\geq0$. Therefore we can always take $\phi\in[0,\pi]$. If $\phi=0$ or $\pi$ then $z=\bar{z}\in \mathbb{R}\setminus \{0,1\}$. Therefore the decomposition (\ref{3h}) for Euclidean space reads
\begin{align}
V^E=\bigcup_{\substack{z=\bar{z}\in \mathbb{R}\setminus \{0,1\}}}V^E_{z,\bar{z}}\cup\bigcup_{\substack{z\in \mathbb{H}\\\bar{z}=z^*}} V^E_{z,\bar{z}}\equiv V_{\mathbb{R}}^E\cup V_{\mathbb{C}}^E.
\end{align}
Does $\text{Conf}(\mathbb{R}^4)$ act transitively on each $V^E_{re^{i\phi},re^{-i\phi}}$? The answer is easily yes. Given $\boldsymbol{w},\boldsymbol{y}\in V^E_{re^{i\phi},re^{-i\phi}}$, we can always find conformal transformations $A_w,A_y$ which map $\boldsymbol{w}$ and $\boldsymbol{y}$ to the conformal plane $A_w\boldsymbol{w}=A_y\boldsymbol{y}=\bar{\boldsymbol{w}}(r,\phi)$. Then if we let $A\equiv A_w^{-1}A_y$ we see that $\boldsymbol{w}=A\boldsymbol{y}$.
\subsection{Minkowski Space}\label{confplanemink}
The situation is much more subtle in compactified Minkowski space $\mathbb{R}^{1,3}_c$, where the squared differences $x_{ij}^2$ between finite points can be positive or negative. The conformal structure is much richer, and we will see that $z$ and $\bar{z}$ can be complex conjugate pairs, or independent real numbers. Given a finite configuration $\boldsymbol{w}=(x_1,x_2,x_3,x_4)\in V_f$, we define the \textit{kinematics} $k(\boldsymbol{w})$ of $\boldsymbol{w}$ to be the $6-$tuple
\begin{align}
k(\boldsymbol{w})\equiv (x_{12}^2,x_{34}^2,x_{23}^2,x_{14}^2,x_{13}^2,x_{24}^2).
\end{align}
As a slight abuse of terminology, we will also refer to the individual $x_{ij}^2$ as kinematics. We restrict to finite configurations $\boldsymbol{w}$ because the kinematics are not well-defined when at least one of the points is on $U_{\infty}$. We can also take a generic configuration to be finite. We define the \textit{sign} of the kinematics (and similarly for the sign of any ordered tuple)
\begin{align}
\text{sgn}(k(\boldsymbol{w}))\equiv (\hspace{0.05cm}\text{sgn}(x_{12}^2),\hspace{0.1cm}\text{sgn}(x_{34}^2),\hspace{0.1cm}\text{sgn}(x_{23}^2),\hspace{0.1cm}\text{sgn}(x_{14}^2),\hspace{0.1cm}\text{sgn}(x_{13}^2),\hspace{0.1cm}\text{sgn}(x_{24}^2)\hspace{0.05cm}),
\end{align}
where sgn$(x)=+$ if $x>0$ and sgn$(x)=-$ if $x<0$. By non-singularity of the cross-ratios (\ref{11j}) none of the squared differences for finite configurations vanish and so sgn$(x_{ij}^2)\in\{-,+\}$ for each $i<j$. Therefore depending on $\boldsymbol{w}\in V_f$ there are $2^6=64$ possibilities for sgn$(k(\boldsymbol{w}))$. We will concatenate signs as a shorthand, so for example sgn$(1,-3,2,5,5,2)=(+,-,+,+,+,+)\equiv (+-+++\hspace{0.02cm}+)$. We denote by $P_{ij}$ the operator which flips the sign of $x_{ij}^2$, so for example $P_{13} (+++++\hspace{0.02cm}+)=(++++-\hspace{0.02cm}+)$. Moreover we define $P\equiv \prod_{i<j}P_{ij}$ as the operator which flips the sign of all kinematics, so for example $P(+-+-+\hspace{0.05cm}-)=(-+-+-\hspace{0.05cm}+)$.
\newline
\newline
We again ask the question: Does $\text{Conf}(\mathbb{R}^{1,3})$ act transitively on each $V_{z,\bar{z}}$? In this case the answer is \textit{no} in general, and the existence of $A\in \text{Conf}(\mathbb{R}^{1,3})$ connecting finite configurations $\boldsymbol{w},\boldsymbol{y}\in V_{z,\bar{z}}$ depends on $k(\boldsymbol{w})$ and $k(\boldsymbol{y})$. Inversions and SCTs can change the kinematics of a configuration, in such a way that the cross-ratios are fixed. However they can only change the \textit{signs} of the kinematics in a restricted way. For the rest of this section we refine (\ref{3h}) such that $\text{Conf}(\mathbb{R}^{1,3})$ acts transitively on each set in the decomposition. This is possible provided the discrete transformations $\mathcal{P},\mathcal{T}$ are also used.
\newline
\newline
Let $b\in\mathbb{R}^{1,3}$, $\boldsymbol{w}=(x_1,x_2,x_3,x_4)\in V_f$. Under a special conformal transformation $\boldsymbol{w}\rightarrow C_b\boldsymbol{w}$ the kinematics transform
\begin{align}\label{10f}
x_{ij}^2\rightarrow \frac{x_{ij}^2}{\sigma_b(x_i)\sigma_b(x_j)},\hspace{1cm} \text{sgn}(x_{ij}^2) \rightarrow \frac{\text{sgn}(x_{ij}^2)}{\text{sgn}(\sigma_b(x_i)\sigma_b(x_j))},
\end{align}
where $\sigma_b(x)\equiv 1-2b\cdot x+b^2x^2$. If $\sigma_{b}(x_i)=0$ for some $i$ then the image of $x_i$ under $C_{b}$ is on $U_\infty$. Therefore as $b$ changes $\sigma_{b}(x_i)$ passing through $0$ corresponds to $x_i$ `crossing infinity'. For a fixed configuration $\boldsymbol{w}\in V_f$ and $b\in\mathbb{R}^{1,3}$ such that $C_b\boldsymbol{w}\in V_f$ there are up to $2^4=16$ possibilities\footnote{Note that it is not always 16, e.g.\ if the configuration contains $x=0$ then $\sigma_b(x)=1$ for all $b$.} for
\begin{align}\label{3f}
\text{sgn}(\sigma_b(\boldsymbol{w}))\equiv\text{sgn}(\hspace{0.05cm}\sigma_b(x_1),\hspace{0.1cm}\sigma_b(x_2),\hspace{0.1cm}\sigma_b(x_3),\hspace{0.1cm}\sigma_b(x_4)\hspace{0.05cm}),
\end{align}
and up to eight possibilities for the $6-$tuple $\text{sgn}(k_b(\boldsymbol{w}))$, where $k_b(\boldsymbol{w})\equiv$
\begin{align}\label{2f}
(\sigma_b(x_1)\sigma_b(x_2), \sigma_b(x_3)\sigma_b(x_4),\sigma_b(x_2)\sigma_b(x_3),\sigma_b(x_1)\sigma_b(x_4),\sigma_b(x_1)\sigma_b(x_3),\sigma_b(x_2)\sigma_b(x_4)).
\end{align}
These eight possibilities can be deduced easily by calculating $\text{sgn}(k_b(\boldsymbol{w}))$ for each of the sixteen possibilities for (\ref{3f}). Letting $S\equiv(+++++\hspace{0.08cm}+)$, these eight possibilities are
\begin{align}\label{4f}
S,\hspace{1cm} P_{12}P_{14}P_{13}S,\hspace{1cm} P_{12}P_{23}P_{24}S,\hspace{1cm} P_{34}P_{23}P_{13}S,\hspace{1cm} P_{34}P_{14}P_{24}S,
\end{align}
\begin{align*}
P_{23}P_{14}P_{13}P_{24}S,\hspace{1cm} P_{12}P_{34}P_{23}P_{14}S,\hspace{1cm} P_{12}P_{34}P_{13}P_{24}S.
\end{align*}
For example, if $\text{sgn}(\sigma_b(\boldsymbol{w}))=(+-+\hspace{0.08cm}-)$ or $(-+-\hspace{0.08cm}+)$ then $\text{sgn}(k_b(\boldsymbol{w}))=(----+\hspace{0.08cm}+)=P_{12}P_{34}P_{23}P_{14}S$. Denoting these eight elements in the order they appear in (\ref{4f}) by $g_i$ for $i=0,1,\dots, 7$, collecting them in a set $G=\{g_i\}_{i=0}^7$, and introducing the composition law\footnote{For example, $(++---\hspace{0.08cm}-)\cdot (+-+-+\hspace{0.08cm}-)=(+--+-\hspace{0.08cm}+)$.}
\begin{align}
\small \text{sgn}(a_1, a_2, a_3, a_4, a_5, a_6)\cdot \text{sgn}(b_1, b_2, b_3, b_4, b_5,b_6)\equiv \text{sgn}(a_1b_1, a_2b_2, a_3b_3, a_4b_4, a_5b_5, a_6b_6)
\end{align}
\normalsize
then $(G,\cdot)$ is an abelian group of order eight, with identity element $g_0=S$. Note that $g_i^2=g_0$ for all $i=0,\dots,7$, and so we conclude that $G\simeq \mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$. This composition law is compatible with the action of SCTs on any initial kinematics, in the sense that
\begin{align*}
\text{sgn}(k(C_b\boldsymbol{w}))= \text{sgn}(k_b(\boldsymbol{w})) \cdot \text{sgn}(k(\boldsymbol{w})),
\end{align*}
for $b\in\mathbb{R}^{1,3}$ and $\boldsymbol{w},C_b\boldsymbol{w}\in V_f$. Therefore the group $G$ encodes the action of SCTs $C_b$ on $\text{sgn}(k(\boldsymbol{w}))$. We define the eight sets
\begin{align}
K_1\equiv G,\hspace{1cm} K_2\equiv P_{12}G,\hspace{1cm} K_3\equiv P_{23}G,\hspace{1cm} K_4\equiv P_{13}G,
\end{align}
\begin{align*}
\bar{K}_i\equiv PK_i,\hspace{1cm} i=1,2,3,4,
\end{align*}
where $P_{12}G\equiv\{P_{12}g_i\}_{i=0}^7$, and similarly for the others. Each of these sets has eight elements, and together they account for the $8\times 8=64$ different possibilities for $\text{sgn}(k(\boldsymbol{w}))$ for a finite configuration $\boldsymbol{w}$. They are listed explicitly in appendix \ref{kin}. Furthermore they are each invariant under the action\footnote{The action of $G$ on these sets is defined analogously to the action of $G$ on itself.} of $G$, and so the eight sets of signs of kinematics $K_i, \bar{K}_i$ are invariant under SCTs\footnote{A similar argument holds for inversions.}. Therefore given a configuration $\boldsymbol{w}\in V_f$ such that $\text{sgn}(k(\boldsymbol{w}))\in K_i$ (or $\bar{K}_i$) for some $i$ then for any $b\in \mathbb{R}^{1,3}$ such that $C_b\boldsymbol{w}\in V_f$ we have that $\text{sgn}(k(C_b\boldsymbol{w}))\in K_i$ (or $\bar{K}_i$) for the same $i$. It is possible to assign a set $K_i$ (or $\bar{K}_i$) to non-finite configurations. For configurations $\boldsymbol{U}\in V$ which are not finite it is always possible to make an infinitesimal SCT $C_b$ such that $C_b\boldsymbol{U}$ is finite. Then sgn$(k(C_b\boldsymbol{U}))\in K_i$ (or $\bar{K}_i$) is well-defined, and we set sgn($k(\boldsymbol{U}))\in K_i$ (or $\bar{K}_i$) for the same $i$, even though $k(\boldsymbol{U})$ is not well-defined. We define the refined subsets of $V_{z,\bar{z}}$
\begin{align}
V_{i,z,\bar{z}}\equiv\{\boldsymbol{U}\in V_{z,\bar{z}}\hspace{0.1cm} | \hspace{0.1cm} \text{sgn}(k(\boldsymbol{U}))\in K_i\},\hspace{1cm} \bar{V}_{i,z,\bar{z}}\equiv\{\boldsymbol{U}\in V_{z,\bar{z}}\hspace{0.1cm} | \hspace{0.1cm} \text{sgn}(k(\boldsymbol{U}))\in \bar{K}_i\},
\end{align}
which are well-defined and each separately invariant under the conformal group by the above discussion. Investigating the sets $V_{i,z,\bar{z}}$ and $\bar{V}_{i,z,\bar{z}}$ in detail we find that there are different constraints on $z$ and $\bar{z}$ for each $i$. The first important fact is that a configuration $\boldsymbol{U}\in V$ can only have $z,\bar{z}\in\mathbb{C}\setminus\mathbb{R}$ if $\boldsymbol{U}\in \bar{V}_{1,z,\bar{z}}$, and so
\begin{align}\label{6h}
V_{\mathbb{C}}=\bigcup_{\substack{z\in \mathbb{H}\\ \bar{z}=z^*}}\bar{V}_{1,z,\bar{z}}.
\end{align}
Note that $\bar{K}_1$ includes the Euclidean sign assignment $PS=(-----\hspace{0.12cm}-)$, and the seven other possible signs of kinematics are found by acting with $G$ on $PS$. (\ref{6h}) is proven using the analogue of conformal planes in Minkowski space in appendix \ref{confplanepseudoeuc}. Furthermore given $\boldsymbol{U}\in V_{i,z,\bar{z}}$ (or $\bar{V}_{i,z,\bar{z}}$) with $z,\bar{z}\in \mathbb{R}$ then $z$ and $\bar{z}$ are constrained to different intervals of $\mathbb{R}$ depending on $i$ (table \ref{table1}). For example, let $\boldsymbol{U}\in V_{4,z,\bar{z}}$ so that sgn($k(\boldsymbol{U}))\in K_4$. Going through all the possibilities for $\text{sgn}(k(\boldsymbol{U}))$ in table \ref{table7} we deduce that $u(\boldsymbol{U})<0$ and $v(\boldsymbol{U})<0$. Since $u=z\bar{z},\hspace{0.05cm} v=(1-z)(1-\bar{z}),$ and we take $\bar{z}\geq z$ we conclude that $z\in(-\infty,0),\hspace{0.05cm} \bar{z}\in(1,\infty)$. Conversely given a configuration $\boldsymbol{U}\in V$ with a known $z$ and $\bar{z}$ we immediately have information about the signs of the kinematics of $\boldsymbol{U}$. For example, if we are told $\boldsymbol{U}$ has $z=-1,\bar{z}=2$, we see from table \ref{table1} that there are two possibilities $\boldsymbol{U}\in V_{4,-1,2}$ or $\bar{V}_{4,-1,2}$, and so $\text{sgn}(k(\boldsymbol{U}))\in K_{4}$ or $\bar{K}_4$.
\setlength{\extrarowheight}{1.5pt}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|}
\hline
$i$ & $z,\bar{z}$ for non-empty $V_{i,z,\bar{z}},\hspace{0.05cm}\sp\bar{V}_{i,z,\bar{z}}$\\ [0.2ex]
\hline
$1$ & $z,\bar{z}\in (-\infty,0)\hspace{0.1cm}\text{OR}\hspace{0.1cm} (0,1)\hspace{0.1cm}\text{OR}\hspace{0.1cm} (1,\infty)$\\
\hline
$2$ & $z\in(-\infty,0),\bar{z}\in (0,1)$\\
\hline
$3$ & $z\in(0,1),\bar{z}\in (1,\infty)$\\
\hline
$4$ & $z\in(-\infty,0),\bar{z}\in (1,\infty)$\\
\hline
\end{tabular}
\caption{$z,\bar{z}$ such that $V_{i,z,\bar{z}}$ is non-empty for $z,\bar{z}\in\mathbb{R}$. Note for $i=1$ we always take $z\leq \bar{z}$.}
\label{table1}
\end{center}
\end{table}
Therefore we define the sets
\begin{align}
V_1=\bigcup_{\substack{z,\bar{z}\in (-\infty,0)\\z,\bar{z}\in (0,1)\\z,\bar{z}\in (1,\infty)\\z\leq\bar{z}}}V_{1,z,\bar{z}}, \hspace{1cm} \bar{V}_1=\bigcup_{\substack{z,\bar{z}\in (-\infty,0)\\z,\bar{z}\in (0,1)\\z,\bar{z}\in (1,\infty)\\z\leq\bar{z}}}\bar{V}_{1,z,\bar{z}},
\end{align}
\begin{align}
V_2=\bigcup_{\substack{z\in (-\infty,0)\\\bar{z}\in (0,1)}}V_{2,z,\bar{z}}, \hspace{1cm} \bar{V}_2=\bigcup_{\substack{z\in (-\infty,0)\\\bar{z}\in (0,1)}}\bar{V}_{2,z,\bar{z}},
\end{align}
\begin{align}
V_3=\bigcup_{\substack{z\in (0,1)\\\bar{z}\in (1,\infty)}}V_{3,z,\bar{z}}, \hspace{1cm} \bar{V}_3=\bigcup_{\substack{z\in (0,1)\\\bar{z}\in (1,\infty)}}\bar{V}_{3,z,\bar{z}},
\end{align}
\begin{align}
V_4=\bigcup_{\substack{z\in (-\infty,0)\\\bar{z}\in (1,\infty)}}V_{4,z,\bar{z}}, \hspace{1cm} \bar{V}_4=\bigcup_{\substack{z\in (-\infty,0)\\\bar{z}\in (1,\infty)}}\bar{V}_{4,z,\bar{z}},
\end{align}
and deduce the decomposition\footnote{A similar decomposition appeared recently in \cite{qiao2020classification} by Jiaxin Qiao while this was being written.}
\begin{align}\label{7h}
V=\bigcup_{i=1}^4 (V_i\cup\bar{V}_i)\cup V_{\mathbb{C}}.
\end{align}
The question remains: Does $\text{Conf}(\mathbb{R}^{1,3})$ act transitively on each non-empty $V_{i,z,\bar{z}}$ (and $\bar{V}_{i,z,\bar{z}}$)? The answer is yes \textit{up to the discrete transformations $\mathcal{P},\hspace{0.05cm}\mathcal{T}$}. The proof relies on Minkowskian conformal planes.
\newline
\newline
We define the five \textit{Minkowskian conformal plane} configurations $\boldsymbol{w}_{\mathbb{C}}(r,\phi)$ and $\boldsymbol{w}_{bc}(a,\eta)$ for $b,c\in\{+,-\}$ in table \ref{table2}. They are expressed in terms of the unit vectors $e_0=(1,0,0,0)$ and $e_3=(0,0,0,1)$.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Configuration & $(x_1,x_2,x_3,x_4)$ & $z,\bar{z}$\\ [0.2ex]
\hline
\hline
$\boldsymbol{w}_{\mathbb{C}}(r,\phi)$ & $(0,\hspace{0.05cm}\sp r(0,\sin\phi,0,\cos\phi),\hspace{0.05cm}\sp e_3,\hspace{0.05cm}\sp \iota)$ & $re^{i\phi},re^{-i\phi}$\\
\hline
$\boldsymbol{w}_{++}(a,\eta)$ & $(0,\hspace{0.05cm}\sp a(\cosh\eta,0,0,\sinh\eta),\hspace{0.05cm}\sp e_0,\hspace{0.05cm}\sp \iota)$ &$
ae^{-\eta},ae^{\eta}$\\
\hline
$\boldsymbol{w}_{--}(a,\eta)$ & $(0,\hspace{0.05cm}\sp a(\sinh\eta,0,0,\cosh\eta),\hspace{0.05cm}\sp e_3,\hspace{0.05cm}\sp \iota)$ &$ae^{-\eta},ae^{\eta}$\\
\hline
$\boldsymbol{w}_{-+}(a,\eta)$ & $(0,\hspace{0.05cm}\sp a(\sinh\eta,0,0,\cosh\eta),\hspace{0.05cm}\sp e_0,\hspace{0.05cm}\sp \iota)$ &$-ae^{-\eta},ae^{\eta}$\\
\hline
$\boldsymbol{w}_{+-}(a,\eta)$ & $(0,\hspace{0.05cm}\sp a(\cosh\eta,0,0,\sinh\eta),\hspace{0.05cm}\sp e_3,\hspace{0.05cm}\sp \iota)$ &$-ae^{-\eta},ae^{\eta}$\\
\hline
\end{tabular}
\caption{Minkowskian conformal plane configurations.}
\label{table2}
\end{center}
\end{table}
Similarly to the Euclidean case we always find $A\in \text{Conf}(\mathbb{R}^{1,3})$ and possibly a discrete transformation $\mathcal{P},\hspace{0.05cm}\mathcal{T},$ or $\mathcal{P}\mathcal{T}$ to map configurations in the nine sets in (\ref{7h}) to one of the five configurations in table \ref{table2}. The case of $V_{\mathbb{C}}$ is totally analogous to the Euclidean case, and indeed any configuration $\boldsymbol{U}\in V_{\mathbb{C}}$ with $z=re^{i\phi}, \bar{z}=re^{-i\phi}$ can be conformally mapped (no need for discrete transformations) to the pseudo-Euclidean configuration $\boldsymbol{w}_{\mathbb{C}}(r,\phi)\in\bar{V}_{1,z,\bar{z}}$ for $r>0, \phi\in(0,\pi)$, see appendix \ref{confplanepseudoeuc} for some details. The eight remaining sets in (\ref{7h}) can be conformally mapped to the four sets $\boldsymbol{w}_{bc}(a,\eta)$. This is a bit more subtle, and care must be taken to restrict the range of $a$ and $\eta$ in such a way that each $\boldsymbol{U}\in V_{i,z,\bar{z}}$ (or $\bar{V}_{i,z,\bar{z}}$) is mapped to exactly one configuration $\boldsymbol{w}_{bc}(a,\eta)$ for the appropriate $b,c$. This can be done provided the discrete transformations $\mathcal{P},\hspace{0.05cm}\mathcal{T}$ are used. The results are summarised in table \ref{table3}.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$V$ & Mapped to & Configuration bounds & $z,\bar{z}$ bounds\\ [0.2ex]
\hline
$V_{\mathbb{C}}$ & $\boldsymbol{w}_{\mathbb{C}}(r,\phi)$ & $r>0, \phi\in(0,\pi)$ & $z\in \mathbb{H}$, $\bar{z}=z^*$\\
\hline
& & $a\in(0,1),\hspace{0.05cm} e^\eta\in(1,1/a)$ & $0<z\leq\bar{z}<1$ \\
$V_1$ & $\boldsymbol{w}_{++}(a,\eta)$ &$a\in(1,\infty),\hspace{0.05cm} e^\eta\in(1,a)$ & $1<z\leq\bar{z}<\infty$ \\
& & $a\in(-\infty,0),\hspace{0.05cm} e^\eta\in(1,\infty)$ & $-\infty<z\leq\bar{z}<0$\\
\iffalse\hline
$\bar{V}_3$ & $\boldsymbol{w}_{++}(a,\eta)$ &$a\in(0,\infty),e^\eta\in(1,\text{max}(a,1/a))$& $0<z<1<\bar{z}<\infty$ \\
\hline
& & & $a\in(0,1),\hspace{0.05cm} e^\eta\in(1,1/a)$ & $0<z\leq\bar{z}<1$ \\
$\bar{V}_1$ & $\boldsymbol{w}_{--}(a,\eta)$ & $a\in(1,\infty),\hspace{0.05cm} e^\eta\in(1,a)$ & $1<z\leq\bar{z}<\infty$ \\
& & & $a\in(-\infty,0),\hspace{0.05cm} e^\eta\in(1,\infty)$ & $-\infty<z\leq\bar{z}<0$\\ \fi
\hline
$V_{2}$ & $\boldsymbol{w}_{-+}(a,\eta)$ & $a\in(0,\infty),e^{\eta}\in(0,1/a)$&$-\infty<z<0<\bar{z}<1$\\
\hline
$V_{3}$ & $\boldsymbol{w}_{--}(a,\eta)$ &$a\in(0,\infty),e^\eta\in(\text{max}(a,1/a),\infty)$& $0<z<1<\bar{z}<\infty$ \\
\hline\iffalse
$\bar{V}_{4}$ & $\boldsymbol{w}_{-+}(a,\eta)$ &$-ae^{-\eta},ae^{\eta}$ & $a\in(0,\infty),e^{-\eta}\in(0,1/a)$&$-\infty<z<0<\bar{z}<1$\\
\hline
$\bar{V}_{2}$ & $\boldsymbol{w}_{+-}(a,\eta)$ &$-ae^{-\eta},ae^{\eta}$ &\\\fi
\hline
$V_{4}$ & $\boldsymbol{w}_{+-}(a,\eta)$ & $a\in(0,\infty),e^{\eta}\in(1/a,\infty)$&$-\infty<z<0<1<\bar{z}<\infty$\\
\hline
\end{tabular}
\caption{Conformal plane structure for $V_{\mathbb{C}}$ and $V_i$. For $\bar{V}_i$ the only change is that the signs on $\boldsymbol{w}_{bc}(a,\eta)$ reverse. For example, $\bar{V}_1$ gets mapped to $\boldsymbol{w}_{--}(a,\eta)$, and all other columns with respect to $V_1$ are unchanged.}
\label{table3}
\end{center}
\end{table}
\newline
We explain the argument for $V_2$, so let $\boldsymbol{w}=(x_1,x_2,x_3,x_4)\in V_2$. Without loss of generality we can take $\boldsymbol{w}$ to be a finite configuration. If $\boldsymbol{w}$ is not a finite configuration we can first make a small SCT such that it is finite. As in the Euclidean case we define the conformal transformation\footnote{We can make $A_1$ explicitly connected to the identity by making an extra inversion and using $C_b=\mathcal{I}T_{-b}\mathcal{I}$} $A_1\equiv T_{-\mathcal{I}x_1}\mathcal{I}T_{-x_4}$ and act on $\boldsymbol{w}$
\begin{align}
\boldsymbol{w}\rightarrow \boldsymbol{w}_1\equiv A_1\boldsymbol{w}=(0,y_2,y_3,\iota),
\end{align}
where $y_2\equiv A_1x_2=\mathcal{I}(x_{24})-\mathcal{I}(x_{14}),\hspace{0.05cm}\sp y_3\equiv A_1x_3=\mathcal{I}(x_{34})-\mathcal{I}(x_{14})$. The stabiliser of $0$ and $\iota$ is generated by dilatations and Lorentz rotations $SO^+(1,3)\subset SO^+(2,4)$. How we proceed next depends on the signs of the remaining free kinematics. We compute
\begin{align}
(y_{2}^2,y_{3}^2,y_{23}^2)=\left(\frac{x_{12}^2}{x_{14}^2x_{24}^2},\frac{x_{13}^2}{x_{14}^2x_{34}^2},\frac{x_{23}^2}{x_{24}^2x_{34}^2}\right),
\end{align}
where we used the property that under inversions the kinematics transform
\begin{align}
\mathcal{I}:x_{ij}^2\longrightarrow \frac{x_{ij}^2}{x_i^2x_j^2}.
\end{align}
Since $\boldsymbol{w}\in V_2$ we have information about the signs of the kinematics, namely that $\text{sgn}(k(\boldsymbol{w}))\in K_2$ (see table \ref{table7}). For example, we could have $\text{sgn}(k(\boldsymbol{w}))=P_{12}S=(-++++\hspace{0.05cm}\sp+)$. In this case we have that
\begin{align}\label{8h}
\text{sgn}(y_{2}^2,y_{3}^2,y_{23}^2)=(-++).
\end{align}
In fact going through all the cases in table \ref{table7} it turns out that (\ref{8h}) holds for any $\text{sgn}(k(\boldsymbol{w}))\in K_2$, and hence $\boldsymbol{w}\in V_2$. We have that $y_3^2=c^2>0$, so we can use a Lorentz rotation $L$ to map $y_3\rightarrow (\pm c,0,0,0)$ and then rescale to $\pm e_0$. Defining $A_2\equiv D_{1/c}L$ we map
\begin{align}
\boldsymbol{w}_1\rightarrow \boldsymbol{w}_2^\pm\equiv A_2\boldsymbol{w}_1=(0,\hspace{0.05cm}\sp z_2,\hspace{0.05cm}\sp\pm e_0,\hspace{0.05cm}\sp\iota),
\end{align}
where $z_2\equiv A_2y_2$. The stabiliser of $0, \pm e_0,$ and $\iota$ is generated by rotations $SO(3)$ acting on the Euclidean coordinates of $\mathbb{R}^{1,3}$. We use an $SO(3)$ rotation $R$ to eliminate the second and third component of $z_2$, so $z_2\rightarrow (z_2^0,0,0,z_2^3)$. We know from (\ref{8h}) that $(z_2^0)^2-(z_2^3)^2<0$, so we map
\begin{align}\label{9h}
\boldsymbol{w}_2^{\pm}\rightarrow \bar{\boldsymbol{w}}^{\pm}(a,\eta)\equiv R\boldsymbol{w}_{2}^\pm=(0,\hspace{0.05cm}\sp a(\sinh\eta,\sp0,\sp0,\hspace{0.05cm}\cosh\eta),\hspace{0.05cm}\sp\pm e_0,\hspace{0.05cm}\sp\iota),
\end{align}
for some $a\in \mathbb{R}\setminus \{0\}, \eta\in \mathbb{R}$. We compute the conformal invariants of $
\bar{\boldsymbol{w}}^{\pm}(a,\eta)$ to be
\begin{align}\label{10h}
z=\text{min}(-ae^{\mp\eta},ae^{\pm\eta}),\hspace{1cm} \bar{z}=\text{max}(-ae^{\mp\eta},ae^{\pm\eta}).
\end{align}
Note we have the issue that starting with a configuration $\boldsymbol{w}\in V_2$ we have two possible configurations (\ref{9h}) we can end up with, $\bar{\boldsymbol{w}}^{+}(a,\eta)$ and $\bar{\boldsymbol{w}}^-(a,\eta)$. Moreover we know from table \ref{table1} that $z\in (-\infty,0), \bar{z}\in (0,1)$. We need our final configuration $\boldsymbol{w}_{-+}(a,\eta)$ to contain each possible set of conformal invariants exactly once. This is because we want to be sure we are mapping all $\boldsymbol{w}\in V_{2,z,\bar{z}}$ to the exact same configuration, to prove that $\text{Conf}(\mathbb{R}^{1,3})$ acts transitively on $V_{2,z,\bar{z}}$. A priori there could be multiple configurations with the same $z,\bar{z}$ in our final $\boldsymbol{w}_{-+}(a,\eta)$. To resolve these issues we need to use the discrete transformations $\mathcal{P}$ and $\mathcal{T}$. We first note that $\bar{\boldsymbol{w}}^+(a,\eta)=\mathcal{T}\bar{\boldsymbol{w}}^-(a,-\eta)$. Therefore if we end up at $\bar{\boldsymbol{w}}^-(a,\eta)$ we simply apply a time-reversal transformation and relabel $\eta\rightarrow \eta'\equiv -\eta$ to arrive at $\bar{\boldsymbol{w}}^+(a,\eta)$. We can invert (\ref{10h}) to recover $a$ and $\eta$ for $\bar{\boldsymbol{w}}^+(a,\eta)$
\begin{align}{\label{11h}}
a=\pm\sqrt{-z\bar{z}},\hspace{1cm} \eta=\pm\frac{1}{2}\log(-\bar{z}/z).
\end{align}
We see from (\ref{11h}) that indeed multiple configurations $\bar{\boldsymbol{w}}^+(a,\eta)$ give the same $z,\bar{z}$, which we want to avoid. We note that $\bar{\boldsymbol{w}}^+(a,\eta)=\mathcal{P}\bar{\boldsymbol{w}}^+(-a,-\eta)$. Therefore any configuration with $a<0$ we can apply a parity transformation and relabel $\eta\rightarrow \eta'=-\eta$ at no cost, so we can assume $a>0$. In this case we have that
\begin{align}\label{12h}
z=-ae^{-\eta},\hspace{1cm} \bar{z}=ae^{\eta},\hspace{1cm} a=\sqrt{-z\bar{z}},\hspace{1cm} \eta=\frac{1}{2}\log(-\bar{z}/z).
\end{align}
From (\ref{12h}) we see that we have exactly one configuration $\bar{\boldsymbol{w}}^{+}(a>0,\eta)\equiv \boldsymbol{w}_{-+}(a,\eta)$ for each $z,\bar{z}$, as required. To get $z,\bar{z}$ in the correct range $-\infty<z<0<\bar{z}<1$, or equivalently enforce the $y_{23}^2>0$ condition of (\ref{8h}), there is a further constraint $e^{\eta}<1/a$. The final configuration $\boldsymbol{w}_{-+}(a,\eta)$ in terms of $z,\bar{z}$ is
\begin{align}
\boldsymbol{w}_{-+}(z,\bar{z})=\left(0,\hspace{0.05cm}\sp \left(\frac{\bar{z}+z}{2},0,0,\frac{\bar{z}-z}{2}\right),\hspace{0.05cm}\sp e_0,\hspace{0.05cm}\sp \iota\right),
\end{align}
\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}[thick,scale=1, every node/.style={scale=0.81}]
\draw [red,dotted,fill=green] (-5.5,0.7)-- (-4.75,1.45) -- (-1.8,-1.5) -- (-2.5,-2.3) --cycle ;
\draw[->] (-6.25,0.7)--(-1.5,0.7);
\draw[->] (-5.5,-2)--(-5.5,2.5);
\draw[thick] (1.33,0) ellipse (1.3 and 1);
\draw[dotted,red] (-6,1.2)--(-2.5,-2.3);
\draw[dotted,red] (-6,0.2)--(-4.2,2);
\draw[dotted,purple] (-6,2.7)--(-1.8,-1.5);
\draw[dotted,purple] (-6,1.7)--(-5,2.7);
\filldraw[blue] (-5.5,2.2) circle (1.3pt);
\filldraw[blue] (-5.5,0.7) circle (1.3pt);
\filldraw[blue] (-3.5,-0.7) circle (1.3pt);
\draw[thick] (0,0) arc (180:-180:2);
\draw[red,thick] (2.62,0) to[out=-50,in=-130] (4,0);
\draw[red,dotted] (2.62,0) to[out=50,in=130] (4,0);
\draw[red,thick] (2.1,0.8) to[out=0,in=-20] (3.05,1.7);
\draw[red,dotted] (3.05,1.7) to[out=180,in=160] (2.1,0.8);
\draw[red,dotted] (2.1,-0.8) to[out=0,in=20] (3.05,-1.7);
\draw[red,thick] (3.05,-1.7) to[out=180,in=-160] (2.1,-0.8);
\draw[red,thick] (1.3,1) to[out=10,in=-10] (1.2,1.85);
\draw[red,dotted] (1.3,1) to[out=170,in=190] (1.2,1.85);
\draw[red,dotted] (1.3,-1) to[out=10,in=-10] (1.2,-1.85);
\draw[red,thick] (1.3,-1) to[out=170,in=140] (1.2,-1.82);
\filldraw[blue] (0.02,0) circle (1.3pt);
\footnotesize
\node at (-0.6,0){$x_4=\iota$};
\node at (-5.8,2.2){$x_3$};
\node at (-5.86,0.83){$x_1$};
\node at (-3.2,-0.7){$x_2$};
\node at (-1.3,0.715){$3$};
\node at (-5.495,2.72){$0$};
\normalsize
\end{tikzpicture}
\end{center}
\caption{Minkowskian conformal plane configuration $\boldsymbol{w}_{-+}(a,\eta)$ for $V_2$, $z\in (-\infty,0),\hspace{0.05cm}\bar{z}\in(0,1)$. $x_2$ can be anywhere in green region depending on $z,\bar{z}$.}
\label{figg22}
\end{figure}\noindent and is shown in figure \ref{figg22}.
\newline
These arguments can be repeated analogously for the other sets $V_{i,z,\bar{z}}$ and $\bar{V}_{i,z,\bar{z}}$ to show that up to the discrete transformations $\mathcal{P},\mathcal{T}$ all configurations in these sets are conformally equivalent to a single Minkowskian conformal plane configuration $\boldsymbol{w}_{b,c}(a,\eta)$ or $\boldsymbol{w}_\mathbb{C}(r,\phi)$, as summarised in table \ref{table3}. Note that for $\bar{V}_{1,z,\bar{z}}$ there are two cases, one where $z,\bar{z}\in\mathbb{R}$ and one where $z,\bar{z}\in\mathbb{C}\setminus \mathbb{R}$. Therefore up to the discrete transformations $\mathcal{P},\mathcal{T}$ $\text{Conf}(\mathbb{R}^{1,3})$ acts transitively on each non-empty $V_{i,z,\bar{z}}$ and $\bar{V}_{i,z,\bar{z}}$. Given configurations $\boldsymbol{U}_1,\boldsymbol{U}_2\in V_{i,z,\bar{z}}$ (or $\bar{V}_{i,z,\bar{z}}$) we deduce from the above the existence of $A_i\in \text{Conf}(\mathbb{R}^{1,3})$ and discrete transformations $\mathcal{D}_i\in\{\mathbb{I},\mathcal{P},\mathcal{T},\mathcal{P}\mathcal{T}\}$ for $i=1,2$ such that $\mathcal{D}_1A_1\boldsymbol{U}_1=\mathcal{D}_2A_2\boldsymbol{U}_2=\bar{\boldsymbol{w}}(z,\bar{z})$ for exactly one Minkowskian conformal plane configuration $\bar{\boldsymbol{w}}(z,\bar{z})$ given in table \ref{table2}. Then if we let $A\equiv (\mathcal{D}_1A_1)^{-1}\mathcal{D}_2A_2$ we see that $\boldsymbol{U}_1=A\boldsymbol{U}_2$.
\newline
\newline
Overall we have shown that all configurations $\boldsymbol{U}\in V_{z,\bar{z}}$ with $z,\bar{z}\in\mathbb{C}\setminus\mathbb{R}$ are conformally equivalent, and that for $\boldsymbol{U}\in V_{z,\bar{z}}$ with $z,\bar{z}\in\mathbb{R}$ there are two conformal equivalence classes of configurations (up to $\mathcal{P},\hspace{0.05cm}\mathcal{T}$) exchanged by a reversal of kinematics $V_{i,z,\bar{z}}\rightarrow \bar{V}_{i,z,\bar{z}}=PV_{i,z,\bar{z}}$ for a fixed $i$.
\section{Symmetries of the Box Integral}\label{symmetry}
Let $\boldsymbol{w}=(x_1,x_2,x_3,x_4)\in V_f$ be a finite configuration. The dual conformal box integral in Minkowski space is defined
\begin{align}\label{5b}
I(\boldsymbol{w})=\int\frac{d^4x_5}{i\pi^2}\frac{x_{13}^2x_{24}^2}{(x_{15}^2+ i\epsilon)(x_{25}^2+ i\epsilon)(x_{35}^2 + i\epsilon)(x_{45}^2+ i\epsilon)},
\end{align}
where $+i\epsilon$ indicates the Feynman prescription for propagators in Minkowski space ($\epsilon$ is always taken to be a positive infinitesimal quantity). The chosen prescription tells us on which side of the branch cut we must evaluate any multivalued functions that appear upon computing the integral. It is important to take $\boldsymbol{w}$ to be a finite configuration because the integral is not well-defined when any of the points\footnote{The only exception is the case where a single point is at $\iota$. In this case the integral is well-defined.} $x_i$ are on $U_{\infty}$. In the next section we will consider specific lightlike limits of (\ref{5b}) such that two of the external points are sent to $\mathscr{I}$. However the result depends in general on the direction these points are sent to $\mathscr{I}$, so that it is not possible to define the integral if one or more points are exactly on $\mathscr{I}$. Configurations with one or more points on $\mathscr{I}$ are singular and are exactly the configurations where the value of (\ref{5b}) can change under infinitesimal SCTs.
\newline
\newline
We study the symmetries of the integral (\ref{5b}) directly in Minkowski space. There are two main different representations useful for studying these symmetries, given in \cite{Duplan_i__2002}. The first is the Feynman parametric representation
\begin{align}\label{9b}
I(\boldsymbol{w})= I'(x_{ij}^2)\equiv x_{13}^2x_{24}^2\int_0^1 d\boldsymbol{y}\frac{\delta(1-y_1-y_2-y_3-y_4)}{(\sum_{j<k}y_jy_kx_{jk}^2+ i\epsilon)^2},
\end{align}
where $\int_0^{1}d\boldsymbol{y}\equiv \prod_{i=1}^4\int_0^1 dy_i$. Note that $I'$ is manifestly a function of the kinematics $x_{ij}^2$. We also see that
\begin{align}\label{8b}
I'(-x_{ij}^2)=I'(x_{ij}^2)^*.
\end{align}
Another useful expression for (\ref{5b}) is the Mellin representation $I'(x_{ij}^2)=$
\begin{align}\label{10b}
\frac{1}{(2\pi i)^2}\int_{C_1}ds\int_{C_{2}}ds'(\Gamma_{s,s'})^2 \left(\frac{x_{12}^2\!+\!i\epsilon}{x_{13}^2\!+\!i\epsilon}\right)^{-s}\left(\frac{x_{34}^2\!+\!i\epsilon}{x_{24}^2\!+\!i\epsilon}\right)^{-s}\left(\frac{x_{14}^2\!+\!i\epsilon}{x_{13}^2\!+\!i\epsilon}\right)^{-s'}\left(\frac{x_{23}^2\!+\!i\epsilon}{x_{24}^2\!+\!i\epsilon}\right)^{-s'},
\end{align}
where $\Gamma_{x,y}\equiv \Gamma(x)\Gamma(y)\Gamma(1-x-y)$. The complex power is $z^s\equiv\exp(s\log z)$, where the logarithm has a branch cut on the negative real axis. $C_j$ are contours in the complex plane
\begin{align}
C_j=\{\gamma_j+it\hspace{0.05cm}|\hspace{0.05cm} t\in(-\infty,\infty)\},
\end{align}
where $\gamma_1,\gamma_2$ satisfy $0<\gamma_1,\gamma_2<1,\hspace{0.05cm} 0<\gamma_1+\gamma_2<1$. The corresponding Euclidean versions of (\ref{9b}) and (\ref{10b}) are found by setting $\epsilon=0$.
\subsection{Conformal Transformations}\label{confbox}
$\text{Conf}(\mathbb{R}^{1,3})$ acts on $\mathbb{R}^{1,3}_c$ by translations, proper orthochronous Lorentz rotations, dilatations, and SCTs. Since (\ref{9b}) and (\ref{10b}) are functions of only the kinematics $x_{ij}^2$, which are Poincaré invariants, it is clear that (\ref{5b}) is invariant under Poincaré transformations
\begin{align}
I(\boldsymbol{w})=I(T_a\boldsymbol{w})=I(L\boldsymbol{w}),
\end{align}
for all $a\in\mathbb{R}^{1,3},\hspace{0.1cm} L\in SO^+(1,3)$. It is also clear that (\ref{5b}) is invariant under dilatations $x_i\rightarrow D_\lambda x_i\equiv \lambda x_i$, since for all $\lambda>0$ we have
\begin{align}
I(D_\lambda\boldsymbol{w})=\int \frac{d^4x_5}{i\pi^2}\frac{\lambda^2x_{13}^2\lambda^2x_{24}^2}{\prod_{j=1}^4((\lambda x_j-x_5)^2+ i\epsilon)}
\end{align}
\begin{align*}
=(x_5=\lambda x_5')=\int \frac{\lambda^4 d^4x_5'}{i\pi^2\lambda^8}\frac{\lambda^4x_{13}^2x_{24}^2}{\prod_{j=1}^4(( x_j-x_5')^2+ i\frac{\epsilon}{\lambda^2})}
\end{align*}
\begin{align*}
=\int \frac{ d^4x_5'}{i\pi^2}\frac{x_{13}^2x_{24}^2}{\prod_{j=1}^4(( x_j-x_5')^2+ i\epsilon')}=I(\boldsymbol{w}),
\end{align*}
because $\epsilon'\equiv \frac{\epsilon}{\lambda^2}$ is still an infinitesimal positive quantity. This invariance under dilatations is also easily proven from representations (\ref{9b}) and (\ref{10b}). In Euclidean signature (\ref{3t}) the box integral is also checked to be invariant under SCTs $C_b$, and so the integral is fully conformally invariant. In Minkowski space however this invariance is broken by the IR regulator $i\epsilon$. Exactly how the invariance is broken is most easily seen in terms of the Mellin representation (\ref{10b}). For $b\in \mathbb{R}^{1,3}$ we have that $I'(x_{ij}^2)\rightarrow I'(C_bx_{ij}^2)=$
\begin{align}\label{13j}
\frac{1}{(2\pi i)^2}\int_{C_1}ds\int_{C_{2}}ds'(\Gamma_{s,s'})^2 \left(\frac{x_{12}^2\!+\!i\epsilon_{12}}{x_{13}^2\!+\!i\epsilon_{13}}\right)^{-s}\left(\frac{x_{34}^2\!+\!i\epsilon_{34}}{x_{24}^2\!+\!i\epsilon_{24}}\right)^{-s}\left(\frac{x_{14}^2\!+\!i\epsilon_{14}}{x_{13}^2\!+\!i\epsilon_{13}}\right)^{-s'}\left(\frac{x_{23}^2\!+\!i\epsilon_{23}}{x_{24}^2\!+\!i\epsilon_{24}}\right)^{-s'},
\end{align}
\begin{align}
\epsilon_{ij}\equiv \epsilon\sigma_b(x_i)\sigma_b(x_j),
\end{align}
where (\ref{10f}) was used and $\sigma_b(x)=1-2b\cdot x+b^2x^2$. From (\ref{13j}) we see the extent to which conformal invariance is broken is encoded in the signs of the infinitesimal imaginary parts $i\epsilon_{ij}$. Given a finite configuration $\boldsymbol{w}=(x_1,x_2,x_3,x_4)$ there is a possible branch jumping of the integral under \textit{finite} special conformal transformations $C_b$ for which at least one of the $\epsilon_{ij}<0$. $\epsilon_{ij}$ changes from positive to negative when $C_bx_k$ crosses infinity for some $k\in\{i,j\}$ and the kinematics $x_{kl}^2$ for $l\neq k$ change sign, i.e.\ the kinematic region changes. In computing (\ref{13j}) the $i\epsilon_{ij}$ select the correct branch of the logs/dilogs which appear in the computation, after which the result depends only on the cross-ratios $u$ and $v$, or equivalently $z$ and $\bar{z}$. This computation was completed for arbitrary kinematics in \cite{Duplan_i__2002}, and we specialise it in section \ref{branches} to give the box integral as a function of $z$ and $\bar{z}$ in each kinematic region $\text{sgn}(k(\boldsymbol{w}))$. While from (\ref{13j}) there are naively eight values of the integral which can be reached under SCTs, corresponding to the eight possible sign assignments for $\epsilon_{ij}$, it is actually up to four. The dependence of the box integral on only the conformal invariants $z$ and $\bar{z}$ as well as the kinematic region $\text{sgn}(k(\boldsymbol{w}))$ can be thought of as \textit{pseudo}-conformal invariance.
\newline
\newline
As for discrete conformal transformations, the parity and time reversal maps $\mathcal{P},\mathcal{T}:\mathbb{R}^{1,3}\rightarrow \mathbb{R}^{1,3}$ are also symmetries of the box integral. $\mathcal{P}$ and $\mathcal{T}$ do not change the kinematics $x_{ij}^2$, so
\begin{align}
I(\boldsymbol{w})=I(\mathcal{P}\boldsymbol{w})=I(\mathcal{T}\boldsymbol{w})=I(\mathcal{P}\mathcal{T}\boldsymbol{w}).
\end{align}
It is also useful to keep in mind how $z,\bar{z}$, and $I(\boldsymbol{w})$ respond to permutations $\sigma\in S_4$ of the external points. This is discussed in appendix \ref{permutation}.
\subsection{Branches of the Minkowski Box Integral}\label{branches}
We write the result of \cite{Duplan_i__2002} for $I(\boldsymbol{w})$ in all kinematic regions explicitly in terms of the conformal invariants $z,\bar{z}$. We first note that given a finite configuration $\boldsymbol{w}\in V_f$ with real $z,\bar{z}$ it is always possible to find a permutation $\sigma\in S_4$ such that $\sigma\boldsymbol{w}$ either has $z,\bar{z}\in (0,1)$, or $z\in (-\infty,0),\bar{z}\in(0,1)$. Since $I(\boldsymbol{w})$ transforms at most by a conformally invariant constant\footnote{Since it is conformally invariant, $I(\boldsymbol{w})$ and $I(\sigma\boldsymbol{w})$ have the same qualitative behaviour under SCTs.} under this permutation (table \ref{table5}), we can focus our attention on \textit{restricted} configurations $\boldsymbol{w}$ with $z,\bar{z}\in\mathbb{C}$, $z,\bar{z}\in (0,1)$, or $z\in (-\infty,0),\bar{z}\in(0,1)$ so that $\boldsymbol{w}$ is either in $V_{\mathbb{C}},V_1,\bar{V}_1, V_2,$ or $\bar{V}_2$ as defined in section \ref{confplanemink}. We define four functions $f_i(z,\bar{z})$ by
\begin{align}
f_1(z,\bar{z})=1,\hspace{1cm} f_2(z,\bar{z})=\log(z/\bar{z}),\hspace{1cm} f_3(z,\bar{z})=\log(\frac{1-z}{1-\bar{z}}),
\end{align}
\begin{align}
f_4(z,\bar{z})=2\text{Li}_2(z)-2\text{Li}_2(\bar{z})+\log(z\bar{z})\log(\frac{1-z}{1-\bar{z}}),
\end{align}
where we take $z,\bar{z}\in \mathbb{C}\setminus [1,\infty),$ $\bar{z}\geq z$ if $z,\bar{z}\in \mathbb{R}$, and $z\in\mathbb{H},\bar{z}=z^*$ if $z,\bar{z}\in\mathbb{C}\setminus\mathbb{R}$. Note these are the same functions which appear from Yangian invariance in \cite{Loebbert_2020} and $f_1,f_2,f_3$ are essentially the functions arising from period contours in \cite{hodges2010box}. We evaluate arguments on the branch cut of the logarithms (the negative real axis) slightly above the cut, so we implicitly take $\log(z)\rightarrow\log(z+i\epsilon)$. Due to the way we organised things, we never need to consider arguments on the branch cut $(1,\infty)$ of $\text{Li}_2(z)$. Note that $f_4$ is the Bloch-Wigner function (\ref{4t}) when $\bar{z}=z^*$. In this section, given a finite configuration $\boldsymbol{w}$, we abbreviate the kinematic region $k\equiv \text{sgn}(k(\boldsymbol{w}))$. For any restricted configuration we can write the box integral (\ref{5b}) in terms of $f_i(z,\bar{z})$
\begin{align}\label{4i}
I(z,\bar{z},k)=\sum_{i=1}^4\frac{c_i^kf_i(z,\bar{z})}{z-\bar{z}},
\end{align}
where $c_i^k\in\mathbb{C}$ depend on the kinematic region $k$. We specialise the result of \cite{Duplan_i__2002} to restricted configurations, i.e.\ those in $V_{\mathbb{C}}$, $V_1$ with $z,\bar{z}\in (0,1)$, $\bar{V}_1$ with $z,\bar{z}\in(0,1)$, $V_2$, and $\bar{V}_2$.
\begin{table}[h!]
\begin{center}
\begin{tabular}{||c|c|c|c||}
\hline
$k$ & $I(z,\bar{z},k)$ \\ [0.5ex]
\hline
$(+++++\hspace{0.1cm}+),(-++--\hspace{0.1cm}+)$, &\\
$(-+-++\hspace{0.1cm}-)$, & $\frac{f_4}{z-\bar{z}}$\\
$(+--+-\hspace{0.1cm}+),(+-+-+\hspace{0.1cm}-)$&\\
\hline
$(++---\hspace{0.1cm}-)$ &$\frac{f_4-2\pi i f_3}{z-\bar{z}}$\\
\hline
$(--++-\hspace{0.1cm}-)$ & $\frac{f_4+2\pi i f_2}{z-\bar{z}}$\\
\hline
$(----+\hspace{0.1cm}+)$ & Missing region\\
\hline
\end{tabular}
\caption{$I(z,\bar{z},k)$ for $0<z\leq\bar{z}<1$, $k\in K_1$.}
\label{table8}
\end{center}
\end{table}
\begin{table}[h!]
\begin{center}
\begin{tabular}{||c|c|c|c||}
\hline
$k$ & $I(z,\bar{z},k)$ \\ [0.5ex]
\hline
$(-----\hspace{0.1cm}-),(+--++\hspace{0.1cm}-)$, &\\
$(+-+--\hspace{0.1cm}+)$, & $\frac{f_4}{z-\bar{z}}$\\
$(-++-+\hspace{0.1cm}-),(-+-+-\hspace{0.1cm}+)$&\\
\hline
$(--+++\hspace{0.1cm}+)$ &$\frac{f_4+2\pi i f_3}{z-\bar{z}}$\\
\hline
$(++--+\hspace{0.1cm}+)$ & $\frac{f_4-2\pi i f_2}{z-\bar{z}}$\\
\hline
$(++++-\hspace{0.1cm}-)$ & $\frac{f_4+2\pi i(f_2-f_3-2\pi i f_1)}{z-\bar{z}}$\\
\hline
\end{tabular}
\caption{$I(z,\bar{z},k)$ for $0<z\leq\bar{z}<1$ or $z\in\mathbb{H},\bar{z}=z^*$, $k\in \bar{K}_1$.}
\label{table9}
\end{center}
\end{table}
\newline
We take a configuration $\boldsymbol{w}\in V_1$ with $0<z\leq\bar{z}<1$, so that $k\in K_1$ (see table \ref{table7}). Firstly we conjecture that such configurations $\boldsymbol{w}$ cannot have $k=k^*\equiv (----+\hspace{0.05cm}+)$, when there is a priori no reason this is not possible. We demonstrate this fact numerically in appendix \ref{missing}. We call this the `missing' kinematic region. Note it is possible for a configuration $\boldsymbol{w}\in V_1$ to have $k=(----++)$ if $z,\bar{z}\in(-\infty,0)$ or $z,\bar{z}\in (1,\infty)$. It is also possible to pick kinematics $x_{ij}^2$ such that $k=(----+\hspace{0.1cm}+)$ and $z,\bar{z}\in (0,1)$, however we conjecture such kinematics cannot be realised by configurations $\boldsymbol{w}\in V_f$. We list the results for the box integral $I(z,\bar{z})$ as a function of the kinematic region $k\in K_1$ in table \ref{table8}. Since Conf$(\mathbb{R}^{1,3})$ acts transitively\footnote{Up to $\mathcal{P},\hspace{0.05cm} \mathcal{T}$ which doesn't change $I(\boldsymbol{w})$.} on each $V_{1,z,\bar{z}}$ we see that given any configuration $\boldsymbol{w}\in V_1$ with $z,\bar{z}\in (0,1)$ one can reach \textbf{three} branches of the box integral using SCTs. If $\boldsymbol{w}\in V_{\mathbb{C}}$ or $\bar{V}_1$ with $0<z\leq\bar{z}<1$, then $k\in \bar{K}_1$. In this case the kinematics reversed version of the missing region $Pk^*=(++++-\hspace{0.08cm}-)$ with $z,\bar{z}\in(0,1)$ is realisable, although numerically it is found to be much rarer than the other regions (appendix \ref{missing}). We list the results for the box integral $I(z,\bar{z})$ as a function of the kinematic region $k\in \bar{K}_1$ in table \ref{table9}. In this case starting with any configuration $\boldsymbol{w}\in V_{\mathbb{C}}$ or $\bar{V}_1$ with $0<z\leq\bar{z}<1$ one can reach \textbf{four} branches of the box integral with SCTs. For a fixed $z,\bar{z}\in (0,1)$ there are \textbf{six} possible values of the box integral, corresponding to the four functions in table \ref{table9} and the two differing functions in table \ref{table8}. Note that when $f_1$ appears there is no symmetry under $z\leftrightarrow\bar{z}$ and our ordering of $z,\bar{z}$ is important.
\newline
\begin{table}[h!]
\begin{center}
\begin{tabular}{||c|c|c|c||}
\hline
$k$ & $I(z,\bar{z},k)$ \\ [0.5ex]
\hline
$(-++++\hspace{0.1cm}+),(---+-\hspace{0.1cm}+),(--+-+\hspace{0.1cm}-)$ & $\frac{f_4}{z-\bar{z}}$\\
\hline
$(+++--\hspace{0.1cm}+),(++-++\hspace{0.1cm}-),(-+---\hspace{0.1cm}-)$ & $\frac{f_4-2\pi i f_3}{z-\bar{z}}$\\
\hline
$(+---+\hspace{0.1cm}+)$ & $\frac{f_4-2\pi i f_2}{z-\bar{z}}$\\
\hline
$(+-++-\hspace{0.1cm}-)$ & $\frac{f_4+2\pi i(f_2-f_3-2\pi i f_1)}{z-\bar{z}}$\\
\hline
\end{tabular}
\caption{$I(z,\bar{z},k)$ for $z\in(-\infty,0),\hspace{0.05cm} \bar{z}\in(0,1)$, $k\in K_2$.}
\label{table10}
\end{center}
\end{table}
\begin{table}[h!]
\begin{center}
\begin{tabular}{||c|c|c|c||}
\hline
$k$ & $I(z,\bar{z},k)$ \\ [0.5ex]
\hline
$(---++\hspace{0.1cm}-),(--+--\hspace{0.1cm}+),(+-+++\hspace{0.1cm}+)$ & $\frac{f_4}{z-\bar{z}}$\\
\hline
$(+----\hspace{0.1cm}-),(+++-+\hspace{0.1cm}-),(++-+-\hspace{0.1cm}+)$ & $\frac{f_4-2\pi i f_3}{z-\bar{z}}$\\
\hline
$(-+--+\hspace{0.1cm}+)$ & $\frac{f_4-2\pi i f_2}{z-\bar{z}}$\\
\hline
$(-+++-\hspace{0.1cm}-)$ & $\frac{f_4+2\pi i(f_2-f_3-2\pi i f_1)}{z-\bar{z}}$\\
\hline
\end{tabular}
\caption{$I(z,\bar{z},k)$ for $z\in(-\infty,0),\hspace{0.05cm} \bar{z}\in(0,1)$, $k\in
\bar{K}_2$.}
\label{table11}
\end{center}
\end{table}
\noindent We list the corresponding results for configurations $\boldsymbol{w}\in V_f$ with $z\in(-\infty,0), \bar{z}\in(0,1)$, so that $k\in K_2$ or $k\in\bar{K}_2$, in tables \ref{table10} and \ref{table11} respectively. In both cases starting with any finite configuration $\boldsymbol{w}\in V_2$ (or $\bar{V}_2$) one can reach \textbf{four} branches of the box integral using SCTs. Since the functions in tables \ref{table10} and \ref{table11} overlap given any finite configuration with $z\in(-\infty,0), \bar{z}\in(0,1)$ there are \textbf{four} possible values for the box integral.
\section{Double Infinity Configurations}\label{doubleinf}
In this section we demonstrate that the integral (\ref{5b}) can be computed for most kinematic regions with real $z,\bar{z}$ by considering special `double infinity' configurations. We consider eight configurations $\boldsymbol{X}^{ab}, \boldsymbol{Y}^{ab}$ of four points in Minkowski space, where $a,b\in\{+,-\}$, defined by the Hermitian matrices (related to usual Minkowski vectors by (\ref{30a}))
\begin{align}\label{1c}
X_1^{ab}=\begin{pmatrix}0&0\\0&0 \end{pmatrix},\hspace{0.1cm} \hspace{0.1cm} X_2^{ab}=\begin{pmatrix}\xi_+&0\\0&\eta_b \end{pmatrix},\hspace{0.1cm}\spa X_3^{ab}=\begin{pmatrix}1&0\\0&1 \end{pmatrix}, \hspace{0.1cm}\spa X_4^{ab}=\begin{pmatrix}\eta_a&0\\0&\xi_- \end{pmatrix},
\end{align}
\begin{align}\label{2c}
Y_1^{ab}=\begin{pmatrix}0&0\\0&0 \end{pmatrix},\hspace{0.1cm} \hspace{0.1cm} Y_2^{ab}=\begin{pmatrix}\xi_+&0\\0&-\eta_b \end{pmatrix},\hspace{0.1cm}\spa Y_3^{ab}=\begin{pmatrix}1&0\\0&-1 \end{pmatrix}, \hspace{0.1cm}\spa Y_4^{ab}=\begin{pmatrix}\eta_a&0\\0&-\xi_- \end{pmatrix},
\end{align}
where $\xi_+,\xi_-\in \mathbb{R}\setminus\{0,1\}$ and we take the limit $\eta_\pm\rightarrow\pm\infty$. For each $a,b$ we see that $X_1^{ab}$ and $Y_1^{ab}$ correspond to the origin in Minkowski space, while $X_3^{ab}$ and $Y_3^{ab}$ correspond to the unit vectors $e_0=(1,0,0,0)$ and $e_3=(0,0,0,1)$ respectively. The remaining points live on $\mathscr{I}^+$ or $\mathscr{I}^-$ depending on the choice of $a$ or $b$, and are parametrised by a degree of freedom $\xi_\pm$. The configurations $\boldsymbol{X}^{+-}$ and $\boldsymbol{X}^{--}$ are visualised on an extended Penrose diagram in figure \ref{figg20}.
\begin{figure}[!htb]
\centering
\begin{minipage}{.5\textwidth}
\centering
\begin{tikzpicture}[thick,scale=0.7, every node/.style={scale=0.8}]
\node[inner sep=0pt] (russell) at (0,0)
{\includegraphics[width=.8\textwidth]{Xpm.pdf}};
\node at (-.61,0.21){$x_1$};
\node at (2.8,-1.3){$x_2$};
\node at (-.4,1.8){$x_3$};
\node at (3,1){$x_4$};
\end{tikzpicture}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\begin{tikzpicture}[thick,scale=0.7, every node/.style={scale=0.8}]
\node[inner sep=0pt] (russell) at (0,0)
{\includegraphics[width=.8\textwidth]{Xmm.pdf}};
\node at (-.61,0.21){$x_1$};
\node at (2.8,-1.1){$x_2$};
\node at (-.4,1.8){$x_3$};
\node at (-1.2,-2.6){$x_4$};
\end{tikzpicture}
\end{minipage}
\caption{Left: Configuration $\boldsymbol{X}^{+-}$ --- $x_4$ and $x_2$ on $\mathscr{I}^+(\theta=0)$ and $\mathscr{I}^-(\theta=0)$. Right: Configuration $\boldsymbol{X}^{--}$ --- $x_4$ and $x_2$ on $\mathscr{I}^-(\theta=\pi)$ and $\mathscr{I}^-(\theta=0)$. In terms of unitary matrices $\boldsymbol{X}^{--}$ and $\boldsymbol{X}^{+-}$ coincide due to the identification $\mathscr{I}^+\sim \mathcal{A}\mathscr{I}^-$ (see section \ref{confcompact}).}
\label{figg20}
\end{figure}
\noindent In the limit $\eta_\pm\rightarrow \pm\infty$ $\boldsymbol{X}^{ab}$ correspond to the same configuration in terms of unitary matrices $\boldsymbol{U}_{\boldsymbol{X}}$, and similarly $\boldsymbol{Y}^{ab}$ correspond to the same configuration $\boldsymbol{U}_{\boldsymbol{Y}}$
\begin{align}\label{1t}
\boldsymbol{U}_{\boldsymbol{X}}=\left(\begin{pmatrix}1&0\\0&1\end{pmatrix},\begin{pmatrix}\frac{i-\xi_+}{i+\xi_+}&0\\0&-1\end{pmatrix},\begin{pmatrix}i&0\\0&i\end{pmatrix},\begin{pmatrix}-1&0\\0&\frac{i-\xi_-}{i+\xi_-}\end{pmatrix}\right),
\end{align}
\begin{align}
\boldsymbol{U}_{\boldsymbol{Y}}=\left(\begin{pmatrix}1&0\\0&1\end{pmatrix},\begin{pmatrix}\frac{i-\xi_+}{i+\xi_+}&0\\0&-1\end{pmatrix},\begin{pmatrix}i&0\\0&-i\end{pmatrix},\begin{pmatrix}-1&0\\0&\frac{i+\xi_-}{i-\xi_-}\end{pmatrix}\right).
\end{align}
Note that even though each $\boldsymbol{X}^{ab}$ corresponds to the same configuration in terms of unitary matrices $\boldsymbol{U}_{\boldsymbol{X}}$, the value of the box integral $I(\boldsymbol{X}^{ab})$ depends on $a,b$, i.e.\ the direction $X_2$ and $X_4$ are sent to $\mathscr{I}$ (and similarly for $\boldsymbol{Y}^{ab}$). This is because the box integral is singular for configurations with points on $\mathscr{I}$, and we should think of $X_2,\hspace{0.05cm} Y_2$ and $X_4,\hspace{0.05cm} Y_4$ as being slightly off $\mathscr{I}$ so that $\mathscr{I}^+$ and $\mathscr{I}^-$ are distinguished. We take the kinematics to be large, but finite. We compute the kinematics $k(\boldsymbol{X}^{ab})$ and $k(\boldsymbol{Y}^{ab})$ in the limit $\eta_\pm\rightarrow \pm\infty$ to be
\begin{align}
k(\boldsymbol{X}^{ab})=\left(\hspace{0.05cm}\xi_+\eta_b,\hspace{0.05cm}\sp \eta_a(\xi_--1),\hspace{0.05cm}\sp \eta_b(\xi_+-1),\hspace{0.05cm}\sp\xi_-\eta_a,\hspace{0.05cm}\sp 1,\hspace{0.05cm}\sp-\eta_a\eta_b\hspace{0.05cm}\right),
\end{align}
\begin{align}\label{3c}
k(\boldsymbol{Y}^{ab})=-k(\boldsymbol{X}^{ab}).
\end{align}
We also compute the conformal cross-ratios in the limit
\begin{align}
u(\boldsymbol{X}^{ab})= u(\boldsymbol{Y}^{ab})=\xi_+(1-\xi_-),\hspace{1cm} v(\boldsymbol{X}^{ab})= v(\boldsymbol{Y}^{ab})=\xi_-(1-\xi_+),
\end{align}
so that the cross-ratios $u$ and $v$ are the same for each of the eight configurations $\boldsymbol{X}^{ab}$ and $\boldsymbol{Y}^{ab}$. Written in terms of $z$ and $\bar{z}$ the cross-ratios for each configuration are
\begin{align}
z=\text{min}(\xi_+,1-\xi_-),\hspace{1cm} \bar{z}=\text{max}(\xi_+,1-\xi_-),
\end{align}
since by convention we take $\bar{z}\geq z$. Note that since $\xi_+,\xi_-\in \mathbb{R}\setminus\{0,1\}$, the configurations (\ref{1c}) and (\ref{2c}) can be tuned to give any real $z$ and $\bar{z}$ with $\bar{z}\geq z$. Depending on $\xi_+,\hspace{0.05cm} \xi_-$ and the choice of configuration $\boldsymbol{X}^{ab}$ or $\boldsymbol{Y}^{ab}$ the kinematics of the configuration can vary widely. In fact most configurations $\boldsymbol{w}$ with real $z$ and $\bar{z}$ and a given $\text{sgn}(k(\boldsymbol{w}))$ can be conformally mapped to one of the double infinity configurations $\boldsymbol{w}_\infty$, i.e.\ (\ref{1c}) or (\ref{2c}) (possibly permuted) with the same $z$ and $\bar{z}$ and the \textit{same signs of kinematics} so that $I(\boldsymbol{w})=I(\boldsymbol{w}_\infty)$ by pseudo-conformal invariance. Specialising to the restricted kinematics considered in tables \ref{table8}$-$\ref{table11}, the only kinematic regions which cannot be realised by a double infinity configuration are the missing kinematic region $k^*$ described in section \ref{branches}, and the corresponding region obtained by reversing all kinematics $Pk^*$ (which can be realised by finite configurations). For example, we show how to realise all kinematic regions in $V_2$ in table \ref{table77}.
\begin{table}[h!]
\begin{center}
\begin{tabular}{||c|c|c|c||}
\hline
Kinematic region $k$ & $\boldsymbol{w}_\infty$ & $\xi_+,\xi_-$ range \\ [0.5ex]
\hline
$(-++++\hspace{0.1cm}+)$& $\boldsymbol{X}^{+-}$ & $\xi_+\in(0,1),\hspace{0.05cm}\xi_-\in(1,\infty)$\\
\hline
$(+++--\hspace{0.1cm}+)$&$\boldsymbol{Y}^{++}$ &$\xi_+\in(-\infty,0),\hspace{0.05cm} \xi_-\in(0,1)$\\
\hline
$(++-++\hspace{0.1cm}-)$&$\boldsymbol{X}^{++}$ &$\xi_+\in(0,1),\hspace{0.05cm}\xi_-\in(1,\infty)$\\
\hline
$(---+-\hspace{0.1cm}+)$&$\boldsymbol{Y}^{--}$ &$\xi_+\in(-\infty,0),\hspace{0.05cm} \xi_-\in(0,1)$\\
\hline
$(--+-+\hspace{0.1cm}-)$&$\boldsymbol{X}^{--}$ &$\xi_+\in(0,1),\hspace{0.05cm}\xi_-\in(1,\infty)$\\
\hline
$(-+---\hspace{0.1cm}-)$&$\boldsymbol{Y}^{+-}$ &$\xi_+\in(-\infty,0),\hspace{0.05cm} \xi_-\in(0,1)$\\
\hline
$(+---+\hspace{0.1cm}+)$&$\boldsymbol{X}^{-+}$ &$\xi_+\in(0,1),\hspace{0.05cm}\xi_-\in(1,\infty)$\\
\hline
$(+-++-\hspace{0.1cm}-)$&$\boldsymbol{Y}^{-+}$ &$\xi_+\in(-\infty,0),\hspace{0.05cm} \xi_-\in(0,1)$\\
\hline
\end{tabular}
\caption{How to realise all kinematic regions in $V_2$ ($z\in(-\infty,0),\bar{z}\in(0,1)$) with double infinity configurations. No need for permutations in this case.}
\label{table77}
\end{center}
\end{table}What makes these $\boldsymbol{w}_\infty$ configurations particularly interesting is that the box integral (\ref{5b}) with these configurations can be calculated directly in Minkowski space, i.e.\ without reference to Feynman parameters or a Mellin representation. For each configuration the integral depends only on the choice of $\xi_+$ and $\xi_-$ and so we define
\begin{align}
I^{ab}_{\boldsymbol{X}}(\xi_+,\xi_-)\equiv I(\boldsymbol{X}^{ab}),\hspace{1cm} I^{ab}_{\boldsymbol{Y}}(\xi_+,\xi_-)\equiv I(\boldsymbol{Y}^{ab}),
\end{align}
where how we make sense of the limit $\eta_{\pm}\rightarrow\pm\infty$ in the calculation will be described shortly. First we note that we can use symmetries to reduce the number of integrals to compute from eight to two. We can immediately combine (\ref{8b}) and (\ref{3c}) to conclude
\begin{align}\label{4c}
I^{ab}_{\boldsymbol{Y}}(\xi_+,\xi_-)=I^{ab}_{\boldsymbol{X}}(\xi_+,\xi_-)^*.
\end{align}
We can also note relations between the configurations (\ref{1c}) using permutations, discrete transformations, and translations. We have
\begin{align}\label{5c}
\mathcal{P}\circ(24)\boldsymbol{X}^{+-}=\boldsymbol{X}^{-+} (\xi_+\leftrightarrow \xi_-),
\end{align}
\begin{align}\label{6c}
T_{\mathbb{I}_2}\circ\mathcal{P}\mathcal{T}\circ(13)\boldsymbol{X}^{--}=\boldsymbol{X}^{++} (\xi_\pm \rightarrow 1-\xi_\pm).
\end{align}
Since the box integral is fully invariant under each of the transformations used in (\ref{5c}) and (\ref{6c}) we see that
\begin{align}\label{7c}
I^{-+}_{\boldsymbol{X}}(\xi_+,\xi_-)=I^{+-}_{\boldsymbol{X}}(\xi_-,\xi_+),\hspace{1cm} I^{++}_{\boldsymbol{X}}(\xi_+,\xi_-)=I^{--}_{\boldsymbol{X}}(1-\xi_+,1-\xi_-).
\end{align}
In section \ref{doubleinfcalc} we describe our results for $I^{+-}_{\boldsymbol{X}}(\xi_+,\xi_-)$ and $I^{--}_{\boldsymbol{X}}(\xi_+,\xi_-)$, leaving the details of the calculation to appendix \ref{doubleinfcalcdetails}. We can then use (\ref{4c}) and (\ref{7c}) to recover the value of the integral for each of the configurations (\ref{1c}) and (\ref{2c}).
\subsection{Double Infinity Box Integral}\label{doubleinfcalc}
We write the integration variable $x_5$ in spherical coordinates
\begin{align}
x_5=(t,r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta),\hspace{1cm}
X_5=\begin{pmatrix}t+r\cos\theta&re^{-i\phi}\sin\theta\\re^{i\phi}\sin\theta&t-r\cos\theta\end{pmatrix},
\end{align}
with $r>0, \phi\in[0,2\pi), \theta\in[0,\pi]$. We then consider the two cases described above. We consider the configuration $\boldsymbol{X}^{+-}$ defined in (\ref{1c}). The Lorentz squares $x_{i5}^2$ for $i=1,2,3,4$ are calculated in the appropriate limit, for example
\begin{align}
x_{25}^2=|X_2-X_5|=\det\begin{pmatrix}\xi_+-t-r\cos\theta&-re^{-i\phi}\sin\theta\\-re^{i\phi}\sin\theta&\eta_--t+r\cos\theta\end{pmatrix}
\end{align}
\begin{align*}
=\eta_-(\xi_+-t-r\cos\theta)+O(\eta_-^0).
\end{align*}
Letting $x=\cos\theta$, for this configuration the box integral becomes $I_{\boldsymbol{X}}^{+-}(\xi_+,\xi_-)\simeq$
\begin{align}\label{12i}
\frac{2i}{\pi}\int_{x,r,t} \frac{r^2 \eta_+\eta_-}{(t^2-r^2+i\epsilon)((t-1)^2-r^2+i\epsilon)\eta_-(\xi_+-t-rx+\frac{i\epsilon}{\eta_-})\eta_+(\xi_--t+rx+\frac{i\epsilon}{\eta_+})},
\end{align}
where $\int_{x,r,t}\equiv\int_{-1}^1 dx\int_{0}^{\infty}dr\int_{-\infty}^\infty dt $ and the trivial $\phi$ integration has been performed. Note that in the limit $\eta_{\pm}\rightarrow \pm\infty$ the integral is well-defined, as factors of $\eta_\pm$ cancel in numerator and denominator. We define an $\epsilon'$ by $\frac{\epsilon}{\eta_{\pm}}\sim\pm \epsilon'$, and throughout the calculation we treat $\epsilon'\ll\epsilon$ in the limit $\eta_{\pm}\rightarrow \pm\infty$. This is essential to properly regulate the poles of the integrand. We describe the details of the remaining calculation in appendix \ref{doubleinfcalcdetails}. The answer can be expressed in terms of a set of four `basis' integrals
\begin{align}
h_1(a,b)\equiv (a-b)\int_0^1dx\frac{1}{(x-a)(x-b)},
\end{align}
\begin{align}
h_2(a,b)\equiv (a-b)\int_0^1dx\frac{\log(x)}{(x-a)(x-b)},
\end{align}
\begin{align}
h_3(a,b)\equiv (a-b)\int_0^1dx\frac{\log(1-x)}{(x-a)(x-b)},
\end{align}
\begin{align}
h_4(a,b)\equiv (a-b)\int_0^1dx\frac{\log(1+x)}{(x-a)(x-b)},
\end{align}
which are all expressible as simple combinations of logs and dilogs of $a$ and $b$, found easily, e.g.\ using
\texttt{Mathematica}. These are all well-defined in our case as each $a,b$ will have a small imaginary part. The final result for the integral is
\begin{align}\label{7i}
I_{\boldsymbol{X}}^{+-}(\xi_+,\xi_-)=\frac{1}{\Box\xi-1}\Bigg(\log(1/2)(h_1(r_{1+},r_{2+})-h_1(-r_{1-},-r_{2-}))
\end{align}
\begin{align*}
+\log(-1/2-i\epsilon)(-h_1(-r_{1+},-r_{2+})+h_1(r_{1-},r_{2-}))-\log(-\Delta\xi/2+i\epsilon)h_1(s_{1+}^2,s_{2+}^2)
\end{align*}
\begin{align*}
+\log(\Delta\xi/2-i\epsilon)h_1(s_{1-}^2,s_{2-}^2)+\frac{1}{2}(h_2(s_{1+}^2,s_{2+}^2)-h_2(s_{1-}^2,s_{2-}^2))+\Big(\log(\xi_+-i\epsilon)\times
\end{align*}
\begin{align*}
(h_1(-r_{1-},-s_{1-}) -h_1(r_{1+},s_{1+}))+\log(-\xi_--i\epsilon)(h_1(r_{2-},s_{1-})-h_1(-r_{2+},-s_{1+}))
\end{align*}
\begin{align*}
+h_3(r_{1+},s_{1+})+h_3(-r_{2+},-s_{1+})-h_4(-r_{1-},-s_{1-})-h_4(r_{2-},s_{1-})-(\xi_{\pm}\rightarrow 1-\xi_{\mp})\Big)\Bigg),
\end{align*}
where $\Box\xi=\xi_++\xi_-, \Delta\xi=\xi_+-\xi_-$, and $r_{i\pm}$ and $s_{i\pm}$ for $i=1,2$ are functions of $\xi_+,\xi_-$ defined in (\ref{u1}) and (\ref{u2}). (\ref{7i}) is invariant under $\xi_\pm\rightarrow 1-\xi_\mp$. Note that the overall prefactor agrees with our expectation
\begin{align}
\frac{1}{\Box\xi-1}=\pm\frac{1}{z-\bar{z}}.
\end{align}
For the configuration $\boldsymbol{X}^{--}$ the integral becomes $ I_{\boldsymbol{X}}^{--}(\xi_+,\xi_-)=$
\begin{align}
\frac{2i}{\pi}\int_{x,r,t} \frac{r^2}{(t^2-r^2+i\epsilon)((t-1)^2-r^2+i\epsilon)(\xi_+-t-rx-i\epsilon')(\xi_--t+rx-i\epsilon')}.
\end{align}
The final result for the integral is
\begin{align}\label{8i}
I_{\boldsymbol{X}}^{--}(\xi_+,\xi_-)=\frac{1}{\Box\xi-1}\Bigg(\log(1/2)(h_1(r_{1+},r_{2+})-h_1(-r_{1-},-r_{2-}))+\log(-1/2-i\epsilon) \times
\end{align}
\begin{align*}
(h_1(r_{1-},r_{2-})-h_1(-r_{1+},-r_{2+}))+\Big(-\log(\xi_+-i\epsilon)h_1(r_{1+},s_{1+}) +\log(\xi_--i\epsilon)h_1(r_{2+},s_{1+})
\end{align*}
\begin{align*}
+\log(-1+\xi_+-i\epsilon)h_1(-r_{1+},s_{2+})+\log(-1+\xi_--i\epsilon)h_1(-r_{2+},s_{2+})
\end{align*}
\begin{align*}
+h_3(r_{1+},s_{1+})-h_3(-r_{1+},s_{2+})+h_4(-r_{2+},s_{2+})-h_4(r_{2+},s_{1+})+(\xi_{+}\leftrightarrow \xi_{-})\Big)\Bigg),
\end{align*}
and is invariant under $\xi_+\leftrightarrow\xi_-$. Using (\ref{4c}) and (\ref{7c}) an expression for the box integral (\ref{5b}) for each of the eight configurations in (\ref{1c}) and (\ref{2c}) can be recovered from (\ref{7i}) and (\ref{8i}). These results were numerically verified in each accessible kinematic region with tables \ref{table8}$-$\ref{table11} as well as with the package \cite{vanOldenborgh:1990yc}. Note there are numerical issues when $\Delta\xi=0,\Box\xi=0, 1, 2$, $\xi_+=1/2,$ or $\xi_-=1/2$. These are easily remedied however by adding a small correction to $\xi_+$ or $\xi_-$.
\subsection{Numerical Example}\label{numeric}
\begin{figure}[!htb]
\centering
\begin{minipage}{.5\textwidth}
\centering
\begin{tikzpicture}[thick,scale=0.7, every node/.style={scale=0.8}]
\node[inner sep=0pt] (russell) at (0,0)
{\includegraphics[width=.8\textwidth]{pic8.pdf}};
\node at (-.61,0.21){$x_1$};
\node at (-1.8,2){$x_2$};
\node at (-.35,1.75){$x_3$};
\node at (-2.65,-1.15){$x_4$};
\end{tikzpicture}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\begin{tikzpicture}[thick,scale=0.7, every node/.style={scale=0.8}]
\node[inner sep=0pt] (russell) at (0,0)
{\includegraphics[width=.8\textwidth]{pic9.pdf}};
\node at (-.61,0.21){$x_1$};
\node at (2.3,-1.6){$x_2$};
\node at (-.35,1.75){$x_3$};
\node at (1.45,2.35){$x_4$};
\end{tikzpicture}
\end{minipage}
\caption{Configurations $\boldsymbol{w}$ (left) and $\boldsymbol{y}$ (right). The blue and green curves show the trajectories of $x_2$ and $x_4$ respectively under the SCT $C_{(0,0,0,b)}$ for $b\in\mathbb{R}$. $x_4$ crosses infinity again at $b=3/2+\epsilon$.}
\label{figg28}
\end{figure}\noindent As a concrete example, consider the configuration $\boldsymbol{w}\equiv \boldsymbol{X}^{-+}(\xi_+=1/4,\xi_-=2/3)$. For this configuration we have $z=1/4,\bar{z}=1/3$, $\text{sgn}(k(\boldsymbol{w}))=(++--+\hspace{0.11cm}+)$, and $\boldsymbol{w}\in \bar{V}_1$. The value of the box integral is $I^{-+}_{\boldsymbol{X}}(1/4,2/3)\simeq 5.88-21.69i$, which can be calculated using (\ref{7i}) and (\ref{7c}), table \ref{table9}, or for example the package \cite{vanOldenborgh:1990yc}. Under the infinitesimal SCT $C_{(0,0,0,\epsilon)}$ $\boldsymbol{w}$ is mapped to $\boldsymbol{y}\equiv\boldsymbol{X}^{+-}(\xi_+=1/4,\xi_-=2/3)$ with $\text{sgn}(k(\boldsymbol{y}))=(--+++\hspace{0.05cm}+)$ and $I^{+-}_{\boldsymbol{X}}(1/4,2/3)\simeq 5.88-8.88i$. Under the SCT $C_{(0,0,0,3/2+\epsilon)}$ $\boldsymbol{w}$ is mapped to a configuration $\boldsymbol{z}$ with $\text{sgn}(k(\boldsymbol{z}))=(-+-+-\hspace{0.12cm}+)$ and $I(\boldsymbol{z})\simeq 5.88$. This example shows three of the four branches of the box integral accessible in $\bar{V}_{1,1/4,1/3}$ (table \ref{table9}). In figure \ref{figg28} we show the configurations $\boldsymbol{w}$ and $\boldsymbol{y}$ and the orbits of $x_2$ and $x_4$ under the SCT parametrised by $(0,0,0,b)$.
\section{Conclusions}
Due to the causality-violating nature of the conformal group, the one-loop massless box integral is not fully conformally invariant in Minkowski space. We classified conformally equivalent configurations of four points in compactified Minkowski space, and found a much richer structure than the corresponding Euclidean classification. In Minkowski space the conformal invariants $z$ and $\bar{z}$ are not enough to decide if two configurations are conformally equivalent, and further information about the kinematic invariants $x_{ij}^2$ is needed. It is worth noting that a result similar in spirit appeared recently in \cite{qiao2020classification}, although the classifications are slightly different and serve different purposes. We rewrote the result \cite{Duplan_i__2002} for the box integral explicitly in terms of the conformal invariants $z$ and $\bar{z}$ in each kinematic region. Combining this and our classification of conformally equivalent configurations we conclude that using SCTs up to four branches of the box integral can be reached starting from any configuration, and up to six values of the integral for a given $z,\bar{z}$. There are two qualitatively different behaviours for the branch structure, one which appears in tables \ref{table8},\ref{table9} and one in tables \ref{table10},\ref{table11}. One interesting fact is that the set $V_1$ of conformally equivalent configurations contains only \textit{three} branches of the box integral. This is due to the `missing' kinematic region, the non-existence of which in general we were so far only able to establish numerically. We studied a range of double infinity configurations in Minkowski space, and found that the vast majority of kinematic regions for configurations with $z,\bar{z}\in\mathbb{R}$ can be realised. We were able to calculate the box integral for these configurations directly in Minkowski space. By pseudo-conformal invariance we can use this calculation to find the value of the box integral for all finite configurations with real $z,\bar{z}$ except for a single special case (the kinematics reversed version of the missing region, which is numerically found to be rare). One disadvantage of the results (\ref{7i}) and (\ref{8i}) is their relative non-transparency. While they are numerically easy to work with, it is technically difficult to analytically reduce them to the form (\ref{4i}) for the different kinematic regions, although we found total numerical agreement. In any case it is interesting that it is even possible to calculate the box integral directly in Minkowski space for such a large class of kinematics.
\newline
\newline
There are a number of interesting directions for further research. Since the box integral is the simplest of a large family of (locally) dual conformal integrals, one could study the extent to which dual conformal invariance is broken in other examples. We expect the situation to be similar for other four-point conformal integrals, although the number of branches reachable via SCTs may change. At higher points it would be interesting to see how the classification of conformally equivalent configurations changes, and what kind of missing kinematic regions appear. Some questions have been answered in the case of null hexagon configurations in \cite{dorn2012conformal}. At four points we found that the kinematic regions which are not realised by double infinity configurations can be realised by special triple and quadruple infinity configurations \cite{workinprogress}. These configurations are interesting because together with the double infinity configurations they can generate configurations with any $z,\bar{z}$ and any allowed $\text{sgn}(k(\boldsymbol{w}))$. However, it is much trickier to calculate the box integral in these configurations directly in Minkowski space because of nontrivial $\phi,\sin\theta$ dependence and square roots in denominators. Therefore we are unsure if such configurations can be useful for calculations. It would be interesting to study `higher infinity' configurations at higher points and check if all possible kinematic regions can always be realised.
\section*{Acknowledgements}
We thank Lance Dixon, Julian Miczajka, and Lorenzo Quintavalle for helpful discussions. Thanks to Julian and Lorenzo for comments on the manuscript. MS thanks Ewha Womans University for hospitality and his host Changrim Ahn for inspiring discussions. MS thanks the Brain Pool Program of the Korean National Research Foundation (NRF) under grant 2-2019-1283-001-1 for generous support. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 764850 “SAGEX”.
|
1,108,101,563,240 | arxiv | \section{Introduction}
Both special and general relativity are pillars of our modern understanding of reality. They were fundamentally classical theories, as quantum mechanics has only been introduced decades after their construction. Being constructed without the constraints of quantum consistency, they became also particularly difficult to reconcile with the quantum structures. Special relativity has been successfully reconciled with quantum mechanics via quantum field theory and the process of renormalisation. General relativity resisted a similar approach due to its non-renormalisability and its ill UV behaviour[1-4]. However, various dualities have shown that classical gravitational solutions encode precisely properties of quantum field theories in different regimes. One can therefore ask if there is some quantum "remnant" in general relativity that makes such descriptions possible. The ER=EPR proposal links two black holes connected via a wormhole to quantum entanglement [5]. The AdS/CFT duality links a classical anti-de-Sitter solution of general relativity to a highly quantum conformal field theory in a different regime. Lacking a proper axiomatisation of quantum mechanics, one has to rely on the most common quantum properties that differ from classical physics in order to derive what the quantum "remnants" could be that can be found in general relativity. There are several ways in which quantum mechanics differs from classical mechanics. First of all, observables are expanded to the status of operators acting on a Hilbert space of states. The outcome of measurements is not anymore one single possible result, but instead one has to consider all possible outcomes and to propagate them accordingly via a wavefunction or, further on in quantum field theory, via a path integral quantisation prescription. As observables become operators, one can define a notion of compatibility between them, and hence a notion of commutativity. Incompatible observables will not commute, leading to what we understand as quantum fluctuations. The state of the system is described by means of a state of maximal knowledge [6], through a wavefunction, and not by means of a state of complete absolute knowledge, as was the case in classical mechanics. Perfect knowledge was assumed in classical mechanics even at the statistical level, where fluctuations were caused by subjective limitations and not by fundamental uncertainty. Finally, probably one of the main properties of quantum mechanics that I believe will resist most reductionist attempts implicit to axiomatisation, is the non-cartesianity of the spaces of states. While in classical physics, an object is constructed from individual separated pieces that are placed in a whole and allowed to interact in a rather newtonian way, and we can always separate the system back into its individual pieces without losing any information, this becomes impossible in quantum mechanics, where the global wavefunction of the whole system encodes more information than any individual wavefunction associated with any of the pieces. Moreover, the difference between the state space of the combined system and that of the individual pieces grows exponentially, leading to the tensorial pairing available in quantum mechanics. In essence, we have a tension between a local and a global interpretation of physics, local referring to the underlying subsystems, and global referring to the over-arching combined system. Another aspect that is common to quantum computation but not so much to classical computation is the concept of reversibility. Quantum processes need to be reversible. In classical mechanics a bit of memory can be turned on or off without considering its history. This is impossible in quantum mechanics as the qubit operations need to be reversible. We find this in another situation in classical physics, namely in the construction of spacetime worldsheets or worldlines and in the cobordism category that defines their interaction [7]. This is why a potential functorial relation between the category of worldsheet cobordisms and the category of Hilbert spaces with their operations has been postulated. In quantum information we will need ancilla qubits to encode certain states with desired properties. This is necessary because in general we cannot act on a qubit while ignoring its history or "un-doing" actions on qubits. This property will become important in what follows. There are at least three domains where the global vs. local tension becomes interesting. First, it is renormalisation. The idea of renormalisation is based on the decoupling of scales. Basically, the attempt at renormalisation is an attempt at systematically decoupling the different scales of energy such that one can construct a meaningful theory at a certain scale, while incorporating the other scales in some effective manner. The effective contribution of the smaller scales usually manifests itself as flow equations in the parameters of the effective theory. That is also why we need to apply certain operations to the couplings of our theory as we move from one scale to the next (the well known renormalisation (semi)-group equations). The redefinitions of parameters and terms in the case of the standard model interactions, are finite and preserve the predictive power of the theories. They do represent an elegant way of connecting different scales, and of combining subsystems and constituents into a larger global system. The problem of describing strongly entangled/coupled systems in this way is not fully resolved in more than one dimension [8]. The origin of this problem resides in the exponential complexity of the many body wave function. I discussed previously the various results emerging from a topological analysis of the renormalisation group and the problem of whether a fibre bundle approach to the RG transformation could be a suitable approach is still open [9]. In any case, the main problem originates from the fact that indeed the transition between scales is not as simple as one may think. While we can renormalise all interactions of the standard model, adding gravity to the system makes the renormalisation procedure ill defined. Indeed, as is well known, the renormalisation group is not actually a group, it is more like a semi-group, as its inverse is not defined. Indeed, in that sense if one takes into account topological features of the renormalisation one is well advised to consider principal bundles where a preferred choice of identity is non-existent. The topological non-triviality of such a scale transformation would imply the non-existence of a trivial cartesian product between domains of scale, and hence would lead to scale interference, which is expected in string theory due to the T-duality and in the standard model via the flow of couplings [10]. Of course the two effects are apparently rather different. In any case, they are unified by the idea that the global structure cannot directly be un-paired into constituent pieces.
The other domain where the local/global tension emerges is of course quantum mechanics itself, where we have entanglement which is a property of a combined super-system, which has information on the subsystems that cannot be recovered when analysing the subsystems separately.
The third domain where we have a local/global problem is gauge theory. There we try to develop theories with consistent interactions, only to find that the interactions emerge from a fibre bundle approach based on a structure group which we associate to a gauge symmetry, and that this bundle encodes significant global information. In fact, a measure of curvature is what makes interactions possible, hence a global property, which cannot be detected at each point in space separately. Indeed, one of the first principles of general relativity was that there always exists a transformation that makes the spacetime locally flat, hence a freely falling observer, unless having access to some global information would not know it is in free fall. In order to obtain a theory of gravity, one had to incorporate this global information by means of curvature and that had to be defined by means of a connection on the fibre bundle. It comes at no surprise that quantum wavefunctions and quantum fields also have fibre-bundle interpretations, however they are less often used due to us being accustomed with other tools through which we link local and global structure, for one, the partition function and its path integral approach to quantisation, and the combination of phases of wavefunctions. However, if we admit that an essential property of quantum mechanics is that it takes into account non-cartesianity and hence that separation of a quantum system in local subsystems is basically always incomplete due to entanglement, we have to also admit that both the other problems also have a quantum nature. While nobody questions the idea that renormalisability is a feature of quantum field theory, the fact that general relativity incorporates a similar tension between local and global structures as the one found in quantum mechanics did not seem to have occurred until now. If this is accepted, then general relativity already has a property of quantum mechanics, namely non-cartesianity, incorporated in its construction: one cannot infer the same information from local measurements as one can from global ones. Also, one cannot annihilate a qubit neglecting its history, this being true both in quantum mechanics and in worldsheet physics. It simply so happens that the mathematics chosen to describe this tension was based on fibre bundles with connections, and this geometric approach is used more often in the general relativity context, albeit trying to solve the same problem as the partition function and the path integral quantisation tries to solve in quantum field theory. This property is of course not the sole property an axiomatisation of quantum mechanics would contain. Indeed, in quantum mechanics we have to expand our observables from objects that can take one single possible outcome and hence imply absolute knowledge, to operator-valued observables which rely on several possible outcomes having various probabilities of emerging at different times when the experiment is repeated. But this is also a method through which one gains access to the global structure of the state space, where different potential outcomes combine to generate a statistics that takes such potential events into account. Hence, even this construction was required because one had to expand from a local viewpoint of outcomes to one that involved several successive repetitions that contribute and interfere to produce the overall outcome. Such observables may not commute, meaning the information they encode globally is not locally compatible. But a very similar aspect occurs in general relativity. In fact the connection is used to compare space-time separated objects in a consistent way, while the curvature of a space being defined by the non-commuting property of covariant derivatives resulting from the existence of a non-trivial connection. We use the non-commuting property of connections (or covariant derivatives) to get a measure for some global property, namely curvature. This is surely not the sole global property one can consider, but if one considers only point particles, one is less sensitive to more interesting non-local properties of spacetime. There should be analogues of such non-local properties also in quantum mechanics. So, here we have, yet another property that general relativity shares with quantum mechanics, and it also originates from the local/global tension that appears in both domains. One attempt would be to construct two categories, one of quantum physics, and the other of general relativity, and to notice how both could be related and what functors one could develop to relate them. Indeed, this would show that the two are in many ways similar. One could therefore say that general relativity is already quantum to some extent, or that it preserves some quantum remnants from its overarching quantum sibling, potentially string theory. One may be surprised therefore at the difficulty of completing the UV domain of quantum gravity. After all, other theories that share similar properties have a well defined UV limit, say quantum chromodynamics (where of course the problems appear in the low energy limit where the couplings are strong, but that is another subject not to be discussed here, but think at asymptotic freedom and asymptotic safety). The solution to the UV problem of quantum gravity comes from incorporating extended objects instead of points. Of course extended objects would make the connection between local and global properties more amenable to a construction that is based on quantum mechanics. Once this is realised one can continue with the tools and methods of string theory and describe quantum gravity. However, various limits in string theory give us links between so called classical general relativity solutions and non-stringy quantum field theories. This can only be possible if general relativity retains some of the quantum properties of string theory.
\section{T-duality as a result of a higher-entangled state}
We expect to have a functorial relation between general relativity and quantum mechanics, although it is not fully understood what exactly the functor should be. We do understand, following J. Baez and his categorial approach to the two theories, that the non-cartesianity of the two categories, that of the Hilbert spaces and the linear operators between them, and that of smooth manifolds and cobordisms are linked by some functor that allows us to interpret operations in one theory by means of analogue operations in the other. This approach leads us to the understanding that there is a deeper connection between the two, that we didn't fully explore until now. However, what seems to be particularly intriguing is how the two theories deal with global structure and particularly how they transition from local to global structures. While it is true that both do this and hence their structures are linked precisely by this, the mathematical tools used to link local and global structures were so different that a unification is still lacking. String theory, surprisingly has one component that makes this connection apparent, namely T-duality, which in some flux compactifications indeed links two string theories with different topology, and particularly, also links scales, transforming the trajectory of scale transformations from a straight line to a circle. In fact string theory solves many of the conundrums of quantum gravity, so much so that it has convinced many (and certainly this author) that if it is not the final theory of quantum gravity, it certainly walks the right path. In general T-duality links a theory with big spacetime radius with another one with small radius. In the case of one single compactified dimension with radius $R$ all physical properties of an interacting theory remain unchanged if one replaces
\begin{equation}
R\rightarrow \frac{\alpha}{R}
\end{equation}
Of course this works if the dilaton field is transformed by
\begin{equation}
\phi\rightarrow \phi- log(R/\sqrt{\alpha '})
\end{equation}
This can be expanded to the case of toroidal compactifications for a constant metric and an antisymmetric tensor as
\begin{equation}
\begin{array}{c}
(g+b)\rightarrow(g+b)^{-1}\\
\phi \rightarrow \phi - \frac{1}{2} log(det(g+b))
\end{array}
\end{equation}
The $T$-duality is generally regarded as somewhat mysterious. However, considering the connection between the fibre-bundle construction of gauge theories and of gauge connection on one side and the quantum mechanical entanglement as well as the emergence of spacetime (and in general geometry) from entanglement, the idea that the small and large scales of a theory with interactions should generate equivalent theories is not as surprising anymore. If indeed we have a functorial connection between quantum mechanics and gauge theory, as I expect to be the case, T-duality linking vastly different metric scales is just the reflection of the relation between long and short length entanglement. Being fundamentally a correlation and taking into account the global structure of the base manifold, entanglement should have the same properties, linking vastly different scales, hence T-duality should be nothing but a reflection of this nature of entanglement for sigma-models. Moreover, the causal structure, and hence the concept of spacetime interval is altered by a T-dual approach.
Focusing on the nature of the pairing in quantum mechanics and gauge theory and on the functorial connection between the two, it is interesting to try to analyse T-duality from a quantum mechanics point of view, allowing for its quantum nature to emerge. Let us focus on the quantum result that the state space of a composed system is not reducible to the cartesian product of the state spaces of its subsystems. That implies that there exists a global structure to be accounted for. The non-trivial fibre bundles describe precisely how such a local cartesian product fails to be replicated globally. Fascinatingly enough when deriving the T-duality one obtains an exact analogue of quantum entanglement for gauge theories (or sigma models). Let us consider a manifold $M$ with metric $g_{ij}$ with $i,j=0,...,d-1$, the usual antisymmetric tensor $b_{ij}$ and the dilaton field $\phi(x_{i})$. The $\sigma$-model associated to it has at least one symmetry associate to an (in this case abelian) isometry on the metric. The procedure of obtaining the T-duality is to first gauge the isometry group, introducing the gauge field variables $A$. The field is required to be flat by means of a Lagrange multiplier $\xi\cdot dA$. We then perform two equivalent operations differing simply through the order in which they are performed. First we integrate over $\xi$, obtain a $\delta$ function $dA$ on the measure and the result of our constraint, namely that $A=d\psi$ is a pure gauge is being implemented. If we fix $\psi=0$ we recover the original model. Next we first integrate the gauge field $A$. We don't have a gauge kinetic term, hence we simply perform a Gaussian integration and obtain a Lagrangian depending on the original variables and the auxiliary variable $\xi$. We continue by fixing the gauge and obtain the dual action. It is important to highlight the aspects that are based on a quantum information understanding of the procedure leading to this duality. The novelty here lies in the interpretation, namely that T-duality can be seen as an effect of quantum entanglement in a categorial sense, applied to higher structures, leading to "entangled theories". This interpretation may shed new light on the origins of dualities in gauge and string theories. The first step is to lift the field space to a higher space, incorporating the gauge fields associated to the isometries of the metric. Then we require that for the resulting field, the gauge connection is flat, this condition being implemented by an additional field introduced in the Lagrange constrained as an auxiliary field, or an extended form of an ancilla quibit, encoding a technique of fixing part of the global structure. We take the inspiration from control ancilla-mediated quantum computation. In fact the method generating T-duality (at least for the abelian case) is analogous to the manipulation of registers via ancilla qubits leading to an entangled system. Basically in the first part we implement the construction of an ancilla entangled with the first set of fields in our initial theory. Another ancilla comes on the other side of the theory in the form of our condition of having a pure gauge field and a flat connection, implemented via another auxiliary field. Now, we integrate / measure the ancillas erasing the information regarding what theory is "entangled" to what ancilla. We obtain a fully entangled state between four objects. If next, we implement a "measurement" on each ancilla separately, we obtain an entangled state between the two regions. In quantum information the protocol is relatively simple. We construct the ancillas as
\begin{equation}
\ket{C_{\alpha}^{\pm}}=\mathcal{N}(\ket{\alpha}\pm \ket{-\alpha})
\end{equation}
with $\mathcal{N}_{\pm}=\frac{1}{\sqrt{2(1\pm e^{-2 |\alpha|^{2}})}}$. The two qubit measurement process of the two ancillas generated as a two-qubit measurement produces the global state, while the single qubit ancilla measurements make the entanglement manifest. What we obtain is an entangled state between the two sides with unit probability.
The procedure is as follows: first we generate local entanglement between two non-interacting qubits, using ancilla qubits. The four qubits are initialised for example in the $\ket{+}$ state. Next a CPHASE gate is applied to the first qubit and its ancilla, followed by a Hadamard rotation of the first qubit. The same is applied on the second qubit and its ancilla. This generates an entangled state of the first qubit with its ancilla and the second qubit and its ancilla
\begin{equation}
\begin{array}{c}
\ket{e_{1}}=\frac{1}{\sqrt{2}}(\ket{f,f}+\ket{a_{1},a_{1}})\\
\\
\ket{e_{1}}=\frac{1}{\sqrt{2}}(\ket{s,s}+\ket{a_{2},a_{2}})\\
\end{array}
\end{equation}
We then perform a two qubit measurement on the two ancillas. The result is a four qubit entangled state that, according to the outcome of the measurement is either
\begin{equation}
\frac{1}{\sqrt{2}}(\ket{f,f,s,s} + \ket{a_{1},a_{1},a_{2},a_{2}})
\end{equation}
or
\begin{equation}
\frac{1}{\sqrt{2}}(\ket{f,a_{1},s,a_{2}} + \ket{a_{1},f,a_{2},s})
\end{equation}
Performing single qubit measurements on the ancillas, the first and the second qubits become projected onto a particular entangled state, for example
\begin{equation}
\ket{\Psi}=\frac{1}{\sqrt{2}}(\ket{+,+}+p\ket{-,-})
\end{equation}
This has led to an entangled state between the first and the second qubit.
While this procedure is not fundamentally new, it is important to understand how it is realised in a categorial interpretation of entanglement looking at gauge theories instead of quantum qubits.
First, gauge theories have an entanglement remnant from their construction as fibre bundles that implement, generally non-trivial, topological structures that are not describable globally by means of cartesian products of local patches.
Indeed, if one considers the procedure of generating a gauge field, hence "gauging a symmetry" one basically defines a gauge connection on a fibre bundle, and in order to do that, one requires the gauge group, which defines the global structure. This makes the patches that one wishes to connect on the base manifold, fundamentally impossible to decouple and decompose in separated, disjoint objects that would retain the global information. This is where the tension between local and global structures emerges and why we share the fibre bundle construction both in the case of gauge theories and in the case of a geometrical interpretation of quantum wavefunctions. The fact that our basis manifold in our derivation of T-duality has a gauge symmetry that we wish to gauge, introduces a global structure which we can use to create "entangled" theories, in a categorial sense. In this case however, we introduce a gauge field that is associated to a flat connection, and in order to do that we introduce also another field that implements this constraint. These are our ancilla constructions. We perform then two operations in different orders. What those realise is to eliminate the information between the two regions of the "entangled" theory that would be associated to each theory, and hence "entangles" the two sides of our construction. Then we Integrate over the respective ancillas individually, obtaining two theories that are dual. The two dual parts of the theory are represented by the two higher-entangled theories, hence it comes as no surprise that the resolution of one provides insight on the solutions of the other.
Let us rephrase: We have two qubits that are not entangled. We bring in two ancillas, they represent two auxiliary fields in our construction, both having the role of gauging a symmetry of the worldsheet and implementing a constraint on the global structure (imposing a flat connection). By the way they are introduced they generate global information shared by the two sides of our to-be duality. Our flat connection condition generates the globally shared information. Now we can decide to integrate out and eliminate the two ancillas, each at a time, leading to two theories, which continue to carry the global information shared by the previous conditions implemented by the ancillas but now being directly entangled. This results in two theories that are said to be T-dual. This therefore shows that, in this interpretation of the gauging procedure, the resulting duality is basically a higher form of entanglement.
It is not trivial to construct this type of higher entanglement. We need to give a pertinent definition of the operations we find in designing quantum information circuits that play a role in the operations we can perform on gauge theories and $\sigma$-models. This amounts to a consistent part of the argument being constructed by means of, what mathematicians would call, abstract nonsense, and as is usually the case with abstract nonsense, the results are deep and far reaching. In general quantum algorithms ancilla qubits are required because they need to maintain some level of reversibility of the computation. Indeed, quantum computation must be reversible, something that it has in common with the category of cobordisms in the worldsheet description of spacetime. If one creates a state one cannot simply turn it off as is the case in classical computation, where the effects can be eliminated by simply setting a variable to its natural zero state. The same is valid for spacetime worldsheets. Their history is preserved no matter what we decide to do at some point, and hence one cannot simply "undo" the effects encoded in its history by turning a spacetime worldsheet "off". Ancilla qubits can be used to preserve the global information over entangled states, while performing operations on the original states themselves, using local operations. Without such ancilla qubits, such operations would not be allowed by means of local computation. In an analogous sense let us consider what we are performing in the construction of T-dual theories. Considering the process of introducing the auxiliary gauge fields as analogues to ancillary qubits, we introduce them to encode the gauge symmetry while enforcing a special global structure, namely that the overall curvature vanishes. We prepare and preserve this by means of another auxiliary variable (analogue to a second ancilla qubit). However, while we keep the curvature term zero, and we expect the fields to be pure gauge, there exists global information that has been encoded and shared between the two dual partners. The ancilla qubits preserve it in a sense analogous to entanglement, just applied at a higher level, between two dual theories. The process of "measurement" is basically the result of integrating out the respective fields hence the process of "undoing" the local structures created before via the ancillas. While we undo the local actions, we preserve the "entanglement" which is recovered globally as two dual theories.
Hence T-duality, appears to be some higher form of entanglement which can be interpreted by means of quantum circuits in some higher category where a functorial relation between the quantum circuits and gauge theories is manifest.
\section{conclusion}
Quantum information theory has universal validity. By this we usually mean that we can create entangled states of objects and perform universal computations on them. However, there exists a more general notion of universality, one that is related to a categorial and functorial interpretation of quantum information that remains valid not only at the level of entangled photons or electrons, but also at a more abstract and hence more fundamental level, for example in the case of categories that have similar properties as quantum mechanics and hence are, form an axiomatic point of view, already "quantum". An axiomatisation of quantum mechanics has not been performed, but is deeply desired. While not having such a formulation, I assumed that non-separability of spaces of states in simple cartesian products of states is a fundamental aspect of quantum mechanics, and it is valid at the level of the fibre bundles which are also known to encode how a cartesian product fails to be cartesian due to global information. In reality, entangled states are exactly what remains if one eliminates from a quantum space of states all the states that are separable. Hence, this property seems to be rather general. By using this admittedly heuristic approach, one can implement quantum information operations on structures that are more general than our usual quantum states. In fact it is not known if there are other physical systems except the quantum ones that exhibit such non-separability properties. Hence, at least at the level of conjecture, one can assume this to be a fundamental aspect of a system being quantum. On the other side, most likely more axioms will be needed to properly define quantum mechanics. I expect however that this structure defined here will be maintained even when such axioms are added in a consistent way. However, it is also possible that by giving up on some or all of the other axioms, new more general versions of "quantum mechanics" will emerge. I do not know whether those will be relevant in any physical sense. Here, I used the analogue of ancilla qubits to explain a well known procedure of introducing auxiliary fields in a gauge theory and integrating them away, in the same way in which we use ancillas in quantum mechanics to introduce operations we wish to annihilate after their use. Of course, those operations will leave their global, locally undetectable, traces on the combined system, which are here interpreted as a T-duality.
There is much work left. First I did not discuss the non-abelian case of the manifold isometry, which creates its own complications. Then, it is strongly desirable to refine the concepts and to give a proper definition to the respective operations done, as well as a stronger connection between the gauge theory tricks involved and ancilla qubit constructions in quantum information. At the same time it would be desirable to show some universal properties that would encode more general principles of dualities between various $\sigma$-models and to properly encapsulate them in a categorial approach. If anything, I hope this article will spark new research in the connection between gauge and string theory dualities on one side and quantum information and its categorial interpretations on another side.
|
1,108,101,563,241 | arxiv | \section{Introduction}
Power supply ripple at harmonics of the mains power frequency (50~Hz) has been systematically observed in the transverse beam spectrum of the Large Hadron Collider (LHC) since the start of its operation. In a previous paper \cite{previous}, we investigated the source of the perturbation based on a systematic analysis of experimental observations. It was clearly shown that the 50~Hz harmonics are not an artifact of the instrumentation system but correspond to actual beam oscillations.
Two regimes of interest have been identified in the transverse beam spectrum: a series of 50~Hz harmonics extending up to approximately 3.6~kHz, referred to as the \textit{low-frequency cluster}, and a cluster of 50~Hz harmonics centered around the alias of the betatron frequency, i.e., $f_{\rm rev}-f_x$, where $f_{\rm rev}$=11.245~kHz is the LHC revolution frequency and $f_x$ the betatron frequency, namely the \textit{high-frequency cluster}. Both clusters are the result of dipolar beam excitations and it was demonstrated with dedicated experiments that the source of the low-frequency cluster are the eight Silicon Controlled Rectifier (SCR) power supplies of the main dipoles \cite{SCR, SCR2}. Table~\ref{tab:50Hz} presents the frequency and the amplitude of the 12 most important 50~Hz harmonics in the transverse spectrum of Beam 1 during collisions as observed in the Q7 pickup of the transverse damper Observation Box (ADTObsBox) \cite{ADT, ADT3, ADT4}. The offsets induced in Beam 2 due to the power supply ripple are also depicted, which are approximately lower by a factor of two compared to Beam 1. Based on the source, the 50~Hz harmonics are expected to be present in the future operation of the accelerator.
\begin{table*}
\caption{\label{tab:50Hz}%
The beam offsets observed in the Q7 pickup of the transverse damper for the 12 largest 50~Hz harmonics in the horizontal spectrum of Beam 1 (B1) and 2 (B2) during collisions.}
\begin{ruledtabular}
\begin{tabular}{cccccccc}
\textrm{Frequency (kHz)}& Cluster &
\textrm{B1 offset (nm)} & \textrm{B2 offset (nm)} & \textrm{Frequency (kHz)}& Cluster &
\textrm{B1 offset (nm)} & \textrm{B2 offset (nm)}\\
\colrule
7.6 & High& 166.2 & 76.51 & 7.7 & High& 109.75 & 51.38\\
1.2 & Low& 98.63 & 71.87 & 7.65 & High& 97.4 & 51.97 \\
7.8 & High& 92.75 & 42.78 & 2.95 & Low& 91.62 & 32.21\\
2.35 & Low& 88.48 & 29.61 & 7.9 & High& 82.23 & 25.46 \\
7.5 & High& 77.82 & 35.08 & 2.75 & Low& 67.43 & 31.77 \\
7.85& High & 67.37 & 18.86 & 0.6 & Low &54.14 & 14.33 \\
\end{tabular}
\end{ruledtabular}
\end{table*}
In the presence of strong non-linearities such as beam-beam interactions and fields of non-linear magnets, a modulation in the dipole strength due to power supply ripple results in a resonance condition \cite{tomas1, tomas2}:
\begin{equation}
\label{eq:tune}
l \cdot Q_x + m\cdot Q_y + p \cdot Q_p = n
\end{equation}
where $l, m,p,n$ are integers, $Q_x$ and $Q_y$ are the horizontal and vertical tunes, respectively, and $Q_p$ is the power supply ripple tune. Power supply ripple can impact the beam performance by introducing resonances in addition to the ones excited due to the non-linearities of the lattice. These resonances can increase the tune diffusion of the particles in the distribution, which may lead to the reduction of the beam lifetime. The aim of the present paper is to determine whether the 50~Hz harmonics act as a limiting factor to the beam performance of the LHC, as well as its future upgrade, the High-Luminosity LHC (HL-LHC) \cite{Apollinari:2284929}, using single-particle tracking simulations with power supply ripple in the form of dipolar excitations. The impact of power supply ripple that results in a tune modulation is discussed in a previous paper \cite{kostoglou2020tune}.
First, the motion of a particle in the presence of a modulated dipolar field error is described using a linear formalism (Section~\ref{app:modulated_dipole}). Second, single-tone dipolar excitations are considered in the tracking simulations to define a minimum power supply ripple threshold that results in a reduction of the Dynamic Aperture (DA) in the presence of strong non-linearities such as long-range and head-on beam-beam encounters, chromatic arc sextupoles and Landau octupoles. Then, the impact of the harmonics on the beam performance in terms of tune diffusion, DA and lifetime is discussed in Sec.~\ref{Sec:simulations} by including in the simulations a power supply ripple spectrum as acquired from experimental observations. Finally, the impact of controlled dipolar excitations on the beam lifetime, conducted with the transverse damper, is described in Section~\ref{Sec:adt_excitations}, which provides a tool for the validation of the DA simulations in the presence of power supply ripple.
\section{Linear formalism for a modulated dipolar field error}
\label{app:modulated_dipole}
In a circular accelerator, the kick related to a modulated dipolar field error \(\Theta_p\) with a deflection \(\theta_{p}\) and a frequency of \(Q_{p}\) oscillations per turn can be represented in normalized phase space as:
\begin{equation}
\bar{P}_n = \begin{pmatrix} 0 \\ \Theta_{p}(n) \end{pmatrix} = \begin{pmatrix} 0 \\ \sqrt{\beta} \theta_{p} \cos{(2 \pi Q_p n)} \end{pmatrix},
\label{appendix:eq_1}
\end{equation}
where \(n\) is the turn considered and $\beta$ is the $\beta$-function at the location of the perturbation.
In the linear approximation, considering only the horizontal motion and assuming that the source and the observation point are situated in the same location, the vector representation of the position and momentum at turn \(N\) is:
\begin{equation}
\bar{X}_N=
\left(\begin{array}{c}
\bar{x}_N
\\
\bar{x}_N'
\end{array}\right)=
\sum_{n=-\infty}^{N}M^{N-n}\bar{P}_n,
\label{appendix:eq_2}
\end{equation}
where \(M\) is the linear rotation equal to:
\begin{equation}
M^{N}=\left(\begin{array}{cc}{\cos (2 \pi Q N)} & {\sin (2 \pi Q N)} \\ {-\sin (2 \pi Q N)} & {\cos (2 \pi Q N)}\end{array}\right)
\label{appendix:eq_3}
\end{equation}
with \(Q\) representing the machine betatron tune. Combining Eq.~\eqref{appendix:eq_1}, \eqref{appendix:eq_2} and \eqref{appendix:eq_3} and assuming that the perturbation is present from $n \rightarrow-\infty$, yields:
\begin{align}
& \bar{x}_N= \sqrt{\beta} \theta_{p} \sum_{n=-\infty}^{N} \cos{(2 \pi Q_p n)} \sin{\left(2 (N-n) \pi Q\right)}, \nonumber \\
& \bar{x}'_N= \sqrt{\beta} \theta_{p} \sum_{n=-\infty}^{N} \cos{(2 \pi Q_p n)} \cos{\left(2 (N-n) \pi Q \right)}.
\label{appendix:eq_4}
\end{align}
In Eq.~\eqref{appendix:eq_4}, using Euler's formula and writing the sum expressions in terms of geometric series yields:
\begin{widetext}
\begin{equation}
{
\begin{aligned}
{\bar x}_N =\sqrt{\beta} \theta_p \lim_{k\to-\infty} \Bigr(&\frac{\cos (2 \pi Q_p N) \sin(2 \pi Q)}{2(\cos(2\pi Q_p)-\cos(2\pi Q))} \\
&+\frac{- \cos (2 \pi Q_p (k-1) ) \sin (2 \pi Q (k - N)) + \cos (2 \pi Q_p k ) \sin (2 \pi Q(k - 1 - N) )}{2(\cos(2\pi Q_p)-\cos(2\pi Q))} \Bigl) \;\;,\\
{\bar x'}_N = \sqrt{\beta} \theta_p \lim_{k\to-\infty} \Bigr(&\frac{\cos (2 \pi Q_p (N+1) )-\cos (2 \pi Q_p N )\cos (2 \pi Q)}{2(\cos{(2\pi Q_p)}-\cos{(2\pi Q)})}\\
&+\frac{\cos (2 \pi Q_p (k-1)) \cos (2 \pi Q (k-N))+\cos (2 \pi Q_p k) \cos (2 \pi Q (k-1-N))}{2(\cos{(2\pi Q_p)}-\cos{(2\pi Q)})} \;\;.\\
\label{eq:sum2}
\end{aligned}}
\end{equation}
\end{widetext}
The limit of the terms in the numerator of Eq.~\eqref{eq:sum2} involving $k$ in the argument of the trigonometric functions is bounded but indeterminate. Using the Ces\`aro mean the trigonometric products involving $k$ as an argument in Eq.~\eqref{eq:sum2} is equal to zero and the expressions for the position and momentum simply become:
\begin{align}
\label{eq:sumsol}
{\bar x}_N &= \sqrt{\beta} \theta_p \frac{\cos (2 \pi Q_p N) \sin(2 \pi Q) }{2(\cos{(2\pi Q_p)}-\cos{(2\pi Q)})}, \\
{\bar x'}_N &= \sqrt{\beta} \theta_p \frac{\cos(2 \pi Q_p N) \cos(2 \pi Q)-\cos (2 \pi (N+1) Q_p)}{2(\cos{(2\pi Q_p)}-\cos{(2\pi Q)})}. \nonumber
\end{align}
In physical coordinates, the maximum offset computed from Eq.~\eqref{eq:sumsol} is:
\begin{equation}
|x_{\text{max}, N}|=\left\lvert \frac{\beta \theta_{p}\sin{(2\pi Q)}}{2 \left( \cos{(2\pi Q_{p})}-\cos(2\pi Q) \right) }\right\rvert.
\label{appendix:eq_6}
\end{equation}
\section{Simulations of 50~Hz harmonics with beam-beam interactions}
\label{Sec:simulations}
Tracking simulations are performed using the single-particle symplectic code SixTrack \cite{sixtrack, sixtrack2}. A distribution of particles is tracked in the element-by-element LHC and HL-LHC lattice at top energy, including important non-linearities such as head-on and long-range beam-beam encounters, sextupoles for chromaticity control and Landau octupoles for the mitigation of collective instabilities. For the LHC case, the chromaticity in the horizontal and vertical plane is $Q'$=15 and the amplitude detuning coefficients $\alpha_{n_1 n_2}=\partial Q_{n_1} / \partial(2 J_{n_2})$ with $n_1, n_2 \in \{x,y\}$ are:
\begin{eqnarray}
\alpha_{\rm xx}=16.8\times 10^{4} \ \text{m}^{-1} , \alpha_{\rm xy} = -11.8\times 10^{4} \ \text{m}^{-1}, \nonumber \\
\alpha_{\rm yy} =16.5\times 10^{4} \ \text{m}^{-1}
\end{eqnarray}
as computed from the PTC-normal module of the MAD-X code \cite{madx}.
The beam-beam interactions are simulated with the weak-strong approximation. The maximum duration of the tracking is \(\rm 10^6\) turns that corresponds to approximately 90 seconds of beam collisions in operation. The main parameters used in simulations for LHC and HL-LHC are presented in Table~\ref{tab:table_parameters_collision}. For the LHC, the simulated conditions refer to the start of the collisions. The simulated conditions in the HL-LHC case correspond to the nominal operational scenario at the end of the $\beta^*$-leveling, taking place after several hours from the start of the collisions \cite{HL_LHC_scenarios}.
\begin{table*}
\caption{\label{tab:table_parameters_collision}%
The LHC and HL-LHC parameters at top energy used in the tracking simulations. The LHC parameters correspond to the start of collisions, while the HL-LHC parameters refer to the end of the $\beta^*$-leveling as envisaged in the nominal scenario \cite{HL_LHC_scenarios}.
}
\begin{ruledtabular}
\begin{tabular}{lcc}
\textrm{Parameters (unit)}&
\textrm{LHC (values)} & \textrm{HL-LHC (values)}\\
\colrule
Beam energy (TeV) & 6.5 & 7\\
Bunch spacing (ns) & 25 & 25\\
RMS bunch length (cm) & 7.5 & 7.5 \\
Bunch population (protons) & $1.25 \times 10^{11}$ & $1.2\times 10^{11}$\\
Beam-beam parameter $\xi_{x,y} $ & 7.6$\times 10^{-3}$ & 5.8$\times 10^{-3}$ \\
Horizontal tune \(Q _x\) &0.31 & 0.31/0.315\footnote{\label{note1}Nominal/optimized working point.} \\
Vertical tune \(Q_y\) &0.32/0.315\textsuperscript{\ref{note1}} & 0.32\\
Horizontal and vertical normalized emittance (\(\rm \mu m \ rad\)) & 2.0 & 2.5\\
Horizontal and vertical chromaticity & 15 & 15\\
Octupole current (A) & 550 & -300\\
IP1/5 Half crossing angle (\(\rm \mu rad\)) & 160 & 250\\
Horizontal and vertical IP1/5 \(\beta^*\) (cm) & 30 & 15\\
Total RF voltage (MV) & 12 & 16 \\
Synchrotron frequency (Hz) & 23.8 & 23.8 \\
Revolution frequency (kHz) & 11.245 & 11.245\\
Relative momentum deviation $\delta p /p$ & 27$\times 10^{-5}$ & 27$\times 10^{-5}$ \\
Horizontal \(\beta\)-function at Q7 pickup (m) & 105.4 & 102.3\\
\end{tabular}
\end{ruledtabular}
\end{table*}
In a previous paper \cite{previous}, it was demonstrated that, as far as the low-frequency cluster is concerned, the power supply ripple is distributed in all eight sectors. An accurate representation of the power supply ripple propagation across the chains of the LHC dipoles requires a model of the transfer function as a transmission line for all the spectral components in the low-frequency cluster, similarly to the studies performed for the SPS \cite{Burla:263879}. Furthermore, as the exact mechanism of the high-frequency cluster is not yet clearly identified, an accurate transfer function is not known at the moment.
To simplify these studies, the distributed power supply ripple is projected to a single location with an equivalent impact on the beam's motion. Specifically, a horizontal modulated dipolar source is included at the location of the Q7 pickup of the transverse damper, which coincides with the observation point. To simulate the dipolar excitation, the strength of a horizontal kicker is modulated with a sinusoidal function:
\begin{equation}
\label{eq:mod}
\Delta k (t) = \theta_p \cos{(2\pi f_p t)},
\end{equation}
where $t$ is the time, $\theta_p$ the deflection, $f_p=Q_p \cdot f_{\rm rev}$ the power supply ripple frequency and $f_{rev}$ the revolution frequency listed in Table~\ref{tab:table_parameters_collision}. In this way, the contribution of all the dipoles is represented by a single, equivalent kick and the offsets observed in the LHC spectrum are reproduced in the simulations. The maximum offset induced on the particle's motion is computed from Eq.~\eqref{appendix:eq_6} with the horizontal $\beta$-function listed in Table~\ref{tab:table_parameters_collision}.
The simulations are then repeated for the HL-LHC case. The need to perform projections for the HL-LHC is justified by the fact that no modifications are envisaged for the power supplies of the main dipoles. Consequently, based on the source, the 50~Hz harmonic are expected to also be present in the HL-LHC era. In the following sections, the HL-LHC studies are based on the power supply ripple spectrum acquired experimentally from LHC, although it is expected that the foreseen upgrade of the transverse damper system will lead to a more efficient suppression of the harmonics.
\subsection{Impact of single-tone ripple spectrum on the Dynamic Aperture}
As a first step, we consider individual tones in the power supply ripple spectrum for increasing values of the deflection $\theta_p$. For each study, a different combination of the frequency $f_p$ and the amplitude of the excitation $\theta_p$ is selected. For each case, the minimum DA in the initial configuration space, i.e., the initial horizontal $x$ and vertical $y$ displacements is compared to the value derived in the absence of power supply ripple. In these simulations, the tune modulation due to the coupling of the synchrotron oscillations to the betatron motion for a chromaticity and a relative momentum deviation listed in Table~\ref{tab:table_parameters_collision} is also included.
The scan in the ripple parameter space (\(f_p, \theta_p\)) is performed to define the most dangerous tones of the low and high-frequency cluster, i.e., the frequencies that, for a constant excitation amplitude, have a maximum impact on the DA. Then, the offset induced in the beam motion at the location of observation point is estimated from Eq.~\eqref{appendix:eq_6} with the parameters $f_p$, $\theta_p$ and the $\beta$-function of Table~\ref{tab:table_parameters_collision}. A threshold for the minimum offset at the Q7 pickup that leads to a reduction of DA is determined.
Figure~\ref{fig:heatmap} presents the ripple frequency as a function of the offset at the observation point. The harmonics of the low and the high-frequency cluster that reside in the vicinity of \(f_x\) and \(f_{\rm rev}-f_x\) have been selected for the analysis. A color code is assigned to the reduction of the minimum DA to illustrate the regimes with a negligible (blue) and significant (yellow) impact.
\begin{figure}
\includegraphics[width = \columnwidth]{heat_LHC.png}
\caption{\label{fig:heatmap} The ripple frequency as a function of the horizontal offset in the Q7 pickup, as computed from the deflection and Eq.~\eqref{appendix:eq_6}. A color code is assigned to the reduction of the minimum DA from the value computed in the absence of power supply ripple.}
\end{figure}
From the scan, it is evident that the ripple frequencies with the largest impact are the ones that reside in the proximity of the tune and its alias. For the LHC, an offset threshold of \(\rm 0.4 \ \mu m\) is defined, while this limit reduces to \(\rm 0.2 \ \mu m\) for the HL-LHC. For comparison, the maximum excitation observed experimentally due to the 50~Hz lines is approximately 0.16~\(\rm \mu m\) as presented in Table~\ref{tab:50Hz}.
\subsection{Frequency Map Analysis with a realistic power supply ripple spectrum}
A more significant impact on the beam performance is anticipated in the presence of multiple ripple tones, similar to the experimental observations of the 50~Hz harmonics, than considering individual tones due to the resonance overlap \cite{intro1994_1, SPS2_tune}. To this end, a realistic power supply ripple spectrum must be included in the simulations. Similarly to Table~\ref{tab:50Hz}, the offsets and the frequencies of the 40 largest 50~Hz harmonics are extracted from the horizontal beam spectrum as observed during operation. For each frequency, the equivalent kick at the location of the Q7 pickup is computed from Eq.~\eqref{appendix:eq_6} and it used as an input for the power supply ripple simulations.
Figure~\ref{fig:spectrum} shows the horizontal spectrum of Beam 1 from the experimental observations (black) centered around the low (left) and high (right) frequency cluster. A single particle is tracked in the LHC lattice including the most important 50~Hz harmonics. The output of the simulations (green) is also illustrated in Fig.~\ref{fig:spectrum}. The comparison of the two is a sanity check illustrating the good agreement between the simulated and the experimental beam spectrum. A similar agreement (between simulated and expected beam spectrum) is found for the HL-LHC case, were the equivalent kicks have been recomputed due to a small variation of the \(\beta\)-function as shown in Table~\ref{tab:table_parameters_collision}.
\begin{figure}
\includegraphics[width = \columnwidth]{Simulations.png}
\caption{\label{fig:spectrum} The spectrum centered around the low (left) and high (right) frequency cluster as acquired experimentally from the horizontal plane of Beam 1 (black) and from tracking simulations (green).}
\end{figure}
The studies are organized as follows. First, a study in the absence of power supply ripple is performed that is used as a baseline. Second, the largest 50~Hz harmonics of the low-frequency cluster are considered. Then, a separate study is conducted including only the most important harmonics of the high-frequency cluster. Last, both regimes are included in the simulations.
Particles extending up to 6 \(\rm \sigma\) in the initial configuration space are tracked for $\rm 10^4$ turns in the LHC and HL-LHC lattice. The particles are placed off-momentum as shown in Table~\ref{tab:table_parameters_collision}, without however considering the impact of synchrotron oscillations. A Frequency Map Analysis (FMA) is performed for each study \cite{fma1, fma2, fma3, fma4}. The turn-by-turn data are divided into two groups containing the first and last 2000 turns, respectively. The tune of each particle is computed for each time interval using the Numerical Analysis of Fundamental Frequencies (NAFF) algorithm \cite{NAFF1, NAFF, PhysRevAccelBeams.22.071002}. Comparing the variation of the tune of each particle across the two time spans reveals information concerning its tune diffusion rate $D$ that is defined as:
\begin{equation}
\label{eq:diffusion}
D = \sqrt{ (Q_{x_1}- Q_{x_2})^2 + (Q_{y_1}- Q_{y_2})^2 },
\end{equation}
where $Q_{j_1}, Q_{j_2}$ with $j \in \{x,y\}$ denote the tunes of each particle in the first and second time interval, respectively.
Figure~\ref{fig:fmas_LHC} illustrates the frequency maps (left panel) and the initial configuration space (right panel) for the four studies in the LHC. For the frequency map, the particle tunes of the second interval are plotted and a color code is assigned to logarithm of the tune diffusion rate $D$ to distinguish the stable particles (blue) from the ones with an important variation of their tune (red) due to the resonances. The gray lines denote the nominal resonances, i.e., the resonances that intrinsically arise from the non-linear fields such as non-linear magnets and beam-beam effects without power supply ripple.
\begin{figure*}
\subfloat{\subfigimg[width = 0.49\textwidth]{ \textbf{a)}}{LHC_fma_without.png} \label{subfig:FMA_without}}
\subfloat{\subfigimg[width = 0.49\textwidth]{\textbf{b)}}{LHC_fma_low.png} \label{subfig:FMA_low}} \\
\subfloat{\subfigimg[width = 0.49\textwidth]{\textbf{c)}}{LHC_fma_high.png} \label{subfig:FMA_high}}
\subfloat{\subfigimg[width = 0.49\textwidth]{\textbf{d)}}{LHC_fma_all.png} \label{subfig:FMA_all}} \\
\caption{\label{fig:fmas_LHC} The frequency maps (left) and the initial configuration space (right) color-coded with the logarithm of the tune diffusion rate (Eq.~\eqref{eq:diffusion}) for the LHC simulation parameters of Table~\ref{tab:table_parameters_collision} (a) in the absence of power supply ripple , (b) with the 50~Hz harmonics of the low and (c) high-frequency cluster and (d) combining both regimes. The gray lines represent the nominal resonances and the black and blue lines illustrate the resonances (Eq.~\eqref{eq:low} and \eqref{eq:high}) excited due to power supply ripple.}
\end{figure*}
In the absence of power supply ripple (Fig.~\ref{subfig:FMA_without}), an important impact is observed due to the third-order resonance ($\text{3} \cdot Q_y \rm =\text{1}$), which is in the vicinity of tune footprint and it affects particles at large amplitudes (red).
From the FMAs, it is observed that the dipolar power supply ripple results in an increase of the particles' diffusion rate, first, by enhancing the strength of some of the nominal resonances and second, through the excitation of resonances in addition to the nominal ones as described in Eq.~\eqref{eq:tune}. Including the low-frequency cluster (Fig.~\ref{subfig:FMA_low}), the resonances with the largest impact are:
\begin{equation}
\label{eq:low}
Q_x = Q_p.
\end{equation}
As the power supply ripple is injected in the horizontal plane, these resonances appear as vertical lines in the tune domain in a location equal to the excitation frequency (black). Due to the coupling of the transverse planes, the strongest dipolar excitations are also visible in the vertical plane, appearing as horizontal lines in the frequency maps.
The excitations of the high-frequency cluster appear as aliases (blue). For instance, the excitation at 7.8~kHz in Fig.~\ref{subfig:FMA_high} is located at $f_{\rm rev} \rm -7.8~kHz$. For the high-frequency cluster (Fig.~\ref{subfig:FMA_high}), the most critical resonances are:
\begin{equation}
\label{eq:high}
Q_x = 1-Q_p.
\end{equation}
For the 50~Hz harmonics in the vicinity of the betatron tune and its alias these additional resonances are located inside the beam's footprint. As clearly shown in the $x-y$ plane (right panel), the existence of such resonances impacts both the core and the tails of the distribution. Reviewing the impact of both clusters (Fig.~\ref{subfig:FMA_all}) indicates that the main contributors to the increase of tune diffusion rate are the spectral components in the high-frequency cluster. Similar results are obtained for the HL-LHC case.
\subsection{Intensity evolution simulations}
\label{sec:losses}
Quantifying the impact of the power supply ripple and the non-linearities on the intensity evolution requires tracking a distribution with realistic profiles in all planes that are similar to the ones observed experimentally. If impacted by resonances, particles at the tails of the distribution close to the limit of DA diffuse and will eventually be lost. Therefore, a detailed representation of the tails of the distribution is needed along with a realistic longitudinal distribution that extends over the whole bucket height.
The initial conditions form a 4D round distribution, which is sampled from a uniform distribution, extending up to 7 \(\rm \sigma \) both in the initial configuration space, as well as in the initial phase space. In the longitudinal plane, the relative momentum deviation of the particles is a uniform distribution that extends up to the limit of the bucket height (not the value presented in Table~\ref{tab:table_parameters_collision}).
To reduce the statistical variations, $\rm 25 \times 10^{4}$ particles are tracked in the LHC lattice including synchrotron oscillations. In the post-processing, a weight is assigned to each particle according to its initial amplitude, as computed from the Probability Density Function (PDF) of the final distribution that needs to be simulated. The weight of each particle is:
\begin{equation}
w = \frac{\prod_{j=1}^{3} \text{PDF}(r_j, \sigma_j) }{\sum_i w_i},
\end{equation}
where $j$ iterates over the three planes, $r_j = x_{j}^2 + p_{j}^2$ with $x_j, p_j$ the initial normalized position and momentum coordinates, $\sigma_j$ represents the RMS beam size in the $j$-plane and the denominator denotes the normalization with the sum of the weights of all the particles.
Experimental observations have shown that the tails of the LHC bunch profiles are overpopulated in the transverse plane and underpopulated in the longitudinal plane compared to a normal distribution \cite{profiles, papadopoulou2018impact}. For an accurate description of the bunch profiles, the PDF of the q-Gaussian distribution is employed \cite{tsallis1, tsallis2, wolfram}:
\begin{equation}
\label{eq:pdf}
\begin{array}{cc}
\text{PDF}(r, \sigma) = \Bigg\{ &
\begin{array}{cc}
\frac{e^{-\frac{r^2}{2 \sigma ^2}}}{\sqrt{2 \pi } \sigma } & q=1 \\
\frac{\sqrt{q-1} \Gamma \left(\frac{1}{q-1}\right) \left(\frac{(q-1) r^2}{2 \sigma ^2}+1\right)^{\frac{1}{1-q}}}{\sqrt{2 \pi } \sigma \Gamma \left(-\frac{q-3}{2 (q-1)}\right)} & 1<q<3 \\
\frac{\sqrt{1-q} \Gamma \left(\frac{3}{2}+\frac{1}{1-q}\right) \left(\frac{(q-1) r^2}{2 \sigma ^2}+1\right)^{\frac{1}{1-q}}}{\sqrt{2 \pi } \sigma \Gamma \left(1+\frac{1}{1-q}\right)} & q<
\end{array}
\\
\end{array}
\end{equation}
The parameter $q$ and the ratio of the RMS beam size of the Gaussian ($q$=1) to the q-Gaussian distribution $\sigma_{\rm G} / \sigma_{\rm qG}$ are presented in Table \ref{tab:qgauss} for the longitudinal and the transverse plane (assuming the same profiles for the horizontal and vertical plane).
\begin{table}
\caption{\label{tab:qgauss}%
The parameters of the q-Gaussian PDF shown in Eq.~\eqref{eq:pdf} for the longitudinal and transverse plane \cite{profiles, papadopoulou2018impact}.
}
\begin{ruledtabular}
\begin{tabular}{ccc}
\textrm{Parameters}&
\textrm{Longitudinal plane} & \textrm{Transverse plane}\\
\colrule
$q$ & 0.88 & 1.15 \\
$\sigma_{\rm G} / \sigma_{\rm qG}$ & 0.95 & 1.05
\end{tabular}
\end{ruledtabular}
\end{table}
Furthermore, a mechanical aperture is defined in the post-processing at 6.6 \(\rm \sigma\) to simulate the effect of the primary collimators on the transverse plane based on the settings of the 2018 operation. Particles beyond this threshold are considered lost and their weight is set to zero.
Figure~\ref{fig:losses} presents the intensity evolution without power supply ripple (black), including the 50~Hz harmonics of the low (blue) or high (orange) frequency cluster and considering both regimes (red). The results indicate that the presence of the 50~Hz harmonics leads to a reduction of the beam intensity, which is already visible with a tracking equivalent to 90 seconds of beam collisions. The results also show that, for the time span under consideration, the high-frequency cluster is the main contributor to the decrease of the beam intensity. A similar effect is observed when considering the same power supply ripple spectrum for the HL-LHC case with the high-frequency cluster acting as the main contributor.
\begin{figure}
\includegraphics[width = \columnwidth]{low_high_all.png}
\caption{\label{fig:losses} Intensity evolution without power supply ripple (black), considering only the low (blue) or high (orange) frequency cluster and including the 50~Hz harmonics in both regimes (red).}
\end{figure}
To quantify the impact of the 50~Hz harmonics on the beam performance, the lifetime $\tau$ is computed from the intensity evolution $I$ as:
\begin{equation}
\label{eq:lifetime}
I(t) = I_0 \cdot e^{-\frac{t}{\tau}},
\end{equation}
where $I_0$ is the initial intensity and $t$ the time. An approximation of the instantaneous lifetime is estimated by fitting the exponential decay of the intensity with a sliding window, with each step consisting of a few thousand turns.
Figure~\ref{fig:losses_b1b2} illustrates the intensity evolution in the absence of power supply ripple (black), including the power supply ripple spectrum from the experimental observations of Beam 1 (blue) and 2 (red). The green line indicates the fits of Eq.~\eqref{eq:lifetime} to compute the lifetime evolution.
\begin{figure}
\includegraphics[width = \columnwidth]{life_fit_115.png}
\caption{\label{fig:losses_b1b2} Intensity evolution in the absence of power supply ripple (black), including the power supply ripple spectrum of Beam 1 (blue) and 2 (red). The green line indicates the fits of the exponential decay of the intensity to compute lifetime (Eq.~\eqref{eq:lifetime}).}
\end{figure}
As shown in Table~\ref{tab:50Hz}, the fact that the power supply ripple spectrum of Beam 2 is lower by approximately a factor of two compared to Beam 1 results in an asymmetry of the intensity evolution between the two beams. Starting from $\tau=28.4$~h in the absence of power supply ripple, the lifetime reduces to $\tau=27.4$~h when including the ripple spectrum of Beam 2 in the simulations and $\tau=22.3$~h when considering the spectrum of Beam 1. During the operation of the accelerator in run 2 (2015-2018), it has been observed that the lifetime of Beam 1 was systematically lower than the lifetime of Beam 2 \cite{b1life1}. It is the first time that simulations reveal that, amongst other mechanisms, the 50~Hz harmonics can contribute to this effect.
\section{Simulation benchmark with controlled excitations}
\label{Sec:adt_excitations}
During the latest LHC run in 2018, controlled dipolar excitations were applied on the beam using the transverse damper kicker. The goal of the experiment was to study the impact of dipolar excitations at various frequencies and amplitudes and to validate our simulation framework in a controlled manner. The experiments were performed at injection energy.
Experimentally, some of the excitations led to a significant reduction of the beam lifetime. To retrieve the initial deflection applied from the transverse damper, the offset and the frequency are extracted from the calibrated ADTObsBox beam spectrum. For instance, Fig.~\ref{fig:excitation_2.5kHz} shows the horizontal spectrum of Beam 1 during a controlled excitation at 2.5~kHz (green star-shaped marker) for a single bunch. The gray vertical lines illustrate the multiples of 50~Hz. Then, using the offset and the frequency, the equivalent kick is computed from Eq.~\ref{appendix:eq_6}. This procedure is repeated for all the excitations applied during the experiments.
\begin{figure}
\includegraphics[width = \columnwidth]{Fill7342_exc.png}
\caption{\label{fig:excitation_2.5kHz} The horizontal spectrum of a single bunch (black) during a controlled dipolar excitation at 2.5~kHz (green marker) performed using the transverse damper. The vertical lines represent the multiples of 50~Hz.}
\end{figure}
The impact of the excitations on the beam lifetime is compared against the DA scans computed with tracking studies in the presence of dipolar power supply ripple. The important parameters of the tracking simulations at injection energy are summarized in Table~\ref{tab:injection_parameters}.
\begin{table}
\caption{\label{tab:injection_parameters}%
The LHC parameters at injection energy used in the simulations with dipolar power supply ripple.
}
\begin{ruledtabular}
\begin{tabular}{cc}
\textrm{Parameters (unit)}&
\textrm{Values} \\
\colrule
Beam energy (GeV) & 450 \\
Bunch spacing (ns) & 25 \\
RMS bunch length (cm) & 13 \\
RF voltage (MV) & 8 \\
Betatron tunes (\(Q _x, Q_y\)) & (0.275, 0.295) \\
H/V normalized emittance (\(\rm \mu m \ rad\)) & 2.5 \\
H/V chromaticity & 15 \\
Octupole current (A) & 20 \\
Bunch population (protons) & $1.15\times 10^{11}$ \\
Horizontal \(\beta\)-function at Q7 (m) & 130.9 \\
\end{tabular}
\end{ruledtabular}
\end{table}
In each study, a different combination of the excitation frequency and amplitude is selected. In particular, considering a constant excitation frequency, the value of the deflection is increased and the minimum DA is computed for each case. A ripple amplitude threshold is defined as a function of the excitation frequency. Beyond this limit, a reduction of DA is expected.
Figure~\ref{fig:heat_injection} presents the frequency of the excitation as a function of the deflection. A color code is assigned to the minimum DA to distinguish the regime where the power supply ripple has no significant impact (blue) from the one where a significant reduction of DA (red) is observed in the simulations.
\begin{figure}
\includegraphics[width =\columnwidth]{MD_B1H_heat2.png}
\caption{\label{fig:heat_injection} The frequency of the excitation as a function of the deflection. A color code is assigned to the minimum DA computed with tracking simulations including power supply ripple. The star-shaped markers present the equivalent kicks, as computed from the beam spectrum and Eq.~\eqref{appendix:eq_6} during the controlled excitations with the transverse damper. The red markers indicate that the excitation had an impact on the beam lifetime. The blue markers denote the excitations that did not affect the beam lifetime.}
\end{figure}
In Fig.~\ref{fig:heat_injection}, the star-shaped markers denote the experimental excitation kicks and frequencies. A color code is assigned to the markers that allows distinguishing the excitations that had no impact on lifetime experimentally (blue) from those that lead to a lifetime drop (red). Although an excitation at 8.1~kHz was performed experimentally, the position measurements at this time were not stored and a star-shaped marker is not included.
The comparison between experimental observations and the power supply ripple threshold defined by DA simulations yields a fairly good agreement between the two for the majority of the excitations, taking into account the simplicity of the machine model and the absence of effects such as linear and non-linear imperfections in the simulations. This comparison provides a validation of our simulation framework including power supply ripple. The method to benchmark simulations and experimental findings presented in this section is not only limited to the studies of the 50~Hz harmonics but can be used to validate the tracking results for different types of power supply ripple and noise effects.
\section{Conclusions}
The purpose of the current paper was to determine whether the 50~Hz harmonics perturbation is a mechanism that can impact the beam performance during the LHC operation. To this end, single-particle tracking simulations were performed in the LHC and HL-LHC lattice including important non-linear fields such as head-on and long-range beam-beam interactions, chromatic sextupoles and Landau octupoles.
Including a realistic power supply ripple spectrum in the simulations shows that the 50~Hz harmonics increase the tune diffusion of the particles through the excitation of additional resonances, a mechanism that was clearly shown with frequency maps. The high-frequency cluster is identified as the main contributor.
From simulations that correspond to 90 seconds of beam collisions, the increase of the tune diffusion due to the 50~Hz harmonics results in a lifetime reduction of 21\% with respect to the lifetime in the absence of power supply ripple. Based on these results, it is concluded that the 50~Hz harmonics have an impact on the beam performance during the LHC operation. Mitigation measures should be incorporated in the future operation of the accelerator to suppress the high-frequency cluster from the beam motion.
Due to the asymmetry of the power supply ripple spectrum between Beam 1 and 2 by a factor of approximately two, the simulations illustrate a clear discrepancy in the intensity evolution of the two beams. Including realistic bunch profiles, the estimated lifetime is $\tau=22.3$~h and $\tau=27.4$~h for Beam 1 and 2, respectively. An important lifetime asymmetry between the two beams has been observed since the beginning of run 2 and it is the first time that tracking simulations show that power supply ripple can contribute to this effect.
In the context of this study, a general simulation framework has been developed. This paper illustrated a method to define an acceptable power supply ripple threshold for operation through DA scans. The results of these scans were bench-marked against experimental observations with controlled excitations using the transverse damper. Finally, a method to determine the intensity evolution with weighted distributions and realistic bunch profiles has been demonstrated. The analysis methods presented in this paper can be applied to studies of other types of power supply ripple and noise effects.
\begin{acknowledgments}
The authors gratefully acknowledge H.~Bartosik, M.~C.~Bastos, O.~S.~Brüning, R.~T.~Garcia, M.~Martino and S.~Papadopoulou for valuable suggestions and discussions on this work. We would like to thank D.~Valuch and M.~Soderen for all transverse damper related measurements and experiments.
\end{acknowledgments}
\section{Introduction}
Power supply ripple at harmonics of the mains power frequency (50~Hz) has been systematically observed in the transverse beam spectrum of the Large Hadron Collider (LHC) since the start of its operation. In a previous paper \cite{previous}, we investigated the source of the perturbation based on a systematic analysis of experimental observations. It was clearly shown that the 50~Hz harmonics are not an artifact of the instrumentation system but correspond to actual beam oscillations.
Two regimes of interest have been identified in the transverse beam spectrum: a series of 50~Hz harmonics extending up to approximately 3.6~kHz, referred to as the \textit{low-frequency cluster}, and a cluster of 50~Hz harmonics centered around the alias of the betatron frequency, i.e., $f_{\rm rev}-f_x$, where $f_{\rm rev}$=11.245~kHz is the LHC revolution frequency and $f_x$ the betatron frequency, namely the \textit{high-frequency cluster}. Both clusters are the result of dipolar beam excitations and it was demonstrated with dedicated experiments that the source of the low-frequency cluster are the eight Silicon Controlled Rectifier (SCR) power supplies of the main dipoles \cite{SCR, SCR2}. Table~\ref{tab:50Hz} presents the frequency and the amplitude of the 12 most important 50~Hz harmonics in the transverse spectrum of Beam 1 during collisions as observed in the Q7 pickup of the transverse damper Observation Box (ADTObsBox) \cite{ADT, ADT3, ADT4}. The offsets induced in Beam 2 due to the power supply ripple are also depicted, which are approximately lower by a factor of two compared to Beam 1. Based on the source, the 50~Hz harmonics are expected to be present in the future operation of the accelerator.
\begin{table*}
\caption{\label{tab:50Hz}%
The beam offsets observed in the Q7 pickup of the transverse damper for the 12 largest 50~Hz harmonics in the horizontal spectrum of Beam 1 (B1) and 2 (B2) during collisions.}
\begin{ruledtabular}
\begin{tabular}{cccccccc}
\textrm{Frequency (kHz)}& Cluster &
\textrm{B1 offset (nm)} & \textrm{B2 offset (nm)} & \textrm{Frequency (kHz)}& Cluster &
\textrm{B1 offset (nm)} & \textrm{B2 offset (nm)}\\
\colrule
7.6 & High& 166.2 & 76.51 & 7.7 & High& 109.75 & 51.38\\
1.2 & Low& 98.63 & 71.87 & 7.65 & High& 97.4 & 51.97 \\
7.8 & High& 92.75 & 42.78 & 2.95 & Low& 91.62 & 32.21\\
2.35 & Low& 88.48 & 29.61 & 7.9 & High& 82.23 & 25.46 \\
7.5 & High& 77.82 & 35.08 & 2.75 & Low& 67.43 & 31.77 \\
7.85& High & 67.37 & 18.86 & 0.6 & Low &54.14 & 14.33 \\
\end{tabular}
\end{ruledtabular}
\end{table*}
In the presence of strong non-linearities such as beam-beam interactions and fields of non-linear magnets, a modulation in the dipole strength due to power supply ripple results in a resonance condition \cite{tomas1, tomas2}:
\begin{equation}
\label{eq:tune}
l \cdot Q_x + m\cdot Q_y + p \cdot Q_p = n
\end{equation}
where $l, m,p,n$ are integers, $Q_x$ and $Q_y$ are the horizontal and vertical tunes, respectively, and $Q_p$ is the power supply ripple tune. Power supply ripple can impact the beam performance by introducing resonances in addition to the ones excited due to the non-linearities of the lattice. These resonances can increase the tune diffusion of the particles in the distribution, which may lead to the reduction of the beam lifetime. The aim of the present paper is to determine whether the 50~Hz harmonics act as a limiting factor to the beam performance of the LHC, as well as its future upgrade, the High-Luminosity LHC (HL-LHC) \cite{Apollinari:2284929}, using single-particle tracking simulations with power supply ripple in the form of dipolar excitations. The impact of power supply ripple that results in a tune modulation is discussed in a previous paper \cite{kostoglou2020tune}.
First, the motion of a particle in the presence of a modulated dipolar field error is described using a linear formalism (Section~\ref{app:modulated_dipole}). Second, single-tone dipolar excitations are considered in the tracking simulations to define a minimum power supply ripple threshold that results in a reduction of the Dynamic Aperture (DA) in the presence of strong non-linearities such as long-range and head-on beam-beam encounters, chromatic arc sextupoles and Landau octupoles. Then, the impact of the harmonics on the beam performance in terms of tune diffusion, DA and lifetime is discussed in Sec.~\ref{Sec:simulations} by including in the simulations a power supply ripple spectrum as acquired from experimental observations. Finally, the impact of controlled dipolar excitations on the beam lifetime, conducted with the transverse damper, is described in Section~\ref{Sec:adt_excitations}, which provides a tool for the validation of the DA simulations in the presence of power supply ripple.
\section{Linear formalism for a modulated dipolar field error}
\label{app:modulated_dipole}
In a circular accelerator, the kick related to a modulated dipolar field error \(\Theta_p\) with a deflection \(\theta_{p}\) and a frequency of \(Q_{p}\) oscillations per turn can be represented in normalized phase space as:
\begin{equation}
\bar{P}_n = \begin{pmatrix} 0 \\ \Theta_{p}(n) \end{pmatrix} = \begin{pmatrix} 0 \\ \sqrt{\beta} \theta_{p} \cos{(2 \pi Q_p n)} \end{pmatrix},
\label{appendix:eq_1}
\end{equation}
where \(n\) is the turn considered and $\beta$ is the $\beta$-function at the location of the perturbation.
In the linear approximation, considering only the horizontal motion and assuming that the source and the observation point are situated in the same location, the vector representation of the position and momentum at turn \(N\) is:
\begin{equation}
\bar{X}_N=
\left(\begin{array}{c}
\bar{x}_N
\\
\bar{x}_N'
\end{array}\right)=
\sum_{n=-\infty}^{N}M^{N-n}\bar{P}_n,
\label{appendix:eq_2}
\end{equation}
where \(M\) is the linear rotation equal to:
\begin{equation}
M^{N}=\left(\begin{array}{cc}{\cos (2 \pi Q N)} & {\sin (2 \pi Q N)} \\ {-\sin (2 \pi Q N)} & {\cos (2 \pi Q N)}\end{array}\right)
\label{appendix:eq_3}
\end{equation}
with \(Q\) representing the machine betatron tune. Combining Eq.~\eqref{appendix:eq_1}, \eqref{appendix:eq_2} and \eqref{appendix:eq_3} and assuming that the perturbation is present from $n \rightarrow-\infty$, yields:
\begin{align}
& \bar{x}_N= \sqrt{\beta} \theta_{p} \sum_{n=-\infty}^{N} \cos{(2 \pi Q_p n)} \sin{\left(2 (N-n) \pi Q\right)}, \nonumber \\
& \bar{x}'_N= \sqrt{\beta} \theta_{p} \sum_{n=-\infty}^{N} \cos{(2 \pi Q_p n)} \cos{\left(2 (N-n) \pi Q \right)}.
\label{appendix:eq_4}
\end{align}
In Eq.~\eqref{appendix:eq_4}, using Euler's formula and writing the sum expressions in terms of geometric series yields:
\begin{widetext}
\begin{equation}
{
\begin{aligned}
{\bar x}_N =\sqrt{\beta} \theta_p \lim_{k\to-\infty} \Bigr(&\frac{\cos (2 \pi Q_p N) \sin(2 \pi Q)}{2(\cos(2\pi Q_p)-\cos(2\pi Q))} \\
&+\frac{- \cos (2 \pi Q_p (k-1) ) \sin (2 \pi Q (k - N)) + \cos (2 \pi Q_p k ) \sin (2 \pi Q(k - 1 - N) )}{2(\cos(2\pi Q_p)-\cos(2\pi Q))} \Bigl) \;\;,\\
{\bar x'}_N = \sqrt{\beta} \theta_p \lim_{k\to-\infty} \Bigr(&\frac{\cos (2 \pi Q_p (N+1) )-\cos (2 \pi Q_p N )\cos (2 \pi Q)}{2(\cos{(2\pi Q_p)}-\cos{(2\pi Q)})}\\
&+\frac{\cos (2 \pi Q_p (k-1)) \cos (2 \pi Q (k-N))+\cos (2 \pi Q_p k) \cos (2 \pi Q (k-1-N))}{2(\cos{(2\pi Q_p)}-\cos{(2\pi Q)})} \;\;.\\
\label{eq:sum2}
\end{aligned}}
\end{equation}
\end{widetext}
The limit of the terms in the numerator of Eq.~\eqref{eq:sum2} involving $k$ in the argument of the trigonometric functions is bounded but indeterminate. Using the Ces\`aro mean the trigonometric products involving $k$ as an argument in Eq.~\eqref{eq:sum2} is equal to zero and the expressions for the position and momentum simply become:
\begin{align}
\label{eq:sumsol}
{\bar x}_N &= \sqrt{\beta} \theta_p \frac{\cos (2 \pi Q_p N) \sin(2 \pi Q) }{2(\cos{(2\pi Q_p)}-\cos{(2\pi Q)})}, \\
{\bar x'}_N &= \sqrt{\beta} \theta_p \frac{\cos(2 \pi Q_p N) \cos(2 \pi Q)-\cos (2 \pi (N+1) Q_p)}{2(\cos{(2\pi Q_p)}-\cos{(2\pi Q)})}. \nonumber
\end{align}
In physical coordinates, the maximum offset computed from Eq.~\eqref{eq:sumsol} is:
\begin{equation}
|x_{\text{max}, N}|=\left\lvert \frac{\beta \theta_{p}\sin{(2\pi Q)}}{2 \left( \cos{(2\pi Q_{p})}-\cos(2\pi Q) \right) }\right\rvert.
\label{appendix:eq_6}
\end{equation}
\section{Simulations of 50~Hz harmonics with beam-beam interactions}
\label{Sec:simulations}
Tracking simulations are performed using the single-particle symplectic code SixTrack \cite{sixtrack, sixtrack2}. A distribution of particles is tracked in the element-by-element LHC and HL-LHC lattice at top energy, including important non-linearities such as head-on and long-range beam-beam encounters, sextupoles for chromaticity control and Landau octupoles for the mitigation of collective instabilities. For the LHC case, the chromaticity in the horizontal and vertical plane is $Q'$=15 and the amplitude detuning coefficients $\alpha_{n_1 n_2}=\partial Q_{n_1} / \partial(2 J_{n_2})$ with $n_1, n_2 \in \{x,y\}$ are:
\begin{eqnarray}
\alpha_{\rm xx}=16.8\times 10^{4} \ \text{m}^{-1} , \alpha_{\rm xy} = -11.8\times 10^{4} \ \text{m}^{-1}, \nonumber \\
\alpha_{\rm yy} =16.5\times 10^{4} \ \text{m}^{-1}
\end{eqnarray}
as computed from the PTC-normal module of the MAD-X code \cite{madx}.
The beam-beam interactions are simulated with the weak-strong approximation. The maximum duration of the tracking is \(\rm 10^6\) turns that corresponds to approximately 90 seconds of beam collisions in operation. The main parameters used in simulations for LHC and HL-LHC are presented in Table~\ref{tab:table_parameters_collision}. For the LHC, the simulated conditions refer to the start of the collisions. The simulated conditions in the HL-LHC case correspond to the nominal operational scenario at the end of the $\beta^*$-leveling, taking place after several hours from the start of the collisions \cite{HL_LHC_scenarios}.
\begin{table*}
\caption{\label{tab:table_parameters_collision}%
The LHC and HL-LHC parameters at top energy used in the tracking simulations. The LHC parameters correspond to the start of collisions, while the HL-LHC parameters refer to the end of the $\beta^*$-leveling as envisaged in the nominal scenario \cite{HL_LHC_scenarios}.
}
\begin{ruledtabular}
\begin{tabular}{lcc}
\textrm{Parameters (unit)}&
\textrm{LHC (values)} & \textrm{HL-LHC (values)}\\
\colrule
Beam energy (TeV) & 6.5 & 7\\
Bunch spacing (ns) & 25 & 25\\
RMS bunch length (cm) & 7.5 & 7.5 \\
Bunch population (protons) & $1.25 \times 10^{11}$ & $1.2\times 10^{11}$\\
Beam-beam parameter $\xi_{x,y} $ & 7.6$\times 10^{-3}$ & 5.8$\times 10^{-3}$ \\
Horizontal tune \(Q _x\) &0.31 & 0.31/0.315\footnote{\label{note1}Nominal/optimized working point.} \\
Vertical tune \(Q_y\) &0.32/0.315\textsuperscript{\ref{note1}} & 0.32\\
Horizontal and vertical normalized emittance (\(\rm \mu m \ rad\)) & 2.0 & 2.5\\
Horizontal and vertical chromaticity & 15 & 15\\
Octupole current (A) & 550 & -300\\
IP1/5 Half crossing angle (\(\rm \mu rad\)) & 160 & 250\\
Horizontal and vertical IP1/5 \(\beta^*\) (cm) & 30 & 15\\
Total RF voltage (MV) & 12 & 16 \\
Synchrotron frequency (Hz) & 23.8 & 23.8 \\
Revolution frequency (kHz) & 11.245 & 11.245\\
Relative momentum deviation $\delta p /p$ & 27$\times 10^{-5}$ & 27$\times 10^{-5}$ \\
Horizontal \(\beta\)-function at Q7 pickup (m) & 105.4 & 102.3\\
\end{tabular}
\end{ruledtabular}
\end{table*}
In a previous paper \cite{previous}, it was demonstrated that, as far as the low-frequency cluster is concerned, the power supply ripple is distributed in all eight sectors. An accurate representation of the power supply ripple propagation across the chains of the LHC dipoles requires a model of the transfer function as a transmission line for all the spectral components in the low-frequency cluster, similarly to the studies performed for the SPS \cite{Burla:263879}. Furthermore, as the exact mechanism of the high-frequency cluster is not yet clearly identified, an accurate transfer function is not known at the moment.
To simplify these studies, the distributed power supply ripple is projected to a single location with an equivalent impact on the beam's motion. Specifically, a horizontal modulated dipolar source is included at the location of the Q7 pickup of the transverse damper, which coincides with the observation point. To simulate the dipolar excitation, the strength of a horizontal kicker is modulated with a sinusoidal function:
\begin{equation}
\label{eq:mod}
\Delta k (t) = \theta_p \cos{(2\pi f_p t)},
\end{equation}
where $t$ is the time, $\theta_p$ the deflection, $f_p=Q_p \cdot f_{\rm rev}$ the power supply ripple frequency and $f_{rev}$ the revolution frequency listed in Table~\ref{tab:table_parameters_collision}. In this way, the contribution of all the dipoles is represented by a single, equivalent kick and the offsets observed in the LHC spectrum are reproduced in the simulations. The maximum offset induced on the particle's motion is computed from Eq.~\eqref{appendix:eq_6} with the horizontal $\beta$-function listed in Table~\ref{tab:table_parameters_collision}.
The simulations are then repeated for the HL-LHC case. The need to perform projections for the HL-LHC is justified by the fact that no modifications are envisaged for the power supplies of the main dipoles. Consequently, based on the source, the 50~Hz harmonic are expected to also be present in the HL-LHC era. In the following sections, the HL-LHC studies are based on the power supply ripple spectrum acquired experimentally from LHC, although it is expected that the foreseen upgrade of the transverse damper system will lead to a more efficient suppression of the harmonics.
\subsection{Impact of single-tone ripple spectrum on the Dynamic Aperture}
As a first step, we consider individual tones in the power supply ripple spectrum for increasing values of the deflection $\theta_p$. For each study, a different combination of the frequency $f_p$ and the amplitude of the excitation $\theta_p$ is selected. For each case, the minimum DA in the initial configuration space, i.e., the initial horizontal $x$ and vertical $y$ displacements is compared to the value derived in the absence of power supply ripple. In these simulations, the tune modulation due to the coupling of the synchrotron oscillations to the betatron motion for a chromaticity and a relative momentum deviation listed in Table~\ref{tab:table_parameters_collision} is also included.
The scan in the ripple parameter space (\(f_p, \theta_p\)) is performed to define the most dangerous tones of the low and high-frequency cluster, i.e., the frequencies that, for a constant excitation amplitude, have a maximum impact on the DA. Then, the offset induced in the beam motion at the location of observation point is estimated from Eq.~\eqref{appendix:eq_6} with the parameters $f_p$, $\theta_p$ and the $\beta$-function of Table~\ref{tab:table_parameters_collision}. A threshold for the minimum offset at the Q7 pickup that leads to a reduction of DA is determined.
Figure~\ref{fig:heatmap} presents the ripple frequency as a function of the offset at the observation point. The harmonics of the low and the high-frequency cluster that reside in the vicinity of \(f_x\) and \(f_{\rm rev}-f_x\) have been selected for the analysis. A color code is assigned to the reduction of the minimum DA to illustrate the regimes with a negligible (blue) and significant (yellow) impact.
\begin{figure}
\includegraphics[width = \columnwidth]{heat_LHC.png}
\caption{\label{fig:heatmap} The ripple frequency as a function of the horizontal offset in the Q7 pickup, as computed from the deflection and Eq.~\eqref{appendix:eq_6}. A color code is assigned to the reduction of the minimum DA from the value computed in the absence of power supply ripple.}
\end{figure}
From the scan, it is evident that the ripple frequencies with the largest impact are the ones that reside in the proximity of the tune and its alias. For the LHC, an offset threshold of \(\rm 0.4 \ \mu m\) is defined, while this limit reduces to \(\rm 0.2 \ \mu m\) for the HL-LHC. For comparison, the maximum excitation observed experimentally due to the 50~Hz lines is approximately 0.16~\(\rm \mu m\) as presented in Table~\ref{tab:50Hz}.
\subsection{Frequency Map Analysis with a realistic power supply ripple spectrum}
A more significant impact on the beam performance is anticipated in the presence of multiple ripple tones, similar to the experimental observations of the 50~Hz harmonics, than considering individual tones due to the resonance overlap \cite{intro1994_1, SPS2_tune}. To this end, a realistic power supply ripple spectrum must be included in the simulations. Similarly to Table~\ref{tab:50Hz}, the offsets and the frequencies of the 40 largest 50~Hz harmonics are extracted from the horizontal beam spectrum as observed during operation. For each frequency, the equivalent kick at the location of the Q7 pickup is computed from Eq.~\eqref{appendix:eq_6} and it used as an input for the power supply ripple simulations.
Figure~\ref{fig:spectrum} shows the horizontal spectrum of Beam 1 from the experimental observations (black) centered around the low (left) and high (right) frequency cluster. A single particle is tracked in the LHC lattice including the most important 50~Hz harmonics. The output of the simulations (green) is also illustrated in Fig.~\ref{fig:spectrum}. The comparison of the two is a sanity check illustrating the good agreement between the simulated and the experimental beam spectrum. A similar agreement (between simulated and expected beam spectrum) is found for the HL-LHC case, were the equivalent kicks have been recomputed due to a small variation of the \(\beta\)-function as shown in Table~\ref{tab:table_parameters_collision}.
\begin{figure}
\includegraphics[width = \columnwidth]{Simulations.png}
\caption{\label{fig:spectrum} The spectrum centered around the low (left) and high (right) frequency cluster as acquired experimentally from the horizontal plane of Beam 1 (black) and from tracking simulations (green).}
\end{figure}
The studies are organized as follows. First, a study in the absence of power supply ripple is performed that is used as a baseline. Second, the largest 50~Hz harmonics of the low-frequency cluster are considered. Then, a separate study is conducted including only the most important harmonics of the high-frequency cluster. Last, both regimes are included in the simulations.
Particles extending up to 6 \(\rm \sigma\) in the initial configuration space are tracked for $\rm 10^4$ turns in the LHC and HL-LHC lattice. The particles are placed off-momentum as shown in Table~\ref{tab:table_parameters_collision}, without however considering the impact of synchrotron oscillations. A Frequency Map Analysis (FMA) is performed for each study \cite{fma1, fma2, fma3, fma4}. The turn-by-turn data are divided into two groups containing the first and last 2000 turns, respectively. The tune of each particle is computed for each time interval using the Numerical Analysis of Fundamental Frequencies (NAFF) algorithm \cite{NAFF1, NAFF, PhysRevAccelBeams.22.071002}. Comparing the variation of the tune of each particle across the two time spans reveals information concerning its tune diffusion rate $D$ that is defined as:
\begin{equation}
\label{eq:diffusion}
D = \sqrt{ (Q_{x_1}- Q_{x_2})^2 + (Q_{y_1}- Q_{y_2})^2 },
\end{equation}
where $Q_{j_1}, Q_{j_2}$ with $j \in \{x,y\}$ denote the tunes of each particle in the first and second time interval, respectively.
Figure~\ref{fig:fmas_LHC} illustrates the frequency maps (left panel) and the initial configuration space (right panel) for the four studies in the LHC. For the frequency map, the particle tunes of the second interval are plotted and a color code is assigned to logarithm of the tune diffusion rate $D$ to distinguish the stable particles (blue) from the ones with an important variation of their tune (red) due to the resonances. The gray lines denote the nominal resonances, i.e., the resonances that intrinsically arise from the non-linear fields such as non-linear magnets and beam-beam effects without power supply ripple.
\begin{figure*}
\subfloat{\subfigimg[width = 0.49\textwidth]{ \textbf{a)}}{LHC_fma_without.png} \label{subfig:FMA_without}}
\subfloat{\subfigimg[width = 0.49\textwidth]{\textbf{b)}}{LHC_fma_low.png} \label{subfig:FMA_low}} \\
\subfloat{\subfigimg[width = 0.49\textwidth]{\textbf{c)}}{LHC_fma_high.png} \label{subfig:FMA_high}}
\subfloat{\subfigimg[width = 0.49\textwidth]{\textbf{d)}}{LHC_fma_all.png} \label{subfig:FMA_all}} \\
\caption{\label{fig:fmas_LHC} The frequency maps (left) and the initial configuration space (right) color-coded with the logarithm of the tune diffusion rate (Eq.~\eqref{eq:diffusion}) for the LHC simulation parameters of Table~\ref{tab:table_parameters_collision} (a) in the absence of power supply ripple , (b) with the 50~Hz harmonics of the low and (c) high-frequency cluster and (d) combining both regimes. The gray lines represent the nominal resonances and the black and blue lines illustrate the resonances (Eq.~\eqref{eq:low} and \eqref{eq:high}) excited due to power supply ripple.}
\end{figure*}
In the absence of power supply ripple (Fig.~\ref{subfig:FMA_without}), an important impact is observed due to the third-order resonance ($\text{3} \cdot Q_y \rm =\text{1}$), which is in the vicinity of tune footprint and it affects particles at large amplitudes (red).
From the FMAs, it is observed that the dipolar power supply ripple results in an increase of the particles' diffusion rate, first, by enhancing the strength of some of the nominal resonances and second, through the excitation of resonances in addition to the nominal ones as described in Eq.~\eqref{eq:tune}. Including the low-frequency cluster (Fig.~\ref{subfig:FMA_low}), the resonances with the largest impact are:
\begin{equation}
\label{eq:low}
Q_x = Q_p.
\end{equation}
As the power supply ripple is injected in the horizontal plane, these resonances appear as vertical lines in the tune domain in a location equal to the excitation frequency (black). Due to the coupling of the transverse planes, the strongest dipolar excitations are also visible in the vertical plane, appearing as horizontal lines in the frequency maps.
The excitations of the high-frequency cluster appear as aliases (blue). For instance, the excitation at 7.8~kHz in Fig.~\ref{subfig:FMA_high} is located at $f_{\rm rev} \rm -7.8~kHz$. For the high-frequency cluster (Fig.~\ref{subfig:FMA_high}), the most critical resonances are:
\begin{equation}
\label{eq:high}
Q_x = 1-Q_p.
\end{equation}
For the 50~Hz harmonics in the vicinity of the betatron tune and its alias these additional resonances are located inside the beam's footprint. As clearly shown in the $x-y$ plane (right panel), the existence of such resonances impacts both the core and the tails of the distribution. Reviewing the impact of both clusters (Fig.~\ref{subfig:FMA_all}) indicates that the main contributors to the increase of tune diffusion rate are the spectral components in the high-frequency cluster. Similar results are obtained for the HL-LHC case.
\subsection{Intensity evolution simulations}
\label{sec:losses}
Quantifying the impact of the power supply ripple and the non-linearities on the intensity evolution requires tracking a distribution with realistic profiles in all planes that are similar to the ones observed experimentally. If impacted by resonances, particles at the tails of the distribution close to the limit of DA diffuse and will eventually be lost. Therefore, a detailed representation of the tails of the distribution is needed along with a realistic longitudinal distribution that extends over the whole bucket height.
The initial conditions form a 4D round distribution, which is sampled from a uniform distribution, extending up to 7 \(\rm \sigma \) both in the initial configuration space, as well as in the initial phase space. In the longitudinal plane, the relative momentum deviation of the particles is a uniform distribution that extends up to the limit of the bucket height (not the value presented in Table~\ref{tab:table_parameters_collision}).
To reduce the statistical variations, $\rm 25 \times 10^{4}$ particles are tracked in the LHC lattice including synchrotron oscillations. In the post-processing, a weight is assigned to each particle according to its initial amplitude, as computed from the Probability Density Function (PDF) of the final distribution that needs to be simulated. The weight of each particle is:
\begin{equation}
w = \frac{\prod_{j=1}^{3} \text{PDF}(r_j, \sigma_j) }{\sum_i w_i},
\end{equation}
where $j$ iterates over the three planes, $r_j = x_{j}^2 + p_{j}^2$ with $x_j, p_j$ the initial normalized position and momentum coordinates, $\sigma_j$ represents the RMS beam size in the $j$-plane and the denominator denotes the normalization with the sum of the weights of all the particles.
Experimental observations have shown that the tails of the LHC bunch profiles are overpopulated in the transverse plane and underpopulated in the longitudinal plane compared to a normal distribution \cite{profiles, papadopoulou2018impact}. For an accurate description of the bunch profiles, the PDF of the q-Gaussian distribution is employed \cite{tsallis1, tsallis2, wolfram}:
\begin{equation}
\label{eq:pdf}
\begin{array}{cc}
\text{PDF}(r, \sigma) = \Bigg\{ &
\begin{array}{cc}
\frac{e^{-\frac{r^2}{2 \sigma ^2}}}{\sqrt{2 \pi } \sigma } & q=1 \\
\frac{\sqrt{q-1} \Gamma \left(\frac{1}{q-1}\right) \left(\frac{(q-1) r^2}{2 \sigma ^2}+1\right)^{\frac{1}{1-q}}}{\sqrt{2 \pi } \sigma \Gamma \left(-\frac{q-3}{2 (q-1)}\right)} & 1<q<3 \\
\frac{\sqrt{1-q} \Gamma \left(\frac{3}{2}+\frac{1}{1-q}\right) \left(\frac{(q-1) r^2}{2 \sigma ^2}+1\right)^{\frac{1}{1-q}}}{\sqrt{2 \pi } \sigma \Gamma \left(1+\frac{1}{1-q}\right)} & q<
\end{array}
\\
\end{array}
\end{equation}
The parameter $q$ and the ratio of the RMS beam size of the Gaussian ($q$=1) to the q-Gaussian distribution $\sigma_{\rm G} / \sigma_{\rm qG}$ are presented in Table \ref{tab:qgauss} for the longitudinal and the transverse plane (assuming the same profiles for the horizontal and vertical plane).
\begin{table}
\caption{\label{tab:qgauss}%
The parameters of the q-Gaussian PDF shown in Eq.~\eqref{eq:pdf} for the longitudinal and transverse plane \cite{profiles, papadopoulou2018impact}.
}
\begin{ruledtabular}
\begin{tabular}{ccc}
\textrm{Parameters}&
\textrm{Longitudinal plane} & \textrm{Transverse plane}\\
\colrule
$q$ & 0.88 & 1.15 \\
$\sigma_{\rm G} / \sigma_{\rm qG}$ & 0.95 & 1.05
\end{tabular}
\end{ruledtabular}
\end{table}
Furthermore, a mechanical aperture is defined in the post-processing at 6.6 \(\rm \sigma\) to simulate the effect of the primary collimators on the transverse plane based on the settings of the 2018 operation. Particles beyond this threshold are considered lost and their weight is set to zero.
Figure~\ref{fig:losses} presents the intensity evolution without power supply ripple (black), including the 50~Hz harmonics of the low (blue) or high (orange) frequency cluster and considering both regimes (red). The results indicate that the presence of the 50~Hz harmonics leads to a reduction of the beam intensity, which is already visible with a tracking equivalent to 90 seconds of beam collisions. The results also show that, for the time span under consideration, the high-frequency cluster is the main contributor to the decrease of the beam intensity. A similar effect is observed when considering the same power supply ripple spectrum for the HL-LHC case with the high-frequency cluster acting as the main contributor.
\begin{figure}
\includegraphics[width = \columnwidth]{low_high_all.png}
\caption{\label{fig:losses} Intensity evolution without power supply ripple (black), considering only the low (blue) or high (orange) frequency cluster and including the 50~Hz harmonics in both regimes (red).}
\end{figure}
To quantify the impact of the 50~Hz harmonics on the beam performance, the lifetime $\tau$ is computed from the intensity evolution $I$ as:
\begin{equation}
\label{eq:lifetime}
I(t) = I_0 \cdot e^{-\frac{t}{\tau}},
\end{equation}
where $I_0$ is the initial intensity and $t$ the time. An approximation of the instantaneous lifetime is estimated by fitting the exponential decay of the intensity with a sliding window, with each step consisting of a few thousand turns.
Figure~\ref{fig:losses_b1b2} illustrates the intensity evolution in the absence of power supply ripple (black), including the power supply ripple spectrum from the experimental observations of Beam 1 (blue) and 2 (red). The green line indicates the fits of Eq.~\eqref{eq:lifetime} to compute the lifetime evolution.
\begin{figure}
\includegraphics[width = \columnwidth]{life_fit_115.png}
\caption{\label{fig:losses_b1b2} Intensity evolution in the absence of power supply ripple (black), including the power supply ripple spectrum of Beam 1 (blue) and 2 (red). The green line indicates the fits of the exponential decay of the intensity to compute lifetime (Eq.~\eqref{eq:lifetime}).}
\end{figure}
As shown in Table~\ref{tab:50Hz}, the fact that the power supply ripple spectrum of Beam 2 is lower by approximately a factor of two compared to Beam 1 results in an asymmetry of the intensity evolution between the two beams. Starting from $\tau=28.4$~h in the absence of power supply ripple, the lifetime reduces to $\tau=27.4$~h when including the ripple spectrum of Beam 2 in the simulations and $\tau=22.3$~h when considering the spectrum of Beam 1. During the operation of the accelerator in run 2 (2015-2018), it has been observed that the lifetime of Beam 1 was systematically lower than the lifetime of Beam 2 \cite{b1life1}. It is the first time that simulations reveal that, amongst other mechanisms, the 50~Hz harmonics can contribute to this effect.
\section{Simulation benchmark with controlled excitations}
\label{Sec:adt_excitations}
During the latest LHC run in 2018, controlled dipolar excitations were applied on the beam using the transverse damper kicker. The goal of the experiment was to study the impact of dipolar excitations at various frequencies and amplitudes and to validate our simulation framework in a controlled manner. The experiments were performed at injection energy.
Experimentally, some of the excitations led to a significant reduction of the beam lifetime. To retrieve the initial deflection applied from the transverse damper, the offset and the frequency are extracted from the calibrated ADTObsBox beam spectrum. For instance, Fig.~\ref{fig:excitation_2.5kHz} shows the horizontal spectrum of Beam 1 during a controlled excitation at 2.5~kHz (green star-shaped marker) for a single bunch. The gray vertical lines illustrate the multiples of 50~Hz. Then, using the offset and the frequency, the equivalent kick is computed from Eq.~\ref{appendix:eq_6}. This procedure is repeated for all the excitations applied during the experiments.
\begin{figure}
\includegraphics[width = \columnwidth]{Fill7342_exc.png}
\caption{\label{fig:excitation_2.5kHz} The horizontal spectrum of a single bunch (black) during a controlled dipolar excitation at 2.5~kHz (green marker) performed using the transverse damper. The vertical lines represent the multiples of 50~Hz.}
\end{figure}
The impact of the excitations on the beam lifetime is compared against the DA scans computed with tracking studies in the presence of dipolar power supply ripple. The important parameters of the tracking simulations at injection energy are summarized in Table~\ref{tab:injection_parameters}.
\begin{table}
\caption{\label{tab:injection_parameters}%
The LHC parameters at injection energy used in the simulations with dipolar power supply ripple.
}
\begin{ruledtabular}
\begin{tabular}{cc}
\textrm{Parameters (unit)}&
\textrm{Values} \\
\colrule
Beam energy (GeV) & 450 \\
Bunch spacing (ns) & 25 \\
RMS bunch length (cm) & 13 \\
RF voltage (MV) & 8 \\
Betatron tunes (\(Q _x, Q_y\)) & (0.275, 0.295) \\
H/V normalized emittance (\(\rm \mu m \ rad\)) & 2.5 \\
H/V chromaticity & 15 \\
Octupole current (A) & 20 \\
Bunch population (protons) & $1.15\times 10^{11}$ \\
Horizontal \(\beta\)-function at Q7 (m) & 130.9 \\
\end{tabular}
\end{ruledtabular}
\end{table}
In each study, a different combination of the excitation frequency and amplitude is selected. In particular, considering a constant excitation frequency, the value of the deflection is increased and the minimum DA is computed for each case. A ripple amplitude threshold is defined as a function of the excitation frequency. Beyond this limit, a reduction of DA is expected.
Figure~\ref{fig:heat_injection} presents the frequency of the excitation as a function of the deflection. A color code is assigned to the minimum DA to distinguish the regime where the power supply ripple has no significant impact (blue) from the one where a significant reduction of DA (red) is observed in the simulations.
\begin{figure}
\includegraphics[width =\columnwidth]{MD_B1H_heat2.png}
\caption{\label{fig:heat_injection} The frequency of the excitation as a function of the deflection. A color code is assigned to the minimum DA computed with tracking simulations including power supply ripple. The star-shaped markers present the equivalent kicks, as computed from the beam spectrum and Eq.~\eqref{appendix:eq_6} during the controlled excitations with the transverse damper. The red markers indicate that the excitation had an impact on the beam lifetime. The blue markers denote the excitations that did not affect the beam lifetime.}
\end{figure}
In Fig.~\ref{fig:heat_injection}, the star-shaped markers denote the experimental excitation kicks and frequencies. A color code is assigned to the markers that allows distinguishing the excitations that had no impact on lifetime experimentally (blue) from those that lead to a lifetime drop (red). Although an excitation at 8.1~kHz was performed experimentally, the position measurements at this time were not stored and a star-shaped marker is not included.
The comparison between experimental observations and the power supply ripple threshold defined by DA simulations yields a fairly good agreement between the two for the majority of the excitations, taking into account the simplicity of the machine model and the absence of effects such as linear and non-linear imperfections in the simulations. This comparison provides a validation of our simulation framework including power supply ripple. The method to benchmark simulations and experimental findings presented in this section is not only limited to the studies of the 50~Hz harmonics but can be used to validate the tracking results for different types of power supply ripple and noise effects.
\section{Conclusions}
The purpose of the current paper was to determine whether the 50~Hz harmonics perturbation is a mechanism that can impact the beam performance during the LHC operation. To this end, single-particle tracking simulations were performed in the LHC and HL-LHC lattice including important non-linear fields such as head-on and long-range beam-beam interactions, chromatic sextupoles and Landau octupoles.
Including a realistic power supply ripple spectrum in the simulations shows that the 50~Hz harmonics increase the tune diffusion of the particles through the excitation of additional resonances, a mechanism that was clearly shown with frequency maps. The high-frequency cluster is identified as the main contributor.
From simulations that correspond to 90 seconds of beam collisions, the increase of the tune diffusion due to the 50~Hz harmonics results in a lifetime reduction of 21\% with respect to the lifetime in the absence of power supply ripple. Based on these results, it is concluded that the 50~Hz harmonics have an impact on the beam performance during the LHC operation. Mitigation measures should be incorporated in the future operation of the accelerator to suppress the high-frequency cluster from the beam motion.
Due to the asymmetry of the power supply ripple spectrum between Beam 1 and 2 by a factor of approximately two, the simulations illustrate a clear discrepancy in the intensity evolution of the two beams. Including realistic bunch profiles, the estimated lifetime is $\tau=22.3$~h and $\tau=27.4$~h for Beam 1 and 2, respectively. An important lifetime asymmetry between the two beams has been observed since the beginning of run 2 and it is the first time that tracking simulations show that power supply ripple can contribute to this effect.
In the context of this study, a general simulation framework has been developed. This paper illustrated a method to define an acceptable power supply ripple threshold for operation through DA scans. The results of these scans were bench-marked against experimental observations with controlled excitations using the transverse damper. Finally, a method to determine the intensity evolution with weighted distributions and realistic bunch profiles has been demonstrated. The analysis methods presented in this paper can be applied to studies of other types of power supply ripple and noise effects.
\begin{acknowledgments}
The authors gratefully acknowledge H.~Bartosik, M.~C.~Bastos, O.~S.~Brüning, R.~T.~Garcia, M.~Martino and S.~Papadopoulou for valuable suggestions and discussions on this work. We would like to thank D.~Valuch and M.~Soderen for all transverse damper related measurements and experiments.
\end{acknowledgments}
|
1,108,101,563,242 | arxiv | \section{Introduction}
An interesting problem of QCD is to understand the behaviour of the
theory in the Regge limit of large energy, fixed momentum transfer. In
the Leading Logarithmic Approximation the leading pole in the $C=+1$
channel is the famous BFKL pomeron \cite{BFKL}. Later this was
generalized to the channel odd under charge conjugation ($C=-1$) ---
the odderon \cite{odd}. In contrast to the BFKL case, however,
the value of the intercept remained unknown despite the discovery of
conformal symmetry and integrals of motion \cite{lipq}.
The hard odderon may be observed in such processes as $\gamma^*
\gamma^*\longrightarrow\eta_c+X$, $\gamma^* p\longrightarrow\eta_c+X$, $\gamma^*\gms\longrightarrow \eta_c\eta_c$.
A number of recent papers \cite{phen} calculated the cross sections
in the approximation where the odderon was modelled by an exchange of three
(non-reggeized) gluons. The cross sections were small, and hence it is
interesting to see if reggeization could lead to a significant
enhancement at high energies, which could perhaps be investigated
experimentally.
Recently substantial progress \cite{lip,fadkor} has been made with the
reduction of the problem to solving the Baxter equation
\begin{equation}
\label{e.baxter}
(\lambda+i)^3 Q(\lambda+i)+(\lambda-i)^3 Q(\lambda-i)=(2\lambda^3+q_2\lambda+q_3)Q(\lambda)
\end{equation}
for physical values of the constants of motion $q_2$ and $q_3$.
The intercept then follows through a logarithmic derivative \cite{fadkor}.
A number of approximation techniques for solving this equation have been
tried \cite{fadkor,KorQ}. In our work \cite{usI} an exact method of solving the
Baxter equation for arbitrary values of the parameters $q_2$ and $q_3$
was developed. This reduced the problem to finding the physical
spectrum of $q_3$ ($q_2$ is fixed by group theory to be $q_2=h(1-h)$
where $h$ is the conformal weight).
Ideally one could do this directly, by requiring that the solution of
the Baxter equation $Q(\lambda)$, gives rise to a normalizable, single-valued
wavefunction (see below).
However, the explicit relation between wavefunctions and the
functional forms of the solutions $Q(\lambda)$ is very indirect
\cite{fadkor}. In particular it is still unknown
how does the
normalizability requirement explicitly look like in terms of $Q(\lambda)$.
In our second paper \cite{usII} we pursued a more direct approach,
using standard physical requirements which are imposed on the wave
function to find the allowed values of $q_3$ and the explicit form of
the wave function. This form can be useful when considering the
coupling of the odderon to external probes.
In
this talk we would like to present the main results of these papers
and give our conclusions on the intercept of the odderon.
\section{Solution of the Baxter equation}
The Baxter equation poses a number of
difficulties. It is a nonlocal functional equation which possesses
polynomial solutions only for integer values of the conformal weight
$h=n>3$. In contrast to the BFKL case there is no easy analytical
continuation to the physically most interesting case of $h=1/2$.
Quasiclassical methods \cite{KorQ} involve an expansion
in $1/h$ and use powerful methods to continue to $h=1/2$, the
precision is, however, difficult to control.
Our aim was to find an expression for the solution directly for
$h=1/2$ (or indeed also for more general $h$) in a form which could be
numerically calculated to an arbitrary precision.
The starting point was the contour integral representation used in
\cite{fadkor}:
\begin{equation}
\label{e.int}
Q(\lambda)=\int_C \frac{dz}{2\pi i} z^{-i\lambda-1}(z-1)^{i\lambda-1} \tilde{Q}(z)
\end{equation}
where $\tilde{Q}(z)$ satisfies a $3^{rd}$ order differential equation $D\tilde{Q}(z)=0$.
The expression (\ref{e.int}) satifies then the Baxter equation only if the contour
$C$ is choosen as such that the boundary terms arising from
integration by parts cancel out. It turns out that for $h=1/2$ it is
impossible to choose such a contour because the curve always ends up
on a different sheet of the Riemann surface of the integrand.
To remedy this we extended the
ansatz (\ref{e.int}) to a sum of two integrals \cite{Acta,usI}:
\begin{equation}
\label{e.twoint}
\int_{C_{I}}\frac{dz}{2\pi i}K(z,\lambda) \tilde{Q}_1(z)+
\int_{C_{II}}\frac{dz}{2\pi i} K(z,\lambda) \tilde{Q}_2(z)
\end{equation}
where the contours are independent (see {\em e.g.} figures in
\cite{Acta,usI}).
The functions $\tilde{Q}_1(z)$ and $\tilde{Q}_2(z)$ are both solutions of
$D\tilde{Q}(z)=0$, and thus depend initially on 6
free parameters.
We now impose the condition of cancellation of boundary terms. This
gives 3 equations, leaving us with $6-3=3$ parameters. Now one notes
\cite{usI} that since $D\tilde{Q}(z)=0$ has a solution holomorphic
at infinity, it gets integrated out to zero in each of the integrals
in (\ref{e.twoint}). This leaves us with only $3-2=1$ parameter which
is just an irrelevant normalization. The above procedure leads
therefore to a unique solution within our ansatz (\ref{e.twoint}).
This solution can be calculated to an arbitrary precision by
including a sufficient number of terms in power series expansions of
the $\tilde{Q}_i(z)$'s.
This being done we will move on, in the next section, to consider the
problem of finding the physical values of $q_3$.
\section{Quantization of $q_3$}
The wavefunction $\Psi(z,\overline{z})$ can be decomposed into the following sum:
\begin{equation}
\label{e.decomp}
\Psi(z,\overline{z})=\sum_{i,j} \overline{u}_i(\overline{z}) A^{(0)}_{ij} u_j(z),
\end{equation}
where $u_i(z)$ and $\overline{u}_j(\overline{z})$ are eigenfunctions (analytic in the whole
complex plane apart from some cuts) of the $3^{rd}$
order ordinary differential operators $\hat{q}_3$ and
$\hat{\overline{q}}_3$ with eigenvalues $q_3$ and $\overline{q}_3$.
We will later also
consider the $(-q_3,-\overline{q}_3)$ sector which is necessary for Bose
symmetry of the wavefunction.
We impose the following obvious physical requirements for the
wavefunction:
(1) $\Psi(z,\overline{z})$ must be single-valued,
(2) $\Psi(z,\overline{z})$ must be normalizable and
(3) $\Psi(z,\overline{z})$ must satisfy Bose symmetry.
The crucial assumption is the first one. It turns out that
normalizability follows automatically (apart from the unphysical case
of $q_3=0$), and Bose symmetry is easy to implement.
We must examine the requirement of single valuedness near the singular
points $z=0$ and $z=1$ ($z=\infty$ follows --- see discussion in \cite{usII}).
Using the asymptotic behaviour of the $u_i$'s near $z=0$ ($u_1(z)\sim
z^{1/3}$, $u_2(z)\sim z^{5/6}$, $u_3(z)\sim z^{5/6}\log z +z^{-1/3}$)
it is clear that the coefficient matrix $A^{(0)}$ of (\ref{e.decomp}) must
have the form
\begin{equation}
\label{e.form}
A^{(0)}=\left( \begin{array}{ccc}
\alpha & 0 & 0 \\
0 & \beta & \gamma \\
0 & \gamma& 0 \\
\end{array} \right).
\end{equation}
We thus have initially 3 parameters $\alpha$, $\beta$ and $\gamma$. Now we
must impose the same requirement around $z=1$. To this end we note
that the solutions $v_i(z)\equiv u_i(1-1/z)$ have similar asymptotics
around $z=1$ to the asymptotics of $u_i$'s around $z=0$. One can
numerically calculate the analytical continuation matrices
$u_i(z)=\Gamma_{ij} v_j(z)$.
We must now reexpress the wavefunction $\Psi(z,\overline{z})$ in terms of the $v_i$'s
and require that the transformed coefficient matrix
$A^{(1)}=\overline{\Gamma} A^{(0)}\Gamma$ has the same form as
(\ref{e.form}). This leads to a
number of linear homogeneous equations for $\alpha$, $\beta$ and
$\gamma$.
Because the number of equations is greater than 3, the
existence of a nonzero solution fixes both the parameters of the
wavefunction $\alpha$, $\beta$, $\gamma$ {\em and} the allowed values of
$q_3$. If the coefficient matrices $A^{(0)}$ and $A^{(1)}$ coincide,
the requirement of Bose symmetry boils down to adding the
corresponding wavefunction in the $(-q_3,-\overline{q}_3)$ sector \cite{usII}.
\begin{equation}
\Psi(z,\overline{z})=\Psi_{q_3,q_3^*}(z,\overline{z})+\Psi_{-q_3,-q_3^*}(z,\overline{z})
\end{equation}
\section{Results}
In \cite{usII} a number of possible solutions were found. The
requirement of Bose symmetry picked out just those lying on the
imaginary axis. We may now plug in those values of $q_3$ into our
solution of the Baxter equation to yield the intercepts corresponding
to those states. The results are summarized below:
\begin{center}
\begin{tabular}{|c|cc|cc|} \hline\hline
{\em No.} & \multicolumn{2}{c|}{ $q_3$ }
& \multicolumn{2}{c|}{ $\epsilon_3$} \\
\hline
1 & \multicolumn{2}{c|}{$0.20526 i$}
& \multicolumn{2}{c|}{$-0.49434$} \\
2 & \multicolumn{2}{c|}{$2.34392 i$}
& \multicolumn{2}{c|}{$-5.16930$} \\
3 & \multicolumn{2}{c|}{$8.32635 i$} & \multicolumn{2}{c|}{$-7.70234$} \\
\hline\hline
\end{tabular}
\end{center}
We are thus led to conclude that the odderon state with the highest
intercept has $q_3=\pm 0.20526i$. The intercept of this state reads
\begin{equation}
\alpha_O=1-0.24717\f{\alpha_s N_c}{\pi}=1-0.16478\cdot\f{3\alpha_s N_c}{2\pi}
\end{equation}
and the wavefunction parameters are $\alpha=0.7096$, $\beta=-0.6894$ and $\gamma=0.1457$.
Recently this result has been confirmed by M.A. Braun \cite{braunlast} who
redid his earlier work on the variational functional using our wave function,
and obtained after a formidable numerical calculation
$\alpha_O=1-0.16606\cdot\f{3\alpha_s N_c}{2\pi}$,
in excellent agreement with our result following from the Baxter
equation. A second check of our results follows from the new symmetry
of the odderon discovered by Lipatov \cite{Lipsym} which forces the
wavefunction parameters to be related by $\gamma=|q_3|\alpha$ which indeed
is verified in our case.
A more precise check can be made using the very recent asymptotic
formula \cite{Lipsym} for the energy expressed in terms of $q_3$
derived by Lipatov
for large $q_3$. We cannot use it for the ground state but for the
higher states identified here one obtains $\epsilon_3=-5.16956$ and
$\epsilon_3=-7.70234$ (for states with $q_3=\pm 2.34392i$ and $q_3=\pm
8.32635i$ respectively) which is in perfect agreement with our earlier results
obtained using the Baxter equation.
The states with higher $q_3$ give subleading contributions as
expected. Also a change in $h$ from $h=1/2$ to $h=1/2+i\nu$ also
lowers the energy. This observation is consistent with the expectation
that the dominant contribution should come from the $h\leftrightarrow
1-h$ symmetric point $h=1/2$ investigated in this work.
Some further confirmations and results are
presented in \cite{braunlast} and in \cite{Rostwor}.
As a final point we note that our results suggest that the predictions
\cite{phen} for odderon mediated processes will not be enhanced at
high energies. It remains to see if the specific form of the wave
function might lead to some interesting effects through the coupling
of the ground state of the odderon found here, to external probes.
\section*{Acknowledgments}
This work is supported by the Polish Committee for Scientific Research
under grant no. PB 2P03B08614 and PB 2P03B04412.
|
1,108,101,563,243 | arxiv | \section{Introduction}
The emergence of classical dynamics from the underlying quantum
behaviour still continues to be one of the foremost questions in
physics today. Two ingredients essential to this transition are (i)
the establishing of suitable ``classical'' correlations between
appropriate coordinates and their conjugate momenta (ie. classical
equations of motion) and (ii) the suppression or destruction of
quantum correlations. The need for the former is self-evident while
the latter is necessary to explain the general absence of macroscopic
quantum superpositions. A paradigm which primarily addresses (ii) and
which attempts to describe this transition is {\em Decoherence}. In
this school of thought, Decoherence is the general term which
describes a dynamical process whereby the quantum correlations in the
system in question are
diminished or destroyed. These processes can be split into those which
deviate from ordinary quantum mechanics, eg. GRW theory and quantum
mechanical spontaneous localisation QMSL
\cite{GHIRARDI:1986}, and those processes which stay within
the confines of quantum mechanics. We will only consider the latter in
this paper. The two primary paradigms which attempt to describe the
transition from quantum to classical using pure quantum mechanics
are Environmental
Induced Superselection (EIS) and Consistent Histories (CH)
\cite{ZUREK:1991,HARTLE:1993a}.
Although much has been written concerning these two theories,
calculations comparing the two in non-trivial models are lacking. Even
in relatively
simple models the computations quickly become lengthy and difficult
to interpret.
In this paper we examine the dynamics of a simple quantum optical
system from the viewpoints of both EIS and CH. Upon comparison we
find that they generally agree, that is, if EIS occurs in a particular
basis then histories in that basis become more consistent.
In section II we derive the general solution to the quantum
optical master equation in the Markov regime for a system
comprising of an harmonic oscillator linearly coupled to an infinite
bath of harmonic oscillators with the usual quantum optical
approximations. We solve for the evolution of the reduced density
matrix of the system in the position, momentum, number and coherent
state bases. In section III we discuss
EIS in these bases using a general measure of ``diagonality''
which quantifies the peaking of the density matrix about the diagonal.
This measure is sensitive to the coherence of the system state.
We solve for the general behaviour of this measure in the position
basis for an arbitrary
initial system state and examine the particular case of a mixture of
two coherent states. We find that EIS occurs to some extent
in all of the above bases with
$\rho_{reduced}$ becomming more ``diagonal'' with time and
increasing bath temperature.
In section IV we calculate the decoherence functional of CH for two-time
projected coarse-grained histories on phase-space.
As in all previous calculations, we use
quasi-projectors, that is, operators which are complete but are not
exclusive, to define the coarse-grained histories on phase-space. We
then compute the degree of consistency and peaking of the decoherence
functional using measures advanced by Dowker and Halliwell
\cite{DOWKER:1992}. We find that the consistency and peaking behave as
expected except for an anomalous {\em decrease} in consistency with
increased initial grain size. This behaviour is also found in all
previous calculations. We also adopt the above proposed measure for
diagonality in the EIS framework to quantify the ``coherence'' of the
decoherence functional. We find a similar dependence of this measure on
the model parameters to that displayed by the the Dowker-Halliwell measure.
In the conclusion we discuss the results of the EIS and CH
calculations and the anomalous dependence of the consistency in the CH
picture on the initial graining. We also discuss the legitimacy of
using the ``coherence'' measure in the CH picture. We argue that in
the case of a set of many alternatives, this ``coherence'' measure
gives a good indication that the probability sum rule is met to the
given degree.
Before examining the dynamics of the quantum optical master equation
we first make a few remarks concerning both the EIS and CH scenarios.
\subsection{Environment Induced Superselection}
Environmentally Induced Superselection was primarily developed by
Zurek in the early 1980's \cite{ZUREK:1981} in relation to quantum
non-demolition measurements of gravitational wave antennas. Since then,
the theory has steadily evolved to encompass more complicated and
realistic models. Essentially, the system in question is coupled to a
large external bath via a particular interaction with Hamiltonian
$H_{int}$. If the interaction is such that the reduced density matrix of
the system becomes approximately diagonal in a particular basis in a
time much less than the relaxation time then we say EIS has occurred.
The process is aimed primarily at removing the quantum coherence from
the system. The system's reduced density matrix in this particular
basis is approximately diagonal and is given a purely ignorance
interpretation. That is, the system is now in one of the states on the
diagonal but we do not know which one.
A number of calculations have
shown that in some cases EIS is very successful and can damp away the
quantum superpositions very quickly indeed \cite{JOOS:1985,ZUREK:1991}.
Other models have shown however, that the spectral properties of
environment play a significant role in determining the
decoherence time \cite{HU:1992,PAZ:1992}. In this paper we will work with the
standard ohmic environment.
It is also the case that the initial conceptions of EIS have changed.
The concept of a pointer basis $|\Lambda\rangle$,
the particular basis in which the
reduced density matrix becomes diagonal, strictly holds only in those
models for which the interaction Hamiltonian commutes with the
number operator of the pointer basis
$\tilde{\Lambda}|\Lambda\rangle=\Lambda|\Lambda\rangle$. For more
realistic cases and for the models treated in this paper, no exact
pointer basis exist. Work on a suitable generalization of the concept
of a pointer basis, that is, a system basis which is least effected by the
interaction with the bath, is currently being pursued
\cite{ZUREK:1993,ZUREK:1993a}. Also, there does not appear to be a general
concrete description concerning the ``diagonality'' of a reduced
density matrix in a particular basis. Some measures have been advanced
for gaussian states. For particular
non-gaussian states
a fringe visibility measure based on the appearance of
the Wigner function has been proposed \cite{PAZ:1992}.
This however, appears to quantify
the amount of interference in the state rather than the coherence. EIS
is the process whereby the reduced state of the system
deviates more and more from a description as a single ray in Hilbert space.
Such a process reduces the interferences present in the state. However, pure
states, with unitary evolution, can display little interference.
Such processes do {\rm not} display EIS. With this in mind
we propose and investigate a
measure of the ``diagonality'' of the state in a number of bases in a
quantum optical model. We find that the coherence of the reduced
density matrix of the system decays and gives clear evidence of EIS in
a number of bases. However, the degree of diagonality depends on the
basis chosen. It also depends on the model parameters in an expected
manner, becoming more diagonal with increasing bath temperature.
We
have not examined the attainment of (i) above in the EIS framework.
This is because there appears to be some dispute concerning the
quantification of classical correlations in the state. Previously,
some have examined the Wigner function of the state for peaks
\cite{HABIB:1990}. However, the Wigner function is not a true probability
distribution and does not correspond to the results of any conceivable
measurement on the system. The $Q$ function of quantum optics {\em
does} correspond to the result of a particular measurement scheme
\cite{BRAUNSTEIN:1991}.
More generally, there exist true probability distribution functions
corresponding to more general measurements \cite{CAVES:1986}.
To establish the existence and to quantify ``classical'' correlations
one should examine such distributions. Work towards this goal is in
progress \cite{ANDERSON:1993}.
Finally, the ignorance interpretation of the resulting diagonal reduced
density matrix seems part and parcel of EIS. This feature of the
framework has
surprisingly not attracted much attention in the literature. If EIS is to
describe a truly ontological reality then Zurek's arguments may not suffice
\cite{DESPAGNAT:1990}. It is clearly capable of describing an
empirical reality \cite{DESPAGNAT:1990} but are it's aspirations
higher than just such a description?
\subsection{Consistent Histories}
In order not to confuse the reader with various different meanings of
the term decoherence,
we will refrain from the more usual nomenclature of ``Decohering
Histories''
and
use the more descriptive terminology -- ``Consistent Histories''.
Consistent Histories was initially researched
by Griffiths and Omn\'{e}s \cite{GRIFFITHS:1984,OMNES:1992} and later on
independently rediscovered and greatly expanded on by Gell-Mann and
Hartle \cite{GELL-MANN:1990}. This paradigm is more ambitious than that
of EIS as it attempts to describe the emergence of the complete
quasi-classical world of familiar experience from the underlying
quantum world. Central to this framework is the notion of
coarse-grained histories, each with an assigned probability. A primary
goal is the identification of complete sets of coarse-grained
histories which exactly satisfy the probability sum rules. Although
much has been written concerning this theory, there are only a few
calculations on realistic models \cite{DOWKER:1992,HARTLE:1993a}.
Quantum optics is an avenue where both quantum and
classical behaviour are accessible to the experimentalist. We will
in particular look at coarse-grained histories on phase-space itself. Previous
calculations have concentrated on position projected histories with
momenta information gained through time-of-flight. In this paper we
look for consistent histories in {\em both} position and momentum.
We also note that the interpretation of the
diagonal decoherence functional is typically a relative state
interpretation and thus the interpretational problems concerning which history
is realized are moot (however see \cite{DESPAGNAT:1990}).
\section{Quantum Optical Master Equation}
We begin by looking at the optical master equation for a system
linearly coupled via the usual quantum optical coupling to a bath of
harmonic oscillators \cite{GARDINER:QUANTUM_NOISE,TWAMLEY:1993a}. For
completeness and to relate this to earlier work let us first consider
a system with a Hamiltonian
\begin{equation} H/\hbar=\omega (t)\,a^{\dagger}
a+f_{1}(t) \,a+f^{*}_{1}(t)\,a^{\dagger}+
f_{2}(t)\,a^{2}+f^{*}_{2}(t)\,a^{\dagger}{}^{2}\;\;.\label{eq0.1}
\end{equation}
In the quantum optical regime we can write the master equation for
the reduced density matrix of the system as
\cite{GARDINER:QUANTUM_NOISE}
\begin{equation}
\dot{\rho}=-\frac{i}{\hbar}[H,\rho]+\Lambda\rho={\cal L}\rho\;\;,
\label{eq1.1}\end{equation}
\begin{equation}
{\Lambda}\rho=
\frac{\gamma(\bar{n}+1)}{2}\left\{[a ,\rho a^{\dagger} ]+
[a\rho,a^{\dagger} ]\right\}
+\frac{\gamma\bar{n}}{2}\left\{[a^{\dagger} ,\rho a ]+[a^{\dagger}\rho
,a ]\right\}\;\;.\label{eq1.2}
\end{equation}
This may be converted into a partial differential equation for a
Quasi-Distribution Function (QDF), $W(\alpha ,\alpha^{*}, s)$
(or $W(\alpha,s)$ for short) where
\cite{CAHILL:1969}
\begin{equation}
W(\alpha,\alpha^{*},s,t)=\frac{1}{\pi}\int\,
\chi(\xi,s,t)e^{\alpha\xi^{*}-\alpha^{*}\xi}\,d^{2}\xi\;\;,\label{eq2}
\end{equation}
\begin{equation}
\chi(\xi,s,t)={\rm Tr}[\rho(t) {\rm
D}(\xi,s)]\;\;,\label{eq3}\end{equation}
\begin{equation}
{\rm D}(\xi,s)=\exp (\xi a^{\dagger}-\xi^{*}a+\frac{1}{2}s|\xi
|^{2})\;\;,
\label{eq4}\end{equation}
where we have the normalization $\int W(\alpha,s,t)d^{2}\alpha/\pi=1$.
The family of distribution functions $W(\alpha,s,t)$ encompasses
the more familiar distributions. For $s=1$, $W(\alpha,+1,t)=P(\alpha,t)$ is
the P distribution of Quantum Optics. For $s=0$,
$W(\alpha,0,t)=W(\alpha,t)$ is the familiar Wigner distribution while for
$s=-1$, $W(\alpha,-1,t)=Q(\alpha,t)$ is the Q or Husimi distribution. The
different distributions correspond to different operator orderings.
From (\ref{eq1.1}) we can derive a partial differential equation for
both $W(\alpha,s)$ and $\chi(\xi,s)$ from first principles using
equations (\ref{eq2}, \ref{eq3}, \ref{eq4}). However, using the general rules
derived in Vogel and Risken \cite{VOGEL:1989} we can easily obtain
\widetext\begin{eqnarray}
&&\dot{W}(\alpha,s,t)= \nonumber\\
&&-\frac{i}{\hbar}\left\{
[f_{1}+2\alpha f_{2}+\alpha^{*}\omega ]\partial_{\alpha^{*}}
-[f_{1}^{*}+2\alpha^{*}f_{2}^{*}+\alpha\omega]\partial_{\alpha}
-s[f_{2}\partial^{2}_{\alpha\alpha^{*}}-f_{2}^{*}
\partial^{2}_{\alpha\alpha^{*}}]\right\}W(\alpha,s,t)\\
&&+\frac{\gamma}{2}\left\{\partial_{\alpha}\alpha+
\partial_{\alpha^{*}}\alpha^{*}
+[2\bar{n}+1-s]\partial^{2}_{\alpha\alpha^{*}}\right\}W(\alpha,s,t)
\;\;,\label{eq6}\end{eqnarray}
\begin{eqnarray}
&&\dot{\chi}(\xi,s,t)=\nonumber\\
&&-\frac{i}{\hbar}\left\{\omega[\xi^{*}\partial_{\xi^{*}}-\xi\partial_{\xi}]
-[\xi f_{1}+\xi^{*}f_{1}^{*}]
-s[\xi^{2}f_{2}-\xi^{*\,2}f_{2}^{*}]
+2[f_{2}\partial_{\xi^{*}}-f_{2}^{*}\partial_{\xi}]\right\}\chi(\xi,s,t)\\
&&+\frac{\gamma}{2}\left\{(s-1-2\bar{n})\xi\xi^{*}-\xi^{*}\partial_{\xi^{*}}
-\xi\partial_{\xi}\right\}\chi(\xi,s,t)\;\;,\label{eq7}\end{eqnarray}
\narrowtext
where we have dropped the $t$ dependence of $f_{i}$ and $\omega$.
We see that the PDE for $\chi$ is first order. The general solution
for this model was treated in \cite{TWAMLEY:1993a}. We will instead
specialize to the case of a single harmonic oscillator with constant
frequency linearly coupled to the bath oscillators. Going to the
interaction picture we get
\narrowtext
\begin{equation}\dot{W}(\alpha,s,t)=\frac{\gamma}{2}
\left\{\partial_{\alpha}\alpha
+\partial_{\alpha^{*}}\alpha^{*}+[2\bar{n}+1-s]\partial^{2}_{\alpha\alpha^{*}}
\right\}W(\alpha,s,t)\;\;,\label{eq8}\end{equation}
\begin{equation}\dot{\chi}(\xi,s,t)=-\frac{\gamma}{2}\left\{
(2\bar{n}+1-s)\xi\xi^{*}+\xi^{*}\partial_{\xi^{*}}+\xi\partial_{\xi}\right\}
\chi(\xi,s,t)\;\;.\label{eq9}\end{equation}
Using the method of characteristics we will solve for the dynamics of
this model for an initial state
$\rho=\sum_{\alpha,\beta}N_{\beta\alpha}|\alpha\rangle\langle\beta|$
where ${\rm
Tr}\rho=\sum_{\alpha\beta}N_{\beta\alpha}\langle\beta|\alpha\rangle=1$.
Thus $\rho$ is a general superposition of coherent states at $t=0$.
(Coherent states are eigenstates of the lowering operator of the
oscillator algebra i.e. $a|\alpha\rangle=\alpha|\alpha\rangle$). Since
the coherent states are overcomplete any initial state may be
represented thus. Let us solve (\ref{eq9}) for $\chi(\xi,s=-1,t)$ when
$\rho (t=0)=|\alpha_{0}\rangle\langle\beta_{0}|$. From (\ref{eq3}) and
using $Q(\alpha)=\langle\alpha|\rho|\alpha\rangle = W(\alpha,-1)$ we get
\begin{equation}
\chi(\xi,-1,t)=\int\,\frac{d^{2}\alpha}{\pi}e^{\xi\alpha^{*}-\xi^{*}\alpha}\,
Q(\alpha,t)\;\;.\label{chi_def}\end{equation}
Using this we can calculate the form of the characteristic function
$\chi$ at
$t=0$ to be
\begin{equation}
\chi(\xi,-1,t=0)=\langle\beta_{0}|\alpha_{0}\rangle\exp
[-|\xi|^{2}-\xi^{*}\alpha_{0}+\beta_{0}^{*}\xi]\;\;.\label{eq10}\end{equation}
For $s=-1$ we can rewrite (\ref{eq9}) as
\begin{equation}
\chi_{,t}+\frac{\gamma}{2}a\chi_{,a}+\frac{\gamma}{2}b\chi_{,b}
=-\gamma(1+\bar{n})ab\chi\;\;,
\label{eq12}\end{equation}
where $a=\xi$ and $b=\xi^{*}$ are now treated as independent
variables.
Looking at the characteristics of (\ref{eq12}) we see that
\begin{equation}
dt=\frac{da}{\gamma a/2}=\frac{db}{\gamma b/2}=
\frac{d\chi}{-\gamma (1+\bar{n})ab\chi}\;\;.
\label{eq13}\end{equation}
Integrating this we obtain three constants,
\begin{equation}
k_{1}=\frac{2}{\gamma }\ln a-t\;,\qquad
k_{2}=\frac{2}{\gamma }\ln b-t\;,\nonumber\end{equation}\begin{equation}
k_{3}=-\frac{\ln \chi}{1+\bar{n}}-ab\;\;.\label{eq13a}\end{equation}
Solving for $a,b$ and $\chi$ in terms of $k_{i}$ and inserting these
into (\ref{eq10}) at $t=0$ gives a relationship between the $k_{i}$
which is valid for $t\geq 0$. Re-inserting the $t$ dependence back into
the $k_{i}$ and going back to the Heisenberg picture yields
\begin{equation}
\chi(\xi,s=-1)=\langle\beta_{0}|\alpha_{0}\rangle
\exp
[-\kappa(t)|\xi|^{2}+\beta_{0}^{*}(t)\xi-\alpha_{0}(t)\xi^{*}]\;\;,
\label{eq14}\end{equation}
where $\kappa(t)=1+\bar{n}(1-e^{-\gamma t})$,
$\beta_{0}(t)=\beta_{0}e^{-(\gamma /2+i\omega)t}$ and
$\alpha_{0}(t)=\alpha_{0}e^{-(\gamma /2+i\omega)t}$.
Inverting (\ref{chi_def}) to obtain the $Q$ distribution we get
\begin{eqnarray}
Q(\alpha,t) & = &
\int\,\chi(\xi,\xi^{*})\,e^{\alpha\xi^{*}-\alpha^{*}\xi}
\frac{d^{2}\xi}{\pi}\nonumber\\
& = & \frac{\langle\beta_{0}|\alpha_{0}\rangle}{\kappa(t)}\exp\,
\left[-\frac{1}{\kappa(t)}(\alpha-\beta_{0}(t))^{*}
(\alpha-\alpha_{0}(t))\right]\;\;,\label{eq15}\end{eqnarray}
where we have used (\ref{appendix1}) from Appendix A.
To obtain the off-diagonal elements
$\langle\tilde{\alpha}|\rho(t)|\tilde{\beta}\rangle$ we use the
identity
\begin{equation}\rho = \int\frac{d^{2}\xi}{\pi}
e^{-\xi a^{\dagger}}e^{\xi^{*} a} \chi(\xi,\xi^{*})\;\;,
\label{eq17}\end{equation}
and inserting (\ref{eq14}) we get
\begin{eqnarray}
&&\langle\tilde{\alpha}|\rho(t)|\tilde{\beta}\rangle=\nonumber\\
&&\frac{\langle\beta_{0}|\alpha_{0}\rangle}{\kappa(t)}
\exp\,\left[-\frac{1}{\kappa(t)}(\tilde{\alpha}-\beta_{0}(t))^{*}
(\tilde{\beta}-\alpha_{0}(t))\right]\langle\tilde{\alpha}|
\tilde{\beta}\rangle
\;\;.\label{eq18}\end{eqnarray}
Using equation (\ref{eq18}) we can now examine the
general case where $\rho(t=0)=\sum_{\alpha_{0}\beta_{0}}\,
N_{\beta_{0}\alpha_{0}}|\alpha_{0}\rangle\langle\beta_{0}|$
and can evaluate $\rho(t)$ to be
\begin{eqnarray}
\rho(t) &=& \int\frac{d^2\alpha d^2\beta}{\pi^2}\,
\langle\alpha|\rho|\beta\rangle|
\alpha\rangle\langle\beta| \;\;,\nonumber\\
\langle\alpha|\rho(t)|\beta\rangle&=&\langle\alpha|e^{\int{\cal L}dt}\rho(0)
|\beta\rangle\;\;,\end{eqnarray}
giving
\widetext
\begin{equation}
\rho(t)=
\sum_{\alpha_{0}\beta_{0}}\frac{N_{\beta_{0}\alpha_{0}}}
{\kappa(t)}\langle\beta_{0}|\alpha_{0}\rangle
\int\,\frac{d^2\alpha d^2\beta}{\pi^2}
\langle\alpha|\beta\rangle
\exp\,
\left[
-\frac{1}{\kappa(t)}
(\alpha-\alpha_{0}(t))^{*}(\beta-\beta_{0}(t))
\right]\,
|\alpha\rangle\langle\beta|
\;\;,\label{eq20}\end{equation}
where ${\cal L}$ is the Markov super-operator (\ref{eq1.1}).
With this the dynamics is completely solved. However, to compare the decay of
off-diagonal elements of the reduced density matrix in other bases we will
also compute $\rho$ in the position $x$, momentum $p$ and number
basis.
\widetext
To compute the off-diagonal elements in the position basis we again
use equation (\ref{eq17}), giving
\begin{equation}
\langle x|\rho(t)|y\rangle=\langle x|\int\frac{d^2\xi}{\pi}
e^{-\xi
a^{\dagger}}e^{\xi^{*}a}\chi(\xi,t)|y\rangle\;\;,\label{xy}\end{equation}
where $\chi(\xi,t)$ is given in (\ref{eq14}). From Appendix B we have
\begin{equation}
\langle x|e^{-\xi a^{\dagger}}e^{\xi^{*}a}|y\rangle =
\exp\,\left[ \frac{|\xi|^2}{2}+\frac{\xi^{*\,2}-\xi^{2}}{4}+
x\sqrt{\frac{\omega}{2\hbar}}(\xi^{*}-\xi)\right]\,
\delta(\xi_{x}+\sqrt{\frac{\omega}{2\hbar}}(x-y))\sqrt{\frac{\omega}{2\hbar}}
\;\;.\label{xy1}\end{equation}
For
$\rho(t=0)=\sum_{\alpha_{0}\beta_{0}}\,N_{\beta_{0}\alpha_{0}}|
\alpha_{0}\rangle\langle\beta_{0}|$
we can perform the gaussian integrations in (\ref{xy}) and use the result
(\ref{xy1}) to finally get
\begin{eqnarray}
&&\langle x|\rho(t)|y\rangle = \nonumber\\
&& \sqrt{\frac{2}{\pi\sigma_{+}^{2}}}\sum_{\alpha_{0}\beta_{0}}
N_{\beta_{0}\alpha_{0}}\langle\beta_{0}|\alpha_{0}\rangle\times\nonumber\\
&&
\exp\,\left[
-\frac{1}{2\sigma_{-}^{2}}X_{-}^{2}+\frac{1}{\sqrt{\sigma_{-}\sigma_{+}}}
(\alpha_{0}(t)-\beta_{0}^{*}(t))X_{-}-\frac{1}{2\sigma_{+}^{2}}
(X_{+}-\sqrt{\sigma_{+}\sigma_{-}}
(\alpha_{0}(t)+\beta_{0}^{*}(t)))^{2}\right]\;\;,
\label{eq21}\end{eqnarray}
where
\begin{equation}
\sigma_{+}^{2}=\frac{2\hbar}{\omega}(2\kappa(t)-1)\;,\qquad\qquad
\sigma_{-}^{2}=\frac{2\hbar}{\omega(2\kappa(t)-1)}\;\;,\label{sigma_def}
\end{equation}
and $X_\pm=x\pm y$.
Using similar methods we can calculate the elements in the momentum
basis to be
\begin{eqnarray}
&&\langle p_1 | \rho(t) | p_2 \rangle = \nonumber \\
&&
\frac{1}{\hbar\omega}\sqrt{\frac{\tilde{\sigma}_{-}^{2}}{2\pi}}\sum_{\alpha_0
\beta_0}
N_{\beta_0 \alpha_0}\langle \beta_0 | \alpha_0 \rangle\nonumber\\
&&\exp\,\left[
-\frac{1}{2\tilde{\sigma}_{-}^{2}}P_{-}^2
+\frac{i}{\sqrt{\tilde{\sigma}_{-}\tilde{\sigma}_{+}}}
(\beta_{0}^{*}(t)+\alpha_{0}(t))P_{-} -\frac{1}{2\tilde{\sigma}_{+}}(
P_{+}-i\sqrt{\tilde{\sigma}_{-}\tilde{\sigma}_{+}}
(\alpha_{0}(t)-\beta_{0}^{*}(t)))^{2}\right]\;\;,
\label{eq22}\end{eqnarray}
where
\begin{equation}
\tilde{\sigma}_{+}^{2}=2\hbar\omega(2\kappa(t)-1)\;,\qquad\qquad
\tilde{\sigma}_{-}^{2}=\frac{2\hbar\omega}{2\kappa(t)-1}\;\;,\end{equation}
and $P_\pm=p_1\pm p_2$.
To calculate the elements of $\rho$ in the number basis is slightly
tedious.
Noting that
\begin{equation}
\langle n | \rho | m \rangle=
\int\frac{d^2\alpha d^2\beta}{\pi^2}\langle n| \alpha\rangle\langle
\alpha|\rho|\beta
\rangle\langle\beta| m\rangle\;,\qquad\qquad\langle n|\alpha\rangle=
e^{-|\alpha|^2/2}\alpha^n/\sqrt{n!}\;\;,\label{eq23}\end{equation}
and the identity
\begin{equation}
A^n=\int\frac{d^2\alpha}{\pi}\alpha^n \, e^{-|\alpha|^2+A\alpha^{*}}=
\int\frac{d^2\alpha}{\pi}\alpha^{*\,n} \, e^{-|\alpha|^2+A\alpha}\;\;,
\end{equation}
we find, with some work, that
\begin{eqnarray}
\langle n | \rho(t) | m \rangle= & & \nonumber\\
& &\frac{1}{\kappa(t)}\sum_{\alpha_0 \beta_0}N_{\beta_0 \alpha_0}
\langle \beta_0|\alpha_0\rangle
e^{-\frac{1}{\kappa(t)}\alpha_{0}^{*}(t)\beta_{0}(t)}
\,\Pi^{nm}(\alpha_0,\beta_0,t)\left(\frac{\alpha_0^*(t)}
{\kappa(t)}\right)^m\left(\frac{\beta_0(t)}{\kappa(t)}\right)^n\;\;,
\label{eq24}\end{eqnarray}
where
\begin{equation}
\Pi^{nm}(\alpha_0,\beta_0,t)\equiv
\sum_{s=0}^{{\rm min}(n,m)}\frac{ n!m!}{s!(n-s)!(m-s)!}\left(
\frac{\kappa(t)(\kappa(t)-1)e^{\kappa(t)t}}{\beta_0\alpha_0^*}\right)^s\;\;.
\label{eq24a}\end{equation}
We note that even in the case where
$\rho(t=0)=|\alpha_0\rangle\langle\alpha_0|$ equations (\ref{eq24}) and
(\ref{eq24a}) do not reduce to a simple tractable forms.
In what
follows we will not make much use of equations (\ref{eq24}) and
(\ref{eq24a}) except to note that they become diagonal as $t\rightarrow\infty$.
Finally we note that the results for $\langle x|\rho| y\rangle$ found here
agree with those found in \cite{SAVAGE:1985}.
\narrowtext
\section{Environment Induced Superselection}
We now wish to examine the behaviour of the off-diagonal elements of
$\rho$ in a number of different bases and to determine whether EIS
occurs. There have been a number of proposals for the measure
of the degree of diagonality of the density matrix
\cite{MORIKAWA:1990,LAFLAMME:1993}.
These have mostly been defined however for a
gaussian $\rho$. Here we expand these definitions to a measure
$\Omega_{\alpha}$ of the ``diagonality'' of a general density matrix
in a particular basis $\alpha$ ($\alpha$ here denotes a general basis
i.e. x, p, n or coherent state).
From recent work
\cite{ALBRECHT:1992,ALBRECHT:1991,ZUREK:1993a,ZUREK:1993} there appears to
exist some competition between the attainment of a stable predictable pointer
basis and the attainment of EIS type decoherence.
In this paradigm one must be careful not to confuse the loss of
coherence with the loss of interference.
In this paper we interpret EIS as a process whereby the {\em coherence} of the
state is lost.
From Zurek's earlier work the signature
of decoherence was the vanishing of the {\em coefficients} of the
off-diagonal elements of the reduced density matrix of the system in a
given basis. The
decay of the interference in a particular basis
was due to the decay of these coefficients. However, the loss of
interference can also be brought about by
sending the overlap of the off-diagonal states in that basis to
zero.
For example, in Young's slits, when one sets the slit
separation to be very large in comparison to the screen distance, the
magnitude of the interference terms decays due to the vanishing of the
overlap of these off-diagonal elements in the position basis.
(There may, of course, be significant overlap in other bases, eg. the momentum
basis.) In this case
the loss of interference
is {\em not} due to the decay of the coefficients associated
with these off-diagonal elements. The state remains pure, no
coherences have been destroyed yet the quasi-probability distribution
function {\em tends} towards (but is never exactly)
that displayed by a true probability
distribution. Recently decoherence in the EIS paradigm has been
defined as the absence of
interference fringes in the Wigner function in the special case of a
superposition of two gaussians \cite{PAZ:1992}.
These interference fringes decay in
magnitude for large separations of the two gaussians. They never
disappear except in the case of infinite separation. The fringes also
vanish dynamically with time if one introduces a coupling with an
infinite bath. In the latter case coherence has been lost and
the off-diagonal coefficients decay while in the former case the
interference terms vanish because the overlap of the off-diagonal elements
in the chosen basis
goes to zero. Thus, while the loss of coherence necessitates the loss
of interference the loss of interference need not result in a loss of
coherence. In this paper we follow the spirit of earlier work and define
Environment Induced Superselection as the dynamical process whereby the
reduced state of the system deviates from a single ray in Hilbert
space and moves towards a description as an improper mixture
(for more information on improper mixtures see \cite{DESPAGNAT:CONCEPTUAL}).
The most obvious measure of the deviation of a given state $\rho$ from
being a ray in Hilbert space is the statistical distance between the
state $\rho$ and the closest mixed state
\cite{HUBNER:1992,BURES:1969,CAVES:1993}
\begin{equation}
D_{1}(\rho_{1})\equiv \min_{\rho_{m}\in {\rm diag}\,\rho}\,
d^{2}_{B}(\rho_1 ,\rho_{m})\;\;,\end{equation}
where
\begin{equation}
d_{B}^{2}(\rho_1,\rho_2)=2(1-{\rm Tr}\sqrt{
\rho_{1}^{1/2}\rho_{2}\rho_{1}^{1/2}})\;\;,\label{eq25}
\end{equation}
is the Bures metric on the state space of all pure and impure states
$\rho$ \cite{BURES:1969}. Equation (\ref{eq25}) is very difficult to
evaluate in general. Instead we will generalize a measure of coherence
for gaussian $\rho$ first
introduced by Morikawa \cite{MORIKAWA:1990} and used later in
\cite{BRANDENBERGER:1990,LAFLAMME:1993}. For $\rho$ of the form
\begin{equation}
\langle \bar{x}|\rho|x\rangle =
D \exp\left[
-A(x-\bar{x})^2-B(x+\bar{x})^2+iC(x-\bar{x})(x+\bar{x})\right]\;\;,
\end{equation}
Morikawa defined a measure of coherence to be
\begin{equation} {\rm QD}\equiv
\frac{A}{B}\;\;.\label{eq26}\end{equation}
When ${\rm QD}>1$ the magnitude of the elements of the density matrix
in the position basis are peaked preferentially along the diagonal
$x=\bar{x}$. For $\rho$ pure, ${\rm QD}=1$. The natural generalization
of this measure to non-gaussian $\rho$ is to compute the ratio of the
off-diagonal variance to the on-diagonal variance of the squared
magnitude of $\rho$ in a specified basis
$\hat{\eta}|\eta\rangle=\eta|\eta\rangle$, that is
\begin{equation}
\Omega_{\eta}\equiv\frac{\int\int\,d\eta_{1}\,d\eta_{2}\,
\langle\eta_1|\rho|\eta_2\rangle\langle\eta_2|\rho|\eta_1\rangle
|\eta_1-\eta_2|^{2}}
{\int\int\,d\eta_1d\eta_2\,\langle\eta_1|\rho|\eta_2\rangle
\langle\eta_2|\rho|\eta_1\rangle|\eta_1+\eta_2-\langle
\eta_1+\eta_2\rangle|^{2}}\;\;.\label{eq27}\end{equation}
For $\rho$ gaussian and $|\eta\rangle=|x\rangle$, $\Omega_{x}=1/{\rm
QD}$ from (\ref{eq26}). For pure states,
$\rho=|\psi\rangle\langle\psi|$ and
$\Omega_{\eta}=1\;\;\forall\,\eta$. This makes sense as
no pure state undergoing unitary evolution can lose it's coherence.
The offset in the denominator gives the variance of
${\cal S}\equiv|\langle\eta_1|\rho|\eta_2\rangle|^2$ about the mean
$\langle\eta\rangle$. Distributions of $\cal S$ which are concentrated
along $\eta_{-}=\eta_1-\eta_2=0$ result in $\Omega_{\eta}<1$. As $\cal
S$ becomes more concentrated along $\eta_{-}=0$, $\Omega_{\eta}$
decreases to zero. From (\ref{eq27}) the value of $\Omega_{\eta}$
depends on the basis chosen. This we expect as some basis will
display a greater peaking about their diagonal than others. The
advantage of the dimensionless measure (\ref{eq27}) is that one can
compare $\Omega_{\eta}$ and $\dot{\Omega}_{\eta}$ for various bases
$|\eta\rangle$. One can also consider bases which are discrete. In
this case the integrals in (\ref{eq27}) become discrete sums.
Heuristically the measure $\Omega_{\eta}$ tells us how damped the
off-diagonal coherences are in the basis $|\eta\rangle$. For pure
states, no matter what the configuration, these ``off-diagonal''
coherences never vanish. For every contribution to the ``on-diagonal''
elements of $\rho$ there are equal contributions to the
``off-diagonal'' elements. A classic example of this is the
superposition of two localized coherent states
$|\psi\rangle=|\alpha\rangle+|-\alpha\rangle$. For a pictorial
description of $\cal S$ for this state see Zurek \cite{ZUREK:1991}.
In this example the heights of the four peaks in Zurek's plots do not
change with increasing $|\alpha|^2$. No coherence is lost, however the
interference diminishes.
A quantification of the amount of interference present in a state is
generally given by the magnitude of a typical off-diagonal element.
For the special case of two superposed gaussians a measure of the
fringe visibility of the ripples of the Wigner function has been
proposed \cite{PAZ:1992}. Another quantification of the
degree of ``off-diagonality'' would be
\begin{equation}
\hat{\Omega}_{\eta}\equiv\frac{1}{{\rm Tr}\,\rho^{2}}
\int\int\,d\eta_1 d\eta_2\,
\langle\eta_1|\rho|\eta_2\rangle\langle\eta_2|\rho|\eta_1\rangle
|\eta_1-\eta_2|^2\;\;.\label{eq35}\end{equation}
This gives a measure of the mean squared variance of the distribution
$|\rho(\eta_1,\eta_2)|^2$ along the off-diagonal. Because this measure
has dimensions of $|\eta|^2$, comparisons of $\hat{\Omega}$
between bases are made complicated through the different
length scales associated with each basis. For a comparison between
bases one should use the dimensionless measure (\ref{eq27}). We also
note that while the discrete version of (\ref{eq35}) vanishes for
$\rho\rightarrow$diagonal (eg. spin 1/2 particle), for
\begin{eqnarray}
\rho &=&|\psi\rangle\langle\psi|\;\;,\nonumber\\
2|\psi\rangle &=&|\alpha\rangle+|-\alpha\rangle\;\;,\label{state}
\end{eqnarray}
the continuous version
of (\ref{eq35}) evaluated in the coherent state basis increases like
$|\alpha|^2$.
Although (\ref{eq27}) and (\ref{eq35}) seem to be natural
candidates for a measure of ``diagonality'' in the EIS sense we see
that the prototypical state (\ref{state}) is not diagonal nor even
approximately diagonal in the coherent state basis. It may be said
that since the ripples in the Wigner function decrease in size as
$|\alpha|^2\rightarrow\infty$ one should treat this state in this
limit as ``interferenceless'' or ``classical'', This I believe is
incorrect. The Wigner function for this state
always possesses negative regions for all
$|\alpha|\neq\infty$.
The interference represented by these would be manifested
in the expectation values of observables which have support in these
regions. The effects of EIS are to cause these negative regions to
disappear in a finite time. However, as pointed out by Bell,
positivity of the Wigner function is a necessary but {\em not}
sufficient condition that the Wigner distribution correctly represents a true
probability distribution \cite{BELL:SPEAKABLE}. This connection will
be made clearer in \cite{TWAMLEY:1993d}. Finally we can use the
relation
\begin{equation}
\langle x|\rho| y\rangle=\int\,dp\,e^{ip(y-x)/\hbar}
W(p,\frac{x+y}{2})\;\;,\label{eq30}\end{equation}
to rewrite (\ref{eq35}) as
\widetext\begin{eqnarray}
&&\hat{\Omega}_{\eta}=\frac{1}{\int\,dpdx\,W^2(p,x)}\times\nonumber\\
&&\left\{
1+8\hbar\omega\int\,dpdx\left[
W(p,x)\left( \frac{d^2}{\omega^2dx^2}+\frac{d^2}{dp^2}\right)W(p,x)-
\left(\frac{d}{\omega
dx}W\right)^2-\left(\frac{d}{dp}W\right)^2\right]
\right\}\;\;,\label{eq30a}\end{eqnarray}\narrowtext
where we have used
\begin{equation}
{\rm
Tr}\,\rho^2=\int\,\frac{d^2\alpha}{\pi}\,W^{2}(\alpha,\alpha^{*})=
\int\,\frac{dpdx}{2\pi\hbar}W^{2}(p,x)\;\;.\end{equation}
In what follows we will use $\Omega_{\eta}$, (\ref{eq27}) as a measure of
diagonality. Taking $\rho(t=0)=|\alpha_0\rangle\langle\alpha_0|$ we can
compute
\begin{equation}
\Omega_{x}(t)=\Omega_{p}(t)=\Omega_{\alpha}^2=
\frac{\sigma_{-}^2}{\sigma_{+}^2}=\frac{1}{(2\kappa(t)-1)^2}
\;\;.\label{EIS}\end{equation}
Recalling that $\kappa(t)=1+\bar{n}(1-e^{-\gamma t})$, (\ref{eq14}),
this result
would indicate that we obtain a greater degree of diagonalisation
in the $x$ and $p$ basis than in the coherent state basis $\alpha$. At
$t=0$, since the system begins in a pure state, $\Omega=1$ for all
bases. Thus Environmental Induced Superselection has occurred in all
three bases. From (\ref{eq24a}), $|\langle n+m|\rho(t)|n-m\rangle|\sim
e^{-m\gamma t}$ and thus $\rho$ also diagonalises in the number basis,
however the exact form of $\Omega_{n}$ is quite complicated and will
not be given here. For the
case of a completely general initial state we only quote the result in
the position basis. For
$\rho(t=0)=\sum\,N_{ij}|\alpha_{j}\rangle\langle\alpha_{i}|$ one finds
\widetext\begin{eqnarray}
&&\Omega_{x}=\nonumber\\
&&\frac{\sigma_{-}^2}{\sigma_+^2}\left\{\frac{
\sum_{ijkl}\,N_{ij}N_{kl}\langle\alpha_i|\alpha_j\rangle
\langle\alpha_k|\alpha_l\rangle e^{A_+^2\sigma_+^2+A_-^2\sigma_-^2+B}
\left[1+2A_-^2\sigma_-^2\right]}
{\sum_{ijkl}\,N_{ij}N_{kl}\langle\alpha_i|\alpha_j\rangle
\langle\alpha_k|\alpha_l\rangle e^{A_+^2\sigma_+^2+A_-^2\sigma_-^2+B}
\left[1+2A_+^2\sigma_+^2+8\langle x\rangle(\frac{\langle
x\rangle}{\sigma_+^2}-A_+)\right]}\right\}\;\;,\label{19a}\end{eqnarray}
\narrowtext
where
\begin{eqnarray}
A_-&=&\frac{-1}{2\sqrt{\sigma_-\sigma_+}}
\left[\alpha_i^*(t)-\alpha_k^*(t)+\alpha_l(t)-\alpha_j(t)\right]\;\;,\\
A_+&=&\frac{\sqrt{\sigma_+\sigma_-}}{2\sigma_+^2}
\left[\alpha_i^*(t)+\alpha_k^*(t)+\alpha_j(t)+\alpha_l(t)\right]\;\;,
\end{eqnarray}
and
\begin{equation}
B=-\left[(\alpha_i^*(t)+\alpha_j(t))^2+(\alpha_k^*(t)+\alpha_l(t))^2\right]
\left(\frac{\sigma_-}{2\sigma_+}\right)\;\;.\end{equation}
A similar result holds for $\Omega_p$ where we replace $A_\pm$ with
\begin{eqnarray}
A_+ &=&
\frac{i}{2\sqrt{\sigma_+\sigma_-}}\left[\alpha_i^*(t)-\alpha_k^*(t)
+\alpha_j(t)-\alpha_l(t)\right]
\;\;,
\nonumber\\
A_- & = & \frac{-i\sqrt{\sigma_+\sigma_+}}{2\sigma_+^2}
\left[\alpha_i^*(t)+\alpha_k^*(t)-\alpha_j(t)-\alpha_l(t)\right]\;\;.
\end{eqnarray}
To obtain a clearer understanding of $\Omega$ let us compute
$\Omega_x$ for the particular state
$\rho_2=
\frac{1}{2}|\alpha_0\rangle\langle\alpha_0|+\frac{1}{2}|-\alpha_0\rangle
\langle -\alpha_0|$ where
$\alpha_0=(\omega\bar{x}+i\bar{p})/\sqrt{2\hbar\omega}$. Denoting
$\rho_1=|\alpha_0\rangle\langle\alpha_0|$ we find that
$\Omega_{2\,x}=\Omega_{1\,x}\hat{\Omega}_{2\,x}$ where
$\Omega_{1\,x}=\sigma_{-}^2/\sigma_{+}^2$, (from above) and
\begin{equation}
\hat{\Omega}_{2\,x}=\left[
\frac{1+\frac{1}{2}(1-\frac{2\bar{p}^2\sigma_-^2}{\hbar^2})
e^{-4\bar{x}^2/\sigma_+^2-\bar{p}^2\sigma_-^2/\hbar^2}}
{1+\frac{8\bar{x}^2}{\sigma_+^2}+
e^{-4\bar{x}^2/\sigma_+^2-\bar{p}^2\sigma_-^2/\hbar^2}}\right]\;\;,
\label{omegax}\end{equation}
where $\sigma_\pm$ are defined in (\ref{xy1}) and evaluated at $t=0$.
Since $\hat{\Omega}_{2\,x}\le 1$ we see that the coherence of a
mixture of two Glauber states is less than the coherence of just one.
The \mbox{$x$-coherence} of the mixture $\rho_2$ decreases inversely as
$(\bar{x}/\sigma_{+})^2$. Thus packets which are separated in
position by amounts significantly greater than their half widths
possess lower $x$-coherence. Also $\hat{\Omega}_{2\,x}$ depends very
weakly on the momentum $\bar{p}$ except for small $\bar{x}$ where the
overcompleteness property of the Glauber states cause
interference-like behaviour in the $x$ basis. A plot of (\ref{omegax})
is given in Fig 1.
The dynamical evolution of $\Omega_{2\,x}$ is
given again by equation (\ref{omegax}) with $\sigma_{\pm}(t)$ given
by (\ref{xy1}) while
\begin{eqnarray}
\bar{x}(t)&=&e^{-\gamma t/2}\left[\bar{x}(0)\cos \omega
t+\frac{\bar{p}(0)}{\omega} \sin \omega t\right]\;\;,\\
\bar{p}(t)&=&e^{-\gamma t/2}\left[\bar{p}(0)\cos\omega t-\omega
\bar{x}(0)\sin\omega t\right]
\;\;.\\
\end{eqnarray}
Essentially, the evolution begins at $t=0$
with the two Glauber states located symmetrically about the origin at
positions $\pm\alpha_0\in\Bbb{C}$. Since
the Glauber states are overcomplete there is some overlap between the
two. As time progresses, the state, represented loosely as two
superposed gaussians in the Wigner distribution, spirals in towards
the origin while the half widths of each gaussian spreads with time
for nonzero $\bar{n}$. The system motion complicates the
interpretation and instead, one can return to the interaction picture.
In the interaction picture, for a fixed choice of $\alpha_0(t=0)$ where
$\bar{x}\gg\sqrt{2\hbar/\omega}$,
$\Omega_{2\,x}$ decreases below the value $\Omega_{1\,x}$ due to the
large separation in position between the two gaussians. However, as
$t\rightarrow\infty$, $\hat{\Omega}_{2\,x}\rightarrow 1$ and thus the
coherence of $\rho_2(t)$ tends to the same value as the coherence of
$\rho_1(t)$ in this limit. This is not surprising as the final states
are the same. As the bath temperature is raised the time for
$\Omega_{2\,x}$ to reach a given value decreases since the
gaussian's half widths become larger and thus $\Omega_{1\,x}$
decreases faster. Also, we note that when the $\bar{x}$ separation is
large, the state may possess a {\em lower} coherence than in the
equilibrium state at $t=\infty$. This phenomena is also seen is the
relaxation behaviour of squeezed states in a thermal bath.
The coherence behaviour of extended quantum states
and the relation of the $\Omega$ measure of coherence to more
information theoretic measures will be treated in another paper
\cite{TWAMLEY:1993d}.
\begin{center}
\graphics{myfig1.ps}{5}{7}{-2}{-10}{.5}
\end{center}
\begin{center}
{\parbox{15cm}{\small\center Fig 1.
Surface plot of $\Omega_x$ for state $\rho_2$ with $\omega=1$ and $\hbar=1$.
}}
\end{center}
To conclude and summarize: we have described the dynamics of the
reduced density matrix of a harmonic oscillator linearly coupled to an
infinite bath of oscillators in the quantum optical regime. We have
expressed the results in the position, momentum, number and coherent
state basis. After considering a number of proposals for a measure of
the ``diagonality'' or degree of coherence of the state $\rho$ we have
advanced $\Omega_\eta$ (\ref{eq27}) as a natural measure. We
calculated this quantity in the above mentioned bases for an initial
Glauber state and found that coherence was lost in each basis. The coherence
loss rate for this initial state was found to be identical in the $x$
and $p$ basis while the loss rate in the coherent state basis was
smaller. Expressions for the coherence of a general initial state were
found in both the $x$ and $p$ bases. Finally, the position coherence for
an initial mixture of two coherent states was calculated and was found
to be more ``diagonal'' than that of a single coherent state,
especially of the position separation between the two coherent states is large.
Although the measure $\Omega_\eta$ may seem to be a ``natural''
generalization of the early definitions \cite{MORIKAWA:1990} it would be
interesting to understand the information theoretic aspects of such
measures. Of more interest would be the link between the
``diagonality'' of $\rho$ and the representation of $\rho$ by a true
probability distribution. This work is in progress \cite{TWAMLEY:1993d}.
\section{Consistent Phase-Space Histories}
Consistent Histories is an alternate paradigm developed by Hartle
and Gell-Mann \cite{HARTLE:1993a,HARTLE:1989,OMNES:1992,GELL-MANN:1992d}
which attempts to describe the
emergence of the ``Quasiclassical'' world from the quantum world. The
primary ingredients of this paradigm are coarse grained histories and
the decoherence functional. Essentially, a coarse grained history is a
partial description of the temporal evolution of the total
system+environment where one has ``coarse grained'' away information
either by (i) ignoring some of the coordinates for all time (ie. the
environment) (ii) only describing the configuration of the priveliged
coordinates at certain times (iii) projecting the
privileged system coordinates at specified time on to quantities of
interest eg. binning, averaging etc. The resulting coarse grained
histories may be described in some cases (those treated here)
by history independent
temporally ordered strings of projectors $P^\alpha_i$, where the
$P_i^\alpha$ are a complete and exclusive set of projectors on to a set
of alternatives labeled by $\{\alpha\}$ and where $P_i^\alpha$ is the
ith projector in the set $P^{\{\alpha\}}$. In the past, calculations
have concentrated on sequences on projectors in position using the
Calderia-Legget model \cite{DOWKER:1992,PAZ:1992a}. In this section we
expand these calculations to projectors on phase-space and consider
coarse grained phase-space
histories of the privileged system harmonic oscillator
in the quantum optical regime. In the Consistent Histories paradigm a
probability is assigned to a coarse grained history via the rule
\begin{equation}
P(h)\equiv {\rm Tr}[P_{h}\rho P_h^\dagger ]\;\;,\label{cheq1}
\end{equation}
where $P_h$ is the temporally ordered string of projectors
corresponding to the coarse grained history $h$ ie.
\begin{equation}
P_h\equiv
P_{i_N}^{\alpha_N}{\cal U}(\Delta t_N)
P_{i_{N-1}}^{\alpha_{N-1}}
{\cal U}(\Delta t_{N-1})\cdots P_{i_1}^{\alpha_1}
\;\;,\label{cheq2}
\end{equation}
and where $\cal U$ is the propagator for the full system-environment.
We now assume that these projectors factor out the environment and can thus be
written in the form $P_i^\alpha=\tilde{P}_{i\,S}^\alpha\otimes I_{E}$.
In the Markov regime the dynamics of system's reduced density operator
is purely local in time and $\rho_{ S}\equiv {\rm Tr}_{ E}\,\rho$
satisfies
partial differential equation which is first order in time and contains
only system quantities.
In this regime
we can take the trace in (\ref{cheq1}) over
then environment and arrive at
\begin{equation}
P(h)={\rm Tr}_S\left[\tilde{P}^{\alpha_{N}}_{i_{N}}\,
e^{\int_{t_{N-1}}^{t_{N}}\,{\cal
L}dt}\left[\tilde{P}_{i_{N-1}}^{\alpha_{N-1}}\cdots\rho_{S}\cdots\right]\right]
\;\;,\end{equation}
where $\tilde{P}$ acts only on the system's Hilbert space,
$\rho_{S}$ is the reduced density operator for the system and we have
again used (\ref{eq1.1}) to evolve the system's reduced density operator
from $t_{i-1}$ to $t_i$. A similar method is used in \cite{PAZ:1993}.
Thus in the Markov regime the effects of the coupling to the
environment are taken care of through the superoperator $\cal L$ and
the calculation (\ref{cheq1}) reduces to a calculation on the system
only.
At the heart of this formalism is the requirement that these
probabilities satisfy the sum rule
\begin{equation}
P(h)=\sum_{\tilde{h}\in h}\,P(\tilde{h})\;\;,\label{cheq3}
\end{equation}
where $\tilde{h}$ is a finer graining of $h$. For the sum rule
(\ref{cheq3}) to hold with the graining $\tilde{h}$ we must have
\begin{equation}
{\rm Re}\,{\rm D}[\tilde{h}^\prime,\tilde{h}]=0\qquad\forall\;
\tilde{h}^\prime,\;\tilde{h}\in h\;\;,\tilde{h}^\prime\neq\tilde{h}\;\;,
\label{cheq4}\end{equation}
where
\begin{equation}
{\rm D}[\tilde{h}^\prime,\tilde{h}]={\rm Tr}[P_{\tilde{h}}\rho
P_{\tilde{h}^\prime}^\dagger ]\;\;,\label{decoherence_functional}
\end{equation}
is known as the Decoherence Functional \cite{HARTLE:1989}.
Condition (\ref{decoherence_functional}) is often termed {\em Weak
Consistency}. A stronger, although sometimes argued more physical,
condition is
\begin{equation}
{\rm D}[\tilde{h}^\prime,\tilde{h}]=0\qquad\forall\;
\tilde{h}^\prime,\;\tilde{h}\in h\;\;,\tilde{h}^\prime\neq\tilde{h}\;\;,
\end{equation}
and is called {\em Medium Consistency}.
A major goal in this
approach is the identification of all those sets of histories which
are exactly consistent and from this, an identification of those
histories which yield a ``Quasiclassical'' world. We do not wish to
give an in-depth review of this formalism and refer the reader the
excellent articles of Gell-Mann and Hartle
\cite{HARTLE:1993a,HARTLE:1989,OMNES:1992,GELL-MANN:1992d}.
It has been recognized that sets of histories which are {\em exactly}
consistent may be characterized by very exotic and un-physical system
variables \cite{ALBRECHT:1992}. To examine the consistency properties of
histories parametrised by (perhaps) more interesting and physical
variables (ie. position and momentum) a measure of the degree of
consistency of a set of histories $\{\tilde{h}\}$ has been advanced by
Dowker and Halliwell \cite{DOWKER:1992}
\renewcommand{\arraystretch}{.5}
\begin{equation}
\epsilon\equiv
\begin{array}{c}
{\rm max} \\
\tilde{h}^\prime,\; \tilde{h}\in\{h\}\\
\tilde{h}^\prime\neq\tilde{h}
\end{array}\frac{{\rm Re}{\rm D}
[\tilde{h}^\prime,\tilde{h}]}
{\sqrt{{\rm D}[\tilde{h},\tilde{h}]{\rm D}
[\tilde{h}^\prime,\tilde{h}^\prime]}}\;\;.\label{epsilon}\end{equation}
As $\{\tilde{h}\}$ becomes more consistent $\epsilon$ decreases.
Dowker and Halliwell have also argued that in general $\epsilon$
decreases as the graining size is increased. Only for very artificial
grainings, where the phases of the constituent histories in
the coarser graining add constructively, does $\epsilon$ remain
unchanged or actually increase. This point will be important later.
As mentioned above, all previous calculations have looked at histories on
position space. However we normally consider classical dynamics as
operating within phase-space. It would be of great interest to see if
phase-space histories of the system become more consistent with the
addition of an interaction with an external bath. In the previous
section on Environmental Induced Superselection we saw that the
quantum optical model gave EIS in both the $x$ and $p$ bases. Here we wish
to see whether phase-space histories for the privileged harmonic
oscillator displays consistency and if so, to what degree? How does
the degree of consistency
depend on the model parameters ie. $\gamma$, $\bar{n}$ graining
etc? In the following we will consider two-time projected histories on
the phase-space of the privileged oscillator. We will again couple the
system to an external bath and use the solutions of the master
equation (\ref{eq1.1}) found in section II to evolve $\rho_{S}$ between
projections. We will calculate the Decoherence Functional and the
degree $\epsilon$ of consistency of these histories. This $\epsilon$
will be a function of the equilibrium number of photons per mode in the
bath ($\bar{n}=1/(\exp\frac{\hbar\omega}{kT}-1)$), the time $t$, the
initial state of the system $\rho(t=0)$ and the phase-space projectors
chosen (ie. the graining). We will choose gaussian type ``projectors''
on phase-space and will find that $\epsilon$ decreases with time $t$,
increasing bath temperature $T$ and final grain size $\sigma_2$ but
anomalously decreases with initial grain size $\sigma_1$. This
anomalous behaviour can also be seen in other calculations of
Decoherence Functionals for similar models
\cite{DOWKER:1992,PAZ:1992a}. We conclude that the introduction of the
bath causes the phase-space histories to become more consistent and
less peaked about the classical path. Finally we note that Consistent
Histories calculations tend to give rather lengthy and hard to
interpret results. We have found that working with this model has
resulted in a more transparent description.
\subsection*{Two-Time Phase Space Histories}
We wish to calculate the decoherence functional for two-time projected
histories on phase-space, that is
\begin{eqnarray}
{\rm D}[\tilde{h},\tilde{h}^\prime] &=&
{\rm D}[\alpha_3,\alpha_2,\alpha_1,\sigma_2,\sigma_1,t]\nonumber\\
& =&
{\rm Tr}\left[P(\alpha_3,\sigma_2)\,e^{\int_{t_1}^{t_2}\,{\cal L}dt}
\left[P(\alpha_2,\sigma_1)\rho(t_1)P^\dagger(\alpha_1,\sigma_1)\right]\right]
\;\;,\label{cheq6}\end{eqnarray}
where the coarse grained history is defined by a projector
$P(\alpha,\sigma)$ on the phase-space of the system oscillator at
$\alpha\in\Bbb{C}$
of ``width'' $\sigma$. The trace in
(\ref{cheq6}) is over the system's Hilbert space. We will assume,
as in \cite{DOWKER:1992,PAZ:1992a}, that $P$ is a {\em
quasi-projector}.
In particular we will take $P$ to be gaussian with
\begin{equation}
P(\alpha,\sigma)=\frac{1}{\sigma^2}\int\,\frac{d^2\beta}{\pi}
e^{-\frac{|\beta-\alpha|^2}{\sigma^2}}|\beta\rangle\langle\beta|\;\;,
\label{cheq7}
\end{equation}
where $|\beta\rangle$ is a Glauber coherent state. These projectors are
complete ie. $\int\,d^2\alpha/\pi\,P(\alpha,\sigma)=1$, but are not
exclusive
\begin{equation}
P(\alpha_1,\sigma)P(\alpha_2,\sigma)\neq
\delta(\alpha_1-\alpha_2)P(\alpha_2,\sigma)\;\;.\label{not_projectors}
\end{equation}
Using such gaussian ``quasi-projectors'' makes the calculation analytically
tractable. In fact, since true projectors on phase-space do not exist
\cite{OMNES:1989b} the best one can do is to use such complete but
nonexclusive ``quasi-projectors''. Indeed, these types of
``quasi-projections'' play a significant role in the more rigorous ``Effects
and Operations'' theory of quantum measurement
\cite{CAVES:1986}. To create other quasi-projectors which are more
localized in phase-space one could use instead the marginally
overcomplete set of coherent states $\{|\alpha\rangle\}_{VN}$ where
one places an ordinary Glauber coherent state $|\alpha\rangle$ at each
point of a \mbox{von Neumann} lattice in phase space
\cite{PERELOMOV:1971,BARGMANN:1971}.
We shall calculate (\ref{cheq6}) for an arbitrary $\rho(t_1)$ by
decomposing $\rho(t_1)$ into coherent states and using (\ref{eq20})
to evolve to $t_2$. We will then consider a $\rho(t_1)$ resulting from
the propagation from a time $t_0<t_1$ of $\rho(t_0)$ which we shall
choose to be $\rho(t_0)=|\alpha_0\rangle\langle\alpha_0|$. We will
obtain the decoherence functional ${\rm D}$ and from this
calculate the degree of consistency $\epsilon$ and peaking $\cal P$
about the classical trajectory. We will also calculate $\Omega_h$, a
measure similar to (\ref{eq27}), which measures the coherence in
${\rm D}$ when considered as a density matrix. We find that
$\epsilon$ and $\Omega_h$ behave similarly with respect to their
dependence on the parameters in the model.
To begin, we take
$\rho(t_1)=\sum_{ij}N_{ij}|\alpha_j\rangle\langle\alpha_i|$ where we
have normalized according to
$\sum_{ij}N_{ij}\langle\alpha_j|\alpha_i\rangle=1$. Using (\ref{appendix1}),
\begin{equation}
\int\,\frac{d^2\alpha}{\pi} e^{-A|\alpha|^2+B\alpha^*+C\alpha}=
\frac{1}{A}e^{\frac{BC}{A}}\;\;,\label{cheq9}\end{equation}
and the definition of a coherent state $|\alpha\rangle=e^{\alpha
a^\dagger -\alpha^* a}|0\rangle$ we can compute
\begin{equation}
P(\alpha_2,\sigma_1)\rho(t_1)P^\dagger (\alpha_1,\sigma_1)=
\frac{1}{s_1^2}\sum_{ij}{\bbox{\Bbb F}}
(\alpha_1,\alpha_2,\alpha_i,\alpha_j,\sigma_1)|\zeta_j\rangle\langle\eta_i|
\;\;,\label{cheq10}\end{equation}
where
\widetext
\begin{equation}
s_1=1+\sigma_1^2\;,\qquad\qquad
s_1\zeta_j=\sigma_1^2\alpha_j+\alpha_2\;,\qquad\qquad
s_1\eta_i=\sigma_1^2\alpha_i+\alpha_1\;\;,\label{cheq11}\end{equation}
and
\begin{eqnarray}
{\bbox{\Bbb F}}(\alpha_1,\alpha_2,\alpha_i,\alpha_j,\sigma_1)=
\exp\left\{-\frac{1}{2s_1^2}
\right.&&
\left[ (|\alpha_i|^2+|\alpha_j|^2+|\alpha_1|^2+|\alpha_2|^2)
(1+2\sigma_1^2)\right.\nonumber\\
&&\left.\left.+(\alpha_2^*\alpha_j+\alpha_1\alpha_i^*)(2+3\sigma_1^2)
+\sigma_1^2(\alpha_2\alpha_j^*+\alpha_1^*\alpha_j)\right]\right\}\;\;.
\label{cheq12}\end{eqnarray}
Using (\ref{eq20}) we can evolve (\ref{cheq10}) from $t_1$ to
$t_2=t_1+\Delta t_2$ and denoting this by $\rho_{eff}(t_2)$ we get
\begin{equation}
\rho_{eff}(t_2)=\int\,\frac{d^2\alpha d^2\beta}{\pi^2}\,
\langle\alpha|\rho_{eff}(t_2)|\beta\rangle\,|\alpha\rangle\langle\beta|\;\;,
\label{cheq13}\end{equation}
where
\begin{eqnarray}
&&\langle\alpha|\rho_{eff}(t_2)|\beta\rangle=\nonumber\\
&&\frac{1}{s_1^2}\sum_{ij}
\frac{{\bbox{\Bbb F}}(\alpha_1,\alpha_2,\alpha_i,\alpha_j,\sigma_1)}
{\kappa(\Delta t_2)}
\exp\,\left\{
-\frac{1}{\kappa(\Delta t_2)}\left(\alpha-\eta_i(\Delta t_2)\right)^*
\left(\beta-\zeta_j(\Delta t_2)\right)\right\}
\langle\alpha|\beta\rangle\langle\eta_i|\zeta_j\rangle\;\;,\label{cheq14}
\end{eqnarray}
where $\zeta_j(t)\equiv e^{-(\gamma/2+i\omega)t}\zeta_j$ and
$\eta_i(t)\equiv e^{-(\gamma+i\omega)t}\eta_i$.
The final decoherence functional is thus
\begin{equation}
{\rm D}(\alpha_3,\alpha_2,\alpha_1,\sigma_2,\sigma_1,t_2)=
\frac{1}{\sigma_2^2}\int\frac{d^2\gamma}{\pi}e^{-\frac{|\alpha_3-\gamma|^2}
{\sigma_2^2}}\,\langle\gamma|\rho_{eff}(t_2)|\gamma\rangle\;\;,\label{cheq15}
\end{equation}
and is thus a gaussian smearing of the Q function of
$\rho_{eff}(t_2)$. Note that although Q functions are strictly positive for
true density matrices, when $\alpha_1\neq\alpha_2$, $\rho_{eff}(t_2)$
is {\em not} a true density matrix and thus ${\rm D}
[\alpha_3,\alpha_2,\alpha_1\neq\alpha_2]$ may be complex. To
calculate (\ref{cheq15}) is quite tedious but one finally arrives at
\begin{eqnarray}
{\rm D} = && \sum_{ij}\frac{N_{ij}}{\sigma_1^2}
{\bbox{\Bbb F}}(\alpha_2,\alpha_1,\alpha_i,\alpha_j,\sigma_1)
\frac{\langle\eta_i|\zeta_j\rangle}{s_2}\times\nonumber\\
&&\exp\left\{
-\frac{|\alpha_3|^2}{\sigma_2}
-\frac{\eta_i^*(\Delta t_2)\zeta_j(\Delta t_2)}{\kappa_2}
+\frac{(\kappa_2\alpha_3+\sigma_2^2\zeta_j(\Delta t_2))
(\kappa_2\alpha_3+\sigma_2^2\eta_i(\Delta t_2))^*}
{s_2\sigma_2^2\kappa_2 }\right\}\;\;,\label{cheq16}\end{eqnarray}
where $\kappa_2\equiv\kappa(\Delta t_2)$ and
$s_2\equiv\kappa(\Delta t_2)+\sigma_2^2$. For simplicity we will now
turn to the interaction picture and consider our projectors to be
stationary in this picture ie. $\alpha_1\rightarrow\alpha_1
e^{-i\omega t}$. With this, all $\omega$ dependence drops out of
(\ref{cheq16}). With a bit of rearranging (\ref{cheq16}) can be
written in the more convenient form
\widetext
\begin{eqnarray}
{\rm D} =
\sum_{ij}\frac{N_{ij}}{s_1^2s_2}\exp &&\left\{
-\frac{|\alpha_i|^2}{2}-\frac{|\alpha_j|^2}{2}
+\frac{(s_1-1)^2}{s_1^2}\alpha_i^*\alpha_jG
+\alpha_i^*{\bbox{\Bbb E}}(\alpha_1,\alpha_2,\alpha_3)
+\alpha_j{\bbox{\Bbb E}}^*(\alpha_2,\alpha_1,\alpha_3)\right.\nonumber\\
&&\left.
-\frac{|\alpha_3|^2}{s_2}-\frac{|\alpha_1|^2}{s_1}-\frac{|\alpha_2|^2}{s_1}
+\frac{\alpha_1^*\alpha_2}{s_1^2}G
+(\alpha_3^*\alpha_2+\alpha_3\alpha_1^*)\frac{e^{-\gamma \Delta
t/2}}{s_1s_2}
\right\}\;\;,\label{cheq17}\end{eqnarray}\narrowtext
where $G\equiv 1-e^{-\gamma \Delta t_2}/s_2$ and
\begin{equation}
{\bbox{\Bbb E}}(\theta_1,\theta_2,\theta_3)\equiv
\frac{\theta_1}{s_1}+\frac{\sigma_1^2\theta_2G}{s_1^2}
+\frac{\sigma_1^2\theta_3e^{-\gamma\Delta t_2/2}}{s_1s_2}\;\;.
\label{cheq19}\end{equation}\narrowtext
We are now ready to choose
$\rho(t_0)=|\alpha_0\rangle\langle\alpha_0|$. From (\ref{eq1.1})
we have
\begin{equation}
\rho(t_1)=\int\,\frac{d^2\alpha_j d^2\alpha_i}{\pi^2}\,
\langle\alpha_j|\,e^{\int_{t_0}^{t_1}\,{\cal
L}dt}\rho(t_0)|\alpha_i\rangle\,
|\alpha_j\rangle\langle\alpha_i|\;\;,\end{equation}
and using (\ref{eq20}) yields
\widetext\begin{equation}
\rho(t_1)=\int\,\frac{d^2\alpha_jd^2\alpha_i}{\kappa(\Delta t_1)\pi^2}\,
e^{-\frac{1}{\kappa(\Delta t_1)}
(\alpha_j-\alpha_0e^{-\gamma \Delta t_1/2})^*
(\alpha_i-\alpha_0r^{-\gamma \Delta t_1/2})}
\langle\alpha_j|\alpha_i\rangle\,|\alpha_j\rangle\langle\alpha_i|\;\;,
\label{cheq20}\end{equation}
where $\Delta t_1=t_1-t_0$.
Since the action of the decoherence functional is linear we can use
(\ref{cheq17}) to obtain the action on
$|\alpha_j\rangle\langle\alpha_i|$ and can then
perform the integrations in (\ref{cheq20}) to
compute the complete decoherence functional. We finally get
\begin{equation}
{\rm D}=\frac{1}{s_2\Upsilon}\exp\{ -A|\Delta_1|^2
-B|\Delta_2|^2 +C(\Delta_1^*\Delta_2+\Delta_1\Delta_2^*) -D|z|^2-i{\rm
Im}{\bbox{\Bbb A}}\}\;\;,\label{cheq21}\end{equation} where the
definitions of the various quantities appearing in (\ref{cheq21}) are
\widetext
\begin{equation}
\begin{tabular}{ll}
$\Upsilon=s_1^2\kappa_1-\Phi (s_1-1)^2G\;,\qquad\qquad$&
$4\Upsilon A=2s_1-1+\frac{1}{s_2}e^{-\gamma \Delta t_2}(1-\Phi)\;,$\\
$\Phi=\bar{n}(1-e^{-\gamma \Delta t_1})\;,$ & \\
$ \Delta_1^*=\alpha_1+\alpha_2-2\alpha_0e^{-\gamma \Delta t_1/2}\;,$ &
$s_2\Upsilon B=s_1^2-\Phi\;,$\\
$\Delta_2^*=\alpha_3-\alpha_0e^{-\gamma(\Delta t_2+\Delta t_1)/2}\;,$ &
\\
$ z=\alpha_1-\alpha_2 \;,$& $2s_2\Upsilon C=(s_1-\Phi)
e^{-\gamma\Delta t_2/2}\;,$\\
$G=1-\frac{e^{\gamma \Delta t_2}}{s_2}\;,$ & \\
$ s_1=1+\sigma_1^2\;,$ &
$4\Upsilon D=1+2s_1-\frac{1}{s_2}e^{-\gamma \Delta t_2}
+\Phi\left[4-e^{-\gamma \Delta t_2}\frac{(3-2s_1)}{s_2}\right]\;,$\\
$ s_2=\sigma_2^2+\kappa_2 \;,$ & \\
$ \kappa_2=1+\bar{n}(1-e^{-\gamma \Delta t_2}) \;,$ &
$ 2\Upsilon{\bbox{\Bbb A}}=z\left\{
\left[ 1-\frac{1}{s_2}e^{-\gamma\Delta t_2}(1+\Phi)\right]\Delta_1^*
+\frac{2}{s_2}[s_1+\Phi]e^{-\gamma\Delta t_2/2}\Delta_2^*\right\}\;,$\\
$\kappa_1=1+\bar{n}(1-e^{-\gamma \Delta t_1})\;.$ & \\
\end{tabular}\label{cheq22}\end{equation}
\narrowtext
Thus $\Delta_1$ and $\Delta_2$ are the deviations of the projector
positions at times $t_1$ and $t_2$ from the classical path of the
damped coherent state $|\alpha_0\rangle$. The diagonal elements of
${\rm D}$, ie. $z=0$, are peaked along this classical path
($\Delta_1=\Delta_2=0$). We also note that because of the
nonexclusivity of these quasi-projectors the diagonal elements of
the decoherence functional do not sum exactly to one. For a
decoherence functional of the form (\ref{cheq21}) it easy to show
that the degree $\epsilon$ of consistency defined in (\ref{epsilon}) is
\begin{equation}
\epsilon = \max_{\alpha_3\in\Bbb{C}}
\, e^{(C-D)|z|^2}\,\cos\,
{\bbox{\Bbb A}}\;\;.\label{cheq23}\end{equation}
Thus the maximum departure from perfect consistency between a pair of
histories occurs for (i) those histories for which $\alpha_3$ is such
that $|\cos\,{\bbox{\Bbb A}}|=1$ (ii) for pairs of histories
$h,\;h^\prime$ where the quasi-projectors $P(\alpha_1,\sigma_1)$ and
$P(\alpha_2,\sigma_1)$ are separated by their full width at half
maximum (FWHM)
ie. $z=1+\sigma_1^2=s_1$. From (i), (ii) and (\ref{cheq23}) we get
\begin{eqnarray}
\epsilon&=&\exp\left\{-\frac{
s_1}{2\Upsilon}\left[1-\frac{1}{s_2}e^{-\gamma\Delta
t_2}+\Phi\left(2+e^{-\gamma\Delta
t_2}\frac{(s_1-1)}{s_2}\right)\right]\right\}\nonumber\\
&\equiv&
\exp\{-\bar{\epsilon}(\sigma_1,\sigma_2,\gamma,\Delta t_2,\Delta
t_1,\bar{n})\}\;\;.\label{cheq24}\end{eqnarray}
For $\bar{\epsilon}\gg 1$ we achieve good consistency. To examine the
degree of peaking about the classical path we follow Dowker and
Halliwell \cite{DOWKER:1992} and
look at the product of the determinant of the
coefficients of $\Delta_i$ with the FWHM of the initial and final
projectors, that is ${\cal P}\sim (AB-C^2)s_1(1+\sigma_2^2)$. We get
\begin{equation}
{\cal P}=\frac{s_1(1+\sigma_2^2)}{4s_2\Upsilon^2}\left[
(s_1^2-\Phi)(2s_1-1)-\Phi e^{-\gamma\Delta t_2}\frac{(s_1-1)^2}{s_2}
\right]\;\;.\label{cheq25}\end{equation}
For ${\cal P}\gg 1$ we get significant peaking about the classical
path. Equations (\ref{cheq21}, \ref{cheq22}, \ref{cheq24}, \ref{cheq25})
are the main results of this section.
Let us examine the short and long time behaviour of $\bar{\epsilon}$ and
$\cal P$. For $t_2\sim t_1\sim 0$ we have
\begin{equation}
\bar{\epsilon}\approx \frac{1}{2s_1}[1-\frac{1}{s_2}]\;\;.
\label{cheq26}
\end{equation}
Thus as the final graining size $\sigma_2$ increases the degree of
consistency increases. This is in agreement with the general
arguments of Dowker and Halliwell. However, as the initial graining
size $s_1$ is increased, the degree of consistency drops. This
anomalous behaviour is also seen in other calculations
\cite{DOWKER:1992,PAZ:1992a}. In the long time limit, with $\Delta
t_2=\Delta t_1=\tau$ and letting $\tau\rightarrow\infty$ we have
\begin{equation}
\bar{\epsilon}\approx
\frac{(2\bar{n}+1)s_1}{2[s_1^2+\bar{n}(2s_1-1)]}\;\;,\label{cheq27}
\end{equation}
which again decreases with increasing $s_1$. However, the dependence
on bath temperature, through $\bar{n}$, is now apparent and thus the
consistency increases with increasing $T$. In the short time limit the
peaking is
\begin{equation}
{\cal P}\approx\frac{2s_1-1}{4s_1}\;\;,\label{cheq28}\end{equation}
while for long times
\begin{equation}
{\cal P}\approx\frac{s_1(s_1^2-\bar{n})(2s_1-1)}{2[s_1^2+\bar{n}(2s_1-1)]^2}
\left(\frac{1+\sigma_2^2}{1+\bar{n}+\sigma_2^2}\right)\;\;.\label{cheq29}
\end{equation}
The short time and long time limits of $\cal P$ display the expected
behaviour with the peaking getting better with increasing graining size
and decreasing bath temperature and time. Plots of the long time
behaviors of $\bar{\epsilon}$ and $\cal P$ are shown in Fig 2.
\begin{center}
(a)\hspace{6cm}(b)\\
\graphics{myfig2.ps}{5}{7}{-2}{-8}{.4}
\graphics{myfig3.ps}{5}{7}{1}{-8}{.4}
\end{center}
\begin{center}
{\parbox{15cm}{\small Fig 2.
Dependencies of the
degree of consistency and degree of peaking in the long time limit
on the bath temperature
(through $\bar{n}$) and
initial grain size $s_1$.
Graph (a) plots (\ref{cheq27}) as a function of $s_1$ and $\bar{n}$,
while graph (b) plots (\ref{cheq29}). We see that increasing the bath
temperature (which increases $\bar{n}$) increases consistency while
decreasing peaking. Increasing the initial grain size $s_1$ {\em
decreases} consistency and increases peaking
}}
\end{center}
We can also examine the degree of consistency by considering the decoherence
functional as a density matrix in the space of the coarse grained
histories $\{h\}$. With this viewpoint we can adopt the definition
(\ref{eq27}) to give a measure of the concentration of ${\rm
D}[h,h^\prime]$ about the diagonal $h=h^\prime$. For a
decoherence functional of the form
\mediumtext\begin{equation}
{\rm D}\sim N\exp\left[
-A|\Delta_1|^2-B|\Delta_2|^2+C(\Delta_1^*\Delta_2+\Delta_1\Delta_2^*)
-D|z|^2-i{\rm Im}\,z {\bf\Delta}\right]\;\;,\end{equation}
we can calculate
\begin{equation}
\Omega_h=\max_{\Delta_2\in\Bbb{C}}\,\frac{
\int\,d^2\Delta_1d^2z\,
{\rm D}[\Delta_2,\Delta_1,z]{\rm D}[\Delta_2,\Delta_1,-z]|z|^2}
{\int\,d^2\Delta_1d^2z\,
{\rm D}[\Delta_2,\Delta_1,z]{\rm D}
[\Delta_2,\Delta_1,-z]|\Delta_1-\langle\Delta_1\rangle|^2}\;\;.\label{cheq30}
\end{equation}\narrowtext
We can motivate this measure by noting that for each value of
$\Delta_2$, ${\rm D}[\Delta_2,\Delta_1,z]$ is very like a
density matrix with the on-diagonals labeled by $\Delta_1$ and the
off-diagonals labeled by $z$. Computing (\ref{cheq30}) for a given
$\Delta_2$ gives a measure of the degree of coherence between all the
possible pairs of histories which have $P(\alpha_3,\sigma_2)$ as their
final quasi-projector. The maximum of (\ref{cheq30}) over $\Delta_2$
gives the largest violation of consistency for the complete set of
histories labeled by both $\Delta_2$ and $\Delta_1$. Before computing
(\ref{cheq30}) we first obtain $\langle \Delta_1\rangle$ for a fixed
$\Delta_2\neq 0$ to be
\begin{eqnarray}
\langle\Delta_1\rangle&=&\int
\frac{d^2\Delta_1}{2\pi}\,N\exp\left[
-A|\Delta_1|^2
-B|\Delta_2|^2
+C(\Delta_1^*\Delta_2+\Delta_1\Delta_2^*)\right]\Delta_1\nonumber\\
&=&\frac{NC\Delta_2}{A^2}e^ {-(AB-C^2)|\Delta_2|^2/A}\;\;,\end{eqnarray}
which gives $\langle\Delta_1\rangle=0$ when $\Delta_2=0$ as expected.
After some calculation one obtains the maximum violation of
consistency to simply be
\begin{eqnarray}
\Omega_h &=& \frac{A}{D}\;\;,\nonumber\\
& = & \frac{\left[2s_1-1+\frac{1}{s_2}e^{-\gamma\Delta t_2}(1-\Upsilon)
\right]}{\left[
1+2s_1-\frac{1}{s_2}e^{-\gamma \Delta t_2}+\Upsilon\left(
4-\frac{1}{s_2}[3-2s_1]e^{-\gamma \Delta t_2}\right)\right]
}\;\;.\label{cheq31}
\end{eqnarray}
In the short and long time limits we get
\begin{eqnarray}
\Omega_h(t_1\sim t_2\rightarrow 0) & = &
\frac{2s_1-(1-\frac{1}{s_2})}
{2s_1+(1-\frac{1}{2_s})}\;\leq \;1\;\;,\label{cheq32}\\
\Omega_h(t_1\sim t_2\rightarrow \infty ) &=&
\frac{2s_1-1}{2s_1+1+4\bar{n}}\;\leq\;
\Omega_h(t_1\sim t_2\rightarrow 0)\;\leq\; 1\;\;.
\label{cheq33}\end{eqnarray}
From (\ref{cheq33}) we see that even when $\bar{n}=0$ the coherence
decays. This is due to the vacuum fluctuations. It is interesting to
note that all dependence on $s_2$ drops out of these expressions in
the long time limit.
Plots of $\Omega_h$ for $\Delta t_1=\Delta
t_2=\tau$ (\ref{cheq31}), are shown in Fig 3. Very
similar dependence on the model parameters is shown between $\Omega_h$
and $\bar{\epsilon}$, in particular the anomalous $s_1$ behaviour.
\begin{center}
(a)\hspace{6cm}(b)\\
\graphics{myfig4.ps}{5}{7}{-2}{-8}{.4}
\graphics{myfig5.ps}{5}{7}{1}{-8}{.4}
\end{center}
\begin{center}
{\parbox{15cm}{\small Fig 3.
Plots of $\Omega_h$, (\ref{cheq31}) as a function of $\bar{n}$, $s_1$ and
$s_2$. We have set $\gamma=1$ and $\Delta t_1=\Delta t_2=\tau =0.2\;$.
Graph (a)
shows that consistency decreases with increasing initial grain size $s_1$.
}}
\end{center}
To conclude and summarize: we have examined the Consistent Histories framework
within a quantum optical model using two-time projected phase-space
histories. In the Markovian regime the bath can be traced away and
it's effect completely absorbed into the non-unitary evolution
super-operator $\cal L$. This is used to evolve the reduced density
matrix of the system between projections. Since no true phase-space
projectors exist we use ``quasi-projectors'' which are gaussian
convolutions of coherent states. Although these quasi-projectors are
complete, they are not totally exclusive and therefore do not
exactly meet the specifications of this paradigm. They, however, do
provide a tractable model and have been used successfully in other
calculations. Moreover, such quasi-projectors arise naturally in
more rigorous theories of quantum measurement. We first calculated the
``action'' of the decoherence functional for these coarse grained
histories on the state $|\alpha_j\rangle\langle\alpha_i|$ at $t_1$.
With this information, the action of the decoherence functional on any
state at $t_1$ can be evaluated through a decomposition of $\rho(t_1)$
into coherent states. This was done for the particular case of
$\rho(t_0)=|\alpha_0\rangle\langle\alpha_0|$ where $t_0 \leq t_1$. The
resulting decoherence functional was exponential in form and peaked on the
classical path of the damped oscillator. Using the measures of Dowker
and Halliwell the degree of consistency and the degree of peaking about
the classical path were calculated and the short and long time limits
found. The behaviour of these quantities were mostly as expected: the
consistency gets better with time, increasing bath temperature and final
grain size while the peaking is better with increased final and
initial grain size and small bath temperature. The consistency {\em
decreased} with increased initial grain size. We also calculated $\Omega_h$, a
measure similar to the one proposed in the EIS section of this paper,
which quantifies the amount of coherence in the decoherence functional
when the decoherence functional is considered as a density matrix on
the space of coarse grained histories. This measure displayed similar
behaviour to the measure of consistency as advanced by Dowker and
Halliwell.
\section{Conclusion}
Upon comparison we see that both EIS and consistency among phase-space
histories improves as the bath temperature increases. Altering the
coupling strength $\gamma$, changes only the time derivatives of the
degree of EIS or
consistency achieved and not their actual values. A time for
``decoherence'' may be set equal to the time needed for $\Omega_\eta$ to
reach some predetermined value ie. $\Omega_\eta (t_d)=.5$, say.
The anomalous dependence of $\bar{\epsilon}$ on $s_1$ occurs in all
the two-time projected history calculations to date. For this to occur
in one model for a special type of graining might be understandable
within the arguments given by Dowker and Halliwell concerning the
constructive interference of the constituent histories in a
very special coarse-graining. Since the behaviour is seen in three
different calculations with two different models
\cite{DOWKER:1992,PAZ:1992a} one suspects that there is some feature
common to all that is upsetting the expected behaviour of increased
consistency with increased $s_1$. An approximation common to all of
these calculations is the use of ``quasi-projectors''. It may be that
the non-exclusivity of these projectors is somehow overcounting the
contributions of ``off-diagonal'' histories in the decoherence
functional. It is thus necessary to resort to true projectors. Since
there do not exist true projectors on phase-space one must consider
true projectors on to other spaces. An obvious choice is either the
position, momentum or number basis. Choosing the latter is the most
computationally efficient and was treated in \cite{TWAMLEY:1993b}. The
results were very surprising and highlighted the differences between
the two paradigms, EIS and CH.
Essentially the goal of EIS is the damping of the coherences present
in the system's quantum state. The goal of CH, however, is the
attainment of histories whose probabilities satisfy the sum rule. For
this to occur the quantum {\em interferences} between the individual
histories must vanish. The damping of quantum coherence and vanishing
of quantum interferences need not be synonymous. The decay of
coherence necessitates the vanishing of interference but not visa
versa. This is clearly shown in the double slit examples of Gell-Mann
and Hartle. When the slit separation is much greater than the
slit/screen separation the resulting coarse-grained histories are very
consistent. However, if the separate beams are then later on
recombined somehow, they will display interference. One then says
that the histories which include this recombination are no longer
consistent. Thus, histories which may begin very consistent, may,
later on, become very inconsistent and visa versa. This is so, even
though the evolution can be strictly unitary! From the viewpoint of
EIS, the coherence of the state is not lost under unitary evolution
and thus EIS does not occur. However, if one now inserts the Young's
slits in a bath of electrons \cite{HARTLE:1989}, the resulting
histories are even more consistent and furthermore they cannot
usually be made less consistent through recombination etc. This is so
because the reduced state of the light beams has lost some of it's
coherence to the bath of electrons. Thus, both CH and EIS has occurred in the
reduced state of the light beams. Since the measure $\Omega_\eta$ was
introduced in section III as a measure of coherence, its value as a measure of
consistency (which involves the vanishing of interferences)
is slightly suspect. However, we would argue that
the vanishing of interference between
{\em all} pairs of possible histories in a set of very many alternatives
necessitates the vanishing of interference between neighboring
histories. For this to happen, as in the Young Slit case for small
slit separation, one must have a loss of coherence (usually by
introducing a bath).
Finally, it would be of greater interest to have a more information
theoretic formulation of the concept of ``diagonalisation'' and
consistency. The introduction of a bath causes the reduced quantum
state to lose it's non-local correlations preferentially in certain
bases. To link this concept with that of a pointer basis - that basis
least effected through the bath interaction - would be of interest.
Also to see how the decohered state (either via EIS or CH) satisfies
the Bell inequalities would give, perhaps, a more physical measure of
what ``diagonality'' really means. This work is in progress
\cite{TWAMLEY:1993d}.
Also, the quantification of classical correlations via information
theory in the CH approach is also important \cite{HALLIWELL:1993}.
In this paper we have shown that in this relatively simple model of a
quantum open system EIS and CH seem to behave similarly, becoming
better with increasing bath temperature and time. The dependence of the
consistency on graining also behaves almost as expected. This example
shows how the two paradigms are similar.
Other calculations also demonstrate the differences between the two
\cite{TWAMLEY:1993b,LAFLAMME:1993}. Clearly, if both paradigms are to
describe an ontology it is essential to discover situations where
they differ and to experimentally determine which is correct.
\acknowledgments
The author thanks
F. Dowker,
S. Habib,
B. L. Hu
R. Laflamme,
A. Matacz,
J. McCarthy,
G. Milburn,
J. P. Paz,
M. Simpson,
W. Unruh and
W. Zurek
for interesting discussions. We also thank the University of
British Colombia, Los Alamos National Laboratories and the University
of Maryland for their kind hospitality during the course of this research.
|
1,108,101,563,244 | arxiv | \section{Introduction}
The case for linking Sgr~A*, the radio source at the center of the
Milky Way, with a supermassive black hole is very strong. Mass
estimates inferred from stellar orbits, proper motion studies that
indicate Sgr~A* is nearly motionless, VLBI observations that reveal it
is ultracompact, and short-timescale variability from radio to X-rays
all point towards Sgr~A*'s association with a
$\sim4\times10^6~M_\odot$ black hole \citep[and references
therein]{reid2009}. At a distance of $\sim8~$kpc, the Schwarzschild
radius of this black hole subtends $R_\mathrm{Sch} \sim10~\mu$as,
making the apparent size of its event horizon the largest that we know
of. VLBI at (sub)millimeter wavelengths is ideally suited to
observing Sgr~A* on these angular scales. Previous 1.3~mm VLBI
detections of Sgr~A* on a Hawaii-Arizona baseline established the
existence of coherent structures on scales of a few $R_\mathrm{Sch}$
\citep{doeleman2008}.
Current 1.3~mm VLBI observations can be used to address two
fundamental questions concerning the nature of Sgr~A*. The first is
whether the accretion flow surrounding Sgr~A* exhibits an expected
``shadow'' feature that occurs due to the strong gravitational lensing
near the black hole. Emission from the accretion flow is
preferentially lensed onto the last photon orbit, resulting in a
relatively dim central region encircled by a brighter annulus
\citep{falcke2000}. A second question is whether the flaring behavior
exhibited by Sgr~A* has its origins in compact structures that arise
near the black hole event horizon. Broadband flares on timescales
ranging from minutes to hours are well-documented
\citep{marrone2008,yusefzadeh2009,doddseden2009} and imply
time-variable structures in the innermost accretion region.
If small-scale variable structures are present, 1.3~mm VLBI can
sensitively monitor the changing morphology of Sgr~A* using
non-imaging techniques with time resolutions of tens of seconds
\citep{doeleman2009,fish2009b}.
We report on new 1.3~mm VLBI observations of Sgr~A* using a
four-telescope array. These observations confirm event horizon scale
structure within Sgr~A*, impose new constraints on accretion models
for Sgr~A*, and reveal time-dependent variability on scales of a few
$R_\mathrm{Sch}$.
\section{Observations}
\label{observations}
Sgr~A* and several calibrator sources were observed with four
telescopes at three observatories: the James Clerk Maxwell Telescope
(JCMT; henceforth also J) on Mauna Kea in Hawaii, the Arizona Radio
Observatory's Submillimeter Telescope (ARO/SMT; S) in Arizona, and two
telescopes of the Combined Array for Research in Millimeter-wave
Astronomy (CARMA; C and D, located $\sim60$~m apart) in California.
On Mauna Kea, the Submillimeter Array (SMA) housed the VLBI recording
system and synthesized the hydrogen maser based VLBI reference used at
the JCMT. Masers at all sites were checked against ultra-stable
crystals; combined losses due to maser instabilities and local
oscillator decoherence are estimated to be $\lesssim5$\%.
Observations occurred over three nights: 2009 April 5--7 (days
95--97). Sources were observed in left circular polarization in two
480~MHz bandwidths centered at 229.089 and 229.601~GHz (low and high
bands). Data recorded at all sites was shipped to MIT Haystack
Observatory in Westford, Massachusetts for processing on the Mark4
VLBI correlator. Once correlated, data for each scan (typically
10--15 minutes) were corrected for coherence losses due to atmospheric
turbulence and searched for detections using methods detailed in
\citet{doeleman2001, doeleman2008}. Atmospheric coherence times
ranged from a few to $\sim20$~s, depending on weather conditions at
each telescope.
\section{Calibration}\label{calibration}
The VLBI correlation coefficient for each baseline was multiplied by
the geometric mean of the System Equivalent Flux Density (SEFD) of
both antennas.
The SEFD is a product of antenna gain (Jy/K) and the opacity-corrected
system temperature, which
was measured just prior to each VLBI scan using a
vane calibration technique that corrects for the atmosphere. For the
JCMT and ARO/SMT, antenna gains were determined from observations of
planets at several points during the multiple day campaign, and the
gains were observed to be stable. Relative gains for the two CARMA
dishes were estimated using observations taken by CARMA in
interferometric array mode before each VLBI scan, and the gains were
then set to a common flux scale using planet scans at the end of each
night.
The flux densities of all VLBI targets (Sgr~A*, 1924$-$292, M87,
3C273, 3C345, 1733$-$130, 3C279, 0854$+$201) were measured with CARMA.
For Sgr~A*, data with baselines shorter than 20~k$\lambda$ were
discarded to filter out extended emission in the Galactic center. The
measured flux densities of all sources increased from day 95 to day 96
and from day 96 to day 97. We attribute this systematic trend to
errors in the planet calibrations made shortly after sunrise, when
antenna focus, pointing offsets, and atmospheric coherence typically
change. The flux density measured for the calibrator 1924$-$294,
observed over the same time and elevation ranges as Sgr~A*, was 9.95,
10.21, and 10.75~Jy on days 95, 96, and 97. We normalized CARMA gains
to a constant flux density of 10.25~Jy on all three days. The
resulting measured flux densities for Sgr A* are 3.03, 3.16, and
3.61~Jy on days 95, 96, and 97, respectively. We adopt these fluxes
for all subsequent analysis.
As shown in the upper panels of Figures~\ref{fig-1921-scans} and
\ref{fig-sgra-scans}, there are still noticeable variations in the
correlated flux densities even after renormalizing the day-to-day flux
scales. These residual calibration errors and amplitude variation can
be corrected for by making three simplifying assumptions that allow us
to use standard ``self-calibration" techniques.
First, the flux densities of detections in the low and high bands,
which differ by only 0.2\% in frequency, are assumed to be equal in
each scan. Second, flux densities on the SC and SD baselines are
assumed to be equal. While one could in principle require that JC and
JD flux densities be equal as well, the signal-to-noise ratio (SNR) is
generally much lower on the JC and JD baselines than on the shorter
VLBI baselines (SC and SD), since both 1924$-$292 and Sgr~A* are more
resolved on longer baselines. Third, CARMA antenna gains are adjusted
to make the correlated flux density on the CD baseline (with a fringe
spacing measured in arcseconds) equal to the total flux density
measured each night by CARMA. This final constraint enforces a
constant source flux density over the duration of each night of
observation. While some of the observed variation in Sgr~A* over the
course of a night may be due to intrinsic variability, the 1924$-$292
data exhibit similar scatter, suggesting that calibration errors may
dominate over source variability. Combined, these assumptions result
in a closed-form solution for gain-correction coefficients for
telescopes C, D, and S in each band. Henceforth, we will use the term
``gain-corrected'' to refer to flux densities that have been
multiplied by these gain-correction coefficients. We note that if the
total flux density (CD) is varied, the SJ flux densities are unchanged
while other flux densities vary as the square root of the factor.
The quasar 1924$-$292 was easily detected on all baselines
(Table~\ref{table-detections}). On each scan, low-band and high-band
fluxes after a-priori calibration track each other consistently
(Figure~\ref{fig-1921-scans}). After gain correction assuming a total
flux density of $10.25$~Jy, the data from all three days are highly
consistent with one another. The SC and SD baselines show consistent
variation in the correlated flux density that is repeated each day.
The long-baseline detections (SJ, JC, and JD) also show day-to-day
repeatable behavior, indicating detection of stable source structure
presumably associated with a jet \citep{shen1997}. The consistency of
these data demonstrates the validity of the gain-correction technique.
Based on the statistics of the data on scans of 1924$-$292, systematic
errors are estimated to be $\sim5$\%.
\section{Results}\label{results}
\begin{figure*}
\resizebox{0.31\hsize}{!}{\includegraphics{fig1a.eps}}
\resizebox{0.31\hsize}{!}{\includegraphics{fig1b.eps}}
\resizebox{0.31\hsize}{!}{\includegraphics{fig1c.eps}}\\
\resizebox{0.31\hsize}{!}{\includegraphics{fig1d.eps}}
\resizebox{0.31\hsize}{!}{\includegraphics{fig1e.eps}}
\resizebox{0.31\hsize}{!}{\includegraphics{fig1f.eps}}
\caption{\emph{Top}: Observations of $1924-292$. Detections are
color-coded by baseline. Squares indicate low-band data and circles
indicate high-band data. The day 97 a-priori calibration is
noticeably poorer at CARMA. \emph{Bottom}: Same, after gain
correction. Errorbars include a 5\% systematic component. The
gain-corrected data exhibit much higher day-to-day repeatability.
The red dashed line shows the assumed CD flux (the associated flux
scale uncertainty is $\sim5$\%, modulo uncertainties in planet
fluxes). Gain-corrected SC and SD data are equal by definition.
\label{fig-1921-scans}
}
\end{figure*}
\begin{figure*}
\resizebox{0.31\hsize}{!}{\includegraphics{fig2a.eps}}
\resizebox{0.31\hsize}{!}{\includegraphics{fig2b.eps}}
\resizebox{0.31\hsize}{!}{\includegraphics{fig2c.eps}}\\
\resizebox{0.31\hsize}{!}{\includegraphics{fig2d.eps}}
\resizebox{0.31\hsize}{!}{\includegraphics{fig2e.eps}}
\resizebox{0.31\hsize}{!}{\includegraphics{fig2f.eps}}
\caption{Same as Figure~\ref{fig-1921-scans} but for Sgr~A*. Only
scans for which detections are seen in both bands on the CD, SC, and
SD baselines are shown.
\label{fig-sgra-scans}
}
\end{figure*}
\begin{figure*}
\resizebox{0.31\hsize}{!}{\includegraphics{fig3a.eps}}
\resizebox{0.31\hsize}{!}{\includegraphics{fig3b.eps}}
\resizebox{0.31\hsize}{!}{\includegraphics{fig3c.eps}}\\
\resizebox{0.31\hsize}{!}{\includegraphics{fig3d.eps}}
\resizebox{0.31\hsize}{!}{\includegraphics{fig3e.eps}}
\resizebox{0.31\hsize}{!}{\includegraphics{fig3f.eps}}
\caption{Correlated gain-corrected flux density plots. Solid lines
show best-fit circular Gaussian models of the compact emission in
Sgr~A* and are consistent with a source size of approximately
43~$\mu$as on all days. The dotted lines show the best ring model
fits to Sgr~A* data.
\label{fig-corr}
}
\end{figure*}
We report the first 1.3~mm VLBI detections of Sgr~A* on
Hawaii-California baselines with correlated flux densities for several
scans of $\gtrsim400$~mJy on the JC and JD baselines during day 96
(Figure~\ref{fig-sgra-scans}). Nondetections on the JC and JD
baselines on day 95 are attributable to the higher opacity at the JCMT
on that day.
The robust detections on the long (Hawaii-Arizona and
Hawaii-California) baselines confirm the detection of event horizon
scale structure reported in \citet{doeleman2008}.
Because Sgr~A* was detected on baselines between all three sites, we
are able to measure closure phase: the sum of interferometric phase
around a closed triangle of baselines. This quantity is relatively
immune to calibration errors and in general provides important
constraints on source structure. On the CARMA-ARO/SMT-JCMT triangle,
the phase closures on Sgr A* for day 96 (5 independent measurements)
were computed using 10-second coherent integrations, which were
averaged over full 10-minute VLBI scans to increase SNR
\citep{rogers1995}. Because of the relatively low SNR on the long
baselines to the JCMT, it is only possible to say that the closure
phases are consistent with a value of zero with a variation of
$\pm40^\circ$.
\subsection{Variability}
\label{variability}
The arcsecond-scale flux density of Sgr~A* on days 95 and 96 is the
same to within uncertainties in the flux scale, but on day 97 the flux
density of Sgr~A* was $\sim17$\% higher. This brightening is
accompanied by changes on VLBI scales as well. First, the SC/CD and
SD/CD flux density ratios are higher on day 97 than on days 95 and 96
(Figure~\ref{fig-sgra-scans}). Second, there are more
Hawaii-California detections during day 97, and the apparent flux
densities on these baselines are on average higher than on day 96.
Third, the flux densities on the SJ baseline are also larger on day
97. These differences are consistent with an episode of variability
in Sgr~A* during which the small-scale structure increased in flux
density between days 96 and 97.
\subsection{Geometrical Models of the Structure of Sgr~A*}\label{structure}
Though our data are better calibrated than in the previous epoch
\citep{doeleman2008}, the structure of Sgr~A* is poorly constrained
because millimeter VLBI detections of Sgr~A* remain limited in terms
of baseline length and orientation. As a result, many models can be
made to fit the data: extended double sources, large rings,
combinations of large- and small-scale components, etc.
Nevertheless, with the caveat that this small dataset should not be
overinterpreted, it is instructive to investigate the two classes of
models originally considered by \citet{doeleman2008} to fit the 1.3~mm
VLBI data obtained in 2007: circular Gaussians and rings. All of the
2007 data points could be fitted by a single Gaussian component. In
contrast, we note a loss of $\sim1$~Jy of correlated flux density
between the connected-element (CD) and SC/SD baselines
(Figure~\ref{fig-corr}). In the context of Gaussian models of
emission on $R_\mathrm{Sch}$ scales, this suggests the existence of
additional variable structure on scales between those probed by the
SC/SD (a few hundred microarcseconds) and the CD (a few arcseconds)
baselines.
We adopt this assumption to estimate the size of the inner accretion
flow in Sgr~A*. Effectively, this reduces to fitting all of the VLBI
data excluding the CD data points. For the Gaussian model, the best
fits imply a flux density of $2.07^{+0.14}_{-0.15}$~Jy and a size of
$41.3^{+5.4}_{-4.3}~\mu$as (FWHM; errors are $3\sigma$) on day 95 and
$2.07^{+0.19}_{-0.19}$~Jy and $44.4^{+3.0}_{-3.0}~\mu$as on day 96
(Figure~\ref{fig-corr} and Table~\ref{table-fits}). These values are
consistent with the single compact component Gaussian fit of
\citet{doeleman2008}, who estimated a flux density of $2.4 \pm 0.5$~Jy
and a size of $43^{+14}_{-8}~\mu$as (before deconvolution of the
expected interstellar scattering; $37~\mu$as unscattered) for the
230~GHz emission. On day 97, the best fit model has a much higher
flux density of $2.85^{+0.29}_{-0.28}$~Jy but a similar FWHM of
$42.6^{+3.1}_{-2.9}~\mu$as. Despite the increase in flux density
observed on day 97, the diameter of the fitted compact component in
Sgr~A* on that day is identical (to within the errors) to the values
for the size obtained on days 95 and 96.
Ring models with three parameters (inner radius, outer radius, and
flux density) can also fit the VLBI data (Figure~\ref{fig-corr} and
Table~\ref{table-fits}). However, no single set of ring model
parameters consistently fits the data on all three days, which would
suggest that the size and structure of Sgr~A* are variable within the
context of ring models. This stands in contrast with the Gaussian
model, for which all epochs of data are consistent with a uniform size
despite differences in the flux density. Longer-wavelength VLBI
observations are inconclusive as to whether a significant correlation
exists between the flux density and size of the emission in Sgr~A*
\citep{bower2004,lu2010}. However, the size of the emission at these
wavelengths is dominated by interstellar scattering effects.
Future 1.3~mm VLBI observations with higher sensitivity, sufficient to
robustly measure the closure phase, will be an important discriminant
between these and other models. For example, an elliptical Gaussian
distribution will result in zero closure phase on any triangle of
baselines, while a ring model can result in closure phases of 180
degrees depending on the orientation and length of the array
baselines. The ring models shown in Figure~\ref{fig-corr} all have
closure phases of zero on the CARMA-ARO/SMT-JCMT triangle of
baselines, consistent with the measured closure phases
(Section~\ref{results}). However, a ring model with a null near
$3.4$~$G\lambda$ between the CARMA-JCMT and ARO/SMT-JCMT baselines,
would produce a closure phase of 180 degrees, which is strongly ruled
out by the April 2009 data. Measurement on an intermediate baseline
in the $1-2$~$G\lambda$ range would provide a powerful discriminant
between large classes of geometrical models.
\section{Discussion}
\subsection{Implications for Accretion and Flare Models}
The flux density of Sgr~A* on VLBI scales is seen to increase from day
95/96 to day 97. During the first $\sim1$ hour on day 97, when the
atmosphere at CARMA was relatively stable, the data are consistent
with a constant flux density, suggesting that the flux density
increased before observations on day 97 but held steady at a higher
level than on the previous nights. This behavior is consistent with
other (sub)millimeter observations, which show variability punctuated
with periods when the flux density is stable
\citep[e.g.,][]{marrone2006,yusefzadeh2009,kunneriath2010}.
The flux density increase appears to be due to an event that
establishes a new steady-state in Sgr~A*. If instead the flux density
increase is due to a short-duration event that concluded before the
start of observations on day 97, the unchanging size of the compact
region (as implied by Gaussian models in Section~\ref{structure}) and
the timescale over which the compact flux density is seen to be
constant limits the expansion speed of the region to be highly
nonrelativistic \citep[$v \lesssim0.05 c$, consistent with][]{lu2010}
and much lower than the sound speed ($c/\sqrt{3}$)
\citep{marrone2008}, in contrast with relativistic jet models
\citep[e.g.,][]{falcke2009}. While a low expansion speed is predicted
by adiabatically-expanding plasmon models \citep{yusefzadeh2009},
these models also predict an increase in source size. Our
observations detect Sgr~A* after the increase in flux density has
occurred, but we do not find evidence of an increase in source size as
predicted by adiabatic expansion. Future, more sensitive observations
of Sgr~A* before, during, and after a flare event will be necessary to
more fully understand the mechanism responsible for variability in
Sgr~A*.
\subsection{Physical Models of the Structure of Sgr~A*}\label{physical}
Numerous physically-motivated models have been advanced to explain the
multiwavelength emission from Sgr~A*
\citep[e.g.,][]{falcke2000,yuan2003,markoff2007,noble2007}. In recent
years, several efforts to constrain disk models using 1.3~mm VLBI
observations have placed limits on model parameters such as the
inclination of the disk and the spin of the black hole. Radiatively
inefficient accretion flow models and some general relativistic
magnetohydrodynamic simulations found that the \citet{doeleman2008}
data were already sufficient to rule out a low-inclination (i.e.,
nearly face-on) disk \citep{broderick2009,moscibrodzka2009}.
Low-inclination disk models are qualitatively similar to the ring
model in Section~\ref{structure}, with a ``null'' in the correlated
flux density as a function of baseline length whose depth decreases
with increasing disk inclination \citep{dexter2009,fish2009}. The
location of this null as a function of baseline length is set by the
ratio of the mass of the black hole and the distance to Sgr~A*, which
is determined to within a few percent \citep{ghez2008,gillessen2009}.
Emission in high-inclination models is dominated by the highly
Doppler-boosted approaching side of the disk. High-inclination disk
models exhibit a monotonic decrease in flux density as baselines
approach the length of the SJ baseline, with long-baseline correlated
flux densities highly dependent on the orientation of the disk in the
plane of the sky. The decline of correlated flux density with
baseline length from Hawaii-California to Hawaii-Arizona baselines
requires that the inclination be $\gtrsim40\degr$ in several disk
models that were consistent with the 2007 data
\citep{dexter2009,broderick2009}. The impact of the 2009
Hawaii-California detections on the 2007 fits is to strengthen the
constraint on inclination in these models (e.g., Broderick et al.,
private communication).
\section{Summary}
VLBI observations of Sgr~A* at 1.3mm wavelength in 2009 have robustly
confirmed the detection of Schwarzschild radius scale structures
reported in \citet{doeleman2008}. On the third of three days of
observations, the total flux density of Sgr~A* was observed to have
increased by $\sim17$\%, indicating an episode of variability similar
to those described in multi-wavelength monitoring campaigns
\citep[e.g.,][]{marrone2008,yusefzadeh2009}. The VLBI data support a
picture in which this flux density increase is due to a brightening of
structure on scales of only a few $R_\mathrm{Sch}$. Having achieved
the dual technical milestones of successfully detecting closure phase
and developing robust methods of amplitude calibration, it is clear
that future (sub)millimeter VLBI observations with higher sensitivity
and many more baselines will be able to powerfully constrain models of
Sgr~A* on event horizon scales.
\acknowledgments
High-frequency VLBI work at MIT Haystack Observatory is supported by
grants from the National Science Foundation (NSF). ARO receives
partial support from the NSF ATI program. The Submillimeter Array is
a joint project between the Smithsonian Astrophysical Observatory and
the Academia Sinica Institute of Astronomy and Astrophysics and is
funded by the Smithsonian Institution and the Academia Sinica.
Funding for ongoing CARMA development and operations is supported by
the NSF and the CARMA partner universities. DEB acknowledges support
from the NSF Research Experiences for Undergraduates program.
{\it Facilities:} \facility{CARMA ()}, \facility{HHT ()},
\facility{JCMT ()}, \facility{SMA ()}, \facility{CSO ()}
\pagebreak
|
1,108,101,563,245 | arxiv | \section{Motivations}
The PGPfone project \cite{pgpfone}, developed by Boulder Software
Engineering, provides high-quality secure voice communications over
ordinary phone lines. Implicit in this project, as in any security
project, is the need to keep the (voice) data and the keys used for
scrambling/encrypting the data secure from eavesdroppers---or from
hostile listeners who may be able to do more than simply tap
telephones. In normal operations, Alice can simply telephone
Bob using PGPfone. The program will negotiate an encryption
``key'' between Alice
and Bob and transform the data so that any eavesdropper cannot understand
their conversation without the key. Although this works against passive
eavesdropping, it is not secure against a more powerful adversary. If
the hostile Mike, who works at the hotel where Alice is staying, can
arrange to intercept or reroute her calls, he can arrange that her call
goes to him, while he himself places the call to Bob. Mike's computer
can then
negotiate two keys (one with Bob and one with Alice) while Mike
connects the two conversations to each other, and notes everything
said.
Figure~\ref{figure:manmiddle} illustrates this basic
``man-in-the-middle'' attack.
However, if
each call uses a separate encryption key (as in PGPfone), and if Alice
and Bob can confirm that they are using the same key, they can
be relatively confident that there is no Mike in the middle.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\psfig{figure=nemlapfig.eps,height=6.00cm}
\end{center}
\caption{\label{figure:manmiddle}Man-in-the-middle attack scenario}
\end{figure}
Reading long binary (or even hexadecimal) strings
over the phone is, however, tedious and error-prone. There are many other
applications that might require this sort of data exchange. For instance,
one standard method of confirming that a key you obtained is valid is by
calculating a ``key fingerprint'' and checking it against the owner.
From the owner's perspective, it is difficult to remember the string
of random characters that comprise the fingerprint, and from the reader's
perspective, it is imperative that the words be understood properly
at the other end of a telephone line. Some other applications are
reading signatures or remembering keys---and in all cases, it would
be more efficient and accurate to use some sort of encoding for compression
and error-checking.
\section{Alphabet Design}
PGPfone's designed solution to this problem is to develop a word
list, styled after
the traditional military or pilot's alphabet (alpha, bravo, charlie, \ldots),
with each word representing some
fixed number of bits.
The problem of developing such a linguistic encoding for data exchange
is unusual in that, unlike most NLP projects, ``language'' here
is one of the independent variables that can be manipulated at the
engineer's whim. The words used in developing this alphabet, if
properly chosen, can not
only provide compression, but can also provide some error
prevention, error detection, and a considerable human factors advantage.
On the other hand, ``proper chosing'' will in turn be helped by
an efficient, accurate, and
numerical model of the desired properties.
For example, the length of the word list, obviously, will determine some of its
attributes. A small list (for example, sixteen words) would provide
no compression over reading hexadecimal numbers, but could still provide
some degree of errorproofing by removing potentially confusing tuples
like five/nine, B/C/D/E,
and so forth. A list of 64 words would allow about 30\% compression (over
hexadecimal digits)
in terms of number of words, but the complete list would be much harder
for a human to memorize.
For situations where humans are required to generate
keys (and/or responses) from memory, this would be an unreasonable expectation.
However, in PGPfone, all keys are generated and stored by the computers,
and the only job for a human is to read a series of words presented by
the computer; thus, there is no need for a human to ever memorize
the complete list.
Because these keys are going to be generated automatically from a
list known only to
a computer, we opted to use lists of 256 words, allowing each word to
represent a byte. Using larger lists would obviously allow significantly
better compression, but require considerably more (computer) memory
to store the word table. For example, two lists of 256 words can be
stored in only 5 kilobytes of memory. A larger list (two bytes
per word) would require nearly 650 kilobytes of memory, as well as
a word vocabulary larger than most native English speakers' productive
vocabulary.
From a human factors perspective,
an ideal word list consists of short,
easily recognizable and easily pronounceable words with easily distinguished
prefixes and a minimum of phonetic confusibility or bad associations.
We chose to approach this task as a selection problem---from a much
larger list,
select words with appropriately chosen
characteristics.
For this project, we used the Moby Pronunciator database, distributed
by Grady Ward,
which contains nearly 200,000 word/pronunciation pairs.
In some characteristics, such as ``short'', the selection process is
a trivial task. In others, such as ``no bad associations'', this is
nearly impossible to perform automatically and it was recognized that this
would need to be done by hand. The main technical difficulty that
we considered to be solvable by computer occurred in the representation
of ``phonetic confusibility.''
The ideal metric for phonetic confusibility for this project would
be capable of accepting any two speaker-independent representations of
word pronunciations and returning, as a distance, an accurate measure
of the probability
of one word being confused for the other in an isolated word context.
In practical terms, this is probably an unachievable goal. For example,
part of the mathematical definition of distance includes the notion of
symmetry, that if one word has a fixed probability of confusion with
the second word, the second word has an identical probability of
being confused with the first. One may naively expect this to fail.
Even a vaguely correct metric could have far wider applications than the
simple PGPfone alphabet, however. For instance, Nakisa and Hahn
\cite{nakisa-hahn} describe a model for the German plural system based
on the notion of mapping novel words to appropriate inflectional
categories based on the phonetic properties of the word. In one experiment,
they simply calculated the ``nearest neighbor'' using a Euclidean distance
of a 240-element feature representation. Clearly, the more accurate
the distance representation used, the more confidence one can have in
their (psycholinguistic) conclusions. Furthermore, an accurate
confusibility measure could have important engineering applications, for
example to the development of case-based reasoning tools for
text to speech system or speech synthesis. Finally, because mathematical
tools like this permit language and speech to be the object of manipulation
instead of mere study, this has wider applications in any
controlled-language situation, for example the development of simplified
language for MT projects or the production of distinctive brand
or product names.
\section{Linguistic distances}
Our approach to the problem of phonetic confusibility is a variant
of the work of Miller and Nicely \cite{miller-nicely}. In particular, words are
ordered strings of phonemes instead of acoustic signals, phonemes
in turn contain features [such as those enumerated in
Ladefoged \cite{ladefoged}], and individual
phonemes can be meaningfully compared by comparing their features.
It is further assumed that
the phonetic distance between
two words can be
approximated by some function of the differences between
the phonemes that comprise the words.
It should be noted immediately that this is only one of
many possible approaches. For example, Lindblom \cite{lindblom} measured
vowel similarity
based on formant frequency, and in particular the F1 and F2
frequencies. However, this approach is less satisfactory for several
reasons : sounds, especially consonants, show
much more word-to-word variance than
phonetic features; although the phonetic transcription of a given
word does not often vary from person to person, the actual acoustic
signal does; and thirdly, the simple task of mapping from sound to lexeme
is itself a hard problem, while taking little if anything away from
the difficulties in designing a distance metric. Other approaches have
been proposed using ``autosegmental phonology'' as suggested by
Goldsmith \cite{goldsmith} to compress word representations into feature change
sets, at the expense of synchronization data. Strictly speaking,
Miller and Nicely \cite{miller-nicely} and more recently
Bell\footnote{Alan Bell, 1995. Personal communication.} use a more
introspective/scientific approach than simply calculating
mathematical distances, instead directly examining people's
perceived distances, which may or may not exactly map onto a feature-based
metric---but this approach requires either extensive lab-work to validate,
or a willingness to rely on pure introspection without validation.
Unfortunately, the chosen approach almost immediately encounters severe
representational difficulties at a number of levels. For instance,
phonologists and phoneticians usually use
feature representations designed to represent differences
important to the production of a sound. The amount of detail, and
hence importance, thus changes with the degree of variability in
a feature. Sounds with voiceless stop
consonants can be produced at many different locations ranging from the
lips to the very back of the throat. Voicing, on the
other hand, is either present
or absent; no known language makes a distinction between voiceless,
strongly-voiced, and weakly-but-still-voiced consonants. However,
Miller and Nicely \cite{miller-nicely} indicate that voicing is one
of the most salient and robust features of English consonants; in other
words that /d/ is
more likely to be misheard as /g/ than as /t/ (under most circumstances).
Generalizing this, we have the unfortunate result that phoneme pairs may
differ in several unimportant features and yet sound closer than another
pair that differ only in one extremely salient aspect.
Furthermore, the relative salience of features varies wildly depending upon
the sort of noise in which the signal is embedded. Given that
system designers have no idea of the conditions under which people may
use a telephone, the best one can reasonably do is to make assumptions; in
this case, we assume white noise.
Although the Miller-Nicely confusion matrices provide exact data that
could be used to balance some features, they don't provide enough data
for our purposes. The
study only incorporated differences between some English consonants and
no vowel distinctions at all. Because of the absence of such data,
standard automatic feature weighting or pattern recognition
techniques seemed inapplicable;
instead, we relied on the balance information from Miller-Nicely,
applied as best we could to the entire featural universe.
Ladefoged \cite{ladefoged} proposes a more extensive
list of features that allow for all sounds of English, consonants and
vowels alike, to be presented and distinguished on a uniform scale. As
discussed above, this list provides no data on salience, but with
appropriate judgements (and
some coercion of scales between vowels and consonants), the various
features can be
approximately balanced to the Miller-Nicely data.
Using this method, the perceptual
difference between two comparable phonemes can be measured as the number of
bits that differ in the two representations.
The final representation developed for the PGPfone alphabet is attached as
table~\ref{table:features}. Multivalued features, such as place, were
represented as ``thermometer codes''. Binary features, such as voicing,
were of course merely on or off, replicated enough times to achieve the
desired weight. The final representation requires 26 bits per phoneme.
\begin{table}
\centering
\begin{tabular}{|lll|}
\hline
Feature name & Sample & Number of bits \\
\hline \hline
Place of articulation & /d/ vs /g/ & 7 \\
Manner of articulation & /l/ vs /t/ & 6 \\
Height of articulation & /i/ vs /$\epsilon$/ & 5 \\
Voicing & /z/ vs /s/ & 4 \\
Syllabic & vowels vs. cons. & 1 \\
Nasal & /n/ vs /d/ & 1 \\
Lateral & /l/ vs /r/ & 1 \\
Roundedness & (various) & 1 \\
Sibilant & /s/ vs /f/ & not used \\
\hline
\end{tabular}
\caption{\label{table:features}Phoneme coding for PGPfone alphabet}
\end{table}
Even granting the viability and success of a phoneme-by-phoneme
perceptual distance metric, there are difficulties in its extension
to full-word distances, and here theory provides less support than
might be wished. For example, if each phoneme were weighted equally
and could be directly compared with a single other phoneme, the
difference between two words can be as simple as the sum of the
phoneme differences. However, some phonemes are clearly more
salient than others. On a gross level, the stressed syllables of
a word pair are intuitively of much greater salience than the unstressed
ones. Furthermore, psycholinguistic results like
Slobin \cite{slobin} or Derwing and Neary \cite{derwing-nearey}
suggest that onsets
are more salient than codas. These
results unfortunately provide little suggestion about whether a simple
weighting will
address this disparity, or what weights would be most appropriate.
The approach taken in PGPfone was a simple one; the preceding consonant cluster
and vowel(s) of the stressed syllable were given twice normal weight,
as was the (word-)onset phoneme.
A similar problem arises with non-aligned or non-existent sounds.
For example, should the word /bEst/ be treated as most similar to
/bEts/, /bEt/, or /bEs/? Derwing and Neary \cite{derwing-nearey}
present a few primitive
metrics to address this question, based on primarily on a notion
of sequences of identical vs. nonidentical phonemes. A more sophisticated
approach could use the notion of ``edit-distance'' as typified by Myers
\cite{myers},
but only with an accurate measure of the perceived difference between a
sound and its absence, or in other words, a featural representation for
silence. The representation of silence has produced some interesting
opinions [for example, Cottrell and Plunkett \cite{cottrell-plunkett}
presented silence as a
voiced, nasal, sibilant, back vowel], but little agreement or theory.
This problem can be reduced by the use of templates. For instance, if all
the words in
a study are of the form CVC, then there need be no
representation of silence as all phonemes are aligned directly. If the
words can be coerced into such a form, for example by elimination of
words with consonant clusters, then fewer sound/silence comparisons are
necessary. For
independent reasons (discussed below), the PGPfone list demands
words with small consonant clusters and of a particular syllabic
structure. The list to be selected from was filtered before the
selection process began to eliminate unsuitable words with long consonant
clusters. By increasing
the strength of the prefiltering, one
can restrict attention to words where comparisons are meaningful, or
phrased another way, one can greatly limit the damage done by a bad
representation of silence.
Similarly, by careful use of duplicate sounds, some of the silence/sound
comparisons can be avoided. Vowel sounds can be lengthened
or shortened almost at will, thus, vowel blends (such as /Oi/)
are compared with ``pure'' vowels such as /i/ by the simple technique
of presenting the pure vowel twice (/ii/) and comparing---and thus
/Oi/ is accurately represented as mid-way
between /O/ (/OO/) and /i/ (/ii/). Similar
tricks could be used to tease apart different consonant clusters; for
example, fricative (but not stop) consonants could be extended as
vowels above, or some simple combinatorics might apply to compare all
possible alignments and select one.
One final concern for the PGPfone distance metric touches on the
incorporation of additional, non-linguistic features. For example,
it would be nice if the final words had distinguishable orthographic prefixes,
to make it easier for keyboard entry and similar (non-linguistic)
processing tasks.
Obviously, paying attention to such things will, in theory, negatively
impact the linguistic quality of the final solution but result in
a better system overall. For the PGPfone list, orthographic spread
was achieved by appending to each phonological representation an
ASCII representation of the first two {\em characters} of the word.
Once the representation is in place, the actual distance was calculated
as the number of bits that differ between two word representations.
The word level distance metric (for the two-syllable template)
is attached as table~\ref{table:wordcoding}. The three syllable template,
of course, is similar except for the additional consonant and vowel,
and a slight increase in complexity in the representation of the
stressed syllable.
\begin{table}
\centering
\begin{tabular}{|ll|}
\hline
Phonetic aspect & Number of bits \\
\hline \hline
Onset consonant(s) & 78 \\
First syllable vowel & 52 \\
Middle consonant(s) & 52 \\
Second syllable vowel & 52 \\
Final consonant(s) & 52 \\
Initial characters & 12 \\
Stressed vowel & 52 \\
Stress pattern & 7 \\
\hline
\end{tabular}
\caption{\label{table:wordcoding}Word coding for PGPfone alphabet}
\end{table}
\section{Engineering Aspects}
The ultimate test of any representation is the quality of solutions
it permits. A good solution for the PGPfone list involves several
additional qualities than simply an accurate distance representation,
as detailed in this section.
Humans, when reading sequences, tend to make different errors than simple
bit-flips (misreadings), so error detection and recovery is a bit different
than simply correcting bits. Instead, humans tend to
either omit, duplicate, or switch (adjacent) words, rather than misread
them. Furthermore, humans tend not to be able to do complex Boolean
arithmetic in their heads, and so full error correction is usually not
practical. Stewart\footnote{Zhahai Stewart. 1991. Personal communication
to Philip Zimmermann} suggested a clever way to
allow human-like errors to be easily detected. By building {\em two} lists
instead of one, and alternating the lists from which the words in the
sequence come, one can easily spot any such errors by noticing that
two successive words come from the same list. This assumes, of course,
that the (listening) human can tell from which list a word has been drawn.
The two lists for PGPfone are obviously different in that one consists only
of two syllable words, and the other of three.
Similarly, the lists should consist of words that are easily pronounceable
and easily readable. Words with multiple spellings or multiple pronunciations
(including cases like Polish, the nationality, vs. polish, the
cleaning product) are perilous because they may be read or transcribed
incorrectly---and accordingly were deleted from the lists without
consideration. Similarly, any words for which we had evidence of
significant phonetic variations (e.g. tomato) were removed. Furthermore,
as there might be a significant pool of list users with difficulty with
some sounds or clusters, any hard-to-pronounce words, defined
as words incorporating any non-English sounds or lengthy consonant
or vowel clusters, were also eliminated.
Unfortunately, the filtering methods chosen do not readily solve the
problem of dialect or language variance. As an obvious example,
the words in the Moby Pronunciator database are given with their
pronunciations {\em in an American, and specifically Pacific,
accent.} Although the California accent is relatively neutral within
the United States, it's certainly not neutral or standard worldwide.
Even within the United States, dialect differences can make even as
fundamental questions such as the number of syllables problematic.
Words containing semivowels such as ``oilcan'' and ``fragile''
have been suggested
as being three syllable words in some dialects of (American) English,
and as two syllable words in others, although the actual phonetic
data on this is unclear. Further afield, native language
differences can also raise difficulties. Japanese and Chinese, for
example, are notorious for not distinguishing the ``lateral'' feature
between /l/ and /r/; the phoneme weighting of this feature has been
artificially reduced, but not eliminated, in an effort to balance its
relevance to the English speaking community and its irrelevance to
(parts of) the Pacific rim. To attempt to solve this in the filtering
process, for example by eliminating all words with semivowels or
lateral consonants, would have resulted in a list of candidate words
too small to be useful.
The most difficult aspect of the list to control was unfortunately
one of the more important; the final lists should contain words with
appropriate associations. One of the goals was to develop a word list
that would inspire a certain amount of confidence in the security of
the overall product. The standard military/pilots' alphabet, for example,
has a certain ``coolness,'' the same mystique that applies to a
child's Captain Midnight secret decoder rings. An ideal list would
capture the same indefinable feeling. And although it proved difficult
enough to banish repugnant words (for instance, the computer
selected ``nigger'' from the dictionary in an early test), there seems
no automated procedure for detecting all and only ``cool'' words.
We were forced to rely on what ad hoc principles we could identify. The
standard pilots' alphabet, for instance, contains familiar but
uncommon words; several words do not appear in the Brown corpus at all,
while no word listed has a frequency of 85 occurrences or more.
So for the PGPfone list, words
that were too unfamiliar or too common were eliminated.
The filtering to get appropriate consonant clusters
seemed to help here as well. Empirically, nouns seemed better
than verbs, which in turn seemed better than adjectives, but all three were
substantially better than the rest of English, but appropriate
databases were not available to automate and make use of that
observation. In general,
noninflected words seem stronger than their inflected variants.
In the end, we
were forced to rely on human judgement, generating a list, blue-pencilling
or modifying
words that we found inappropriate, then using the survivors as the base
for another list.
Once the selection and measuring criteria are available, the actual selection
of the list is, technologically speaking, near-trivial. Because of
the high dimensionality of the search space, direct solution of the
best subset in the candidate was held to be infeasible. (Such algorithms
tend to be either polynomial in the dimension of space to be searched, or
exponential in the number of elements in the candidate list.)
Instead, we opted to
use a standard multivariate approximation technique to find an
acceptable partial solution that could be refined as needed. Several
algorithms could be used for this; for example, tabu search, a
recent variation on hill climbing with momentum, was briefly considered
but also rejected due to the high dimensionality. Simulated annealing
is another standard optimization technique, but lacks a strong enough
element of incremental learning. As the initial stage in simulated
annealing is typically the ``melting'' of the entire knowledge base,
any useful knowledge from an initial approximation will be entirely
lost on the second and subsequent attempts at a solution.
For these reasons, we used a simple
genetic algorithm \cite{GA} to evolve a
near-optimum (sub)set of the candidates
such that the smallest distance between any pair was maximized.
Genetic algorithms (GAs) have been widely used as a general-purpose black-box
optimization algorithm, and their use here has no wider implications
beyond simply being a known, uncontroversial, and effective method
of solving optimization problems. Specifically,
the GA generated a population of random 256-word
subsets of the
candidate list. Subsets were permitted to ``breed'' by trading some of
their members, and the daughter subsets were evaluated (using the distance
metric described above) to determine the
closest pairwise distance. Successful children were allowed to
be fruitful and multiply, while the losers in the genetic sweepstakes were
simply dropped from the population. After several hundred generations,
the top candidate was then edited as described above, and the surviving words
were used as
a fixed and unchanging part of the entire population for the next run of
the GA selection program.
It proved necessary as well to cross-check the list pairs. For example,
the word ``guitar'' is phonetically distinct in English (being one of
the few words where a hard g preceeds a short i). Unfortunately, the
word ``guitarist'' is phonetically distinct for the same reason. Because
the comparison scheme used was template based, there was no easy way
to automatically compare words from the two lists and calculate a
numeric distance. Instead words from one list which were derivationally
related to words from the other list were individually inspected, and
usually the less ``cool'' element of the relational pairs (most often
the base or uninflected form) was hand-eliminated.
After several runs, when a final, accepted list had been agreed upon,
the words in each list were alphabetized without regard to case
and used to represent byte values from 0 to 255. Some sample words from
the middle of the lists are here attached as table~\ref{table:words}.
\begin{table}
\centering
\begin{tabular}{|lll|lll|} \hline
Number & 2 syllable & 3 syllable & Number & 2 syllable & 3 syllable \\
\hline \hline
111 & glucose & hesitate & 116 & guidance & impartial \\
112 & goggles & hideaway & 117 & hamlet & impetus \\
113 & goldfish & holiness & 118 & highchair & inception \\
114 & granny & hurricane & 119 & hockey & indigo \\
115 & gremlin & hydraulic & 120 & hotdog & inertia \\
\hline
\end{tabular}
\caption{\label{table:words}Sample words from the PGPfone list}
\end{table}
Figure~\ref{figure:inuse} shows an example of the list in use in a
nonPGPfone context. The large block of nearly opaque text is
a cryptographic key for a program called ``PGP.'' Using this key,
anyone can send secret mail to the author. The block of hexadecimal
digits, the ``key fingerprint'' can be used to quickly confirm that the
key has been received properly. It can easily be seen that the same
function can be performed more accurately, quickly, and memorably by
the encoded fingerprint.
\input{minipage}
\section{Implications and Conclusions}
The final alphabet as distributed in PGPfone appears to work well enough
for the purpose for which it was designed; our feedback has generally been
positive, and suggested improvements tend to be matters of opinion on
single words rather than major changes to the underlying structure or
model. This work does strongly suggest the need for further work on
the development of word-scale phonological confusibility models. The
alphabet itself might have been made much stronger if we had been able to take
several dozen subjects into a phonetics laboratory and test the weightings
we conjectured above. Fundamental data on the salience of various
word-level characteristics is available only in a very sketchy manner
(and likely to vary significantly with language anyway.)
Clearly, a full evaluation of this work requires some empirical
checking, which at this point has not yet been done. Although informal
tests show that the words are understood, the degree of confusibility
has not been rigorously tested. There are many open questions that
are grounds for future work. How confusible {\em are} the words?
Does the actual transmission channel correspond to the assumptions
used in the feature weights? Do the assumptions of a reasonable,
obvious, and unique pronunciation fail when the reader is not a native
English speaker?
Although this problem may seem artificial in many regards, it lends itself
well to treatment as a touchstone problem for many speech/language
generation problems. The difficulty we encountered with the representation
of consonant clusters mimics the difficulties other researchers such as
Cottrell and Plunkett \cite{cottrell-plunkett} or MacWhinney \cite{macwhinney}
have had with the learning
and representation
of sound patterns in language acquisition tasks. Particularly in situations
such as neural networks or supervised learning, where a distance measure
is used to direct the system to its new state, an accurate distance measure
is more a necessity than a convenience. An accurate statistical analysis
of the effectiveness and salience of various feature-based representations
may shed light to bridge the sound/phoneme gap---as well as help with the
(word) segmentation problem and provide fundamental evidence about the
psychological reality of phonemes and phonetic features.
From an engineering perspective, an accurate way of measuring perceptual
distance could help in any situation where language must be engineered to
fit a particular need. This could be of use, for instance, in sublanguage
selection and generation, or more prosaically to help with the creation
of novel and distinctive product and service names.
This work illustrates several basic principles that a reasonable metric
should follow:
\begin{itemize}
\item First, that standard feature sets
do not accurately reflect the perceived salience of
various features.
\item Second, that feature differences are a significant but not
all-encompassing
part of the perceived differences among words; superphonemic attributes
such as stress and onset must also be taken into account.
\item Third, that templates are best used to control the sorts of
comparisons and measurements taken, but that using them will greatly
restrict the overall validity of the measurements.
\end{itemize}
There are almost certainly other principles that could be found and
added to this list. It is hoped that future work, whether in the
context of PGPfone 2.0 or other unrelated projects, will be able to
extend this list of principles to a full theory of isolated word perception.
|
1,108,101,563,246 | arxiv | \section{STATE OF THE ART}
\IEEEPARstart{W}{ith} the advent of fifth generation (5G) RANs, networks are required to handle an ever growing amount of diverse devices. All these connections, especially with the rise in IoT device usage, will have different requirements on reliability, latency, etc. compared to current 4G cellular networks. This will determine a heavy demand on current telecom companies and infrastructures. Solutions to meet these new demands have already been researched, such as heterogeneous networks, however these might not be sufficient to cover the increasing demands in all situations.
In emergency situations, such as disaster response/recovery, the deployment of terrestrial infrastructures is economically unfeasible or impractical. This is why, in this proposal, the idea of leveraging unmanned aerial vehicles (UAVs/drones) as mobile base stations is considered a promising and feasible way to overcome these setbacks.
\subsection{UAVs TECHNOLOGY}
The current state of the art for UAVs feature diverse type of drones depending on their own operational features\cite[p.~1-4]{UAVNetworks}. The two main categories are HAP (high altitude platform) and LAP (low altitude platform). In each of these categories, we can further divide UAVs depending on their physical features, which are, multirotors and fixed wing type aircrafts. As mentioned before, each of these platforms have their advantages, for instance HAPs have longer endurance than LAPs, while LAPs have lower operational cost than HAPs. Furthermore, fixed wings have greater speed and payload than multirotors, while multirotors operate with ease in confined or small environments. We will discuss in particular how to exploit each UAVs type advantage in the proposal section.
In both cases the main concern is energy consumption. Ideally, in an aerial network, in fact, there is the need to stay airborne as long as possible. This is not always achievable and definitely a difficult task, but the literature provides different solutions to do so. First of all, from my personal experience in the field, it is mandatory to correctly power and size the UAVs in order to have the correct power-to-weight ratio, hence not wasting resources that could be used otherwise. This seems trivial, but examples showed that it is not always as obvious as it should be. Second, the correct choice for battery chemical technology should be considered. In fact, nowadays there is a widespread application of LiPo batteries, which have an exceptional stress resistance and power-to-weight ratio, but might not be the best choice for our UAVs. LiFe or LiIon batteries, in fact, definitely provide less stress resistance and a lot less instant current than their LiPo counterparts, but are a much more energy dense, so if appropriately provisioned for the specific equipment, they will outlast any LiPo counterpart.
On the other hand, a completely different approach could be adopted, as we will see in the proposal section, such as using a multirotor/fixed wing hybrid or even a glow powered multirotor.
All the aforementioned options, though, are examples of small and lightweight UAVs, but we don't want to rule out the possibility of integrating military-grade high-powerer options. In fact, the latest drone from Leonardo, the Falco Xplorer, might check all the boxes for our required characteristics due to its 24 hour of flight time and 350kg of payload\cite{Leonardo}.
\subsection{SDN EXAMPLE TECHNOLOGY}
The other drawback of airborne drones is that they have a limited payload so the base transceiver station (BTS) must be as small as possible. Because of this, current techniques regarding software defined networks (SDN) should be used and exploited as much as possible. An actual real world example of how this can be done is Athonet’s CUBE or BACKPACK\cite{Athonet}. These two technologies are relatively small: the former lighter than 30Kg and circa 12Kg the latter, and therefore transportable. The main idea behind them, as aforementioned, is that of providing a mobile virtual network operator (MVNO) by providing a centralized Mobile Core (EPC/NC) with proven roaming capability to national MNOs. This specific solution can interface with current existing national MNOs, so it definitely guarantees the crucial aspect of the integration between aerial and terrestrial networks. It is not the only suitable solution, since, despite their fairly small dimensions, it can still be too big for a single (or a fleet) of UAVs, hence a hybrid approach could be adopted to achieve our goal.
\subsection{COMMUNICATION BETWEEN DRONES AND TERRESTRIAL BTS}
Finally, the other necessary feature of this research proposal is how to be able to communicate with terrestrial BTS or orbital satellites needed to forward network traffic according to network topology. Wireless communication should be carefully chosen in order to avoid common problems related to wireless network and be capable of working without interfering with UAV’s radio and telemetry up link and down link. In this case, the current state of the art provides enough options and techniques to achieve this purpose.
Strictly speaking of frequencies, these should be carefully chosen also depending on each countries regulations, which might be strict boundaries. It seems that mmWave frequencies are currently the way to go: since they range between 30 and 300GHz, they allow using a frequency space which is free to use in most countries, thus not requiring special licences as of now. These frequencies, though, are pretty sensitive to path loss and losses typical of wireless links. On the upside though, mmWave frequencies allow using lower and hence more robust frequencies for the actual UAV radio link. In fact, the whole 2.4GHz band is available to be exploited for this specific use. This band is in fact extremely common in the field of RC air crafts\cite{RCprotocols}, supporting a wide variety of proprietary protocols based on DSSS/FHSS modulation techniques. We can even lower the frequency further to achieve longer ranges, like what has been done by TBS’s Crossfire and FrSky R9, which operates at 868/910MHz, or going to 433MHz as with the Dragonlink RC or even combining those two techniques, as done by JETI radio systems.
\section{Research Proposal}
\IEEEPARstart{D}{epending} on the actual use case needed, the accurate choice of UAVs should be carried out. Usually, as previously briefly presented, the standard calls for a fixed-wing for high endurance operations and a multirotor for high-flexibility scenarios. My idea is that combining these two aspects would enable network operators to exploit the best of both worlds.
\subsection{UAV design characteristics}
In order to do this, we could build a long-endurance gasoline powered UAV, like that proposed by Tao Pang, Kemao Peng, Feng Lin and Ben M.\cite{GasMultirotor}. This is definitely a good idea, but I would not proceed this way, since having a gasoline powered motor further complicates the whole system and is certainly less ideal in urban scenarios because of pollution (noise and air). In my opinion, the ideal is combining their variable propeller pitch approach with that of the wing-multirotor hybrid Songbird\cite{Songbird}. This way we can build a variable pitch multirotor which can, depending on the specific use case or weather conditions, transform like what’s done by USAF Ospreys into a flying wing.
\begin{figure}[H]
\centering
\includegraphics[width=3in,height=2in,keepaspectratio]{songbird_drone.png}
\label{fig_songbird}
\centering
\caption{The Songbird UAV}
\end{figure}
As design characteristics for this project I would recommend following these guidelines:
\begin{itemize}
\item the airframe should be made by carbon-fiber, hollow fiberglass structures or other light materials (e.g. titanium/graphene) in order to keep make it as light as possible, maximizing the payload.
\item the added wing surface can be exploited to fit solar panels to recharge batteries while in flight\cite[p.~24]{UAVComms}.
\item the motors should have a low kV value, to be able to use highest voltage batteries in order to achieve maximum power with the lowest current consumption, hence having more reliable batteries.
\item strictly speaking of batteries the UAVs should use LiIon cells, to exploit their higher energy density than standard LiFe and LiPo batteries.
\item the airframe should have a structure that can glide as much as possible as this can favor huge power savings by exploiting weather conditions such as ascending thermics.
\item equipping each UAV with an autonomous driving system, such as a small ML-base guidance system.
\end{itemize}
\subsection{UAVs Guidance system}
The use of ML can be exploited for different reasons: to calculate the best and most power efficient flying pattern that can be followed to still guarantee connectivity in the specific area; be able to coordinate a swarm of UAVs in case the payload for the mobile base station is not enough and has to be divided between multiple UAVs and finally to guarantee a “safe” (or the safest) behaviour in case of emergencies or equipment faults.
The UAVs operation will be coordinated by a unique point for their specific area of operation. This point of control corresponds to the terrestrial BTS which will act as a controller both for the drone network and as a relay for the main network. This BTS could also be built like a tower, where UAVs currently not in operation can be stored. This way the UAVs have a single place where they can be stored as backups or where they can automatically land for maintenance or battery recharging. Keeping backups in the (relatively) close vicinity will guarantee continuity of service in case something unexpected happens, and the ability of base stations to handle handoffs will allow for hot swaps of the UAVs when they need maintenance.
\subsection{Network operation}
Now it’s time to consider the technologies needed for the specific network operation. As previously discussed, the use of SDN techniques is fundamental to enable a base station to be carried on UAVs, since it will cut power and weight constraints a lot by virtualizing a network as much as possible. The current state of the art solutions, though, are still too bulky to effectively address the problem, so different approaches should be considered. For instance, instead of fitting a drone with an entire BTS, we could use only (one or more) femtocells. This approach will definitely be the best to develop on drones since its power and payload requirements are easily handled by drones, but still there are all the common limitations due to the use of femtocells and the need of a common BTS. These limitations however can be overcome by extending the transmission ranges of the femtocells with the appropriate techniques (to be discussed later) and by having the correct number of femtocells to allow for the appropriate coverage.
Another option, since there is not actually the need of creating a full-fledged network only with drones, is to have the UAVs acting as SDN switches on data plane for collecting context information in a distributed way, while the ground BSs (or even the satellites BS) are controllers gathering data and making control decisions on network functions and resource allocation. Helped by SDN, network reconfiguration and resource allocation among a swarm of UAVs can be conducted in a more flexible way.
In both these hypothesis UAVs will act as a relay between the UEs and the “main” BTS, which (in these case) is still terrestrial, but will be smaller and cheaper for establishing the infrastructure, since it relies on the UAVs to handle part of the computation and network operation.
\subsection{Transmission techniques}
Finally, to be able to carry out the task proposed in the aforementioned approaches, it is necessary to extend and reinforce the transmission ranges and techniques. In particular, the use of mmWave frequencies to communicate between the UAV and the UEs needs some additional measures, because of the nature of mmWave frequencies\cite[p.~16]{UAVComms}. Being high frequencies and short wavelengths, in fact, means that they are extremely susceptible to problematics like propagation loss. A possible solution to this is using a combination of beamforming techniques, which can be exploited to construct a narrow directional beam and overcome the high path loss or additional losses caused by atmospheric absorption and scattering. The fact that on a UAV network, the BTS is moving further complicates this task and calls for a better approach. Beamforming should in fact be made more efficient by combining tracking equipment so that the transmitter and receiver can move with a greater degree of freedom, hence guaranteeing that their movement is always relative to the position of the user's equipment, thus making micro corrections in correlation with UAV movements. Different techniques have been studied, relying on measures of link quality, like RSSI, to properly adapt to movement. Furthermore I think that preparing different antenna combinations (e.g. arrays of microstrip + horn + helix) could also be an option, so that the airborne BTS can also dynamically change the actual hardware transmitting to be able to better beamform and carry out its task.
\section{BUSINESS ANALYSIS}
\IEEEPARstart{T}{he} possible applications of UAVs networks are numerous and varied.\cite[p.~34]{UAVComms}\cite[p.~5-8]{UAVNetworks}
\subsection{Disaster Response and Recovery}
The first and most useful is that of a rapid network deployment in case of natural disasters or terrorist attacks. Studies have shown in fact how, after some large scale natural events, the network continuity was disrupted. Taking hurricane Katrina as an example, the aftermath showed that approximately 25\% of subnets were inferred as unreachable, and while 62\% of unreachable subnets were small subnets at the edges of networks, the majority (73\%) of unreachable subnets lasted longer than four weeks. This is even more evident, showing how a natural disaster like Katrina causes extreme damage on networks, but is even more indicative on how long it took to effectively recover the normal network functionality. Since network-service disruption is inevitable in these extreme cases, it is crucial to have effective techniques to overcome this. Guaranteeing network continuity is imperative nowadays: system downtime results in loss of revenue that might even bring a business venture to the brink of bankruptcy\cite{SystemDowntime}. Small business especially rely every day more and more on cloud-based applications, so the loss or excessive sluggishness of network connectivity can cripple their entire operations, while bigger businesses, like Amazon/Google/Facebook, will suffer a lot more from the longer-term effect of brand damage, like what happened in the earliest stages of AWS, when for different reasons, the company produced an uptime of 99.99\%, 52.56 minutes of system downtime over the course of a year, which, given today standards, is unacceptable for many customers. Finally the possible data loss and/or exposure to attacks resulting from network continuity problems is something that is a threat to all business ventures. Nowadays, in an increasingly interconnected world, the impact of network-service disruption goes beyond a strictly business perspective, and can in fact result in heavy losses of human lives. Having a reliable network becomes crucial to coordinating rescue operations in case of natural disasters, so, being able to quickly re-instate the lost infrastructures using a UAV network to provide coverage will definitely prove valuable for rescuing people in distress.
\subsection{Network Establishment Feasibility Study}
Another interesting application of UAVs established network is that of studying the feasibility and profitability of a terrestrial network. This application is particularly useful in rural areas where the “standard” network infrastructure is not present or is not able to guarantee coverage. If, in fact, a network operator decides to expand its current network, it could use a UAV network as a temporary solution to probe the specific area and further model if establishing a proper terrestrial network infrastructure in the specific area makes sense both business and cost wise. Plus, using this approach to probe the area of interest could also further be explored and expanded in case in that area there is a business interest but establishing a terrestrial network would be too costly. In this case, in fact, the UAVs network could be kept there for the time being, thus providing coverage, and only once the area is ready and big enough to build a terrestrial network, switch to it.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{target_specific_serving.png}
\label{fig_target_specific_serving}
\centering
\caption{UAVs as aerial BSs serving a target area}
\end{figure}
\subsection{Traffic Offloading}
One of the most useful applications of mobile aerial networks is that of providing traffic offloading for already established networks. Thinking, for instance, of large events hosted in a particular area, for example Olympic games, or political elections, which can call for a temporary needed increase in network capacity, the ability of UAVs network to be quickly deployed can supply this much needed capacity increase. The UAV network can in fact act as a back-haul for whatever type of connection is needed, both cellular data or WLAN, guaranteeing a flexible way to provide momentary and on demand relief for terrestrial networks.
\subsection{Target-specific CDN and Caching}
Another pretty useful aspect of having a mobile UAV network is that of using UAVs and Content Delivery Network nodes. Wireless data traffic has been increasing in recent years due to the greater demand of content-centric communications (such as video and music streaming). This determines a heavy burden in current back-haul links for this type of content, plus, since mobile users are constantly moving, a more flexible caching strategy is preferred. Thus, using UAVs BTS can be used to dynamically cache the popular contents, track the mobility patterns of wireless devices and then effectively serve them, reducing transmission latency and alleviating the traffic load, especially on back-haul links. In UAV-assisted caching, the more interesting contents can be directly cached in the UAV-BSs and then distributed to users.
\subsection{Target-specific Edge Computing}
Similar to this application is that of using UAVs BTS as edge computing devices to offload tasks and relieve computational burdens from the computing server. By adopting a distributed computing architecture with drones as its node, it is in fact possible to move the computation closer to the clients and so reduce network resource demands needed to communicate back and forth between the clients and the required computation node. By adopting a strategy similar to that of the aforementioned CDN/caching purpose, the drone with the required computation node can be kept mobile, instead of fixed as it is now, and shift its position according to which geographical area needs that specific node the most, hence providing lower response time to the final client.
Both of these two techniques will be definitely useful in dramatically increasing the throughput of a network, providing not only relief to the terrestrial network, but also better overall quality of service and quality of experience for the final client.
\section{Conclusion}
The application of UAVs networks are not limited only to the aforementioned scenarios, their actual application possibilities are endless and will evolve even more as newer technologies will push the current boundaries even further. Consider all the possible applications regarding UAV-to-UAV communication for the creation of ad-hoc networks, or even the possibility of directly connecting UAVs to satellites, enabling Satellite-to-UAV communications. With this research we have only scratched the surface, as this is only the beginning of a new generation of networks.
\ifCLASSOPTIONcaptionsoff
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1,108,101,563,247 | arxiv | \subsubsection*{\bibname}}
\bibliographystyle{apalike}
\usepackage[vlined,ruled,linesnumbered]{algorithm2e}
\begin{document}
\title{Growing Representation Learning}
\author{Ryan King, Bobak Mortazavi }
\maketitle
\begin{abstract}
Machine learning continues to grow in popularity due to its ability to learn increasingly complex tasks. However, for many supervised models, the shift in a data distribution or the appearance of a new event can result in a severe decrease in model performance. Retraining a model from scratch with updated data can be resource intensive or impossible depending on the constraints placed on an organization or system. Continual learning methods attempt to adapt models to new classes instead of retraining. However, many of these methods do not have a detection method for new classes or make assumptions about the distribution of classes. In this paper, we develop an attention based Gaussian Mixture, called GMAT, that learns interpretable representations of data with or without labels. We incorporate this method with existing Neural Architecture Search techniques to develop an algorithm for detection new events for an optimal number of representations through an iterative process of training a growing. We show that our method is capable learning new representations of data without labels or assumptions about the distributions of labels. We additionally develop a method that allows our model to utilize labels to more accurately develop representations. Lastly, we show that our method can avoid catastrophic forgetting by replaying samples from learned representations.
\end{abstract}
\section{Introduction}
Machine learning algorithms are powerful tools for a variety of data tasks including classification, compression, denoising, and generation. However, the performance on many state-of-the-art methods \cite{} is a result of fixed models and static datasets. When these models are deployed, they are susceptible to shifts in data distributions \cite{QuioneroCandela2009DatasetSI,DBLP:journals/corr/abs-2107-13964} and perform poorly on new events or classes. In these situations, a new model could be trained with the addition of the new data. However, this process becomes more resource intensive every time a new model needs to be trained. In more constrained situations, old data may not be available due to storage issues or privacy.
In the real world, data changes either with the appearance of new events or shifts in data distribution. Continual Learning (CL) attempts to adapt models to changes in data without forgetting previously learned information \cite{MCCLOSKEY1989109}.
Many CL methods exists for adapting models to new events \cite{}. In these methods, models are trained with a subsample of the classes that are available from each dataset. Then a new label is added, and the model is trained with the new data. If a user is aware of a new event, this can be done. However, these methods do not include mechanism for detecting when new events have occurred. A combination of clustering and cluster scoring, such as silhouette score, could be used to identify new events but this requires access to old data which may not be available. Dynamic Gaussian Mixture Models (GMM) have be proposed \cite{DBLP:journals/corr/abs-1808-06508} that form new gaussian mixtures based on a splitting indicator. In the unsupervised setting, these GMMs need to make assumptions about the number of data representations in a data set and often constrain them to be uniform.
In this work, we propose a continual learning algorithm that gradually grows as new representations of data appear which works with or without labels. We accomplish this by first developing an attention mechanism that learns to reconstruct its input data from a set of learned representations of the data we call prototypes. We utilize the weights generated by our attention mechanism to provide interpretability and match prototypes to labels when provided. We then show a method of determining the optimal number of prototypes through an iterative process of training and splitting. During the splitting, we create a network morphism \cite{DBLP:journals/corr/WeiWRC16} of our attention mechanism that can be used to determine the potential improvement from a split and the optimal prototype to be split. The contributions of this paper can be summarized as follows:
\begin{itemize}
\item A novel attention mechanism for learning data representations that we call Gaussian Mixture ATtention (GMAT).
\item A continuous learning scheme that splits our model to an optimal number of representations through NAS.
\item A soft label matching method that allows us to pair learned representations with user inputs.
\end{itemize}
\section{Related Work}
\subsection{Continual Learning}
Data in the real world is always changing either through changes in data distribution \cite{QuioneroCandela2009DatasetSI,DBLP:journals/corr/abs-2107-13964} or through the appearance of new tasks. For machine learning models, this raises the question of what the best policy is for changing models when new data appears. A simple solution would be to retrain the model from scratch with a combination of the data from the old and new tasks. However, this requires organizations to constantly increase their data storage capabilities which may not be feasible. An online embedded model, such as in robotics \cite{DBLP:journals/corr/abs-1907-00182}, a system may not have the ability to store any information. Additional privacy constraints may exist that require data to be destroyed a certain time after its creation \cite{HIPAA}. Continual Learning (CL) attempts to provide a solution to these constraints by updating a model to new information.
In CL, as a model is adapted to each new task, old representations have to be remembered without old data. The loss of old representations or previously learned information is called Catastrophic Forgetting \cite{MCCLOSKEY1989109}. One method of combating catastrophic forgetting is called generative replay \cite{DBLP:journals/corr/ShinLKK17,DBLP:journals/corr/abs-1808-06508}. When a adapt a model to new data, generative replay creates represents of old data based on previously learned representations of the data. In \cite{DBLP:journals/corr/ShinLKK17}
\subsection{Gaussian Mixtures}
One way of representing data different classes of data is with a Gaussian Mixture (GM) \cite{reynolds2009gaussian} where the entire data is represented by multiple Gaussian Distributions. Gaussian Mixture Variational Autoencoders have been developed \cite{DBLP:journals/corr/DilokthanakulMG16,DBLP:journals/corr/abs-1910-14481} to learn the parameters of each Gaussian Distributions without labels. However, these methods assume a uniform distribution of labels. In real world data, the actual distribution may be difficult or impossible to determine. In some domains, it's possible that minority classes are more important to classify than the majority classes. An example of this would be in the medical domain where the detection of uncommon health issues need to be detected.
\subsection{Progressive Expansion}
One method of dealing with new tasks is to dynamically expand a model so that new parameters can learn each specific task. In \cite{pmlr-v22-zhou12b}, labeled data is used to train a discriminator and generator. Then a collection of hard examples is collected for data points who's objective function was greater than a predetermined threshold. The model is then expanded based on the number of hard examples collected. In the updated model, new parameters are updated using the hard examples while old parameters are fixed. They additionally, provide a merging method to constrain the size of the network.
A similar method is used in \cite{DBLP:journals/corr/RusuRDSKKPH16} where the network is updated if the objective function from a new task is higher than some threshold. When the objective function is too high, $k$ units are added to each layer of the network and trained with a sparse regularization to control the growth of the network.
In \cite{DBLP:journals/corr/DraelosMLVCJA16}, an AutoEncoder (AE) is trained using a layerwise reconstruction error. When the network is trained on a new task, the reconstruction error provides an indicator for network growth. When the reconstruction error is large enough, new nodes are added to each level of the AE to allow it to represent the new data. In this method and in \cite{DBLP:journals/corr/RusuRDSKKPH16}, additional nodes may become entangled making it difficult to develop any type of interpretability for each new task.
Neural Architecture Search (NAS) is a field of Automated Machine Learning (AutoML) that attempts to automatically find the optimal neural architecture for a given dataset. Recent methods in NAS \cite{DBLP:journals/corr/abs-1910-02366,DBLP:journals/corr/abs-1910-03103} have shown that many local minimums are high dimensional saddle points. They escape these saddle points by creating a network morphic model \cite{DBLP:journals/corr/WeiWRC16} where potential splitting parameters are represented by multiple parameters while producing the same output for every input. With this representation of their model, they are able to calculate a splitting indicator which can be used to determine the strength of a split at each candidate position and the direction of a split. They show that this method can be coupled with an interpretable machine learning method \cite{DBLP:journals/corr/DraelosMLVCJA16} to find the optimal number of representations in a labeled dataset.
\section{Methods}
In this section, we break down the components of our method. Specifically, we describe a set of parameters called prototypes and how we train them to represent data. We then describe a hierarchical method for represent our prototypes. Finally, we describe the method we use to grow the number of prototypes.
\subsection{Prototype}
Given some input data $\mathbf{x} \in \mathbf{R}^{N \times L}$, where we would like to learn how to best represent the data. We choose to model our data as a Mixture of Gaussian \cite{reynolds2009gaussian} defined by:
$$p(x |\lambda) = \sum_{i=1}^M w_i g(x|\mu_i, \sigma_i)$$
Where $M$ is the number of prototype in our mixture, $w_i$ is the weight of each Gaussian distribution, $\mu_i$ is the mean, $\sigma_i$ is the standard deviation and $\lambda$ is the collection of $w$, $\mu$, and $\sigma$ associated with each individual Gaussian. Weights are constrained so that $\sum w_i = 1$. This makes the probability of $x \in \mathbf{x}$ the weighted average of Gaussian Distributions in the mixture. In our implementation, we consider each Gaussian Distribution a prototype.
Traditionally, the values of $\mu$ and $\sigma$ are the output of a trained linear layer. In each of our prototypes, we choose to represent $\mu$ and $\sigma$ as trainable parameters. To train these parameters to represent the data, we attempt to reconstruct the data with a combination of each prototype. We accomplish this by first taking the Mahaloanobis Distance \cite{mahalanobis1936generalized} between $X$ and $P$ defined as:
\begin{equation}
D(x) = \sqrt{(\mathbf{x} - \mathbf{\mu_i})^{\mathbf{T}}\Sigma_i^{-1}(\mathbf{x} - \mathbf{\mu_i})}
\end{equation}
With these distances, we consider data points that are close to the mean of a particular prototype to be highly represented by that prototype. With this intuition, we use the softmin equation to determine what percentage of each prototype, $\alpha$, should be used to reconstruct the data point:
\begin{equation}
w_i(\mathbf{x}) = \frac{e^{-D(x_i)}}{\sum_{j=1}^L{e^{-D(x_j)}}}
\end{equation}
Once we have derived the weights, we can take the weighted average, $\mathbf{z}_i$, of the prototypes to reconstruct the data point.
\begin{equation} \label{eq:attn}
\mathbf{z}_i(\mathbf{x}) = w_i(\mathbf{x}) \cdot P
\end{equation}
This mechanism, which we call GMAT, acts similar to soft attention where a importance weight is applied to each feature. However, in our method we take the softmin of the L2-norm to enable our attention mechanism create weight based on a prototypes closeness to a particular distributions mean. A graphic representation of our attention block is in Figure \ref{fig:attnblock}
\begin{figure}[h!]
\includegraphics[scale=0.6]{./Figures/PrototypeAttention.jpg}
\caption{A visual illustration of our GMAT mechanism. We calculate the weights by first measuring the distance between a data point $\mathbf{x}$ and each prototype distribution. The softmin is then appplied to the distances to determine the weight of each distribution in the Gaussian Mixture. The weighted sum of the prototypes is used to reconstruct the value $\mathbf{x}$}
\label{fig:attnblock}
\end{figure}
Given this setup, we'd like to reconstruct a data point based on some combination of the prototypes. We measure the distance between $x$ and each prototype as the Mean Square Error given as:
$$\mathcal{L}_{recon}(\mathbf{x}, \mathbf{z}) = \frac{1}{N} \sum_{i = 1}^{N}||x_i - z_i||^2_2$$
Where $\mu_i$ is the mean of each prototype and $\Sigma_i$ is the covariance matrix generated by diagonalizing $\sigma^2_i$. Since prototype that are close to a point provide a better representation of that point, we want them to be weighted more heavily. With this intuition, we use the softmin of the distances to generate the weights for each prototype. We use the reparameterization trick \cite{kingma2014autoencoding} by randomly sample a point from each of our prototypes. Weights from our softmin are applied to the randomly sampled points.
We make further use of the importance weights in our modified KL Divergence term. We start by calculating the weighted mean, $\mu_i$ and variance, $\sigma^2$, with $w_i(\mathbf{x})$ for each prototype:
\begin{equation} \label{eq:w_mean}
\hat{\mu}_i = \frac{\sum_{j=1}^N w(\mathbf{x})_{i,j} \mathbf{x}}{\sum_{j=1}^N w(\mathbf{x})_{i,j}}
\end{equation}
\begin{equation} \label{eq:w_var}
\hat{\sigma}_i^2 = \frac{\sum_{j=1}^N w(\mathbf{x})_{i,j} (\mathbf{x}_i - \mu_i(\mathbf{x}))^2}{\sum_{j=1}^N w(\mathbf{x})_{i,j}}
\end{equation}
We use equations \ref{eq:w_mean} and \ref{eq:w_var} to create a target Normal Distribution, $P_i$, of $x$ for each prototype:
\begin{equation} \label{eq:w_var}
\mathbf{P} = \mathcal{N}(\hat{\mu}_i, \hat{\sigma}_i^2)
\end{equation}
We then use this distribution to calculate the KL Divergence between the our prototypes, $\mathbf{Q}_i$, and the weighted distribution of the data.
\begin{equation} \label{eq:w_var}
\mathbf{KL}(\mathbf{P}_i || \mathbf{Q}_i)
\end{equation}
We additionally add two terms, equations \ref{eq:R1} and \ref{eq:R2}, to our loss function that increase the interpretability and certainty of our model.
\begin{equation} \label{eq:R1}
\mathcal{L}_{R_1} = -\frac{1}{N} \sum_{i=1}^{N} \mathbf{y}_i log(\mathbf{Z}_{[i,:]})
\end{equation}
\begin{equation} \label{eq:R2}
\mathcal{L}_{R_2} = -\frac{1}{M} \sum_{j=1}^{M} \mathbf{y}_j log(\mathbf{Z}_{[:,j]})
\end{equation}
Where $z_i \in \mathbf{Z}$, $y_i$ and $y_j$ is a one hot encoding of the batch-wise argmax and prototype-wise argmax of $\mathbf{Z}$. Both of these equations treat the weights from $z_i$ as probabilities. In equation \ref{eq:R1}, we take the Negative Log Likelihood in an attempt to avoid saddle points where the model may want to use a combination of prototypes to reconstruct the output. Equation \ref{eq:R2} increases the certainty that each prototype is used to reconstruct the data.
\subsection{Progressive Splitting}
We have a way of training our model to learn the weights of each prototype. We would now like to learn the optimal number of prototypes in our data. In order to accomplish, we develop an iterative process of training the parameters of our network, testing for potential growth and splitting if necessary. The algorithm for our splitting method can be seen in Algorithm \ref{algo_splitting}. In previous sections we discuss how our GMAT mechanism is trained. This section will be broken down into the steps following training which include the creation of a network morhpic model to test for growth and a splitting indicator for growth.
\begin{algorithm}
\caption{Optimal Splitting}
\SetKwData{Left}{left}
\SetKwData{This}{this}
\SetKwData{Up}{up}
\SetKwFunction{Union}{Union}
\SetKwFunction{FindCompress}{FindCompress}
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}
\Input{\{X, Y\}}
initialize $\theta$ \\
S = $\infty$ \\
\While{ $S > \epsilon$}{
\While{ not converged }{
train($\theta$, X, Y)
}
for
$S = \nabla L(X, \theta)$
}
\label{algo_splitting}
\end{algorithm}
\subsubsection{Network Morphism}
In order to test our for potential growth, we create a network morphism \cite{DBLP:journals/corr/WeiWRC16} of our GMAT mechanism by creating a set of potential splitting position we call dummies with the constraint that the new network structure $G$ must perform the same mapping as our original GMAT mechanism $F$. For both our linear and binary tree structure, we create $G$ by copying each prototype in $F$. For our linear structure, we simply copy the value of $\mathbf{\mu}$ for each prototype.
In order to prove that the addition of new prototypes results in an network morphic representation of our prototype layer, we provide the following proof. Given that for $\hat{x} = \{x, x\}, \hat{P} = \{P, P\}$.
\begin{equation}
\sum_{m \in M} \frac{P_m e^{x_m}}{\sum_{j \in M} e^{x_j}} = \sum_{m \in 2M} \frac{\hat{P}_m e^{\hat{x}_m}}{\sum_{j \in 2M} e^{\hat{x}_j}}\\
\end{equation}
\begin{equation}
\sum_{m \in 2M} \frac{\hat{P}_m e^{\hat{x}_m}}{\sum_{j \in 2M} e^{\hat{x}_j}}
\end{equation}
Since $\hat{x} = \{x, x\}$, it is equivalent to write the following:
\begin{equation}
\sum_{m \in 2M} \frac{\hat{P}_m e^{\hat{x}_m}}{2\sum_{j \in M} e^{x_j}}
\end{equation}
The same claim can be made for the outer summation where $\hat{x} = \{x, x\}$ and $\hat{P} = \{P, P\}$ giving the following:
\begin{equation}
2\sum_{m \in M} \frac{P_m e^{x_m}}{2\sum_{j \in M} e^{x_j}}
\end{equation}
The final reduction gives the following:
\begin{equation}
\sum_{m \in M} \frac{P_m e^{x_m}}{\sum_{j \in M} e^{x_j}} = \sum_{m \in M} \frac{P_m e^{x_m}}{\sum_{j \in M} e^{x_j}}
\end{equation}
With this isomorphic model representation, we can determine which additional branch would produce the best improvement. We do this by taking the average magnitude of the gradient from a pass through a dataset. By doing this, we
\subsubsection{Splitting Indicator}
With our network morphic structure, we are now able to represent each prototype as multiple prototypes. We would now like to understand how much each prototype in $G$ can potential help to increase the performance of $F$. Given the models parameters from $G$, $\hat{\theta}$, we calculate the average direction $D$ of the gradient to determine the splitting direction:
$$D(x,\hat{\theta}) = \frac{1}{N} \sum_{i=1}^{N} \nabla L(\mathbf{x}_i, \hat{\theta})$$
We use the magnitude of the gradient of $G$ as an indicator of the strength, $S$, of each split:
$$S(x,\hat{\theta})=||D(x,\theta)||^2_2$$
As prototypes are split to create new representations of the data, the magnitude of the splitting indicator decreases. We use $S$ as an indicator of when to stop split. We choose a threshold parameter $\epsilon$ to indicate when splitting should stop.
\subsection{Model Architecture}
Our GMAT mechanism can be used by itself to detect Gaussian Distributed classes from streams of data. However, we include an encoder $E(x)$ and decoder $D(z)$ so that the dimensionality of the input data can be reduced. We show in the section on Label Matching that the encoder can be used to map inputs to their respective average latent representations with the use of labels. For
A visual representation of our model architecture is shown in Figure \ref{fig:modelarch}
\begin{figure}[h!]
\includegraphics[scale=0.7]{./Figures/GMAT_Arch.png}
\caption{A visual representation of our model architecture. Input data is fed in a decoder which produces a latent representation. The latent representation is used as the input to our GMAT mechanism. GMAT reconstructs the representation for a combination of prototype distributions to produce $z$. This becomes the input for a decoder that reproduces $x$.}
\label{fig:modelarch}
\end{figure}
An appropriate encoder and decoder architecture can be chosen based on the input data. In our experiments we use a linear architecture to show how or model does on two simulated datasets. Additionally, we use a convolutional architecture in our experiments on MNIST.
\subsection{Generative Replay}
As our model adapts to new data, we need to remember previously learned representations. Generative Replay methods \cite{DBLP:journals/corr/ShinLKK17,DBLP:journals/corr/abs-1808-06508} have been successful in combating catastrophic forgetting by training on generated representations of old data. In DGR \cite{DBLP:journals/corr/ShinLKK17}, a generator is used to replay old representations of data while a solver is trained to classify. Other adaptations have been developed \cite{DBLP:journals/corr/abs-1808-06508} that use the decoder of a VAE architecture to generate samples from a Gaussian Mixture prior called Mixture Generative Replay MGR. We develop an adaptation of MGR using our prototypes. During training, we add generated into our dataset and treat it as training data. We generate data from our decoder by uniformly sampling each prototype learned from previous tasks. Samples from each prototype are randomly selected based on the $\mu$ and $\sigma$ describing their distribution.'
\subsection{Label Matching}
We provide an unsupervised method for learning an optimal number of representations of data. It's possible that data has labels or labels can be generated as prototypes become interpretable. However, the number of labels that are in a dataset may not match the number of prototypes currently trained. In the opposite situation, we may have a label for each prototype but we want to see if new representations of data exist. Since we can potentially have a different number of prototypes and labels, we would provide a soft labeling mechanism that works with our attention mechanism to match prototypes to labels.
We need each class of data to be Gaussian Distributed in order for it to be represented by a prototype. We utilize any provided labels to compute the per batch average of each class. We then measure the distance of each point to each average representation in the latent space.
$$d(x, L_{avg}) = ||\mathbf{x} - L_{avg}^i||^2_2$$
We use the softmin function similar to the way prototype importance is determined in the unsupervised setting.
$$\frac{e^{d(x_i)}}{\sum_{j=1}^L e^{d(x_j)}}$$
We treat the output of the softmin function as the probability that each data point, $E(x_i)$, belongs to each latent representation $l_n$. With these probabilities and provided labels, we calculate the classification loss using Cross Entropy. When label matching is being used, the encoder learns to map each $x_i$ into clusters centered around their latent average.
\section{Experiments}
We create our model and implement our splitting method using PyTorch \cite{NEURIPS2019_9015}. Our model, methods and experiments can be found at github.com. For all of our
\subsection{Datasets and Evaluation}
MNIST is an image dataset of handwritten digits from 0 to 9. We use this dataset throughout our experiments for supervised, unsupervised and continual learning. We use decode the means of each prototype to get an understanding of the data each distribution is representing. For all of our experiments with MNITS, we use a 4 layer CNN for our encoder and decoder with channel sizes of 32, 32, 32, and 10.
We evaluate each of our methods using the Normalized Mutual Information score on unsupervised tasks:
$$NMI(Y, C) = \frac{2 \times I(Y, C)}{[H(Y) + H(C)]}$$
Where $Y$ are the true labels, $C$ are the predicted labels, $H$ is the entropy, and $I(Y, C)$ is the mutual information between $Y$ and $C$. We also calculate the accuracy for comparison with other supervised methods.
\subsection{Simulated Dataset}
We use two simulated datasets to provide some intuition behind what each part of our model is doing. The first is a mixture of isotropic Gaussian Distributed classes. We train our GMAT mechanism alone on this dataset to show that it is capable of identifying Gaussian Distributed clusters from streams of data. The second dataset is the half moon dataset which we use with our entire model. We use this dataset to show that labels can be used to help our prototypes identify non-Gaussian distributed classes. We generate both of these datasets using the scikit-learn library \cite{scikit-learn}. We use a simple linear Autoencoder with a single hidden layer of size 100 and a latent size of 2 for easy visualization. Details about the data generation and training for each dataset can be found in Appendix... In Figure \ref{fig:sim_data} we show that the the mean of each prototype is centered in each cluster and the confidence intervals are drawn around those points.
\begin{figure}[h!]
\includegraphics[scale=0.8]{./Figures/sim_data.png}
\caption{GMAT trained on simulated dataset. In this figure, each color represent a different class of data. The means of our prototype distributions are represented by 'x' which are centered on each class.}
\label{fig:sim_data}
\end{figure}
\subsection{Optimal Splitting}
We train our model on every class in the MNIST dataset starting with 1 prototype distribution and allowing our network to split until an optimal NMI is reached. We collect the average magnitude per gradient and NMIs at each iteration and plot them in Figure \ref{fig:mags_per_iter}
bar graph with error
\subsection{Supervised Learning}
Incremental class learning and incremental task learning \cite{DBLP:journals/corr/abs-1904-07734,DBLP:journals/corr/abs-1810-12488} are popular methods of evaluating a CL methods. Both methods evaluate a models ability to learn new data without forgetting previously learned information by training a model by incrementally training a model on a subset of . In incremental class learning, a model is To evaluate our model, we use the SplitMNIST dataset, where the MNIST dataset is divided into 5 separate datasets by class (0/1, 2/3, 4/5, 6/7, 8/9). We train each iteration for up to 500 epochs with a patience of 50 epochs without improvement in the loss function. We compare our method with other Continuous Learning methods on the same task and report the results in Table \ref{tab:inc_class}. The prototypes resulting from our supervised method are shown in Figure \ref{fig:sup_prot} \footnote{Results are taken from \cite{DBLP:journals/corr/abs-1910-14481}, \cite{DBLP:journals/corr/abs-1810-12488}, and \cite{DBLP:journals/corr/abs-1904-07734}}.
\begin{figure}[h!]
\includegraphics[scale=1.25]{./Figures/10_prototypes.png}
\caption{Decoded means of each prototype trained with labels.}
\label{fig:sup_prot}
\end{figure}
\subsection{Unsupervised Clustering}
We test our method on it's ability to cluster data without labels by training our model on the MNIST dataset. We initialize our model with a single prototype and allow is to continue splitting for 30 iterations. We collect the splitting strength at each iteration and the NMI. We report the values in \ref{fig:unsup_bar}
We found that as the strength of our splitting indicator decreases, the increase in the NMI between iterations also decreases.
\begin{figure}[h!]
\includegraphics[scale=0.75]{./Figures/unsup_mag.png}
\caption{Unsupervised splitting}
\label{fig:unsup_bar}
\end{figure}
\subsubsection*{\bibname}}
\begin{document}
\onecolumn
\aistatstitle{Instructions for Paper Submissions to AISTATS 2022: \\
Supplementary Materials}
\section{FORMATTING INSTRUCTIONS}
To prepare a supplementary pdf file, we ask the authors to use \texttt{aistats2022.sty} as a style file and to follow the same formatting instructions as in the main paper.
The only difference is that the supplementary material must be in a \emph{single-column} format.
You can use \texttt{supplement.tex} in our starter pack as a starting point, or append the supplementary content to the main paper and split the final PDF into two separate files.
Note that reviewers are under no obligation to examine your supplementary material.
\section{MISSING PROOFS}
The supplementary materials may contain detailed proofs of the results that are missing in the main paper.
\subsection{Proof of Lemma 3}
\textit{In this section, we present the detailed proof of Lemma 3 and then [ ... ]}
\section{ADDITIONAL EXPERIMENTS}
If you have additional experimental results, you may include them in the supplementary materials.
\subsection{The Effect of Regularization Parameter}
\textit{Our algorithm depends on the regularization parameter $\lambda$. Figure 1 below illustrates the effect of this parameter on the performance of our algorithm. As we can see, [ ... ]}
\vfill
\end{document}
|
1,108,101,563,248 | arxiv | \section{Dynamical density functional theory}\label{ddft}
The time evolution of the density distribution $\rho(\boldsymbol{r},t)$ in a fluid system of particles is given by the continuity equation
\begin{align}
\frac{\partial \rho(\boldsymbol{r},t)}{\partial t} = - \nabla \cdotp \vec{J},
\label{continuity}
\end{align}
where $\vec{J}$ is the current of particles. This equation merely expresses the fact that the number of particles is conserved in the system.
Although formal expression for the current can be derived, it is often necessary to make approximations to find solutions to this equation. Dynamical density functional theory provides such an approximation for $\vec{J}$ which has been found to be rather accurate.\cite{marconi1999dynamic,marconi2000dynamic}
DDFT approximates the current $\vec{J}$ as
\begin{align}
\vec{J} = -\beta \Bar{\Bar{\boldsymbol{D}}} \cdot \rho(\boldsymbol{r},t) \nabla \frac{\delta \boldsymbol{F}[\rho]}{\delta \rho(\boldsymbol{r},t)},
\label{flux}
\end{align}
where $\beta = 1/(k_B T)$, $\Bar{\Bar{\boldsymbol{D}}}$ is an (arbitrary) diffusion tensor and $\boldsymbol{F}$ is the equilibrium Helmholtz free energy functional.
In the absence of external potential, the Helmholtz free energy functional is given as follows:
\begin{align}
\boldsymbol{F}[\rho(\boldsymbol{r},t)] = k_B T \int d\boldsymbol{r} \rho(\boldsymbol{r},t)\left[\ln (\rho(\boldsymbol{r},t) \Lambda^3)- 1\right] + \boldsymbol{F}_{\mathrm{ex}}[\rho(\boldsymbol{r},t)].
\label{functional}
\end{align}
The first term is the ideal gas free energy and $\Lambda$ is the thermal wavelength.
$\boldsymbol{F}_{\mathrm{ex}}[\rho]$ is the excess free energy; that is, the contribution due to the interactions between the particles.
The main assumption underlying DDFT is that the correlations between the particles when the fluid is out of equilibrium are
the same as in an equilibrium fluid with the same one-body density profile $\rho(\boldsymbol{r},t)$.
This is a major assumption which cannot be justified \emph{a priori}.
Nevertheless, DDFT has proven highly successful in describing the approach of a system towards equilibrium from a nonequilibrium initial state.\cite{archer2014solidification}
It is remarkable that DDFT can successfully describe the steady states of systems that are continuously driven out of equilibrium, such as sheared colloidal
suspensions \cite{chakrabarti2003dynamical,brader2011density} and active Brownian particles. \cite{menzel2016dynamical,sharma2017brownian}
The functional derivative in Eq. \eqref{flux} can be interpreted as the chemical potential $\mu(\boldsymbol{r},t)$ acting on a particle located at ($\boldsymbol{r},t)$.
One can then calculate the net driving force on the particle as the spatial gradient of the chemical potential, $\nabla \mu(\boldsymbol{r},t)$, and obtain the current $\vec{J} = -\beta \Bar{\Bar{\boldsymbol{D}}} \cdot \rho(\boldsymbol{r},t) \nabla \mu(\boldsymbol{r},t)$. The chemical potential obtained from Eq. \eqref{flux} has two contributions
\begin{align}
\frac{\delta \boldsymbol{F}[\rho]}{\delta \rho(\boldsymbol{r})} \equiv \mu(\boldsymbol{r},t)= \mu_{\mathrm{id}} + \mu_{\mathrm{ex}},
\label{fderiv}
\end{align}
where
\begin{align}
\mu_{\mathrm{id}} = k_B T\ln \left[\rho(\boldsymbol{r}) \Lambda^3\right]
\end{align}
is the ideal gas contribution to the chemical potential, and
\begin{align}
\mu_{\mathrm{ex}} = \frac{\delta \boldsymbol{F}_{\mathrm{ex}}[\rho]}{\delta \rho(\boldsymbol{r})} \equiv -k_B Tc^{(1)}(\boldsymbol{r}),
\label{c1}
\end{align}
is the contriubution to the chemical potential comming from the interactions between the particles, where $c^{(1)}(\boldsymbol{r})$ is the one-body direct correlation function.\cite{evans1979nature}
The functional derivative of $c^{(1)}(\boldsymbol{r})$ with respect to $\rho(\boldsymbol{r})$ yields the Ornstein-Zernike pair direct correlation function of the fluid \cite{hansen1990theory}
\begin{align}
k_B T c^{(2)}(\boldsymbol{r}, \boldsymbol{r}') \equiv -\frac{\delta^2\boldsymbol{F}_{\mathrm{ex}}[\rho]}{\delta \rho(\boldsymbol{r}) \delta \rho(\boldsymbol{r}')},
\label{c2}
\end{align}
which is one of the key quantities of interest in liquid state theories.\cite{hansen1990theory}
Below we show how $c^{(2)}(\boldsymbol{r}, \boldsymbol{r}') $ appears naturally in description of the dynamics of spinodal decomposition.
\section{Spinodal decomposition}\label{spinodal_decomposition}
In this section, we apply the DDFT in Sec. \ref{ddft} to fluid spinodal decomposition.
When a colloidal fluid is quenched to a state point inside the binodal, the fluid undergoes phase seperation into, for instance,
a liquid and a gas phase. There are two mechanisms of phase seperation.
The mechanism, referred to as nucleation occurs when droplets of one phase form in the other phase.\cite{gunton1983dynamics,onuki2002phase}
Phase separation proceeds by nucleation when the fluid is quenched to state point that lies inside the binodal but close to it.
This region corresponds to the region lying within the spinodal and binodal lines (see Fig. \ref{phasediagram}).
The other mechanism of phase seperation is called spinodal decomposition and it occurs when the state point lies well inside the binodal.
Spinodal decomposition is characterized by the exponential growth of density fluctuations of certain wavelengths.\cite{gunton1983dynamics,onuki2002phase}
Experimentally, however, there is not a sharp distinction between regions where phase separation occurs via nucleation and via spinodal decomposition.\cite{archer2004dynamical}
Nevertheless, it is generally accepted that for a deep quench into the coexistence region, the phase seperation occurs via spinodal decomposition.
In a fluid undergoing spinodal decomposition three different regimes can be distinguished. For early times after the quench, the amplitude of the density fluctuations are small and
theories linear in the density fluctuations, such as the well-known Cahn--Hilliard theory, \cite{cahn1959free,cahn1961spinodal} provide a good description of this early stage of
spinodal decomposition. At intermediate times the density fluctuations can be large, but sharp interfaces between domains of
gaslike and liquidlike regions have still not formed.\cite{dhont1996spinodal}
At long times sharp interfaces develop between domains of liquid and gas.
The Allen-Cahn theory explicitly takes into account the dynamics of the interfaces, and successfully describes the dynamics of spinodal decomposition in this regime.\cite{allen1979microscopic}
In this paper, we focus only on the short- and intermediate-time dynamics of spinodal decomposition.
\begin{figure}[t]
\centering
\vspace{-1cm}
\includegraphics[height=10cm]{phasediagram.pdf}
\caption{(a) The phase diagram of the fluid system composed of particles interacting via pair potential that is infinitely repulsive for $r <\sigma$ and is attractive (Yukawa) for $r >\sigma$.
The packing fraction is denoted as $\eta = \pi\rho_b\sigma^3/6$ where $\sigma$ is the hard-sphere diameter of a particle and $\rho_b$ is the bulk number density.
The parameter $k_B T \sigma^3/a$ is the reduced temperature. The parameter $a$ governs the energy scale of the attractive interaction between particles (Eq. \eqref{vat}).
The arrow shows the quench from state point A to state point B in the coexistence region.
The state point B correponds to $k_B T \sigma^3/a = 0.05$ and $\eta = 0.2$.
All the results presented below correspond to the dynamics of the phase separation evolving from this initial state point.
(b) The function $-k^2(1-\rho_b\hat{c}^{(2)}(k))$ is shown for the case of isotropic diffusion.
This function describes the rate of growth of the density fluctuations in Eq. \eqref{fourier} at short times.
For $k<k_c\approx 0.8$, where this function is positive, the density fluctuations grow exponentially.
For $k>k_c$, any fluctuation is damped. The critical wavenumber $k_c$ depends on how far the state point lies inside the coexistence region.
}
\label{phasediagram}
\end{figure}
\begin{figure*}[t]
\centering
\begin{tabular}{ccccc}
\hspace{-0.6cm} \includegraphics[width = 0.247\textwidth]{kxkz_D100.pdf} & \hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{kxkz_D99.pdf} &\hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{kxkz_D97.pdf} &\hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{kxkz_D95.pdf} \\
\hspace{-0.6cm} \includegraphics[width = 0.247\textwidth]{kxkz_D8.pdf} & \hspace{-0.3cm} \includegraphics[width = 0.22\textwidth]{kxkz_D7.pdf} &\hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{kxkz_D6.pdf} &\hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{kxkz_D5.pdf}\\
\hspace{-0.6cm} \includegraphics[width = 0.247\textwidth]{kxkz_D4.pdf} & \hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{kxkz_D3.pdf} &\hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{kxkz_D2.pdf} &\hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{kxkz_D1.pdf}
\end{tabular}
\caption{Colorplots of $\hat \rho(\boldsymbol{k})$ shown in the $k_x$-$k_z$ plane at time $\tilde t = \sigma^2 t/D_{zz} = 69$ with $D_{zz}=1$
for different values of $D\leq1$ (see legend).
The top row of figures correspond to small anisotropy whereas the bottom row to large anisotropy.
In the case of isotropic diffusion $\hat \rho(\boldsymbol{k}) = \hat \rho(k)$.
Fourier components with wavenumbers $k<k_c$ have the largest magnitude.
Fourier components outisde the range $k<k_c$ also grow due to the nonlinear coupling between different Fourier components.
However, with increasing anisotropy, the growth of the Fourier components outside the $k<k_c$ region becomes increasingly suppressed.
When the anisotropy is large ($D<0.5$) only the Fourier components along the $(0,0,k_z)$ direction show any significant growth.
}
\label{kxkzplanes}
\end{figure*}
\begin{figure*}[t]
\centering
\begin{tabular}{ccccc}
\hspace{-0.6cm} \includegraphics[width = 0.247\textwidth]{rhokxkz_D100.pdf} & \hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{rhokxkz_D99.pdf} &\hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{rhokxkz_D97.pdf} &\hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{rhokxkz_D95.pdf} \\
\hspace{-0.6cm} \includegraphics[width = 0.247\textwidth]{rhokxkz_D8.pdf} & \hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{rhokxkz_D7.pdf} &\hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{rhokxkz_D6.pdf} &\hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{rhokxkz_D5.pdf}\\
\hspace{-0.6cm} \includegraphics[width = 0.247\textwidth]{rhokxkz_D4.pdf} & \hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{rhokxkz_D3.pdf} &\hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{rhokxkz_D2.pdf} &\hspace{-0.1cm} \includegraphics[width = 0.22\textwidth]{rhokxkz_D1.pdf}
\end{tabular}
\caption{Colorplots of $\tilde \rho(\boldsymbol{r},t) = \rho(\boldsymbol{r},t)-\rho_b$ shown in the $x$-$z$ plane at time $\tilde t = \sigma^2 t/D_{zz} = 69$ with $D_{zz}=1$
for different values of $D\leq 1$ (see legend). These density distributions are the result of an inverse Fourier transformation of the results in Fig. \ref{kxkzplanes}.
The top row of figures correspond to small anisotropy whereas the bottom row to large anisotropy.
In the case of isotropic diffusion, $\tilde \rho(\boldsymbol{r},t)$ grows isotropically from the origin.
The radially symmetrical distribution becomes practically one dimensional in the $z$-direction, even for relatively small anisotropy as can be seen in for $D=0.8$.
Anisotropy significantly slows down the dynamics; the magnitude of the density fluctuations decreases strongly with increasing anisotropy.
}
\label{rhokxkzplanes}
\end{figure*}
The approach that we follow is the same as in Ref. \cite{archer2004dynamical}
We consider a homogeneous fluid that has been rapidly quenched to the region of the phase diagram inside the spinodal.
We consider small density fluctuations $\tilde \rho(\boldsymbol{r},t) = \rho(\boldsymbol{r},t)-\rho_b$ about the bulk fluid density, $\rho_b$, and obtain an equation for the growth of these fluctuations.
We perform a Functional Taylor expansion of the excess free energy in Eq. \eqref{functional} about the bulk density truncated at quadratic order:\cite{evans1979nature,evans1979spinodal}
\begin{align}
\boldsymbol{F}_{\mathrm{ex}}[\rho(\boldsymbol{r})] &= \boldsymbol{F}_{\mathrm{ex}}[\rho_b] + \int d\boldsymbol{r} \left. \frac{\delta \boldsymbol{F}_{\mathrm{ex}}[\rho]}{\delta \rho(\boldsymbol{r})} \right |_{\rho_b}\tilde \rho(\boldsymbol{r},t) \nonumber \\ &+ \frac{1}{2}\int d\boldsymbol{r} \int d\boldsymbol{r}' \left. \frac{\delta^2\boldsymbol{F}_{\mathrm{ex}}[\rho]}{\delta \rho(\boldsymbol{r}) \delta \rho(\boldsymbol{r}')}\right |_{\rho_b}\tilde \rho(\boldsymbol{r},t)\tilde \rho(\boldsymbol{r}',t).
\label{taylor}
\end{align}
Using Eqs. \eqref{c1} and \eqref{c2}, Eq. \eqref{taylor} can be rewritten as
\begin{align}
\boldsymbol{F}_{\mathrm{ex}}[\rho(\boldsymbol{r})] &= \boldsymbol{F}_{\mathrm{ex}}[\rho_b] + \mu_{\mathrm{ex}} \int d\boldsymbol{r} \tilde \rho(\boldsymbol{r},t) \nonumber \\&-\frac{k_B T}{2}\int d\boldsymbol{r} \int d\boldsymbol{r}' c^{(2)}(|\boldsymbol{r} -\boldsymbol{r}'|)\tilde\rho(\boldsymbol{r},t)\tilde \rho(\boldsymbol{r}',t),
\label{taylor2}
\end{align}
where we have used that in a bulk homogeneous fluid $\left. c^{(2)}(\boldsymbol{r}, \boldsymbol{r}') \right |_{\rho_b} = c^{(2)}(|\boldsymbol{r} - \boldsymbol{r}'|;\rho_b)$.
Using Eq. \eqref{taylor2} to calculate the flux in Eq. \eqref{flux}, we get from the continuity equation \eqref{continuity} the following equation for the time evolution of the density
\begin{align}
\frac{\partial \rho(\boldsymbol{r},t)}{\partial t} &= \nabla \cdotp \Bar{\Bar{\boldsymbol{D}}} \cdotp \rho(\boldsymbol{r},t) \nabla \ln[\rho(\boldsymbol{r},t) \Lambda^3] \\ &- \nabla \cdotp \Bar{\Bar{\boldsymbol{D}}} \cdotp \rho(\boldsymbol{r},t) \nabla \int d\boldsymbol{r}' c^{(2)}(|\boldsymbol{r} - \boldsymbol{r}'|) \tilde \rho(\boldsymbol{r}',t),
\label{spinodal1}
\end{align}
where we have used that $\nabla \mu_{\mathrm{ex}} = 0$ in a bulk homogeneous fluid.
The equation for the density fluctuations $\tilde \rho(\boldsymbol{r},t)$ is obtained from Eq. \eqref{spinodal1} as
\begin{align}
\frac{\partial \tilde\rho(\boldsymbol{r},t)}{\partial t} &= \nabla \cdotp \Bar{\Bar{\boldsymbol{D}}} \cdotp \nabla \tilde\rho(\boldsymbol{r},t) \\ &- \rho_b \nabla \cdotp \Bar{\Bar{\boldsymbol{D}}} \cdotp \nabla \int d\boldsymbol{r}' c^{(2)}(|\boldsymbol{r} - \boldsymbol{r}'|) \tilde \rho(\boldsymbol{r}',t) \nonumber \\ &- \nabla \cdotp \Bar{\Bar{\boldsymbol{D}}} \cdotp \tilde\rho(\boldsymbol{r},t) \nabla \int d\boldsymbol{r}' c^{(2)}(|\boldsymbol{r} - \boldsymbol{r}'|) \tilde \rho(\boldsymbol{r}',t).
\label{spinodal2}
\end{align}
To analyze the unstable modes of the density fluctuations we consider Eq. \eqref{spinodal2} in Fourier space.
The Fourier transformation of the density fluctuations is defined as
$\hat \rho(\boldsymbol{k},t) = \int d\boldsymbol{r} \tilde \rho(\boldsymbol{r},t) \exp(-i \boldsymbol{k} \cdotp \boldsymbol{r})$ and of the direct pair correlation function as $\hat c^{(2)}(\boldsymbol{k}) = \int d\boldsymbol{r} c^{(2)}(\boldsymbol{r}) \exp(-i \boldsymbol{k} \cdotp \boldsymbol{r})$.
With these definitions, the DDFT equation in Fourier space for the chosen quadratic approximation to $\boldsymbol{F}_{\mathrm{ex}}[\rho(\boldsymbol{r})]$ becomes
\begin{align}
\frac{\partial \hat \rho(\boldsymbol{k},t) }{\partial t} &= -(\boldsymbol{k} \cdot \Bar{\Bar{\boldsymbol{D}}} \cdot \boldsymbol{k})\left(1 -\rho_b \hat c^{(2)}(\boldsymbol{k})\right) \hat \rho(\boldsymbol{k},t) \nonumber \\ &+\frac{1}{(2\pi)^3}\int d\boldsymbol{k}' (\boldsymbol{k} \cdot \Bar{\Bar{\boldsymbol{D}}} \cdot \boldsymbol{k}')\hat \rho(\boldsymbol{k} - \boldsymbol{k}',t) \hat c^{(2)}(\boldsymbol{k}') \hat \rho(\boldsymbol{k}',t).
\label{fourier}
\end{align}
\noindent
The first term is linear in $\hat \rho(\boldsymbol{k},t)$ and governs the short-time evolution of the density fluctuations.
The second term is nonlinear in $\hat \rho(\boldsymbol{k},t)$ and captures the coupling between different Fourier components of the density fluctuations.
The corresponding equation for a system with an isotropic diffusion rate $D_0$ can be obtained as a special case of Eq. \eqref{fourier} by replacing $\boldsymbol{k} \cdot \Bar{\Bar{\boldsymbol{D}}} \cdot \boldsymbol{k}'$
with $D_0 \boldsymbol{k} \cdot \boldsymbol{k}'$.
In this case, the diffusion rate $D_0$ simply appears as a scale on the right hand side of Eq. \eqref{fourier} and can be removed by redefining the time variable; this simplification is not possible in the case of an anistropic diffusion tensor.
The short-time evolution of the density fluctuations is obtained by neglecting the second term on the right hand side of Eq. \eqref{fourier}.
The resulting equation is a linear equation and can be solved for $\hat \rho(\boldsymbol{k},t)$ as
\begin{align}
\hat \rho(\boldsymbol{k},t) = \hat \rho(\boldsymbol{k},0) \exp\left[-\frac{\boldsymbol{k} \cdot \Bar{\Bar{\boldsymbol{D}}} \cdot \boldsymbol{k}}{S(k)}t\right],
\label{rhoexponential}
\end{align}
where $S(k) = (1-\rho_b \hat c^{(2)}(k) )^{-1}$ and $k=|\boldsymbol{k}|$.
This linear theory predicts that (a) different Fourier components $\hat \rho(\boldsymbol{k},t)$ evolve independently and
(b) $\hat \rho(\boldsymbol{k},t)$ grows exponentially in time.
For an equilibrium homogeneous fluid at a state point outside the spinodal, $S(k)$ is the well-known structure factor of the fluid.\cite{hansen1990theory}
Since for an equilibrium fluid $S(k) >0$ for all $\boldsymbol{k}$, it follows that for state point outside the spinodal all Fourier modes decay in time implying that the fluid is stable to small perturbations.
However, the situation is different for state point chosen inside spinodal.
For a chosen $\boldsymbol{F}_{\mathrm{ex}}$, $S(k)$ can become less than zero for $k< k_c$ where the value of $k_c$ depends on how deep the fluid is quenched inside the spinodal.
This implies that the Fourier components with $k < k_c$ will grow exponentially in time.
For a chosen $\boldsymbol{F}_{\mathrm{ex}}$, one can determine the phase diagram and chose the state point inside the spinodal.
It is then straightforward to determine the Fourier modes that will grow exponentially.
\begin{figure}[t]
\centering
\vspace{-1cm}
\includegraphics[width=\columnwidth]{rhokz_small_anisotropy.pdf}
\caption{Fourier components of the density fluctuations $\tilde \rho(k_z,t)$ along the $k_z$-direction are shown for different times $\tilde t = \sigma^2 t/D_{zz}$ with $D_{zz}=1$.
The diffusion in the $x$-$y$ plane is slightly slower than in the $y$-direction with $D_{xx}=D_{yy} = 0.95$ in (a), 0.97 in (b), 0.99 in (c),
and the isotropic case of $D_{xx} = D_{yy} = D_{zz} = 1$ in (d).
Even small anisotropy significantly reduces the nonlinear contribution, i.e., the coupling of different Fourier components,
as can be seen in the slow growth of those $k$-components
which lie inside the stable region ($k>k_c \approx 0.8$).
The dashed lines show the exponential growth predicted by the linear theory (Eq. \eqref{rhoexponential}).
}
\label{rhokz}
\end{figure}
\section{Results and discussion}\label{results}
It is clear from Eq. \eqref{fourier} that in order to study the dynamics of spinodal decomposition one only needs to specify the direct pair correlation function.
At this point it becomes neccessary to assume an approximate functional for the excess Helmholtz energy which we take the same as in Ref. \cite{archer2004dynamical}
The interested reader can find the details in Ref. \cite{archer2004dynamical} This model system exhibits fluid-fluid (gas-liquid) phase transition.
Briefly, the system is three dimensional and is composed of particles interacting via pair potential which is infinitely
repulsive for $|\boldsymbol{r}|\leq \sigma$ and attractive for $|\boldsymbol{r}| > \sigma$. The attractive part of the potential has a Yukawa form
\begin{align}
v_{at}(\boldsymbol{r}) = -\frac{a \exp(-|\boldsymbol{r}|/\sigma)}{4\pi \sigma^2 |\boldsymbol{r}|},
\label{vat}
\end{align}
where $a$ is a positive parameter that determines the strength of the attraction.
Treating the attractive interactions in a mean-field fashion, the following expression is obtained for the direct pair correlation function in Fourier space:
\begin{align}
\hat c^{(2)}(k) = \hat c_{\mathrm{PY}}(k) + \frac{\beta a}{1 + (k\sigma)^2},
\end{align}
where $\hat c_{\mathrm{PY}}(k)$ is the Percus-Yevick approximation for the hard-sphere pair direct correlation function. \cite{hansen1990theory}
The phase diagram of the fluid system under consideration is shown in Fig. \ref{phasediagram}(a). All the results below are shown for the quench corresponding to sudden cooling from state point $A$ outside the spinodal to the point $B$ inside the spinodal. In Fig. \ref{phasediagram}(b) we plot the function $-k^2/S(k) \equiv -k^2(1-\rho_b \hat c^{(2)}(k))$ which is the argument of the exponential in Eq. \eqref{rhoexponential} corresponding to the special case of isotropic diffusion with $\Bar{\Bar{\boldsymbol{D}}}=\mathbb{1}$. As can be seen in Fig. \ref{phasediagram}(b), this function is positive for the wavenumbers $k<k_c$ where $k_c$ is determined by the condition $\rho_b\hat c^{(2)}(k_c) =1$.
\begin{figure}[t]
\centering
\vspace{-1cm}
\includegraphics[width=\columnwidth]{rhokz_large_anisotropy.pdf}
\caption{Same as in Fig. \ref{rhokz}, but for longer times and larger anisotropy, $D_{xx}=D_{yy} = 0.1$ in (a), 0.2 in (b), 0.3 in (c), and 0.4 in (d).
Anisotropic diffusion strongly reduces the growth of $k_z$-modes in the $k_z>k_c$ region.
This is apparent in the relatively smaller second peak located at $k_z \sigma \approx 1.1$.
The dashed lines show the exponential growth predicted by the linear theory (Eq. \eqref{rhoexponential}).
The agreement with the predictions of the linear theory is nearly perfect implying that the nonlinear contribution to the growth of density fluctuations is negligible.
}
\label{rhokz_strong_anisotropy}
\end{figure}
\begin{figure*}[t]
\centering
\vspace{-1cm}
\begin{tabular}{cc}
\includegraphics[width=\columnwidth]{rhokxy_large_anisotropy1.pdf} &
\hspace{-1cm} \includegraphics[width=\columnwidth]{rhokxy_large_anisotropy2.pdf}
\end{tabular}
\caption{Fourier components of the density fluctuations $\tilde \rho(k_x,t)$ along the $(k_x,0,0)$ direction is shown for times $\tilde t = \sigma^2 t/D_{zz}$ with $D_{zz}=1$ The diffusion in the $x$-$y$ plane is much slower with $D_{xx}=D_{yy} = 0.1$ in (a,e), 0.2 in (b,f), 0.3 in (c,g), and 0.4 in (d,h).
Figures in the left column (a-d) show the Fourier components for times $\tilde t = 49$, 51, 53, and 55 whereas the
figures in the right column (e-h) show the same for times $\tilde t =68$, 70, 72, and 74. Note that the scale of y-axis in (a-d) is much smaller than in (e-h).
Acording to the linear instability analysis the fastest growing Fourier mode is the one with $k_x\sigma \approx 0.5$ which should grow exponentially at short times.
The dashed lines show the predicted exponential growth from the linear theory (Eq. \eqref{rhoexponential}).
In (a), which corresponds to the smallest value of $D_{xx}$, there is no peak at $k_x\sigma \approx 0.5$.
The peak becomes increasingly visible in going from (a) to (d).
The growth of density fluctuations in the $x$-$y$ plane is predominantly driven by the nonlinear contributions coming from the coupling of different Fourier components.
This is evident in the figures (e-h), where after sufficiently long time, there is only a single peak located at $k_x \sigma \approx 0.17$.
}
\label{rhokxy}
\end{figure*}
\begin{figure}[t]
\centering
\vspace{-1cm}
\includegraphics[width=\columnwidth]{rhokz_dxyfixed.pdf}
\caption{Fourier components of the density fluctuations $\rho(k_z,t)$ along the $k_z$-direction is shown for times $\tilde t = \sigma^2 t/D_{xx}$ with $D_{xx} = 1$. The diffusion in the $z$-direction is slightly slower with $D_{zz}= 0.95$ in (a), 0.97 in (b), 0.99 in (c), and 1 in (d).
Though anisotropic diffusion reduces the growth of $k_z$-modes outside the $k_z < k_c$ region, as in Fig. \ref{rhokz}, the effect is comparatively weaker.
The dashed lines show the predicted exponential growth from the linear theory (Eq. \eqref{rhoexponential}).
}
\label{rhokz_zslow}
\end{figure}
To model anisotropic diffusion, we consider the following diagonal diffusion tensor:
\begin{align}
\Bar{\Bar{\boldsymbol{D}}} = \left(\begin{array}{ccc} D_{xx}=D & 0 & 0\\ 0 & D_{yy}=D & 0\\ 0 & 0 & D_{zz} \end{array}\right),
\end{align}
where $D$ and $D_{zz}$ are parameters. Note that such tensor is cylindrically symmetric.
We use the approximate form for $\hat c^{(2)}(k)$ as input to Eq. \eqref{fourier} and calculate the time evolution of density fluctuations for the state point $k_B T \sigma^3/a = 0.05$
and $\eta= 0.2$ (See Fig. \ref{phasediagram}).
This state point lies well inside the spinodal region. We assume that at time $t=0$ all the Fourier components have the value $\hat \rho(\boldsymbol{k},0) = 10^{-8}$.
In real space this corresponds to a single perturbation at the origin in the otherwise uniform density profile.
The cylindirical symmetry of the diffusion tensor together with the chosen intial conditions imply that $\tilde \rho(\boldsymbol{k},t)$ is completely described by $\tilde \rho(\{k_x,0,k_z\},t)$.
In Fig. \ref{kxkzplanes}, we show $\tilde \rho(\{k_x,0,k_z\})$ at time $\tilde t = \sigma^2 t/D_{zz} = 69$ for different values of $D$.
Diffusion along the $z$-direction is fixed to $D_{zz} = 1$.
The choice of $\tilde t = 69$ corresponds to intermediate-time dynamics of spinodal decomposition in an isotropic system ($\Bar{\Bar{\boldsymbol{D}}}=\mathbb{1}$).
In the case of isotropic diffusion $\tilde \rho(\{k_x,0,k_z\})$ has circular contours in the $k_x$-$k_z$ plane.
The Fourier components with wavenumbers $k<k_c$ exhibit strong growth with the fastest growing mode at $k\sigma \approx 0.5$.
Fourier components with wavenumbers outside the instability region also grow due to the nonlinear coupling of different Fourier components.
It is clear from the figure \ref{kxkzplanes} that anisotropy strongly impacts the nonlinear coupling. The growth along the 'slow' directions is strongly suppressed.
The suppression seems to be significant even for relatively small anisotropy ($0.8<D<1$). For large anisotropy ($D<0.5$), only the Fourier components along the $(0,0,k_z)$ direction show
any significant growth. In Fig. \ref{rhokxkzplanes}, we plot the real-space density (fluctuation) distribution obtained from inverting the Fourier transform of Fig. \ref{kxkzplanes}.
The density distribution is spherical symmetrical for $D=\mathbb{1}$.
The radially symmetrical distribution becomes practically one-dimensional in the $z$-direction even for relatively small anisotropy as can be seen in Fig. \ref{kxkzplanes} for $D=0.8$.
What is more interesting is how anisotropy slows down the dynamics; at a given instant of time, the magnitude of the density fluctuations is much smaller than that of the isotropic diffusion case.
In order to better understand the dynamics at different times, we focus on the time evolution of Fourier components $\tilde \rho(k_z,t)$ along the $(0,0,k_z)$ direction, which we refer to as the $k_z$-direction.
Although a complete description of the decompostion dynamics can only be obtained by considering the full $\tilde \rho(\boldsymbol{k},t)$,
by selectively focusing on the dynamics along $k_z$-direction, one can clearly see the effect of anisotropy on the intermediate-time dynamics of spinodal decomposition.
We first consider the case of small anisotropy.
The diffusion along the $z$-axis is fastest with $D_{zz} = 1$ and the diffusion in the $x$-$y$ plane is slightly slower with $D_{xx} = D_{yy} = D$.
In Fig. \ref{rhokz} we plot the Fourier components of the density fluctuation along the $k_z$-direction for different values of $D$.
We also show the results ignoring the nonlinear contribution. On neglecting the nonlinear contribution, only the $k_z$-components which lie within the range $k_z <k_c\approx 0.8$,
for which $S(k_z) <0$, grow exponentially. The growth of $k_z$-components outside this range is due to the coupling of Fourier components.
For short times, only the components inside the linear instability region grow. As these modes lying within $k_z <k_c$ grow, they couple to feed the growth of $k_z$ modes lying outisde
the instability region. It is for the growth of these $k_z$ modes lying outside the $k_z <k_c$ region that the effect of anisotropy is most pronounced.
As can be seen in Fig. \ref{rhokz}, the nonlinear contribution to the density fluctuations is significantly reduced in the case of anisotropic diffusion.
Surprisingly, even a relatively small anisotropy can significantly reduce the coupling of different Fourier components.
In case of strongly anisotropic diffusion ($D<0.5$), the nonlinear contribution to the growth of the density fluctuations is highly supressed.
We show in Fig. \ref{rhokz_strong_anisotropy} the Fourier components of the density fluctuations along the $k_z$-direction for small values of $D$.
The diffusion along the $z$-axis remains the fastest. As can be seen in Fig. \ref{rhokz_strong_anisotropy}, there is an additional small peak outside the $k_z <k_c$
region which grows slowly in time. The height of the secondary peak remains negligible in comparison to the main peak for the times considered. The growth of the main peak is accurately described by the linear theory.
It follows that the nonlinear contribution to the growth of density fluctuations within the $k_z <k_c$ is negligible from the Fourier components with $k_z>k_c$.
Of course, if one considers the density fluctuations on longer time scales than those shown in Fig. \ref{rhokz_strong_anisotropy}, the second peak will grow larger.
However, in this case the density fluctuations (for instance in the main peak) will become so large that the Taylor expansion of the Helmholtz free energy functional in Eq. \eqref{taylor}
up to quadratic order is insufficient to describe the spinodal decomposition at intermediate times.
On considering the dynamics in the $k_x$-$k_y$ plane, one finds that the growth of the density fluctuations is predominantly driven by the nonlinear contributions. In Fig. \ref{rhokxy} we plot the Fourier components of the density fluctuations along the $k_x$ direction for same values of $D_{xx}=D_{yy}$ as in Fig. \ref{rhokz_strong_anisotropy}. It is useful to first consider the extreme case of $D_{xx}=0$. In this case, the density fluctuations of the Fourier component with a given wave number $k_x$ can grow only due to the coupling of other Fourier components. Not surprisingly, this implies a complete absence of the linear regime.
When $D_{xx}$ is slightly greater than zero, there will be linear regime which will persist on a time scale governed by $D_{xx}$ and $k_x$, as can be seen in Fig. \ref{rhokxy}.
For $D_{xx} = 0.1$ there is no peak at the location $k_x \sigma \approx 0.5$ for the times considered in Fig. \ref{rhokxy}(a), as predicted by the linear theory.
On increasing $D_{xx}$ the peak becomes increasingly visible at the location $k_x \sigma \approx 0.5$ (see Figs. \ref{rhokxy}(b-d)). For all $D_{xx}$ there is an additional peak that grows at the location $k_x \sigma\approx 0.17$. With increasing time, the nonlinear contribution becomes dominant and there remains a single peak at $k_x \sigma\approx 0.17$ (see Figs. \ref{rhokxy}(e-h)).
Until now we have considered anisotropy of the form such that the diffusion is fastest along the $z$-direction. With increasing anisotropy, the system can be considered to becomes effectively one dimensional.
We now consider another scenario in which we fix $D_{xx} = D_{yy} = 1$ and decrease the diffusion coefficient along the $z$ direction below 1.
The extreme case in this scenario of $D_{zz} =0$ would then correspond to an effectively two-dimensional system. We expect that in this scenario, the anisotropic diffusion would have a weaker effect than the previous scenario.
In Fig. \ref{rhokz_zslow} we plot the Fourier components of the density fluctuation along the $z$-direction for different values of $D_{zz}$. As can be seen in this figure the anisotropy has a much weaker effect than what was observed in Fig. \ref{rhokz}.
This can be qualitatively understood considering that in Fig. \ref{rhokz}, the anisotropy reduces the contribution to the coupling from $k_x$-$k_y$ plane whereas in Fig. \ref{rhokz_zslow} only the contribution from $k_z$-direction is reduced.
\section{Conclusions}\label{conclusions}
We studied spinodal decomposition in a colloidal system composed of particles which diffuse anisotropically in space.
We used DDFT to model the dynamics of spinodal decomposition at short and intermediate times. We found that spatial anisotropy significantly alters the dynamics of spinodal decomposition.
At short times, the main effect of anisotropic diffusion is a trivial modification of the growth rate of density fluctuations.
For later stages of decomposition, anisotropy can significantly reduce the coupling of different Fourier components.
As a consequence of which the growth of density fluctuations at wave numbers that lie inside the range of linear stability is significantly slowed down.
We found that the nonlinear contribution to the density fluctuations is highly sensitive to the degree of anisotropy.
The slow growth of Fourier components outside the range $k < k_c$ is significant even for relatively small anisotropy.
In the case of strongly anisotropic diffusion, the growth of Fourier components outside the range $k < k_c$ is strongly suppressed.
The results presented in this paper show that the dynamics of spinodal decomposition become quite rich when the diffusion is assumed to be anisotropic.
However, there are some aspects that we have not addressed in this paper. For instance, consider the case of strongly anisotropic diffusion (Fig. \ref{rhokz_strong_anisotropy}).
The secondary peak that is located at $k_z \sigma \approx 1.1$ is well seperated from the main peak located at $k_z\sigma \approx 0.5$. It is natural to ask why the $k_z$ components between the two locations do not grow for the times considered in Fig. \ref{rhokz_strong_anisotropy}.
Another aspect that we have not addressed is the quantification of the intermediate time.
This is particularly relevant in the case of strongly anisotropic diffusion because of the possibility of formation of sharp interfaces.
These aspects require more mathematical analysis than presented in this paper and will be investigated in the future.
In this paper we focused on a fluid system that exhibits fluid-fluid phase transition.
There are other dynamical phenomena which are sensitive to spatial anisotropy of diffusion.
For instance, a colloidal system that exhibits freezing transition. In such a system anisotropic diffusion is expected to have important consequences for dynamical pattern formation during
the freezing transition. \cite{archer2014solidification}
Another interesting phenomenon is the laning transition in a sheared colloidal system.\cite{ackerson1988shear,chakrabarti2003dynamical,brader2011density}
It will be very interesting to study these phenomena using DDFT together with anisotropic diffusion. Anisotropic diffusion can be obtained in practise in a system of interacting colloidal
particles subjected to Lorentz forces. It will be very interesting to perform Brownian dynamics simulations of such systems and study the above mentioned dynamical phenomena.
\section*{Conflicts of interest}
There are no conflicts to declare.
\bibliographystyle{apsrev}
|
1,108,101,563,249 | arxiv | \section{Introduction}\label{sec:Intro}
The success of generative models in recent years has caused a paradigm shift in the field of machine learning. Generative modelling is one of the most important types of unsupervised learning, with recent applications in semi-supervised learning as well. It addresses the problem of probability density estimation, which is a core problem in unsupervised learning. Given training data, the primary goal of generative models is to generate new samples or observations having the same or approximately the same distribution as the training data. There are several different categories of generative models, each dealing with a different flavor of density estimation. Some generative models deal with explicit and tractable exact density estimation, such as fully visible belief networks (\citet{Frey}, \citet{Frey1998GraphicalMF}) and nonlinear independent components analysis (\citet{deco1995higher}, \citet{DinhNICE}, \citet{DinhRealNVP}). Some other models deal with explicit but approximate density estimation, such as Variational Autoencoders (\citet{KingmaVAEbook}, \citet{VAETutorial}, \citet{kingma2013fast}, \citet{rezende2014}, \citet{kingmaimproveVar}, \citet{chen2016variational}), Boltzmann Machines (\citet{MassParFahlman}, \citet{AckleyBoltz}, \citet{Hinton-84}, \citet{HintonSejnowski86}) and deep Boltzmann machines (\citet{salakhutdinovDeepBoltzmann}). Finally, some generative models are concerned with implicit density estimation, which are capable of sampling from the estimated probability density without explicitly estimating it. These encompass generative stochastic networks (\citet{bengio2014deep}) and, perhaps the most popular and widely successful generative model in recent years, Generative Adversarial Network (\citet{GoodfellowGAN}, \citet{goodfellow2016nips}, \citet{Radford2016UnsupervisedRL}, \citet{WGAN}, \citet{ImprovedWGANS}, \citet{StyleGAN}, \citet{PacGAN}, \citet{CycleGAN}, \citet{Pix2PixGAN}, \citet{StackGAN}).
In spite of the tremendous amount of success achieved by these generative models, in particular Variational Autoencoders (VAEs) and Generative Adversarial Networks (GANs), they are found to suffer from a few drawbacks. Most important among these drawbacks are mode collapse in GANs and mode mixtures in VAEs. Mode collapse is said to occur when the target distribution of samples is multimodal, but the sample generation procedure fails to produce any sample from one or more modal regions. For example, the MNIST dataset (\citet{MNISTdata}) contains black and white images of handwritten digits from 0 to 9, constituting 10 distinct classes or categories of observations. It is reasonable to believe that the distribution of these images will have 10 distinct modes corresponding to each category of images. When the generative model fails to produce samples corresponding to any particular category (say there are no samples containing the digit 6), an extreme form of mode collapse is said to occur. A slightly weaker form of mode collapse occurs when the proportion of generated samples corresponding to a particular mode is much smaller than the proportion of samples corresponding to the same mode in the observed data. On the other hand, mode mixture is said to occur when the target distribution has its support on a manifold with well-separated modal regions, but the generated samples lie in between these modal regions, corresponding to low probability zones of the target distribution. In most cases, such samples combine characteristics of samples belonging to the separate modes between which they lie, and thus are quite different from the observed data. In the case of MNIST data, a generated sample which looks like a combination of a 5 and 6 is a mixture between the two modes of the target distribution corresponding to the digits 5 and 6.
Recently, these shortcomings have been addressed using the theory of optimal transport in the paper \citet{modecollapsereg}. The generator function of a GAN can be viewed as a composition of an optimal transport map (with the noise distribution as source and a conceptual \say{latent code} distribution) with a decoder neural network. It is observed that this optimal transport map is discontinuous and it leads to the discontinuity of the generator function in GANs. This makes the generator function unfit for modelling using neural networks. Forcefully modelling such a discontinuous function using neural networks creates the problem of mode collapse in GANs. We elaborate on this optimal transport perspective of GANs and its implications in Section \ref{subsec:GAN and the role of optimal transport theory in generative modelling}.
This viewpoint of GANs led to the development of a new generative model, Autoencoder-Optimal Transport (AE-OT) model, proposed in \citet{AEOT}. The AE-OT model comprises of an autoencoder. The encoder network of the autoencoder creates an empirical latent code distribution corresponding to the training data in a latent space, with the latent codes representing the essential features of the observations. An optimal transport map between a noise distribution and the empirical latent code distribution is computed, and then a continuous linear approximation of the optimal transport map is constructed. This continuous function coupled with the decoder neural network of the autoencoder serves the role of the generator function in this model. The model is described in detail in Section \ref{subsec:The AE-OT model}.
However, the AE-OT methodology also suffers from a few drawbacks of its own, and we attempt to understand and illustrate them. The optimal transport map between the noise distribution and the empirical latent code distribution is discontinuous, and maps every possible sample generated from the noise distribution to one of the latent codes corresponding to observed data, which in turn gets mapped to a sample exactly equal to an observed sample, if the autoencoder is trained sufficiently. To generate new samples similar to the observed data without exact reconstruction, the optimal transport map needs to be smoothed and made globally continuous. In the AE-OT model, this is achieved by a piecewise linear extension of the OT map, with the extended map having as its domain a simplicial complex obtained by triangulating the latent codes corresponding to observed data. Then, by a complicated procedure depending upon a user-specified parameter which is difficult to interpret and tune, the regions of discontinuity of the estimated optimal transport map, known as singularity sets, are estimated, and samples from the noise distribution which get mapped to singularity sets are rejected, since these samples are mixtures of modes of the distribution of the observed data. Thus, the AE-OT methodology involves a complicated, unintuitive and computationally expensive method of generating new samples based on triangulations, and wastes computational resources in generating a large number of potential samples which are ultimately discarded by the rejection sampling scheme employed within this methodology. The technical details regarding the triangulation of the latent codes, construction of the piecewise linear extension $\tilde{T}$ and singular set detection are described in detail in Section \ref{subsec:The AE-OT model}, and even more elaborately in \citet{AEOT}.
The main motivation behind the paper \citet{AEOT} is to tackle mode collapse and mode mixture problems in general generative models, not only for GANs, by providing a theoretical justification for these issues and developing a generative model capable of mitigating them. In this paper, we proceed one step further by addressing the drawbacks of AE-OT. We develop a generative model which modifies the generative module of AE-OT in order to improve the sample generation methods followed in \citet{AEOT}, based on ideas of convex smoothing proposed in \citet{nesterov1998introductory} and \citet{Convex}.
Our primary contributions in this paper are:
\begin{itemize}
\item We develop a generative model which produces good quality samples, in the sense that they resemble the observed data and do not suffer from mode collapse and mode mixture.
\item We provide a theoretical validation for the efficacy of the convex smoothed AE-OT model by proving an uniform bound on the error of approximation of the optimal transport map between the noise distribution and the empirical latent code distribution, which serves as a measure of how closely the generated samples resemble the observed samples.
\item We improve upon the sample generation method of the AE-OT model while developing our model by removing the need for rejection sampling, thus saving precious computational time and resources.
\item In contrast to the method of latent vector generation in AE-OT, which ultimately produces linear combinations of encoded latent vectors corresponding to training data, our proposed method is not restricted to producing only linear combinations of latent vectors and potentially allows one to cover the entire manifold support of the distribution of latent vectors defined within the latent space corresponding to the autoencoder used.
\item Our model is dependent upon an user specified parameter controlling the degree of accuracy of our method, ensuring the mitigation of mode-collapse and mode-collapse without exact reconstruction of the training data, and having the additional benefit of being more interpretable and easier to choose than the tuning parameter $\theta$, used in the AE-OT model for controlling the degree of mode-mixture.
\item We propose a strategy for choosing the optimal value of the user-specified parameter based on a two-sample statistical test of equality of the distribution from which samples are generated and the true distribution of the data we intend to generate, based on the generated and observed samples. We show that there is a trade-off between diversity in the generated samples and the degree of similarity between the generated samples and training samples, and our proposed strategy provides an optimal balance between the two extreme scenarios.
\end{itemize}
The organization of the paper is as follows. Section \ref{sec:Intro} provides a brief introduction to generative modelling along with existing models in the literature, with particular focus on the Autoencoder-Optimal Transport model (\citet{AEOT}) and an overview of our contributions in this paper. Section \ref{sec:AE-OT to modified AE-OT Framework} begins with a primer on Generative adversarial networks (GANs) along with the relevant elements of optimal transport theory. We then proceed to discuss the problems faced by GANs, providing the motivation for the development of the AE-OT model. The AE-OT model is discussed next along with its drawbacks. We then discuss our novel contribution in the form of a sample generation procedure based on the idea of convex smoothing (\citet{nesterov1998introductory}, \citet{Convex}) as an alternative to the sample generation method of the AE-OT model, and propose the convex smoothed AE-OT model, along with relevant theoretical justifications. Section \ref{sec:Algorithm} provides the complete algorithm for constructing the convex smoothed AE-OT model and obtaining the generated samples based on input training data. Section \ref{sec:Theoretical validation of the Convex Smoothed AE-OT model} contains a theoretical proof validating the use of the convex smoothed AE-OT model. Section \ref{sec:Experimental results} provides the experimental results obtained on applying the convex smoothed AE-OT model to simulated 2 dimensional multimodal datasets. Section \ref{sec:Conclusion} includes a concluding discussion.
\section{AE-OT to Convex Smoothed AE-OT Framework}\label{sec:AE-OT to modified AE-OT Framework}
In the family of generative models, GANs are among the most successful, being able to generate highly realistic samples, especially in case of image data, and hence they serve as a reference model for sample generation problem. Recently, the theory of optimal transport has been used to provide us a deeper insight into the GAN paradigm.
In the first two subsections, we will discuss the GAN paradigm from the optimal transport viewpoint and the difficulties faced by GANs, which provide us the motivation for developing the AE-OT model. Later, we propose our modification of the sample generation method, along with the complete sample generation algorithm as well as a procedure to choose the optimal level of approximation error to allow.
\subsection{GAN and the role of optimal transport theory in generative modelling}\label{subsec:GAN and the role of optimal transport theory in generative modelling}
Let $o_{1},o_{2},\dots,o_{n}$ denote $n$ observed data points (usually images) belonging to the image space or ambient space $\mathcal{X}$ and let $\eta$ be the the corresponding empirical distribution (true distribution). The manifold distribution hypothesis allows us to imagine a manifold $\Sigma$ within $\mathcal{X}$ on which the data/images reside, and $\eta$ is defined on the manifold support $\Sigma$.
A Generative Adversarial Network (GAN) consists of two components: A generator and a discriminator. We assume $x_{1},x_{2},\dots,x_{N}\sim \mu$ are samples generated from a tractable noise distribution $\mu$ (usually uniform or Gaussian) defined on a low-dimensional space $\mathcal{Z}$ (a latent space encoding latent features or essential characteristics of observed images). The generator neural network G, represented as a function $g_{\gamma}:\mathcal{Z} \rightarrow \mathcal{X}$, of a GAN transforms $x_{i}$'s into $g_{\gamma}(x_{i})$'s in $\mathcal{X}$ to generate new image samples having distribution $\zeta_{\gamma}$ i.e. $$x \sim \mu \rightarrow g_{\gamma}(x) \sim \zeta_{\gamma} \textrm{ where } x \in \mathcal{Z} \textrm{ and } g_{\gamma}(x) \in \mathcal{X} $$
Here $\gamma$ represents the neural network parameters corresponding to the generator network, and hence is used to parametrize both the generator function $g_{\gamma}$ and the empirical distribution of the generated samples $\zeta_{\gamma}$.
The discriminator neural network D works as an adversary and attempts to discriminate between the generated image distribution $\zeta_{\gamma}$ and the true image distribution $\eta$, helping the generator network to learn from the training data. The Jenson Shannon divergence $JS(\eta ||\zeta_{\gamma})$ is used by discriminators in traditional GANs to measure the degree of dissimilarity between the two distributions (\citet{GoodfellowGAN}), while discriminators in Wasserstein GANs use the Wasserstein distance based on $L_{p}$ loss $W_{p}(\eta,\zeta_{\gamma})$ (\citet{arjovsky2017wasserstein}). The paper (\citet{geometricunderstanding}, \citet{geometricview}) shows that GANs try to learn the manifold $\Sigma$, together with the optimal transport map T between $\mu$ and $\eta$ using quadratic loss and a manifold parametrization $g$ which maps local coordinates in the latent space $\mathcal{Z}$ to the manifold $\Sigma$ within $\mathcal{X}$.
Following the papers (\citet{geometricunderstanding}, \citet{geometricview}), the GAN model can be understood, in principle, to accomplish two major tasks:
\begin{enumerate}
\item manifold learning, discovering the manifold structure of the data
\item probability transformation, transforming a white noise to the data distribution.
\end{enumerate}
Accordingly, the generator map $g_{\gamma}:(\mathcal{Z}, \mu) \rightarrow\left(\Sigma, \zeta_{\gamma}\right)$ can be further decomposed into two steps,
$$
g_{\gamma}:(\mathcal{Z}, \mu) \stackrel{T}{\rightarrow}(\mathcal{Z}, \rho) \stackrel{g}{\rightarrow}\left(\Sigma, \zeta_{\gamma}\right)
$$
where $T$ is a transportation map, maps the white noise $\mu$ to $\rho$ in the latent space $\mathcal{Z}, g$ is the manifold parametrization, maps local coordinates in the latent space to the manifold $\Sigma$. Specifically, $g$ gives a local chart of the data manifold $\Sigma$, $\rho=g^{-1}_{\#}\eta$ is determined by the real data distribution $\eta$ and the encoding map $g^{-1}$ and $T$ realizes the probability measure transformation. Hence the generator $g_{\gamma}$ is equal to $g \circ T$. The goal of the GAN model is to find $g_{\gamma},$ such that the generated distribution $\zeta_{\gamma}$ fits the real data distribution $\eta,$ namely
$$
{g_{\gamma}}_{\#} \mu=\eta
$$
Let $c: \mathcal{Z} \times \mathcal{Z} \rightarrow[0, \infty]$ be a measurable loss function: $c(x, y)$ represents the cost of transporting $x$ to $y$ where $x,y \in \mathcal{Z}$. For example, when $\mathcal{Z}=\mathbb{R}^{d},$ we can take $c$ to be the quadratic (or $L_{2}$ ) loss function
$$
c(x, y)=\|x-y\|^{2}
$$
The goal of optimal transport (Monge's problem) is to find a measurable transport map $T: \mathcal{X} \rightarrow \mathcal{Y}$ solving the (constrained) minimization problem
$$
\inf _{T} \int_{\mathcal{X}} c(x, T(x)) d \mu(x) \quad \text { subject to } \quad {T_{\#}}\mu=\rho
$$
where the minimization is over $T$ (a transport map), a measurable map from $\mathcal{Z}$ to $\mathcal{X}$, and ${T_{\#}}\mu$ is the push forward of $\mu$ by $T,$ i.e.,
$$
{T_{\#}}\mu(B)=\mu\left(T^{-1}(B)\right), \quad \text { for all } B \in \mathcal{X}.
$$
\subsection{Problems with GANs: Motivation behind the AE-OT model}\label{subsec:Problems with GANs: Motivation behind the AE-OT model}
GAN training is tricky, unstable and sensitive to hyperparameters. More importantly, they suffer from mode collapse where they learn to generate samples from a subset of modes from among the entire collection of modes in the true data distribution $\eta$. Mode collapse is said to occur also when proportions of generated samples from different modes do not match with the proportions of images belonging to the different modes in $\eta$. In addition, mode mixture may also occur when generated samples fall outside the true data manifold $\Sigma$ in between modal regions.
\citet{modecollapsereg} discusses the following theoretical reasons behind mode collapse and mode mixture. Brenier Theory gives us the following result:
\begin{theorem}\label{thm:Brenier}
Suppose $\mathcal{X}$ and $\mathcal{Y}$ are the Euclidean space $\mathbb{R}^{d}$ and the transportation cost is the quadratic Euclidean distance $c(x, y)=\|x-y\|^{2}$ for every $x \in \mathcal{X},y \in \mathcal{Y}$. Furthermore $\mu$ is absolutely continuous, and both $\mu$ and $\rho$ have finite second order moments, $\int_{\mathcal{X}}|x|^{2} d \mu(x)+\int_{\mathcal{Y}}|y|^{2} d \rho(y)<\infty,$ then there exists a convex function $u: X \rightarrow \mathbb{R},$ the so-called Brenier potential, its
gradient map $T=\nabla u$ gives the solution to the Monge's problem,
$$
{T_{\#}} \mu=\rho
$$
The Brenier potential is unique upto a constant, hence the optimal transportation map is unique.
\end{theorem}
In case of GANs, we have $\mathcal{X}=\mathcal{Y}=\mathcal{Z}$. Further, since $\rho$ is a discrete distribution and $\mu$ is absolutely continuous, discrete Brenier theory under quadratic transportation cost can be used to show the existence of a convex piecewise linear continuous function $u$, the gradient of which is the optimal transport map $T = \nabla u$. Following Section 4 of \citet{modecollapsereg}, we can view this Brenier potential map $u$ geometrically as the upper envelope of a collection of hyperplanes (\citet{geometricview}) and $u$ can be parametrized uniquely upto an additive constant by a parameter $h$, referred to as the height vector (\citet{VarMinkowski}, \citet{AEOT}). $u$ is often referred to as $u_{h}$ using this parametrization. If $y_{1},y_{2},\dots,y_{n}$ are the latent codes obtained from the observed data $o_{1},o_{2},\dots,o_{n}$ using the inverse decoding function i.e. $g^{-1}(o_{i})=y_{i},i=1,2,\dots,n$, then $\rho=\frac{1}{n}\sum_{i=1}^{n}\delta_{y_{i}}$. In such a case, $u_{h}$ can be shown to be of the form
$$
u_{h}(x) = \max _{i=1}^{n}\left\{\pi_{h, i}(x)\right\}=\max _{i=1}^{n} \{{x^{T}}y_{i}+h_{i}\}
$$
where $\pi_{h, i}(x)={x^{T}}y_{i}+h_{i}$ is the hyperplane corresponding to $y_{i}$. Regularity theory of optimal transport given by Cafarelli and Figalli (\citet{modecollapsereg}) states that whenever the support of $\rho$ is non-convex or composed of disconnected components due to multimodality of $\rho$ (induced by multimodality of $\eta$), T is a discontinuous function, and hence $g_{\gamma}$ is discontinuous. Deep Neural Networks (DNN) can only model/approximate continuous functions and $g_{\gamma}$ lies outside the functional space represented using DNNs. This leads to the problems of unstable training, non-convergence of the training process, mode collapse and mode mixture. The regions where the transport map $T = \nabla u$ is discontinuous are referred to as singular sets, which are collection of points where $u$ has a non-unique sub-gradient. Singular sets are characterized by sharp ridges, indicated by large dihedral angles between adjoining hyperplanes. Latent vectors belonging to or close to singular sets correspond to mixtures between modes of the empirical latent code distribution $\rho$, which in turn correspond to images which are mixtures between modes of the distribution observed image distribution $\eta$.
The AE-OT model (\citet{AEOT}) is motivated by these insights. It is proposed as a generative model free from the problems faced by GANs, yet able to generate new images which respect the diversity present in the real-life images and look realistic.
\subsection{The AE-OT model}\label{subsec:The AE-OT model}
The AE-OT model has two major components (Fig. \ref{AE-OTmodel}) :
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{AE-OT_model.PNG}
\caption{AE-OT model (\citet{AEOT})}
\label{AE-OTmodel}
\end{figure}
\begin{enumerate}
\item[i.] \textbf{Autoencoder} (AE) - An autoencoder is used for learning the data manifold $\Sigma$ in the image space $\mathcal{X}$. It learns the essential features of the data through dimensionality reduction. An autoencoder is composed two parts:
\begin{itemize}
\item[a.]An encoder network $\left(f_{\theta}\right)$ which encodes the data manifold from the image space $\mathcal{X}$ to the low-dimensional latent space $\mathcal{Z}$, and map the data distribution $\eta$ to the latent code distribution $\nu$ i.e.
$$f_{\theta}:\mathcal{X}\rightarrow\mathcal{Z}$$ where $\theta$ represents the neural network parameters corresponding to the encoder network of the autoencoder.
Both $\eta$ and $\nu$ are empirical discrete distributions of the form
$$\eta=\frac{1}{n}\sum_{k=1}^{n} \delta_{o_{k}}\textrm{ and }\nu=\frac{1}{n}\sum_{k=1}^{n}\delta_{y_{k}}$$
where $o_{k}$ is the $k$-th observed data point with $y_{k}=f_{\theta}(o_{k})\in\mathcal{Z}$ being the latent representation of $o_{k}, k=1,2,\dots,n$, and $\delta$ is the Dirac function.
Here we note that the empirical latent code distribution $\rho$ is defined implicitly in case of GANs (since there is no explicit encoding network involved), while the empirical latent code distribution $\nu$ is explicitly defined based on the observed data as well as the explicit encoding network.
\item[b.] A decoder network $g_{\xi}$ which maps/decodes the latent codes from $\mathcal{Z},$ back to the ambient/image space $\mathcal{X}$. i.e.
$$g_{\xi}:\mathcal{Z}\rightarrow\mathcal{X}$$ where $\xi$ represents the neural network parameters corresponding to the decoder network of the autoencoder.
\end{itemize}
The encoded latent vectors/representations corresponding to the observed data is interpreted as essential features extracted from the data through dimensionality reduction, by minimizing the reconstruction loss between real images and reconstructed images obtained by passing the real images through the autoencoder. We refer to the review of autoencoders in \citet{geometricunderstanding}, the references therein and \citet{FirstAE},\citet{AEreview} for a detailed exposition on autoencoders.
\item[ii.] \textbf{Optimal transport map} (OT) - New generated images can be obtained by the following steps :
\begin{enumerate}
\item[a.] Generate random samples $$x\sim\mu$$ where $\mu$ is a tractable absolutely continuous noise distribution defined on $\Omega \subset \mathcal{Z}$ (say, uniform or Gaussian).
\item[b.] Compute a probability distribution transformation between $\mu$ and the empirical latent code distribution $\nu$ , which is exactly the semi-discrete optimal transport map T under quadratic loss with $\mu$ as the source distribution and $\nu$ as the target distribution i.e.
$${T_{\#}}\mu=\nu$$
The Brenier potential $u$ corresponding to T can be parametrized uniquely by a \say{height} vector $h$ under a linear restriction, and can be referred to as $u_{h}$. $u_{h}$ is found by a convex optimization process involving Monte-Carlo simulation according to \citet{VarMinkowski} such that $T=\nabla u_{h}$. This is implemented using Algorithm (\ref{Alg.1}) in Section \ref{sec:Algorithm}. AE-OT tries to model the continuous Brenier potential map $u_{h}$ instead of the discontinuous OT map T using deep neural networks and thus potentially avoids the problems that GANs face.
\item[c.] Smooth the optimal transport map T to obtain a continuous map $\tilde{T}$ by extending T to a globally continuous function $\tilde{T}$. The transport map T is piecewise linearly extended to a global continuous map $\tilde{T}$, where the image domain becomes a simplicial complex obtained by triangulating the latent codes $y_{1},y_{2},\dots,y_{n}$.
The technical details regarding the triangulation of the latent codes, construction of the piecewise linear extension $\tilde{T}$ and singular set detection are described in \citet{AEOT} and the reader is strongly advised to refer to it. This construction ensures that mode collapse cannot occur. We will discuss some of these details shortly.
\item[d.] Define
$$\Omega_{k}\left(u\right)\coloneqq\left\{x\in\Omega\subset\mathcal{Z}\mid \dim\left(\partial u(x)\right)=k\right\},k=0,1,2,\dots,\dim(\mathcal{Z})$$ where
$\partial u(x)$ is the collection of sub-gradients of $u$ evaluated at $x$.
Then the singularity set is $$\Omega_{sing}(u)={\bigcup_{k>0}}\Omega_{k}(u)$$
which is essentially the region of discontinuity of the Optimal Transport map T. Detect the singularity set $\Omega_{sing}(u)$ in the source domain $\Omega \subset \mathcal{Z}$ of T. If $x\in\Omega_{sing}(u)$, then $\tilde{T}(x)$ represents a sample which is a mixture between modes of the distribution $\nu$, and consequently $g_{\xi}\circ\tilde{T}(x)$ is a spurious sample representing mixtures between modes in the observed data distribution $\eta$, $g_{\xi}$ being the decoder network. Hence to mitigate mode mixture, samples $x \in \Omega_{sing}(u)$ are rejected. Thus this is a rejection sampling scheme.
\item[e.] Generate the sample image by $g_{\xi} \circ \tilde{T}(x)$ where $g_{\xi}$ is the decoder network.
\end{enumerate}
We now discuss the technicalities involved with the above steps.
The semi-discrete OT map T induces a cell decomposition (a partition) of $\Omega$ of the form $\Omega=\bigcup_{i=1}^{n} W_{i}$. Thus corresponding to every $x \in \Omega$, there exists an $i \in \left\{1,2,\dots,n\right\}$ such that $x \in W_{i}$. Further, for every $i \in \left\{1,2,\dots,n\right\}$, every $x$ belonging to cell $W_{i}$ is mapped to the target $y_{i}$ by the optimal transport map T i.e.
$$T(x) = y_{i} \textrm { if and only if } x \in W_{i}$$
Consequently, we also have that $\mu\left(W_{i}\right)=\frac{1}{n}$.
Under quadratic loss, T is the gradient of the piecewise linear convex Brenier potential $$u_{h}: \Omega \rightarrow \mathbb{R}, u_{h}(x):=\max _{i=1}^{n}\left\{\pi_{h, i}(x)\right\}=\max _{i=1}^{n} \{{x^{T}}y_{i}+h_{i}\}$$ where $\pi_{h, i}(x)={x^{T}}y_{i}+h_{i}$ is the hyperplane corresponding to $y_{i} \in Y$. The projection of the graph of $u_{h}$ decomposes $\Omega$ into cells $W_{i}(h)$, each cell $W_{i}(h)$ is the projection of the supporting plane $\pi_{h, i}(x)$ i.e.
$$
W_{i}(h) = \left\{x \in \Omega \mid \nabla u_{h}(x)=y_{i}\right\}, i=1,2,\dots,n
$$
We often drop the reference to $h$ and refer to $W_{i}(h)$ as $W_{i}$, as in the previous paragraph. The height vector $h$ is the unique minimizer of the following convex energy
$$E(h)=\idotsint_{S_{h}} \sum_{i=1}^{n} w_{i}(v) d v_{i}-\frac{1}{n}\sum_{i=1}^{n} h_{i}
$$
under the linear restriction that $\sum_{i=1}^{n} h_{i}=0$, where $S_{h}=\{v = (v_{1},v_{2},\dots,v_{n})\in \mathbb{R}^{n}\mid 0\leq v_{i} \leq h_{i},i=1,2,\dots,n\}$, $w_{i}(v)$ is the $\mu$-volume of $W_{i}(v)$ i.e. $\mu(W_{i}(v))=w_{i}(v)$ and $v=\left(v_{1},v_{2}.\dots,v_{n}\right)$ being the variable of integration. Following \citet{VarMinkowski}, $E(h)$ can be optimized by gradient descent method. The $\mu$-volume $w_{i}(h)$ of each cell $W_{i}(h),$ is estimated using conventional Monte Carlo method.
To generate new samples, the semi-discrete OT map $T=\nabla u_{h}$ is extended to a piecewise linear (PL) mapping $\tilde{T}$ as follows. By representing the cells $W_{i}(h)$ by their $\mu$ -mass centers as $$c_{i}:=\int_{W_{i}(h)} x d \mu(x)$$ we obtain the point-wise map $f: c_{i} \mapsto y_{i}$.
The Poincaré of the cell decomposition induces a triangulation of the centers $C=\left\{c_{i};i=1,2,\dots,n\right\}:$ if $W_{i} \cap W_{j} \neq \emptyset,$ then $c_{i}$ is connected with $c_{j}$ to form an edge $\left[c_{i}, c_{j}\right] .$ Similarly, if $W_{i_{0}} \cap W_{i_{1}} \cdots \cap W_{i_{k}} \neq \emptyset,$ then there is a $k$ -dimensional simplex $\left[c_{i_{0}}, c_{i_{1}}, \ldots, c_{i_{k}}\right]$. The simplicial complex formed by these simplices is a triangulation of $C$, denoted as $\mathcal{T}(C)$ . A triangulation $\mathcal{T}(\mathcal{Z})$ of $\mathcal{Z}$ is computed similarly.
After drawing a random sample $x\sim\mu,$,with $\mu$ being the noise distribution, one can determine the simplex $\sigma$ in $\mathcal{T}(C)$ containing $x .$ Assuming the simplex $\sigma$ has $d+1$ vertices $\left\{c_{i_{0}}, c_{i_{1}}, \ldots, c_{i_{d}}\right\},$ the barycentric coordinates of $x$ in $\sigma$ is defined as $x=\sum_{k=0}^{d} \lambda_{k} c_{i_{k}},$ and $\sum_{k=0}^{d} \lambda_{k}=1$ with all $\lambda_{k}$ non-negative. Then the generated latent code of $x$ under this piecewise linear map is given by $$\tilde{T}(x)=\sum_{k=0}^{d} \lambda_{k} y_{i_{k}}$$
No modes are lost and mode collapse is avoided since all of the $y_{i}$ s are used to construct the simplicial complex $\mathcal{T}(\mathcal{Z})$ in the support of the target distribution.
During practical implementation, the $\mu$ -mass center $c_{i}$ is approximated by the mean value of all the Monte-Carlo samples inside $W_{i}(h)$ i.e. $$ \hat{c}_{i}=\frac{\sum_{x_{j} \in W_{i}} x_{j}} {\#\left\{x_{j} \in W_{i}\right\}}$$ where $x_{j} \sim \mu,j=1,2,\dots,N_{m}$ and $N_{m}$ is the number of Monte Carlo samples used in estimation. The connectivity information $\mathcal{T}(C)$ is too complicated to construct and to store in high dimensional space, thus $\mathcal{T}(C)$ is not explicitly built.
In practice, the simplex $\sigma \in \mathcal{T}(C)$ containing $x$ is determined as follows: given a random point $x \in \Omega$, evaluate and sort its Euclidean distances to the centers $d\left(x, \hat{c}_{i}\right), i=1,2, \ldots, n$ in the ascending order. Suppose the first $d+1$ items are $\left\{d\left(x, \hat{c}_{i_{0}}\right), d\left(x, \hat{c}_{i_{1}}\right), \ldots, d\left(x, \hat{c}_{i_{d}}\right)\right\},$ then $\sigma$ is formed by $\left\{\hat{c}_{i_{k}}\right\} .$ The barycentric coordinates $\hat{\lambda}_{i_{k}}$ are estimated as $$\hat{\lambda}_{k}=\frac{d^{-1}\left(x, \hat{c}_{i_{k}}\right)} {\sum_{k=0}^{d} d^{-1}\left(x, \hat{c}_{i_{k}}\right)}$$
This constitutes the backbone of the sample generation procedure of the AE-OT model.
However, this may generate some spurious samples, when some of the $x$'s randomly generated from $\mu$ fall inside the singular set $\Omega_{sing}$, leading to mode mixture. To mitigate this problem, one needs to detect the singular set $\Omega_{sing}$ and remove the samples falling inside it.
If there are multiple modes in the target distribution or the support of the target distribution of the optimal transport map T is concave, then, according to Figalli's theory, there will be singular sets $\Omega_{sing} \subset \Omega,$ where the Brenier potential $u_{h}$ is continuous but not differentiable, making its gradient map, i.e. the transport map $T=\nabla u_{h}$, discontinuous. In case of multimodality, which is the most common situation that occurs in practice, $\Omega \backslash \Omega_{sing}$ will consist of as many connected components as the number of modes, each of them mapped to a single mode. $\Omega_{sing}$ consists of codimension 1 facets of cells. If $W_{i}(h) \cap W_{j}(h) \subset \Omega_{sing},$ then the dihedral angle between two supporting planes $\pi_{h, i}$ and $\pi_{h, j}$ of $u_{h}$ is prominently large. Therefore, on the graph of Brenier potential, we pick the pairs of facets whose dihedral angles are larger than a given threshold, the projection of their intersection gives a co-dimension 1 cell in the singular set $\Omega_{sing}$. During the generation process, if a random sample $x$ is around $\Omega_{sing},$ it will be mapped by $\tilde{T}$ to the gaps among the modes. When generating new latent codes, we reject such samples, and this helps to prevent the mode mixture phenomenon.
Specifically, given $x \sim \mu$, we can detect if it belongs to the singular set by checking the angles $\theta_{i_{k}}$ between $\pi_{i_{0}}$ and $\pi_{i_{k}}, k=1,2, \ldots, d$ as $$\theta_{i_{k}}=\left\langle y_{i_{0}}, y_{i_{k}}\right\rangle /\left\|y_{i_{0}}\right\| \cdot\left\|y_{i_{k}}\right\|$$ If all of the angles $\theta_{i_{k}}$ is larger than a threshold $\hat{\theta},$ we say $x$ belongs to the singular set and just reject it. Or else we select a subset $\left\{\pi_{i_{k}}\right\}$ with $\theta_{i_{k}} \leq \hat{\theta},$ denoted as $\left\{\pi_{\hat{i}_{k}}, k=0,1, \ldots, d_{1}\right\} .$ Then we can compute $\lambda_{k}=d^{-1}\left(x, \hat{c}_{i_{k}}\right) / \sum_{j=0}^{d_{1}} d^{-1}\left(x, \hat{c}_{i_{j}}\right)$ and $\tilde{T}(x)=\sum_{k=0}^{d_{1}} \lambda_{k} T\left(\hat{c}_{\hat{i}_{k}}\right) .$ Intuitively, $\tilde{T}(\cdot)$
smooths the discrete function $T(\cdot)$ in regions where latent codes are dense and keeps the discontinuity of $T(\cdot)$ where latent codes are very sparse. In this manner AE-OT avoids generating spurious latent code and thus improves the quality of generated samples.
We would like to focus on a few problems of the AE-OT sample generation method:
\begin{itemize}
\item Detection of the singular set $\Lambda$ is based on the observation that sharp ridges between adjoining hyperplanes of $\hat{u_{h}}$ is indicated by large dihedral angles, and a thresholding parameter $\theta$ is used to determine which values of the angles should be considered prominently large, acting as a tuning parameter to be determined separately for different datasets. This is however just a proxy for finding the points of non-differentiability of $u_{h}$ in absence of any direct or better method. There is no principled method for finding an optimal value of $\theta$ to be used except for trying out different values and seeing which one gives best results, which is computationally expensive. There is no data-dependent intuition regarding what should be a good choice of $\theta$.
\item A very computationally expensive rejection sampling scheme has been proposed to ensure mode mixture does not occur in the generated samples, which must be implemented for every choice of the threshold $\theta$ that we want to try out. It is very difficult to get large number of generated samples using this rejection sampling scheme, since a large of samples is rejected in practice, even when the choice of $\theta$ is close to optimal.
\item Computing and storing the entire connectivity information corresponding to the simplicial complex $\mathcal{T}(C)$ is infeasible, even for moderately large dimensions of $\mathcal{Z}$. The algorithm approximates the true simplicial complex $\mathcal{T}(C)$ by constructing a simplicial complex having simplices of maximum degree $d$ (a $d$ dimensional simplex is defined using $d+1$ points). Although not explicitly stated in \citet{AEOT}, choice of $d$ is important to ensure the approximation is sufficiently accurate; too small a value of $d$ will lead to loss in accuracy of approximation while too large a value of $d$ leads to an approximation which cannot be computed and stored in practice due to computational limitations.
\item An additional source of error that creates a difference between theory and practice is that the barycentric coordinates $\lambda_{k}, k=0,1,\dots,d$ need to be estimated since exact computation is again infeasible due to computational limitations.
\end{itemize}
\end{enumerate}
\subsection{Our modification: Convex Smoothed AE-OT model}
We were inspired to develop a generative model which borrows largely from the AE-OT model, but makes improvements to the sample generation method of AE-OT based on the idea of smoothing the convex Brenier potential function $u_{h}$. The idea for smoothing the convex function $u_{h}$ is based on the idea of smoothing non-smooth convex estimators proposed in \citet{Convex}. The smoothed function is convex, Lipschitz continuous and differentiable everywhere, gives rise to an optimal transport map $\hat{T}$ that approximates T and is continuous everywhere. Further, T can be represented by a deep neural network with sufficient expressibility to arbitrary accuracy. We can control the degree of approximation based on an uniform error bound $\epsilon $ that is user-specified and is an interpretable parameter, unlike the tuning parameter $\theta$ discussed here.
The use of $\tilde{T}(\cdot)$ is to primarily smooth the discrete function $T(\cdot)$ and allow us to generate new samples. We are motivated by this idea of smoothing, but we smooth $\hat{u_{h}}$ to remove non-differentiability of the function at certain points. On obtaining the gradient of this smoothed Brenier potential function, we automatically obtain a function capable of transforming any random sample $x$ from the noise distribution $\mu$ into a latent vector $z$ in $\mathcal{Z}$.
We propose to approximate the piecewise affine function $u_{h}(x)=\underset{i=1,2,\dots,n}{\max}x^{T}y_{i}+h_{i}$ or more precisely $u_{\hat{h}(x)}$ ($\hat{h}$ is a the estimate of $h$ obtained using Algorithm 1 of \citet{AEOT}; here we consider that we either know the true $h$ or are able to estimate $h$ using $\hat{h}$ very accurately, so we will refer to $u_{h}$ only in our discussion) by a convex smooth differentiable function to a sufficient degree of accuracy, say $\hat{u_{h}}(x)$. This accuracy is defined by a uniform bound $\epsilon$ on the difference of the true and approximated functions i.e.
$\sup _{\mathrm{x}}\left|u_{h}(x)-\hat{u_{h}}(x)\right| \leq \epsilon$). $\hat{u_{h}}(x)$ can play the role of the Brenier potential function so that the gradient of this approximated function will be the optimal transport map between the noise distribution and an appropriate approximation of the discrete empirical distribution of the embedded latent vectors in the latent space.
Here a question may arise as to whether the gradient $\hat{T}$ of this smooth convex approximated function $\hat{u_h}(x)$ is indeed an optimal transport map, since all functions do not qualify to be optimal transport maps. In this respect we refer to a result in Brenier (\citet{brenier1987polar}), originally proved by Ryff (\citet{ryff1965orbits}), which basically states that any convex function is an optimal transport map between two distributions under quadratic loss. In Section \ref{sec:Theoretical validation of the Convex Smoothed AE-OT model} of this paper, we also investigate how close this OT map $\hat{T}$ is close to the true OT map $T$, even though they are fundamentally different due to the former being a continuous function while the latter being a discontinuous one.
If this can be done, then the semi-discrete optimal transport problem between a continuous noise distribution and a discrete empirical distribution on observed latent codes is now transformed to an optimal transport problem between two continuous distributions, the source distribution being the noise distribution as before but the target distribution is a continuous approximation (hopefully good) of the discrete distribution.
\subsubsection{Justification of the modification}
Following the development in section 3.2 of the paper \cite{Convex} based on the convex optimization theory of \cite{nesterov1998introductory}, we have that
$$u_{h}(x_{j})=\underset{i=1,2,\dots,n}{\max}x_{j}^{T}y_{i}+h_{i}=\underset{\Delta_{n}}{\sup}\sum_{i=1}^{n}w_{i}\left(x_{j}^{T}y_{i}+h_{i}\right)= \underset{\Delta_{n}}{\sup} \langle Az_{j}^{T},\textbf{w} \rangle$$
where $$\Delta_{n}=\lbrace{ \textbf{w}=(w_{1},w_{2},\dots,w_{n}) \in \mathbb{R}^{n}:\sum_{i=1}^{n}w_{i}=1 , w_{i}\geq 0, i=1,2,\dots,n\rbrace}$$ and
$z_{j}$ is the $j$-th row of Z i.e. $z_{j} = (1,x_{j})^{T}$.
We require the notion of a proximity function. A proximity function (or prox function) $\rho(.)$ defined on $\Delta_{n}$ is a continuous strongly convex function with strong convexity parameter $m=1$ i.e. $$\rho(y) \geq \rho(x)+\nabla \rho(x)^{T}(y-x)+\frac{1}{2}\|y-x\|_{2}^{2}$$ for any $x,y \in \Delta_{n}$.
Let us define $\hat{u_{h}}(x;\tau)=\underset{\Delta_{n}}{\sup} \langle Az^{T},\textbf{w}\rangle - \tau\rho\textbf(w)$ where $z=(1,x)^{T}$. Often we will drop reference to $\tau$ when it is understood. The following results are the basis of the proposed method:
\begin{lemma}
For any fixed $\tau>0,$ the function $\hat{u_{h}}(x;\tau)$ is convex and is continuously differentiable in z. Its gradient is given by $\nabla \hat{u_{h}}(x;\tau) =A^{T} \hat{\mathbf{w}}^{\tau},$ where $$\hat{\mathbf{w}}^{\tau} \in \argmax _{\mathbf{w} \in \Delta_{n}}\{\langle Az^{T}, \mathbf{w}\rangle-\tau \rho(\mathbf{w})\} $$ Furthermore, the gradient map $\mathrm{z} \mapsto \nabla \hat{u_{h}}(x;\tau)$ is Lipschitz continuous with parameter $\frac{\|A\|^{2}}{\tau}$.
\end{lemma}
\begin{lemma}
For any $\tau \geq 0,$ the perturbation $\hat{u_{h}}(x;\tau)$ of $\hat{u_{h}}(x;0)=u_{h}(x)$ satisfies the following
uniform bound over z:
$$
u_{h}(x)-\tau \hspace{2pt}\sup _{\mathbf{w} \in \Delta_{n}} \rho(\mathbf{w}) \leq \hat{u_{h}}(x;\tau) \leq \hat{u_{h}}(x;0) = u_{h}(x)
$$
\end{lemma}
We initially test our idea using a particular choice of the proximity function, namely the entropy prox function. The entropy prox function on the unit simplex $\Delta_{n}$ is given by $\rho(\mathbf{w})=$ $\sum_{i=1}^{n} w_{i} \log \left(w_{i}\right)+\log n$.
For the entropy prox function, we are able to obtain a closed form solution for $\hat{u_{h}}(x;\tau)$. We have that
\begin{align}
&\underset{\textbf{w}\in \Delta_{n}}{\argmax} \sum_{i=1}^{n}w_{i}\left(x_{j}^{T}y_{i}+h_{i}\right)-\tau \left(\sum_{i=1}^{n} w_{i} \log \left(w_{i}\right)+\log n\right)\\
&= \left( \frac{\exp c_{1}}{\sum_{i=1}^{n}\exp c_{i}},\frac{\exp c_{2}}{\sum_{i=1}^{n}\exp c_{i}},\dots,\frac{\exp c_{n}}{\sum_{i=1}^{n}\exp c_{i}} \right)
\end{align}
where $c_{i} = \frac{\mathbf{y}_{i}^{T} \mathbf{x}+h_{i}}{\tau}$. Hence we have
$$\hat{u_{h}}(x;\tau)=\tau \log \left(\sum_{i=1}^{n} \exp \left(\frac{\mathbf{y}_{i}^{T} \mathbf{x}+h_{i}}{\tau}\right)\right)-\tau \log n$$ and
$$\nabla \hat{u_{h}}(x;\tau) = \frac{\sum_{i=1}^{n}y_{i}\exp \hspace{2pt}c_{i}}{\sum_{i=1}^{n}\exp \hspace{2pt}c_{i}}$$
Choosing $\tau=\frac{\epsilon}{\log n}$, we get the optimal transport map as $$\hat{T}(x)=\nabla \hat{u_{h}}(x;\tau) = \frac{\sum_{i=1}^{n}y_{i}\exp \hspace{2pt}c_{i}}{\sum_{i=1}^{n}\exp \hspace{2pt}c_{i}}$$
Thus given any noise sample $x$, $\hat{T}(x)$ is the generated latent vector.
\section{Algorithm }\label{sec:Algorithm}
The algorithm to compute the Optimal transport map for our modified AE-OT model is exactly the same as Algorithm \ref{Alg.1} as proposed in \citet{AEOT}. However the algorithm for latent code generation by smoothing the semi-discrete OT map is different and will replace Algorithm 2 (\citet{AEOT}) of the AE-OT methodology (dealing with piecewise linear extension of the Semi-Discrete Optimal Transport Map) for generating new latent codes. We provide the algorithm here as Algorithm \ref{Alg.1} for sake of completeness.
\begin{algorithm}
\caption{Semi-Discrete OT Map}
\label{Alg.1}
\begin{algorithmic}[1]
\Require Latent codes $Y=\left\{y_{i}\right\}_{i \in \mathcal{I}},$ empirical latent code distribution $\nu=\frac{1}{|\mathcal{I}|} \sum_{i \in \mathcal{I}} \delta_{y_{i}},$ number of Monte Carlo samples $N,$ positive integer $s$
\Ensure Optimal transport map $T$ ( ).
\State Initialize $h=\left(h_{1}, h_{2}, \ldots, h_{|\mathcal{I}|}\right) \leftarrow(0,0, \ldots, 0)$
\Repeat
\State Generate $N$ uniformly distributed samples $\left\{x_{j}\right\}_{j=1}^{N}$
\State Calculate $\nabla E=\left(\hat{w}_{i}(h)-\nu_{i}\right)^{T}$
\State Update $h$ by Adam algorithm with $\beta_{1}=0.9, \beta_{2}=0.5$
\State $h= h-\operatorname{mean}(h)$
\If{$E(h)$ has not decreased for $s$ steps}
\State $N \gets N \times 2$
\EndIf
\Until{Converge}
\State OT $\operatorname{map} T(\cdot) \leftarrow \nabla\left(\max _{i}\left(\cdot, y_{i}\right\rangle+h_{i}\right)$
\end{algorithmic}
\end{algorithm}
Let $P$ be a matrix of dimension $n$ $\times$ $d$ (where $n$ is the number of observed latent vectors embedded in the latent space, and $d$ is the dimension of the latent space) having the embedded latent vector $y_{i}$ as its $i$-th row. Let $h=(h_{1},h_{2},\dots,h_{n})$ be the same as in the AE-OT methodology.
We find out the optimal value of $h$ first using Algorithm 1 under the AE-OT methodology as before. One point we would like to mention is that a natural stopping criterion for Algorithm 1 to terminate is when the energy function $E$ either does not change for a few steps, or the successive reductions in its value is very small. However that involves the calculation of $E$ at each step. To avoid the additional computational burden, we can alternatively use the norm of the gradient to specify a stopping criterion. When the value of $E$ is near a local minimum, the gradient should be very small and consequently the norm of the gradient should be close to zero. So we terminate Algorithm 1 when the norm of the gradient is sufficiently small (say less than 0.002).
Having obtained the optimal $h$, we define the matrix A as $A = \left[h,P\right]$ i.e. stacking $h$ and $P$ horizontally. Let us say we want to generate $N$ samples. Then we draw $N$ i.i.d. samples $x_{1},x_{2},\dots,x_{N}$ from the noise distribution and define Q to be a matrix whose $i$-th row is $x_{i}$. We append a column of ones at the left of this matrix Q to obtain $Z$ which is a $N$ $\times$ $d+1$ matrix. Next we obtain $I=AZ^{T}$ which is a $n$ $\times$ $N$ dimensional matrix with $(i,j)$-th element equal to $x_{j}^{T}y_{i}+h_{i}$. Then we obtain $I_{scaled}=\frac{1}{\tau}I$ and apply the softmax function over each column of $I_{scaled}$ to obtain $W$. The softmax function is defined as $$\sigma(\boldsymbol{t})=\left(\frac{\exp \hspace{2pt}t_{1}}{\sum_{i=1}^{n}\exp \hspace{2pt}t_{i}},\frac{\exp \hspace{2pt}t_{2}}{\sum_{i=1}^{n}\exp \hspace{2pt}t_{i}},\dots,\frac{\exp \hspace{2pt}t_{n}}{\sum_{i=1}^{n}\exp \hspace{2pt}t_{i}}\right)$$
Here, W is the matrix of optimized weights with the weights corresponding to the noise sample $x_{j}$ in the $j$-th column of W. Then we obtain $G=A^{T}W$ which gives the matrix of gradients with respect to each column of Z. Removing the first row of G we obtain the matrix of generated samples $X_{gen}$, with the $j$-th column being the generated sample corresponding to $x_{j}$.
The proposed modified algorithm is summarised below in Algorithm \ref{Alg.New}.
\begin{algorithm}
\caption{Generate Latent Code}
\label{Alg.New}
\begin{algorithmic}[1]
\Require 1. Optimal value of $h=(h_{1},h_{2},\dots,h_{n})$ from Algorithm 1 of the AE-OT algorithm\newline
2. Number of samples to generate $N$\newline
3. Noise distribution to sample from: $\nu$\newline
4. Matrix P of dimension $n$ $\times$ $d$ (where $n$ is the number of observed latent vectors embedded in the latent space, and $d$ is the dimension of the latent space) having the embedded latent vector $y_{i}$ as its $i$-th row\newline
5. Uniform error bound on approximating true Brenier potential $\epsilon$
\Ensure Generated latent code $X_{gen}$.
\State Define the matrix A as $A = \left[h,P\right]$ i.e. stacking $h$ and $P$ horizontally.
\For{i in 1,2,\dots,N}
\State Sample $x_{i} \sim \nu$
\EndFor
\State Define Q to be a matrix whose $i$-th row is $x_{i}$ $i=1,2,\dots,N$
\State Append a column of ones at the left of this matrix Q to obtain $Z$, which will be a $N$ $\times$ $d+1$ matrix.
\State Compute $I=AZ^{T}$ which is a $n$ $\times$ $N$ dimensional matrix with $(i,j)$-th element equal to $x_{j}^{T}y_{i}+h_{i}$.
\State Define $\tau=\frac{\epsilon}{\log n}$
\State Compute $I_{scaled}=\frac{1}{\tau}I$
\State Apply the softmax function over each column of $I_{scaled}$ to obtain $W$
\State Compute $G=A^{T}W$.
\State Remove the first row of G to obtain the $d \times N$matrix of generated samples $X_{gen}$, with the $j$-th column being the generated sample corresponding to $x_{j}$, $j=1,2,\dots,N$.
\end{algorithmic}
\end{algorithm}
\subsection{Optimal choice of $\epsilon$}
Algorithm \ref{Alg.New}. requires a user specified hyperparameter $\epsilon$, which represents the uniform error bound on the approximation of the true Brenier potential function $u_{h}(x)$ by the estimate $\widehat{u_{h}(x)}$, since
$$
\sup _{\mathrm{x}}\left|u_{h}(x)-\hat{u_{h}}(x)\right| \leq \epsilon
$$
One might be tempted to choose $\epsilon$ as small as possible, in order to ensure that the error in approximation is minimized. However, such an approach, in the limit when $\epsilon$ tends to 0, will lead to a scenario where, irrespective of the sample $x$ generated from the noise distribution $\mu$, the latent vector $\hat{T}(x)$ will be exactly equal to one of the observed latent vectors $y_{1},y_{2},\dots,y_{n}$. Although this leads to the the generated latents and hence the generated images to have exactly the same distribution as the observed images, it defeats our purpose of generating \say{new} samples. On the other hand, a large value of $\epsilon$ would lead to generation of samples very dissimilar from the observed data.
To mitigate this problem and provide a reasonable choice for $\epsilon$ which provides a trade-off between the two extreme scenarios, one may use the following strategy:
Choose a sequence of $\epsilon$ values, varying from extremely large to extremely small. For each choice of $\epsilon$, generate $n$ samples $t_{1},t_{2},\dots,t_{n}$ based on Algorithms \ref{Alg.1} and \ref{Alg.New}. One then has two discrete (multivariate) distributions in hand: the distribution $\phi=\frac{1}{n}\sum_{l=1}^{n}\delta_{t_{l}}$ of the generated samples and the distribution $\eta=\frac{1}{n}\sum_{l=1}^{n}\delta_{o_{l}}$ of the observed samples.
We assume that the generated and observed samples are drawn from underlying distributions $\mathcal{P}$ and $\mathcal{Q}$, respectively. A statistical test of similarity of these two distributions $\mathcal{P}$ and $\mathcal{Q}$ based on $\phi$ and $\eta$ would provide a measure of similarity between the two distributions, by means of the computed p-value. A very large p-value indicates a large degree of similarity between the two distributions and we expect to obtain such large p-values corresponding to extremely small values of $\epsilon$. On the other hand, a very small p-value will indicate a large degree of dissimilarity between the two distributions, and we expect to obtain such small p-values corresponding to extremely large values of $\epsilon$. We fix a threshold $\alpha$ for the p-value (equivalent to fixing the significance level of the test) to reasonably indicate the point of transition from dissimilarity to similarity of the two distributions based on the sequence of $\epsilon$ values. A reasonable choice of $\epsilon$ would be one which leads to a p-value approximately equal to $\alpha$.
A good and popular choice of a statistical test of equality of multivariate distributions is the Maximum Mean Discrepancy (MMD) Test (\citet{MMDtest}), which has been used for comparing the the generated sample distribution to a reference distribution such as the observed sample distribution in order to assess the performance of generative models like GANs (for e.g. in \citet{ModelMMDtest}). At a high level, the test is based on maximizing the difference between the expectation of a suitable function evaluated on the two datasets separately, and rejecting the null hypothesis of equality if the difference is significantly large. A brief introduction to the MMD test is given following this subsection.
\subsubsection{The Maximum Mean Discrepancy Test}
Let $k$ be the kernel of a reproducing kernel Hilbert space (RKHS) $\mathcal{H}_{k}$ of functions on the space $\mathcal{X}$ of observed and generated samples. $k$ is assumed to be measurable and bounded, $\sup _{x \in \mathcal{X}} k(x, x)<\infty$. The Maximum Mean Discrepancy (MMD) in $\mathcal{H}_{k}$ between the two distributions $\mathcal{P}$ and $\mathcal{Q}$ over $\mathcal{X}$ is defined in the following manner (\citet{MMDtest}):
$$
\operatorname{MMD}_{k}^{2}(\mathcal{P}, \mathcal{Q}):=\mathbb{E}_{t, t^{\prime}}\left[k\left(t, t^{\prime}\right)\right]+\mathbb{E}_{i, i^{\prime}}\left[k\left(i, i^{\prime}\right)\right]-2 \mathbb{E}_{t, i}[k(t, i)]
$$
where $t, t^{\prime} \stackrel{\text { iid }}{\sim} \mathcal{P}$ and $i, i^{\prime} \stackrel{\text { iid }}{\sim} \mathcal{Q} .$
Given the empirical distributions $\phi$ and $\eta$ corresponding to the $t_{l}$'s and the $o_{l}$'s, respectively, an unbiased estimator of $\operatorname{MMD}(\mathcal{P}, \mathcal{Q})$ with nearly minimal variance among unbiased estimators is
$$
\widehat{\mathrm{MMD}}_{\mathrm{U}}^{2}(\phi, \eta):=\frac{1}{\left(\begin{array}{c}
n \\
2
\end{array}\right)} \sum_{l \neq l^{\prime}} k\left(t_{l}, t_{l^{\prime}}\right)+\frac{1}{\left(\begin{array}{c}
n \\
2
\end{array}\right)} \sum_{m \neq m^{\prime}} k\left(o_{m}, o_{m^{\prime}}\right)-\frac{2}{\left(\begin{array}{c}
n \\
2
\end{array}\right)} \sum_{l \neq m} k\left(t_{l}, o_{m}\right)
$$
Following \citet{MMDtest}, we conduct a hypothesis test with null hypothesis $H_{0}: \mathcal{P}=\mathcal{Q}$ and alternative $H_{1}: \mathcal{P} \neq \mathcal{Q},$ using test statistic $n \widehat{\mathrm{MMD}}_{\mathrm{U}}^{2}(\phi, \eta) .$ For the chosen significance level $\alpha,$ we choose a test threshold $c_{\alpha}$ and reject $H_{0}$ if $n \widehat{\mathrm{MMD}}_{\mathrm{U}}^{2}(\phi, \eta)>c_{\alpha}$
Under $H_{0}: \mathcal{P}=\mathcal{Q}, n \widehat{\mathrm{MMD}}_{\mathrm{U}}^{2}(\phi, \eta)$ converges asymptotically to a distribution that depends on the unknown distribution $\mathcal{P}$; we thus cannot evaluate the test threshold $c_{\alpha}$ in closed form. We instead estimate a data-dependent threshold $\hat{c}_{\alpha}$ via permutation, thus using a bootstrap/ permutation test. This gives us a distribution-free test.
Let $T$ and $I$ represent the collection of generated and observed samples respectively. The permutation test involves randomly partitioning the data $T \cup I$ into $T^{\prime}$ and $I^{\prime}$ many times (with $\phi^\prime$ and $\eta^\prime$ representing the corresponding empirical distributions), evaluating $n \widehat{M M D}_{U}^{2}\left(\phi^{\prime}, \eta^{\prime}\right)$ on each split, and estimating the $(1-\alpha)$-th quantile $\hat{c}_{\alpha}$ from these samples.
In practice we use the implementation of the MMD test available in the Python package \texttt{alibi-detect} (\citet{alibi}) (Documentation available at \url{https://docs.seldon.io/projects/alibi-detect/en/stable/methods/mmddrift.html})
\paragraph{Choice of kernel function:} The test requires the choice of a kernel function for comparing the similarity of samples from the generated collection and the observed collection. Many kernels, including the popular Gaussian Radial Basis Function (RBF) kernel, are characteristic, which implies that the MMD is a metric, and in particular that $\operatorname{MMD}_{k}(\mathcal{P},\mathcal{Q})=0$ if and only if $\mathcal{P}=\mathcal{Q},$ so that tests with any characteristic kernel are consistent. The RBF kernel is given by,
$$
k\left(t, i\right)=\exp \left(-\frac{\left\|t- o\right\|^{2}}{2 \sigma^{2}}\right)
$$
However different characteristic kernels will yield different test powers for finite sample sizes. In this paper, we stick to using the RBF kernel, with the kernel bandwidth $\sigma$ chosen as the median of the $L_{2}$ norms of the pairwise differences between the $t_{l}$'s and the $o_{l}$'s i.e. $\sigma = \textrm{median }\left\{\|t_{l}-o_{m}\|_{2};l,m=1,2,\dots,n\right\}$
\paragraph{Obtaining the optimal $\mathbf{\epsilon}$ :} Let the initial sequence of length $s$ of possible $\epsilon$ values be $\epsilon_{1,1},\epsilon_{1,2},\dots,\epsilon_{1,s}$, arranged in increasing order. Corresponding to $\epsilon_{1,l}$ , $l \in {1,2,\dots,s}$, we obtain the collection of generated samples $T_{1,l}$ and the corresponding empirical distribution of generated samples $\phi_{1,l}$. If $\eta$ is the empirical distribution of the observed samples, we perform the permutation test based on the MMD test statistic computed using $\phi_{1,l}$ and $\eta$, and obtain the corresponding p-value $\textrm{\textbf{pval}}_{1,l}$, as described earlier. If any one of the p-values is approximately equal to the chosen significance level $\alpha$, then the corresponding $\epsilon$ value is the optimal choice. Otherwise, assume that there exists $l_{1}^{*},l_{2}^{*} \in {1,2,\dots,s}$ such that $\textrm{\textbf{pval}}_{1,l_{1}^{*}}>\alpha$ and $\textrm{\textbf{pval}}_{1,l_{2}^{*}}<\alpha$. Then the optimal choice of $\epsilon$ i.e. $\epsilon_{opt}$ belongs to the interval $\left(\epsilon_{1,l_{1}},\epsilon_{1,l_{2}}\right)$. Then consider the sequence of $\epsilon$ values $\epsilon_{2,m}=\epsilon_{1,l_{1}^{*}}+m\times\frac{\epsilon_{1,l_{2}^{*}}-\epsilon_{1,l_{1}^{*}}}{10},m=1,2,\dots,10$. Again, we obtain the generated samples $T_{2,m}$ corresponding to the $\epsilon_{2,m}$'s, perform the MMD tests and calculate the p-values $\textrm{\textbf{pval}}_{2,m}$'s. If any one of the p-values is approximately equal to the chosen significance level $\alpha$, then the corresponding $\epsilon$ value is the optimal choice $\epsilon_{opt}$. Otherwise, we continue to proceed in a similar manner till such an $\epsilon$ is obtained. Whether any of the p-values are approximately equal to the desired $\alpha$ value can be checked by specifying a tolerance threshold $\delta$ such that if the p-value \textbf{pval} is such that $|\textrm{pval}_{l}-\alpha|<=\delta$, then the corresponding choice of $\epsilon$ is declared to be the optimal choice. If there are multiple such choices, then any one of them might be used, as it makes little difference in practice.
The proposed procedure is summarised below in Algorithm \ref{Alg. eps}.
\begin{algorithm}[H]
\caption{Obtain Optimal choice of $\epsilon$ and Generated Latent Codes}
\label{Alg. eps}
\begin{algorithmic}[1]
\Require 1. Optimal value of $h=(h_{1},h_{2},\dots,h_{n})$ from Algorithm 1 of the AE-OT algorithm\newline
2. Number of samples to generate $n$\newline
3. Noise distribution to sample from: $\nu$\newline
4. Matrix P of dimension $n$ $\times$ $d$ (where $n$ is the number of observed latent vectors embedded in the latent space, and $d$ is the dimension of the latent space) having the embedded latent vector $y_{i}$ as its $i$-th row\newline
5. List of uniform error bounds on approximating true Brenier potential E=$\left\{\epsilon_{l},l=1,2,\dots,s\right\}$\newline
6. Chosen significance level $\alpha$\newline
7. Tolerance threshold for p-value $\delta$
\Ensure Optimal choice of the uniform error bound $\epsilon_{opt}$ and the corresponding Generated latent code $X_{gen}$.
\State Set $\epsilon_{opt}=0$, $\textrm{\textbf{pval}} = 1$, $\textrm{\textbf{pval}}_{lower} = 1$, $\textrm{\textbf{pval}}_{upper} = 1$
\While{$|\textrm{\textbf{pval}}-\alpha|>\delta$}
\State Set $s=\textrm{card}(E)$
\For{l in 1,2,\dots,s}
\State Run Algorithm \ref{Alg.New} with parameters $h,n,\nu$, P and $\epsilon_{l}$ to generate latent codes $T_{l}$
\State Perform the MMD test based on $T_{l}$ and $I$. Store the p-value obtained $\textrm{pval}_{l}$.
\If{$|\textrm{pval}_{l}-\alpha|<=\delta$}
\State Set $\textrm{\textbf{pval}} = \textrm{pval}_{l}$
\State $\epsilon_{opt}=\epsilon_l$
\State \textbf{break}
\ElsIf{$\textrm{pval}_{l}<\alpha$}
\State Set $\epsilon_{lower} = \epsilon_l$
\Else{$\textrm{ pval}_{l}>\alpha$}
\State Set $\epsilon_{upper} = \epsilon_l$
\EndIf
\EndFor
\If{$|\textrm{pval}_{l}-\alpha|>\delta$}
\State Set E = $\left\{\epsilon_{m}=\epsilon_{lower}+m\times\frac{\epsilon_{upper}-\epsilon_{lower}}{10},m=1,2,\dots,10\right\}$
\EndIf
\EndWhile
\State Run Algorithm \ref{Alg.New} with parameters $h,n,\nu$, P and $\epsilon_{opt}$ to generate latent codes $X_{gen}$
\end{algorithmic}
\end{algorithm}
\section{Theoretical validation of the Convex Smoothed AE-OT model}\label{sec:Theoretical validation of the Convex Smoothed AE-OT model}
Following the discussion in Section \ref{sec:AE-OT to modified AE-OT Framework}, the true Brenier potential map $u(.)$ corresponding to the true optimal transport map $T=\nabla u$ between $\mu$, the noise distribution, and $\nu$, the empirical latent code distribution is parametrized by a \say{height} vector $h=(h_{1},h_{2},\ldots,h_{n})$ and is of the form,
\begin{equation*}\label{true u}
u_{h}(\mathbf{x})=\underset{i=1,2,\dots,n}{\max}\left\{\mathbf{x}^{T}\mathbf{y}_{i}+h_{i}\right\}
\end{equation*}
For a given dataset, once the autoencoder has been trained, $y_{1},y_{2},\dots,y_{n}$ are known constants and the height vector $h=(h_{1},h_{2},\ldots,h_{n})$ is an unknown parameter with linear restriction $\sum_{i=1}^{n}h_{i}=0$.
Based on the entropy prox(imity) function and the theory of smoothing a convex non-smooth function as presented in \citet{nesterov1998introductory}, the smooth approximation of $u_{h}(x)$ is given by
\begin{equation*}\label{estimated u}
\begin{aligned}
\widehat{u_{h}}(\mathbf{x};\tau)&=\tau \log \left(\sum_{i=1}^{n} \exp \left(\frac{\mathbf{x}^{T} \mathbf{y}_{i}+h_{i}}{\tau}\right)\right)-\tau \log n \\
&= \tau \hspace{2pt}\log \left(\sum_{i=1}^{n} \exp c_{i}\right)-\tau \hspace{2pt} \log n
\end{aligned}
\end{equation*}
where $c_{i} = \frac{\mathbf{x}^{T} \mathbf{y}_{i}+h_{i}}{\tau}$ and $\tau$ is a quantity controlling the degree of accuracy of the approximation. Following Equation (21) of \citet{Convex}, we have an uniform error bound as follows:
\begin{equation*}\label{error bound}
\sup _{\mathbf{x}}\left|u_{h}(\mathbf{x})-\widehat{u_{h}}(\mathbf{x};\tau)\right| \leq \tau \hspace{2pt} \log n
\end{equation*}
If the user specified upper bound on $\sup _{\mathbf{x}}\left|u_{h}(\mathbf{x})-\widehat{u_{h}}(\mathbf{x};\tau)\right|$ is $\epsilon$, then $\tau$ is chosen to be any positive real number less than or equal to $\frac{\epsilon}{\log n}$. For definiteness, we set $\tau = \frac{\epsilon}{\log n}$.
Under the assumption that the true height vector $h$ is recovered by Algorithm \ref{Alg.1}, the aforementioned result thus provides a bound on the error of approximation of the true Brenier potential map $u_{h}(x)$ by the smooth approximate Brenier potential map $\widehat{u_{h}}(x;\tau)$ constructed in the convex smoothed AE-OT model, with the bound being a decreasing function of $\tau$, and hence a decreasing function of $\epsilon$. Our objective is to obtain a similar result regarding the accuracy of approximation of the true OT map $\nabla u_{h}(x)$ by the approximate OT map $\nabla \widehat{u_{h}}(x;\tau)$.
In this section, we prove a bound on the error of approximation of the true OT map $\nabla u_{h}(x)$ by the approximate OT map $\nabla \widehat{u_{h}}(x;\tau)$ constructed in the convex smoothed AE-OT model. More specifically, we prove the following result:
\begin{theorem}
Let the $d$-dimensional noise distribution $\mu$ of the Convex Smoothed AE-OT model have convex and bounded support $\mathcal{X}$. Further, assume that Algorithm \ref{Alg.1} is able to recover the true value of the parameter $h$. Then, for any $\mathbf{x} \in \mathcal{X}$,
\begin{equation*}\label{subsec:final result}
\|\nabla u_{h}(\mathbf{x})-\nabla \widehat{u_{h}}(\mathbf{x})\|_{\mathrm{L}^{2}} \leq K \times \left(\log {n}\right)^{1 / 2} \times \tau^{1 / 2}
\end{equation*}
where $K$ is a constant which depends only on $\mathcal{X}$ and is independent of $\tau$.
\end{theorem}
Theorem \ref{subsec:final result} thus proves that the $L^{2}$ norm of the difference between the OT map $\nabla u_{h}(x)$ and the smoothed OT map $\nabla \widehat{u_{h}}(x;\tau)$ is bounded by a decreasing function of the error bound $\tau$, and hence by a decreasing function of $\epsilon$, as shown later by substituting $\tau=\frac{\epsilon}{\log n}$.
In order to prove Theorem \ref{subsec:final result}, we require the following result (Proposition 3.7) from \citet{Gradientdiffnorm}:
\begin{proposition}\label{sec:Proposition for final result}
Let $f$ and $g$ be convex functions on a bounded convex set $\mathcal{X},$ then
\begin{equation*}\label{result}
\|\nabla f-\nabla g\|_{\mathrm{L}^{2}} \leq 2 C_{\mathcal{X}}\|f-g\|_{\infty}^{1 / 2}\left(\|\nabla f\|_{\infty}^{1 / 2}+\|\nabla g\|_{\infty}^{1 / 2}\right)
\end{equation*}
where $C_{\mathcal{X}}$ depends only on $\mathcal{X}$.
\end{proposition}
We now proceed to prove Theorem \ref{subsec:final result}.
\begin{proof}[Proof of Theorem \ref{subsec:final result}]
We observe that both $u_{h}(\mathbf{x})$ and $\widehat{u_{h}}(\mathbf{x};\tau)$ ($\tau>0$) are convex functions. Convexity of $u_{h}(\mathbf{x})$ follows from the fact that $u_{h}(\mathbf{x})$ is a piecewise linear function and the convexity of $\widehat{u_{h}}(\mathbf{x};\tau)$ ($\tau>0$) follows from Section 3 of \citet{Convex}. Further, the domain $\mathcal{X}$ of both $u_{h}(\mathbf{x})$ and $\widehat{u_{h}}(\mathbf{x};\tau)$ ($\tau>0$) is assumed to be convex and bounded. Hence, if we choose $f=u_{h}(\mathbf{x})$ and $g=\widehat{u_{h}}(\mathbf{x};\tau)$, the conditions for applying Proposition \ref{sec:Proposition for final result} are satisfied.
Now, the gradient of $u_{h}(\mathbf{x})$ is given by $$\nabla u_{h}(\mathbf{x}) = y_{m}$$ where $m \in \left\{1,2,\ldots,n\right\}$ is such that $\mathbf{x}^{T}\mathbf{y}_{i}+h_{i}$ is maximized over all $i \in \left\{1,2,\dots,n\right\}$ when $i=m$.
Again, the gradient of $\widehat{u_{h}}(\mathbf{x};\tau)$ is given by
$$
\nabla \widehat{u_{h}}(\mathbf{x};\tau) = \left(\frac{\sum_{i=1}^{n}y_{i1}\exp\hspace{2pt}c_{i}}{\sum_{i=1}^{n}\exp c_{i}},\frac{\sum_{i=1}^{n}y_{i2}\exp c_{i}}{\sum_{i=1}^{n}\exp c_{i}},\dots,\frac{\sum_{i=1}^{n}y_{id}\exp c_{i}}{\sum_{i=1}^{n}\exp c_{i}}\right)
$$ where $\mathbf{y}_{i}=\left(y_{i1},y_{i2},\dots,y_{id}\right)$ is the $i$-th training latent code i.e. the encoding of the $i$-th training data point in the latent space $\mathcal{Z}$. Observe that, for any fixed $\mathbf{x}$, $$ \|\nabla u_{h}(\mathbf{x})\|_{\infty} = \underset{i=1,2,\dots,d}{\max}\left|y_{mi}\right|$$ where $m \in \left\{1,2,\ldots,n\right\}$ is such that $\mathbf{x}^{T}\mathbf{y}_{i}+h_{i}$ is maximized over all $i \in \left\{1,2,\dots,n\right\}$ when $i=m$. Given the training dataset and after the autoencoder has been trained, this is a non-negative constant independent of $\tau$ (and hence $\epsilon$), say $k_{1}$. Further, we have that $k_{1}\leq \underset{i,j=1,2,\dots,d}{\max}\left|y_{ij}\right|=k$, say.
For any fixed $\mathbf{x}$ and $\epsilon$ (hence fixed $\tau$), we have that $$ \|\nabla \widehat{u_{h}}(\mathbf{x;\tau})\|_{\infty} = \underset{k=1,2,\dots,d}{\max}\left|\frac{\sum_{i=1}^{n}y_{ik}\exp c_{i}}{\sum_{i=1}^{n}\exp c_{i}}\right|$$ We note that, for any k, $\frac{\sum_{i=1}^{n}y_{ik}\exp c_{i}}{\sum_{i=1}^{n}\exp c_{i}}$, is a weighted average of $y_{ik}$'s with non-negative weights, and hence must satisfy$$m_{k}=\underset{i=1,2,\dots,d}{min}y_{ik}\leq \frac{\sum_{i=1}^{n}y_{ik}\exp c_{i}}{\sum_{i=1}^{n}\exp c_{i}}\leq \underset{i=1,2,\dots,d}{\max}y_{ik}=M_{k}$$ Then $ \|\nabla \widehat{u_{h}}(\mathbf{x;\tau})\|_{\infty}$ is less than or equal to maximum over all the $\left|m_{k}\right|$'s and $\left|M_{k}\right|$'s, taken together, say $k_{2}$. Given the training dataset and after the autoencoder has been trained, this is a non-negative constant independent of $\tau$ (and hence $\epsilon$). In particular, we observe that $k_{2}\leq \underset{i,j=1,2,\dots,d}{\max}\left|y_{ij}\right|=k$.Thus we obtain that $$\|\nabla u_{h}(\mathbf{x})\|_{\infty}^{1\slash2}+\|\nabla \widehat{u_{h}}(\mathbf{x})\|_{\infty}^{1\slash2}\leq k_{1}^{1\slash2} + k_{2}^{1\slash2}\leq 2k^{1\slash2}$$
Following Lemma 2 in Section 3 of \citet{Convex}, using the inequality, we have that $$u_{h}(\mathbf{x})\geq \nabla \widehat{u_{h}}(\mathbf{x;\tau})$$ for any $\tau \geq 0$.
From Equation \eqref{error bound}, which is a restatement of Equation (21) of \citet{Convex}, we have that $$\|u_{h}(\mathbf{x})-\widehat{u_{h}}(\mathbf{x})\|_{\infty}=\sup _{\mathbf{x}}\left|u_{h}(\mathbf{x})-\widehat{u_{h}}(\mathbf{x};\tau)\right| \leq \tau \log n$$
Based on the above observations and using Proposition \ref{sec:Proposition for final result}, we have, for any $\mathbf{x} \in \mathcal{X}$,
\begin{equation*}\label{final result}
\|\nabla u_{h}(\mathbf{x})-\nabla \widehat{u_{h}}(\mathbf{x})\|_{\mathrm{L}^{2}} \leq 2 C_{\mathcal{X}}\left(\tau \log n\right)^{1 / 2}\times 2k^{1 / 2}=K \times \left(\log n\right)^{1 / 2} \times \tau^{1 / 2}
\end{equation*}
where $C_{\mathcal{X}}$ depends only on $\mathcal{X}$ and is independent of $\tau$ (and hence $\epsilon$), and $K=4C_{\mathcal{X}}k^{1 / 2}$ is a constant independent of $\tau$ (and hence $\epsilon$).
Substituting $\tau=\frac{\epsilon}{log\hspace{2pt}n}$, we have, for any $\mathbf{x} \in \mathcal{X}$
\begin{equation*}\label{last result}
\|\nabla u_{h}(\mathbf{x})-\nabla \widehat{u_{h}}(\mathbf{x})\|_{\mathrm{L}^{2}} \leq \frac{K}{\left(\log n\right)^{1 / 2}} \times \epsilon^{1 / 2}.
\end{equation*}
\end{proof}
Thus as $\epsilon$ decreases, the $L^{2}$ norm of the difference in gradients of the true Brenier potential and the smoothed approximate Brenier potential i.e. the $L^{2}$ norm of the difference between the true OT map $T=\nabla u_{h}(.)$ and the approximate OT map $\hat{T} = \nabla \hat{u_{h}}(.)$ becomes smaller. This implies that for a sufficiently small value of the error bound $\epsilon$, not only the Brenier potentials, but the OT maps themselves becomes closer to each other. However, the important property of continuity of the smoothed OT map is preserved as long as $\epsilon>0$. The choice of $\mathcal{X}$ in Theorem \ref{subsec:final result} is immaterial as long as it is compact and bounded. In particular, we have assumed the noise distribution $\mu$ to be a $d$-dimensional uniform distribution with support $\left[-1,1\right]^{d}$, and hence the theorem applies to the scenario we are interested in the Convex Smoothed AE-OT model.
\section{Experimental results}\label{sec:Experimental Results}
\label{sec:Experimental results}
To validate that our proposed algorithm works in practice, we conduct a series of experiments. We want to study whether our proposed algorithm is able to deal with the problems of mode collapse and mode mixture, and generate high quality samples closely resembling the observed data, with good generalization power i.e. not reconstructing the training data exactly.
First, we compare the performance of both the AE-OT model and our proposed algorithm on two toy datasets consisting of 2-dimensional data points, 2D-ring and 2D-grid, consisting of observations simulated from a mixture of 8-Gaussian and 25-Gaussian distributions, respectively, following the authors of \citet{AEOT}. Descriptions of these datasets are given in the Appendix along with other relevant details regarding the training of the AE-OT model and our proposed model. These are ideal datasets for testing the relative performance of the two models, since both the datasets are multimodal in nature. Since these are 2-dimensional datasets, it does not make sense to embed the data in a latent space and then decode the latent codes to generate new samples i.e. it is not necessary to use an autoencoder. Instead we are able to generate samples directly in this case.
\subsection{2-D Toy Datasets}
\subsubsection{Convex Smoothed AE-OT algorithm performance for optimal $\epsilon$}
We report the results of applying the Convex Smoothed AE-OT on the 2-dimensional datasets 8Gaussian and 25Gaussian.
\begin{itemize}
\item For initially testing out ideas, we ran simulations on 2-D examples similar to what we have done for AE-OT, following Section 3.2 of the paper \citet{Convex} using the entropy prox function (since it has a closed form for $\hat{u_{h}}(x)$ and more importantly for its gradient, thus requiring no additional computational expense).
\item Initial experiments show that the uniform bound on error made in approximating $u_{h}(x)$ by $\hat{u_{h}}(x)$, denoted by $\epsilon$, can be used to specify how closely we want $u_{h}(x)$ to be approximated. There is a very simple dependence of this uniform error bound on the regularization hyperparameter $\tau$ used in the approximation. For the entropy prox function the choice $\tau=\frac{\epsilon}{\log n}$ (n is the number of observed latent vectors) yields a uniform error bound of $\epsilon$.
\item It is observed that mode collapse does not occur for any value of $\epsilon$, and the $\epsilon$ value is inversely related to the degree of mode mixture in the generated samples.
\item For sufficiently small choices of $\epsilon$ (in the order of $10^{-4}$ or less), we observe that the generated samples cover all the modes of the observed data i.e. there is no mode collapse, and there is no mode mixture also. For larger values of $\epsilon$, with the lowering of the accuracy of approximation, mode mixture occurs. So we observe that at least for these 2 datasets, the proposed modification of the AE-OT methodology works very well.
\item We use the proposed procedure of choosing the optimal $\epsilon$ value based on the MMD test, obtaining the \say{best} choice of $\epsilon$ for these 2 datasets to be in the order of $10^{-1}$. Since we observe that mode collapse does not occur for any value of $\epsilon$, we can view the procedure of choosing the optimal $\epsilon$ value as a procedure to decide what constitutes an acceptable level of mode mixture in the generated samples for the dataset at hand.
\end{itemize}
We obtained results for both the datasets varying $\epsilon$ from $10^{-6}$ to $100$, incrementing by a factor of 10. In addition, while choosing the optimal $\epsilon$ using the MMD test, we obtain generate samples corresponding to additional $\epsilon$ values as dictated by Algorithm \ref{Alg. eps}.
Results for the optimal choice of $\epsilon$ for the 8-Gaussian and 25-Gaussian datasets are displayed here, while those corresponding to $\epsilon$ values $10^{-6}$, $10^{-5}$, $10^{-3}$ and $10^{-2}$ are given in the Appendix (\ref{Appendix:Convex_AE-OT_2d}).
Since there are $n=256$ training samples, we generate an equal number of samples using the Convex Smooth AE-OT model, and perform the two-sample permutation test for equality of observed sample distribution and generated sample distribution
based on the MMD test statistic, using 1000 permutations in case of both the datasets. We choose the significance level to be $\alpha=0.05$ in each case and declare an $\epsilon$ value as optimal if it is within $\alpha+\pm \delta$ where we set $\delta=0.01$.
Following this procedure, the optimal choice of $\epsilon$ for the 8-Gaussian dataset based on the two sample MMD test is obtained to be 0.6, while that for the 25-Gaussian dataset is obtained to be 0.8. The corresponding p-values based on the permutation tests were 0.054 in both cases.
The generated samples together with the observed data are as follows:
\begin{figure}[H]
\begin{subfigure}{0.8\textwidth}
\includegraphics[width=\textwidth]{8gaussians_kernel_best_eps_0.6.png}
\caption{8Gaussians Dataset $\epsilon=0.6$}
\label{8gaussianeps0.6}
\end{subfigure}
\begin{subfigure}{0.8\textwidth}
\includegraphics[width=\textwidth]{25gaussians_kernel_best_eps_0.8.png}
\caption{25Gaussians Dataset $\epsilon=0.8$}
\label{25gaussianeps0.8}
\end{subfigure}
\end{figure}
\textbf{Comments:} We observe that the generated samples cover all the modes of the observed data and hence the phenomenon of mode collapse is mitigated here. Further, all the generated samples are mixtures of two nearest modes and falls close to the approximate manifold defined by the observed data.
\subsubsection{AE-OT results for varying $\theta$}
To appreciate the efficacy of the Convex Smoothed AE-OT model, it is required to compare its performance with that of the AE-OT model itself on the 8-Gaussian and 25-Gaussian datasets. We provide the results obtained using the AE-OT model along with the relevant discussions in the Appendix (\ref{Appendix:AE-OT_2d}).
\section{Conclusion}\label{sec:Conclusion}
As seen in the Experimental results section (Section \ref{sec:Experimental results}), our proposed generative model - Convex Smoothed AE-OT, produces affirmative results. We improve upon the original AE-OT model (\citet{AEOT}) with regards to its sample generation algorithm, while ensuring that mode collapse is absent and mode mixture is present only upto an allowable level in the generated samples. In addition to empirically validating the efficacy of the proposed model, we provide a theoretical justification for the approximated OT map $\hat{T}$ for being close to the true OT map $T$. Our current efforts are aimed at applying the Convex Smoothed AE-OT model to benchmark Image datasets and evaluating its performance.
\section*{Acknowledgements}
The author is extremely grateful to Prof. Bodhisattva Sen for guiding her in the development of this paper.
\bibliographystyle{chicago}
|
1,108,101,563,250 | arxiv | \section{Introduction and statement of the result}
The local discrepancy $\Delta(\alpha,\beta,\mathcal{P})$ of an $N$-element point set $\mathcal{P}=\{\boldsymbol{x}_0,\dots,\boldsymbol{x}_{N-1}\}$ in the unit square $\left[0,1\right)^2$ is defined as
\[ \Delta(\alpha,\beta,\mathcal{P})=A(\left[0,\alpha\right)\times \left[0,\beta\right),\mathcal{P})-N\alpha\beta \]
for $\alpha, \beta \in \left(0,1\right]$. In this definition $A(\left[0,\alpha \right)\times \left[0,\beta\right),\mathcal{P})$ is the number of indices $0\leq n\leq N-1$ satisfying $\boldsymbol{x}_n \in \left[0,\alpha\right)\times \left[0,\beta\right)$. The $L_p$ discrepancy of a point set $\mathcal{P}$ in $\left[0,1\right)^2$ is defined as
\[L_p(\mathcal{P})=\frac{1}{N}\left(\int_{0}^{1}\int_{0}^{1} |\Delta(\alpha,\beta,\mathcal{P})|^p \,\mathrm{d}\alpha \,\mathrm{d}\beta\right)^{\frac{1}{p}} \]
for $p\in[1,\infty)$. For $p\to \infty$ we obtain the notable star discrepancy. In this work we do not study this kind of discrepancy directly, but it should be mentioned that there is a remarkable asymptotic relation between the $L_p$ discrepancy and the star discrepancy (see \cite{Hor}). The $L_p$ discrepancy is a quantitative measure for the irregularity of distribution of a point set $\mathcal{P}$ in $[0,1)^2$, see e.g. \cite{DT97,kuinie,mat}. It is also related to the worst-case integration error of a quasi-Monte Carlo rule, see e.g. \cite{DP10,LP14,Nied92}. It is well known that for every $p\in [1,\infty)$ there exists a constant $c_p > 0$ with the following
property: for the $L_p$ discrepancy of any point set $\mathcal{P}$ consisting of $N$ points in $[0, 1)^2$ we
have
\begin{equation} \label{roth} L_p(\mathcal{P}) \geq c_p \frac{\sqrt{\log{N}}}{N}, \end{equation}
where $\log$ denotes the natural logarithm.
This was first shown by Roth \cite{Roth} for $p = 2$ and hence for all $p \in [2,\infty)$ and later by
Schmidt \cite{schX} for all $p\in(1,2)$. The case $p=1$ was verified by Hal\'{a}sz \cite{hala}.
Here we consider digit shifted Hammersley point sets. Let therefore $m$ be a positive integer and $\boldsymbol{\sigma}=(\sigma_1,\sigma_2,\dots,\sigma_m) \in \{0,1\}^m$ a dyadic shift. We define the point set
$$ \mathcal{H}_m(\boldsymbol{\sigma}):=\left\{\left(\frac{t_m}{2}+\frac{t_{m-1}}{2^2}+\dots+\frac{t_1}{2^m},\frac{s_1}{2}+\frac{s_2}{2^2}+\dots+\frac{s_m}{2^m}\right):
t_1,\dots,t_m \in \{0,1\}\right\}, $$
where $s_j=t_j \oplus \sigma_j$ for all $j\in \{1,\dots,m\}$ (the operation $\oplus$ denotes addition modulo 2). The point set $\mathcal{H}_m(\boldsymbol{\sigma})$ contains $2^m$ elements. We obtain the classical Hammersley point set $\mathcal{H}_m$ with $2^m$ points by choosing $\boldsymbol{\sigma}=(0,0,\dots,0)$. Additionally, we define
the $m$-tuple $\boldsymbol{\sigma}^{\ast}=(\sigma_1^{\ast},\sigma_2^{\ast},\dots,\sigma_m^{\ast})$ by $\sigma_j^{\ast}=\sigma_j \oplus 1$ for all
$j\in \{1,\dots,m\}$. Then we introduce the symmetrized Hammersley point set $\mathcal{H}_m^{{\rm sym}}(\boldsymbol{\sigma})$ as
$$ \mathcal{H}_m^{{\rm sym}}(\boldsymbol{\sigma}):=\mathcal{H}_m(\boldsymbol{\sigma})\cup \mathcal{H}_m(\boldsymbol{\sigma}^{\ast}). $$
This point set has $2^{m+1}$ elements and can be regarded as symmetrized, since $\mathcal{H}_m^{{\rm sym}}(\boldsymbol{\sigma})$ may also
be written as the union of $\mathcal{H}_m(\boldsymbol{\sigma})$ with the point set
$$ \left\{ \left(x,1-\frac{1}{2^m}-y\right): (x,y)\in \mathcal{H}_m(\boldsymbol{\sigma}) \right\}. $$
Figure~\ref{examples} shows examples of two symmetrized Hammersley point sets.
\begin{figure}[ht] \label{examples}
\centering
{\includegraphics[width=60mm]{shift00000000.pdf}}
\hspace{.1in}
{\includegraphics[width=60mm]{shift01010101.pdf}}
\caption{The symmetrized Hammersley point sets $\mathcal{H}_8^{{\rm sym}}(\boldsymbol{\sigma}_i)$ for $i=1,2$, where $\boldsymbol{\sigma}_1=(0,0,0,0,0,0,0,0)$ and $\boldsymbol{\sigma}_2=(0,1,0,1,0,1,0,1)$. The $L_2$ discrepancy is $0.00255571\dots$ in both cases.}
\label{f1}
\end{figure}
The concept of symmetrizing point sets plays an important role in finding point sets with the optimal order
of $L_p$ discrepancy in the sense of \eqref{roth}. Davenport \cite{daven} used this method in 1956 to construct for the first
time a two-dimensional point set with an $L_2$ discrepancy of order $\mathcal{O}\left(\sqrt{\log{N}}/N\right)$, and therefore showing that the lower bound \eqref{roth} is sharp for $p=2$. For this reason, the symmetrization method we use here is often referred to as Davenport's reflection principle.
It is known that $L_p(\mathcal{H}_m)$ is only of order $\mathcal{O}((\log{N})/N)$ for all $p\in[1,\infty)$ (see \cite{Pill}). However, in \cite[Theorem 2]{HKP14} it was shown with tools from harmonic analysis (the Haar function system and the Littlewood-Paley inequality) that the symmetrized Hammersley point set achieves an $L_p$ discrepancy of order $\mathcal{O}\left(\sqrt{\log{N}}/{N}\right)$
for all $p\in [1,\infty)$ independently of the shift $\boldsymbol{\sigma}$. This order is best possible in the sense of \eqref{roth}. For the case $p=2$, this result follows already from \cite[Theorem 2]{lp} for the slightly different definition of a symmetrization of the classical Hammersley point set $\mathcal{H}_m$, namely
$$ \widetilde{\mathcal{H}}_m^{{\rm sym}}:=\mathcal{H}_m \cup \left\{ \left(x,1-y\right): (x,y)\in \mathcal{H}_m \right\}. $$
The previously mentioned results have the drawback that they do not deliver an exact value for the implied constant of the leading term of the $L_2$ discrepancy. The aim of this paper is to show an exact formula for the $L_2$ discrepancy of $\mathcal{H}_m^{{\rm sym}}(\boldsymbol{\sigma})$, which gives not only a concrete constant, but also demonstrates that $L_2(\mathcal{H}_m^{{\rm sym}}(\boldsymbol{\sigma}))$ solely depends on the number of elements $N=2^{m+1}$ and not on the shift $\boldsymbol{\sigma}$ whatsoever.
\begin{theorem} \label{Theo} Let $m\in\mathbb{N}$ and $\boldsymbol{\sigma} \in \{0,1\}^m$. Then we have
$$ (2^{m+1}L_2(\mathcal{H}_m^{{\rm sym}}(\boldsymbol{\sigma})))^2=\frac{m}{24}+\frac{11}{8}+\frac{1}{2^{m}}-\frac{1}{9\cdot 2^{2m+1}}, $$
which can be displayed in terms of the number of elements $N=2^{m+1}$ as
$$ L_2(\mathcal{H}_m^{{\rm sym}}(\boldsymbol{\sigma}))=\frac{1}{N}\left(\frac{\log{N}}{24\log{2}}+\frac{4}{3}+\frac{2}{N}-\frac{2}{9N^2}\right)^{\frac12}.
$$
\end{theorem}
We derive the following corollary on the point set $\widetilde{\mathcal{H}}_m^{{\rm sym}}(\boldsymbol{\sigma})$ defined as the union of $\mathcal{H}_m(\boldsymbol{\sigma})$
with the point set
$ \left\{ \left(x,1-y\right): (x,y)\in \mathcal{H}_m(\boldsymbol{\sigma}) \right\}. $
This point set also has $2^{m+1}$ elements, where some points might coincide.
\begin{corollary} Let $m\in\mathbb{N}$ and $\boldsymbol{\sigma} \in \{0,1\}^m$. Then we have with $N=2^{m+1}$
$$ L_2(\widetilde{\mathcal{H}}_m^{{\rm sym}}(\boldsymbol{\sigma}))=\frac{1}{N}\sqrt{\frac{\log{N}}{24\log{2}}}+\mathcal{O}\left(\frac{1}{N}\right). $$
\end{corollary}
\begin{proof} From \cite[Lemma 4]{HKP14} we have the relation
$$ \left| L_2(\widetilde{\mathcal{H}}_m^{{\rm sym}}(\boldsymbol{\sigma}))-L_2(\mathcal{H}_m^{{\rm sym}}(\boldsymbol{\sigma})) \right|\leq \frac{1}{2^{m+1}}=\frac{1}{N}. $$
Together with Theorem~\ref{Theo} this inequality yields the result.
\end{proof}
The proof of Theorem~\ref{Theo} relies strongly on techniques developed and employed in the papers \cite{Kri2,Kri1,Lar,Pill}. The methods and results of \cite{Kri2}, where the $L_2$ discrepancy of $\mathcal{H}_m(\boldsymbol{\sigma})$ was computed exactly,
are particularly important in order to prove the theorem. We comment on those results in Remark~\ref{remrem}, Lemma~\ref{l2shift} and Remark~\ref{optimalshift}. The fact that we can write the symmetrized Hammersley point set as a union of two shifted Hammersley point sets allows us to employ the same techniques in this paper. The reader is invited to compare Theorem~\ref{Theo} to the result of Kritzer and Pillichshammer as stated in Lemma~\ref{l2shift}.
\begin{remark} \rm \label{remrem} Theorem~\ref{Theo} shows that we cannot expect a lower $L_2$ discrepancy by first shifting the classical Hammersley point set and then symmetrizing it. We can therefore simply symmetrize the classical Hammersley point set itself. This is a remarkably easy construction of a point set with very low $L_2$ discrepancy. However, the coefficient of the leading term $\sqrt{\log{N}}/N$ of $L_2(\mathcal{H}_m^{{\rm sym}}(\boldsymbol{\sigma}))$ is $\sqrt{1/(24\log{2})}\approx 0.2451\dots$, which is slightly higher than for the shifted Hammersley point set
$\mathcal{H}_m(\boldsymbol{\sigma})$ under the condition that the number of ones and zeros in $\boldsymbol{\sigma}$ is more or less balanced. In this case $\mathcal{H}_m(\boldsymbol{\sigma})$ achieves an $L_2$ discrepancy of optimal order of magnitude in $N$ as shown by Kritzer and Pillichshammer in \cite{Kri2, Kri1}. The coefficient of the leading term of $L_2(\mathcal{H}_m(\boldsymbol{\sigma}))$ is then $\sqrt{5/(192\log{2})}\approx 0.1938\dots$ (see also Lemma~\ref{l2shift} and Remark~\ref{optimalshift}). The smallest known leading constant is achieved for the $L_2$ discrepancy of digit scrambled Hammersley point sets in base $22$ and has the value $\sqrt{278629/(2811072\log{22})}\approx 0.1790\dots$, as shown in \cite{FPPS09}.
\end{remark}
\begin{remark} \rm A further exact formula for the $L_2$ discrepancy of a symmetrized point set was discovered in
\cite{bil}. There the authors considered symmetrized Fibonacci lattice point sets, whose $L_2$ discrepancy is also of order $\mathcal{O}\left(\sqrt{\log{N}}/N\right)$. The leading term has a complicated form, but numerical results indicate that the $L_2$ discrepancy of these point sets has a constant around $0.176\dots$ This would be slightly better than the result for digit scrambled Hammersley point sets mentioned in Remark~\ref{remrem}.
\end{remark}
\section{Auxiliary results}
Throughout this paper, we call a real number $\alpha \in \left[0,1\right)$ $m$-bit if it is contained in the set
$\mathbb{Q}(2^m):=\{0,\frac{1}{2^m},\dots,\frac{2^m-1}{2^m}\}$. Hence, $\alpha$ is of the form $\alpha=\frac{\alpha_1}{2}+\dots+\frac{\alpha_m}{2^m}$, where $\alpha_j \in \{0,1\}$ for all $j \in \{1,\dots,m\}$. We also set $\mathbb{Q}^{\ast}(2^m):=\mathbb{Q}(2^m)\setminus \{0\}$. We write $\Delta_1(\alpha,\beta)$ for the local discrepancy of $\mathcal{H}_m(\boldsymbol{\sigma})$, $\Delta_2(\alpha,\beta)$ for the local discrepancy of $\mathcal{H}_m(\boldsymbol{\sigma}^{\ast})$ and $\Delta_{{\rm sym}}(\alpha,\beta)$ for the local discrepancy of $\mathcal{H}_m^{{\rm sym}}(\boldsymbol{\sigma})$.
The first lemma, which gives an exact formula for the local discrepancy of $\mathcal{H}_m(\boldsymbol{\sigma})$, can be derived
from a result of Larcher and Pillichshammer in \cite{Lar} and was first stated explicitely in \cite[Lemma 1]{Kri1}. The second assertion in this lemma is a consequence of the fact that the components of all elements in $\mathcal{H}_m(\boldsymbol{\sigma})$ are $m$-bit as it has already been pointed out in \cite[Remark 3]{Kri1}.
Here and in the following, $\|x\|:=\min_{z \in \mathbb{Z}}|x-z|$ denotes the distance to the nearest integer of a real number $x$.
\begin{lemma} \label{AllgemeineFormel} For the local discrepancy $\Delta(\alpha,\beta)$ of $\mathcal{H}_m(\boldsymbol{\sigma})$ we have
\begin{enumerate}
\item $ \Delta(\alpha,\beta)=\sum_{u=0}^{m-1} \|2^u \beta\|(-1)^{\sigma_{u+1}} (\alpha_{m-u}\oplus\alpha_{m+1-j(u)})$ for $m$-bit numbers $\alpha=\frac{\alpha_1}{2}+\dots+\frac{\alpha_m}{2^m}$ and $\beta=\frac{\beta_1}{2}+\dots+\frac{\beta_m}{2^m}$ (we set $\alpha_{m+1}=0$), where $j(u)$ for $0\leq u\leq m-1$ is defined as
\[ j(u)=\begin{cases}
0 & \text{if } u=0, \\
0 & \text{if } \alpha_{m+1-j}=\beta_j \oplus \sigma_j \text{\, for \,} j=1,\dots,u, \\
\max\{j \leq u: \alpha_{m+1-j} \neq \beta_j \oplus \sigma_j \} & \text{otherwise.}
\end{cases}
\]
\item $\Delta(\alpha,1)=0$ for $m$-bit $\alpha$ and $ \Delta(\alpha,\beta)=\Delta(\alpha(m),\beta(m))+2^m(\alpha(m)\beta(m)-\alpha\beta)$ for arbitrary $\alpha,\beta \in (0,1]$, where $\alpha(m)$ and $\beta(m)$ are the smallest $m$-bit numbers greater than or equal to $\alpha$ or $\beta$, respectively. (For $\alpha, \beta > 1-2^{-m}$ we choose $\alpha(m)=1$ and $\beta(m)=1$, respectively.)
\end{enumerate} \end{lemma}
\begin{lemma} \label{symlemma} For all $\alpha,\beta\in (0,1]$ we have $ \Delta_{{\rm sym}}(\alpha,\beta)=\Delta_1(\alpha,\beta)+\Delta_2(\alpha,\beta). $
\end{lemma}
\begin{proof}
We have
\begin{align*}
\Delta_{{\rm sym}}(\alpha,\beta)=& A(\left[0,\alpha\right)\times \left[0,\beta\right),\mathcal{H}_m^{{\rm sym}}(\boldsymbol{\sigma}))-2^{m+1}\alpha\beta \\
=& A(\left[0,\alpha\right)\times \left[0,\beta\right),\mathcal{H}_m(\boldsymbol{\sigma}))+ A(\left[0,\alpha\right)\times \left[0,\beta\right),\mathcal{H}_m(\boldsymbol{\sigma}^{\ast}))
-2^m\alpha\beta-2^m\alpha\beta \\
=& \Delta_1(\alpha,\beta)+\Delta_2(\alpha,\beta)
\end{align*}
for all $\alpha,\beta\in (0,1]$.
\end{proof}
Throughout the next lemma, we always write $j_1(u)$ if the function $j(u)$ appearing in the first part of Lemma~\ref{AllgemeineFormel} refers to $\Delta_1(\alpha,\beta)$ and $j_2(u)$ if it refers to $\Delta_2(\alpha,\beta)$.
\begin{lemma} \label{Alphalemma}
Let $\alpha=\frac{\alpha_1}{2}+\dots+\frac{\alpha_m}{2^m}$ and $\beta=\frac{\beta_1}{2}+\dots+\frac{\beta_m}{2^m}$ be $m$-bit.
\begin{enumerate}
\item For $u_1,u_2 \in \{0,\dots,m-1\}$ with $u_1\neq u_2$ we have
$$ \sum_{\alpha\in\mathbb{Q}^{\ast}(2^m)}(\alpha_{m-u_1}\oplus \alpha_{m+1-j_1(u_1)})(\alpha_{m-u_2}\oplus \alpha_{m+1-j_2(u_2)})=2^{m-2}. $$
\item For $u \in \{0,\dots,m-1\}$ we have
\begin{align*} \sum_{\alpha\in\mathbb{Q}^{\ast}(2^m)}&(\alpha_{m-u}\oplus \alpha_{m+1-j_1(u)})(\alpha_{m-u}\oplus \alpha_{m+1-j_2(u)}) \\
=& \begin{cases} 2^{m-u-1} & \mbox{if } u\in \{0,1\}, \\
2^{m-u-1}\left(1+\sum_{j=1}^{u-1}2^j((\gamma_j \oplus 1)\gamma_u+\gamma_j (\gamma_u \oplus 1))\right) & \mbox{if } u\in\{2,\dots,m-1\}.
\end{cases} \end{align*}
In the last expression, we define $\gamma_j:=\beta_j\oplus \sigma_j$ for all $j\in\{1,\dots,m-1\}$.\end{enumerate}
\end{lemma}
\begin{proof}
We mention that Pillichshammer showed in \cite[Lemma 2]{Pill} the formula
$$ \sum_{\alpha\in\mathbb{Q}^{\ast}(2^m)}\prod_{i=1}^{k}\left(\alpha_{m-u_i}\oplus \alpha_{m+1-j(u_i)}\right)=2^{m-k} $$
for an integer $1\leq k \leq m-1$ and numbers $u_1,\dots,u_k\in \{0,\dots,m-1\}$ with $u_i\neq u_j$ for $1\leq i\neq j \leq k$, where $j(u)$ refers to the local discrepancy of the classical Hammersley point set.
By studying his proof, one sees that the argumentation does not change at all if we replace some of the $j(u_i)$
appearing in the formula by $j_1(u_i)$ and the others by $j_2(u_i)$, and thus we obtain the claimed identity stated in the first item of this lemma by choosing $k=2$ and replacing $j(u_1)$ by $j_1(u_1)$ and $j(u_2)$ by $j_2(u_2)$.
We show the second item.
For $u=0$ we have $j_1(u)=0$ and $j_2(u)=0$ by definition and hence
\begin{align*}
\sum_{\alpha\in\mathbb{Q}^{\ast}(2^m)}(\alpha_m\oplus\alpha_{m+1})(\alpha_m\oplus\alpha_{m+1})=\sum_{\alpha_1,\dots,\alpha_m=0}^{1}\alpha_m= \sum_{\alpha_1,\dots,\alpha_{m-1}=0}^{1}1=2^{m-1}=2^{m-u-1}.
\end{align*}
If $u=1$, we use the fact that $j_1(1)$ and $j_2(1)$ only depend on $\alpha_m$ and write
\begin{align*}
\sum_{\alpha\in\mathbb{Q}^{\ast}(2^m)}&(\alpha_{m-1}\oplus\alpha_{m+1-j_1(1)})(\alpha_{m-1}\oplus\alpha_{m+1-j_2(1)}) \\
=& \sum_{\alpha_m=0}^{1}\left(\sum_{\alpha_1,\dots,\alpha_{m-1}=0}^{1}(\alpha_{m-1}\oplus\alpha_{m+1-j_1(1)})(\alpha_{m-1}\oplus\alpha_{m+1-j_2(1)})\right) \\
=& 2^{m-2}\sum_{\alpha_m=0}^{1}\left(\alpha_{m+1-j_1(1)}\alpha_{m+1-j_2(1)}+(\alpha_{m+1-j_1(1)}\oplus 1)(\alpha_{m+1-j_2(1)}\oplus 1)\right).
\end{align*}
We have to distinguish between the cases $\alpha_m=\gamma_1$ and $\alpha_m=\gamma_1\oplus 1$. In the first case we obviously have
$j_1(1)=0$ and $j_2(1)=1$ whereas in the second case we have $j_1(1)=1$ and $j_2(1)=0$. We conclude
\begin{align*}
2^{m-2}&\sum_{\alpha_m=0}^{1}\left(\alpha_{m+1-j_1(1)}\alpha_{m+1-j_2(1)}+(\alpha_{m+1-j_1(1)}\oplus 1)(\alpha_{m+1-j_2(1)}\oplus 1)\right) \\
=& 2^{m-2}\sum_{\alpha_m=\gamma_1}(\alpha_m\oplus 1)+2^{m-2}\sum_{\alpha_m=\gamma_1\oplus 1}(\alpha_m\oplus 1) \\
=& 2^{m-2}(\gamma_1\oplus 1)+2^{m-2}\gamma_1=2^{m-2}=2^{m-u-1}.
\end{align*}
We turn to the case $u \in \{2,\dots,m-1\}$. Since $j_1(u)$ and $j_2(u)$ only depend on $\alpha_{m+1-u},\dots,\alpha_m$
but not on $\alpha_1,\dots,\alpha_{m-u}$, we observe that
\begin{align*}
&\sum_{\alpha\in\mathbb{Q}^{\ast}(2^m)} (\alpha_{m-u}\oplus \alpha_{m+1-j_1(u)})(\alpha_{m-u}\oplus \alpha_{m+1-j_2(u)}) \\
&= \sum_{\alpha_1,\dots,\alpha_m=0}^{1}(\alpha_{m-u}\oplus \alpha_{m+1-j_1(u)})(\alpha_{m-u}\oplus \alpha_{m+1-j_2(u)}) \\
=&\sum_{\alpha_{m+1-u},\dots,\alpha_{m}=0}^{1}\left(\sum_{\alpha_1,\dots,\alpha_{m-u}=0}^{1}(\alpha_{m-u}\oplus \alpha_{m+1-j_1(u)})(\alpha_{m-u}\oplus \alpha_{m+1-j_2(u)})\right) \\
=& 2^{m-u-1}\sum_{\alpha_{m+1-u},\dots,\alpha_{m}=0}^{1}\left(\alpha_{m+1-j_1(u)}\alpha_{m+1-j_2(u)}+(\alpha_{m+1-j_1(u)}\oplus 1)(\alpha_{m+1-j_2(u)}\oplus 1)\right) \\
=& 2^{m-u-1}\sum_{j_1=0}^{u-1}\sum_{\substack{\alpha_{m+1-u},\dots,\alpha_{m}=0 \\ j_1(u)=j_1}}^{1}\left(\alpha_{m+1-j_1}\alpha_{m+1-j_2(u)}+(\alpha_{m+1-j_1}\oplus 1)(\alpha_{m+1-j_2(u)}\oplus 1)\right) \\
&+ 2^{m-u-1}\sum_{j_2=0}^{u-1}\sum_{\substack{\alpha_{m+1-u},\dots,\alpha_{m}=0 \\ j_2(u)=j_2}}^{1}\left(\alpha_{m+1-j_1(u)}\alpha_{m+1-j_2}+(\alpha_{m+1-j_1(u)}\oplus 1)(\alpha_{m+1-j_2}\oplus 1)\right) \\
=:&\, T_1+T_2.
\end{align*}
One might wonder why the sums over $j_1$ and $j_2$ end in $u-1$ instead of $u$ and why they do not coincide. The reason is that $j_1(u)\in \{0,\dots,u-1\}$ implies $j_2(u)=u$ and $j_2(u)\in \{0,\dots,u-1\}$ implies $j_1(u)=u$. This can be seen as follows: $j_1(u)\in \{0,\dots,u-1\}$ implies $a_{m+1-u}=\gamma_u$, because otherwise we would have $j_1(u)=u$. But from the fact that $a_{m+1-u}=\gamma_u\neq \gamma_u\oplus 1$, we immediately derive $j_2(u)=u$. The other way round can be explained analogously. This means that the case $j_2(u)=u$ is actually contained in the sum over $j_1$ and reversely. We find
\begin{align*}
T_1=& 2^{m-u-1}\sum_{\substack{\alpha_{m+1-u}=\gamma_u \\ \vdots \\ \alpha_{m-1}=\gamma_{2} \\ \alpha_{m}=\gamma_{1}}}\left(\alpha_{m+1}\alpha_{m+1-j_2(u)}+(\alpha_{m+1}\oplus 1)(\alpha_{m+1-j_2(u)}\oplus 1)\right) \\
&+ 2^{m-u-1}\sum_{j_1=1}^{u-1}\sum_{\alpha_{m-j_1+2},\dots,\alpha_{m}=0}^{1} \\ &\sum_{\substack{\alpha_{m+1-u}=\gamma_u \\ \vdots \\ \alpha_{m-j_1}=\gamma_{j_1+1} \\ \alpha_{m+1-j_1}=\gamma_{j_1}\oplus1}}\left(\alpha_{m+1-j_1}\alpha_{m+1-j_2(u)}+(\alpha_{m+1-j_1}\oplus 1)(\alpha_{m+1-j_2(u)}\oplus 1)\right) \\
=& 2^{m-u-1}\left(\gamma_u \oplus 1\right)
+ 2^{m-u-1}\sum_{j_1=1}^{u-1}2^{j_1-1}\left((\gamma_{j_1}\oplus1)\gamma_u+\gamma_{j_1}(\gamma_u\oplus 1)\right).
\end{align*}
Similarly we argue that
$$ T_2=2^{m-u-1}\gamma_u
+ 2^{m-u-1}\sum_{j_2=1}^{u-1}2^{j_2-1}\left((\gamma_{j_2}\oplus1)\gamma_u+\gamma_{j_2}(\gamma_u\oplus 1)\right). $$
Adding $T_1$ and $T_2$ completes the proof of the second item of this lemma.
\end{proof}
\begin{lemma} \label{Betalemma} Let $\beta$ be $m$-bit.
\begin{enumerate}
\item For $u_1,u_2 \in \{0,\dots,m-1\}$ with $u_1\neq u_2$ we have
$$ \sum_{\beta\in\mathbb{Q}^{\ast}(2^m)}\|2^{u_1}\beta\|\|2^{u_2}\beta\|=\frac{2^m}{2^4}. $$
\item For $u \in \{0,\dots,m-1\}$ we have
$$ \sum_{\beta\in\mathbb{Q}^{\ast}(2^m)}\|2^{u}\beta\|^2=\frac{2^{2m}+2^{2u+1}}{3\cdot 2^{m+2}}. $$
\end{enumerate}
\end{lemma}
\begin{proof} The first formula follows from \cite[Lemma 3 a)]{Pill} and the second one
is \cite[Lemma 3 b)]{Pill}.
\end{proof}
We introduce the parameter $l=l(\boldsymbol{\sigma}):=|\{i\in\{1,\dots,m\}:\sigma_i=0\}|$, i. e. $l$ is the number of components of $\boldsymbol{\sigma}$ which are equal to zero. We use this notation for the rest of this paper.
\begin{lemma} \label{gemischt}
We have
$$ \frac{1}{2^{2m}}\sum_{\alpha,\beta\in\mathbb{Q}^{\ast}(2^m)}\Delta_1(\alpha,\beta)\Delta_2(\alpha,\beta)=-\frac{m^2}{64}-\frac{l^2}{16}+\frac{lm}{16}-\frac{m}{192}-\frac{5}{144}-\frac{1}{9\cdot 2^{2m+2}}. $$
\end{lemma}
\begin{proof}
In this proof we write for the sake of simplicity
$ A(\alpha,\beta,\boldsymbol{\sigma},u):=\alpha_{m-u}\oplus \alpha_{m+1-j(u)},$
where we emphasize the dependence of $j(u)$ on $\alpha$, $\beta$ and $\boldsymbol{\sigma}$.
With the first point of Lemma~\ref{AllgemeineFormel} we get
\begin{align*}
\frac{1}{2^{2m}}&\sum_{\alpha,\beta\in\mathbb{Q}^{\ast}(2^m)}\Delta_1(\alpha,\beta)\Delta_2(\alpha,\beta) \\
=&
\frac{1}{2^{2m}}\sum_{\alpha,\beta\in\mathbb{Q}^{\ast}(2^m)}\left(\sum_{u_1=0}^{m-1}\|2^{u_1}\beta\|(-1)^{\sigma_{u_1+1}}A(\alpha,\beta,\boldsymbol{\sigma},u_1)\right) \\ &\hspace{100pt}\times
\left(\sum_{u_2=0}^{m-1}\|2^{u_2}\beta\|(-1)^{\sigma_{u_2+1}^{\ast}}A(\alpha,\beta,\boldsymbol{\sigma}^{\ast},u_2)\right) \\
=&
-\frac{1}{2^{2m}}\sum_{\alpha,\beta\in\mathbb{Q}^{\ast}(2^m)}\left(\sum_{u_1=0}^{m-1}\|2^{u_1}\beta\|(-1)^{\sigma_{u_1+1}}A(\alpha,\beta,\boldsymbol{\sigma},u_1)\right) \\ &\hspace{100pt}\times
\left(\sum_{u_2=0}^{m-1}\|2^{u_2}\beta\|(-1)^{\sigma_{u_2+1}}A(\alpha,\beta,\boldsymbol{\sigma}^{\ast},u_2)\right) \\
=& -\frac{1}{2^{2m}}\sum_{\alpha,\beta\in\mathbb{Q}^{\ast}(2^m)}\left(\sum_{\substack{u_1=0 \\ \sigma_{u_1+1}=0}}^{m-1}\|2^{u_1}\beta\|A(\alpha,\beta,\boldsymbol{\sigma},u_1)\right)
\left(\sum_{\substack{u_2=0 \\ \sigma_{u_2+1}=0}}^{m-1}\|2^{u_2}\beta\|A(\alpha,\beta,\boldsymbol{\sigma}^{\ast},u_2)\right) \\
&+\frac{1}{2^{2m}}\sum_{\alpha,\beta\in\mathbb{Q}^{\ast}(2^m)}\left(\sum_{\substack{u_1=0 \\ \sigma_{u_1+1}=0}}^{m-1}\|2^{u_1}\beta\|A(\alpha,\beta,\boldsymbol{\sigma},u_1)\right)
\left(\sum_{\substack{u_2=0 \\ \sigma_{u_2+1}=1}}^{m-1}\|2^{u_2}\beta\|A(\alpha,\beta,\boldsymbol{\sigma}^{\ast},u_2)\right) \\
&+\frac{1}{2^{2m}}\sum_{\alpha,\beta\in\mathbb{Q}^{\ast}(2^m)}\left(\sum_{\substack{u_1=0 \\ \sigma_{u_1+1}=1}}^{m-1}\|2^{u_1}\beta\|A(\alpha,\beta,\boldsymbol{\sigma},u_1)\right)
\left(\sum_{\substack{u_2=0 \\ \sigma_{u_2+1}=0}}^{m-1}\|2^{u_2}\beta\|A(\alpha,\beta,\boldsymbol{\sigma}^{\ast},u_2)\right) \\
&-\frac{1}{2^{2m}}\sum_{\alpha,\beta\in\mathbb{Q}^{\ast}(2^m)}\left(\sum_{\substack{u_1=0 \\ \sigma_{u_1+1}=1}}^{m-1}\|2^{u_1}\beta\|A(\alpha,\beta,\boldsymbol{\sigma},u_1)\right)
\left(\sum_{\substack{u_2=0 \\ \sigma_{u_2+1}=1}}^{m-1}\|2^{u_2}\beta\|A(\alpha,\beta,\boldsymbol{\sigma}^{\ast},u_2)\right) \\
=:&-R_1+R_2+R_3-R_4.
\end{align*}
With the first part of Lemma~\ref{Alphalemma} and Lemma~\ref{Betalemma} we obtain
\begin{align*} R_2=&\frac{1}{2^{2m}}\sum_{\substack{u_1=0 \\ \sigma_{u_1+1}=0}}^{m-1}\sum_{\substack{u_2=0 \\ \sigma_{u_2+1}=1}}^{m-1}\sum_{\beta\in\mathbb{Q}^{\ast}(2^m)}\|2^{u_1}\beta\|\|2^{u_2}\beta\|
\sum_{\alpha\in\mathbb{Q}^{\ast}(2^m)}A(\alpha,\beta,\boldsymbol{\sigma},u_1)A(\alpha,\beta,\boldsymbol{\sigma}^{\ast},u_2) \\
=& \frac{1}{2^{2m}}\sum_{\substack{u_1=0 \\ \sigma_{u_1+1}=0}}^{m-1}\sum_{\substack{u_2=0 \\ \sigma_{u_2+1}=1}}^{m-1}\frac{2^m}{2^4}2^{m-2}
=\frac{1}{64}l(m-l).
\end{align*}
In the same way we show $ R_3=\frac{1}{64}l(m-l)$. To calculate $R_1$ and $R_4$, we need to distinguish between
the cases where $u_1=u_2$ and where $u_1 \neq u_2$. This leads to
\begin{align*} R_1=&\frac{1}{2^{2m}}\underbrace{\sum_{\substack{u_1=0 \\ \sigma_{u_1+1}=0}}^{m-1}\sum_{\substack{u_2=0 \\ \sigma_{u_2+1}=0}}^{m-1}}_{u_1 \neq u_2} \frac{2^m}{2^4}2^{m-2} \\ &+\frac{1}{2^{2m}}\sum_{\substack{u=0 \\ \sigma_{u+1}=0}}^{m-1}\sum_{\beta\in\mathbb{Q}^{\ast}(2^m)}\|2^{u}\beta\|^2\sum_{\alpha\in\mathbb{Q}^{\ast}(2^m)}A(\alpha,\beta,\boldsymbol{\sigma},u)A(\alpha,\beta,\boldsymbol{\sigma}^{\ast},u) \\
=& \frac{1}{64}l(l-1)+\frac{1}{2^{2m}}\sum_{\substack{u=0 \\ \sigma_{u+1}=0}}^{m-1}\sum_{\beta\in\mathbb{Q}^{\ast}(2^m)}\|2^{u}\beta\|^2
\sum_{\alpha\in\mathbb{Q}^{\ast}(2^m)}A(\alpha,\beta,\boldsymbol{\sigma},u)A(\alpha,\beta,\boldsymbol{\sigma}^{\ast},u).
\end{align*}
Similarly, we obtain
$$ R_4=\frac{1}{64}(m-l)(m-l-1)+\frac{1}{2^{2m}}\sum_{\substack{u=0 \\ \sigma_{u+1}=1}}^{m-1}\sum_{\beta\in\mathbb{Q}^{\ast}(2^m)}\|2^{u}\beta\|^2
\sum_{\alpha\in\mathbb{Q}^{\ast}(2^m)}A(\alpha,\beta,\boldsymbol{\sigma},u)A(\alpha,\beta,\boldsymbol{\sigma}^{\ast},u). $$
Adding $R_1$ to $R_4$ yields
\begin{align*}\frac{1}{2^{2m}}&\sum_{\alpha,\beta\in\mathbb{Q}^{\ast}(2^m)}\Delta_1(\alpha,\beta)\Delta_2(\alpha,\beta) \\
=& -\frac{1}{64}(m^2+4l^2-4lm-m) \\&-\frac{1}{2^{2m}}\sum_{u=0}^{m-1}\sum_{\beta\in\mathbb{Q}^{\ast}(2^m)}\|2^{u}\beta\|^2
\sum_{\alpha\in\mathbb{Q}^{\ast}(2^m)}A(\alpha,\beta,\boldsymbol{\sigma},u)A(\alpha,\beta,\boldsymbol{\sigma}^{\ast},u).
\end{align*}
Hence, our final task is to compute the last expression in the above line.
We employ the second part of Lemma~\ref{Alphalemma} and Lemma~\ref{Betalemma} to obtain
\begin{align*}
\frac{1}{2^{2m}}&\sum_{u=0}^{m-1}\sum_{\beta\in\mathbb{Q}^{\ast}(2^m)}\|2^{u}\beta\|^2\sum_{\alpha\in\mathbb{Q}^{\ast}(2^m)}A(\alpha,\beta,\boldsymbol{\sigma},u)A(\alpha,\beta,\boldsymbol{\sigma}^{\ast},u)\\
=&\frac{1}{2^{2m}}\sum_{u=0}^{m-1}2^{m-u-1}\sum_{\beta\in\mathbb{Q}^{\ast}(2^m)}\|2^u\beta\|^2\\
&+\frac{1}{2^{2m}}\sum_{u=2}^{m-1}2^{m-u-1}\sum_{\beta\in\mathbb{Q}^{\ast}(2^m)}\|2^u\beta\|^2 \sum_{j=1}^{u-1}2^j((\gamma_j \oplus 1)\gamma_u+\gamma_j (\gamma_u \oplus 1)) \\
=&\frac{1}{2^{m+1}}\sum_{u=0}^{m-1}2^{-u}\frac{2^{2m}+2^{2u+1}}{3\cdot 2^{m+2}} \\
&+ \frac{1}{2^{m+1}}\sum_{u=2}^{m-1}2^{-u}\sum_{j=1}^{u-1}2^j\sum_{\beta\in\mathbb{Q}^{\ast}(2^m)}\|2^u\beta\|^2
(\beta_j \oplus \sigma_j \oplus 1)(\beta_u \oplus \sigma_u) \\
&+ \frac{1}{2^{m+1}}\sum_{u=2}^{m-1}2^{-u}\sum_{j=1}^{u-1}2^j\sum_{\beta\in\mathbb{Q}^{\ast}(2^m)}\|2^u\beta\|^2
(\beta_j \oplus \sigma_j)(\beta_u \oplus \sigma_u \oplus 1)=:\Sigma_1+\Sigma_2+\Sigma_3. \\
\end{align*}
Finding the value of $\Sigma_1$ is a matter of straightforward calculation. We have
$$\Sigma_1=\frac{1}{12}\left(1-\frac{1}{2^{2m}}\right).$$
For $\Sigma_2$ we find
\begin{align*}
\Sigma_2=&\frac{1}{2^{m+1}}\sum_{u=2}^{m-1}2^{-u}\sum_{j=1}^{u-1}2^j\sum_{\substack{\beta_1,\dots,\beta_{j-1},\beta_{j+1},\dots,\beta_{u-1}=0 \\ \beta_j=\sigma_j\\ \beta_u=\sigma_u \oplus 1}}^{1}\sum_{\beta_{u+1},\dots,\beta_m=0}^{1}\|2^u\beta\|^2.
\end{align*}
We remark at this point that $\|2^u\beta\|^2$ only depends on $\beta_{u+1},\dots,\beta_m$. Hence,
$$ \sum_{\beta_{u+1},\dots,\beta_m=0}^{1}\|2^u\beta\|^2=2^{-u}\sum_{\beta_{1},\dots,\beta_m=0}^{1}\|2^u\beta\|^2
= 2^{-u}\sum_{\beta\in\mathbb{Q}^{\ast}(2^m)}\|2^u\beta\|^2=2^{-u}\frac{2^{2m}+2^{2u+1}}{3\cdot 2^{m+2}}.$$
We arrive at
\begin{align*}
\Sigma_2=&\frac{1}{2^{m+1}}\sum_{u=2}^{m-1}2^{-u}\sum_{j=1}^{u-1}2^j2^{u-2}2^{-u}\frac{2^{2m}+2^{2u+1}}{3\cdot 2^{m+2}} \\
=&\frac{1}{2^{m+1}}\sum_{u=2}^{m-1}2^{-u}(2^u-2)2^{u-2}2^{-u}\frac{2^{2m}+2^{2u+1}}{3\cdot 2^{m+2}} \\
=& \frac{m}{96}-\frac{7}{288}+\frac{1}{9\cdot 2^{2m+1}}. \end{align*}
It is clear that $\Sigma_3=\Sigma_2$. Thus, after adding all the results the proof of the lemma is finally complete.
\end{proof}
For the proof of Theorem~\ref{Theo}, we will also need an exact formula for the $L_2$ discrepancy of $\mathcal{H}_m(\boldsymbol{\sigma})$. Such a formula was presented in \cite[Theorem 1]{Kri2}.
\begin{lemma}[Kritzer and Pillichshammer] \label{l2shift} Let $m\in\mathbb{N}$ and $\boldsymbol{\sigma} \in \{0,1\}^m$. We have
$$ (2^mL_2(\mathcal{H}_m(\boldsymbol{\sigma})))^2=\frac{m^2}{64}-\frac{19m}{192}-\frac{lm}{16}+\frac{l^2}{16}+\frac{l}{4}+\frac38+\frac{m}{16\cdot 2^m}-\frac{l}{8\cdot 2^m}+\frac{1}{4\cdot 2^m}-\frac{1}{72\cdot 4^m}. $$
\end{lemma}
\begin{remark} \label{optimalshift} \rm It follows from Lemma~\ref{l2shift} that the optimal choice for $l$ is $\left\lceil\frac{m-5}{2}+\frac{1}{2^m}\right\rceil$, which leads to
$$(2^mL_2(\mathcal{H}_m(\boldsymbol{\sigma})))^2=\frac{5m}{192}+\mathcal{O}(1) $$
(see also \cite[Corollary 1]{Kri2}).
This means that we achieve the optimal order of $L_2$ discrepancy for $\mathcal{H}_m(\boldsymbol{\sigma})$ in this case. In \cite[Theorem 1]{HKP14} it was shown that
we achieve the optimal order of $L_p$ discrepancy for all $p\in[1,\infty)$ if and only if $|2l-m|=\mathcal{O}(\sqrt{m})$.
\end{remark}
\section{Proof of Theorem~\ref{Theo}}
We apply Lemma~\ref{symlemma} to write
\begin{align} (2^{m+1}L_2(\mathcal{H}_m^{{\rm sym}}(\boldsymbol{\sigma})))^2=& \int_{0}^{1} \int_{0}^{1} (\Delta_{{\rm sym}}(\alpha,\beta))^2 \,\mathrm{d}\alpha \,\mathrm{d}\beta \nonumber \\ \nonumber
=&\int_{0}^{1} \int_{0}^{1} (\Delta_1(\alpha,\beta))^2 \,\mathrm{d}\alpha \,\mathrm{d}\beta+\int_{0}^{1} \int_{0}^{1} (\Delta_2(\alpha,\beta))^2 \,\mathrm{d}\alpha \,\mathrm{d}\beta\\ \nonumber
&+2\int_{0}^{1} \int_{0}^{1} \Delta_1(\alpha,\beta)\Delta_2(\alpha,\beta) \,\mathrm{d}\alpha \,\mathrm{d}\beta\\ \nonumber
=&(2^{m}L_2(\mathcal{H}_m(\boldsymbol{\sigma})))^2+(2^{m}L_2(\mathcal{H}_m(\boldsymbol{\sigma}^{\ast})))^2 \\ \label{letztesintegral}
&+2\int_{0}^{1} \int_{0}^{1} \Delta_1(\alpha,\beta)\Delta_2(\alpha,\beta) \,\mathrm{d}\alpha \,\mathrm{d}\beta.
\end{align}
We know the values of $(2^mL_2(\mathcal{H}_m(\boldsymbol{\sigma})))^2$ and $(2^mL_2(\mathcal{H}_m(\boldsymbol{\sigma}^{\ast})))^2$ already from Lemma~\ref{l2shift} (where in the latter
case we have to insert $m-l$ instead of $l$ in this formula). This yields
\begin{align*}
(2^mL_2(\mathcal{H}_m(\boldsymbol{\sigma})))^2+(2^mL_2(\mathcal{H}_m(\boldsymbol{\sigma}^{\ast})))^2=\frac{m^2}{32}+\frac{l^2}{8}-\frac{lm}{8}+\frac{5m}{96}+\frac34+\frac{1}{2^{m+1}}-\frac{1}{9\cdot 2^{2m+2}}.
\end{align*} \\
We split the integrals in \eqref{letztesintegral} in four parts:
\begin{align*} \int_{0}^{1} \int_{0}^{1} \Delta_1(\alpha,\beta)\Delta_2(\alpha,\beta) \,\mathrm{d}\alpha \,\mathrm{d}\beta=&
\int_{0}^{1-2^{-m}} \int_{0}^{1-2^{-m}} \Delta_1(\alpha,\beta)\Delta_2(\alpha,\beta) \,\mathrm{d}\alpha \,\mathrm{d}\beta \\&+\int_{0}^{1-2^{-m}} \int_{1-2^{-m}}^{1} \Delta_1(\alpha,\beta)\Delta_2(\alpha,\beta) \,\mathrm{d}\alpha\,\mathrm{d}\beta \\
&+ \int_{1-2^{-m}}^{1} \int_{0}^{1-2^{-m}} \Delta_1(\alpha,\beta)\Delta_2(\alpha,\beta) \,\mathrm{d}\alpha\,\mathrm{d}\beta \\ & +\int_{1-2^{-m}}^{1} \int_{1-2^{-m}}^{1} \Delta_1(\alpha,\beta)\Delta_2(\alpha,\beta) \,\mathrm{d}\alpha \,\mathrm{d}\beta \\ =:&I_1+I_2+I_3+I_4. \end{align*}
We can calculate $I_2$, $I_3$ and $I_4$ with aid of the second part of Lemma~\ref{AllgemeineFormel}. Since this proceeds analogously as in the
proof of \cite[Theorem 1]{Kri1}, we only give the results. We have
$$ I_2=I_3=\frac{25}{36\cdot 2^m}-\frac{5}{9\cdot 4^m}-\frac{25}{36\cdot 4^m}+\frac{2}{3\cdot 8^m}-\frac{1}{9\cdot 16^m} $$ and
$$ I_4=\frac{7}{6\cdot 4^m}+\frac{1}{9\cdot 16^m}-\frac{2}{3\cdot 8^m}. $$
It remains to evaluate $I_1$. We use the second part of Lemma~\ref{AllgemeineFormel} to obtain
\begin{align*}
I_1=& \int_{0}^{1-2^{-m}} \int_{0}^{1-2^{-m}}(\Delta_1(\alpha(m),\beta(m))+2^m(\alpha(m)\beta(m)-\alpha\beta)) \\
&\hspace{65 pt} \times (\Delta_2(\alpha(m),\beta(m))+2^m(\alpha(m)\beta(m)-\alpha\beta)) \,\mathrm{d} \alpha \,\mathrm{d} \beta \\
=&\int_{0}^{1-2^{-m}} \int_{0}^{1-2^{-m}}\Delta_1(\alpha(m),\beta(m))\Delta_2(\alpha(m),\beta(m))\,\mathrm{d} \alpha \,\mathrm{d} \beta \\
&+2^m\int_{0}^{1-2^{-m}} \int_{0}^{1-2^{-m}}\Delta_1(\alpha(m),\beta(m))(\alpha(m)\beta(m)-\alpha\beta)\,\mathrm{d} \alpha \,\mathrm{d} \beta \\
&+2^m\int_{0}^{1-2^{-m}} \int_{0}^{1-2^{-m}}\Delta_2(\alpha(m),\beta(m))(\alpha(m)\beta(m)-\alpha\beta)\,\mathrm{d} \alpha \,\mathrm{d} \beta \\
&+2^{2m}\int_{0}^{1-2^{-m}} \int_{0}^{1-2^{-m}}(\alpha(m)\beta(m)-\alpha\beta)^2\,\mathrm{d} \alpha \,\mathrm{d} \beta=S_1+S_2+S_3+S_4. \end{align*}
The value of $S_4$ can be calculated in a straightforward way and is
$$ S_4=-\frac{1}{72\cdot 16^m}(2^m-1)^2(32\cdot 2^m-25\cdot4^m-8). $$
The expression $S_2$ was computed in the proof of \cite[Theorem 1]{Kri2} and is given by
$$ S_2=2^{m-1}\frac{2^{m+1}-1}{4^m}\left(\frac{l(\boldsymbol{\sigma})}{8}-\frac{m}{16}\right). $$
Analogously, we have
$$ S_3=2^{m-1}\frac{2^{m+1}-1}{4^m}\left(\frac{l(\boldsymbol{\sigma}^{\ast})}{8}-\frac{m}{16}\right), $$
where $l(\boldsymbol{\sigma}^{\ast})$ is the number of components in $\boldsymbol{\sigma}^{\ast}$ which are equal to zero.
Since we obviously have $l(\boldsymbol{\sigma}^{\ast})=m-l(\boldsymbol{\sigma})$, we find $S_2+S_3=0$.
So far we have
$$ I_1=S_1-\frac{1}{72\cdot 16^m}(2^m-1)^2(32\cdot 2^m-25\cdot 4^m-8). $$
But since
\begin{align*} S_1=&\sum_{a,b=1}^{2^m-1}\int_{\frac{a-1}{2^m}}^{\frac{a}{2^m}}\int_{\frac{b-1}{2^m}}^{\frac{b}{2^m}}\Delta_1\left(\frac{a}{2^m},\frac{b}{2^m}\right)\Delta_2\left(\frac{a}{2^m},\frac{b}{2^m}\right)\,\mathrm{d} \alpha \,\mathrm{d} \beta \\
=&\frac{1}{2^{2m}}\sum_{\alpha,\beta\in\mathbb{Q}^{\ast}(2^m)}\Delta_1(\alpha,\beta)\Delta_2(\alpha,\beta),
\end{align*}
we also know the value of $S_1$ from Lemma~\ref{gemischt}. Putting all results together, we obtain the claimed formula in Theorem~\ref{Theo}. $\hfill \Box \vspace{0.5cm}$
\bigskip
\noindent {\bf Acknowledgments.} The author would like to thank Friedrich Pillichshammer for valuable suggestions to improve the presentation.
|
1,108,101,563,251 | arxiv | \section{Introduction}
\label{sec:Introduction}
Although two-dimensional $\phi^4$ theory has a simple Lagrangian, the
spectrum of the theory has some very interesting behavior. Even without
the introduction of a negative bare-mass squared, the theory exhibits
symmetry breaking at sufficiently strong coupling~\cite{Chang}, signaled by a
degeneracy of the lowest massive states with the vacuum. Here we
consider a light-front Hamiltonian calculation of the spectrum
at and below this critical coupling and compare with previous calculations
in both light-front~\cite{VaryHari} and equal-time
quantization~\cite{RychkovVitale,LeeSalwen,Sugihara,SchaichLoinaz,Bosetti,Milsted}.
In particular, we explain why the two quantizations yield different
results for the critical value of the bare dimensionless coupling.
Light-front quantization~\cite{DLCQreview} is in general a convenient approach to
the nonperturbative solution of quantum field theories~\cite{Vary,ArbGauge},
and provides an alternative to lattice~\cite{Lattice} and Dyson--Schwinger methods~\cite{DSE}.
Light-front coordinates~\cite{Dirac} offer a clean separation between external
and internal momenta, and the quantization can keep the vacuum trivial,
so that it does not mix with the massive states. The wave functions
of a Fock-state expansion are then well defined and lay an intuitive
foundation for calculation of observables directly in terms of matrix
elements of operators. Moreover, the formulation
is in Minkowski space rather than the Euclidean space of lattice gauge theory
and Dyson--Schwinger equations.
In two dimensions, light-front quantization uses
a time coordinate $x^+\equiv t+z$ and a spatial coordinate
$x^-\equiv t-z$. The conjugate momentum variables are $p^-\equiv E-p_z$
and $p^+\equiv E+p_z$, respectively. The fundamental eigenvalue problem
for an eigenstate of mass $M$ is
${\cal P}^-|\psi(P^+)\rangle=\frac{M^2}{P^+}|\psi(P^+)\rangle$, where
${\cal P}^-$ is the light-front Hamiltonian and $P^+$ is the total
light-front momentum of the state. We solve this eigenvalue
problem by expanding $|\psi(P^+)\rangle$ in a Fock basis
of momentum and particle-number eigenstates and then expanding
the Fock-space wave functions in terms of fully symmetric,
multivariate polynomials~\cite{GenSymPolys}. The problem is
made finite by truncation in both the Fock-space and
polynomial basis sets.
The use of a polynomial basis for the wave functions has significant
advantages over the more common discretized light-cone quantization (DLCQ)
approach~\cite{PauliBrodsky,DLCQreview}. One purpose of the present
work is to illustrate this. In DLCQ, which relies on a trapezoidal
approximation to integral operators, endpoint corrections associated
with zero modes~\cite{ZeroModes} are normally dropped,\footnote{The
standard DLCQ approach of neglecting zero modes can be modified to
include them~\protect\cite{DLCQzeromodes}.} which delays convergence,
whereas a basis-function method can be tuned to keep the correct
endpoint behavior of the wave functions. Also, the discretization
grid of DLCQ forces a particular allocation of computational
resources to each Fock sector, without regard to the importance
of one sector over another; a basis-function approach allows
the allocation to be adjusted sector by sector, to optimize
computation resources with respect to convergence. The particular
polynomial basis that we use~\cite{GenSymPolys} is specifically
symmetric with respect to interchange of the identical bosons, so
that no explicit symmetrization is required.
The remainder of the paper is structured as follows. Section~\ref{sec:EVProblem}
describes the eigenvalue problem that we solve, with details
of the coupled integral equations for the Fock-state wave functions.
In Sec.~\ref{sec:Mass} we discuss the difference in mass
renormalizations for light-front and equal-time quantization
and provide a scheme for calculation. Our results are presented
and discussed in Sec.~\ref{sec:Results}, with a brief summary
provided in Sec.~\ref{sec:Summary}. Details of the numerical
calculation are given in an Appendix.
\section{Light-front eigenvalue problem}
\label{sec:EVProblem}
From the Lagrangian for two-dimensional $\phi^4$ theory
\begin{equation}
{\cal L}=\frac12(\partial_\mu\phi)^2-\frac12\mu^2\phi^2-\frac{\lambda}{4!}\phi^4,
\end{equation}
where $\mu$ is the mass of the boson and $\lambda$ is the coupling constant,
the light-front Hamiltonian density is found to be
\begin{equation}
{\cal H}=\frac12 \mu^2 \phi^2+\frac{\lambda}{4!}\phi^4.
\end{equation}
The mode expansion for the field at zero light-front time is
\begin{equation} \label{eq:mode}
\phi(x^+=0,x^-)=\int \frac{dp}{\sqrt{4\pi p}}
\left\{ a(p)e^{-ipx^-/2} + a^\dagger(p)e^{ipx^-/2}\right\},
\end{equation}
where for convenience we have dropped the $+$ superscript
and will from here on write light-front momenta such as
$p^+$ as just $p$.
The creation operator $a^\dagger(p)$ satisfies the commutation
relation
\begin{equation}
[a(p),a^\dagger(p')]=\delta(p-p'),
\end{equation}
and builds $m$-constituent Fock states from the Fock vacuum $|0\rangle$ in the form
\begin{equation}
|y_iP;P,m\rangle=\frac{1}{\sqrt{m!}}\prod_{i=1}^m a^\dagger(y_iP)|0\rangle.
\end{equation}
Here $y_i\equiv p_i/P$ is the longitudinal momentum fraction for the $i$th constituent.
The light-front Hamiltonian is
${\cal P}^-={\cal P}^-_{11}+{\cal P}^-_{22}+{\cal P}^-_{13}+{\cal P}^-_{31}$,
with
\begin{eqnarray} \label{eq:Pminus11}
{\cal P}^-_{11}&=&\int dp \frac{\mu^2}{p} a^\dagger(p)a(p), \\
\label{eq:Pminus22}
{\cal P}^-_{22}&=&\frac{\lambda}{4}\int\frac{dp_1 dp_2}{4\pi\sqrt{p_1p_2}}
\int\frac{dp'_1 dp'_2}{\sqrt{p'_1 p'_2}}
\delta(p_1 + p_2-p'_1-p'_2) \\
&& \rule{2in}{0mm} \times a^\dagger(p_1) a^\dagger(p_2) a(p'_1) a(p'_2),
\nonumber \\
\label{eq:Pminus13}
{\cal P}^-_{13}&=&\frac{\lambda}{6}\int \frac{dp_1dp_2dp_3}
{4\pi \sqrt{p_1p_2p_3(p_1+p_2+p_3)}}
a^\dagger(p_1+p_2+p_3)a(p_1)a(p_2)a(p_3), \\
\label{eq:Pminus31}
{\cal P}^-_{31}&=&\frac{\lambda}{6}\int \frac{dp_1dp_2dp_3}
{4\pi \sqrt{p_1p_2p_3(p_1+p_2+p_3)}}
a^\dagger(p_1)a^\dagger(p_2)a^\dagger(p_3)a(p_1+p_2+p_3).
\end{eqnarray}
The subscripts indicate the number of creation and annihilation operators
in each term.
The Fock-state expansion of an eigenstate can be written
\begin{equation} \label{eq:FSexpansion}
|\psi(P)\rangle=\sum_m P^{\frac{m-1}{2}}\int\prod_i^m dy_i
\delta(1-\sum_i^m y_i)\psi_m(y_i)|y_iP;P,m\rangle,
\end{equation}
where $\psi_m$ is the wave function for $m$ constituents.
Because the terms of ${\cal P}^-$ change particle
number by zero or by two, the eigenstates can be separated
according to the oddness or evenness of the number of constituents.
Therefore, the first sum in (\ref{eq:FSexpansion}) is restricted to
odd or even $m$. We will consider only the lowest mass eigenstate
in each case, though the methods allow for calculation of higher
states.
The light-front Hamiltonian eigenvalue problem
${\cal P}^-|\psi(P)\rangle=\frac{M^2}{P}|\psi(P)\rangle$
reduces to a coupled set of integral equations for the
Fock-state wave functions:
\begin{eqnarray}
\lefteqn{m\frac{\mu^2}{y_1 P}\psi_m(y_i)
+\frac{\lambda}{4\pi P}\frac{m(m-1)}{4\sqrt{y_1y_2}}
\int\frac{dx_1 dx_2 }{\sqrt{x_1 x_2}}\delta(y_1+y_2-x_1-x_2) \psi_m(x_1,x_2,y_3,\ldots,y_m)}&&
\nonumber \\
&& +\frac{\lambda}{4\pi P}\frac{m}{6}\sqrt{(m+2)(m+1)}\int \frac{dx_1 dx_2 dx_3}{\sqrt{y_1 x_1 x_2 x_3}}
\delta(y_1-x_1-x_2-x_3)\psi_{m+2}(x_1,x_2,x_3,y_2,\ldots,y_m) \nonumber \\
&& +\frac{\lambda}{4\pi P}\frac{m-2}{6}\frac{\sqrt{m(m-1)}}{\sqrt{y_1y_2y_3(y_1+y_2+y_3)}}
\psi_{m-2}(y_1+y_2+y_3,y_4,\ldots,y_m)=\frac{M^2}{P}\psi_m(y_i).
\end{eqnarray}
We have used the symmetry of $\psi_m$ to collect exchanged momenta in
the leading arguments of the function, with appropriate $m$-dependent
factors in front of each term. The equations are simplified further
by the introduction of a dimensionless coupling
\begin{equation}
g\equiv\frac{\lambda}{4\pi\mu^2}
\end{equation}
and by multiplying the set by $P/\mu^2$, to obtain
\begin{eqnarray} \label{eq:coupledsystem}
\lefteqn{\frac{m}{y_1}\psi_m(y_i)
+\frac{g}{4}\frac{m(m-1)}{\sqrt{y_1y_2}}
\int\frac{dx_1 dx_2 }{\sqrt{x_1 x_2}}\delta(y_1+y_2-x_1-x_2) \psi_m(x_1,x_2,y_3,\ldots,y_m)}&&
\nonumber \\
&& +\frac{g}{6}m\sqrt{(m+2)(m+1)}\int \frac{dx_1 dx_2 dx_3}{\sqrt{y_1 x_1 x_2 x_3}}
\delta(y_1-x_1-x_2-x_3)\psi_{m+2}(x_1,x_2,x_3,y_2,\ldots,y_m) \nonumber \\
&& +\frac{g}{6}\frac{(m-2)\sqrt{m(m-1)}}{\sqrt{y_1y_2y_3(y_1+y_2+y_3)}}
\psi_{m-2}(y_1+y_2+y_3,y_4,\ldots,y_m)=\frac{M^2}{\mu^2}\psi_m(y_i).
\end{eqnarray}
It is this set of equations that we solve numerically, as described in the
Appendix. Our approach takes advantage of the new set of multivariate
polynomials that is fully symmetric on the hypersurface $\sum_i y_i=1$
defined by momentum conservation~\cite{GenSymPolys}. This allows independent
tuning of resolutions in each Fock sector, so that, unlike discrete
light-cone quantization (DLCQ)~\cite{PauliBrodsky,DLCQreview}, sectors
with lower net probability need not overtax computational resources.
Also, within each Fock sector, the use of a polynomial basis has improved
convergence compared to DLCQ~\cite{GenSymPolys}, at least partly because
DLCQ misses contributions from zero modes~\cite{ZeroModes}
associated with integrable singularities at $y_i=0$.
In addition to calculation of the spectrum, it is possible to calculate the expectation
value for the field $\phi$ when the odd and even states are degenerate.
At degeneracy, the two states mix, and the expectation value for $\phi$
comes from cross terms, the matrix element between the odd and even
eigenstates. Let $|\widetilde\psi(P')\rangle$ be the state with an
odd number of constituents, and $|\psi(P)\rangle$ be the state for
an even number. At degeneracy, the eigenstate can be a linear
combination of these, and the desired matrix element for the
field is
\begin{eqnarray}
\lefteqn{\langle\widetilde\psi(P')|\phi(0,x^-)|\psi(P)\rangle=
\sum_m \frac{P^{\prime\,m/2-1}}{P^{(m-1)/2}}\int\prod_j^{m+1} dy'_j\delta(1-\sum_j y'_j)
\frac{\sqrt{m+1}}{\sqrt{4\pi y'_1 P'}}} && \nonumber \\
&& \rule{1in}{0mm} \times \delta(y'_1+P/P'-1)e^{i(P'-P)x^-/2} \widetilde\psi_{m+1}(y'_j)\psi_m(y'_2,...,y'_{m+1}) \\
&& \rule{0.5in}{0mm}+\sum_m \frac{P^{m/2-1}}{P^{\prime\,(m-1)/2}}\int\prod_j^{m+1} dy_j\delta(1-\sum_j y_j)
\frac{\sqrt{m+1}}{\sqrt{4\pi y_1 P}}\delta(y_1+P'/P-1) \nonumber \\
&& \rule{1in}{0mm} \times e^{-i(P'-P)x^-/2} \widetilde\psi_m(y_2,\ldots,y_{m+1})\psi_{m+1}(y_j), \nonumber
\end{eqnarray}
where again we have taken advantage of the wave-function symmetry to
arrange for all but the first constituent to be spectators. In the
limit $P'\rightarrow P$, this expression reduces to
\begin{eqnarray} \label{eq:vev}
\lefteqn{\langle\widetilde\psi(P)|\phi(0,x^-)|\psi(P)\rangle=
\frac12\sum_m\frac{\sqrt{m+1}}{\sqrt{4\pi P}}\int \prod_{i=2}^{m+1} dy_i \delta(1-\sum_i y_i)}&& \\
&& \rule{0.5in}{0mm} \times \left[\widetilde\phi_{m+1}(y_2,\ldots,y_{m+1})\psi_m(y_2,\ldots,y_{m+1})
+\widetilde\psi_m(y_2,\ldots,y_{m+1})\phi_{m+1}(y_2,\ldots,y_{m+1}) \right], \nonumber
\end{eqnarray}
with
\begin{equation}
\phi_{m+1}(y_2,\ldots,y_{m+1})\equiv \lim_{y_1\rightarrow0}\frac{1}{\sqrt{y_1}}\psi_{m+1}(y_1,y_2,\ldots,y_m)
\end{equation}
and $\widetilde\phi$ defined analogously.
Thus the expectation value depends upon zero modes~\cite{ZeroModes}.
In our basis function approach, the zero-momentum limit can be taken
explicitly.
\section{Mass renormalization}
\label{sec:Mass}
One of the advantages of light-front quantization is the absence of
vacuum-to-vacuum graphs~\cite{DLCQreview}. However, to compare
results found in equal-time quantization at equivalent values
of the bare parameters in the Lagrangian, this absence must be
taken into account. In particular, the bare mass in $\phi^4_{1+1}$ theory
is renormalized by tadpole contributions in equal-time quantization
but not in light-front quantization, and the two different masses
are related by~\cite{SineGordon}
\begin{equation}
\mu_{\rm LF}^2=\mu_{\rm ET}^2
+\lambda\left[\langle 0|:\frac{\phi^2}{2}:|0\rangle
-\langle 0|:\frac{\phi^2}{2}:|0\rangle_{\rm free}\right].
\end{equation}
The vacuum expectation values (vev) of $\phi^2$ resum the tadpole
contributions; the subscript {\em free} indicates the vev with
$\lambda=0$. This distinction between bare masses in the
two quantizations implies that the dimensionless coupling $g=\lambda/4\pi\mu^2$
is also not the same. Estimates of the critical coupling must
then be adjusted for the difference if they are to be compared.
Of course, if one compares results only for physical quantities,
the two quantizations should match immediately. However, this
is not straightforward in the case of the critical coupling, where
the physical mass scale goes to zero.
With the tadpole contribution re-expressed as a vev, we
can calculate the contribution in light-front quantization
and avoid doing a second, equal-time calculation. The
vev is regulated by point splitting, and a sum over a
complete set states is introduced, to obtain
\begin{equation}
\langle 0|:\frac{\phi^2}{2}:|0\rangle\rightarrow
\frac12\langle 0|\phi(\epsilon^+,\epsilon^-)\int_0^\infty dP\sum_n|\psi_n(P)\rangle
\langle\psi_n(P)|\phi(0,0)|0\rangle.
\end{equation}
Each mass eigenstate $|\psi_n(P)\rangle$ is expanded in terms of Fock states
and wave functions, just as in (\ref{eq:FSexpansion}), with
the Fock wave functions $\psi_{nm}(y_i)$ now defined with an additional index $n$
for the particular eigenstate. Because the $\phi$ field changes particle number
by one, only one-particle Fock states will contribute to the sum over $n$,
with amplitude $\psi_{n1}$. For the free case, the only contribution
to the sum is the one-particle state $a^\dagger(P)|0\rangle$.
The individual matrix elements are readily calculated. At $x^+=0$ the
field is given by (\ref{eq:mode}), and the matrix element is
\begin{equation}
\langle\psi_n(P)|\phi(0,0)|0\rangle
=\langle 0|\psi_{n1}^*a(P)\int\frac{dp}{\sqrt{4\pi p}}a^\dagger(p)|0\rangle
=\frac{\psi_{n1}^*}{\sqrt{4\pi P}}
\end{equation}
At $x^+=\epsilon^+$, the field is
\begin{equation}
\phi(\epsilon^+,\epsilon^-)=e^{i{\cal P}^-\epsilon^+/2}\phi(0,\epsilon^-)e^{-i{\cal P}^-\epsilon^+/2}.
\end{equation}
The matrix element is
\begin{eqnarray}
\langle 0|\phi(\epsilon^+,\epsilon^-)|\psi_n(P)\rangle
&=&\langle 0|e^{i0\epsilon^+}\int\frac{dp}{\sqrt{4\pi p}}a(p)
e^{-ip\epsilon^-/2}e^{-iM_n^2\epsilon^+/2P}\psi_{n1}a^\dagger(P)|0\rangle \nonumber \\
&=&\frac{\psi_{n1}}{\sqrt{4\pi P}}e^{-i(P\epsilon^- +M_n^2\epsilon^+/P)/2},
\end{eqnarray}
with $M_n$ the mass of the $n$th state. The corresponding matrix elements
for the free case are
\begin{equation}
\langle0|a(P)\phi(0,0)|0\rangle
=\langle 0|a(P)\int\frac{dp}{\sqrt{4\pi p}}a^\dagger(p)|0\rangle
=\frac{1}{\sqrt{4\pi P}}
\end{equation}
and
\begin{eqnarray}
\langle 0|\phi(\epsilon^+,\epsilon^-)a^\dagger(P)|0\rangle
&=&\langle 0|e^{i0\epsilon^+}\int\frac{dp}{\sqrt{4\pi p}}a(p)
e^{-ip\epsilon^-/2}e^{-i\mu^2\epsilon^+/2P}a^\dagger(P)|0\rangle \nonumber \\
&=&\frac{1}{\sqrt{4\pi P}}e^{-i(P\epsilon^- +\mu^2\epsilon^+/P)/2}.
\end{eqnarray}
The combination of these matrix elements yields the two vev's:
\begin{equation}
\langle 0|:\frac{\phi^2}{2}:|0\rangle=\frac12\sum_n\int_0^\infty dP \frac{|\psi_{n1}|^2}{4\pi P}
e^{-i(P\epsilon^- +M_n^2\epsilon^+/P)/2}
\end{equation}
and
\begin{equation}
\langle 0|:\frac{\phi^2}{2}:|0\rangle_{\rm free}=\frac12\int_0^\infty dP \frac{1}{4\pi P}
e^{-i(P\epsilon^- +\mu^2\epsilon^+/P)/2}.
\end{equation}
The completeness of the eigenstates allows the introduction of the sum $1=\sum_n|\psi_{n1}|^2$
into the free vev, so that the difference can be written
\begin{equation}
\langle 0|:\frac{\phi^2}{2}:|0\rangle-\langle 0|:\frac{\phi^2}{2}:|0\rangle_{\rm free}
=\sum_n \frac{|\psi_{n1}|^2}{8\pi}\int_0^\infty \frac{dP}{P} e^{-iP\epsilon^-/2}
\left[e^{-i\frac{M_n^2\epsilon^+}{2P}}-e^{-i\frac{\mu^2\epsilon^+}{2P}}\right].
\end{equation}
With the change of variable $P=2z\epsilon^+$ and the introduction of a convergence
factor $e^{-\eta z}$, this expression becomes
\begin{equation}
\langle 0|:\frac{\phi^2}{2}:|0\rangle-\langle 0|:\frac{\phi^2}{2}:|0\rangle_{\rm free}
=\sum_n \frac{|\psi_{n1}|^2}{8\pi}\int_0^\infty \frac{dz}{z} e^{iz(-\epsilon^2+i\eta)}
\left[e^{-i\frac{M_n^2}{4z}}-e^{-i\frac{\mu^2}{4z}}\right].
\end{equation}
Each term is an integral representation~\cite{GradshteynRyzhik} of the modified
Bessel function $K_0$:
\begin{equation}
\int_0^\infty \frac{dx}{x}\exp\left(i\frac{\alpha}{2}\left[x-\frac{\beta^2}{x}\right]\right)=2K_0(\alpha\beta),
\end{equation}
with positive imaginary parts for $\alpha$ and $\alpha\beta^2$.
The difference of vev's then becomes
\begin{equation}
\langle 0|:\frac{\phi^2}{2}:|0\rangle-\langle 0|:\frac{\phi^2}{2}:|0\rangle_{\rm free}
=\sum_n \frac{|\psi_{n1}|^2}{4\pi}\left[K_0(M_n\sqrt{-\epsilon^2+i\eta})
-K_0(\mu\sqrt{-\epsilon^2+i\eta})\right].
\end{equation}
As $-\epsilon^2+i\eta$ goes to zero, the only contribution from $K_0$ is a simple
logarithm, i.e. $K_0(z)\rightarrow-\ln(z/2)-\gamma$, leaving
\begin{equation} \label{eq:Delta}
\langle 0|:\frac{\phi^2}{2}:|0\rangle-\langle 0|:\frac{\phi^2}{2}:|0\rangle_{\rm free}
=-\sum_n \frac{|\psi_{n1}|^2}{4\pi}\ln\frac{M_n}{\mu_{\rm LF}}\equiv -\Delta/4\pi,
\end{equation}
with $\mu$ written as $\mu_{\rm LF}$ to emphasize that it is the light-front bare mass.
Within the context of our numerical calculation, the terms of the sum can be computed by
fully diagonalizing the matrix representation of the Hamiltonian ${\cal P}^-$.
The bare masses in the two quantizations are then related by
\begin{equation}
\mu_{\rm LF}^2=\mu_{\rm ET}^2-\frac{\lambda}{4\pi}\Delta \;\;
\mbox{or} \;\;
\frac{\mu_{\rm ET}^2}{\mu_{\rm LF}^2}=1+g_{\rm LF}\Delta.
\end{equation}
The bare-mass ratio is what connects the dimensionless couplings and masses
obtained in the two quantizations:
\begin{equation} \label{eq:couplingratio}
g_{\rm ET}=\frac{g_{\rm LF}}{\mu_{\rm ET}^2/\mu_{\rm LF}^2}=\frac{g_{\rm LF}}{(1+g_{\rm LF}\Delta)}\;\;
\mbox{and} \;\;
\frac{M^2}{\mu_{\rm ET}^2}=\frac{M^2/\mu_{\rm LF}^2}{\mu_{\rm ET}^2/\mu_{\rm LF}^2}
=\frac{1}{1+g_{\rm LF}\Delta}\frac{M^2}{\mu_{\rm LF}^2},
\end{equation}
which we can use to compare the values obtained.
\section{Results and discussion}
\label{sec:Results}
With the numerical methods discussed in the Appendix, we have solved the
eigenvalue problem for the lowest odd and even states. The mass values
for different Fock-space truncations are shown in Figs.~\ref{fig:extrap-odd}
and \ref{fig:extrap-even}. The error bars are estimated based on
the extrapolations in basis size. The seven/eight-body truncations yield
results that are the same as the five/six-body results, to within the
error estimates, which means that convergence in the Fock-state
expansion has been achieved.
\begin{figure}[ht]
\vspace{0.2in}
\centerline{\includegraphics[width=15cm]{extrap-odd.eps}}
\caption{\label{fig:extrap-odd}
Lowest mass eigenvalue for odd numbers of constituents.
Results are shown for different Fock-space truncations
to three, five, and seven constituents. Also plotted
are results for the leading light-front coupled-cluster (LFCC)
approximation and for a sector-dependent modification of
the three-body truncation. The errors are estimated
from extrapolations in polynomial basis size.
}
\end{figure}
\begin{figure}[ht]
\vspace{0.2in}
\centerline{\includegraphics[width=15cm]{extrap-even.eps}}
\caption{\label{fig:extrap-even}
Same as Fig.~\ref{fig:extrap-odd} but for even numbers
of constituents, with Fock-space truncations at four,
six and eight.
}
\end{figure}
In the odd case, we also show results from
the leading light-front coupled-cluster (LFCC)
approximation~\cite{LFCC,LFCCphi4}\footnote{There are sign errors in Eq.~(B4)
of \protect\cite{LFCCphi4}. The signs of the $A$ and $B$ terms should be
reversed; however, the computations were done with the correct signs.}
and a modification of the three-body truncation that includes sector-dependent
bare masses~\cite{Wilson,hb,Karmanov,SecDep}. The LFCC calculation includes
a partial summation over all
higher Fock states. The sector-dependent calculation uses the physical
mass in the upper Fock sector, where there can be no self-energy correction.
Both of these alternatives require solution of a three-body problem
and yield results much better than the simple three-body truncation,
with the LFCC approximation doing much better than the sector-dependent
approach.
The relative Fock-sector probabilities for the odd case are plotted
in Fig.~\ref{fig:relprob}. These ratios are computed as
\begin{equation}
R_m\equiv\frac{1}{|\psi_1|^2}\int\left[\prod_{i=1}^{m-1}dy_i\right]|\psi_m(y_i)|^2
=\frac{1}{|\psi_1|^2}\sum_{ni}|c_{ni}^{(m)}|^2,
\end{equation}
where the last expression is in terms of the basis-function
expansion coefficients, as defined in (\ref{eq:expansion}).
These ratios show that the probability for each
Fock sector decreases by an order of magnitude when the number of
constituents goes up by two.
\begin{figure}[ht]
\vspace{0.2in}
\centerline{\includegraphics[width=15cm]{relprob.eps}}
\caption{\label{fig:relprob}
Relative Fock-sector probabilities for the lowest mass eigenstate with
odd numbers of constituents.
}
\end{figure}
The apparent convergence of the Fock-state expansion is somewhat deceptive.
With the bare mass fixed as the same in all Fock sectors, the higher
Fock sectors are suppressed by the large invariant mass of each
Fock state, which is of order $m\mu$ for the sector with $m$
constituents. For weak to moderate couplings this is not a
particular concern, but for strong coupling, approaching the
critical value, one expects much larger contributions from
higher Fock states. This would be best modeled by sector-dependent
masses.\footnote{An alternative is the LFCC method~\protect\cite{LFCCphi4},
which automatically uses the physical mass for the kinetic
energy contributions to the wave equations.} However, to
use sector-dependent masses requires renormalization to
physical observables, which would greatly complicate any
comparison with the published results for equal-time
quantization. Therefore, for purposes of the the present
work, we retain a fixed bare mass.
Based on the results for the masses as a function of the
coupling, we can estimate a critical coupling as the value
at which the lowest masses reach zero. This intersection
is illustrated in Fig.~\ref{fig:critcoup}, where the
lowest mass-squared values are plotted as well as four
times the odd-eigenstate mass squared. Because there
are no bound states in this theory the lowest even-eigenstate
mass should be equal to this; the difference in the plot
is another measure of the numerical and truncation errors.
From this plot, we estimate the critical value of the
dimensionless coupling to be 2.1$\pm$0.05. For comparison,
we list in Table~\ref{tab:critcoup} this and values
from other computations, as gathered in \cite{RychkovVitale};
however, because the definitions of dimensionless couplings vary,
the table uses the definition $\bar{g}\equiv\lambda/24\mu^2$,
which is just $\frac{\pi}{6}g$.
\begin{figure}[ht]
\vspace{0.2in}
\centerline{\includegraphics[width=15cm]{critcoup2.eps}}
\caption{\label{fig:critcoup}
The lowest masses for the odd and even cases, as used
to estimate the critical coupling, including a plot
of the threshold for two-particle states at four
times the mass-squared of the odd case.
}
\end{figure}
\begin{table}[ht]
\caption{\label{tab:critcoup}
Comparison of critical coupling values, adapted
from~\protect\cite{RychkovVitale}, with a slightly
different definition of the dimensionless
coupling $\bar{g}=\frac{\pi}{6}g$.
The first two values were computed
in light-front quantization and the remainder in equal-time
quantization; the results are comparable but only after the
different mass renormalizations are taken into account,
as discussed in the text.}
\begin{center}
\begin{tabular}{lll}
\hline \hline
Method & $\bar{g}_c$ & Reported by \\
\hline
Light-front symmetric polynomials & $1.1\pm0.03$ & this work \\
DLCQ & 1.38 & Harindranath \& Vary~\protect\cite{VaryHari} \\
Quasi-sparse eigenvector & 2.5 &
Lee \& Salwen~\protect\cite{LeeSalwen} \\
Density matrix renormalization group & 2.4954(4) &
Sugihara~\protect\cite{Sugihara} \\
Lattice Monte Carlo & 2.70$\left\{\begin{array}{l} +0.025 \\ -0.013\end{array}\right.$ &
Schaich \& Loinaz~\protect\cite{SchaichLoinaz} \\
& $2.79\pm0.02$ & Bosetti {\em et al.}~\protect\cite{Bosetti} \\
Uniform matrix product & 2.766(5) &
Milsted {\em et al.}~\protect\cite{Milsted} \\
Renormalized Hamiltonian truncation & 2.97(14) &
Rychkov \& Vitale~\protect\cite{RychkovVitale} \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
The results listed in Table~\ref{tab:critcoup}
imply a systematic difference between equal-time and light-front values
for the critical coupling, which is exactly what should be expected,
based on the difference in mass renormalizations discussed in
Sec.~\ref{sec:Mass}. To quantify this difference, we extended the
diagonalization of the Hamiltonian matrix to include the entire
spectrum and computed the shift $\Delta$, defined in (\ref{eq:Delta}).
The results are plotted in Fig.~\ref{fig:shift}, along with
extrapolations of fits to the shifts for coupling values below 1.
Higher coupling values are not used in the fits because the lack of
sector-dependent mass renormalization does not allow a
reasonable approximation to the wave functions. The one-particle
sector should become less and less probable for the lowest eigenstate,
as the critical coupling is approached and as its mass approaches
zero; instead, the one-particle probability remains finite and
the product $|\psi_{11}|\ln(M_1^2)$ diverges. To judge the
value of the coupling where this effect becomes noticeable,
we studied the behavior of the dimensionless mass $M^2/\mu_{\rm ET}^2$,
as predicted by (\ref{eq:couplingratio}), as a function of
the coupling, which we plot in Fig.~\ref{fig:ETmass}. For
coupling values above 1, the mass begins to increase rather
than decrease; this incorrect behavior is the precursor of
the divergence at the critical coupling; we also see that
the convergence with respect to the polynomial basis becomes
somewhat worse at these larger coupling values.
The estimated value of the shift at the critical
coupling 2.1, based on the two extrapolations,
is $\Delta(g=2.1)=-0.47\pm0.12$. The value is
from the higher-order extrapolation, with the
lower-order extrapolation used to indicate the
error. From the latest equal-time value
for the critical coupling~\cite{RychkovVitale},
$g_{\rm ETc}=\frac{6}{\pi}2.97=5.67$, we extract
a shift of $(g_{\rm LFc}/g_{\rm ETc}-1)/g_{\rm LFc}=-0.30$,
which is consistent with the estimated value of the shift.
\begin{figure}[ht]
\vspace{0.2in}
\centerline{\includegraphics[width=15cm]{shift.eps}}
\caption{\label{fig:shift}
The renormalization shift $\Delta$, defined in Eq.~(\ref{eq:Delta})
of the text, as a function of the square of the dimensionless coupling $g$.
The points displayed are obtained as extrapolations in the polynomial
basis size. The lines are linear and quadratic fits to shifts below
$g=1$, extrapolated to the region of the critical coupling.}
\end{figure}
\begin{figure}[ht]
\vspace{0.2in}
\centerline{\includegraphics[width=15cm]{ETmass.eps}}
\caption{\label{fig:ETmass}
Lowest equal-time mass eigenvalues for odd numbers of constituents
plotted versus the dimensionless light-front coupling $g$.
Each mass value is obtained from the light-front masses and
the computed shift $\Delta$ according to Eq.~(\ref{eq:couplingratio})
of the text. Different points at the same $g$ value correspond to
different truncations of the polynomial basis size.
}
\end{figure}
\section{Summary}
\label{sec:Summary}
We have developed a high-order method
for ($1+1$)-dimensional light-front theories
that is distinct from DLCQ~\cite{PauliBrodsky,DLCQreview}. It is based on fully symmetric
multivariate polynomials~\cite{GenSymPolys} that respect the momentum conservation constraint
and allows separate tuning of resolutions
in each Fock sector. This method could be combined with transverse
discretization or basis functions for numerical solution of
($3+1$)-dimensional theories.
As an illustration, the method has been applied here
to $\phi^4_{1+1}$ theory. The lowest mass eigenvalues
have been computed, as shown in Figs.~\ref{fig:extrap-odd} and \ref{fig:extrap-even};
they converge rapidly with respect to the Fock-space truncation.
The odd case includes comparison with the light-front coupled-cluster
(LFCC) method~\cite{LFCC,LFCCphi4},
which indicates that the LFCC method combined with symmetric polynomials shows
promise for rapid convergence.
Either approach can also be applied to the negative-mass
squared case, where the symmetry breaking is explicit.
From the behavior of the mass eigenstates with respect to coupling strength,
as shown in Fig.~\ref{fig:critcoup}, we have extracted an estimate of
the critical coupling for $\phi^4_{1+1}$ theory with positive mass squared.
Above this coupling, the symmetry is broken. With mass renormalization
properly taken into account, as discussed in Sec.~\ref{sec:Mass},
the value obtained ($g_c=2.1\pm0.05$ or $\bar{g}_c=1.1\pm0.03$) is
comparable to values obtained in equal-time quantization.
The calculation can be improved by invocation of sector-dependent mass
renormalization~\cite{Wilson,hb,Karmanov,SecDep}, so that higher Fock
states can make a significant contribution as the coupling approaches
the critical value. However, any comparison with equal-time quantization
will then require the use of physical quantities as reference points,
rather than a direct comparison of bare couplings.
\acknowledgments
This work was supported in part by
the Minnesota Supercomputing Institute through
grants of computing time and (for M.B.) supported in part by
the US DOE under grant number FG03-95ER40965.
|
1,108,101,563,252 | arxiv | \section{Introduction}
Cones of divisors play an essential role in describing the birational geometry of a smooth complex projective variety $X$. A key feature of these cones is their interplay with cones of curves via duality statements. The dual of the nef cone and the pseudo-effective cone of divisors were determined by \cite{kleiman66} and \cite{bdpp04} respectively. We consider the third cone commonly used in birational geometry: the movable cone of divisors.
\begin{defn}
Let $X$ be a smooth projective variety over $\mathbb{C}$. The movable cone $\overline{\mathrm{Mov}}^{1}(X) \subset N^{1}(X)$ is the closure of the cone generated by classes of effective Cartier divisors $L$ such that the base locus of $|L|$ has codimension at least $2$. We say a divisor is movable if its numerical class lies in $\overline{\mathrm{Mov}}^{1}(X)$.
\end{defn}
\begin{defn}
Let $X$ be a smooth projective variety over $\mathbb{C}$. We say that an irreducible curve $C$ on $X$ is movable in codimension $1$, or a $\mathrm{mov}^{1}$-curve, if it deforms to cover a codimension $1$ subset of $X$.
\end{defn}
It is natural to guess that a divisor $L$ is movable if and only if it has non-negative intersection with every $\mathrm{mov}^{1}$-curve. This is false, as demonstrated by \cite{payne06} Example 1. Nevertheless, Debarre and Lazarsfeld have asked whether one can formulate a duality statement for movable divisors and $\mathrm{mov}^{1}$-curves. This has been accomplished for toric varieties in \cite{payne06} and for Mori Dream Spaces in \cite{choi10} by taking other birational models of $X$ into account. Our main theorem proves an analogous statement for all smooth varieties.
Before stating this theorem, we need to analyze the behavior of the movable cone under birational transformations. Suppose that $\phi: Y \to X$ is a birational map of smooth projective varieties and that $L$ is a movable divisor on $X$. It is possible that $\phi^{*}L$ is not movable -- for example, some $\phi$-exceptional centers could be contained in the base locus of $L$. The following definition from \cite{nakayama04} allows us to quantify the loss in movability.
\begin{defn}
Let $X$ be a smooth projective variety over $\mathbb{C}$ and let $L$ be a pseudo-effective $\mathbb{R}$-divisor on $X$. Fix an ample divisor $A$ on $X$. For any prime divisor $\Gamma$ on $X$ we define
\begin{equation*}
\sigma_{\Gamma}(L) = \inf\{\mathrm{mult}_{\Gamma}(L') | L' \geq 0 \textrm{ and } L' \sim_{\mathbb{R}} L + \epsilon A \textrm{ for some } \epsilon > 0 \}
\end{equation*}
where $\sim_{\mathbb{R}}$ denotes $\mathbb{R}$-linear equivalence. As demonstrated by \cite{nakayama04} III.1.5 Lemma, $\sigma_{\Gamma}$ is independent of the choice of $A$.
\end{defn}
Suppose that $E$ is an exceptional divisor for a birational map $\phi: Y \to X$. The $\mathbb{R}$-divisor $\sigma_{E}(\phi^{*}L)E$ represents the ``extra contribution'' from $E$ to the non-movability of $\phi^{*}L$. By subtracting these contributions, we can understand the geometry of the original divisor $L$.
\begin{defn} \label{movtrans}
Let $X$ be a smooth projective variety over $\mathbb{C}$ and let $L$ be a pseudo-effective $\mathbb{R}$-divisor on $X$. Suppose that $\phi: Y \to X$ is a birational map from a smooth variety $Y$. The movable transform of $L$ on $Y$ is defined to be
\begin{equation*}
\phi^{-1}_{\mathrm{mov}}(L) := \phi^{*}L - \sum_{E \, \, \phi \mathrm{-exceptional}} \sigma_{E}(\phi^{*}L)E.
\end{equation*}
\end{defn}
Note that the movable transform is not linear and is only defined for pseudo-effective divisors. We can now state our main theorem.
\begin{thrm} \label{maintheorem}
Let $X$ be a smooth projective variety over $\mathbb{C}$ and let $L$ be a pseudo-effective $\mathbb{R}$-divisor. $L$ is not movable if and only if there is a $\mathrm{mov}^{1}$-curve $C$ on $X$ and a birational morphism $\phi: Y \to X$ from a smooth variety $Y$ such that
\begin{equation*}
\phi^{-1}_{mov}(L) \cdot \widetilde{C} < 0
\end{equation*}
where $\widetilde{C}$ is the strict transform of a generic deformation of $C$.
\end{thrm}
There does not seem to be an easy way to translate Theorem \ref{maintheorem} into a statement involving only intersections on $X$. This is a symptom of the fact that the natural operation on movable divisors is the push-forward and not the pull-back.
The proof of Theorem \ref{maintheorem} is accomplished by reinterpreting the orthogonality theorem of \cite{bdpp04} and \cite{bfj09} using the techniques of \cite{lehmann10}.
\begin{exmple}
For surfaces Theorem \ref{maintheorem} reduces to the usual duality of the nef and pseudo-effective cones.
\end{exmple}
\begin{exmple}
Suppose that $X$ is a smooth Mori dream space and $L$ is an $\mathbb{R}$-divisor on $X$. By running the $L$-MMP as in \cite{hk00}, we obtain a small modification $\phi: X \dashrightarrow X'$, a morphism $f: X \to Z$, and an ample $\mathbb{R}$-divisor $A$ on $Z$ such that
\begin{equation*}
\phi_{*}^{-1}L \equiv f^{*}A
\end{equation*}
where $\phi_{*}^{-1}$ denotes the strict transform.
Let $W$ be a smooth variety admitting birational maps $\psi: W \to X$ and $\psi': W \to X'$. Using \cite{nakayama04} III.5.5 Proposition, one easily verifies that
\begin{equation*}
\psi^{-1}_{mov}(L) \equiv \psi'^{*}(\phi_{*}^{-1}L).
\end{equation*}
Thus Theorem \ref{maintheorem} implies the statements of \cite{payne06} and \cite{choi10}: for a smooth toric variety or Mori Dream Space $X$, a divisor class is movable iff its strict transform class on every $\mathbb{Q}$-factorial small modification $X'$ has non-negative intersection with every $\mathrm{mov}^{1}$-curve on $X'$.
\end{exmple}
\begin{exmple}
Suppose that $X$ is a smooth projective variety with $K_{X}$ numerically trivial. \cite{choi10} explains how to apply techniques of the minimal model program to analyze $\overline{\mathrm{Mov}}^{1}(X)$. Just as before, a divisor class is movable if and only if its strict transform class on every $\mathbb{Q}$-factorial small modification has non-negative intersection with every $\mathrm{mov}^{1}$-curve. When $X$ is hyperk\"ahler, \cite{huybrechts03} and \cite{boucksom04} show that in fact it suffices to consider small modifications that are also smooth hyperk\"ahler varieties.
More generally, \cite{choi10} shows that small modifications can detect certain regions of $\overline{\mathrm{Mov}}^{1}(X)$ by using the minimal model program.
\end{exmple}
We will also prove a slightly stronger version of Theorem \ref{maintheorem} that involves the non-nef locus $\mathbf{B}_{-}(L)$ of $L$ (which will be defined in Definition \ref{nonnefdef}). Although the non-nef locus represents the ``obstruction'' to the nefness of $L$, it is not true that $\mathbf{B}_{-}(L)$ is covered by curves $C$ with $L \cdot C < 0$. However, Proposition \ref{mainprop} formulates a birational version of this negativity using the movable transform.
Finally, we will use Proposition \ref{mainprop} to understand $k$-movability for $k>1$. Define the $k$-movable cone of $X$ to be the closure of the cone in $N^{1}(X)$ generated by effective Cartier divisors whose base locus has codimension at least $k-1$. We say that a divisor is $k$-movable if its numerical class lies in the $k$-movable cone. Note that the $1$-movable cone is just $\overline{\mathrm{Mov}}^{1}(X)$.
Debarre and Lazarsfeld have asked whether there is a duality between the $k$-movable cone of divisors and the closure of the cone of irreducible curves that deform to cover a codimension $k$ subset (for $0 < k < \dim X$). Corollary \ref{maincor} constructs a birational version of this duality. Again, this generalizes results for toric varieties in \cite{payne06} and for Mori dream spaces in \cite{choi11}.
\section{Background}
Throughout $X$ will denote a smooth projective variety over $\mathbb{C}$. We use the notations $\sim, \sim_{\mathbb{Q}}, \sim_{\mathbb{R}}, \equiv$ to denote respectively linear equivalence, $\mathbb{Q}$-linear equivalence, $\mathbb{R}$-linear equivalence, and numerical equivalence of $\mathbb{R}$-divisors. The volume of an $\mathbb{R}$-divisor $L$ is
\begin{equation*}
\mathrm{vol}_{X}(L) = \limsup_{m \to \infty} \frac{h^{0}(X,\lfloor mL \rfloor)}{m^{\dim X}}.
\end{equation*}
\subsection{Divisorial Zariski decomposition}
Let $L$ be a pseudo-effective $\mathbb{R}$-divisor on a smooth projective variety $X$. Recall that for a prime divisor $\Gamma$ on $X$ we have defined
\begin{equation*}
\sigma_{\Gamma}(L) = \inf\{\mathrm{mult}_{\Gamma}(L') | L' \geq 0 \textrm{ and } L' \sim_{\mathbb{R}} L + \epsilon A \textrm{ for some } \epsilon > 0 \}.
\end{equation*}
where $A$ is any fixed ample divisor. \cite{nakayama04} III.1.11 Corollary shows that there are only finitely many prime divisors $\Gamma$ on $X$ with $\sigma_{\Gamma}(L) > 0$, allowing us to make the following definition.
\begin{defn}[\cite{nakayama04} III.1.16 Definition] \label{zardecom}
Let $L$ be a pseudo-effective $\mathbb{R}$-divisor on $X$. Define
\begin{equation*}
N_{\sigma}(L) = \sum \sigma_{E}(L) E \qquad \qquad P_{\sigma}(L) = L - N_{\sigma}(L)
\end{equation*}
The decomposition $L = N_{\sigma}(L) + P_{\sigma}(L)$ is called the \emph{divisorial Zariski
decomposition} of $L$.
\end{defn}
Note that for a birational morphism $\phi: Y \to X$ we have $\phi_{mov}^{-1}(L) = P_{\sigma}(\phi^{*}L) + \phi_{*}^{-1}N_{\sigma}(L)$ where $\phi_{*}^{-1}$ denotes the strict transform. The divisorial Zariski decomposition is closely related to the non-nef locus of $L$.
\begin{defn} \label{nonnefdef}
Let $X$ be a smooth projective variety and let $L$ be a pseudo-effective $\mathbb{R}$-divisor on $X$. We define the $\mathbb{R}$-stable base locus of $L$ to be the subset of $X$ given by
\begin{equation*}
\mathbf{B}_{\mathbb{R}}(L) = \bigcup \{ \mathrm{Supp}(L') | L' \geq 0 \textrm{ and }L' \sim_{\mathbb{R}} L \}.
\end{equation*}
The non-nef locus of $L$ is then defined to be
\begin{equation*}
\mathbf{B}_{-}(L) = \bigcup_{A \textrm{ ample }\mathbb{R}\textrm{-divisor}} \mathbf{B}_{\mathbb{R}}(L+A).
\end{equation*}
\end{defn}
The following proposition records the basic properties of the divisorial Zariski decomposition.
\begin{prop}[\cite{nakayama04} III.1.14 Proposition, III.2.5 Lemma, V.1.3 Theorem]
Let $X$ be a smooth projective variety and let $L$ be a pseudo-effective $\mathbb{R}$-divisor.
\begin{enumerate}
\item $P_{\sigma}(L)$ is a movable $\mathbb{R}$-divisor. In particular for any prime divisor $E$ the restriction $P_{\sigma}(L)|_{E}$ is pseudo-effective.
\item If $\phi: Y \to X$ is a birational morphism of smooth varieties and $\Gamma$ is a prime divisor on $Y$ that is not $\phi$-exceptional, then $\sigma_{\Gamma}(\phi^{*}L) = \sigma_{\phi(\Gamma)}(L)$.
\item The union of the codimension $1$ components of $\mathbf{B}_{-}(L)$ coincides with $\mathrm{Supp}(N_{\sigma}(L))$.
\end{enumerate}
\end{prop}
\subsection{Numerical dimension and orthogonality}
Given a pseudo-effective divisor $L$, the numerical dimension $\nu(L)$ of \cite{nakayama04} and \cite{bdpp04} is a numerical measure of the ``positivity'' of $L$. There is also a restricted variant $\nu_{X|V}(L)$ introduced in \cite{bfj09}; since the definition is somewhat involved, we will only refer to a special subcase using an alternate characterization from \cite{lehmann10}.
\begin{defn} \label{numdimdef}
Let $L$ be a pseudo-effective divisor on $X$. Fix a prime divisor $E$ on $X$ and choose $L' \equiv L$ whose support does not contain $E$. We say $\nu_{X|E}(L) = 0$ if
\begin{equation*}
\liminf_{\phi} \mathrm{vol}_{\widetilde{E}}(P_{\sigma}(\phi^{*}L')|_{\widetilde{E}}) = 0
\end{equation*}
where $\phi: \widetilde{X} \to X$ varies over all birational maps and $\widetilde{E}$ denotes the strict transform of $E$.
\end{defn}
The connection with geometry is given by the following version of the orthogonality theorem of \cite{bdpp04} and \cite{bfj09}.
\begin{thrm}[\cite{bfj09}, Theorem 4.15]
Let $L$ be a pseudo-effective divisor. If a prime divisor $E \subset X$ is contained in $\mathrm{Supp}(N_{\sigma}(L))$ then $\nu_{X|E}(L) = 0$.
\end{thrm}
\begin{proof}
Fix an ample divisor $A$. Choose an $\epsilon > 0$ sufficiently small so that $\mathrm{Supp}(N_{\sigma}(L)) = \mathrm{Supp}(N_{\sigma}(L + \epsilon A))$ and apply \cite{bfj09} Theorem 4.15. The comparison between the numerical dimension of \cite{bfj09} and Definition \ref{numdimdef} is given by \cite{lehmann10} Theorem 7.1.
\end{proof}
\section{Proof}
\begin{proof}[Proof of Theorem \ref{maintheorem}:]
Suppose that $L$ is not movable. Denote by $E$ a fixed divisorial component of $N_{\sigma}(L)$.
Fix a sufficiently general ample divisor $A$ on $X$ and choose $\epsilon$ small enough so that $E$ is a component of $N_{\sigma}(L+\epsilon A)$. Applying the orthogonality theorem of \cite{bdpp04}, we see there is a birational map $\phi: Y \to X$ so that
\begin{enumerate}
\item $\widetilde{E}$ is smooth.
\item $\mathrm{vol}_{\widetilde{E}}(P_{\sigma}(\phi^{*}(L+\epsilon A))|_{\widetilde{E}}) < \mathrm{vol}_{E}(A|_{E}) = \mathrm{vol}_{\widetilde{E}}(\phi^{*}A|_{\widetilde{E}})$.
\item The strict transform of every component of $N_{\sigma}(L)$ is disjoint.
\end{enumerate}
There is a unique expression
\begin{equation*}
P_{\sigma}(\phi^{*}(L+\epsilon A)) = P_{\sigma}(\phi^{*}L) + \phi^{*}A + \alpha(\epsilon)\widetilde{E} + F
\end{equation*}
where $\widetilde{E}$ is the strict transform of $E$, $F$ is an effective divisor with $F \leq N_{\sigma}(\phi^{*}L)$ and the support of $F$ does not contain $E$, and $\alpha(\epsilon)$ is positive and goes to $0$ as $\epsilon$ goes to $0$. By shrinking $\epsilon$ we may ensure that $\alpha(\epsilon) < \sigma_{E}(L)$.
Condition (2) above, along with Lemma \ref{vollemma}, show that the restriction $(P_{\sigma}(\phi^{*}L) + \alpha(\epsilon)\widetilde{E})|_{\widetilde{E}}$ is not pseudo-effective for any $\epsilon > 0$. Since $\alpha(\epsilon) < \sigma_{E}(L)$, we also have that $(P_{\sigma}(\phi^{*}L) + \sigma_{E}(L)\widetilde{E})|_{\widetilde{E}}$ is not pseudo-effective. As the strict transform of components of $N_{\sigma}(L)$ are disjoint, the restriction of $P_{\sigma}(\phi^{*}L) + \phi_{*}^{-1} N_{\sigma}(L)$ to $\widetilde{E}$ is still not pseudo-effective.
By \cite[0.2 Theorem]{bdpp04} there is a curve $\widetilde{C}$ whose deformations cover $\widetilde{E}$ such that $$(P_{\sigma}(\phi^{*}L) + \phi_{*}^{-1} N_{\sigma}(L)) \cdot \widetilde{C} < 0.$$ Since $\widetilde{E}$ is not $\phi$-exceptional, $C = \phi(\widetilde{C})$ is a $\mathrm{mov}^{1}$-curve.
Conversely, if $L$ is movable, then $\phi^{-1}_{mov}(L) = P_{\sigma}(\phi^{*}L)$ is also movable for every $\phi$. Thus every movable transform has non-negative intersection with the strict transform of every $\mathrm{mov}^{1}$-curve general in its family.
\end{proof}
\begin{lem} \label{vollemma}
Let $X$ be a smooth projective variety and let $L$ and $L'$ be pseudo-effective divisors on $X$. Then $\mathrm{vol}_{X}(L + L') \geq \mathrm{vol}_{X}(L)$.
\end{lem}
\begin{proof}
We may assume $L$ is big since otherwise the inequality is automatic. Then for any sufficiently small $\epsilon > 0$ we have
$$\mathrm{vol}_{X}(L+L') = \mathrm{vol}_{X}((1-\epsilon) L + (\epsilon L + L')) \geq (1-\epsilon)^{\dim X}\mathrm{vol}_{X}(L)$$
since $\epsilon L + L'$ is big.
\end{proof}
We now give an alternate formulation of Theorem \ref{maintheorem}.
\begin{prop} \label{mainprop}
Let $X$ be a smooth projective variety and let $L$ be a pseudo-effective $\mathbb{R}$-divisor. Suppose that $V$ is an irreducible subvariety of $X$ contained in $\mathbf{B}_{-}(L)$ and let $\psi: X' \to X$ be a smooth birational model resolving the ideal sheaf of $V$. Then there is a birational morphism $\phi: Y \to X'$ from a smooth variety $Y$ and an irreducible curve $\widetilde{C}$ on $Y$ such that
\begin{equation*}
\phi^{-1}_{mov}(\psi^{*}L) \cdot \widetilde{C} < 0
\end{equation*}
and $\psi \circ \phi(\widetilde{C})$ deforms to cover $V$.
\end{prop}
\begin{proof}
Let $E$ be the $\psi$-exceptional divisor dominating $V$. Since we have $E \subset \mathrm{Supp}(N_{\sigma}(\psi^{*}L))$, we may argue as in the proof of Theorem \ref{maintheorem} for $\psi^{*}L$ and $E$ to find a birational map $\phi$ such that $\phi^{-1}_{mov}(\psi^{*}L)|_{\widetilde{E}}$ is not pseudo-effective.
\cite[2.4 Theorem]{bdpp04} shows that there is some curve $\widetilde{C}$ on $\widetilde{E}$ with $\phi^{-1}_{mov}(\psi^{*}L) \cdot \widetilde{C} < 0$ such that $\widetilde{C}$ deforms to cover $\widetilde{E}$ and is not contracted by any morphism from $\widetilde{E}$ to a variety of positive dimension. Choosing $\widetilde{C}$ on $\widetilde{E}$ to satisfy this stronger property, we obtain the statement of Proposition \ref{mainprop}.
\end{proof}
Proposition \ref{mainprop} shows that the non-nef locus is covered by $L$-negative curves in a birational sense. Alternatively, one can rephrase this result using $k$-movability.
\begin{cor} \label{maincor}
Let $X$ be a smooth projective variety and let $L$ be a pseudo-effective $\mathbb{R}$-divisor. Then $L$ is not $k$-movable if and only if there is a birational morphism $\psi: X' \to X$ from a smooth variety $X'$, a birational morphism $\phi: Y \to X'$ from a smooth variety $Y$, and an irreducible curve $\widetilde{C}$ on $Y$ such that
\begin{equation*}
\phi^{-1}_{mov}(\psi^{*}L) \cdot \widetilde{C} < 0
\end{equation*}
and $\psi \circ \phi(\widetilde{C})$ deforms to cover a $k$-dimensional subset of $V$.
\end{cor}
\begin{proof}
To say that $L$ is not $k$-movable is equivalent to saying that $\mathbf{B}_{-}(L)$ has a component of dimension at least $k$. Apply Proposition \ref{mainprop} to obtain the forward implication. The converse is immediate.
\end{proof}
\begin{rmk}
It is unclear whether Corollary \ref{maincor} is the best formulation possible for the duality of $k$-movable divisors. For Mori Dream Spaces varieties and for $2 < k < \dim X$, \cite{payne06} Theorem 1 and \cite{choi11} Corollary 3 prove a slightly stronger statement. The essential difference is that one does not need to blow-up along top-dimensional components of $\mathbf{B}_{-}(L)$. More precisely, if $L$ is not $k$-movable, one may find a $\mathbb{Q}$-factorial small modification $f: X \dashrightarrow X'$ that is regular at the generic point of a component $V \subset \mathbf{B}_{-}(L)$ of codimension at most $k$ and a family of curves covering the strict transform of $V$ with $f_{*}L \cdot C < 0$. In contrast, Corollary \ref{maincor} may produce a birational map that is not regular at any point of $V$.
\end{rmk}
\nocite{*}
\bibliographystyle{amsalpha}
|
1,108,101,563,253 | arxiv | \section{Introduction}
In this paper we derive the logarithmic versions of several well-known functional inequalities. Some inequalities are obtained with best constants, or with semi-explicit constants, which are useful for some further applications. Our techniques allow us to derive these inequalities in rather general settings, so we will be working in the settings of general Lie groups, as well as on several classes of nilpotent Lie groups, namely, graded and homogeneous Lie groups. Since on stratified Lie groups we also have the horizontal gradient at our disposal, we will also formulate versions of some of the inequalities in the setting of stratified groups, using the horizontal gradient instead of a power of a sub-Laplacian.
In the Euclidean spaces, in one of the Sobolev's pioneering works, Sobolev obtained the following inequality, which at this moment is bearing his name:
\begin{equation}
\|u\|_{L^{p^{*}}(\mathbb{R}^{n})}\leq C \|\nabla u\|_{L^{p}(\mathbb{R}^{n})},
\end{equation}
where $1<p<n$, $p^{*}=\frac{np}{n-p}$ and $C=C(n,p)>0$ is a positive constant. The best constant of this inequality was obtained by Talenti in \cite{T76} and Aubin in \cite{A76}. The Sobolev inequality is one of the most important tools in studying PDE and variational problems.
Folland and Stein extended Sobolev's inequality to general stratified groups (see e.g. \cite{GV}):
if $\mathbb{G}$ is a stratified group and $\Omega\subset \mathbb{G}$ is an open set, then there exists a constant $C>0$ such that we have
\begin{equation}\label{Sobolev-Folland-Stein}
\|u\|_{L^{p^{*}}(\Omega)}\leq C\left(\int_{\Omega}|\nabla_{H} u|^{p}dx\right)^{\frac{1}{p}},\; 1<p<Q,\; p^{*}=\frac{Qp}{Q-p},
\end{equation}
for all $u\in C_{0}^{\infty}(\Omega)$. Here $\nabla_{H}$ is the horizontal gradient and $Q$ is the homogeneous dimension of $\mathbb{G}$.
Inequality \eqref{Sobolev-Folland-Stein} is called the Sobolev or Sobolev-Folland-Stein inequality.
Furthermore, in relation to groups, we can mention
Sobolev inequalities and embeddings on general unimodular Lie groups \cite{VSCC93}, on general locally compact unimodular groups \cite{AR20}, on general noncompact Lie groups \cite{BPTV19,BPV21}, as well as Hardy-Sobolev inequalities on general Lie groups \cite{RY19}.
The Sobolev inequality on graded groups using Rockland operators was proved in \cite{FR17} and the best constant for it was obtained in \cite{RTY20}.
On the other hand, the logarithmic Sobolev inequality was shown to hold on $\mathbb{R}^n$ in the following form:
\begin{equation}\label{ddlogin}
\int_{\mathbb{R}^{n}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{R}^{n})}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{R}^{n})}}\right)dx\leq\frac{n}{p}\log\left(C\frac{\|\nabla u\|^{p}_{L^{p}(\mathbb{R}^{n})}}{\|u\|^{p}_{L^{p}(\mathbb{R}^{n})}}\right).
\end{equation}
We can refer to \cite{Wei78} for the case $p=2$, but to e.g. \cite{DD03} for some history review of cases
$1< p<n$, including the discussion of best constants.
In \cite{Mer08}, the author obtained a logarithmic Gagliardo-Nirenberg inequality. In \cite {FNQ18} and \cite{KRS20} the authors proved the logarithmic Sobolev inequality and the fractional logarithmic Sobolev inequality on the Heisenberg group and on homogeneous groups, respectively. A fractional weighted version of \eqref{ddlogin} on homogeneous groups was proved in \cite{KS20}. In this paper, we prove logarithmic Sobolev inequalities on graded groups and weighted logarithmic Sobolev inequalities on general Lie groups. As applications of these inequalities we show Nash and weighted Nash inequalities on graded and general Lie groups, respectively. The log-Sobolev type inequalities with weights are also sometimes called the log-Hardy inequalities \cite{DDFT10}.
The subject of the logarithmic inequalities has been extensively investigated and it is impossible to give a reasonably complete review of the literature here. We refer to surveys \cite{AB00, GZ03}, to works on relations to other inequalities \cite{BL00, MF93}, coercive inequalities on Carnot groups \cite{Bou21, BZ21a, BZ21b, BZ21c, HZ09}, Nash \cite{Nas58, Bec98, BDS20, OS18} and weighted Nash \cite{BBGM12} inequalities. There are works on the fractional Laplacian \cite{Bec12} as well as on the
logarithmic Sobolev inequalities for the fractional Laplacian \cite{KRS20} in the setting of Folland and Stein (\cite{FS}) homogeneous groups.
Logarithmic inequalities have various applications in different areas, see e.g. \cite{BH99, Car91, FNQ18, Mer08, OW16}.
Explicit constants for some inequalities play an important role (e.g. \cite{T76, Bec75, DD03}), for
best constants on graded Lie groups see \cite{RTY20}.
In view of extending Sobolev inequalities to infinite dimensional spaces, in \cite{Gro75}, L. Gross showed the logarithmic Sobolev inequality in
the following form:
\begin{equation}\label{EQ:Gross}
\int_{\mathbb{R}^{n}}|u(x)|^{2}\log \left(\frac{|u(x)|}{\|u\|_{L^{2}(\mu)}}\right)d\mu(x)
\leq \int_{\mathbb{R}^{n}}|\nabla u(x)|^{2}d\mu(x),
\end{equation}
with the Gaussian measure $d\mu(x)=(2\pi)^{-n/2}e^{-|x|^2/2}dx$.
The importance of \eqref{EQ:Gross} and the use of the Gaussian measure is that it has a uniform constant independent of the dimension, allowing passing to infinite dimensional spaces, and in particular also leading to the ultracontractivity and hypercontractivity properties of the corresponding Markovian semigroups. Due to its importance in many applications, this subject has been intensively investigated, see e.g.
\cite{Ad79, AC79, Ros76, Wei78, Tos97, SZ92}, to mention only very few papers. It is related to uncertainty principles \cite{Bec95} and to
Poincar\'e inequalities \cite{Bec89}, with the
Poincar\'e and logarithmic Sobolev inequalities having different forms \cite{Gen14} and applications, see e.g. \cite{AS94}. We can also refer to the recent work on the
Poincar\'e inequality on filiform type stratified groups \cite{CFZ21}.
In this paper, among other things, we prove the logarithmic Gross inequality on stratified groups, with the gradient $\nabla u$ in \eqref{EQ:Gross} replaced by the horizontal gradient $\nabla_H u$, collecting vector fields from the first stratum. Moreover, {\bf very curiously}, our Gross inequality \eqref{gaus.log.sobi} comes with the Gaussian measure on the first stratum. This appears to be a natural extension of the Euclidean case since such inequality turns out to be equivalent to the $L^2$-log-Sobolev inequality with the horizontal gradient, see Theorem \ref{equiv.thm}.
An interesting result in this perspective is \cite[Theorem 6.3]{HZ09} where Hebisch and Zegarlinski have shown that on stratified groups for probability measures $\mu=e^{-\beta d^p}/Z\, d\lambda$, $\beta\geq 0$, $p\geq 1$, corresponding to smooth homogeneous norms $d(x)$, there is no $L^q$-log-Sobolev inequalities of the type \eqref{EQ:Gross}, for $q\in (1,2]$. Furthermore, they went on to use the heat kernel on the Heisenberg group $\mathbb{H}^n$ to construct a probability measure $\mu$ (with a singular norm) on $\mathbb{H}^n$ such that the $L^2$-log-Sobolev inequality of the Gross type \eqref{gaus.log.sobi} holds on $\mathbb{H}^n$. Our result in Theorem \ref{semi-g} gives such measure on general stratified groups with the probability component on the first stratum, while Theorem \ref{equiv.thm} shows that one can not do better if one wants to start with the inequality
\eqref{cor.log.sobi}, which is clearly true independently of any further choice of measure.
There is an extensive literature on conditions for log-Sobolev inequalities to hold, such as Bakry-Emery and other criteria. Such analysis is not part of this work where we do not work with full probability measures.
As already mentioned, the log-Sobolev inequalities have many applications. For example, in \cite{KRS20} the authors applied the obtained log-Sobolev inequalities for the fractional Laplacian on homogeneous groups, to show blow-up results for some classes of nonlinear PDEs. Applications of the current results are still possible and will be addressed in the sequel.
The methods of this paper allow us to also derive new results for the Euclidean space $\mathbb R^n$, yielding weighted Gross type log-Sobolev inequalities for weights given in terms of arbitrary homogeneous quasi-norms. To give an example, let us consider the family of Euclidean norms on $\mathbb R^n$ given by
$|x|_p=(|x_1|^p+\cdots+|x_n|^p)^{1/p}$ for $1\leq p<\infty$, and by $|x|_\infty=\max\limits_{1\leq j\leq n}|x_j|$, for $x\in\mathbb R^n.$
Let $0\leq \beta <2<n$.
Let $\mu_p$ be the Gaussian measure on $\mathbb{R}^{n}$ given by
$d\mu_p=\gamma_p e^{-\frac{|x|_2^2}{2}}dx$, $1\leq p\leq\infty$, with the normalisation constant $\gamma_p$ given in \eqref{k-w-Rn}, depending on $p$.
Then we have the following weighted Gross-type log-Sobolev inequality:
\begin{equation}
\label{gaus.log.sob-w-Rn-i}
\int_{\mathbb R^n}|x|_p^{-\frac{\beta(n-2)}{n-\beta}}|g|^2\log\left(|x|_p^{-\frac{\beta(n-2)}{2(n-\beta)}}|g|\right)\,d\mu_p \leq \int_{\mathbb R^n} |\nabla g|^2\,d\mu_p,\quad
0\leq \beta <2<n,
\end{equation}
for all $g$ such that $\||x|_p^{-\frac{\beta(n-2)}{2(n-\beta)}}g\|_{L^2(\mu_p)}=1.$
We note that for $\beta=0$ this implies the classical Gross inequality \eqref{EQ:Gross}, and we refer to Corollary \ref{semi-g-w-Rn} for further details.
The case of $\beta=2$ in \eqref{gaus.log.sob-w-Rn-i} corresponds to the logarithmic Hardy inequality and requires very different methods from those developed in this paper; this case will be addressed in \cite{CKR21}. For a discussion of weights in inequalities of the type \eqref{ddlogin}, but only with one weight, we can refer to \cite{Das21}.
The case $p=1$ of the log-Sobolev inequalities in this paper corresponds to the Shannon type inequalities; we will not discuss it here but refer to \cite{CKR21a} for this case in the setting of general homogeneous groups, where it was handled using rather different methods than those in this paper.
To guide the reader through different inequalities in this paper, let us briefly summarise the obtained results. We formulate some of them in a simplified form here, with the full statements given in the sequel of this paper.
\begin{itemize}
\item {\bf Logarithmic inequalities on general Lie groups.}
We will establish several versions of log-Sobolev inequalities on general Lie groups. The only (non-technical) assumption that we impose is that the group is connected, {\em otherwise it can be compact or noncompact, unimodular or non-unimodular, and can have polynomial or exponential volume growth at infinity. }
We refer to Section \ref{SEC:generalLie} for the exact setup and statements, but let us give an example. Namely, let $\mathbb{G}$ be a connected Lie group with a H\"ormander system of vector fields, of local dimension $d$. Then for any $1<p<\infty$ and $0<a<\frac{d}{p}$ we have the log-Sobolev inequality
\begin{equation}\label{sobinnon1-i}
\int_{\mathbb{G}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\lambda)}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\lambda)}}\right) d\lambda(x) \leq \frac{d}{ap}\log\left(C\frac{\|u\|^{p}_{L^{p}_{a}(\lambda)}}{\|u\|^{p}_{L^{p}(\lambda)}}\right),
\end{equation}
where $\lambda$ is the left Haar measure on $\mathbb{G}$, and $L^p_a$
is the Sobolev space associated to the corresponding sub-Laplacian (with drift). In fact, \eqref{sobinnon1-i} is a special case (without weight) of a family of weighted log-Sobolev inequalities given in Theorem \ref{loghardysob}. To give an example, if $q>p>1$, $0\leq \beta < d$ and $0<a<d$ are such that $\frac{1}{p}-\frac{1}{q}=\frac{a}{d}-\frac{\beta}{dq}$, then we have
\begin{multline*}\label{harsobinnon1-i}
\int_{\mathbb{G}}\frac{|x|_{CC}^{-\frac{\beta p}{q}}|u|^{p}}{\||x|_{CC}^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\lambda)}}\log\left(\frac{|x|_{CC}^{-\frac{\beta p}{q}}|u|^{p}}{\||x|_{CC}^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\lambda)}}\right) d\lambda(x) \leq \frac{d-\beta}{ap-\beta}\log\left(C\frac{\|u\|^{p}_{L^{p}_{a}(\lambda)}}{\||x|_{CC}^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\lambda)}}\right),
\end{multline*}
where $|x|_{CC}=d_{CC}(e,x)$ is the Carnot-Carath\'{e}odory distance between $x$ and the identity element $e$ of $\mathbb{G}$. As one can see, \eqref{sobinnon1-i} is a special case of this inequality with $\beta=0$.
The inequality \eqref{sobinnon1-i} for $a\geq \frac{d}{p}$ will be also given in Theorem \ref{logsobcor}. Finally we note that this last estimate also holds true with the Carnot-Carath\'eodory distance replaced by the Riemannian distance, see Remark \ref{REM:CCR}.
\item {\bf Logarithmic inequalities on graded groups.}
We obtain several versions of log-Sobolev (weighted and unweighted), log-Gagliardo-Nirenberg and log-Caffarelli-Kohn-Nirenberg inequalities on graded groups, let us give one example. Let $\mathbb{G}$ be a graded group with homogeneous dimension $Q$, and let $\mathcal{R}$ be a positive Rockland operator of homogeneous degree $\nu$. Let $1<p<\infty$ and $0<a<\frac{Q}{p}.$ Then we have
\begin{equation}\label{a2=0i}
\int_{\mathbb{G}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)dx\leq \frac{Q}{ap}\log\left(A\frac{\|u\|^p_{\dot{L}^p_a(\mathbb{G})}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right),
\end{equation}
with $\|u\|_{\dot{L}^p_a(\mathbb{G})}\equiv
\|\mathcal{R}^\frac{a}{\nu}u\|_{{L}^{p}(\mathbb{G})}$,
for a constant $A$ that can be related to the best constants in the Gagliardo-Nirenberg inequalities, see \eqref{CinfGN}.
This inequality \eqref{a2=0i} can be seen as a refinement of
\eqref{sobinnon1-i}, not only with respect to a better control over a constant under the logarithm, but also because one has the homogeneous Sobolev norm on the right hand side of \eqref{a2=0i}.
\item {\bf Log-Sobolev and Gross inequalities on stratified groups.}
Let $\mathbb{G}$ be a stratified Lie group with homogeneous dimension $Q$. As a special case of the obtained inequalities on graded groups, we have the log-Sobolev inequality
\begin{equation}
\label{cor.log.sobi}
\int_{\mathbb{G}}|u|^2 \log|u|\,dx \leq \frac{Q}{4}\log \left(A \int_{\mathbb{G}}|\nabla_{H}u|^2\,dx \right)\,,
\end{equation}
for every $u$ such that $\|u\|_{L^2(\mathbb{G})}=1$,
and where $\nabla_H$ is the horizontal gradient, and the constant $A$ is related to the best constants in the Gagliardo-Nirenberg inequalities, see
\eqref{Cinf}.
Furthermore, let $n_1$ be the dimension of the first stratum of the Lie algebra of $\mathbb{G}$, and let us write $x \in \mathbb{G}$ as $x=(x',x'') \in \mathbb{R}^{n_1} \times \mathbb{R}^{n-n_1}$, where $n$ is the topological dimension of $\mathbb{G}$. Then the following Gross-type ``semi-Gaussian'' log-Sobolev inequality is satisfied
\begin{equation}
\label{gaus.log.sobi}
\int_{\mathbb{G}}|g|^2\log|g|\,d\mu \leq \int_{\mathbb{G}} |\nabla_{H}g|^2\,d\mu\,,
\end{equation}
for any $g$ such that $\|g\|_{L^2(\mu)}=1,$
where $\mu=\mu_1 \otimes \mu_2$, and $\mu_1$ is the Gaussian measure on $\mathbb{R}^{n_1}$ given by $d\mu_1=\gamma e^{-\frac{|x'|^2}{2}}dx'$, for $x'\in \mathbb{R}^{n_1}$ and $|x'|$ being its Euclidean norm, where
$\gamma:=(4^{-1}Q e^{\frac{2n_1}{Q}{-1}}A)^{Q/2},$
and $\mu_2$ is the Lebesgue measure $dx''$ on $\mathbb{R}^{n-n_1}$.
While both of \eqref{cor.log.sobi} and \eqref{gaus.log.sobi} hold true, they are also {\bf equivalent} to each other, in the sense that one inequality can be shown to imply the other, see Theorem \ref{equiv.thm}. This equivalence is well-known in the Euclidean space, and here it can be also viewed as a justification for considering the Gaussian measure on the first stratum of $\mathbb{G}$. Since in the Euclidean space the first stratum is the whole space, these results recover the well-known Euclidean results as well, with
\eqref{gaus.log.sobi} boiling down to
\eqref{EQ:Gross} in the case of $\mathbb{R}^n$, see the discussion before Theorem \ref{semi-g}.
\item {\bf Weighted Gross log-Sobolev inequalities.}
The weighted log-Sobolev inequalities in Theorem \ref{wsobingrthm} give the following weighted analogue of \eqref{cor.log.sobi}: if
$\mathbb{G}$ is a stratified Lie group with homogeneous dimension $Q$, $|\cdot|$ is an arbitrary homogeneous quasi-norm on $\mathbb{G}$, and $0\leq \beta <2<Q$, then there exists $C>0$ such that the following weighted log-Sobolev inequality holds:
\begin{equation}\label{harsobingr-2-i}
\int_{\mathbb{G}}{|x|^{-\frac{\beta(Q-2)}{Q-\beta}}|u|^{2}}\log\left({|x|^{-\frac{\beta(Q-2)}{2(Q-\beta)}}|u|}\right) dx \leq \frac{Q-\beta}{2(2-\beta)}\log\left(C\int_\mathbb{G} |\nabla_H u|^{2} dx\right),
\end{equation}
for every $u$ such that $\||x|^{-\frac{\beta(Q-2)}{2(Q-\beta)}}u\|_{L^2(\mathbb{G})}=1$.
Consequently, \eqref{harsobingr-2-i} implies the following weighted version of the Gross log-Sobolev inequality:
Let $\mathbb{G}$ be a stratified group with homogeneous dimension $Q$ and let $|\cdot|$ be an arbitrary homogeneous quasi-norm on $\mathbb{G}$. Then for any $0\leq \beta <2<Q$ we have the following weighted ``semi-Gaussian'' log-Sobolev inequality
\begin{equation}
\label{gaus.log.sob-w-i}
\int_{\mathbb{G}}|x|^{-\frac{\beta(Q-2)}{Q-\beta}}|g|^2\log\left(|x|^{-\frac{\beta(Q-2)}{2(Q-\beta)}}|g|\right)\,d\mu \leq \int_{\mathbb{G}} |\nabla_{H}g|^2\,d\mu\,,
\end{equation}
for all $g$ such that $\||x|^{-\frac{\beta(Q-2)}{2(Q-\beta)}}g\|_{L^2(\mu)}=1,$
where $\mu=\mu_1 \otimes \mu_2$, with $\mu_1$ the Gaussian measure on the first stratum of $\mathbb{G}$ given by $d\mu_1=\gamma e^{-\frac{|x'|^2}{2}}dx'$ with $|x'|$ being the Euclidean norm of $x'$,
and $\mu_2$ is the Lebesgue measure $dx''$ on the other strata. In particular, we get \eqref{gaus.log.sobi} for $\beta=0$. We refer to
Theorem \ref{semi-g-w} for more explanations and for the precise value of $\gamma.$ If $\mathbb{G}=\mathbb R^n$ is the Euclidean space, this yields a new family of weighted Gross type log-Sobolev inequalities \eqref{gaus.log.sob-w-Rn-i} on $\mathbb R^n.$
\item {\bf Nash inequality.}
It is well-known in Euclidean spaces that the $L^2$-log-Sobolev inequality is equivalent to the Nash inequality. We show that this remains true also in more generality.
Let $\mathbb{G}$ be a graded group with homogeneous dimension $Q$ and let $\mathcal{R}$ be a positive Rockland operator of homogeneous degree $\nu$. Then, we show an extended Nash inequality
\begin{equation}\label{Nash-final2i}
\|u\|_{L^2(\mathbb{G})}^{2+\frac{4a}{Q}} \leq A_2 \|u\|_{L^1(\mathbb{G})}^{\frac{4a}{Q}}\|u\|^{2}_{\dot{L}^2_a(\mathbb{G})}\,.
\end{equation}
or, equivalently,
\begin{equation}
\label{Nash.gi}
\left[\int_{\mathbb{G}} |u|^2\,dx \right]^{{1+\frac{2a}{Q}}} \leq A_2 \left[ \int_{\mathbb{G}}|u|\,dx\right]^{\frac{4a}{Q}}
\int_{\mathbb{G}} |\mathcal{R}^{\frac{a}{\nu}}u|^2\,dx \,,
\end{equation}
for all $0<a<\frac{Q}{2}$, with constant $A_{2}$ related to the best constants in the Gagliardo-Nirenberg inequalities, see Theorem \ref{THM:Nash}.
In particular, if $\mathbb{G}$ is a stratified Lie group, then for $a=1$ this implies
\begin{equation}
\label{Nasi}
\|u\|_{L^2(\mathbb{G})}^{2+\frac{4}{Q}} \leq A_{2}
\|u\|_{L^1(\mathbb{G})}^{\frac{4}{Q}}
\|\nabla_H u\|_{L^2(\mathbb{G})}^2,
\end{equation}
with the horizontal gradient $\nabla_H.$ For the Euclidean $\mathbb{G}=\mathbb R^n$ this is the full gradient, yielding as a special case of $a=1$ the original Nash inequality in \cite{Nas58} as well as, again for $a=1$, the Nash inequality on stratified groups in \cite{VSCC93}.
As a simple immediate application, in Corollary \ref{cor:par}
we show that if $\Delta_\mathbb{G}$ is a sub-Laplacian on a stratified group $\mathbb{G}$ with $Q\geq 3$, and $u_0\in L^1(\mathbb{G})\cap L^2(\mathbb{G})$ is non-negative $u_0\geq 0$, then the solution $u$ to the heat equation
\begin{equation}
\label{heat.eqi}
\partial_t u=\Delta_\mathbb{G} u,\quad u(0,x)=u_0(x),
\end{equation}
satisfies the time-decay estimate
\begin{equation}
\label{heatesti}
\|u(t,\cdot)\|_{L^2(\mathbb{G})}\leq \left( \|u_0\|_{L^2(\mathbb{G})}^{-\frac{4}{Q}}
+\frac{4}{QA_{2}}\|u_0\|_{L^1(\mathbb{G})}^{-\frac{4}{Q}}t\right)^{-\frac{Q}{4}},
\end{equation}
for all $t\geq 0,$ where $A_{2}$ is a constant in \eqref{Nash.gi}.
Finally, in Theorem \ref{THM:Nash-gen} we show Nash's inequality as well as its weighted versions on general connected Lie groups. Namely, if $d$ is a local dimension of $\mathbb{G}$, and $0\leq \beta <2a<d$,
then the weighted Nash inequality is given by
\begin{equation}\label{Nash-final2th-i}
\||\cdot|_{CC}^{-\frac{\beta(d-2a)}{2(d-\beta)}}u\|_{L^2(\lambda)}^{2+\frac{2(2a-\beta)}{d-\beta}} \leq C \||\cdot|_{CC}^{-\frac{\beta(d-2a)}{2(d-\beta)}}u\|_{L^1(\lambda)}^{\frac{2(2a-\beta)}{d-\beta}}\|u\|^{2}_{L^2_a(\lambda)},
\end{equation}
where $\lambda$ is the left Haar measure on $\mathbb{G}$ and $|x|_{CC}=d_{CC}(e,x)$ is the Carnot-Carath\'{e}odory distance between $x$ and $e$ (but it can be also replaced by the Riemannian distance, see Remark \ref{REM:CCR}).
In particular, for $\beta=0$, we obtain the Nash inequality on general connected Lie groups in the following form:
\begin{equation}\label{Nash-final2th0-i}
\|u\|_{L^2(\lambda)}^{2+\frac{4a}{d}} \leq C \|u\|_{L^1(\lambda)}^{\frac{4a}{d}}\|u\|^{2}_{L^2_a(\lambda)},\quad 0<a<\frac{d}{2}.
\end{equation}
For $a\geq\frac{d}{2}$, we also get the weighted and unweighted versions of the Nash inequality, in particular, for any $q>2$, we have the Nash type inequality
\begin{equation}\label{Nash-final2th0-i2}
\|u\|^{2+\frac{2(q-2)}{q}}_{L^{2}(\lambda)}\leq C\|u\|^{\frac{2(q-2)}{q}}_{L^{1}(\lambda)}\|u\|^{2}_{L^{2}_{a}(\lambda)}, \quad a\geq\frac{d}{2}.
\end{equation}
\end{itemize}
\section{Preliminaries}
In this section, we briefly recall definitions and main properties of the homogeneous, graded and stratified groups.
The comprehensive analysis on such groups has been initiated in the works of Folland and Stein \cite{FS}, but in our exposition below we follow a more recent presentation in the open access book \cite{FR16}.
\subsection{Homogeneous groups} In this sub-section, we give definitions and main properties of homogeneous Lie group.
\begin{defn}[\cite{FS, FR16}, Homogeneous group]
A Lie group (on $\mathbb{R}^{N}$) $\mathbb{G}$ with the dilation
$$D_{\lambda}(x):=(\lambda^{\nu_{1}}x_{1},\ldots,\lambda^{\nu_{N}}x_{N}),\; \nu_{1},\ldots, \nu_{n}>0,\; D_{\lambda}:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N},$$
which is an automorphism of the group $\mathbb{G}$ for each $\lambda>0,$
is called a {\em homogeneous (Lie) group}.
\end{defn}
For simplicity, in this paper we use the notation $\lambda x$ for the dilation $D_{\lambda}(x)$.
We
denote
\begin{equation}
Q:=\nu_{1}+\ldots+\nu_{N},
\end{equation}
the homogeneous dimension of a homogeneous group $\mathbb{G}$.
Let $dx$ denote the Haar measure on $\mathbb{G}$ and let $|S|$ denote the corresponding volume of a measurable set $S\subset \mathbb{G}$.
Then we have
\begin{equation}\label{scal}
|D_{\lambda}(S)|=\lambda^{Q}|S| \quad {\rm and}\quad \int_{\mathbb{G}}f(\lambda x)
dx=\lambda^{-Q}\int_{\mathbb{G}}f(x)dx.
\end{equation}
We also note that from \cite[Proposition 1.6.6]{FR16}, the standard Lebesgue measure $dx$ on $\mathbb{R}^{N}$ is the Haar measure on $\mathbb{G}$.
\begin{defn}[{\cite[Definition 3.1.33]{FR16} or \cite[Definition 1.2.1]{RS19}}]\label{quasi-norm}
For any homogeneous group $\mathbb{G}$ there exist homogeneous quasi-norms, which are continuous non-negative functions
\begin{equation}
\mathbb{G}\ni x\mapsto |x|\in[0,\infty),
\end{equation}
with the properties
\begin{itemize}
\item[a)] $|x|=|x^{-1}|$ for all $x\in\mathbb{G}$,
\item[b)] $|\lambda x|=\lambda|x|$ for all $x\in \mathbb{G}$ and $\lambda>0$,
\item[c)] $|x|=0$ if and only if $x=0$.
\end{itemize}
\end{defn}
They are quasi-norms since the triangle inequality may be satisfied with a constant different from $1.$
\subsection{Graded Lie groups}\label{SEC:graded}
In this subsection, we make a brief summary of the basic definitions and properties of the graded Lie groups.
\begin{defn}[e.g. {\cite[Definition 3.1.1]{FR16}}, graded Lie group and graded Lie algebra]
A Lie algebra $\mathfrak{g}$ is
called {\em graded} if it is endowed with a vector space decomposition (where all but
finitely many of the $V_{j}$'s are $0$)
\begin{equation}
\mathfrak{g}=\oplus_{j=1}^{\infty}V_{j},\,\,\,\,\text{s.t.}\,\,\,[V_{i},V_{j}]\subset V_{i+j}.
\end{equation}
Consequently, a Lie group is called {\em graded} if it is a connected and simply-connected
Lie group whose Lie algebra is graded.
\end{defn}
Before introducing the Rockland operator, we recall the Rockland condition.
By $\pi$ and $\widehat{\mathbb{G}}$ we denote a representation and the unitary dual of $\mathbb{G}$, respectively, and by $\mathcal{H}_{\pi}^{\infty}$ we denote the smooth vectors of the representation $\pi\in\widehat{\mathbb{G}}$. Let us recall the Rockland condition:
\begin{defn}[Rockland condition, see e.g. {\cite[Definition 4.1.1]{FR16}}]
Let $A$ be a left-invariant differential operator on a Lie group $\mathbb{G}$.
Then $A$ satisfies the {\em Rockland condition} when
(Rockland condition) for each representation $\pi\in\widehat{\mathbb{G}}$, except for the trivial representation,
the operator $\pi(A)$ is injective on $\mathcal{H}_{\pi}^{\infty}$, that is,
\begin{equation}
\forall v \in\mathcal{H}_{\pi}^{\infty},\,\,\, \pi(A)v =0 \Rightarrow v = 0.
\end{equation}
\end{defn}
\begin{defn}[Rockland operator, see e.g. {\cite[Definition 4.1.2]{FR16}}]\label{intRockland}
Let $\mathbb{G}$ be a homogeneous Lie group. A {\em Rockland operator} $\mathcal{R}$ on
$\mathbb{G}$ is a left-invariant differential operator which is homogeneous of positive degree
and satisfies the Rockland condition.
\end{defn}
Let us recall a property which connects homogeneous Lie groups and Rockland operators.
\begin{prop}[e.g. {\cite[Proposition 4.1.3]{FR16}}]\label{exishomgr}
Let $\mathbb{G}$ be a homogeneous Lie group. If there exists a Rockland operator on $\mathbb{G}$ then $\mathbb{G}$ is a graded.
\end{prop}
Similarly to the \cite{RTY20}, in this paper we will not be using the representation theoretic interpretation of these operators, so we define Rockland operators as \textit{left-invariant homogeneous hypoelliptic differential operators on $\mathbb{G}$.}
Let $\mathcal{R}$ be a Rockland operator of homogeneous degree $\nu$. Let us recall the definitions of the homogeneous and inhomogeneous Sobolev spaces on graded Lie groups, respectively, endowed with norms:
\begin{equation}
\|f\|_{\dot{L}^p_{a}(\mathbb{G})}:=\|\mathcal{R}^{\frac{a}{\nu}}f\|_{L^{p}(\mathbb{G})},
\end{equation}
and
\begin{equation}
\|f\|_{L^p_{a}(\mathbb{G})}:=(\|f\|^{p}_{L^{p}(\mathbb{G})}+\|\mathcal{R}^{\frac{a}{\nu}}f\|^{p}_{L^{p}(\mathbb{G})})^{\frac{1}{p}}.
\end{equation}
It was shown in \cite{FR16} that while these norms depend on the choice of $\mathcal{R}$, the Sobolev spaces do not. In this paper, we use notation $L^{p}_{a_{1},a_{2}}(\mathbb{G})=\dot{L}^{p}_{a_{1}}(\mathbb{G})\cap \dot{L}^{p}_{a_{2}}(\mathbb{G})$.
\subsection{Stratified Lie groups}\label{defstr}
In this section, we recall the definition of stratified groups (homogeneous Carnot groups) and their basic properties. Let us briefly recall the definition of the stratified Lie group. We refer to e.g. \cite{BLU07}, \cite{FR16} and \cite{RS19} for further discussions in this direction.
\begin{defn} \label{maindef}
A Lie group $\mathbb{G}=(\mathbb{R}^{n},\circ)$ is called a stratified Lie group if it satisfies the following assumptions:
(a) For some natural numbers $n_{1}+...+n_{r}=n$
the decomposition $\mathbb{R}^{n}=\mathbb{R}^{n_{1}}\times...\times\mathbb{R}^{n_{r}}$ is valid, and
for every $\lambda>0$ the dilation $\delta_{\lambda}: \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$
given by
$$\delta_{\lambda}(x)\equiv\delta_{\lambda}(x^{(1)},...,x^{(r)}):=(\lambda x^{(1)},...,\lambda^{r}x^{(r)})$$
is an automorphism of the group $\mathbb{G}.$ Here $x^{(k)}\in \mathbb{R}^{n_{k}}$ for $k=1,...,r.$
(b) Let $n_{1}$ be as in (a) and let $X_{1},...,X_{n_{1}}$ be the left invariant vector fields on $\mathbb{G}$ such that
$X_{k}(0)=\frac{\partial}{\partial x_{k}}|_{0}$ for $k=1,...,n_{1}.$ Then
$${\rm rank}({\rm Lie}\{X_{1},...,X_{n_{1}}\})=n,$$
for every $x\in\mathbb{R}^{n},$ i.e. the iterated commutators
of $X_{1},...,X_{n_{1}}$ span the Lie algebra of $\mathbb{G}.$
\end{defn}
A graded Lie algebra $\mathfrak{g}$ is stratified if $V_{1}$ generates $\mathfrak{g}$ as an algebra. In this case, if $\mathfrak{g}$ is
nilpotent of step $m$ we have
\begin{equation}
\mathfrak{g}=\oplus_{j=1}^{\infty}V_{j},\,\,\,\,\text{s.t.}\,\,\,[V_{i},V_{1}]\subset V_{i+1},
\end{equation}
and the natural dilations $\mathfrak{g}$ are given by
\begin{equation}
D_{r}\left(\sum_{k=1}^{m}X_{k}\right)=\sum_{k=1}^{m}r^{k}X_{k},\,\,\,\,(X_{k}\in V_{k}).
\end{equation}
Consequently, a Lie group is stratified if it is connected and simply-connected
Lie group whose Lie algebra is stratified.
\section{Logarithmic H\"older inequalities}
The aim of this section is to show logarithmic H\"{o}lder's inequality on general measure spaces which will be instrumental in our proof of other logarithmic inequalities. For completeness, we first record a useful auxiliary consequence of the H\"older inequality.
\begin{lem}\label{weightedholder}
Let $\mathbb{X}$ be a measure space. Suppose that $1<p\leq r\leq q\leq\infty$, $a\in[0,1]$, $u\in L^{p}(\mathbb{X})\cap L^{q}(\mathbb{X})$ with
\begin{equation}
\frac{1}{r}=\frac{a}{p}+\frac{1-a}{q}.
\end{equation}
Then we have
\begin{equation}\label{intin}
\|u\|_{L^{r}(\mathbb{X})}\leq \|u\|^{a}_{L^{p}(\mathbb{X})}\|u\|^{1-a}_{L^{q}(\mathbb{X})}.
\end{equation}
\end{lem}
\begin{proof}
From H\"{o}lder's inequality we get
\begin{equation*}
\begin{split}
\|u\|^{r}_{L^{r}(\mathbb{X})}&=\int_{\mathbb{X}}|u(x)|^{r}dx=\int_{\mathbb{X}}(|u(x)|)^{ar}(|u(x)|)^{(1-a)r}dx
\\&\leq\left(\int_{\mathbb{X}}|u(x)|^{p}dx\right)^{\frac{ar}{p}}\left(\int_{\mathbb{X}}|u(x)|^{q}dx\right)^{\frac{(1-a)r}{q}}
\\&=\|u\|^{ar}_{L^{p}(\mathbb{X})}\|u\|^{(1-a)r}_{L^{q}(\mathbb{X})},
\end{split}
\end{equation*}
with
$
\frac{ar}{p}+\frac{(1-a)r}{q}=1,
$
implying \eqref{intin}.
\end{proof}
Now let us show what we can call a logarithmic H\"{o}lder's inequality on general measure spaces. A weighted version of this inequality on homogeneous groups was obtained in \cite[Lemma 3.2]{KRS20}. We now give its general version on measure spaces.
\begin{lem}[Logarithmic H\"older inequality]\label{holder}
Let $\mathbb{X}$ be a measure space. Let $u\in L^{p}(\mathbb{X})\cap L^{q}(\mathbb{X})\setminus\{0\}$ with some $1<p<q< \infty.$
Then we have
\begin{equation}\label{holdernn}
\int_{\mathbb{X}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{X})}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{X})}}\right)dx\leq \frac{q}{q-p}\log\left(\frac{\|u\|^{p}_{L^{q}(\mathbb{X})}}{\|u\|^{p}_{L^{p}(\mathbb{X})}}\right).
\end{equation}
\end{lem}
\begin{proof}
Let us consider the following function
\begin{equation}\label{1/r}
F\left(\frac{1}{r}\right)=\log\left(\|u\|_{L^{r}(\mathbb{X})}\right).
\end{equation}
By using Lemma \ref{weightedholder}, we have
\begin{equation}\label{convex11}
F\left(\frac{1}{r}\right)\leq aF\left(\frac{1}{p}\right)+(1-a)F\left(\frac{1}{q}\right),
\end{equation}
with $a\in[0,1]$ and $\frac{1}{r}=\frac{a}{p}+\frac{1-a}{q}$.
Also, from \eqref{1/r} we have
\begin{equation}\label{convex}
F(r)=r\log\int_{\mathbb{G}}|u(x)|^{\frac{1}{r}}dx,
\end{equation}
and by taking the derivative of \eqref{convex}, we get
\begin{multline}
F'(r)=\log\int_{\mathbb{X}}|u(x)|^{\frac{1}{r}}dx+r\left(\log\int_{\mathbb{X}}|u(x)|^{\frac{1}{r}}dx\right)'_{r}\\
=\log\int_{\mathbb{X}}|u(x)|^{\frac{1}{r}}dx-\frac{1}{r}\frac{\int_{\mathbb{X}}|u(x)|^{\frac{1}{r}}\log |u(x)|dx}{\int_{\mathbb{X}}|u(x)|^{\frac{1}{r}}dx}.
\end{multline}
We know from \eqref{convex11} that $F(r)$ is convex, hence, we have
\begin{equation}
F'(r)\geq\frac{F(r')-F(r)}{r'-r},\,\,\,\,r>r'>0.
\end{equation}
and by taking $r=\frac{1}{p}$ and $r'=\frac{1}{q}$, we get
\begin{equation}\label{convex1}
p\frac{\int_{\mathbb{X}}|u|^{p}\log |u|dx}{\int_{\mathbb{X}}|u(x)|^{p}dx}-\log\int_{\mathbb{X}}|u(x)|^{p}dx\leq \frac{qp}{q-p}\log\left(\frac{\|u\|_{L^{q}(\mathbb{X})}}{\|u\|_{L^{p}(\mathbb{X})}}\right).
\end{equation}
Finally, from this last fact, we establish logarithmic H\"{o}lder's inequality
\begin{equation*}
\begin{split}
p\frac{\int_{\mathbb{X}}|u|^{p}\log |u|dx}{\int_{\mathbb{X}}|u(x)|^{p}dx}&-\log\int_{\mathbb{X}}|u(x)|^{p}dx=p\frac{\int_{\mathbb{X}}|u|^{p}\log |u|dx}{\int_{\mathbb{X}}|u(x)|^{p}dx}-\frac{\int_{\mathbb{X}}|u(x)|^{p}\log\| u\|^{p}_{L^{p}(\mathbb{G})}dx}{\int_{\mathbb{X}}|u(x)|^{p}dx}\\&
=\frac{\int_{\mathbb{X}}|u|^{p}\log |u|^{p}dx}{\int_{\mathbb{X}}|u(x)|^{p}dx}-\frac{\int_{\mathbb{X}}|u(x)|^{p}\log\| u\|^{p}_{L^{p}(\mathbb{X})}dx}{\int_{\mathbb{X}}|u(x)|^{p}dx}\\&
=\int_{\mathbb{X}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{X})}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{X})}}\right) dx\\&
\leq \frac{q}{q-p}\log\left(\frac{\|u\|^{p}_{L^{q}(\mathbb{X})}}{\|u\|^{p}_{L^{p}(\mathbb{X})}}\right),
\end{split}
\end{equation*}
implying \eqref{holdernn}.
\end{proof}
\section{Logarithmic inequalities on graded groups}
\label{SEC:3}
In this section we show analogues of logarithmic inequalities on the graded groups. First we recall a few facts that will be of importance later.
Let us consider the following Schr\"{o}dinger equation,
\begin{equation}\label{problem1}
\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}(|\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}u|^{p-2}\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}u)+\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}(|\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}u|^{p-2}\mathcal{R}_{1}^{\frac{a_{2}}{\nu_{2}}}u)=|u|^{q-2}u,\,\,u\in L^{p}_{a_{1},a_{2}}(\mathbb{G}),
\end{equation}
where $a_{1}>a_{2}\geq 0$, $1<p<\frac{Q}{a_{1}}$, $\frac{Qp}{Q-a_{2}p}<q<\frac{Qp}{Q-a_{1}p}$, and $L^{p}_{a_{1},a_{2}}(\mathbb{G})=\dot{L}^{p}_{a_{1}}(\mathbb{G})\cap \dot{L}^{p}_{a_{2}}(\mathbb{G}).$
Let us denote the energy functional $I:L^{p}_{a_{1},a_{2}}(\mathbb{G})\rightarrow \mathbb{R}$ and the Nehari functional $J:L^{p}_{a_{1},a_{2}}(\mathbb{G})\rightarrow \mathbb{R}$ acting on $L^{p}_{a_{1},a_{2}}(\mathbb{G})\cap L^{q}(\mathbb{G})$ in the following forms:
\begin{equation}
I(u):=\frac{1}{p}\int_{\mathbb{G}}|\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}u(x)|^{p}dx+\frac{1}{p}\int_{\mathbb{G}}|\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}u(x)|^{p}dx-\frac{1}{q}\int_{\mathbb{G}}|u(x)|^{q}dx,
\end{equation}
and
\begin{equation}
J(u):=\int_{\mathbb{G}}|\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}}u(x)|^{p}dx+\int_{\mathbb{G}}|\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}}u(x)|^{p}dx-\int_{\mathbb{G}}|u(x)|^{q}dx.
\end{equation}
Let us introduce the Nehari manifold
\begin{equation}
\mathcal{N}:=\{u\in L^{p}_{a_{1},a_{2}}(\mathbb{G})\setminus \{0\}:J(u)=0\},
\end{equation}
and we put
\begin{equation}\label{d}
d_0:=\inf \{I(u): u\in \mathcal{N}\}.
\end{equation}
The above notions are related by the following theorem.
\begin{thm}[\cite{RTY20}, Theorem 4.3] \label{THM:RTYb}
Let $a_1>a_2\geq0$, $1<p<\frac{Q}{a_{1}}$ and $\frac{pQ}{Q-a_{2}p} <q< \frac{pQ}{Q-a_{1}p}$. Then, the
Schr\"{o}dinger type equation \eqref{problem1} has a least energy solution $\varphi \in L^{p}_{a_1 ,a_2} (\mathbb{G})$. Moreover, we have $d_0 = I(\varphi)$.
\end{thm}
The Sobolev embeddings on graded Lie groups have been obtained in \cite{FR17}, but here we recall its version with the best constant, which is related to the quantities described in Theorem \ref{THM:RTYb}.
\begin{thm}[\cite{RTY20}, Sobolev inequality]\label{sob} Let $\mathbb{G}$ be a graded group with homogeneous dimension $Q$ and let $\mathcal{R}$ be a positive Rockland operator of homogeneous degree $\nu$.
Let $a>0$, $p\in\left(1,\frac{Q}{a}\right)$ and $p<q<\frac{Qp}{Q-ap}$. Then we have
\begin{equation}\label{sobin1}
\left(\int_{\mathbb{G}}|u(x)|^{q}dx\right)^{\frac{p}{q}} \leq C\int_{\mathbb{G}}\left(|\mathcal{R}^{\frac{a}{\nu}} u(x)|^{p}+|u(x)|^{p}\right)dx,
\end{equation}
for any $u\in L^{p}_{a}(\mathbb{G})$.
Moreover, let $\phi$ be a least energy solution of \eqref{problem1} (with $a=a_{1}$, $\mathcal{R}_{1}=\mathcal{R}$ and $a_{2}=0$) and let $C_{S,\mathcal{R}}$ be the smallest positive constant $C$ in \eqref{sobin1}. Then we have
\begin{equation}\label{bestconsob}
C_{S,\mathcal{R}}=\left(\frac{apq}{apq-Q(q-p)}\int_{\mathbb{G}}|\phi(x)|^{p}dx\right)^{\frac{p-q}{q}}=\left(\frac{pq}{q-p}d_0\right)^{\frac{p-q}{q}},
\end{equation}
where $d_0$ is defined in \eqref{d}.
\end{thm}
First we present the inhomogeneous logarithmic Sobolev inequality on graded groups $\mathbb{G},$ where the Sobolev norm in the estimate is inhomogeneous. The homogeneous norm version will be given in Corollary \ref{COR:Lpspb123} as a consequence of the Gagliardo-Nirenberg inequalities.
\begin{thm}
Let $\mathbb{G}$ be a graded group with homogeneous dimension $Q$.
Let $1<p<\infty$ and $0<a<\frac{Q}{p}$. Then we have
\begin{equation}\label{LogSobolev}
\int_{\mathbb{G}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right) dx \leq \frac{Q}{ap}\log\left(C_{p}\frac{\|u\|^{p}_{L^{p}_{a}(\mathbb{G})}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)
\end{equation}
with the constant $C_{p}=\left( \frac{a}{Qd_0}\right)^{\frac{ap}{Q}}$ for $d_0$ as in \eqref{d}, for any non-trivial $u\in L^{p}_{a}(\mathbb{G})$.
\end{thm}
\begin{proof}
From logarithmic H\"older inequality \eqref{holdernn} we have
\begin{equation}
\int_{\mathbb{G}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)dx\leq \frac{q}{q-p}\log\left(\frac{\|u\|^{p}_{L^{q}(\mathbb{G})}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right).
\end{equation}
By the assumption we have $1\leq p<q=p^{*}=\frac{pQ}{Q-ap}$, that is,
\begin{eqnarray}\label{log.sob.}
\int_{\mathbb{G}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right) dx & \leq & \frac{q}{q-p}\log\left(\frac{\|u\|^{p}_{L^{q}(\mathbb{G})}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)\nonumber\\
& \leq & \frac{q}{q-p} \log\left(C_{S,\mathcal{R}}\frac{\|u\|^{p}_{L^{p}_{a}(\mathbb{G})}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)\,,
\end{eqnarray}
where for the last inequality we have applied \eqref{sobin1}.
Notice that the parameter $q\in \left(p, \frac{Qp}{Q-ap} \right)$ does not appear at any norm involved in \eqref{log.sob.}, one can minimise the quantity $\frac{q}{q-p}$ over $q$ and get
\[
\int_{\mathbb{G}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right) dx \leq \frac{Q}{ap}\log\left(C_{p}\frac{\|u\|^{p}_{L^{p}_{a}(\mathbb{G})}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)\,,
\]
where the minimum has been achieved in the limit case where $q \rightarrow \frac{Qp}{Q-ap}$, and the constant $C_{p}$ is now given by
\[
C_{p}=\limsup_{q \rightarrow \frac{Qp}{Q-ap}} C_{S,\mathcal{R}}=\left( \frac{a}{Qd_0}\right)^{\frac{ap}{Q}}\,,
\]
where the constant $C_{S,\mathcal{R}}$ is given by \eqref{bestconsob}.
\end{proof}
Let us now recall the Gagliardo-Nirenberg inequality on graded groups.
\begin{thm}[\cite{RTY20}, Gagliardo-Nirenberg inequality]\label{GN} Let $\mathbb{G}$ be a graded group with homogeneous dimension $Q$ and let $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ be positive Rockland operators of homogeneous degrees $\nu_{1}$ and $\nu_{2}$, respectively.
Let $a_{1}>a_{2}\geq 0$, $p\in\left(1,\frac{Q}{a_{1}}\right)$ and $\frac{Qp}{Q-a_{2}p}<q<\frac{Qp}{Q-a_{1}p}$. Then we have
\begin{equation}\label{gnin1}
\int_{\mathbb{G}}|u(x)|^{q}dx \leq C\left(\int_{\mathbb{G}}|\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}} u(x)|^{p}dx\right)^{\frac{Q(q-p)-a_{2}pq}{(a_{1}-a_{2})p^{2}}}\left(\int_{\mathbb{G}}|\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}} u(x)|^{p}dx\right)^{\frac{a_{1}pq-Q(q-p)}{(a_{1}-a_{2})p^{2}}},
\end{equation}
for any $u\in \dot{L}^{p}_{a_{1},a_{2}}(\mathbb{G})$.
Moreover, let $\phi$ be a least energy solution of \eqref{problem1} and let $C_{GN,\mathcal{R}_{1},\mathcal{R}_{2},a_{1},a_{2},p,q}$ be the best constant $C$ in \eqref{sobin1}. Then we have
\begin{equation}\label{bestcongn}
\begin{split}
C_{GN,\mathcal{R}_{1},\mathcal{R}_{2},a_{1},a_{2},p,q}&=\frac{(a_{1}-a_{2})pq}{a_{1}pq-Q(q-p)}\left(\frac{Q(q-p)-a_{2}pq}{a_{1}pq-Q(q-p)}\right)^{\frac{a_{2}pq-Q(q-p)}{(a_{1}-a_{2})p^{2}}}\\&
\times \left(\frac{a_{1}pq-Q(q-p)}{(a_{1}-a_{2})(q-p)}d_0\right)^{\frac{p-q}{p}},
\end{split}
\end{equation}
where $d_0$ is defined in \eqref{d}.
\end{thm}
We now have the following logarithmic Gagliardo-Nirenberg inequality.
\begin{thm}\label{THM:GNlog}
Let $\mathbb{G}$ be a graded group with homogeneous dimension $Q$ and let $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ be positive Rockland operators of homogeneous degrees $\nu_{1}$ and $\nu_{2}$, respectively.
Let $a_{1}>a_{2}\geq 0$, $p\in\left(1,\frac{Q}{a_{1}}\right)$ and $\frac{Qp}{Q-a_{2}p}<q<\frac{Qp}{Q-a_{1}p}$. If $a_{2}\neq 0$, then for any $u\not=0$ we have
\begin{multline}\label{LogGN1}
\int_{\mathbb{G}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)dx \\ \leq \frac{q}{q-p}\log\left(C^{\frac{p}{q}}_{GN,\mathcal{R}_{1},\mathcal{R}_{2},a_{1},a_{2},p,q}\frac{\|u\|^{\frac{Q(q-p)-a_{2}pq}{(a_{1}-a_{2})q}}_{\dot{L}^{p}_{a_{1}}(\mathbb{G})}\|u\|^{\frac{a_{1}pq-Q(q-p)}{(a_{1}-a_{2})q}}_{\dot{L}^{p}_{a_{2}}(\mathbb{G})}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right).
\end{multline}
\end{thm}
\begin{proof}
Assume that $a_{2}\neq0$. From assumption $1<p<\frac{Q}{a_{1}}$, we have $1<p<\frac{Q}{a_{1}}<\frac{Q}{a_{2}}$. It means $p<\frac{Qp}{Q-a_{2}p}<q<\frac{Qp}{Q-a_{1}p}$, then $p<q$. Using the Gagliardo-Nirenberg inequality \eqref{gnin1} and the logarithmic H\"{o}lder inequality \eqref{holdernn}, we have
\begin{equation*}
\begin{split}
&\int_{\mathbb{G}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)dx\leq \frac{q}{q-p}\log\left(\frac{\|u\|^{p}_{L^{q}(\mathbb{G})}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)\\&
\leq\frac{q}{q-p}\log\left(C^{\frac{p}{q}}_{GN,\mathcal{R}_{1},\mathcal{R}_{2},a_{1},a_{2},p,q}\frac{\left(\int_{\mathbb{G}}|\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}} u(x)|^{p}dx\right)^{\frac{Q(q-p)-a_{2}pq}{(a_{1}-a_{2})pq}}\left(\int_{\mathbb{G}}|\mathcal{R}_{2}^{\frac{a_{2}}{\nu_{2}}} u(x)|^{p}dx\right)^{\frac{a_{1}pq-Q(q-p)}{(a_{1}-a_{2})pq}}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)\\&
=\frac{q}{q-p}\log\left(C^{\frac{p}{q}}_{GN,\mathcal{R}_{1},\mathcal{R}_{2},a_{1},a_{2},p,q}\frac{\|u\|^{\frac{Q(q-p)-a_{2}pq}{(a_{1}-a_{2})q}}_{\dot{L}^{p}_{a_{1}}(\mathbb{G})}\|u\|^{\frac{a_{1}pq-Q(q-p)}{(a_{1}-a_{2})q}}_{\dot{L}^{p}_{a_{2}}(\mathbb{G})}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right).
\end{split}
\end{equation*}
The proof is complete.
\end{proof}
If $\mathcal{R}$ is a positive Rockland operators of homogeneous degrees $\nu$, then, as usual, we identify
$\|u\|_{\dot{L}^p_a(\mathbb{G})}\equiv
\|\mathcal{R}^\frac{a}{\nu}u\|_{{L}^{p}(\mathbb{G})}$.
\begin{cor}\label{COR:Lpspb123}
Let $\mathbb{G}$ be a graded group with homogeneous dimension $Q$ and let $\mathcal{R}$ be a positive Rockland operator of homogeneous degree $\nu$.
Let $1<p<\infty$ and $0<a<\frac{Q}{p}.$
Then for any $u\not=0$ we have
\begin{equation}\label{a2=0}
\int_{\mathbb{G}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)dx\leq \frac{Q}{ap}\log\left(A\frac{\|u\|^p_{\dot{L}^p_a(\mathbb{G})}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right),
\end{equation}
with $\|u\|_{\dot{L}^p_a(\mathbb{G})}\equiv
\|\mathcal{R}^\frac{a}{\nu}u\|_{{L}^{p}(\mathbb{G})}$,
for
\begin{equation}\label{CinfGN}
A=\inf_{q:\, p<q<\frac{Qp}{Q-ap}}
(C_{GN,\mathcal{R},a ,p,q})^{\frac{ap^2}{Q(q-p)}}\,,
\end{equation}
where $C_{GN,\mathcal{R},a ,p,q}$ is the constant given in \eqref{bestcongn} for $a_1=a,a_2=0$.
In particular, for all $\|u\|_{L^p(\mathbb{G})}=1$ we have
\begin{equation}\label{a2=0u1}
\int_{\mathbb{G}} |u|^{p}\log |u| dx\leq \frac{Q}{ap^2}\log\left(A \int_\mathbb{G} |\mathcal{R}^{\frac{a}{\nu}}u(x)|^p dx \right).
\end{equation}
\end{cor}
\begin{proof}
In the $a_{2}=0$ version of Theorem \ref{THM:GNlog}, for $p<q<\frac{Qp}{Q-ap}$, we get
\begin{equation}
\begin{split}
&\int_{\mathbb{G}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)dx\leq \frac{q}{q-p}\log\left(\frac{\|u\|^{p}_{L^{q}(\mathbb{G})}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)\\&
\leq\frac{q}{q-p}\log\left(C^{\frac{p}{q}}_{GN,\mathcal{R}_{1},a_{1},p,q}\frac{\left(\int_{\mathbb{G}}|\mathcal{R}_{1}^{\frac{a_{1}}{\nu_{1}}} u(x)|^{p}dx\right)^{\frac{Q(q-p)}{a_{1}pq}}\left(\int_{\mathbb{G}}|u(x)|^{p}dx\right)^{\frac{a_{1}pq-Q(q-p)}{a_{1}pq}}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)\\&
=\frac{q}{q-p}\log\left(C^{\frac{p}{q}}_{GN,\mathcal{R}_{1},a_{1},p,q}\frac{\|u\|^{\frac{Q(q-p)}{a_{1}q}}_{\dot{L}^{p}_{a_{1}}(\mathbb{G})}\|u\|^{\frac{a_{1}pq-Q(q-p)}{a_{1}q}}_{L^{p}(\mathbb{G})}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)\\&
=\frac{q}{q-p}\log\left(C^{\frac{p}{q}}_{GN,\mathcal{R}_{1},a_{1},p,q}\frac{\|u\|^{\frac{Q(q-p)}{a_{1}q}}_{\dot{L}^{p}_{a_{1}}(\mathbb{G})}}{\|u\|^{\frac{Q(q-p)}{a_{1}q}}_{L^{p}(\mathbb{G})}}\right).
\end{split}
\end{equation}
Denoting $a=a_1$, we have
\begin{multline*}
\frac{q}{q-p}\log\left(C^{\frac{p}{q}}_{GN,\mathcal{R}_{1},a,p,q}\frac{\|u\|^{\frac{Q(q-p)}{a q}}_{\dot{L}^{p}_{a_{1}}(\mathbb{G})}}{\|u\|^{\frac{Q(q-p)}{a q}}_{L^{p}(\mathbb{G})}}\right)=
\frac{q}{q-p}\log\left(C^{\frac{p}{q}\cdot\frac{aqp}{Q(q-p)}}_{GN,\mathcal{R}_{1},a ,p,q}\frac{\|u\|^{p}_{\dot{L}^{p}_{a}(\mathbb{G})}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right)^{\frac{Q(q-p)}{a q p}} \\
= \frac{Q}{ap}
\log\left(C^{\frac{ap^2}{Q(q-p)}}_{GN,\mathcal{R}_{1},a ,p,q}\frac{\|u\|^{p}_{\dot{L}^{p}_{a}(\mathbb{G})}}{\|u\|^{p}_{L^{p}(\mathbb{G})}}\right).
\end{multline*}
Minimising the constant over the admissible range of $q$, the proof
is complete, with \eqref{a2=0u1} following immediately from \eqref{a2=0}.
\end{proof}
Next, let us show a logarithmic Caffarelli-Kohn-Nirenberg inequality on graded groups. Firstly, from \cite[Theorem 5.9]{RY18b} we recall the following Caffarelli-Kohn-Nirenberg on graded groups:
\begin{thm}
Let $\mathbb{G}$ be a graded Lie group of homogeneous dimension $Q$ and let
$\mathcal{R}$ be a positive Rockland operator of homogeneous degree $\nu$. Let $|\cdot|$ be an arbitrary homogeneous quasi-norm. Let $1<p,r<\infty$, $\delta\in(0,1]$, $q\in(0,\infty)$ with $q\leq\frac{p}{1-\delta}$ for $\delta\neq 1$. Let $0<a r<Q$ and $\beta,\gamma\in \mathbb{R}$ with $\delta q(Q-ar-\beta r)\leq r(Q+q\gamma-q\beta)$ and $\beta(1-\delta)-\delta a\leq \gamma\leq \beta(1-\delta)$. Assume that $\frac{q(\delta Q+r(\beta(1-\delta)-\gamma -a\delta))}{rQ}+\frac{q(1-\delta)}{p}=1$. Then
there exists a positive constant $C$ such that
\begin{equation}\label{CKNgr}
\||x|^{\gamma}u\|_{L^{q}(\mathbb{G})}\leq C\|\mathcal{R}^{\frac{a}{\nu}}u\|^{\delta}_{L^{r}(\mathbb{G})}\||x|^{\beta}u\|^{1-\delta}_{L^{p}(\mathbb{G})},
\end{equation}
for all $u\in \dot{L}^{r}_{a}(\mathbb{G})$.
\end{thm}
Now, in combination with the logarithmic H\"older inequality, our usual strategy implies the logarithmic Caffarelli-Kohn-Nirenberg inequality on graded groups.
\begin{thm}\label{logCKNgrthm}
Let $\mathbb{G}$ be a graded Lie group of homogeneous dimension $Q$ and let
$\mathcal{R}$ be a positive Rockland operator of homogeneous degree $\nu$. Let $|\cdot|$ be an arbitrary homogeneous quasi-norm. Let $1<p,r<\infty$, $\delta\in(0,1]$, $q\in(p,\infty)$ with $q\leq\frac{p}{1-\delta}$ for $\delta\neq 1$. Let $0<a r<Q$ and $\beta,\gamma\in \mathbb{R}$ with $\delta q(Q-ar-\beta r)\leq r(Q+q\gamma-q\beta)$ and $\beta(1-\delta)-\delta a<\gamma\leq \beta(1-\delta)$. Assume that $\frac{q(\delta Q+r(\beta(1-\delta)-\gamma -a\delta))}{rQ}+\frac{q(1-\delta)}{p}=1$. Then we have
\begin{equation}
\int_{\mathbb{G}}\frac{|x|^{\gamma p}|u|^{p}}{\||x|^{\gamma}u\|^{p}_{L^{p}(\mathbb{G})}}\log\left(\frac{|x|^{\gamma p}|u|^{p}}{\||x|^{\gamma}u\|^{p}_{L^{p}(\mathbb{G})}}\right)dx\leq \frac{q}{q-p}\log\left(C\frac{\|\mathcal{R}^{\frac{a}{\nu}}u\|^{\delta p}_{L^{r}(\mathbb{G})}\||x|^{\beta}u\|^{(1-\delta)p}_{L^{p}(\mathbb{G})}}{\||x|^{\gamma}u\|^{p}_{L^{p}(\mathbb{G})}}\right),
\end{equation}
for all non-zero $|x|^{\gamma}u,|x|^{\beta}u, \mathcal{R}^{\frac{a}{\nu}}u\in L^{p}(\mathbb{G})$.
\end{thm}
\begin{proof} The proof is immeediate:
by using $q>p$ and logarithmic H\"{o}lder's inequality, we have
\begin{equation}
\begin{split}
&\int_{\mathbb{G}}\frac{|x|^{\gamma p}|u|^{p}}{\||x|^{\gamma }u\|^{p}_{L^{p}(\mathbb{G})}}\log\left(\frac{|x|^{\gamma p}|u|^{p}}{\||x|^{\gamma }u\|^{p}_{L^{p}(\mathbb{G})}}\right)dx\leq \frac{q}{q-p}\log\left(\frac{\||x|^{\gamma }u\|^{p}_{L^{q}(\mathbb{G})}}{\||x|^{\gamma }u\|^{p}_{L^{p}(\mathbb{G})}}\right)\\&
\stackrel{\eqref{CKNgr}}\leq\frac{q}{q-p}\log\left(C\frac{\|\mathcal{R}^{\frac{a}{\nu}}u\|^{\delta p}_{L^{r}(\mathbb{G})}\||x|^{\beta}u\|^{(1-\delta)p}_{L^{p}(\mathbb{G})}}{\||x|^{\gamma}u\|^{p}_{L^{p}(\mathbb{G})}}\right),
\end{split}
\end{equation}
completing the proof.
\end{proof}
\begin{cor}
Under the same assumptions of Theorem \ref{logCKNgrthm} with $\delta=1$ and $\gamma=0$, we get the log-Sobolev inequalities \eqref{LogSobolev} and \eqref{a2=0}.
\end{cor}
Our next aim is to give a weighted version of the log-Sobolev inequality. For this let us recall the so-called Hardy-Sobolev family of inequalities from
\cite[Theorem 5.1]{RY18b}, interpolating between the Hardy and the Sobolev inequalities.
\begin{thm}[Hardy-Sobolev inequality]\label{THM:HSgraded}
Let $\mathbb{G}$ be a graded Lie group of homogeneous dimension $Q$ and let
$\mathcal{R}$ be a positive Rockland operator of homogeneous degree $\nu$. Let $|\cdot|$ be an arbitrary homogeneous quasi-norm. Let $1<p\leq q<\infty$. Let $0<a p<Q$ and $0\leq \beta< Q$. Assume that $\frac{1}{p}-\frac{1}{q}=\frac{a}{Q}-\frac{\beta}{qQ}$. Then there exists a positive constant $C$ such that
\begin{equation}\label{EQ:HSgrad}
\left\|\frac{u}{|x|^{\frac{\beta}{q}}}\right\|_{L^{q}(\mathbb{G})}\leq C\|\mathcal{R}^{\frac{a}{\nu}}u\|_{L^{p}(\mathbb{G})},
\end{equation}
holds for all $u\in \dot{L}^{p}_{a}(\mathbb{G})$.
\end{thm}
Now let us present the weighted logarithmic Sobolev inequality on graded groups.
\begin{thm}\label{wsobingrthm}
Let $\mathbb{G}$ be a graded Lie group of homogeneous dimension $Q$ and let
$\mathcal{R}$ be a positive Rockland operator of homogeneous degree $\nu$. Let $|\cdot|$ be an arbitrary homogeneous quasi-norm. Let $1<p<\infty$ and $0\leq \beta <a p<Q$. Assume that $\frac{1}{p}-\frac{1}{q}=\frac{a}{Q}-\frac{\beta}{qQ}$. Then we have
\begin{equation}\label{harsobingr}
\int_{\mathbb{G}}\frac{|x|^{-\frac{\beta p}{q}}|u|^{p}}{\||x|^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\mathbb{G})}}\log\left(\frac{|x|^{-\frac{\beta p}{q}}|u|^{p}}{\||x|^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\mathbb{G})}}\right) dx \leq \frac{Q-\beta}{ap-\beta}\log\left(C\frac{\|\mathcal{R}^{\frac{a}{\nu}}u\|^{p}_{L^{p}(\mathbb{G})}}{\||x|^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\mathbb{G})}}\right),
\end{equation}
for all nontrivial $|x|^{-\frac{\beta}{q}}|u|,\mathcal{R}^{\frac{a}{\nu}}u\in L^{p}(\mathbb{G})$.
\end{thm}
\begin{proof}
By assumption $q=\frac{(Q-\beta)p}{Q-ap}$ and by using $0\leq \beta <ap< Q$, we have
\begin{equation}
q-p=\frac{(Q-\beta)p}{Q-ap}-p=\frac{p(Q-\beta-Q+ap)}{Q-ap}=\frac{(ap-\beta)p}{Q-ap}>0.
\end{equation}
It means we have $q>p$ and by using this fact, we compute
\begin{equation}\label{vychetstploghsgr}
0<\frac{q}{q-p}=\frac{\frac{(Q-\beta)p}{Q-ap}}{\frac{(Q-\beta)p}{Q-ap}-p}
=\frac{\frac{Q-\beta}{Q-ap}}{\frac{(Q-\beta)}{Q-ap}-1}
=\frac{Q-\beta}{ap-\beta}.
\end{equation}
We also note that we have
$\frac{1}{p}-\frac{1}{q}= \frac{a}{d}-\frac{\beta}{dq}$ in Theorem \ref{THM:HSgraded}.
By using the logarithmic H\"{o}lder's inequality in Lemma \ref{holder} with $q>p$, we have
\begin{equation*}
\begin{split}
\int_{\mathbb{G}}\frac{|x|^{-\frac{\beta p}{q}}|u|^{p}}{\||x|^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\mathbb{G})}}\log\left(\frac{|x|^{-\frac{\beta p}{q}}|u|^{p}}{\||x|^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\mathbb{G})}}\right) &dx\leq \frac{q}{q-p}\log\left(\frac{\||x|^{-\frac{\beta}{q}} u\|^{p}_{L^{q}(\mathbb{G})}}{\||x|^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\mathbb{G})}}\right)\\&
\stackrel{\eqref{EQ:HSgrad}}\leq \frac{q}{q-p}\log\left(C\frac{\|\mathcal{R}^{\frac{a}{\nu}}u\|^{p}_{L^{p}(\mathbb{G})}}{\||x|^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\mathbb{G})}}\right)\\&
\stackrel{\eqref{vychetstploghsgr}}=\frac{d-\beta}{ap-\beta}\log\left(C\frac{\|\mathcal{R}^{\frac{a}{\nu}}u\|^{p}_{L^{p}(\mathbb{G})}}{\||x|^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\mathbb{G})}}\right),
\end{split}
\end{equation*}
completing the proof.
\end{proof}
\begin{rem}
In particular for $\beta=0$, from \eqref{harsobingr}, we get the logarithmic Sobolev inequality. To be able to use the logarithmic H\"{o}lder's inequality, we need $1<p<q<\infty$ and if $\beta=aq$, then we have $p=q$, and so we can not obtain the logarithmic Hardy inequality by these arguments. Nevertheless, we are still able to prove a version of the logarithmic Hardy inequalities, but by a very different argument, so this will be a subject of another paper \cite{CKR21}.
\end{rem}
\section{Logarithmic inequalities on general Lie groups}
\label{SEC:generalLie}
In this section, we present logarithmic Sobolev and Hardy-Sobolev type inequalities on general Lie groups. In such generality we have a more limited track of constants compared e.g. to the case of graded groups. Moreover, we do not obtain estimates in homogeneous Sobolev spaces but in inhomogeneous ones, associated to H\"ormander's sums of squares with a drift, to make such sub-Laplacian operators self-adjoint, to include the cases when the group is not unimodular. However, if we decide to work with sub-Laplacians and we are not after best constants, very general results are possible, also including weighted versions of these logarithmic estimates.
Let first $\mathbb{G}$ be a noncompact connected Lie group. Assume that $e$ is the identity element of $\mathbb{G}$, and
let $X = \{X_1, . . . , X_n\}$ be a family of linearly independent, left-invariant vector fields on this group satisfying H\"{o}rmander’s condition. We denote by $\mu_{\chi}$ a measure on $\mathbb{G}$, whose density is the continuous positive character $\chi$ of $\mathbb{G}$ with respect to the right Haar measure $\rho$ of $\mathbb{G}$, i.e. $d\mu_{\chi} = \chi d\rho.$ Let us denote by $\delta$ the modular function on $\mathbb{G}$, so that $d\lambda = \delta d\rho$ is the left Haar measure on $\mathbb{G}$. In \cite{HMM05}, the authors showed that the differential operator
\begin{equation}
\Delta_{\chi}=-\sum_{i=1}^{n}\left(X_{j}^{2}+c_{j}X_{j}\right),
\end{equation}
with domain $C_{0}^{\infty}(\mathbb{G})$ on $\mathbb{G}$ is essentially self-adjoint on $L^{2}(\mu_{\chi})$, where $c_{j} = (X_{j}\chi)(e)$, $j = 1, \ldots , n.$
The Sobolev spaces on $\mathbb{G}$ are of the following form:
\begin{equation}
L^{p}_{a}(\mu_{\chi}):=\{u:u\in L^{p}(\mu_{\chi}),\Delta^{\frac{a}{2}}_{\chi}u\in L^{p}(\mu_{\chi}) \},
\end{equation}
endowed with norm
\begin{equation}
\|u\|_{L^{p}_{a}(\mu_{\chi})}:=\|u\|_{L^{p}(\mu_{\chi})}+\|\Delta^{\frac{a}{2}}_{\chi}u\|_{L^{p}(\mu_{\chi})}.
\end{equation}
Similarly, the Sobolev spaces with the left Haar measure $d\lambda$ can be introduced by
\begin{equation}
L^{p}_{a}(\lambda):=\{u:u\in L^{p}(\lambda),\Delta^{\frac{a}{2}}_{\chi}u\in L^{p}(\lambda) \},
\end{equation}
endowed with the norm
\begin{equation*}
\|u\|_{L^{p}_{a}(\lambda)}:=\|u\|_{L^{p}(\lambda)}+\|\Delta^{\frac{a}{2}}_{\chi}u\|_{L^{p}(\lambda)}.
\end{equation*}
The embedding properties of the Sobolev spaces $L^{p}_{a}(\mu_{\chi})$ were established in \cite{BPTV19} and \cite{RY18} by different methods, the latter paper giving also a family of weighted Hardy-Sobolev inequalities for these spaces and including the case of compact Lie groups.
We recall that $X = \{X_1, . . . , X_n\}$ (which is a family of linearly independent, left-invariant vector fields on this group satisfying H\"{o}rmander’s condition) induce the Carnot-Carath\'{e}odory distance $d_{CC}(\cdot, \cdot)$ (shortly, CC-distance). Assume that $B(c_{B},r_{b})$ is a ball with respect to the CC-distance centered at $c_{B}$ with radius $r_{B}$. Also, for simplicity of notation, we will write
\begin{equation}\label{EQ:dcc}
|x|_{CC}=d_{CC}(e,x).
\end{equation}
Let us denote by $V(r):=\rho(B_{r})$ the volume of the ball $B_{r}:=B(e,r)$ with respect to the right Haar measure, then it is well-known that there exist two constants $d=d(\mathbb{G},X)$ and $D=D(\mathbb{G})$ such that
\begin{equation*}
V(r)\approx r^{d},\,\,\,\,\forall r\in (0,1],
\end{equation*}
and
\begin{equation*}
V(r)\lesssim e^{Dr},\,\,\,\,\forall r\in (1,\infty).
\end{equation*}
We say that $d$ and $D$ are local and global dimensions of the metric measure space $(\mathbb{G},d_{CC}, \rho)$, respectively.
If $\mathbb{G}$ is a compact Lie group, it is unimodular, so that we have that the modular function $\delta=1$, and it means that $d\lambda=d\rho$. In addition, if we take the character $\chi=1$, we note that the Sobolev space $L^{p}_{a}(\mu_{\chi})$ coincides with Sobolev space defined by $\Delta_\mathbb{G}=\sum\limits_{i=1}^{n}X_{i}^{2}$. In this case $d$ is the Hausdorff dimension associated to the sub-Laplacian $\Delta_\mathbb{G}.$
Let us now recall the Hardy-Sobolev inequality from \cite{RY18}, for general connected Lie groups. As before, $|x|_{CC}=d_{CC}(x,e)$ is the Carnot-Carath\'{e}odiry distance between $x$ and $e$.
\begin{thm}\label{Harsobinnthm}
Let $\mathbb{G}$ be a connected Lie group. Let $0\leq \beta < d$ and $1< p,q <\infty$.
\begin{itemize}
\item[(a)] If $0< a < d$, then we have
\begin{equation}\label{HSinnon}
\left\|\frac{u}{|x|_{CC}^{\frac{\beta}{q}}}\right\|_{L^{q}(\lambda)}\leq C \|u\|_{L^{p}_{a}(\lambda)},
\end{equation}
for all $q\geq p$ such that $\frac{1}{p}-\frac{1}{q}\leq \frac{a}{d}-\frac{\beta}{dq}$;
\item[(b)] If $\frac{d}{p}\leq a$, then \eqref{HSinnon} holds for all $q\geq p$.
\end{itemize}
\end{thm}
Although in \cite{RY18}, this theorem was formulated for $a<d$ in Part (b), it is clear that if \eqref{HSinnon} holds for some $a$, it also holds for all larger $a$.
Now let us present the logarithmic weighted Sobolev inequality, for general Lie groups (compact and non-compact).
\begin{thm}\label{loghardysob}
Let $\mathbb{G}$ be a connected Lie group.
\begin{itemize}
\item[(a)] If $1< p <\infty$ and $a>0$ is such that $0\leq \beta <ap< d$,
then we have
\begin{equation}\label{harsobinnon1}
\int_{\mathbb{G}}\frac{|x|_{CC}^{-\frac{\beta p}{q}}|u|^{p}}{\||x|_{CC}^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\lambda)}}\log\left(\frac{|x|_{CC}^{-\frac{\beta p}{q}}|u|^{p}}{\||x|_{CC}^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\lambda)}}\right) d\lambda(x) \leq \frac{d-\beta}{ap-\beta}\log\left(C\frac{\|u\|^{p}_{L^{p}_{a}(\lambda)}}{\||x|_{CC}^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\lambda)}}\right),
\end{equation}
where $q=\frac{(d-\beta)p}{d-ap}$.
\item[(b)] If $a\geq\frac{d}{p}$, $0\leq \beta<d$ and $q>p>1$, then we have
\begin{equation}\label{harsobinnon2}
\int_{\mathbb{G}}\frac{|x|_{CC}^{-\frac{\beta p}{q}}|u|^{p}}{\||x|_{CC}^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\lambda)}}\log\left(\frac{|x|_{CC}^{-\frac{\beta p}{q}}|u|^{p}}{\||x|_{CC}^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\lambda)}}\right) d\lambda(x) \leq \frac{q}{q-p}\log\left(C\frac{\|u\|^{p}_{L^{p}_{a}(\lambda)}}{\||x|_{CC}^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\lambda)}}\right).
\end{equation}
\end{itemize}
The constant $C$ in these inequalities may depend on $\mathbb{G}$, on the system of H\"ormander's vector fields, on $p,q,a,\beta$, but not on $u$.
\end{thm}
\begin{proof}
(a) Let us choose $q=\frac{(d-\beta)p}{d-ap}$, then by assumption $0\leq \beta <ap< d$, we have $q>p$ and by using this fact, we compute
\begin{equation}\label{vychetstploghs}
0<\frac{q}{q-p}=\frac{\frac{(d-\beta)p}{d-ap}}{\frac{(d-\beta)p}{d-ap}-p}
=\frac{\frac{d-\beta}{d-ap}}{\frac{(d-\beta)}{d-ap}-1}
=\frac{d-\beta}{ap-\beta}.
\end{equation}
We also note that we have
$\frac{1}{p}-\frac{1}{q}= \frac{a}{d}-\frac{\beta}{dq}$ in Theorem \ref{Harsobinnthm}, (a).
By using logarithmic H\"{o}lder's inequality with $|x|_{CC}^{-\frac{\beta}{q}}|u|\in L^{p}(\lambda)\cap L^{q}(\lambda)$ and $q>p$, we get
\begin{equation}
\begin{split}
\int_{\mathbb{G}}\frac{|x|_{CC}^{-\frac{\beta p}{q}}|u|^{p}}{\||x|_{CC}^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\lambda)}}\log\left(\frac{|x|_{CC}^{-\frac{\beta p}{q}}|u|^{p}}{\||x|_{CC}^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\lambda)}}\right) &d\lambda(x)\leq \frac{q}{q-p}\log\left(\frac{\||x|_{CC}^{-\frac{\beta}{q}} u\|^{p}_{L^{q}(\lambda)}}{\||x|_{CC}^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\lambda)}}\right)\\&
\stackrel{(a)\eqref{HSinnon}}\leq \frac{q}{q-p}\log\left(C\frac{\|u\|^{p}_{L^{p}_{a}(\lambda)}}{\||x|_{CC}^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\lambda)}}\right)\\&
\stackrel{\eqref{vychetstploghs}}=\frac{d-\beta}{ap-\beta}\log\left(C\frac{\|u\|^{p}_{L^{p}_{a}(\lambda)}}{\||x|_{CC}^{-\frac{\beta}{q}} u\|^{p}_{L^{p}(\lambda)}}\right).
\end{split}
\end{equation}
(b) Similarly to the previous case (a) we easily show \eqref{harsobinnon2} by using Theorem \ref{Harsobinnthm}, Part (b).
\end{proof}
\begin{rem}\label{REM:CCR}
According to \cite[Remark 1.2]{RY18}, the statement of Theorem \ref{Harsobinnthm} also holds with the Carnot-Carath\'eodory distance replaced by the Riemannian distance. Consequently,
Theorem \ref{loghardysob} and Theorem \ref{THM:Nash-gen} also hold with the Carnot-Carath\'eodory distance replaced by the Riemannian distance.
\end{rem}
Under assumptions of Theorem \ref{loghardysob} with $\beta=0$, we get the logarithmic Sobolev inequality in the following form:
\begin{cor}\label{logsobcor}
Let $\mathbb{G}$ be a connected Lie group with local dimension $d$.
Then we have the following logarithmic Sobolev inequalities:
\begin{itemize}
\item[(a)] If $1< p <\infty$ and $0<a<\frac{d}{p}$, then we have
\begin{equation}\label{sobinnon1}
\int_{\mathbb{G}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\lambda)}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\lambda)}}\right) d\lambda(x) \leq \frac{d}{ap}\log\left(C\frac{\|u\|^{p}_{L^{p}_{a}(\lambda)}}{\|u\|^{p}_{L^{p}(\lambda)}}\right),
\end{equation}
with $q=\frac{dp}{d-ap}$.
\item[(b)] If $a\geq\frac{d}{p}$ and $q>p>1$, then we have
\begin{equation}\label{sobinnon2}
\int_{\mathbb{G}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\lambda)}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\lambda)}}\right) d\lambda(x) \leq \frac{q}{q-p}\log\left(C\frac{\|u\|^{p}_{L^{p}_{a}(\lambda)}}{\|u\|^{p}_{L^{p}(\lambda)}}\right),
\end{equation}
with the constant $C$ depending on $\mathbb{G}$, on the system of H\"ormander's vector fields, on $p,q,a$, but not on $u$.
\end{itemize}
\end{cor}
\begin{rem}
In Corollary \ref{logsobcor}, if $\mathbb{G}$ is non-compact, the constant $C$ in \eqref{sobinnon1} can be chosen to be of the form $C=A_{1}S(p)^p$, where $A_{1}=A_{1}(\mathbb{G},X)>0$ depends on the group $\mathbb{G}$ and on the H\"ormander's system of vector fields $X$, and $S(p)=\min\left\{\frac{\left(\frac{dp}{d-ap}\right)^{1-\frac{1}{p}}}{p-1},\left(\frac{p}{p-1}\right)^{\frac{1}{p}-\frac{a}{d}}(\frac{dp-d+ap}{d-ap})\right\}$. This follows from the constant presented in \cite[Theorem 3.1]{BPV21}, by taking
$S(p)$ to be $S(p,q)$ in that paper, with
$q=\frac{dp}{d-ap}$. This condition comes from the equality $\frac{1}{p}-\frac{1}{q}= \frac{a}{d}$ with $\beta=0$ in Theorem \ref{Harsobinnthm}, (a).
In particular, it means that the constant $C$ in \eqref{sobinnon1} depends on the group structure, the H\"{o}r\-man\-der system $X$, and $p$.
\end{rem}
We do not have such argument for the constant $C$ in \eqref{sobinnon2}, but, by letting $q\to\infty$ we can make the constant in front of the logarithm as close to 1 as we want, namely:
\begin{cor}\label{COR:apq}
Let $\mathbb{G}$ be a connected Lie group with local dimension $d$. Let $1<p<\infty$ and $a\geq\frac{d}{p}.$ Then for every $\epsilon>0$ there exists $C_\epsilon>0$ such that for all $u\not=0$ we have
\begin{equation}\label{sobinnon2-eps}
\int_{\mathbb{G}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\lambda)}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\lambda)}}\right) d\lambda(x) \leq (1+\epsilon)\log\left(C_\epsilon\frac{\|u\|^{p}_{L^{p}_{a}(\lambda)}}{\|u\|^{p}_{L^{p}(\lambda)}}\right).
\end{equation}
\end{cor}
\section{Nash inequality}
In this section we discuss the Nash inequality on general Lie groups, and on graded groups, and its immediate application to the decay of solutions to the heat equation for the sub-Laplacian on stratified groups. For stratified Lie groups, the Nash inequality would take the form
\begin{equation}
\label{Nas}
\|u\|_{L^2(\mathbb{G})}^{2+\frac{4}{Q}} \leq C
\|u\|_{L^1(\mathbb{G})}^{\frac{4}{Q}}
\|\nabla_H u\|_{L^2(\mathbb{G})}^2.
\end{equation}
For the Euclidean $\mathbb{G}=\mathbb R^n$ and $\nabla_H=\nabla$ being the full gradient, the inequality \eqref{Nas} was established by Nash in \cite{Nas58}, with a simple proof using the Euclidean Fourier analysis, credited to E. M. Stein. It is one of the very useful inequalities for problems in partial differential equations, and can viewed as the limiting $L^1$-case of the family of Gagliardo-Nirenberg inequalities. The best constant $C$ in \eqref{Nas} was determined by Carlen and Loss \cite{CL}, see also
\cite{BDS20} for the further analysis of its extremisers and several different approaches. For sums of squares on unimodular Lie groups, equivalent conditions for the Nash inequality have been investigated e.g. in \cite[Theorem II.5.2]{VSCC93}, in particular, it implies a number of estimates for the heat operator $e^{-t\Delta_\mathbb{G}}$, see e.g. \cite[Theorem II.3.2]{VSCC93}.
Let us show that \eqref{Nas} also allows for an elementary proof by a combination of H\"older and Sobolev inequalities. First, the H\"older inequality implies the inequality
\begin{equation}
\label{EQ:Nash-Holder}
\|u\|_{L^2(\mathbb{G})}^2\leq \|u\|_{L^1(\mathbb{G})}^{\frac{4}{Q+2}}
\|u\|_{L^{\frac{2Q}{Q-2}}(\mathbb{G})}^{\frac{2Q}{Q+2}}.
\end{equation}
Consequently, using Theorem \ref{GN}, we have that
\begin{equation}
\label{EQ:Nash-Holder2}
\|u\|_{L^{\frac{2Q}{Q-2}}(\mathbb{G})}^{2}\leq
C_{S}\|\mathcal{R}^{\frac{1}{\nu}} u\|_{L^2(\mathbb{G})}^2,
\end{equation}
for any Rockland operator $\mathcal{R}$ of degree $\nu$, and
the constant $C_{S}=C^{\frac{Q-2}{Q}}_{GN,\mathcal{R},1,0,2,\frac{2Q}{Q-2}}$ in \eqref{bestcongn}.
Consequently, taking $\mathcal{R}=-\Delta_\mathbb{G}$ to be a sub-Laplacian on $\mathbb{G}$, and using the identity \eqref{lapfor}, we obtain \eqref{Nas}. The above argument also gives \eqref{Nash-final2t} with $a=1$.
On the Euclidean space, again it is well-known that the Nash inequality is equivalent to the $L^2$-log-Sobolev inequality. This is also the case in our setting on general graded Lie groups. In fact, our proof of Corollary \ref{COR:Lpspb123} shows how Nash inequality would imply the $L^2$-log-Sobolev inequality. The converse argument is given in the proof of the next theorem, which gives the Nash inequality as its special case with $a=1$. In particular, if $\mathbb{G}$ is a stratified group, the following theorem with $a=1$ and $\mathcal{R}$ being the sub-Laplacian, and using equality \eqref{lapfor}, implies \eqref{Nas}.
\begin{thm}\label{THM:Nash}
Let $\mathbb{G}$ be a graded group with homogeneous dimension $Q$ and let $\mathcal{R}$ be a positive Rockland operator of homogeneous degree $\nu$. Let $0\leq \beta<2a<Q.$ Then, Nash's inequality is given by
\begin{equation}\label{Nash-final2t}
||\cdot|^{-\frac{\beta(Q-2a)}{2(Q-\beta)}}u\|_{L^2(\mathbb{G})}^{2+\frac{2(2a-\beta)}{Q-\beta}} \leq C \||\cdot|^{-\frac{\beta(Q-2a)}{2(Q-\beta)}}u\|_{L^1(\lambda)}^{\frac{2(2a-\beta)}{Q-\beta}}\|u\|^{2}_{\dot{L}^2_a(\lambda)},
\end{equation}
where $C>0$ is independent of $u$.
In particular, for $\beta=0$ and $0<a<\frac{Q}{2}$, we have
\begin{equation}\label{Nash.g}
\|u\|_{L^2(\mathbb{G})}^{2+\frac{4a}{Q}} \leq A_2 \|u\|_{L^1(\mathbb{G})}^{\frac{4a}{Q}}\|u\|^{2}_{\dot{L}^2_a(\mathbb{G})}\,,
\end{equation}
where $A_2$ is considered as in \eqref{CinfGN} for $p=2$.
\end{thm}
\begin{proof}
Let us choose $q=\frac{2(Q-\beta)}{Q-2a}$ and $v(x)=|x|^{-\frac{\beta}{q}}u(x).$ Observe that $|v|^2 \|v\|^{-2}_{L^2(\mathbb{G})}\,dx$ is a probability measure on $\mathbb{G}$. Under this observation, an application of Jensen's inequality to the convex function $h(u)=\log \frac{1}{u}$ yields
\begin{eqnarray*}
\log \left(\frac{\|v\|^{2}_{L^2(\mathbb{G})}}{\|v\|_{L^1(\mathbb{G})}} \right) & = & h \left( \frac{\|v\|_{L^1(\mathbb{G})}}{\|v\|^{2}_{L^2(\mathbb{G})}}\right)=h \left( \int_{\mathbb{G}}\frac{1}{|v|}|v|^2 \|v\|^{-2}_{L^2(\mathbb{G})}\,dx\right)\\
& \leq & \int_{\mathbb{G}} h \left( \frac{1}{|v|}\right) |v|^2 \|v\|^{-2}_{L^2(\mathbb{G})}\,dx=\int_{\mathbb{G}} \frac{|v|^2}{\|v\|^{2}_{L^2(\mathbb{G})}}\log |v| \,dx\,.
\end{eqnarray*}
The latter, combined with the weighted log-Sobolev inequality in Theorem \ref{wsobingrthm} for $p=2$, using $Q>2a$, i.e., with the estimate
\[
\int_{\mathbb{G}} \frac{|v|^2}{\|v\|^{2}_{L^2(\mathbb{G})}}\log |v| \,dx \leq \log \|v\|_{L^2(\mathbb{G})}+\frac{Q-\beta}{2(2a-\beta)} \log \left( C \frac{\|u\|^{2}_{\dot{L}^2_a(\mathbb{G})}}{\|v\|^{2}_{L^2(\mathbb{G})}}\right)\,,
\]
using properties of the logarithm, gives
\[
\left(1+\frac{Q-\beta}{2a-\beta} \right) \log \|v\|_{L^2(\mathbb{G})} \leq \frac{Q-\beta}{2(2a-\beta)} \log \left(C \|v\|_{L^1(\mathbb{G})}^{\frac{2(2a-\beta)}{Q-\beta}}\|u\|^{2}_{\dot{L}^2_a(\mathbb{G})} \right)\,,
\]
which is equivalent to
\begin{equation}\label{Nash-final2}
\||x|^{-\frac{\beta(Q-2a)}{2(Q-\beta)}}u\|_{L^2(\mathbb{G})}^{2+\frac{2(2a-\beta)}{Q-\beta}} \leq C \||x|^{-\frac{\beta(Q-2a)}{2(Q-\beta)}}u\|_{L^1(\mathbb{G})}^{\frac{2(2a-\beta)}{Q-\beta}}\|u\|^{2}_{\dot{L}^2_a(\mathbb{G})}\,.
\end{equation}
This completes the proof of \eqref{Nash-final2t}. To get the constant $A_2$ in \eqref{Nash.g}, we can do the same argument but use Corollary \ref{COR:Lpspb123} instead of Theorem \ref{wsobingrthm}, with $A_2$ is given by \eqref{CinfGN} for $p=2$.
\end{proof}
As an immediate application of the Nash inequality, one can compute the decay rate for the heat equation for the sub-Laplacian, following Nash's argument. We give a short application just to illustrate such idea, and refer to more general related analysis in the context of symmetric Markovian semigroups to e.g. \cite[Section II.5]{VSCC93}.
\begin{cor}\label{cor:par}
Let $\mathbb{G}$ be a stratified Lie group of homogeneous dimension $Q\geq 3$, and let $\Delta_\mathbb{G}$ be a sub-Laplacian on $\mathbb{G}$.
Let $u_0\in L^1(\mathbb{G})\cap L^2(\mathbb{G})$ be non-negative $u_0\geq 0$.
Then the solution $u$ to the heat equation
\begin{equation}
\label{heat.eq}
\partial_t u=\Delta_\mathbb{G} u,\quad u(0,x)=u_0(x),
\end{equation}
satisfies
\begin{equation}
\label{heatest}
\|u(t,\cdot)\|_{L^2(\mathbb{G})}\leq \left( \|u_0\|_{L^2(\mathbb{G})}^{-\frac{4}{Q}}
+\frac{4}{QA_2}\|u_0\|_{L^1(\mathbb{G})}^{-\frac{4}{Q}}t\right)^{-\frac{Q}{4}},
\end{equation}
for all $t\geq 0,$ where $A_2$ is a constant in \eqref{Nash.g}.
\end{cor}
\begin{proof}
Let $\mathbb{G}$ be a stratified Lie group and let $\Delta_\mathbb{G}$ be a sub-Laplacian on $\mathbb{G}$.
Let $u$ be the solution to the heat equation \eqref{heat.eq}.
If $u_0\in L^1(\mathbb{G})\cap L^2(\mathbb{G})$ is non-negative $u_0\geq 0$, then also $u(t,x)\geq 0$ due to the positivity of the heat kernel $h_t$, see e.g. \cite[p. 48]{VSCC93}. Consequently, as in the Euclidean case, we have the mass conservation
$\|u(t,\cdot)\|_{L^1}=\|u_0\|_{L^1}.$ Indeed, we have
$$
\int_\mathbb{G} u(t,x) dx=\int_\mathbb{G}\int_\mathbb{G} h_t(x y^{-1})u_0(y) dydx=\int_\mathbb{G} u_0(y) dy
$$
in view of the Fubini theorem and the property $\|h_t\|_{L^1}=1.$ Now, multiplying by $u$ the heat equation \eqref{heat.eq}, and integrating over $\mathbb{G}$ we get
\[
\langle \partial_t u(t,\cdot), u(t,\cdot) \rangle_{L^2(\mathbb{G})}+\langle -\Delta_\mathbb{G} u(t,\cdot), u(t,\cdot) \rangle_{L^2(\mathbb{G})}=0\,,
\]
or, in view of \eqref{lapfor},
\[
\frac{d}{dt}\|u(t,\cdot)\|_{L^2(\mathbb{G})}=-2\|\nabla_H u(t,\cdot)\|_{L^2(\mathbb{G})}\,.
\]
If we denote by $y(t):=\|u(t,\cdot)\|_{L^2(\mathbb{G})}$, an application of Nash's inequality \eqref{Nas} (or \eqref{Nash-final2t}) gives
\[
y' \leq -2A_2^{-1}\|u_0\|_{L^1(\mathbb{G})}^{-\frac{4}{Q}}y^{1+\frac{2}{Q}}\,,
\]
which, after integration on $t \geq 0$, implies that the solution to \eqref{heat.eq} satisfies the estimate
\[
\|u(t,\cdot)\|_{L^2(\mathbb{G})}\leq \left(\|u_0\|_{L^2(\mathbb{G})}^{-\frac{4}{Q}}+\frac{4}{QA_2}\|u_0\|_{L^1(\mathbb{G})}^{-\frac{Q}{4}}t \right)^{-\frac{Q}{4}}\,,
\]
and this finishes the proof.
\end{proof}
Let us now give the weighted Nash inequality on general Lie groups, where, as before, $|x|_{CC}=d_{CC}(x,e)$ is the Carnot-Carath\'eodory distance between $x$ and $e$, see \eqref{EQ:dcc} for details. However, we also note that the following estimates also hold true with the Carnot-Carath\'eodory distance replaced by the Riemannian distance, see Remark \ref{REM:CCR}.
\begin{thm}\label{THM:Nash-gen}
Let $\mathbb{G}$ be a connected Lie group with local dimension $d$. Assume that
$0\leq \beta <2a<d$.
Then, the weighted Nash inequality is given by
\begin{equation}\label{Nash-final2th}
\||\cdot|_{CC}^{-\frac{\beta(d-2a)}{2(d-\beta)}}u\|_{L^2(\lambda)}^{2+\frac{2(2a-\beta)}{d-\beta}} \leq C \||\cdot|_{CC}^{-\frac{\beta(d-2a)}{2(d-\beta)}}u\|_{L^1(\lambda)}^{\frac{2(2a-\beta)}{d-\beta}}\|u\|^{2}_{L^2_a(\lambda)}.
\end{equation}
In particular, for $\beta=0$ and $0<a<\frac{d}{2}$, we have the Nash inequality
\begin{equation}\label{Nash-final2th0}
\|u\|_{L^2(\lambda)}^{2+\frac{4a}{d}} \leq C \|u\|_{L^1(\lambda)}^{\frac{4a}{d}}\|u\|^{2}_{L^2_a(\lambda)}.
\end{equation}
Moreover, if $a\geq\frac{d}{2}$ and $0\leq \beta<d$, then for every $q>2$ the weighted Nash inequality is given by
\begin{equation}
\||\cdot|_{CC}^{-\frac{\beta}{q}}u\|^{2+\frac{2(q-2)}{q}}_{L^{2}(\lambda)}\leq C\||\cdot|_{CC}^{-\frac{\beta}{q}}u\|^{\frac{2(q-2)}{q}}_{L^{1}(\lambda)}\|u\|^{2}_{L^{2}_{a}(\lambda)}.
\end{equation}
In particular, for $\beta=0$ and any $a\geq\frac{d}{2}$ and $q>2$, we have the Nash type inequality
\begin{equation}
\|u\|^{2+\frac{2(q-2)}{q}}_{L^{2}(\lambda)}\leq C\|u\|^{\frac{2(q-2)}{q}}_{L^{1}(\lambda)}\|u\|^{2}_{L^{2}_{a}(\lambda)}.
\end{equation}
\end{thm}
\begin{proof}
The proof of this theorem is similar to that of Theorem \ref{THM:Nash}, but instead of Theorem \ref{wsobingrthm} we use Theorem \ref{loghardysob}.
Then, we choose $q=\frac{2(d-\beta)}{d-2a}$. We denote $v(x)=|x|_{CC}^{-\frac{\beta}{q}}u(x).$
Observe that $|v|^{2}=||x|_{CC}^{-\frac{\beta}{q}}u|^2$ and $\frac{|v|^{2}}{\|v\|^{2}_{L^{2}(\lambda)}}d\lambda(x)$ is a probability measure with respect to left Haar measure on $\mathbb{G}$. Under this observation, an application of Jensen's inequality to the convex function $h(v)=\log \frac{1}{v}$ yields
\begin{eqnarray}\label{nashjensen1}
\log \left(\frac{\|v\|^{2}_{L^2(\lambda)}}{\|v\|_{L^1(\lambda)}} \right) & = & h \left( \frac{\|v\|_{L^1(\lambda)}}{\|v\|^{2}_{L^2(\lambda)}}\right)=h \left( \int_{\mathbb{G}}\frac{1}{|v|}|v|^2 \|v\|^{-2}_{L^2(\lambda)}\,d\lambda(x)\right)\\
& \leq & \int_{\mathbb{G}} h \left( \frac{1}{|v|}\right) |v|^2 \|v\|^{-2}_{L^2(\lambda)}\,d\lambda(x)=\int_{\mathbb{G}} \frac{|v|^2}{\|v\|^{2}_{L^2(\lambda)}}\log |v| \,d\lambda(x)\,.\nonumber
\end{eqnarray}
The latter, combined with the log-Hardy-Sobolev inequality in Theorem \ref{loghardysob} for $p=2$, i.e., with the estimate
\[
\int_{\mathbb{G}} \frac{|v|^2}{\|v\|^{2}_{L^2(\lambda)}}\log |v| \,d\lambda(x) \leq \log \|v\|_{L^2(\lambda)}+\frac{d-\beta}{2(2a-\beta)} \log \left( C \frac{\|u\|^{2}_{L^2_a(\lambda)}}{\|v\|^{2}_{L^2(\lambda)}}\right)\,,
\]
where $C>0$ for $p=2$, using properties of the logarithm, gives
\[
\left(1+\frac{d-\beta}{2a-\beta} \right) \log \|v\|_{L^2(\lambda)} \leq \frac{d-\beta}{2(2a-\beta)} \log \left(C \|v\|_{L^1(\lambda)}^{\frac{2(2a-\beta)}{d-\beta}}\|u\|^{2}_{L^2_a(\lambda)} \right)\,,
\]
which is equivalent to
\begin{equation}\label{Nash-final23}
\||\cdot|_{CC}^{-\frac{\beta}{q}}u\|_{L^2(\lambda)}^{\frac{2(2a-\beta)}{d-\beta}+2} \leq C \||\cdot|_{CC}^{-\frac{\beta}{q}}u\|_{L^1(\lambda)}^{\frac{2(2a-\beta)}{d-\beta}}\|u\|^{2}_{L^2_a(\lambda)}\,.
\end{equation}
Let us now prove the case $q>2$ and $a\geq\frac{d}{2}$. Similarly, let us choose $|v|^{2}=||x|_{CC}^{-\frac{\beta}{q}}u|^2$, and observe that $\frac{|v|^{2}}{\|v\|^{2}_{L^{2}(\lambda)}}d\lambda(x)$ is a probability measure with respect to left Haar measure on $\mathbb{G}$.
By combining \eqref{nashjensen1} and \eqref{harsobinnon2}, we have
\begin{equation*}\label{nashjensen2}
\begin{split}
2\log\|v\|_{L^{2}(\lambda)}-\log\|v\|_{L^{1}(\lambda)}&=\log \left(\frac{\|v\|^{2}_{L^2(\lambda)}}{\|v\|_{L^1(\lambda)}} \right) \\&
= h \left( \frac{\|v\|_{L^1(\lambda)}}{\|v\|^{2}_{L^2(\lambda)}}\right)\\&=h \left( \int_{\mathbb{G}}\frac{1}{|v|}|v|^2 \|v\|^{-2}_{L^2(\lambda)}\,d\lambda(x)\right)\\&
\leq \int_{\mathbb{G}} h \left( \frac{1}{|v|}\right) |v|^2 \|v\|^{-2}_{L^2(\lambda)}\,d\lambda(x)\\&
=\int_{\mathbb{G}} \frac{|v|^2}{\|v\|^{2}_{L^2(\lambda)}}\log |v| \,d\lambda(x)\\&
\stackrel{\eqref{harsobinnon2}}\leq \log\|v\|_{L^{2}(\lambda)}+\frac{q}{2(q-2)}\log\left(C\frac{\|u\|^{2}_{L^{2}_{a}(\lambda)}}{\|v\|^{2}_{L^{2}(\lambda)}}\right)\\&
=\frac{q}{2(q-2)}\log\left(C\|u\|^{2}_{L^{2}_{a}(\lambda)}\right)+\log\|v\|^{1-\frac{q}{q-2}}_{L^{2}(\lambda)}\\&
=\frac{q}{2(q-2)}\log\left(C\|u\|^{2}_{L^{2}_{a}(\lambda)}\right)+\left(1-\frac{q}{q-2}\right)\log\|v\|_{L^{2}(\lambda)}.
\end{split}
\end{equation*}
By using this last fact, we have
\begin{equation}
\begin{split}
\left(1+\frac{q}{q-2}\right)\log\|v\|_{L^{2}(\lambda)}\leq \frac{q}{2(q-2)}\log\left(C\|u\|^{2}_{L^{2}_{a}(\lambda)}\|v\|^{\frac{2(q-2)}{q}}_{L^{1}(\lambda)}\right),
\end{split}
\end{equation}
so that we have
\begin{equation}
\|v\|^{\frac{2(q-2)}{q}+2}_{L^{2}(\lambda)}\leq C\|u\|^{2}_{L^{2}_{a}(\lambda)}\|v\|^{\frac{2(q-2)}{q}}_{L^{1}(\lambda)}.
\end{equation}
Finally, this means
\begin{equation}
\||\cdot|_{CC}^{-\frac{\beta}{q}}u\|^{\frac{2(q-2)}{q}+2}_{L^{2}(\lambda)}\leq C\|u\|^{2}_{L^{2}_{a}(\lambda)}\||\cdot|_{CC}^{-\frac{\beta}{q}}u\|^{\frac{2(q-2)}{q}}_{L^{1}(\lambda)}.
\end{equation}
This completes the proof.
\end{proof}
\section{Logarithmic Sobolev and Gross inequalities on stratified groups}
We now give some corollaries of the previous results for the setting of stratified Lie groups. First we note that for $a=1$ and $p=2$, we can rewrite \eqref{a2=0} using the horizontal gradient on $\mathbb{G}.$ Indeed, we use Corollary \ref{COR:Lpspb123} with the sub-Laplacian
$\Delta_\mathbb{G}=-\nabla_H^*\nabla_H$.
\begin{cor}\label{Cor.Log}
Let $\mathbb{G}$ be a stratified Lie group with homogeneous dimension $Q$. Then, the following log-Sobolev inequality is satisfied
\begin{equation}
\label{cor.log.sob}
\int_{\mathbb{G}}|u|^2 \log|u|\,dx \leq \frac{Q}{4}\log \left(A \int_{\mathbb{G}}|\nabla_{H}u|^2\,dx \right)\,,
\end{equation}
for every $u$ such that $\|u\|_{L^2(\mathbb{G})}=1$, where \begin{equation}\label{Cinf}A=\inf_{2<q<\frac{2Q}{Q-2}}(C_{GN, -\Delta_\mathbb{G},1,2,q})^{\frac{4}{Q(q-2)}}\,,
\end{equation}
where $C_{GN, -\Delta_\mathbb{G},1,2,q}$ is the constant given in \eqref{bestcongn} for $a_1=1,a_2=0$ and $p=2$.
\end{cor}
\begin{proof}
Inequality \eqref{a2=0} for $p=2$, $\|u\|_{L^2(\mathbb{G})}=1$, $a_1=1$, and for $\mathcal{R}_1$ being the (positive) sub-Laplacian $-\Delta_\mathbb{G}$ on $\mathbb{G}$ gives
\[
\int_{\mathbb{G}}|u|^2 \log(|u|^2)\,dx \leq \frac{q}{q-2}\log \left(C_{GN, -\Delta_\mathbb{G},1,2,q}^{\frac{2}{q}}\|(-\Delta_{\mathbb{G}})^{1/2}u\|_{L^2(\mathbb{G})}^{\frac{Q(q-2)}{q}} \right)\,,
\]
for any $q\in \left(2,\frac{2Q}{Q-2}\right)$. Rewriting the above inequality as
\[
\int_{\mathbb{G}}|u|^2 \log(|u|^2)\,dx \leq \frac{q}{q-2}\log \left( (C_{GN, -\Delta_\mathbb{G},1,2,q})^{\frac{4}{Q(q-2)}}\|(-\Delta_{\mathbb{G}})^{1/2}u\|_{L^2(\mathbb{G})}^{2}\right)^{\frac{Q(q-2)}{2q}}\,,
\]
and applying the properties of the logarithm we get
\begin{equation}\label{in.cor.log.sob}
\int_{\mathbb{G}}|u|^2 \log(|u|)\,dx \leq \frac{Q}{4}\log \left( (C_{GN, -\Delta_\mathbb{G},1,2,q})^{\frac{4}{Q(q-2)}}\int_{\mathbb{G}}|\nabla_{H}u|^2\,dx\right)\,,
\end{equation}
under the observation that
\begin{equation}\label{lapfor}
\|(-\Delta_\mathbb{G})^{1/2} u\|_{L^2(\mathbb{G})}=\||\nabla_{H}| u\|_{L^2(\mathbb{G})}=\|\nabla_{H}u\|_{L^2(\mathbb{G})}\,,
\end{equation}
where $\nabla_{H}$ denotes the horizontal gradient on $\mathbb{G}$. Finally, to prove \eqref{cor.log.sob}, notice that \eqref{in.cor.log.sob} holds for any $q\in \left(2,\frac{2Q}{Q-2}\right)$.
\end{proof}
It is well-known that in $\mathbb R^n$, the Euclidean log-Sobolev inequality implies the Gross version of such an inequality with the Gaussian measure. We now show that to some extent, this implication continues to hold also on stratified groups. However, instead of the Gaussian measure on the whole space we consider the Gaussian measure on the first stratum. In the case of the Euclidean $\mathbb{G}=\mathbb R^n$, this recovers the above implication since the first stratum is the whole space.
Since in $\mathbb R^n$, we can take $A$ in \eqref{cor.log.sob} to be $A=\frac{2}{\pi e n}$, and we have $Q=n_1=n$, as should be expected, the constant $\gamma$ in \eqref{k} is
$\gamma_{\mathbb R^n}=(2\pi)^{-n/2}$, recovering the well-known Gross inequality \eqref{EQ:Gross} on $\mathbb R^n$.
\begin{thm}\label{semi-g}
Let $\mathbb{G}$ be a stratified group with homogeneous dimension $Q$, topological dimension $n$, and let $n_1$ be the dimension of the first stratum of its Lie algebra, i.e., for $x \in \mathbb{G}$ we can write $x=(x',x'') \in \mathbb{R}^{n_1} \times \mathbb{R}^{n-n_1}$. Then the following ``semi-Gaussian'' log-Sobolev inequality is satisfied
\begin{equation}
\label{gaus.log.sob}
\int_{\mathbb{G}}|g|^2\log|g|\,d\mu \leq \int_{\mathbb{G}} |\nabla_{H}g|^2\,d\mu\,,
\end{equation}
for any $g$ such that $\|g\|_{L^2(\mu)}=1,$
where $\mu=\mu_1 \otimes \mu_2$, and $\mu_1$ is the Gaussian measure on $\mathbb{R}^{n_1}$ given by $d\mu_1=\gamma e^{-\frac{|x'|^2}{2}}dx'$, for $x'\in \mathbb{R}^{n_1}$, and $|x'|$ being the Euclidean norm of $x'$, where
\begin{equation}
\label{k}
\gamma:=(4^{-1}Q e^{\frac{2n_1}{Q}{-1}}A)^{Q/2}\,,
\end{equation}
where $A$ is given in \eqref{Cinf},
and $\mu_2$ is the Lebesgue measure $dx''$ on $\mathbb{R}^{n-n_1}$.
\end{thm}
\begin{proof}
Assume that $g\in L^2(\mu)$ is such that $\|g\|_{L^2(\mu)}=1,$ with $\mu$ as in the conditions of the theorem. Let us define $f(x)$ by $$f(x)=\gamma^{1/2}e^{-\frac{|x'|^2}{4}}g(x),$$
where $\gamma$ is as in \eqref{k}. Then we claim that $f \in L^2(\mathbb{G})$ and $\|f\|_{L^2(\mathbb{G})}=1$.
Indeed, we observe that for $\mu$ as in the hypothesis we get
\begin{equation}
\label{norms1}
1=\|g\|_{L^2(\mu)}^2=\int_{\mathbb{G}}\gamma^{-1} e^{\frac{|x'|^2}{2}}|f(x)|^2\,d\mu=\int_{\mathbb{G}}|f(x)|^2\,dx\,.
\end{equation}
Applying the log-Sobolev inequality \eqref{cor.log.sob} to $f$, we estimate
\begin{eqnarray}\label{thm.eq.glogg}
\int_{\mathbb{G}}|g(x)|^2\log |g(x)|\,d\mu & = & \int_{\mathbb{G}} |f|^2 \log |\gamma^{-1/2}e^{\frac{|x'|^2}{4}}f(x)|\,dx \nonumber\\
& \leq & \frac{Q}{4}\log \left(A \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx\right) \nonumber \\
& & +\log (\gamma^{-1/2})+\int_{\mathbb{G}}\frac{|x'|^2}{4} |f(x)|^2\,dx\,.
\end{eqnarray}
Now, recall that for the $n_1$ elements of the first stratum of the Lie algebra of $\mathbb{G}$ the following formulas are known (see \cite{FR16}):
\[
X_i=\partial_{x_{i}^{'}}+\sum_{j=1}^{n-n_1}p_{j}^{i}(x')\partial_{x''_{j}}\,, \forall i=1,\ldots,n_1\,.
\]
Hence, for each $i=1,\ldots,n_1$, we can calculate
\begin{eqnarray}\label{Xg}
|X_ig(x)|^2 &=& \gamma^{-1} e^{\frac{|x'|^2}{2}} \left|X_if(x)+\frac{x'_{i}}{2}f(x) \right|^{2}\nonumber\\
&=& \gamma^{-1} e^{\frac{|x'|^2}{2}} \left(|X_if(x)|^2+\frac{(x'_{i})^2}{4}|f(x)|^2+{\rm Re} \overline{(X_if(x))}x'_{i}f(x) \right)\,.
\end{eqnarray}
Moreover, notice that for each $x'_{i}$, $i=1,\ldots,n_1$, we have
\begin{eqnarray*}
{\rm Re}\int_{\mathbb{G}}\overline{(\partial_{x'_{i}}f(x))}x'_{i}f(x)\,dx & =& -{\rm Re}\int_{\mathbb{G}}(\partial_{x'_{i}}f(x))x'_{i}\overline{f(x)}\,dx-\int_{\mathbb{G}}|f(x)|^2\,dx\\
&=& -{\rm Re}\int_{\mathbb{G}}\overline{(\partial_{x'_{i}}f(x))}x'_{i}f(x)\,dx-1\,,
\end{eqnarray*}
where the last implies that
\begin{equation}
\label{int.parts1}
{\rm Re}\int_{\mathbb{G}}\overline{(\partial_{x'_{i}}f(x))}x'_{i}f(x)\,dx=-\frac{1}{2}\,.
\end{equation}
Additionally, for $j=1,\ldots,n-n_1$ and for $i=1,\ldots,n_1$, we have
\begin{eqnarray*}
{\rm Re} \int_{\mathbb{G}}p_{j}(x')\overline{(\partial_{x_{j}''}f(x))}x'_{i}f(x)& = & - {\rm Re}\int_{\mathbb{G}}\partial_{x_{j}''}((p_{j}(x')f(x)x'_{i})\overline{f(x)}\,dx\\
&=&- {\rm Re}\int_{\mathbb{G}}p_{j}(x')\overline{(\partial_{x_{j}''}f(x))}x'_{i}f(x)\,dx\,,
\end{eqnarray*}
i.e., we have that
\begin{equation} \label{int.parts2}
{\rm Re} \int_{\mathbb{G}}p_{j}(x')\overline{(\partial_{x_{j}''}f(x))}x'_{i}f(x)\,dx=0\,.
\end{equation}
Hence, by \eqref{int.parts1} and \eqref{int.parts2} we get
\[
\int_{\mathbb{G}}{\rm Re}\overline{(X_if(x))}x'_{i}f(x)\,dx=-\frac{1}{2}\,,
\]
for each $i=1,\ldots,n_1$, so that using the expression \eqref{Xg} we get
\begin{equation}\label{thm.eq.nablag}
\int_{\mathbb{G}}|\nabla_{H}g(x)|^2\,d\mu=\int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx+\int_{\mathbb{G}}\frac{|x'|^2}{4}|f(x)|^2\,dx-\frac{n_1}{2}\,.
\end{equation}
Now, by using \eqref{thm.eq.glogg} and \eqref{thm.eq.nablag}, to show \eqref{gaus.log.sob} it suffices to show that
\[
\frac{Q}{4}\log \left(A \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx\right)+\log (\gamma^{-1/2}) \leq \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx-\frac{n_1}{2}\,,
\]
or equivalently,
\[
\frac{Q}{4}\log \left( A \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx \right)+\log (\gamma^{-1/2}) + \log e^{\frac{n_1}{2}}\leq \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx\,.
\]
Now, we can instead show that
\[
\frac{Q}{4} \log \left(A \gamma^{-\frac{2}{Q}} e^{\frac{n_1}{2}\cdot\frac{4}{Q}} \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx \right)\leq \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx\,,
\]
or that
\[
\log \left(A \gamma^{-\frac{2}{Q}} e^{\frac{2n_1}{Q}} \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx \right)\leq \frac{4}{Q} \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx\,.
\]
Now, since $\log r \leq r-1$, for all $r>0$, it suffices to show that
\[
{e^{-1}}A\gamma^{-\frac{2}{Q}}e^{\frac{2n_1}{Q}} \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx \leq \frac{4}{Q} \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx\,,
\]
{since then we would have
\begin{eqnarray*}
\log \left(A \gamma^{-\frac{2}{Q}} e^{\frac{2n_1}{Q}} \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx \right) & = & \log \left(e^{-1} A\gamma^{-\frac{2}{Q}}e^{\frac{2n_1}{Q}} \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx\right)+1\\
& \leq & e^{-1} A\gamma^{-\frac{2}{Q}}e^{\frac{2n_1}{Q}} \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx\\
& = & \frac{4}{Q} \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx,
\end{eqnarray*}}%
where the last hold true as an equality for the choice of $\gamma$ as in \eqref{k}, and we have shown \eqref{gaus.log.sob}.
\end{proof}
As in the Euclidean setting, cf. e.g. \cite{Bec98}, one can show that ``horizontal'' log-Sobolev inequality \eqref{cor.log.sob} is equivalent to the Gross type ``semi-Gaussian'' log-Sobolev inequality \eqref{gaus.log.sob}.
\begin{thm}\label{equiv.thm}
Let $\mathbb{G}$ and $d\mu$ be as in the hypothesis of Theorem \ref{semi-g}. The following statements are true and imply each other:
\begin{enumerate}
\item[(i)] for $g$ such that $\|g\|_{L^2(\mu)}=1$ we have \begin{equation}
\label{thm.eq.itmq}
\int_{\mathbb{G}}|g|^2 \log |g|\,d\mu \leq \int_{\mathbb{G}} |\nabla_{H}g|^2\,d\mu\,;\end{equation}
\item[(ii)] for $f$ such that $\|f\|_{L^2(\mathbb{G})}=1$ we have \begin{equation}\label{thm.eq.itm2}\int_{\mathbb{G}}|f|^{2}\log |f|\,dx \leq \frac{Q}{4}\log \left(A \int_{\mathbb{G}} |\nabla_{H}f|^2\,dx \right)\,,\end{equation} where $A$ is as in \eqref{Cinf}.
\end{enumerate}
\end{thm}
\begin{proof}
It was already shown in Corollary \ref{Cor.Log} that (ii) holds true, and in
Theorem \ref{semi-g} that (ii) implies (i); so it is enough to show that also (i) implies (ii).
For $\gamma$ as in \eqref{k}, let us define $g$ by
\[
g(x)=\gamma^{-\frac{1}{2}}e^{\frac{|x'|^2}{4}}f(x)\,.
\]
Assuming $\|g\|_{L^2(\mu)}=1$, then we also have that $\|f\|_{L^2(\mathbb{G})}=1$, and, using arguments similar to those developed in Theorem \ref{semi-g}, we see that inequality \eqref{thm.eq.itmq} gives
\begin{equation}\label{before.eps}
\log (\gamma^{-\frac{1}{2}}e^{\frac{n_1}{2}})+\int_{\mathbb{G}}|f|^2\log |f|\,dx \leq \int_{\mathbb{G}} |\nabla_{H}f|^2\,dx\,.
\end{equation}
Now, for any $\epsilon>0$, we have $\|f_\epsilon\|_{L^2(\mathbb{G})}=1$, where $f_\epsilon(x):=\epsilon^{\frac{Q}{2}}f(D_{\epsilon}(x))$, where the mapping $D_\epsilon: \mathbb{G} \rightarrow \mathbb{G}$ stands for the dilations on $\mathbb{G}$. Now, an application of \eqref{before.eps} gives
\begin{eqnarray}\label{after.eps}
\int_{\mathbb{G}}|f|^2\log |f|\,dx & \leq & \epsilon^2 \int_{\mathbb{G}} |\nabla_{H}f|^2\,dx-\frac{Q}{2}\log \epsilon-\log (\gamma^{-\frac{1}{2}}e^{\frac{n_1}{2}})\nonumber\\
& =& \epsilon^2 \int_{\mathbb{G}} |\nabla_{H}f|^2\,dx-\frac{Q}{2}\log \epsilon+\frac{Q}{4} \log \left(\frac{QA}{4e} \right)
\end{eqnarray}
for all $\epsilon>0$.
Now, minimizing the expression on the right-hand side of \eqref{after.eps} over $\epsilon$ we get
\begin{eqnarray*}
\int_{\mathbb{G}}|f|^2\log |f|\,dx & \leq & \frac{Q}{4}+\frac{Q}{4} \log \left(4 \int_{\mathbb{G}}|\nabla_{H}f|^2\,dx \right)- \frac{Q}{4}\log Q+\frac{Q}{4} \log \left(\frac{QA}{4e} \right)\\
& \leq & \frac{Q}{4} \log \left(A \int_{\mathbb{G}}|\nabla_{H}f|^2\,dx \right)\,.
\end{eqnarray*}
This shows that \eqref{thm.eq.itmq} implies \eqref{thm.eq.itm2}.
\end{proof}
\section{Weighted Gross logarithmic Sobolev inequality}
Note that the weighted log-Sobolev inequality also implies its Gross version. In this section we discuss this implication, as well as give examples for it for the Euclidean space, with different choices of homogeneous quasi-norms.
To start with, let us note a special case of the weighted Sobolev inequality in Theorem \ref{wsobingrthm} in the case of $p=2$ and $a=1$, also using the equality \eqref{lapfor}:
let $\mathbb{G}$ be a stratified Lie group of homogeneous dimension $Q$ and let $|\cdot|$ be an arbitrary homogeneous quasi-norm. Assume that $0\leq \beta <2<Q$ and $\frac{1}{p}-\frac{1}{q}=\frac{a}{Q}-\frac{\beta}{qQ}$. Then, as a consequence of Theorem \ref{wsobingrthm}, we have
\begin{equation}\label{harsobingr-2}
\int_{\mathbb{G}}\frac{|x|^{-\frac{2\beta }{q}}|u|^{2}}{\||x|^{-\frac{\beta}{q}} u\|^{2}_{L^{2}(\mathbb{G})}}\log\left(\frac{|x|^{-\frac{2\beta }{q}}|u|^{2}}{\||x|^{-\frac{\beta}{q}} u\|^{2}_{L^{2}(\mathbb{G})}}\right) dx \leq \frac{Q-\beta}{2-\beta}\log\left(C\frac{\|\nabla_H u\|^{2}_{L^{2}(\mathbb{G})}}{\||x|^{-\frac{\beta}{q}} u\|^{2}_{L^{2}(\mathbb{G})}}\right),
\end{equation}
for all nontrivial $|x|^{-\frac{\beta}{q}}|u|,\nabla_Hu\in L^{p}(\mathbb{G})$.
In particular, the condition on $q$ implies that
$q=\frac{2(Q-\beta)}{Q-2},$ so that
$\frac{\beta}{q}=\frac{\beta(Q-2)}{2(Q-\beta)}.$
Altogether, we obtain the following
extension of Corollary \ref{Cor.Log}, which can be viewed as the case of $\beta=0$ of the following statement:
\begin{cor}\label{Cor.Log-w}
Let $\mathbb{G}$ be a stratified Lie group with homogeneous dimension $Q$ and let $|\cdot|$ be an arbitrary homogeneous quasi-norm. Assume that $0\leq \beta <2<Q$. Then, there exists $C>0$ such that the following weighted log-Sobolev inequality holds,
\begin{equation}\label{harsobingr-2}
\int_{\mathbb{G}}{|x|^{-\frac{\beta(Q-2)}{Q-\beta}}|u|^{2}}\log\left({|x|^{-\frac{\beta(Q-2)}{2(Q-\beta)}}|u|}\right) dx \leq \frac{Q-\beta}{2(2-\beta)}\log\left(C\int_\mathbb{G} |\nabla_H u|^{2} dx\right),
\end{equation}
for every $u$ such that $\||x|^{-\frac{\beta(Q-2)}{2(Q-\beta)}}u\|_{L^2(\mathbb{G})}=1$.
\end{cor}
Consequently, the argument similar to that of Theorem \ref{semi-g} gives us its weighted version:
\begin{thm}\label{semi-g-w}
Let $\mathbb{G}$ be a stratified group with homogeneous dimension $Q$, topological dimension $n$, and let $n_1$ be the dimension of the first stratum of its Lie algebra, i.e., for $x \in \mathbb{G}$ we can write $x=(x',x'') \in \mathbb{R}^{n_1} \times \mathbb{R}^{n-n_1}$.
Let $|\cdot|$ be an arbitrary homogeneous quasi-norm on $\mathbb{G}$ and let $0\leq \beta <2<Q$.
Let $|x'|$ denote the Euclidean norm of $x'$, and let $M>0$ be a constant such that we have
\begin{equation}\label{EQ:norms}
|x'|\leq M|x|,
\end{equation}
for the quasi-norm $|x|$, for all $x\in\mathbb{G}.$
Then the following weighted ``semi-Gaussian'' log-Sobolev inequality is satisfied
\begin{equation}
\label{gaus.log.sob-w}
\int_{\mathbb{G}}|x|^{-\frac{\beta(Q-2)}{Q-\beta}}|g|^2\log\left(|x|^{-\frac{\beta(Q-2)}{2(Q-\beta)}}|g|\right)\,d\mu \leq \int_{\mathbb{G}} |\nabla_{H}g|^2\,d\mu\,,
\end{equation}
for all $g$ such that $\||x|^{-\frac{\beta(Q-2)}{2(Q-\beta)}}g\|_{L^2(\mu)}=1,$
where $\mu=\mu_1 \otimes \mu_2$, and $\mu_1$ is the Gaussian measure on $\mathbb{R}^{n_1}$ given by $d\mu_1=\gamma e^{-\frac{|x'|^2}{2}}dx'$, for $x'\in \mathbb{R}^{n_1}$, where the normalization constant $\gamma$ is now given by
\begin{equation}
\label{k-w}
\gamma:= \left(\frac{Q-\beta}{2(2-\beta
)}C e^{(n_1+\frac{M^2}{2})\frac{2-\beta}{Q-\beta}-1}
\right)^{\frac{Q-\beta}{2-\beta}}\,,
\end{equation}
and $\mu_2$ is the Lebesgue measure $dx''$ on $\mathbb{R}^{n-n_1}$.
\end{thm}
We note that the constant $M$ as in \eqref{EQ:norms} exists for any homogeneous quasi-norm $|\cdot|.$ Indeed, on its unit quasi-sphere, $|x'|$ achieves a maximum $M$, so \eqref{EQ:norms} follows by homogeneity, with $M=\max\limits_{|x|=1} |x'|.$ Second, we note that the right hand-side of \eqref{gaus.log.sob-w} does not depend on the choice of a homogeneous quasi-norm $|\cdot|,$ while its left-hand side does depend on it. This is, however, taken care of by the normalisation condition with $\gamma$, which also depends on $|\cdot|.$
\begin{proof}[Proof of Theorem \ref{semi-g-w}]
The proof is similar to that of Theorem \ref{semi-g}, so let us only indicate the differences. We keep the same relation between $f$ and $g$. The inequality
\eqref{thm.eq.glogg} is then replaced by
\begin{eqnarray}\label{thm.eq.glogg-w}
& & \int_{\mathbb{G}}|x|^{-\frac{\beta(Q-2)}{Q-\beta}}|g(x)|^2\log \left(|x|^{-\frac{\beta(Q-2)}{2(Q-\beta)}}|g(x)|\right)\,d\mu \nonumber \\
& = & \int_{\mathbb{G}} |x|^{-\frac{\beta(Q-2)}{Q-\beta}}|f(x)|^2 \log |\gamma^{-1/2}e^{\frac{|x'|^2}{4}}
|x|^{-\frac{\beta(Q-2)}{2(Q-\beta)}}
f(x)|\,dx \nonumber\\
& \leq & \frac{Q-\beta}{2(2-\beta)}\log \left(C \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx\right) \nonumber \\
& & +\log (\gamma^{-1/2})+\int_{\mathbb{G}}\frac{|x'|^2}{4} |x|^{-\frac{\beta(Q-2)}{Q-\beta}}|f(x)|^2\,dx\,,
\end{eqnarray}
where we used inequality \eqref{harsobingr-2}. We then estimate
\begin{multline}\label{EQ:aterm}
\int_{\mathbb{G}}\frac{|x'|^2}{4} |x|^{-\frac{\beta(Q-2)}{Q-\beta}}|f(x)|^2\,dx=\int_{|x|\leq 1} +\int_{|x|\geq 1} \\ \leq
\frac{M^2}{4}\int_{|x|\leq 1} |x|^{-\frac{\beta(Q-2)}{Q-\beta}}|f(x)|^2\,dx+
\int_{|x|\geq 1}
\frac{|x'|^2}{4} |f(x)|^2\,dx
\\ \leq \frac{M^2}{4}+\int_\mathbb{G} \frac{|x'|^2}{4} |f(x)|^2\,dx,
\end{multline}
using that $\||x|^{-\frac{\beta(Q-2)}{2(Q-\beta)}}f\|_{L^2(\mathbb{G})}=1.$
Now, combining the equality \eqref{thm.eq.nablag} together with \eqref{thm.eq.glogg-w} and \eqref{EQ:aterm}, we conclude that to show \eqref{gaus.log.sob-w}, it is enough to show that
\begin{equation}\label{for.k-w}
\log \left(C\gamma^{-\frac{2-\beta}{Q-\beta}}e^{\left(\frac{n_1}{2}+\frac{M^2}{4} \right)\frac{2(2-\beta)}{Q-\beta}} \int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx \right)\leq \frac{2(2-\beta)}{Q-\beta}\int_{\mathbb{G}}|\nabla_{H}f(x)|^2\,dx\,.
\end{equation}
Hence, using arguments similar to those developed in Theorem \ref{semi-g} we can check that for $\gamma$ as in \eqref{k-w} inequality \eqref{for.k-w} is satisfied, and this completes the proof of Theorem \ref{semi-g-w}.
\end{proof}
As a curiosity, let us record a consequence of Theorem \ref{semi-g-w} for the Euclidean space $\mathbb{G}=\mathbb R^n$, in which case $Q=n_1=n$ and $\nabla_H=\nabla$. We also exemplify the fact that we can take any homogeneous quasi-norm on $\mathbb R^n$, so we take a family
$|x|_p=(|x_1|^p+\cdots+|x_n|^p)^{\frac{1}{p}}$ for $1\leq p<\infty$, and $|x|_\infty=\max\limits_{1\leq j\leq n}|x_j|.$ In this case we have \eqref{EQ:norms} with $M_p=\max\{n^{\frac12-\frac1p},1\}.$ As a result, we obtain the following weighted version of the Gross log-Sobolev inequality on $\mathbb R^n$.
\begin{cor}\label{semi-g-w-Rn}
Let $0\leq \beta <2<n$. For $1\leq p\leq\infty$, we denote by $|x|_p$ the $\ell^p$-norm of $x\in \mathbb R^n$, as above.
Let $\mu$ be the Gaussian measure on $\mathbb{R}^{n}$ given by
$d\mu=\gamma e^{-\frac{|x|_2^2}{2}}dx$, with
\begin{equation}
\label{k-w-Rn}
\gamma:=\left(\frac{n-\beta}{2(2-\beta
)}C e^{(n+\frac{M_p^2}{2})\frac{2-\beta}{n-\beta}-1}
\right)^{\frac{n-\beta}{2-\beta}}\,,
\end{equation}
where $C$ is the constant from \eqref{harsobingr-2}, and $M_p=\max\{n^{\frac12-\frac1p},1\}.$
Then the following weighted semi-Gaussian log-Sobolev inequality is satisfied
\begin{equation}
\label{gaus.log.sob-w-Rn}
\int_{\mathbb R^n}|x|_p^{-\frac{\beta(n-2)}{n-\beta}}|g(x)|^2\log\left(|x|_p^{-\frac{\beta(n-2)}{2(n-\beta)}}|g(x)|\right)\,d\mu(x) \leq \int_{\mathbb R^n} |\nabla g(x)|^2\,d\mu(x)\,,
\end{equation}
for all $g$ such that $\||x|_p^{-\frac{\beta(n-2)}{2(n-\beta)}}g\|_{L^2(\mu)}=1.$
\end{cor}
|
1,108,101,563,254 | arxiv | \section{Introduction}
Numerical solutions to the wave equation have been an area of investigation for many decades. The wave equation is ubiquitous in the physical world, arising in acoustics, electromagnetics, and fluid dynamics. Our main interest is in electromagnetic wave propagation in plasmas, which are challenging due not only to the nonlinear coupling of the fields with ionized particles, but also to the disparate time scales introduced by the plasma frequency, which can vary by several orders of magnitude from the speed of light (normalized by an appropriate length scale).
Perhaps to most popular method for kinetic plasma simulations is the particle-in-cell (PIC) method \cite{Birdsall1976}. PIC simulations are comprised of two principle components. The first is the particle push, which relies on a Lagrangian description to move macro-particles in phase space, according to the Vlasov equation. The second component is the field solver, which couples the electromagnetic fields to the moving particles, according to Maxwell's equations. The particles are projected to the mesh, and the fields back to the particles using polynomial interpolation (the so-called PIC weighting).
The field solver is typically built with a finite difference time domain (FDTD), or finite volume time domain (FVTD) algorithm. Plasma problems generally require a solution which allows for time steps large compared to that dictated by the propagation speed. Since explicit schemes must obey the Courant-Friedrichs-Lewy (CFL) stability limit, $\Delta t \leq \Delta x/c $, where $\Delta t$ is the time step, and $\Delta x$ is the width of a mesh cell. To overcome this time step restriction, we propose the use of an implicit method. Several works have implemented Maxwell solvers using an alternate direction implicit (ADI) formulation, \cite{Fornberg,Fornberga,Smithe2009}, resulting in A-stable schemes for which large time steps can be taken. Furthermore, higher orders of temporal accuracy can be achieved by Richardson extrapolation, provided the dispersion error is sufficiently small, so that the coarse and fine solutions are not out of phase. These field solvers are designed for the first order formulation of Maxwell's equations, so that the divergence free nature of the numerical solution can be ensured \cite{Smithe2009}. One drawback of FDTD and FVTD formulations of Maxwell's equations is the inability to accurately describe time dependent point sources, which are generally not collocated on the mesh points. For this reason, we consider integral formulations, for which convolution with a delta function will give exact spatial resolution.
Historically, differential formulations of PDEs were desirable over boundary integral methods, which were computationally expensive. Over the past few decades, fast summation methods, such as the fast multipole method \cite{greengard1987fast}, and the tree-code algorithm \cite{Barnes1986} have been introduced, which reduces the computational complexity of computing particle interactions from $O(N^2)$ to $O(N)$, or $O(N\log N)$. These acceleration methods have been extended to a variety of kernels, and as a result, fast methods for computing boundary integral solutions of many PDEs have been developed, such as the wave equation \cite{Coifman1993, Alpert2000,Alpert2002,Li2006}, Helmholtz equation \cite{Cheng1999}, Poisson equation \cite{Lindsay2001}, and even the modified Helmholtz equation \cite{Gimbutas2002,Li2009,Cheng2006}. This work in this paper was initially motivated by the development of accurate particle methods for simulating bounded plasmas utilizing tree-codes \cite{Christlieb2004,Li2009,Lindsay2001}. However, rather than utilize a tree-code approach, we extend our methods to the multi-dimensional case using an ADI splitting.
Although the MOL$^T$ approach is more commonly used for parabolic problems \cite{Salazar2000, Jia2008}, it has been considered sparsely for the wave equation \cite{Schemann1998}. The work we present here is closely related to an independent set of works recently published \cite{Bruno2010,Lyon2010}, in which the wave equation is solved using a Fourier continuation-ADI (FC-ADI) method. First a semi-discrete boundary value problem (BVP) is formulated using MOL$^T$; next, an ADI splitting results in a series of one-dimensional BVPs, which are solved in turn by employing a Fourier continuation method, which scales as $O(N\log N)$. The FC-ADI method presented is sixth order in space and first order in time, but is subsequently lifted to fourth order in time using Richardson extrapolation, without pollution of the solution due to dispersion error \cite{Lyon2010}. Additionally, the FC-ADI algorithm is unconditionally stable, and can be extended to arbitrary geometries by using periodic extensions of the functions to impose boundary conditions at arbitrary (non-mesh point) locations.
We differ from the work of \cite{Bruno2010,Lyon2010} in that we expand the wave function $u$ using classical polynomial bases, rather than Fourier bases. This method of approach would normally lead to a convolution of complexity $O(N^2)$. However, we derive a fast convolution algorithm that utilizes the analytic properties of the one-dimensional Green's function, which is a decaying exponential. Due to the shift-invariance of the exponential, global convolutions can be decomposed into a local and far-away contribution, using exponential recursion. The result is an $O(N)$ fast convolution algorithm, which can impose boundary conditions at exact boundary values, without the need of the periodic extensions used in \cite{Bruno2010}. In fact, by not relying on FFTs, the convolution remains $O(N)$ even if the mesh spacing is irregular, or boundary points do not lie on the mesh. As shown below, we incorporate Dirichlet, Neumann, periodic and outflow boundary conditions in one spatial dimension with relative ease. Furthermore, due to the ADI splitting, the boundary integrals in higher dimensions are never explicitly formed. Instead, all boundary conditions are implemented by solving one-dimensional (two-point) boundary value problems along ADI lines, and the ADI sweeps couple the information, to implicitly construct the boundary integral. Thus, our algorithm remains $O(N)$ in higher dimensions.
The analytical study of our wave solver was presented in \cite{Causley2013}, where it was shown to be A-stable, and convergent to the wave equation with second order accuracy in space and time, provided that a Lax correction is included to correct for the spatial discretization error. We shall present a slight modification of this result in Section \ref{sec:Fast}, which removes the necessity of the Lax correction, without affecting stability. In short, the polynomial bases we use to approximate the wave function $u$ must be of degree $\geq 2$.
The rest of this paper is laid out as follows. In section \ref{sec:molt}, the semi-discrete boundary integral solution to the wave equation in one spatial dimension is presented. We show how boundary conditions, as well as transmission conditions to a finite domain can be applied. The latter of these is useful in particular for domain decomposition, and thus parallelization of the algorithm, as well as in deriving outflow boundary conditions. In section \ref{sec:Fast}, we present the spatial discretization of the boundary integral solution, and describe how a fast $O(N)$ algorithm can be designed for the fully discrete solution. To avoid the Lax correction used in \cite{Causley2013}, a compact form of Simpson's rule is used for the spatial quadrature. We present the case of a regular grid, as well as unstructured mesh. Next, we incorporate the ADI splitting to solve the wave equation in higher dimensions. In section \ref{sec:ADI}, we propose a splitting in which all spatial derivatives are brought to one side of the equation, so that intermediate variables can no longer be strictly interpreted as intermediate time levels of the solution. As such, we detail the consistent incorporation of boundary integral terms, and the inclusion of sources. In section \ref{sec:results} we demonstrate our solution to be fast, second order accurate, and A-stable, even on non-rectangular domains. Finally, we conclude with several remarks in section \ref{sec:conclusion}.
\section{Semi-discrete solution using MOL$^T$}
\label{sec:molt}
The numerical algorithm is based on the initial boundary value problem for the wave equation in one spatial dimension
\begin{align}
\label{eqn:wave}
\frac{\partial^2 u}{\partial x^2} - \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2}&= -S(x,t), \quad a< x < b, \quad t>0 \\
u(x,0) &= f(x), \quad a < x < b \nonumber \\
u_t(x,0) &= g(x), \quad a < x < b, \nonumber
\end{align}
where $c$ is the propagation speed. The problem is well-posed once consistent boundary conditions are appended. We will consider below several important cases: Dirichlet
\begin{equation}
\label{eqn:Dirichlet}
u(a,t) = U_L(t), \quad u(b,t) = U_R(t),
\end{equation}
Neumann
\begin{equation}
\label{eqn:Neumann}
u_x(a,t) = V_L(t), \quad u_x(b,t) = V_R(t),
\end{equation}
and periodic boundary conditions
\begin{equation}
\label{eqn:Periodic}
u(a,t) = u(b,t), \quad u_x(a,t) = u_x(b,t).
\end{equation}
We also address outflow boundary conditions in 1D, which can be formulated exactly in terms of local differential operators
\begin{equation}
\label{eqn:outflow}
u_t(a,t) -c u_x(a,t) =0, \quad u_t(b,t)+c u_x(b,t) = 0.
\end{equation}
\subsection{Time discretization}
We begin by discretizing $u_{tt}$ using the time-centered finite difference approximation
\[
u_{tt}^n = \frac{u^{n+1}-2u^n+u^{n-1}}{\Delta t^2} - \frac{\Delta t^2}{12} u_{tttt}(x,\eta), \quad \eta \in [t_{n-1},t_{n+1}].
\]
Now, in order to obtain an implicit method, we require that the Laplacian be taken at time $t_{n+1}$. However, since the finite difference stencil is centered, we consider a symmetric 3-point averaging for the Laplacian
\[
\frac{\partial^2}{\partial x^2} u^n = \frac{\partial^2}{\partial x^2} \left( u^n+\frac{u^{n+1}-2u^{n}+u^{n-1}}{\beta^2} \right)-\frac{\Delta t^2}{\beta^2} u_{ttxx}(x,\eta),
\]
where $\beta>0$ is a parameter, and which produces the semi-discrete equation
\begin{equation}
\label{eqn:betaTC}
\left(\frac{\partial^2}{\partial x^2} -\frac{\beta^2}{(c\Delta t)^2} \right)\left(u^n + \frac{u^{n+1}-2u^n+u^{n-1}}{\beta^2} \right) = - \frac{\beta^2}{(c\Delta t)^2}u^n -S(x,t_n).
\end{equation}
This method will result in a purely dispersive scheme, in that the numerical solution does not introduce numerical diffusion. Here $\beta>0$ is a free parameter which can be chosen to reduce the truncation error, but still produce an A-stable scheme. In fact, we have the following
\begin{lemma}
Equation \eqref{eqn:betaTC} is unconditionally stable for $0<\beta \leq 2$. For $\beta = \sqrt{12}$, the scheme is fourth order accurate, but conditionally stable. The optimal choice which maintains A-stability and minimizes the error is $\beta =2$.
\end{lemma}
\begin{proof}
Consider first the stability of the scheme \eqref{eqn:betaTC} with $S = 0$. In order to ensure that the approximation remains finite we take $\beta \neq 0$, and without loss of generality, we assume $\beta>0$. Let $u^n(x) = \rho^n e^{ikx} u^{0}$, and define $\omega = kc$. Substitution and cancellation of the common terms yields the Von-Neumann polynomial
\begin{align*}
&\left(-\left(\frac{\omega}{c}\right)^2 -\frac{\beta^2}{(c\Delta t)^2}\right)\left(\rho +\frac{\rho^2-2\rho+1}{\beta^2}\right) =-\frac{\beta^2}{(c\Delta t)^2}\rho\quad \implies \\
&\rho^2-\left(2-\frac{(\beta\omega\Delta t)^2}{\beta^2+(\omega\Delta t)^2}\right)\rho+1 = 0
\end{align*}
Stability follows from the roots of this quadratic polynomial satisfying $|\rho|\leq 1$, and applying the Schur criterion leads to
\[
\left |2-\frac{(\beta\omega\Delta t)^2}{\beta^2+(\omega\Delta t)^2} \right | \leq 2.
\]
A-stability follows from this inequality being satisfied for all values of $\omega$, and so we find $0<\beta \leq 2$.
Next, we look at the truncation error of the method. Upon substituting the exact solution $u$ into the semi-discrete equation \eqref{eqn:betaTC} and setting $S = 0$, we find the global truncation error
\begin{align*}
\tau := &\left(\frac{\partial^2}{\partial x^2} -\frac{\beta^2}{(c\Delta t)^2} \right)\left(u^n + \frac{u^{n+1}-2u^n+u^{n-1}}{\beta^2} \right) + \frac{\beta^2}{(c\Delta t)^2}u^n \\
= &\frac{\Delta t^2}{\beta^2}u_{ttxx} - \frac{\Delta t^2}{12c^2}u_{tttt}+O(\Delta t^4) \\
= &\frac{(c\Delta t)^2}{12\beta^2}u_{xxxx}\left(12-\beta^2\right) +O(\Delta t^4).
\end{align*}
Thus, the truncation error will be 4th order if we choose $\beta = \sqrt{12}$. However, this is not in the range of A-stability. Observe that as $\beta$ increase, the error constant in the truncation error decreases. Thus, we define the value $\beta=2$ as optimal in the sense that it produces the A-stable scheme with the smallest discretization error.
\end{proof}
\subsection{Integral solution and update equation}
The differential operator which appears in the semi-discrete equation \eqref{eqn:betaTC} is the modified Helmholtz operator, which we define by
\begin{equation}
\label{eqn:HelmholtzL}
\mathcal{L}_\beta[w](x) := \left(\frac{1}{\alpha^2}\frac{\partial^2}{\partial x^2} - 1\right)w(x), \quad \alpha = \frac{\beta}{c\Delta t}.
\end{equation}
A modified Helmoltz equaiton of the form
\begin{equation}
\label{eqn:Helmholtz_Equation}
\mathcal{L}_\beta[w](x) = -u(x), \quad a\leq x \leq b
\end{equation}
can be formally solved by inverting the Helmholtz operator using the Green's function (details can be found in \cite{Causley2013}). We define convolution with this Green's function by the integral operator
\begin{equation}
\label{eqn:Iu}
I[u](x) := \alpha\int_a^b u(y)e^{-\alpha|x-y|}dy, \quad a\leq x \leq b,
\end{equation}
so that the integral solution is given by $\mathcal{L}_\beta^{-1}[u](x) = \frac{1}{2}w(x)$, where
\begin{equation}
\label{eqn:L_Inverse}
w(x):= \underbrace{I[u](x)}_{\text{Particular Solution}}+ \underbrace{\vphantom{I[u](x)} A e^{-\alpha(x-a)} + B e^{-\alpha(b-x)}}_{\text{Homogeneous Solution}}.
\end{equation}
The coefficients $A$ and $B$ of the homogeneous solution are determined by applying boundary conditions. The homogeneous solution \eqref{eqn:L_Inverse} can also be used as a means to incorporate information about $u$ from outside the domain $\Omega = [a,b]$. In this case, the coefficients $A$ an $B$ are enforcing transmission conditions. That is, suppose that the support of $u$ is $\Omega = \mathbb{R}$, so that we have the free space solution to the modified Helmholtz equation
\[
w(x) = \alpha \int_{-\infty}^\infty u^n(y)e^{-\alpha|x-y|}dy.
\]
Now, we shall only ever evaluate $w(x)$ for $a\leq x \leq b$. Then,
\begin{align}
w(x) =& I[u](x) + \alpha\int_{-\infty}^a u^n(y)e^{-\alpha(x-y)}dy + \alpha \int_b^\infty u^n(y) e^{-\alpha(y-x)}dy \nonumber \\
\label{eqn:Transmission}
=& I[u](x) + A^ne^{-\alpha(x-a)} + B^ne^{-\alpha(b-x)},
\end{align}
where the homogeneous coefficients are
\begin{align}
A^n =& \alpha\int_{-\infty}^a u^n(y)e^{-\alpha(a-y)}dy \\
B^n =& \alpha \int_b^\infty u^n(y) e^{-\alpha(y-b)}dy,
\end{align}
and do not depend on $x$.
\begin{remark}
The distinction to be made here is that the integral solution \eqref{eqn:L_Inverse} is valid when the domain $[a,b]$ is the full support of the function $u$, and the homogeneous solution is used to apply boundary conditions. This is in contrast to the integral solution \eqref{eqn:Transmission}, for which the support of $u$ extends beyond $[a,b]$, but we shall only ever evaluate $I[u](x)$ for $a\leq x \leq b$. This latter result will be used below to derive outflow boundary conditions, and, along with the fast convolution algorithm of Section \ref{sec:Fast}, to build an efficient domain decomposition algorithm.
\end{remark}
We now make a few key observations about the particular solution, which will be used extensively in the ensuing discussion. The integral operator can be decomposed into a left and right oriented integral, split at $y=x$ so that
\begin{equation}
\label{eqn:ILR}
I[u](x) = I^L[u](x)+I^R[u](x),
\end{equation}
where
\[
I^L[u](x) = \alpha \int_a^x e^{-\alpha(x-y)}u(y)dy, \quad I^R[u](x)= \alpha\int_{x}^b e^{-\alpha(y-x)}u(y)dy.
\]
We may interpret $I^L$ and $I^R$ as the "characteristics" of $I$, as they independently satisfy first order "initial value problems"
\begin{align}
(I^L)'(y) + \alpha I^L(y) = +\alpha u(y), \quad a<y<x, \quad I^L(a) = 0 \\
(I^R)'(y) - \alpha I^R(y) = -\alpha u(y), \quad x<y<b, \quad I^R(b) = 0,
\end{align}
where the prime denotes spatial differentiation. The solutions to these equations can be found by the integrating factor method. Integrating $I^L$ from $x-\delta_L$ to $x$, and $I^R$ from $x$ to $x+\delta_R$, we find
\begin{align}
\label{eqn:IL_def}
I^L[u](x) &= I^L[u](x-\delta_L) e^{-\alpha \delta_L}+ J^L[u](x), \quad J^L[u](x):= \alpha \int_{0}^{\delta_L} u(x-y) e^{-\alpha y}dy \\
\label{eqn:IR_def}
I^R[u](x) &= I^R[u](x+\delta_R) e^{-\alpha \delta_R}+ J^R[u](x), \quad J^R[u](x):= \alpha \int_{0}^{\delta_R} u(x+y) e^{-\alpha y}dy.
\end{align}
The recursive updates \eqref{eqn:IL_def} and \eqref{eqn:IR_def} are exact (in space), and making $\delta_L$ and $\delta_R$ small (typically, $\delta_L =\delta_R = \Delta x$) effectively localizes the contribution of the integrals. We also observe that combining \eqref{eqn:IL_def} and \eqref{eqn:IR_def}, the total integral operator also satisfies a recursive definition
\begin{equation}
\label{eqn:I_Rec}
I[u](x) = e^{-\alpha \delta_L}I^L[u](x-\delta_L)+e^{-\alpha \delta_R}I^R[u](x+\delta_R) + \alpha \int_{-\delta_L}^{\delta_R} u(x+y) e^{-\alpha |y|}dy.
\end{equation}
In this expression, the remaining integral contains the "local" information of $I[u]$.
We now identify the semi-discrete form of the wave equation \eqref{eqn:betaTC} with the modified Helmholtz equation \eqref{eqn:Helmholtz_Equation} using
\[
w(x) = 2\left(u^n(x)+\frac{u^{n+1}(x)-2u^n(x)+u^{n-1}(x)}{\beta^2}\right), \quad u(x) =\left(u^n(x)+\frac{1}{\alpha^2}S^n(x)\right)
\]
and upon inverting the Helmholtz operator and solving for $u^{n+1}$, find the update equation
\begin{equation}
\label{eqn:Integral_Solution_Full}
u^{n+1} = -(\beta^2-2)u^n-u^{n-1}+\frac{\beta^2}{2}\left(I\left[u^n+\frac{1}{\alpha^2}S^n\right]+A^n e^{-\alpha(x-a)}+B^ne^{-\alpha(b-x)}\right).
\end{equation}
\subsection{Consistency of the Particular Solution} \label{Consistency_Particular_Solution}
In \cite{Causley2013}, quadrature was performed on the convolution integral \eqref{eqn:Iu} using the midpoint and trapezoidal rules. It was shown that a Lax-type correction was required to achieve a consistent numerical scheme for the wave equation, due to coupling between the spatial quadrature error, and the temporal truncation error. This is apparently a difficulty that is intrinsic to using the MOL$^T$ to formulate second order boundary value problems. In particular, a discussion of this same phenomenon, for parabolic problems, is carried out by Bruno and Lyon in Section 4 of \cite{Bruno2010}. It is indicated therein that the numerical solution, which is found using Fourier continuation methods, diverges upon letting $\Delta t \to 0$ when the spatial parameter $h$ held fixed. Bruno and Lyon \cite{Bruno2010} mitigate this difficulty by correcting the solution with a finite difference solver.
We will now show how such difficulties can be avoided. Specifically if $u(x+ y)$ in equation \eqref{eqn:I_Rec} is approximated with a polynomial of degree $p\geq 2$ and integrated analytically, the resulting fully discrete update will be stable and convergent. In the computational algorithm, however the recursive updates \eqref{eqn:IL_def} and \eqref{eqn:IR_def} will be used. Since they are computed separately, we must insist that each of these updates uses the same polynomial, so as to avoid jumps in higher derivatives, which (although they will be $O(\Delta x^p)$ for some $p$), will cause numerical instabilities if $\Delta t \ll \Delta x$.
Our reasoning is as follows. If from the update equation \eqref{eqn:Integral_Solution_Full} we solve for the finite difference approximation of $\frac{1}{c^2}u_{tt}$ and neglect the homogeneous solution, we find
\[
\frac{u^{n+1}-2u^{n}+u^{n-1}}{(c\Delta t)^2} = \frac{\alpha^2}{2}\left(I\left[u^n+\frac{1}{\alpha^2}S^n \right]-2u^n\right),
\]
where again $\alpha = \beta/(c\Delta t)$. Thus, letting $\Delta t \to 0$ on the left hand side is equivalent to $\alpha \to \infty$ on the right hand side, and in order to recover the wave equation \eqref{eqn:wave}, the following limit must hold
\[
\lim_{\alpha \to \infty} \frac{\alpha^2}{2}\left(I\left[u^n+\frac{1}{\alpha^2}S^n \right]-2u^n\right) = u^n_{xx} + S^n,
\]
for $a<x<b$. We justify the neglect of the homogeneous solution in \eqref{eqn:Integral_Solution_Full} because, as $\alpha \to \infty$, both exponentials converge to zero for any $x \neq a,b$. Showing that this limit holds for the semi-discrete solution (i.e. when $u$ is continuous in space) follows from integrating analytically the Taylor expansion of $u(x+y)$ against the Green's function analytically.
We will find need of the following
\begin{lemma}
\label{Exp_Int}
For integers $m\geq 0$ and real $\nu>0$,
\begin{align}
E_{m}(\nu) := \nu\int_0^1 \frac{z^m}{m!} e^{-\nu z}dz = \frac{1}{\nu^m} \left(1-e^{-\nu }P_m(\nu)\right)
\end{align}
where
\[
P_m(\nu) = \sum_{\ell=0}^m\frac{\nu^\ell}{\ell!}
\]
is the Taylor series expansion of order $m$ of $e^{\nu}$.
\end{lemma}
\begin{proof}
The proof of this lemma follows from iterated use of integration by parts.
\end{proof}
We are now prepared to prove the following
\begin{theorem}
\label{Consistency_thm}
Let $u^n(x)$ be a semi-discrete solution given by \eqref{eqn:Integral_Solution_Full}. Then, $u^n(x)$ converges to $u(x,t_n)$ satisfying the wave equation \eqref{eqn:wave}, iff for each $x \in (a,b)$,
\begin{equation}
\label{eqn:Consistency}
\lim_{\alpha \to \infty} \alpha^2\left(I[u^n]-2u^n\right) =2 \frac{\partial^2}{\partial x^2}u^n.
\end{equation}
\end{theorem}
\begin{remark}
This theorem is stated without the inclusion of sources for simplicity. However, all results apply, provided that $S \in L^1([a,b])$.
\end{remark}
\begin{proof}
Upon performing a Taylor expansion of $u^n(x\pm y)$ appearing in the integrals of the recurrence relations \eqref{eqn:IL_def} and \eqref{eqn:IR_def} (with $\delta_L = \delta_R = \delta$), and evaluating with the aid of Lemma \ref{Exp_Int}, we find
\begin{align*}
\alpha \int_{0}^\delta u^n(x\pm y)e^{-\alpha y}dy &= \nu \int_{0}^1 \left(\sum_{m=0}^3 \frac{z^m}{m!}\left(\pm \frac{\nu}{\alpha}\right)^m\frac{\partial^m u^n(x)}{\partial x^m}+O(z^4\delta^4) \right)e^{-\nu z}dz \\
&= \sum_{m=0}^3 \left(\frac{\pm 1}{\alpha}\right)^m \frac{\partial^m u^n(x)}{\partial x^m}\left(1-e^{-\nu}P_m(\nu)\right) + O\left(\frac{1}{\alpha^4}\right)
\end{align*}
where $\nu = \alpha\delta$. When these expansions are summed, all odd powers of $\alpha$ cancel, and the recursive form of the integral \eqref{eqn:I_Rec} becomes
\[
I(x) = 2u + \frac{2}{\alpha^2}\frac{\partial^2u}{\partial x^2}+ e^{-\nu}\left(I^L(x-\delta)+I^R(x+\delta)-2u - \frac{2}{\alpha^2}\frac{\partial^2u}{\partial x^2}\left(1+\nu +\frac{\nu^2}{2}\right)\right) + O\left(\frac{1}{\alpha^4}\right)
\]
and so, since $\alpha^2\left(I[u^n]-2u^n\right) =2 u^n_{xx} +O\left(e^{-\alpha \delta}+\frac{1}{\alpha^2}\right)$, taking the limit produces precisely the desired result \eqref{eqn:Consistency}.
\end{proof}
\begin{remark}
The proof of this theorem, while straightforward, underscores the necessary conditions for a fully discrete algorithm to recover a convergent approximation of the wave equation. That is, in order to recover the consistency condition \eqref{eqn:Consistency}, the first three terms of the Taylor expansion, produced by integrating polynomials up to degree 2 against an exponential, must be computed exactly. We therefore demand that the spatial quadrature used to discretize the integral \eqref{eqn:I_Rec} be exact up to degree $p\geq 2$.
\end{remark}
Consider now a fully discrete solution $u^n_j = u^n(x_j)$, which is obtained by discretizing the integral \eqref{eqn:I_Rec}, to order $p$. Recall that when constructing the numerical algorithm, we will use the recurrence updates for $I^L$ \eqref{eqn:IL_def} and $I^R$ \eqref{eqn:IR_def} to perform the updates. Therefore, to ensure the resulting polynomial approximation for the integral of equation \eqref{eqn:I_Rec} is continuous up to order $p$, we must use the same polynomial to perform quadrature on the integrals in equations \eqref{eqn:IL_def} and \eqref{eqn:IR_def}. Otherwise, the jump in higher derivatives may be of the form $(C_1 \Delta x^p) z$, and upon integrating against the Green's function we would find
\[
\alpha^2\left(I[u^n]-2u^n\right) = C_1 \Delta x^p \alpha +2 u^n_{xx} +O\left(e^{-\alpha \delta}+\frac{1}{\alpha^2}\right),
\]
which, for fixed $\Delta x$, diverges as $\alpha \to \infty$. We formalize this in the following
\begin{corollary}
\label{Consistency_cor}
Let $u^n_j$ be a fully discrete solution, formed by replacing $u^n(x_j\pm y)$ with polynomials of degree $p\geq 2$ appearing in the integrals of equations \eqref{eqn:IL_def} and \eqref{eqn:IR_def}. Then, the discrete analog of equation \eqref{eqn:Consistency}, namely
\begin{equation}
\label{eqn:Consistency_Disc}
\lim_{\alpha \to \infty} \alpha^2\left(I[u^n](x_j)-2u^n_j\right) =2 \frac{\partial^2}{\partial x^2}u^n_{j}+O(\Delta x^p)
\end{equation}
is satisfied for each $x_j \in (a,b)$, iff the resulting polynomial approximation for the integral of equation \eqref{eqn:I_Rec} is continuous up to order $p$.
\end{corollary}
In Section \ref{sec:Fast}, we derive the update equations for the fully discrete numerical scheme, with $p=2$. This removes the necessity for the Lax correction used in \cite{Causley2013}.
\subsection{Boundary conditions}
Before turning our attention to the discretization of the update equation \eqref{eqn:Integral_Solution_Full}, we will discuss the homogeneous part of the integral solution \eqref{eqn:L_Inverse}, which is used to enforce the boundary conditions.
Let us begin with Dirichlet boundary conditions \eqref{eqn:Dirichlet}. Evaluating the semi-discrete solution \eqref{eqn:Integral_Solution_Full} at $x = a$ and $b$, we find
\begin{align*}
U_L(t_{n+1})=-(\beta^2-2)U_L(t_n)-U_L(t_{n-1}) + \frac{\beta^2}{2} \left(I\left[u^n+\frac{1}{\alpha^2}S^n \right](a)+A + B e^{-\alpha(b-a)}\right), \\
U_R(t_{n+1})=-(\beta^2-2)U_R(t_n)-U_R(t_{n-1}) + \frac{\beta^2}{2} \left(I\left[u^n+\frac{1}{\alpha^2}S^n \right](b)+A e^{-\alpha(b-a)} + B\right),
\end{align*}
which, after solving for the unknown coefficients can be written as
\begin{align*}
A^n +\mu B^n &= -w_a^D, \\
\mu A^n + B^n &= -w_b^D,
\end{align*}
with
\begin{align*}
w_a^D &= I\left[u^n+\frac{1}{\alpha^2}S^n \right](a) - \frac{2}{\beta^2}\left(U_L(t^{n+1})+(\beta^2-2)U_L(t^{n}) +U_L(t^{n-1}) \right), \\
w_b^D &= I\left[u^n+\frac{1}{\alpha^2}S^n \right](b) - \frac{2}{\beta^2}\left(U_R(t^{n+1})+(\beta^2-2)U_R(t^{n})+U_R(t^{n-1}) \right),
\end{align*}
and $\mu = e^{-\alpha(b-a)}$. Homogeneous boundary conditions are recovered upon setting $U_L(t) = U_R(t) = 0$. Solving the resulting linear system for the unknowns $A^n$ and $B^n$ gives
\begin{align}
\label{eqn:wh_Dirichlet}
A = -\left(\frac{w_a^D - \mu w_b^D }{1-\mu^2}\right), \quad B = - \left(\frac{w_b^D - \mu w_a^D}{1-\mu^2}\right).
\end{align}
Before considering Neumann conditions, first observe that all dependence on $x$ in the integral solution \eqref{eqn:L_Inverse} is on the Green's function, which is a simple exponential function. Using this, we obtain the following identities
\begin{align}
\label{eqn:DtN}
I'(a) = \alpha I(a), \quad I'(b) = -\alpha I(b).
\end{align}
Now, differentiating the semi-discrete solution \eqref{eqn:Integral_Solution_Full}, and applying the Neumann boundary conditions \eqref{eqn:Neumann} at $x=a$ and $b$ yields
\begin{align*}
V_L(t_{n+1})=-(\beta^2-2)V_L(t_n)-V_L(t_{n-1}) + \alpha \frac{\beta^2}{2} \left(I\left[u^n+\frac{1}{\alpha^2}S^n \right](a)-A + B e^{-\alpha(b-a)}\right), \\
V_R(t_{n+1})=-(\beta^2-2)V_R(t_n)-V_R(t_{n-1}) + \alpha \frac{\beta^2}{2} \left( - I\left[u^n+\frac{1}{\alpha^2}S^n \right](b)-A e^{-\alpha(b-a)} + B\right),
\end{align*}
which, after solving for the unknown coefficients can be written as
\begin{align*}
A^n - \mu B^n &= w_a^N, \\
-\mu A^n + B^n &= w_b^N,
\end{align*}
with
\begin{align*}
w_a^N &= I\left[u^n+\frac{1}{\alpha^2}S^n \right](a) - \frac{2}{\alpha\beta^2}\left(V_L(t^{n+1})+(\beta^2-2)V_L(t^{n}) +V_L(t^{n-1}) \right), \\
w_b^N &= I\left[u^n+\frac{1}{\alpha^2}S^n \right](b) + \frac{2}{\alpha\beta^2}\left(V_R(t^{n+1})+(\beta^2-2)V_R(t^{n})+V_R(t^{n-1})\right).
\end{align*}
Upon solving the linear system we obtain
\begin{align}
\label{eqn:wh_Neumann}
A = \left(\frac{w_a^N +\mu w_b^N}{1-\mu^2}\right), \quad B = \left(\frac{w_b^N+\mu w_a^N}{1-\mu^2}\right).
\end{align}
Finally, we can also impose periodic boundary conditions, by assuming that
\[
u^n(b) = u^n(a), \quad u^n_x(a) = u^n_x(b), \quad n\geq 0.
\]
Enforcing this in the semi-discrete solution \eqref{eqn:Integral_Solution_Full} then yields
\begin{align*}
I[u^n](a) + A + B\mu &= I[u^n](b) + A\mu + B, \\
\alpha \left(I[u^n](a) - A + B\mu\right) &= \alpha\left(- I[u^n](b) - A\mu + B\right),
\end{align*}
where we have used the identity \eqref{eqn:DtN} applied to derivatives of $I$. Solving this linear system is accomplished quickly by dividing the second equation by $\alpha$, and either adding or subtracting it from the first equation, to produce
\begin{align}
\label{eqn:wh_Periodic}
A = \frac{I[u^n](b)}{1-\mu}, \quad B = \frac{I[u^n](a)}{1-\mu}.
\end{align}
\begin{remark}
The cases of applying different boundary conditions at $x=a$ and $b$ are not considered here, but the details follow from an analogous procedure to that demonstrated above.
\end{remark}
\subsection{Outflow boundary conditions}
When computing wave phenomena, whether we are interested in finite or infinite domains, it is often the case that we must restrict our attention to some smaller subdomain $\Omega$ of the problem, which does not include the physical boundaries. We say that $\Omega$ is the \textit{computational domain}, and that the boundary $\partial \Omega$ is the non-physical, or \textit{artificial boundary}. Under these circumstances, it is necessary to enforce an outflow, or non-reflecting boundary condition, which allows the wave to leave the computational domain, but not incur (non-physical) reflections at the artificial boundary.
For the one-dimensional wave equation \eqref{eqn:wave} the exact outflow boundary conditions \eqref{eqn:outflow} turn out to be local in space and time. We emphasize that this is only the case in one spatial dimension, but we shall utilize this fact to obtain an outflow boundary integral solution from the integral equation \eqref{eqn:Transmission}. We extend the support of our function to $(-\infty,\infty)$, and extend the definition of the outflow boundary conditions to the domains exterior to $[a,b]$
\begin{align}
u_t+cu_x = 0, \quad x\geq b, \\
u_t-cu_x = 0, \quad x\leq a.
\end{align}
Next, assume the initial conditions have some compact support; for simplicity we will take this support to be $\Omega_0 = [a,b]$. Then after a time $t=t_n$, the domain of dependence of $u^n(x)$ is $\Omega_t = [a-ct_n,b+c t_n]$, since the propagation speed is $c$. Now the free space solution \eqref{eqn:Transmission} becomes
\begin{align}
w(x) =& \alpha \int_{a-c t_n}^{b+c t_n} e^{-\alpha|x-y|} u^n(y) dy \nonumber \\
=& I[u](x)+ A^ne^{-\alpha(x-a)} + B^ne^{-\alpha(b-x)}
\end{align}
with coefficients
\begin{align}
\label{eqn:A_Out_Def}
A^n &= \alpha\int_{a-ct_n}^{a} e^{-\alpha(a-y)} u^n(y) dy, \\
\label{eqn:B_Out_Def}
B^n &= \alpha\int_{b}^{b+ct_n} e^{-\alpha(y-b)} u^n(y) dy.
\end{align}
At first glance, these coefficients are not at all helpful, as they require computing integrals along spatial domains which not only are outside of the computational domain, but also grow linearly in time. However, we will now make use of the extended boundary conditions to turn these spatial integrals into time integrals, which exist at precisely the endpoints $x=a$ and $b$ respectively. Consider first $x>b$. By assumption, this region contains only right traveling waves, $u(x,t) = u(x-ct)$, and by tracing backward along a characteristic ray we find
\[
u(b+y,t) = u\left(b,t-\frac{y}{c}\right), \quad y>0.
\]
Thus,
\begin{align*}
B^n &= \alpha \int_{0}^{ct_n} e^{-\alpha y} u(b+y,t_n) dy \\
&= \alpha c \int_{0}^{t_n} e^{-\alpha c s} u\left(b,t_n-s\right) ds
\end{align*}
and so $B^n$ is equivalently represented by a convolution in time, rather than space. Now, knowing the history of $u$ at $x=b$ is sufficient to impose outflow boundary conditions. Furthermore, we find in analog to equation \eqref{eqn:IR_def}, a temporal recurrence relation due to the exponential
\begin{align*}
B^n &= \alpha c \int_{0}^{\Delta t} e^{-\alpha c s} u\left(b,t_n-s\right) ds + e^{-\alpha c \Delta t}\left(\int_0^{t_{n-1}} e^{-\alpha c s} u\left(b,t_{n-1}-s\right) ds \right) \\
&= \beta \int_{0}^{1} e^{-\beta z} u\left(b,t_n-z\Delta t\right) dz +e^{-\beta} B^{n-1},
\end{align*}
where $\beta = \alpha c \Delta t$, by definition \eqref{eqn:HelmholtzL}. Thus, the coefficient $B^n$, which imposes an outflow boundary condition at $x=b$, can be computed locally in both time and space. To maintain second order accuracy, we fit $u$ with a quadratic interpolant
\[
u(b,t_n-z\Delta t) \approx p(z) = u^n(b) -\frac{z}{2}\left(u^{n+1}(b)-u^{n-1}(b)\right) + \frac{z^2}{2}\left(u^{n+1}(b)-2u^n(b)+u^{n-1}(b)\right)
\]
and integrate the expression analytically using Lemma \ref{Exp_Int} to arrive at
\begin{equation}
\label{eqn:Outflow_Update_Intermediate}
B^n = e^{-\beta}B^{n-1} +\gamma_0 u^{n+1}(b)+ \gamma_1 u^n(b)+ \gamma_2 u^{n-1}(b)
\end{equation}
where
\begin{align*}
\gamma_0 =& \frac{E_2(\beta)-E_1(\beta) }{2} = \frac{(1-e^{-\beta})}{\beta^2}-\frac{(1+e^{-\beta})}{2\beta} \\
\gamma_1 =& E_0(\beta)-E_2(\beta) = -2\frac{(1-e^{-\beta})}{\beta^2}+ \frac{2}{\beta}e^{-\beta} +1 \\
\gamma_2 =& \frac{E_2(\beta)+E_1(\beta)}{2} = \frac{(1-e^{-\beta})}{\beta^2}+\frac{(1-3e^{-\beta})}{2\beta} - e^{-\beta}.
\end{align*}
In this outflow update equation \eqref{eqn:Outflow_Update_Intermediate}, the quantities $u^{n+1}(b)$ and $B^n$ are both unknown. In order to determine these values, we must also evaluate the update equation for $u^{n+1}$ \eqref{eqn:Integral_Solution_Full} at $x=b$
\[
u^{n+1}(b)+(\beta^2-2)u^{n}(b)+u^{n-1}(b) = \frac{\beta^2}{2}\left(I(b)+A^n \mu + B^n\right), \quad \mu = e^{-\alpha(b-a)}.
\]
We now use these two equations to solve for $u^{n+1}(b)$, and eliminate it from the outflow update equation \eqref{eqn:Outflow_Update_Intermediate}, so that
\begin{equation}
\label{eqn:Outflow_Update_B}
-\Gamma_0 \mu A^n + (1-\Gamma_0)B^n = e^{-\beta}B^{n-1} +\Gamma_0 I(b) +\Gamma_1u^n(b)+ \Gamma_2 u^{n-1}(b)
\end{equation}
where
\[
\Gamma_0 = \frac{\beta^2}{2}\gamma_0, \quad
\Gamma_1 = \gamma_1-\gamma_0(\beta^2-2), \quad
\Gamma_2 = \gamma_2-\gamma_0
\]
\begin{remark}
While this procedure could be avoided by omitting $u^{n+1}(b)$ in the interpolation stencil, it turns out to be necessary to obtain convergent outflow boundary conditions.
\end{remark}
Likewise, upon considering $x<a$, we find
\begin{equation}
\label{eqn:Outflow_Update_A}
(1-\Gamma_0) A^n -\Gamma_0 \mu B^n = e^{-\beta}A^{n-1} +\Gamma_0 I(a) +\Gamma_1u^n(a)+ \Gamma_2 u^{n-1}(a).
\end{equation}
Solving the resulting linear system produces
\begin{equation}
\label{eqn:Outflow_Update}
A^n = \frac{(1-\Gamma_0)w_a^{\text{Out}} +\mu \Gamma_0w_b^{\text{Out}}} {(1-\Gamma_0)^2-(\mu \Gamma_0)^2}, \quad
B^n = \frac{(1-\Gamma_0)w_b^{\text{Out}} +\mu \Gamma_0w_a^{\text{Out}}} {(1-\Gamma_0)^2-(\mu \Gamma_0)^2},
\end{equation}
where
\begin{align*}
w_a^{\text{Out}} &= e^{-\beta}A^{n-1} +\Gamma_0 I(a) +\Gamma_1u^n(a)+ \Gamma_2 u^{n-1}(a), \\
w_b^{\text{Out}} &= e^{-\beta}B^{n-1} +\Gamma_0 I(b) +\Gamma_1u^n(b)+ \Gamma_2 u^{n-1}(b)
\end{align*}
\section{Treatment of point sources, and soft sources}
\label{sec:sources}
We now consider the inclusion of source terms. We are predominantly interested in the case where $S(x,t)$ consists of a large number of time dependent point sources. However, it is often the case that in electromagnetics problems, a soft source is prescribed to excite waves of a prescribed frequency, or range of frequencies, within the domain. A soft source is so named because, although incident fields are generated at a prescribed fixed spatial location, no scattered fields are generated.
The implementation of a soft source $\sigma(t)$ at $x = x_s$ is accomplished by prescribing the source condition
\begin{equation}
\label{eqn:condition}
u(x_s,t) = \sigma(t).
\end{equation}
However, it can be shown that if we set
\begin{align}
S(x,t)= \frac{2}{c}\sigma'(t) \delta(x-x_s)
\end{align}
and insert it into the wave equation \eqref{eqn:wave}, then the soft source condition \eqref{eqn:condition} is satisfied, and the solutions are equivalent. Thus, a soft source is nothing more than a point source, whose time-varying field is integrated by the wave equation.
Upon convolving this source term with the Green's function according to \eqref{eqn:Iu}, we find
\begin{align*}
I\left[\frac{1}{\alpha^2}S\right](x)=& \frac{1}{\alpha}\int_a^b \left(\frac{2}{c}\sigma'(t_n)\delta(x-x_s) \right)e^{-\alpha|x-y|}dy \\
=& \frac{2\Delta t}{\beta} \sigma'(t_n) e^{-\alpha|x-x_s|},
\end{align*}
where the definition of $\alpha = \beta/(c\Delta t)$ has been utilized.
\begin{remark}
It is often the case that taking the analytical derivative $\sigma'(t_n)$ is to be avoided, for various reasons. In this case, any finite difference approximation which is of the desired order of accuracy can be substituted.
\end{remark}
Likewise for general point sources,
\[
S(x,t) = \sum_{i} \tilde{\sigma}_i(t) \delta(x-x_i)
\]
the corresponding form of the source term is
\begin{equation}
\label{eqn:Delta_Source}
I\left[\frac{1}{\alpha^2}S\right](x) = \frac{c \Delta t}{\beta}\sum_i \tilde{\sigma}_i(t_n) e^{-\alpha|x-x_i|}
\end{equation}
Therefore, it suffices to consider delta functions both for the implementation of soft sources, as well as including time dependent point sources.
\section{Spatial discretization}
\label{sec:Fast}
Having considered the homogeneous and source terms, we are now prepared to present a fully discrete numerical solution, defined by discretizing in space the update equation \eqref{eqn:Integral_Solution_Full}. Upon full discretization, computing $I_j \equiv I[u](x_j)$ according to \eqref{eqn:Iu} results in a dense matrix-vector product, which requires $O(N^2)$ operations to compute, where $N$ is the number of spatial grid points. However we will show that by utilizing the recurrence relations for $I^L$ \eqref{eqn:IL_def}, and $I^R$ \eqref{eqn:IR_def}, the particular solution $I = I^L + I^R$ can be formed in $O(N)$ operations, by means of fast convolution. This turns out to be true, even in the case of unstructured grids.
We first discretize the domain $[a,b]$ into $N$ evenly spaced subintervals, of width $\Delta x = (b-a)/N$, and define $x_j = a+j\Delta x$ for $0\leq j \leq N$. Appealing to the recursive definition of the integral operator \eqref{eqn:I_Rec} evaluated at $x_j$ and with $\delta_L=\delta_R = \Delta x$, we make a change of variables $y = z\Delta x$, and find
\begin{align}
\label{eqn:I_Rec_Disc}
I_{j} = e^{-\nu} \left(I^L_{j-1}+I^R_{j+1}\right) + \nu \int_{-1}^{1} u(x_{j}+z\Delta x) e^{-\nu |z|}dz,
\end{align}
where $I_j = I[u](x_j)$, and the discrete integration parameter is
\begin{equation}
\label{eqn:nudef}
\nu = \alpha\Delta x = \frac{\beta\Delta x}{c\Delta t}.
\end{equation}
We now replace $u(x_{j}+ z\Delta x)$ with a polynomial interpolant over the subinterval corresponding to $-1\leq z \leq 1$. We consider polynomials $p_{j}^{(m)}(z)$ of degree $m$, which use successively larger stencils involving the points $u_{j\pm k}$. The recursive updates for $I^L$ and $I^R$ become
\begin{align}
\label{eqn:IL_Rec_Disc}
I^L_{j} =& e^{-\nu} I^L_{j-1} + J^L_j, \quad J^L_j = \nu \int_{0}^{1} u(x_{j}-z\Delta x) e^{-\nu z}dz, \\
\label{eqn:IR_Rec_Disc}
I^R_{j} =& e^{-\nu} I^R_{j+1}+ J^R_j, \quad J^R_j = \nu \int_{0}^{1} u(x_{j}+z\Delta x) e^{-\nu z}dz.
\end{align}
These expressions are still exact, but the remaining integrals must be approximated using quadrature.
\subsection{Compact Simpson's Rule}
Motivated by theorem \ref{Consistency_thm}, we now perform quadrature using second order polynomials $p^{(2)}_j$. To be precise, $u(x_j+z\Delta x)$ is approximated by
\begin{equation}
\label{eqn:p2_def}
p_j^{(2)}(z) = (1-z)u_{j} + z u_{j+1} + \left(\frac{z^2-z}{2}\right)\Delta x^2 u''(\xi_j),
\end{equation}
where
\[
\Delta x^2 u''(\xi_j) =
\begin{cases}
2u_{0}-5u_1+4u_{2}-u_3, \quad &j = 0 \\
u_{j+1}-2u_j+u_{j-1}, \quad &1 \leq j \leq N-1 \\
2u_{N}-5u_{N-1}+4u_{N-2}-u_{N-3}, \quad &j = N,
\end{cases}
\]
so that
\[
u(x_j+z\Delta x) - p_j^{(2)}(z) = O(\Delta x^3).
\]
Now, we replace $u(x_j+z\Delta x)$ with $p_j^2(z)$ in $J^R_j$ of equation \eqref{eqn:IR_Rec_Disc}, and integrate the result analytically. By symmetry, the corresponding polynomial approximation for $u(x_j-z\Delta x)$ is made in $J^L_j$ of equation \eqref{eqn:IL_Rec_Disc}. Making use of Lemma \ref{Exp_Int}, we find
\begin{align}
\label{eqn:IL_update}
J^L_j = Pu_j + Qu_{j-1}+R \Delta x^2 u''(\xi_j) \\
\label{eqn:IR_update}
J^R_j = Pu_j + Qu_{j+1}+R \Delta x^2 u''(\xi_j)
\end{align}
where $I^L_0 = 0$ and $I^R_N = 0$ by definition, and where the coefficients are
\begin{align}
\label{eqn:ddef}
d =& e^{-\nu} \\
P =& E_0(\nu)-E_1(\nu) = 1-\frac{1-d}{\nu} \nonumber \\
Q =& E_1(\nu)= -d+\frac{1-d}{\nu} \nonumber \\
R =& \frac{E_2(\nu)-E_1(\nu)}{2} = \frac{1-d}{\nu^2} - \frac{1+d}{2\nu}. \nonumber
\end{align}
We refer to this method as the compact Simpson's rule, to indicate that a 3-point stencil is used in each of the integrals in $J^L_j$ and $J^R_j$, which are only one computational cell in length. Notice that the only difference between $J^L_j$ and $J^R_j$ is the appearance of $u_{j\pm 1}$ multiplied by the coefficient $Q$, according to the direction of integration. We also emphasize that $I^L_j$ is updated for increasing $j$, and $I^R_j$ for decreasing $j$.
It now remains to show that the presented fully discrete scheme satisfies the consistency condition \eqref{eqn:Consistency_Disc}. First, we combine $I^L_j$ and $I^R_j$ to write the fully discrete update \eqref{eqn:I_Rec_Disc}, which upon simplification of the coefficients \eqref{eqn:ddef} becomes
\begin{align*}
I_j =& d(I^L_{j-1}+I^R_{j+1}) + 2Pu_j +Q(u_{j+1}+u_{j-1})+2R(u_{j+1}-2u_j+u_{j-1}) \\
=& e^{-\nu}(I^L_{j-1}+I^R_{j+1}) + 2(1-e^{-\nu})u_j +\frac{2}{\nu^2}\left(1-\left(1+\nu+\frac{\nu^2}{2}\right)e^{-\nu}\right)(u_{j+1}-2u_j+u_{j-1}).
\end{align*}
If from this expression we solve for the quantity $I_j - 2u_j$ and multiply by $\alpha^2$ (recall, $\nu = \alpha \Delta x$) we have
\[
\alpha^2(I_j-2u_j) = 2\frac{u_{j+1}-2u_j+u_{j-1}}{\Delta x^2}+O\left((\alpha \Delta x)^2e^{-\alpha \Delta x}\right), \quad \alpha\Delta x \gg 1
\]
and letting $\alpha \to \infty$ produces the second order finite difference approximation at $x_j$, confirming the consistency of the scheme.
\begin{remark}
A similar approach could be used to formulate higher order quadrature formulae. In this case, several different stencils for $\Delta x^p u^{(p)}(\xi_j)$ would be required near the boundaries. Development and implementation of consistent quadrature rules is a topic of future work.
\end{remark}
\subsection{Unstructured meshes} \label{sec:unstructured}
We now consider unstructured meshes. More generally, define the partition of $[a,b]$ by the subintervals $[x_{j-1},x_j]$, where $x_0 = a$, $x_N = b$, and $x_{j-1}<x_j$. Likewise, the analogous discrete parameters become
\begin{align}
h_j =& x_j - x_{j-1}, \quad \nu_j = \alpha h_j, \quad d_j = e^{-\nu_j} \\
P_j =& 1-\frac{1-d_j}{\nu_j} \nonumber \\
Q_j =& -d_j+\frac{1-d_j}{\nu_j} \nonumber \\
R_j =& \frac{1}{2\nu_j^2}(2(1-d_j)-\nu_j(1+d_j)). \nonumber
\end{align}
Making use of the recurrence relations \eqref{eqn:IL_def} with $\delta_L = h_j$, and \eqref{eqn:IR_def} with $\delta_R = h_{j+1}$, we obtain
\begin{align}
\label{eqn:ILj}
I^L_j =& d_j I^L_{j -1} + J^L_j, \quad J^L_j =\nu_j \int_0^1 u(x_j-z h_j) e^{-\nu_j z} dz \\
\label{eqn:IRj}
I^R_j =& d_{j+1} I^R_{j+1}+ J^R_j, \quad J^R_j = \nu_{j+1} \int_0^1 u(x_j+z h_{j+1}) e^{-\nu_{j+1} z} dz.
\end{align}
It only remains to construct the polynomial interpolants $p^{(2)}_j(z)$. However, the integration domain is still $0\leq z \leq 1$, and so $p^2_j(z)$ written in the form \eqref{eqn:p2_def} is still valid, as long as we replace $\Delta x$ with $h_j$, and modify stencils to accurately approximate the second derivative at $x_j$. Since this can be done in $O(N)$ operations, the scheme is still fast, and the discretization of $J^L_j$ and $J^R_j$ found in equations \eqref{eqn:IL_update} and \eqref{eqn:IR_update} respectively still apply, so long as the coefficients are replaced with their $j-$dependent counterparts.
\subsection{1D domain decomposition}
\label{sec:DD}
In this section we propose an efficient parallel algorithm for performing the 1D convolution \eqref{eqn:Iu} on a decomposed domain, with the appropriate boundary conditions. If $N$ is the total number of grid points in the domain, and $M$ the number of processors, then both the number of floating point operations and the amount of memory storage scale as $O(N\!+\!M)$. Once the convolution is computed, the solution update \eqref{eqn:Integral_Solution_Full} is completely local and does not require any further communication between the subdomains.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width = \textwidth]{Domain_Decomposition.pdf}
\caption{Decomposition of a 1D domain}
\label{fig:Domain_Decomposition}
\end{center}
\end{figure}
Consider a 1D domain $\Omega = [a,b]$, and decompose it into $M$ subdomains $\Omega_m = [a_m,b_m]$ of generic sizes ($m = 0,1,\dots,M\!-\!1$), as shown in Figure \ref{fig:Domain_Decomposition}. Appealing to the form of the integral solution with transmission coefficients \eqref{eqn:Transmission}, we see that for $x\in \Omega_m$, the integral solution can be written as
\[
I[u](x) = I_m[u](x) + A_m e^{-\alpha (x-a_m)} + B_m e^{-\alpha(b_m-x)}, \quad x \in \Omega_m,
\]
where the "local particular solution" is
\[
I_m[u](x) = \alpha \int\limits_{\Omega_m} u(y) e^{-\alpha |x-y|}dy, \quad
\]
and the local homogeneous coefficients are
\[
A_m = \int_a^{a_m} e^{-\alpha(a_m-y)}u(y)dy, \quad B_m = \int_{b_m}^b e^{-\alpha(y-b_m)}u(y)dy.
\]
We point out here that if $m=0$ or $M-1$, then the coefficient $A_m$ or $B_m$ respectively are used to enforce boundary conditions, rather than transmission conditions. Now, suppose that $u^n(x)$ from the subdomain $\Omega_m$ is copied into local memory on machine $m$, and the integral solution \eqref{eqn:Integral_Solution_Full} is to be computed there, only for $x \in \Omega_m$. Then, the contributions from the total domain $\Omega$, including the boundary conditions, is determined by the scalar values $A_m$ and $B_m$. Additionally, the only information that must be passed to other domains $\Omega_k$ are the scalar values $I_m[u](a_m)$ and $I_m[u](b_m)$.
The idea that underlies the following algorithm is the analogy between a decomposed 1D domain, and a (perhaps non-uniform) "coarse" mesh with $M$ cells: the interfaces between each subdomain $\Omega_m$ are interpreted as mesh nodes $X_m$ (with $m = 0,1,\dots,M$), where
\begin{equation*}
X_0 = a_0, \qquad
X_m = a_m = b_{m-1} \quad{\text{for $m=1,2,\dots M\!-\!1$}}, \qquad
X_M = b_{M-1}.
\end{equation*}
Then, the global solution on the coarse mesh is decomposed into
\[
I[u](X_m) = I^L[u](X_m) + I^R[u](X_m),
\]
with characteristics given by
\begin{align}
I^L[u](X_m) &= e^{-\nu_j}I^L[u](X_{m-1}) + I_m[u](X_m), \\
I^R[u](X_m) &= e^{-\nu_{m+1}}I^R[u](X_{m+1})+ I_{m+1}[u](X_m),
\end{align}
and where $\nu_m = \alpha(X_m-X_{m-1})$. Thus, global particular solution is computed only on the coarse mesh, and is comprised of the scalar values from the local particular solutions, evaluated at the endpoints of their respective domains. Once the global particular solution is computed, the boundary conditions can be applied, and the total integral solution will be known at each $X_m$. Finally, the local homogeneous coefficients $A_m$ and $B_m$ are obtained by
\[
A_m = I^L[u](X_m), \quad B_m = I^R[u](X_{m+1}).
\]
We now have sufficient background for describing algorithm 1 below.
\begin{figure}[htbp]
\begin{center}
\subfigure[Fine-coarse communication]{ \includegraphics[width = .47\textwidth]{Domain_Decomposition_2.pdf} \label{fig:Domain_Decomposition_2}}
\subfigure[Coarse-fine communicatoin]{ \includegraphics[width = .47\textwidth]{Domain_Decomposition_3.pdf} \label{fig:Domain_Decomposition_3}}
\caption{In algorithm 1, the "local particular solutions" $I_m[u](x)$ are found for each $\Omega_m$ and the values $J^L_{m+1} \equiv I_m[u](b_m)$ and $J^R_m\equiv I_m[u](a_m)$ are passed to the coarse grid (a). Once the "global particular solution" and boundary correction is computed on the coarse grid, the transmission coefficients $A_m$ and $B_m$ are passed to $\Omega_m$ (b).}
\end{center}
\end{figure}
\begin{algorithm}[!htp]
\caption{Fast 1D domain decomposition}
\begin{enumerate}[itemsep=2mm]
\item On each subdomain $\Omega_m = [a_m,b_m]$, compute the ``local particular
solution''
\[
I_m[u^n](x) = \alpha \int_{a_m}^{b_m} e^{-\alpha |x-y|} u^n(y) dy, \qquad x\in\Omega_m
\]
by means of the fast convolution algorithm, i.e. equations \eqref{eqn:IL_Rec_Disc} and \eqref{eqn:IR_Rec_Disc};
\item Pass the local values $J^L_{m+1} \equiv I_m[u](b_m)$ and $J^R_m \equiv I_m[u](a_m)$ to the coarse grid (\emph{communication});
\item Compute the contributions from the left and right characteristics on the coarse grid, by means of the recursive relations
\begin{alignat*}{3}
& I^L_0 = 0, \qquad
& & I^L_m = e^{-\nu_{m-1}}I^L_{m-1} + J^L_{m} \quad
& & \text{for $m = 1,2,\dots,M$, and} \\
& I^R_{M} = 0, \qquad
& & I^R_m = e^{-\nu_{m}}I^R_{m+1} + J^R_{m} \quad
& & \text{for $m = M\!-\!1,M\!-\!2,\dots,0$},
\end{alignat*}
where $\nu_j = \alpha(b_{j}-a_j)$ for $j=0,1,\dots,M\!-\!1$;
\item Using the "global particular solution" at the endpoints $I[u](a)=I^R_0$ and $I[u](b)=I^L_{M-1}$, find the global coefficients $A$ and $B$ in accordance with the update equation \eqref{eqn:Integral_Solution_Full}, by imposing the required boundary conditions at $x = a$ and $b$;
\item Compute the local coefficients to contain the global boundary data:
\begin{alignat*}{3}
& A_0 = A, \qquad
& & A_m = I^L_m + A_{m-1} e^{-\nu_{m-1}} \quad
& & \text{for $m = 1,2,\dots,M\!-\!1$, and} \\
& B_{M-1} = B, \qquad
& & B_m = I^R_m + B_{m+1} e^{-\nu_{m}} \quad
& & \text{for $m = M\!-\!2,M\!-\!3,\dots,0$};
\end{alignat*}
\item Send $A_m$ and $B_m$ to the corresponding subdomain $\Omega_m$ (\emph{communication});
\item On each subdomain $\Omega_m = [a_m,b_m]$, construct the local integral solution, by summing the particular and homogeneous solutions
\[ w_{m}[u](x) = I_m[u](x) + A_m e^{-\alpha(x-a_m)} + B_m e^{-\alpha(b_m-x)}, \qquad x\in\Omega_m. \]
\end{enumerate}
\end{algorithm}
\section{Higher spatial dimensions: ADI splitting }
\label{sec:ADI}
The full numerical algorithm in higher dimensions is based on the initial boundary value problem
\begin{align}
\nabla^2 u - \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2} &= -S(r,t), \quad r \in \Omega, \quad t>0 \label{eqn:prob2} \\
u(r,0) &= f(r), \quad r \in \Omega \nonumber \\
u_t(r,0) &= g(r), \quad r \in \Omega \nonumber \\
u(r,t) &= h(r,t), \quad r \in \partial\Omega, \quad t>0. \nonumber
\end{align}
We again utilize MOL$^T$ to perform the temporal discretization. Next we employ ADI splitting to the modified Helmholtz operator
\begin{align*}
\frac{1}{\alpha^2}\nabla^2 -1 &= \frac{1}{\alpha^2}\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right) - 1 \\
&= \left(\frac{1}{\alpha^2}\frac{\partial^2}{\partial x^2} - 1\right) \left(\frac{1}{\alpha^2}\frac{\partial^2}{\partial y^2} - 1\right) \left(\frac{1}{\alpha^2}\frac{\partial^2}{\partial z^2} - 1\right) \\
& \quad +\frac{1}{\alpha^4}\left(\frac{\partial^4}{\partial x^2\partial y^2}+\frac{\partial^4}{\partial x^2\partial z^2}+\frac{\partial^4}{\partial y^2\partial z^2}\right) -\frac{1}{\alpha^6}\frac{\partial^6}{\partial x^2\partial y^2\partial z^2}
\end{align*}
Since $1/\alpha^2 = (c\Delta t)^2/\beta^2$, we have that
\[
\frac{1}{\alpha^2}\nabla^2-1 = \mathcal{L}_{\beta,x}\mathcal{L}_{\beta,y}\mathcal{L}_{\beta,z} + O((c\Delta t)^4),
\]
where $\mathcal{L}_{\beta,x}$, $\mathcal{L}_{\beta,y}$, and $\mathcal{L}_{\beta,z}$ are one-dimensional modified Helmholtz operators \eqref{eqn:HelmholtzL}, applied in the indicated spatial variable. The local truncation error is fourth order, meaning that the scheme is second order accurate. The corresponding two dimensional operator is
\[
\frac{1}{\alpha^2}\nabla^2 - 1 = -\mathcal{L}_{\beta,x}\mathcal{L}_{\beta,y} + O((c\Delta t)^4),
\]
and upon omitting the splitting error we arrive at the 2D PDE
\begin{equation}
\label{eqn:Helmholtz_Equation_2D}
\mathcal{L}_{\beta,x} [\mathcal{L}_{\beta,y}[ u^{n+1}+u^{n-1}+(\beta^2-2)u^n ]](x,y) = \beta^2 u^n(x,y) + (c\Delta t)^2 S^n(x,y),
\end{equation}
analogous to the 1D modified Helmholtz equation \eqref{eqn:Helmholtz_Equation}.
\subsection{Boundary corrections in higher dimensions}
The solution of equation \eqref{eqn:Helmholtz_Equation_2D} will require a consistent implementation of boundary conditions, and inclusion of sources. Define the temporary variables $W(x,y)$ and $Z(x,y)$ such that
\begin{align}
\label{eqn:z_def}
\mathcal{L}_{\beta,x}[W] =& \beta^2 u^n + (c\Delta t)^2 S^n, \quad \mathcal{L}_{\beta,y}[Z] = W.
\end{align}
Thus, the solution $u^{n+1}$ is achieved in two one-dimensional solves, by performing an $x\!-\!y$ sweep
\begin{align}
\label{eqn:w_Solve}
W =& \mathcal{L}_{\beta,x}^{-1}[\beta^2 u^n + (c\Delta t)^2 S^n], \\
\label{eqn:u_Solve}
u^{n+1} =& \mathcal{L}_{\beta,y}^{-1}[W]-u^{n-1}-(\beta^2-2)u^{n}.
\end{align}
The intermediate variable $W(x,y)$ is a boundary integral solution, defined by the 1D result \eqref{eqn:L_Inverse}, where $y$ is a treated as a fixed parameter. The homogeneous solution will also exhibit $y-$dependence through the coefficients $A^n = A^n(y)$ and $B^n = B^n(y)$. For a general domain $\Omega$, the lines $y = y_j$ will intersect with the boundary $\partial \Omega$ at different points $x = a_j, b_j$. Thus, equation \eqref{eqn:w_Solve} is solved by applying boundary conditions at $x = a_j$ and $b_j$.
Likewise, for fixed $x$, the $y-$sweep is performed by constructing 1D boundary integral solution $Z$; and finally we solve for $u^{n+1}$ according to \eqref{eqn:u_Solve}. The boundary conditions are applied at $y = c_k, d_k$, which are defined by the intersection of the boundary $\partial \Omega$ with the line $x = x_k$.
Discretization of $\Omega$ is accomplished by embedding it in a regular Cartesian mesh, and additionally incorporating the termination points of the $x$ and $y$ lines, which will always lie on the boundary, and be within $\Delta x$ (or $\Delta y$, respectively) of the nearest grid point inside $\Omega$. For example, the lines and boundary points for a circle are shown in Figure \ref{fig:sweep}. Since the one-dimensional quadrature formula can incorporate unstructured meshes locally without incurring time step restrictions, it is of no concern to have the boundary lie arbitrarily close to a mesh point. That is, we increase the accuracy without affecting stability by including the boundary points. The implementation of boundary conditions requires knowledge of the temporary variable $Z(x,y)$ (and perhaps its derivatives) at the endpoints of each $x$ line. The general approach for implementation of the ADI scheme is as follows:
\begin{enumerate}
\item
Lay a mesh over the domain $\Omega$. For each horizontal line $y=y_j$, identify the boundary points $x = a_j$ and $b_j$, where $a_j$ and $b_j$ are the points of intersection of $y = y_j$ with the boundary $\partial \Omega.$ Likewise, identify for each vertical line $x = x_k$ the boundary points $y = c_k$ and $d_k$, which are the intersection of $x = x_k$ with the boundary $\partial \Omega.$ A line object is then identified as all regularly spaced points along an $x$ ($y$) line, including it's boundary points. Assume the number of $x$ and $y$ lines are $N_x$ and $N_y$ respectively.
\item
First perform the $x$-sweep. At each time step, construct the temporary variable $W = W(x,y_j)$ defined by \eqref{eqn:w_Solve}, for $1\leq j \leq N_y$. The boundary conditions are imposed at $x = a_j$ and $b_j$.
\item
Next, perform the $y$-sweep. For $1\leq k \leq N_x$, solve for the variable $Z = u^{n+1}+2u^n+(\beta^2-2)u^{n-1}$, according to \eqref{eqn:u_Solve}. The boundary conditions are now applied at $y = c_k$ and $d_k$.
\item
In order to improve the accuracy of the ADI solve, the inversion of the $x$ and $y$ Helmholtz operators is symmetrized, by averaging the results of $x\!-\!y$ and $y\!-\!x$ solves.
\end{enumerate}
The same approach is followed in three dimensions, where the lines are defined by fixing two variables, and similarly finding the endpoints which lie on the boundary $\partial \Omega$ along each line. Symmetrization of the ADI sweeps becomes more tedious, as six different orderings of the ADI sweeps should be taken, and averaged accordingly. However this process still requires $O(N)$ operations.
\begin{figure}[htbp]
\centering
\subfigure[$x$ sweep]{\label{fig:sweep-a} \includegraphics[width=.45\textwidth]{Circle_grid_x_sweep.pdf}}
\subfigure[$y$ sweep]{\label{fig:sweep-b}\includegraphics[width=.45\textwidth]{Circle_grid_y_sweep.pdf}}
\caption{Grids used for the ADI $x$ (a) and $y$ (b) sweeps defined in equations \eqref{eqn:w_Solve} and \eqref{eqn:u_Solve} respectively. The red dots are the intersections of the mesh lines with the boundary, and need not be equally spaced.}
\label{fig:sweep}%
\end{figure}
\section{Numerical Results}
\label{sec:results}
We now demonstrate the ability of our solver to address problems of interest. We first demonstrate the expected second order convergence in both space and time of the algorithm, including the implementation of outflow boundary conditions, and domain decomposition. We next illustrate several two dimensional examples, in non-Cartesian geometries.
\subsection{A one-dimensional example}
We first study the errors produced in 1D by our numerical scheme, as well as those independently produced by our domain decomposition algorithm, and outflow boundary condition implementation. To do this, we construct three separate numerical solutions with the same initial conditions, but with or without domain decomposition, and with or without outflow boundary conditions. The procedure is as follows.
A spatial domain $\Omega = [a,b] \in \mathbb{R}$ is first decomposed into 4 subdomains $\Omega_m = [a_m,b_m],$ for $m=0,1,2,3$. A compactly supported pulse is propagated through the domain, utilizing domain decomposition and imposing outflow boundary conditions, up to time $T = (b-a)/c$, for which the pulse is sure to have left the domain. We also compute independently the numerical solution over $\Omega$, without domain decomposition (but still imposing outflow boundary conditions), so that the difference between the two can be used to measure the error due solely to domain decomposition. A third solution is also constructed, this time over the larger domain $\Omega_{ext} = [a-cT, b+cT]$, that uses neither the domain decomposition nor outflow boundary conditions. The third solution differs from the second only by the numerical reflections caused by outflow for $x \in \Omega$. So in this way, we can independently measure the total discretization errors,
Since the exact solution is known, we can measure the total discretization error, as well as the numerical reflections due independently to the domain decomposition, and outflow boundary conditions.
The initial pulse we use is a Gaussian, given by
\[
u(x,0) = \exp\left(-36\left(\frac{2x-b-a}{b-a}\right)^2\right), \quad \text{and} \quad u_t(x,0) = 0.
\]
The subdomains are discretized using regularly spaced mesh points, as well as Chebyshev mesh points, and also using a varied number of spatial points, so that
\[
x_j = \begin{cases}
a_0 + (b_0-a_0)\frac{j}{N}, \quad &x_j \in \Omega_0 \\
a_1 + (b_1-a_1)\cos\left(\frac{j\pi}{2N}\right), \quad &x_j \in \Omega_1 \\
a_2 + (b_2-a_2)\frac{j}{2N}, \quad &x_j \in \Omega_2 \\
a_3 + (b_3-a_3)\cos\left(\frac{j\pi}{N}\right), \quad &x_j \in \Omega_3.
\end{cases}
\]
Thus, the full domain $\Omega$ is discretized with a total of $6N+1$ unique spatial points, where $N$ is the number of spatial points in $\Omega_0$. The results are displayed in Tables \ref{tab:refinement_1D_1} - \ref{tab:refinement_1D_3}, for increasing $N$, holding the CFL number fixed. For all simulations, $a = -1$, $b = 1$ the propagation speed is $c = 1$, and the final time is $T = 2$. The CFL number is determined using the maximum mesh spacing on the grid, which for $N = 20$, corresponds to $\Delta x = 0.025$. The CFL numbers are set to $0.1$, $1$ and $10$ in Tables \ref{tab:refinement_1D_1}, \ref{tab:refinement_1D_2} and \ref{tab:refinement_1D_3}, respectively. Second order convergence is observed in each case.
\begin{table}[htbp]
\begin{centering}
\begin{tabular}{|c||c|c||c|c||c|c|}
\hline
& \multicolumn{2}{c||}{Domain Decomposition} & \multicolumn{2}{c||}{Outflow } & \multicolumn{2}{c|}{Total Discretization} \\ \hline
$N$ & $L^{2}$ error & $L^{2}$ order& $L^{2}$ error & $L^{2}$ order& $L^{2}$ error & $L^{2}$ order \\ \hline
$20$ & $\num{0.00479840211}$ & $-$ & $\num{0.00125949765}$ & $-$ & $\num{0.00576216479}$ & $-$ \\ \hline
$40$ & $\num{0.00071827454}$ & $2.7399$ & $\num{0.00028492261}$ & $2.1442$ & $\num{0.00140166844}$ & $2.0395$ \\ \hline
$80$ & $\num{0.00009441879}$ & $2.9274$ & $\num{0.00006883611}$ & $2.0493$ & $\num{0.00034557239}$ & $2.0201$ \\ \hline
$120$ & $\num{0.00001220944}$ & $2.9511$ & $\num{0.00001704921}$ & $2.0135$ &$\num{0.00008587556}$ & $2.0087$ \\ \hline
$240$ & $\num{0.00000164072}$ & $2.8956$ & $\num{0.00000425213}$ & $2.0034$ & $\num{0.00002141292}$ & $2.0038$ \\ \hline
$480$ & $\num{0.00000025192}$ & $2.7033$ & $\num{0.00000106236}$ & $2.0009$ & $\num{0.00000534691}$ & $2.0017$ \\ \hline
\end{tabular}
\caption{Refinement study for a 1D Gaussian pulse, demonstrating the convergence of domain decomposition, and outflow boundary conditions. Note that $N$ is the number of spatial points in the first subdomain $\Omega_0$, so the total number of discretization points is $6N+1$. The CFL number is held fixed at $0.1$.}
\label{tab:refinement_1D_1}
\end{centering}
\end{table}
\begin{table}[htbp]
\begin{centering}
\begin{tabular}{|c||c|c||c|c||c|c|}
\hline
& \multicolumn{2}{c||}{Domain Decomposition} & \multicolumn{2}{c||}{Outflow } & \multicolumn{2}{c|}{Total Discretization} \\ \hline
$N$ & $L^{2}$ error & $L^{2}$ order& $L^{2}$ error & $L^{2}$ order& $L^{2}$ error & $L^{2}$ order \\ \hline
$20$ & $\num{0.00197225255}$ & $-$ & $\num{0.00082802594}$ & $-$ & $\num{0.01351638748}$ & $-$ \\ \hline
$40$ & $\num{0.00027765931}$ & $2.8285$ & $\num{0.00021952383}$ & $1.9153$ & $\num{0.00347936511}$ & $1.9578$ \\ \hline
$80$ & $\num{0.00003941848}$ & $2.8164$ & $\num{0.00005718893}$ & $1.9406$ & $\num{0.00087845203}$ & $1.9858$ \\ \hline
$120$ & $\num{0.00000594997}$ & $2.7279$ & $\num{0.00001437569}$ & $1.9921$ & $\num{0.00022054626}$ & $1.9939$ \\ \hline
$240$ & $\num{0.00000101772}$ & $2.5475$ & $\num{0.00000359667}$ & $1.9989$ & $\num{0.00005524280}$ & $1.9972$ \\ \hline
$480$ & $\num{0.00000020255}$ & $2.3290$ & $\num{0.00000089927}$ & $1.9998$ & $\num{0.00001382338}$ & $1.9987$ \\ \hline
\end{tabular}
\caption{Results are the same as for Table \ref{tab:refinement_1D_1}, but with the CFL number fixed at 1.}
\label{tab:refinement_1D_2}
\end{centering}
\end{table}
\begin{table}[htbp]
\begin{centering}
\begin{tabular}{|c||c|c||c|c||c|c|}
\hline
& \multicolumn{2}{c||}{Domain Decomposition} & \multicolumn{2}{c||}{Outflow } & \multicolumn{2}{c|}{Total Discretization} \\ \hline
$N$ & $L^{2}$ error & $L^{2}$ order& $L^{2}$ error & $L^{2}$ order& $L^{2}$ error & $L^{2}$ order \\ \hline
$20$ & $\num{0.00103867308}$ & $-$ & $\num{0.02023754735}$ & $-$ & $ 2.4948 \times 10^{-1}$ & $-$ \\ \hline
$40$ & $\num{0.00003081871}$ & $5.0748$ & $\num{0.01730526450}$ & $0.2258$ & $ 1.4802 \times 10^{-1}$ & $0.7531$ \\ \hline
$80$ & $\num{0.00000180867}$ & $4.0908$ & $\num{0.00483219724}$ & $1.8405$ & $ 5.8893 \times 10^{-2}$ & $1.3296$ \\ \hline
$120$ & $\num{0.00000013251}$ & $3.7707$ & $\num{0.00121302111}$ & $1.9941$ & $\num{0.01720171253}$ & $17756$ \\ \hline
$240$ & $\num{0.00000001894}$ & $2.8066$ & $\num{0.00038659365}$ & $1.6497$ & $\num{0.00448533081}$ & $1.9393$ \\ \hline
$480$ & $\num{0.00000000371}$ & $2.3518$ & $\num{0.00010247849}$ & $1.9155$ & $\num{0.00113647811}$ & $1.9806$ \\ \hline
\end{tabular}
\caption{Results are the same as for Table \ref{tab:refinement_1D_1}, but with the CFL number fixed at 10.}
\label{tab:refinement_1D_3}
\end{centering}
\end{table}
\subsection{Two-dimensional examples}
\subsubsection{Rectangular Cavity With Domain Decomposition}
In this section, we demonstrate the second order convergence of our proposed method, including domain decomposition, for a simple rectangular cavity problem with homogeneous Dirichlet and Neumann boundary conditions. A rectangular domain $\Omega = [-L_{x}/2,L_{x}/2]\times[-L_{y}/2,L_{y}/2]$ is divided into four subdomains with new artificial boundaries along $x=0$ and $y=0$, as in Figure \ref{fig:RC_DD}. Due to the Cartesian geometry of this example, the domain decomposition algorithm we use follows directly from the 1D algorithm we have presented. A more general approach will be required on complex subdomains is, the subject of future investigation.
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=2,>=stealth]
\draw[->] (-0.8,-0.8) -- (-0.6,-0.8)
node[right] {$x$};
\draw[->] (-0.8,-0.8) -- (-0.8,-0.6)
node[above]{$y$};
\draw (-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle;
\draw[dashed] (-1,0)--(1,0);
\draw[dashed] (0,-1)--(0,1);
\draw (1,0) node[anchor=west] {$y=0$}
\draw (0,1) node[anchor=south] {$x=0$}
\end{tikzpicture}
\caption{Rectangular cavity with domain decomposition.}
\label{fig:RC_DD}
\end{figure}
As initial conditions, we choose
\[
u\left(x,y,0\right) =
\begin{cases}
\cos\left(\frac{(2m+1)\pi x}{L_{x}}\right)\cos\left(\frac{(2n+1)\pi y}{L_{y}}\right) & \text{Dirichlet case} \\
\sin\left(\frac{(2m+1)\pi x}{L_{x}}\right)\sin\left(\frac{(2n+1)\pi y}{L_{y}}\right) & \text{Neumann case}
\end{cases}
\]
and
\[
u_{t}\left(x,y,0\right)=0
\]
for $(x,y) \in \Omega$, $m$ and $n$ integers. Exact solutions are well-known in each case. The results of refinement studies are listed in Tables \ref{tab:refinement_RCD} and \ref{tab:refinement_RCN}. The error is the maximum discrete $L^{2}$ error (computed against the exact solution) over all time steps.
\begin{table}[htbp]
\begin{centering}
\begin{tabular}{|c||c|c||c|c||c|c|}
\hline
& \multicolumn{2}{c||}{CFL 0.5} & \multicolumn{2}{c||}{CFL 2} & \multicolumn{2}{c|}{CFL 10} \\ \hline
$\Delta x$ & $L^{2}$ error & $L^{2}$ order& $L^{2}$ error & $L^{2}$ order & $L^{2}$ error & $L^{2}$ order \\ \hline
$1/40$ & $\num{0.0008480082}$ & $-$ & $\num{0.00589191}$ & $-$ & $\num{0.106922697}$ & $-$ \\ \hline
$1/80$ & $\num{0.0001901801}$ & $2.15671$ & $\num{0.00145775}$ & $2.01499$ & $\num{0.033923646}$ & $1.65620$ \\ \hline
$1/160$ & $\num{0.0000448109}$ & $2.08544$ & $\num{0.00036061}$ & $2.01523$ & $\num{0.008868315}$ & $1.93555$ \\ \hline
$1/320$ & $\num{0.0000108611}$ & $2.04468$ & $\num{0.00008955}$ & $2.00965$ & $\num{0.002207988}$ & $2.00592$ \\ \hline
$1/640$ & $\num{0.0000026726}$ & $2.02284$ & $\num{0.00002230}$ & $2.00535$ & $\num{0.000546378}$ & $2.01476$ \\ \hline
\end{tabular}
\caption{Refinement study for rectangular cavity with Dirichlet BC using domain decomposition. Here, $c=1$, $m=n=0$, and $L_{x}=L_{y}=1$.}
\label{tab:refinement_RCD}
\end{centering}
\end{table}
\begin{table}[htbp]
\begin{centering}
\begin{tabular}{|c||c|c||c|c||c|c|} \hline
& \multicolumn{2}{c||}{CFL 0.5} & \multicolumn{2}{c||}{CFL 2} & \multicolumn{2}{c|}{CFL 10} \\ \hline
$\Delta x$ & $L^{2}$ error & $L^{2}$ order& $L^{2}$ error & $L^{2}$ order& $L^{2}$ error & $L^{2}$ order \\ \hline
$1/40$ & $\num{0.000889276}$ & $-$ & $\num{0.00618621}$ & $-$ & $\num{0.1114727}$ & $-$ \\ \hline
$1/80$ & $\num{0.000194862}$ & $2.1902$ & $\num{0.00149418}$ & $2.0497$ & $\num{0.0347712}$ & $1.6807$ \\ \hline
$1/160$ & $\num{0.0000453669}$ & $2.1027$ & $\num{0.00036511}$ & $2.0329$ & $\num{0.0089791}$ & $1.9532$ \\ \hline
$1/320$ & $\num{0.0000109286}$ & $2.0535$ & $\num{0.00009011}$ & $2.0185$ & $\num{0.0022217}$ & $2.0148$ \\ \hline
$1/640$ & $\num{0.0000026809}$ & $2.0273$ & $\num{0.00002237}$ & $2.0098$ & $\num{0.0005480}$ & $2.0192$ \\ \hline
\end{tabular}
\caption{Refinement study for rectangular cavity with Neumann BC using domain decomposition. Here, $c=1$, $m=n=0$, and $L_{x}=L_{y}=1$.}
\label{tab:refinement_RCN}
\end{centering}
\end{table}
\subsubsection{Double Circle Cavity}
In this example, we solve the wave equation with homogeneous Dirichlet boundary conditions on a 2D domain $\Omega$ which is, as in Figure \ref{fig:dblcirc_geom}, the union of two overlapping disks, with centers $P_{1}=\left(- \gamma,0\right)$ and $P_{2}=\left(\gamma,0\right)$, respectively, and each with radius $R$:
\begin{equation} \nonumber
\Omega = \left\{\left(x,y\right) : |\left(x,y\right)-P_{1}| < R \right\}\cup \left\{\left(x,y\right) : |\left(x,y\right)-P_{2}|<R \right\}
\end{equation}
\noindent where $|\left(x,y\right)| = \sqrt{x^{2}+y^{2}}$ is the usual Euclidean vector norm, and $\gamma < R$.
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=2,>=stealth]
\draw[<->] (-1.8,0) -- (1.8,0)
node [above] {$x$};
\draw[<->] (0,-1.1) -- (0,1.1)
node [right] {$y$};
\draw (0,-.8) arc (-126.9:126.9:1);
\draw (0,.8) arc (53.1:306.9:1);
\draw[dashed] (.6,0) -- (.6,1);
\draw (.6,.5) node[anchor=west] {$R$};
\foreach \x in {-.6,.6}
{
\draw (\x,-1pt)--(\x,1pt);
}
\draw (.6,0) node[anchor=north] {$\gamma$};
\draw (-.6,0) node[anchor=north] {$-\gamma$};
\draw (.6,0) node[anchor=south west] {$P_{2}$};
\draw (-.6,0) node[anchor=south east] {$P_{1}$};
\end{tikzpicture}
\caption{Double circle geometry.
\label{fig:dblcirc_geom}
\end{figure}
This geometry is of interest due to, for example, its similarity to that of the radio frequency (RF) cavities used in the design of linear particle accelerators, and presents numerical difficulties due to the curvature of, and presence of corners in, the boundary. Our method avoids the stair-step approximation used in typical finite difference methods to handle curved boundaries, which reduces accuracy to first order and may introduce spurious numerical diffraction.
\begin{figure}[ht!]
\centering
\subfigure[$t=0 $]{\includegraphics[trim={1cm 1cm 1cm 1cm},clip, width=0.3\textwidth]{Two_Circles02.pdf}}
\subfigure[$t=0.5145$]{\includegraphics[trim={1cm 1cm 1cm 1cm},clip, width=0.3\textwidth]{Two_Circles04.pdf}}
\subfigure[$t=0.5145$]{\includegraphics[trim={1cm 1cm 1cm 1cm},clip, width=0.3\textwidth]{Two_Circles06.pdf}}
\subfigure[$t=0.5145$]{\includegraphics[trim={1cm 1cm 1cm 1cm},clip, width=0.3\textwidth]{Two_Circles14.pdf}}
\subfigure[$t=0.9135$]{\includegraphics[trim={1cm 1cm 1cm 1cm},clip, width=0.3\textwidth]{Two_Circles16.pdf}}
\subfigure[$t=1 $]{\includegraphics[trim={1cm 1cm 1cm 1cm},clip, width=0.3\textwidth]{Two_Circles21.pdf}}
\caption{Evolution of the double circle cavity problem.}
\label{fig:dblcirc_plots}
\end{figure}
As initial conditions, we choose
\[
u\left(x,y,0\right) =
\begin{cases}
-\cos^{6}\left(\frac{\pi}{2}\left(\frac{|(x,y)-P_{1}|}{0.8 \gamma}\right)^{2}\right) & |\left(x,y\right)-P_{1}| < 0.8 \gamma \\
\cos^{6}\left(\frac{\pi}{2}\left(\frac{|(x,y)-P_{2}|}{0.8 \gamma}\right)^{2}\right) & |\left(x,y\right)-P_{2}| < 0.8 \gamma \\
0 & \text{otherwise}
\end{cases}
\]
and
\[
u_{t}\left(x,y,0\right)=0
\]
for $(x,y) \in \Omega$. Selected snapshots of the evolution are given in Figure \ref{fig:dblcirc_plots}, and the results of a refinement study are given in Table \ref{tab:refinement_dblcirc}. The discrete $L^{2}$ error was computed against a well-refined numerical reference solution ($\Delta x = \num{0.00021875}$); the error displayed in the table is the maximum over time steps with $t \in [0.28,0.29]$. For this example, $R=0.3$, $\gamma=0.2$, $c=1$, and the CFL is 2.
\begin{table}[htbp]
\begin{centering}
\begin{tabular}{|c|c|c||c|c|} \hline
$\Delta x$ & $\Delta y$ & $\Delta t$ & $L^{2}$ error & $L^{2}$ order \\ \hline
$\num{0.007}$ & $\num{0.0043333333}$ & $\num{0.008666667}$ & $\num{0.006143688}$ & $-$ \\ \hline
$\num{0.0035}$ & $\num{0.0021666667}$ & $\num{0.004333333}$ & $\num{0.001682923}$ & $1.8681$ \\ \hline
$\num{0.00175}$ & $\num{0.0010833333}$ & $\num{0.00216666}$ & $\num{0.000435945}$ & $1.9488$ \\ \hline
$\num{0.000875}$ & $\num{0.0005416667}$ & $\num{0.00108333}$ & $\num{0.000105150}$ & $2.0517$ \\ \hline
\end{tabular}
\caption{Refinement study for the double circle cavity with Dirichlet BC. For the numerical reference solution, $\Delta x = \num{.00021875}$, $\Delta y = \num{.00013542}$, and $\Delta t = \num{.00027083}$.}
\label{tab:refinement_dblcirc}
\end{centering}
\end{table}
\subsubsection{Symmetry on a Quarter Circle}
With the goal of testing the capabilities of our boundary conditions as well as circular geometry, we construct standing modes on a circular wave guide of radius $R$, in two different ways. First, we solve the Dirichlet problem, with initial conditions
\[
u(x,y,0) = J_0\left(z_{20}\frac{r}{R} \right), \quad u_t(x,y,0) = 0,
\]
and exact solution $u = J_0\left(z_{20}\frac{r}{R} \right)\cos\left(z_{20}\frac{ct}{R}\right)$, where $J_0$ is the Bessel function of order 0, and $z_{20} = 5.5218$ is the $2$-nd zero. Secondly, we use the symmetry of the mode to construct the solution restricted to the second quadrant, with homogeneous Neumann boundary conditions taken along the $x$ and $y$ axes.
In both cases, the solution converges to second order. An overlay of the two are shown in Figure \ref{fig:Bessel_Quarter}, demonstrating the close agreement.
\begin{figure}[hb]
\centering
\subfigure[$t = 0.25$]{\includegraphics[width = 0.24\textwidth]{QC_11.pdf}}
\subfigure[$t = 0.50$]{\includegraphics[width = 0.24\textwidth]{QC_21.pdf}}
\subfigure[$t = 0.75$]{\includegraphics[width = 0.24\textwidth]{QC_31.pdf}}
\subfigure[$t = 1.00$]{\includegraphics[width = 0.24\textwidth]{QC_41.pdf}}
\caption{Two separate numerical constructions of a Bessel mode are superimposed, demonstrating that the solution on the quarter circle using Neumann boundary conditions is equivalent to that of the full circle.}
\label{fig:Bessel_Quarter}
\end{figure}
\subsubsection{Periodic Slit Diffraction Grating}
Diffraction gratings are periodic structures used in optics to separate different wavelengths of light, much like a prism. The high resolution that can be achieved with diffraction gratings makes them useful in spectroscopy, for example, in the determination of atomic and molecular spectra. In this example, we apply our method to model an infinite, periodic diffraction grating under an incident plane wave. The purpose of this example is to demonstrate the use of our method with multiple boundary conditions and nontrivial geometry in a single simulation to capture complex wave phenomena.
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=2,>=stealth]
\draw (-1,0)--(-0.2,0);
\draw (0.2,0)--(1,0);
\draw[->] (-0.8,-0.8) -- (-0.6,-0.8)
node[right] {$x$};
\draw[->] (-0.8,-0.8) -- (-0.8,-0.6)
node[above]{$y$};
\draw [decorate,decoration={brace,amplitude=5pt},xshift=0pt,yshift=1pt]
(-0.2,0) -- (0.2,0) node [black,midway,xshift=0,yshift=10pt] {$a$};
\draw [decorate,decoration={brace,amplitude=5pt},xshift=0pt,yshift=12pt]
(-1,0) -- (1,0) node [black,midway,xshift=0,yshift=15pt] {$d$};
\draw[->,decorate,decoration={snake,amplitude=2,post length=.08}]
(0,-.7) -- (0,-.2)
node [right,align=center,midway]
{
{$u_{\mbox{\scriptsize inc}}$}
};
\draw[dashed] (-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle;
\draw[anchor=north] (0,-1) node {Outflow BC};
\draw[anchor=south] (0,1) node {Outflow BC};
\draw[anchor=west] (1,0) node {Periodic BC};
\draw[anchor=east] (-1,0) node {Periodic BC};
\end{tikzpicture}
\caption{Periodic slit diffraction grating geometry}
\label{fig:slit_geom}
\end{figure}
In the next example, we perform a preliminary test of outflow boundary conditions in higher dimensions. While a rigorous analysis of the algorithm is the subject of future work, the results look quite reasonable. Our numerical experiment is depicted in Figure \ref{fig:slit_geom}. An idealized slit diffraction grating consists of a reflecting screen of vanishing thickness, with open slits of aperture width $a$, spaced distance $d$ apart, measured from the end of one slit to the beginning of the next (that is, the periodicity of the grating is $d$).
\begin{figure}[htbp!]
\centering
\subfigure[$t=0.31$]{\includegraphics[trim={1cm 1cm 1cm 1cm},clip, width = .24\textwidth]{Single_Slit_Example2.pdf}}
\subfigure[$t=0.51$]{\includegraphics[trim={1cm 1cm 1cm 1cm},clip, width = .24\textwidth]{Single_Slit_Example3.pdf}}
\subfigure[$t=1.01$]{\includegraphics[trim={1cm 1cm 1cm 1cm},clip, width = .24\textwidth]{Single_Slit_Example4.pdf}}
\subfigure[$t=2.01$]{\includegraphics[trim={1cm 1cm 1cm 1cm},clip, width = .24\textwidth]{Single_Slit_Example6.pdf}}
\caption{Evolution of the slit diffraction grating problem, with aperture width $a=0.1$, grating periodicity $L_{y}=d=1$, and wave speed $c=1$. The CFL is fixed at 2.}
\label{fig:slit_plots}
\end{figure}
We impose an incident plane wave of the form $u_{inc}(x,y,t)=\cos{\left(\omega t + ky\right)}$, where $k = 2 \pi / a$ and $\omega = k/c$, where $c$ is the wave speed. We impose periodic BCs at $x=\pm d/2$ (determining the periodicity of the grating), and homogeneous Dirichlet BCs on the screen. The outflow boundary conditions are imposed at $y=\pm L_{y}/2$. In Figure \ref{fig:slit_plots}, we observe the time evolution of the incident plane wave passing through the aperture, and the resulting interference patterns as the diffracted wave propagates across the periodic boundaries. The outflow boundary conditions allow the waves to propagate outside the domain, with no visible reflections at the artificial boundaries.
\section{Conclusion}
\label{sec:conclusion}
In this paper we have presented a fast, A-stable, second order method for solving the wave equation. Using the Method of Lines Transpose (MOL$^T$), we formulate the semi-discrete problem in one spatial dimension, and solve the resulting boundary value problem using an $O(N)$ fast convolution algorithm. From the the underlying exponential recurrence relation, upon which our algorithm is based, we also develop a means to employ domain decomposition, as well as formulate outflow boundary conditions. Additionally, we address the inclusion of point (delta function) sources into our solver, which is of interest in simulations which couple wave propagation with particle dynamics, i.e. plasma simulations.
We have also extended our fast algorithm to higher spatial dimensions using alternate direction implicit (ADI) splitting, and demonstrated second order convergence of our solver in non-Cartesian geometries, with DIrichlet, Neumann, periodic and outflow boundary conditions. While our efforts to employ domain decomposition in 2D have only consisted of regular Cartesian subdomains, our results are very promising, and this will be investigated in future work.
\bibliographystyle{amsplain}
|
1,108,101,563,255 | arxiv | \section{I. Introduction and method}
\label{sec:intro}
In two recent experiments~\cite{JILA, Aarhus} the energy of an impurity immersed in a Bose-Einstein condensate (BEC) has been measured using RF spectroscopy and a Feshbach resonance to tune the impurity-bath interaction strength. The most striking result is the existence of a well defined quasiparticle peak which, on the attractive branch of the Bose polaron, extends up to the unitary point where the impurity is resonantly interacting with the medium. In the experiment carried out at JILA~\cite{JILA} a low density gas of fermionic $^{40}$K impurities is superimposed to a BEC of $^{87}$Rb atoms, whereas researchers in Aarhus have used two different hyperfine states of bosonic $^{39}$K atoms. Even though the qualitative results are similar in the two experiments, quantitative differences in the energy spectrum arise due to the different mass ratio, $m_B/m_I\simeq2$ in Ref.~\cite{JILA} and $m_B/m_I=1$ in Ref.~\cite{Aarhus} where $m_B$ is the mass of the particles in the BEC and $m_I$ is the mass of the impurity, as well as to the different gas parameter of the medium, respectively $na^3\simeq3\times10^{-5}$ and $na^3\simeq2\times10^{-8}$ in the JILA and Aarhus experiment being $n$ the density and $a$ the scattering length describing interactions within the BEC. Further differences, concerning in particular the width of the spectrum at resonance, probably arise from the coupling to the continuum of excited states which affects the measured response function~\cite{Aarhus}.
The fermionic analogue of this problem, {\it i.e.} an impurity immersed in a Fermi sea, has already been the object of many experimental and theoretical studies~\cite{Fermipolaron}. In particular, on the theory side, the Fermi polaron problem has been addressed using quantum Monte Carlo (QMC) methods which are best suited to deal with the regime of strong correlations between the impurity and the medium. The binding energy of the attractive polaron was calculated exactly by means of the diagrammatic Monte Carlo technique~\cite{Prokofev1, Prokofev2}, whereas projection Monte Carlo methods implementing the fixed-node approximation were used to calculate the equation of state of highly imbalanced Fermi mixtures both with equal~\cite{Pilati08} and unequal masses~\cite{Gezerlis09}.
In a previous study~\cite{Ardila15} we characterized the properties of the Bose polaron using QMC methods in the case where the impurity and the particles in the BEC have the same mass. In particular, we determined the dependence of the binding energy of the impurity resonantly interacting with the bath by varying the gas parameter of the BEC. In the present work we extend the previous calculations by analyzing how this energy changes also as a function of the mass ratio $m_B/m_I$. We report on results ranging from the value $m_B/m_I=2$ of the light impurities used in the JILA experiment~\cite{JILA} to $m_B/m_I=0$ corresponding to the limit of a static impurity. This latter case is an interesting reference problem, which has an exact solution for an impurity in a Fermi sea~\cite{Recati07}, and for the Bose polaron is expected to have a universal dependence on the gas parameter of the bath. We also calculate the binding energy of the attractive branch for the parameters of the JILA experiment in order to allow for a direct comparison.
We calculate the ground state of the following Hamiltonian
\begin{eqnarray}
H&=&-\frac{\hbar^2}{2m_B}\sum_{i=1}^N \nabla_i^2+\sum_{i<j}V_B(r_{ij})
\nonumber\\
&-&\frac{\hbar^2}{2m_I}\nabla_\alpha^2+\sum_{i=1}^NV_I(r_{i\alpha}) \;,
\label{eq:Hamiltonian}
\end{eqnarray}
describing a system of $N$ bosons of mass $m_B$ and one impurity of mass $m_I$. The particle coordinates are denoted by ${\bf r}_\alpha$ and ${\bf r}_i$ ($i=1,\dots,N$) for the impurity and the bosons. The pair potentials $V_B(r)$ and $V_I(r)$ denote, respectively, the boson-boson and the boson-impurity interaction and depend on the distance between the two particles. Similarly to Ref.~\cite{Ardila15}, we model $V_B$ using hard spheres with diameter $a$, which corresponds to the value of the inter-bosons $s$-wave scattering length. For the boson-impurity interaction $V_I$ we use a zero-range potential characterized by the scattering length $b$. The diffusion Monte Carlo (DMC) method used in the simulations has been already described in Ref.~\cite{Ardila15} and we refer the reader to this reference for details on the numerical technique.
The mass-ratio dependence, arising from $m_I\neq m_B$, is accounted for in the DMC simulations by the properly corrected contribution of the impurity kinetic energy. Another slight modification compared to the method described in Ref.~\cite{Ardila15} concerns the trial wave function
\begin{equation}
\psi_T({\bf r}_\alpha,{\bf r}_1,\dots,{\bf r}_N)=\prod_{i=1}^N f_I(r_{i\alpha})\prod_{i<j}f_B(r_{ij}) \;,
\label{trialwf}
\end{equation}
and arises from the zero-range nature of the impurity-boson potential. The boson-boson Jastrow correlation term $f_B$ coincides with the one used in Ref.~\cite{Ardila15}, whereas the impurity-boson function is given by
\begin{eqnarray}
f_I(r)=\begin{cases}
1-\frac{b}{r} & r<\bar{R} \\
B+C\left(e^{-\alpha r}-e^{-\alpha(L-r)}\right) & \bar{R}<r<L/2\;.
\end{cases}
\label{Jastrow}
\end{eqnarray}
Here the parameters $B$ and $C$ are fixed by the continuity condition of the function $f_I$ and its first derivative at the matching point $\bar{R}$. This latter and the parameter $\alpha$ are optimized using a variational procedure on the total energy. It is important to stress that the Jastrow term (\ref{Jastrow}), which is singular in $r=0$, allows one to remove completely the interaction potential $V_I$ from the calculation of the energy and to substitute it with the contact boundary condition $f_I(r)=1-b/r$, fixed by the impurity-boson scattering length $b$. This procedure of replacing a short-range potential with the corresponding Bethe-Peierls boundary conditions has been already introduced in DMC simulations~\cite{Pessoa1, Pessoa2}, where also a particular sampling technique was implemented to handle the zero-range interaction. We find that the branching process in the DMC method~\cite{Mitas11}, by suppressing the weight of configurations with two or more bosons close to the impurity, reduces significantly the problem of large variance affecting variational calculations and allows one to carry out simulations without ``ad hoc'' sampling. We have checked that all the results reported in Ref.~\cite{Ardila15} for the square-well potential $V_I$ with the smallest range are faithfully reproduced by the choice (\ref{Jastrow}) of the trial wave function~\cite{Note1}.
\section{II. Results}
\label{sec:results}
The binding energy of the impurity is defined as the energy difference
\begin{equation}
\mu=E_0(N,1)-E_0(N) \:,
\label{bindener}
\end{equation}
between the ground states of the system with $N$ bosons and the impurity and of the clean system with only $N$ bosons. Both energies are calculated using the DMC method described in Ref.~\cite{Ardila15} for fixed values of $N$ in a box with periodic boundary conditions. The extrapolation to the thermodynamic limit is achieved by ensuring that $N$ is large enough to make finite-size effects negligible. To this aim calculations are repeated for different numbers $N$, typically $N=$ 32, 64 and 128. All energies are measured in units of the following convenient scale provided by the bath
\begin{equation}
\mu_0=\frac{\hbar^2(6\pi^2n)^{2/3}}{2m_B} \;,
\label{scale}
\end{equation}
which corresponds to the Fermi energy of a single-component Fermi gas with the same density and mass.
\begin{table*}
\centering
\caption{$\mu/\mu_{0}$ as a function of the mass ratio $m_{B}/m_{I}$ and the gas parameter $na^{3}$}
\begin{tabular}{|c|c|c|c|c|}
\hline
$m_{B}/m_{I}$ & $na^{3}=5\times10^{-4}$ & $na^{3}=2.66\times10^{-5}$ & $na^{3}=1\times10^{-6}$ & $na^{3}=3\times10^{-7}$\tabularnewline
\hline
$0.0$ & $-0.029(3)$ & $-0.35(5)$ & $-0.39(1)$ & $-0.44(3)$\tabularnewline
\hline
$0.2$ & $-0.037(3)$ & $-0.33(3)$ & $-0.38(4)$ & $-0.34(7)$\tabularnewline
\hline
$0.5$ & $-0.080(3)$ & $-0.47(4)$ & $-0.50(1)$ & $-0.47(1)$\tabularnewline
\hline
$0.8$ & $-0.219(3)$ & $-0.63(6)$ & $-0.70(1)$ & $-0.73(3)$\tabularnewline
\hline
$1.0$ & $-0.317(4)$ & $-0.81(5)$ & $-0.99(5)$ & $-1.06(1)$\tabularnewline
\hline
$1.5$ & $-0.600(1)$ & $-1.21(2)$ & $-1.74(6)$ & $-2.1(1)$\tabularnewline
\hline
$2.0$ & $-0.9 (2)$ & $-1.56(4)$ & $-2.4(5)$ & $-4.4(4 )$\tabularnewline
\hline
\end{tabular}
\label{tab1}
\end{table*}
\begin{figure}
\begin{center}
\includegraphics[width=8.5cm]{Figure1.pdf}
\caption{(color online). Binding energy of the polaron for resonant coupling ($1/b=0$) as a function of the mass ratio $m_B/m_I$. Results corresponding to different values of the interaction strength in the bath are shown. In particular, the value $na^3=2.66\times10^{-5}$ refers to the conditions of the JILA experiment of Ref.~\cite{JILA}.}
\label{fig1}
\end{center}
\end{figure}
In Tab.~\ref{tab1} we report the results of the binding energy $\mu$ at the resonant value of the impurity-boson interaction ($1/b=0$) for different values of the mass ratio $m_B/m_I$ and of the gas parameter $na^3$. The same results are shown in Fig.~\ref{fig1}. The chosen values of the mass ratio range from $m_B/m_I=0$, corresponding to the case of a static impurity with infinite mass, to $m_B/m_I=2$. For lighter impurity masses ($m_B/m_I>2$) the calculation of $\mu$ becomes increasingly difficult because of instabilities towards the formation of cluster states. We notice that, at a given interaction strength within the bath, the binding energy increases in absolute value when the mass ratio increases. This effect is small for mass ratios up to $m_B/m_I=0.5$, while a larger drop of $\mu$ is seen when the impurity gets lighter than the particles in the medium.
The case of a static impurity ($m_B/m_I=0$) resonantly coupled to the medium is particularly interesting. First of all, as visible from the results in Fig.~\ref{fig1}, it applies also to physically relevant situations where $m_I$ is significantly larger than $m_B$. Secondly, it is a reference problem which in the case of the equivalent Fermi polaron, {\it i.e.} a static impurity resonantly coupled to a single-component Fermi sea, features the exact solution $\mu/\mu_0=-1/2$~\cite{Recati07}. The bosonic counterpart is not exactly solvable, nevertheless, since Efimov physics is irrelevant due to the vanishing mass ratio~\cite{Petrov} and also tetramer and larger bound states should not play any role, one expects that $\mu/\mu_0$ is a universal function of the gas parameter in the BEC.
The results for the inverse binding energy $(\mu/\mu_0)^{-1}$ are presented in Fig.~\ref{fig2} as a function of $n^{1/3}a$. One finds that $|\mu|$ increases with decreasing gas parameter, remaining however always smaller than the value 1/2 of the Fermi polaron mentioned above. In the same Fig.~\ref{fig2} we also report the corresponding results for the mass ratio $m_B/m_I=1$~\cite{Ardila15} which exhibit a similar trend as a function of $na^3$, but the value of $\mu$ is about a factor of two larger compared to the static impurity. Noticeably, the resonant Fermi polaron shows a much weaker dependence on the mass ratio as it reaches $\mu/\mu_0\simeq-0.58$ for equal masses~\cite{Lobo06}.
One should also notice that, for both values of mass ratio in Fig.~\ref{fig2}, the dependence of the energy $\mu$ on the interaction strength $na^3$ in the bath is sizeable. Large variations are also predicted for the mass ratio $m_B/m_I=2$, as shown in Fig.~\ref{fig1}. We point out that such a strong dependence is completely underestimated in other theoretical approaches to the Bose polaron problem where interactions within the condensate are either completely neglected, as in the variational approach of Ref.~\cite{DasSarma14}, or produce only small effects, as in the T-matrix approximation of Ref.~\cite{Rath13}. In Ref.~\cite{Bruun1}, instead, the effect of three-body correlations is investigated and a significant decrease of the polaron energy is found as a result of the avoided crossing between the quasiparticle and the deepest Efimov state.
We investigated the presence of cluster states, {\it i.e.} bound states of the impurity with two or more bosons, at resonant coupling both for the mass ratio $m_B/m_I=0$ and $m_B/m_I=1$. While in this latter case we find very shallow bound states of up to five bosons~\cite{Ardila15}, in the case of a static impurity no bound states are found within our model of hard-sphere boson-boson interactions. The role of such cluster states, and in particular of Efimov trimers, becomes important in the proximity of the resonance by adding new length scales to the problem and thereby suppressing the universal character of the polaron energy in terms of the gas parameter of the surrounding BEC~\cite{Bruun1}. We believe that our results for $m_B/m_I=1$ describe correctly the polaron ground state provided the energy of the deepest Efimov trimer is much smaller than $\mu_0$.
We decided to plot the results for the binding energy in Fig.~\ref{fig2} in terms of $(\mu/\mu_0)^{-1}$ in order to discuss more properly the limit $na^3\to0$. For the static impurity at $1/b=0$, one can easily verify that an ideal Bose gas, featuring $a=0$, is unstable against collapse and yields a binding energy $\mu$ which scales with the $N^{1/3}$ law of the number of bosons~\cite{Note2}. Results in Fig.~\ref{fig2} show instead that $1/\mu$ does not scale to zero as $na^3\to0$, provided that one first takes the thermodynamic limit at fixed $na^3$ and only after allows the gas parameter to approach zero. The mobile impurity with mass ratio $m_B/m_I=1$ exhibits a similar behavior (see Fig.~\ref{fig2}). However, in this case, one expects that for small enough densities the energy $\mu$ approaches the binding energy of the deepest cluster state.
\begin{figure}
\begin{center}
\includegraphics[width=8.5cm]{Figure2.pdf}
\caption{(color online). Inverse binding energy of the polaron for resonant coupling ($1/b=0$) as a function of the gas parameter in the bath. Results are shown for a mobile impurity with $m_B/m_I=1$ (see Ref.~\cite{Ardila15}) and for a static impurity ($m_B/m_I=0$). The solid (static impurity) and dashed line (mobile impurity) are polynomial fits to the data including first and second order terms in $n^{1/3}a$.}
\label{fig2}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=8.5cm]{Figure3.pdf}
\caption{(color online). Binding energy of the impurity along the attractive branch as a function of the impurity-bath coupling $1/(n^{1/3}b)$. The mass ratio is $m_B/m_I=2$ and is close to the case of $^{40}$K impurities in a BEC of $^{87}$Rb atoms as in the JILA experiment~\cite{JILA}. The gas parameter has also been chosen to reproduce the experimental value $na^3=2.66\times 10^{-5}$. The experimental points are reproduced from Fig.~(3) in Ref.~\cite{JILA}. The blue dashed line corresponds to the prediction of perturbation theory including first and second order terms. }
\label{fig3}
\end{center}
\end{figure}
Finally, in Fig.~\ref{fig3}, we compare the binding energy obtained from our DMC simulations with the results of the JILA experiment~\cite{JILA}. Both the value of the chosen mass ratio, $m_B/m_I=2$, and of the interaction strength, $na^3=2.66\times10^{-5}$, are close to the parameters of the experiment. The results cover the whole attractive branch of the Bose polaron with $b<0$ from the weakly interacting regime, where $|b|$ is on the order of the interparticle distance $n^{-1/3}$ and good agreement is found with the prediction of second-order perturbation theory~\cite{Note3}, to the resonant point achieved at $1/b=0$. The quantitative agreement between theory and experiment is good, although the large experimental uncertainty does not allow for a stringent comparison. In the case of the Aarhus experiment~\cite{Aarhus}, where $na^3\simeq2\times10^{-8}$ and $m_B/m_I=1$, our results would predict $\mu\simeq-1.2\mu_0$ at $1/b=0$, which is more than a factor two smaller than the measured value. A possible explanation of this discrepancy can come from the coupling to the continuum of excited states above the polaron ground state, which is crucial to give a proper account of the measured spectral response and produces a shift towards higher energies of the signal peak~\cite{Aarhus}. It is however unclear why such effects appear to be less important in the case of the JILA experiment, where the quasiparticle peak remains well defined up to the resonant point~\cite{JILA}.
In conclusion, we calculated the binding energy of the Bose polaron resonantly coupled to the surrounding bath as a function of the mass ratio and of the interaction strength within the medium, thereby extending the analysis of Ref.~\cite{Ardila15} where only the equal-mass case was considered. Furthermore, we determine the equation of state of a static impurity in terms of the gas parameter of the BEC. These results can be useful in understanding to what extent the properties of the Bose polaron are universal functions of the boson-impurity scattering length and of the strength of interactions in the medium. In the case of the recent JILA experiment~\cite{JILA}, our results for the corresponding value of the mass ratio are in good agreement with the measured polaron energy along the attractive branch.
\section{Acknowledgements}
We gratefully acknowledge valuable discussions with G. M. Bruun and M. M. Parish. This work was supported by the QUIC grant of the Horizon2020 FET program and by Provincia Autonoma di Trento.
|
1,108,101,563,256 | arxiv |
\subsection{Experiments}
\paragraph{From MNIST to MNIST-R.}
Additionally to the \emph{MNIST} dataset we created the \emph{MNIST-R} dataset containing all original MNIST images rotated by 30 degrees. We train a simple neural net with two 128- and one 20-width ReLU representations. MNIST is used as source while the performance is measured on MNIST-R (target). Inspired by Heinze-Deml et al.~\cite{Heinze-Deml2017} few samples of the target set are used to improve the performance. We compare the results in Tab.~\ref{fig:MNIST-MNIST-R-DomainAdap} with a baseline model trained on the same amount of samples (20 per class) of the target dataset using data augmentation. Our approach can keep up with data augmentation, respectively even performs better if the imbalance in the amount of samples used during training becomes larger. It can better leverage the structure in the source data and map it to the target domain than simple data augmentation which relies on predefined transformations.
\begin{SCfigure}[1][tb]%
\centering
\includegraphics[width=0.5\textwidth]{figures/mnist_baseline_vs_domainadap.png}
\caption{Accuracy of MNIST with different number of training samples from the target domain (MNIST-R). Our approach (orange) clearly outperforms the simple baseline, especially when only few data from the target domain is provided.}
\label{fig:MNISTvsMNISTSVHN}
\end{SCfigure}
\paragraph{From SVHN to MNIST.}
The Street-View House Number (SVHN) dataset\footnote{\url{http://ufldl.stanford.edu/housenumbers/}, 2020/07/10.}, contains house numbers from Google Street View. The challenge of the SVHN dataset is the structured clutter in the background of images. A Convolutional Neural Network (CNN) with two double-Convolutional layers containing 32 and 64 nodes, respectively, is used. A 20-dim. feature representation on top of this architecture is applied to calculate the affinity loss. Results and comparison to Ganin et al.~\cite{Ganin2017} are shown in Tab.~\ref{tab:domainAdap}. The affinity loss does indeed improve the performance on the target dataset with only a little amount of samples. The performance of our model trained on the SVHN dataset with 10 MNIST samples reaches an accuracy of around 75\% and can be compared with Ganin et al.~\cite{Ganin2017}. In comparison, the baseline model achieves only an accuracy of 3\% (see Fig.~\ref{fig:MNISTvsMNISTSVHN}) on the same data. If there are only a few MNIST samples available, the neural net trained with SVHN and our affinity loss outperforms the baseline model trained solely on the same amount of MNIST samples.
\begin{table}[tb]
\centering
\caption{Domain adaptation on MNIST (source) and MNIST-R (target). Displayed are the accuracies on the MNIST-R dataset after training (training dataset indicated in the header).}
\begin{tabular}{|C{2cm}|C{2.3cm}C{3.5cm}C{3.9cm}|}
\hline
\textbf{\# source samples} &
\textbf{only source dataset}&
\textbf{+ 200 target samples and data aug.} &
\textbf{our approach (+ 200 target samples)}\\ \hline\hline
1,000 & 0.52 & 0.76 & 0.78 \\
10,000 & 0.67 & 0.76 & 0.82 \\
\hline
\end{tabular}
\label{fig:MNIST-MNIST-R-DomainAdap}
\end{table}%
\begin{table}[tb]
\caption{Performance of models trained with the affinity loss on SVHN (source) and MNIST (target). For training a small amount of labeled target samples is used. Ganin et al.~\cite{Ganin2017} uses only unlabeled target samples for training.}
\begin{tabular}{|l|l|C{3.5cm}C{3.5cm}|}
\hline
& & \textbf{accuracy SVHN (source domain)} & \textbf{accuracy MNIST (target domain)}\\\hline\hline
& only SVHN & 0.91 & 0.11 \\\hline
\multirow{2}{1.5cm}{baseline}&+ 100 MNIST samples & 0.91 & 0.11 \\
& + 200 MNIST samples & 0.91 & 0.13 \\\hline
\multirow{3}{1.5cm}{our approach}&+ 10 MNIST samples & 0.91 & 0.75 \\
& + 100 MNIST samples & 0.91 & 0.80 \\
& + 200 MNIST samples & 0.92 & 0.85 \\\hline
&Ganin et al.~\cite{Ganin2017} & - &0.74\\ \hline
\end{tabular}
\label{tab:domainAdap}
\end{table}
\subsection{Fairness Measures}\label{subsection:measures}
There are a lot of different fairness measures used for classification~\cite{Verma2018,Beutel,HardtGoogle,Zafar2016}. Two commonly used ones are summarized in the following. Let $\hat{y}$ be the output of the classifier, $y$ the true label and $z$ the sensitive attribute.
\paragraph{Equality of Opportunity/ Equality Gap.} The most common measure is the so-called equality of opportunity. It is reached if the groups $z_1$ and $z_2$ defined by the sensitive characteristic have equal true positive rates (TPR), i.e., $TPR_{z=z_1} = TPR_{z=z_1}$.
The equality gap is then calculated as
\begin{equation}
{P}(\hat{y} = 1 | z = z_1, y = 1) - {P}(\hat{y} = 1| z = z_2, y = 1) = | TPR_{z=z_1} - TPR_{z=z_2} |.
\end{equation}
\paragraph{Parity Gap.} The parity gap is calculated as independence between prediction $\hat{y}$ and sensitive attribute $z$ for positive predictions, i.e.
\begin{equation}
|P(\hat{y} = 1|Z=z_1) - P(\hat{y} = 1|Z=z_2) |
\end{equation}
For binary case of the sensitive attribute, in medical settings, it is the same as the average treatment effect (ATE)~\cite{Amsterdam2019}.
It is important to note that it is difficult to minimize all fairness metrics at the same time. The appropriate metric depends on the application, but most often the equality of opportunity is targeted. Be aware, that there are some trivial models which yield good results (very small gaps) such as models with very low TPR. For some tasks, the compromise may not even be possible, such as predicting whether someone can give birth. There is a clear causal relationship to the gender; thus, if this information (including implicit information) is removed, it becomes impossible for any classifier to make a correct prediction.
\subsection{Experiment: Adult Dataset} \label{sec:adultExperiment}
The popular \emph{Adult} income dataset\footnote{\url{https://archive.ics.uci.edu/ml/datasets/adult}, 2020/07/10.} from the UCI is used for further experiments and the results are compared with other papers. The task is to predict whether or not an individual is earning more than \$50K per year. The samples are annotated with 14 different attributes from gender and educational level to number of work hours per week. The gender attribute is used as binary sensitive attribute for the affinity loss during training. The dataset is split into 26’049 samples for training, 6,512 samples for validating and 16,281 for testing.
We train our model with a batch size of 512 samples using a network with one hidden layer of 128 and another one with 20 nodes on whose feature representation the affinity loss is calculated. We compare our model with several state-of-the-art approaches
as well as against the baseline. Results are summarized in Tab.~\ref{tab:fairnessAdult}. Our approach achieves a similar fairness level compared to other approaches. Consistently, our feature representation promoted fairness criteria with only a small penalty in accuracy even though the dataset is heavily skewed.
\begin{table}[tb]
\centering
\caption{Our approach compared with a baseline and different other approaches concerning accuracy and fairness on the Adult dataset.}
\begin{tabular}{|l|C{2cm}|C{2cm}C{2cm}|C{3cm}|
\hline
\textbf{approach} & \textbf{accuracy} & \textbf{parity gap} & \textbf{equ. gap} & \textbf{equ. gap (TNR)}\\\hline\hline
{baseline}& 0.85 & 0.18 & 0.088& 0.072\\\hline
{ours} & 0.82& 0.065& 0.02& 0.015 \\\hlin
{Quadrianto'18~\cite{Quadriantoa}} &0.81& - & 0.04 & -\\
{Adel'19~\cite{Adel}} & 0.89& 0.13& - & -\\
{Zemel'13~\cite{Zemel2013}} &0.82 & - & - & -\\
{Quadrianto'17~\cite{Quadrianto}} & 0.84 & - &0.017& -\\
{Beutel'17~\cite{Beutel}} & 0.82 &0.12 & 0.07& 0.04\\
{Louizos'17~\cite{Louizos}} & 0.82 & - & 0.05 & -\\
{Xie'17~\cite{Xie}} & 0.84 & - & - & - \\\hline
\end{tabular}
\label{tab:fairnessAdult}
\end{table}%
\paragraph{Fair Representation.} We train two baseline models once inputting all the features including the sensitive attribute and once removing this attribute. We assume that simply removing the sensitive attribute does not help to omit the gender bias~\cite{Pedreschi2008}. The performances are compared with our fair model and if the sensitive attribute is added back, see Fig.~\ref{fig:CausalityArchitect}. For each model we retrain the last layer (hidden layers are fixed) to predict once the gender of the input samples and once the income. Results are summarized in Tab.~\ref{tab:CausalityGender}. The accuracies of the baseline models lay very close together, which shows that information about the gender attribute is indeed still hidden in the input data. Our model is not able to classify the genders correctly just labeling almost all samples as male. The performance of our model with reattached gender attribute is similar to the baseline models. This shows that the sensitive attribute helps the model to perform better.
The histograms of the fair one-dim. representation (see Fig.~\ref{fig:CausalityArchitect}) for the male and female samples in Fig.~\ref{fig:CausalityHists} show a very similar distribution. Hence, this supports the assumption that our model contains a fair representation of the data.
\begin{table}[t]
\centering
\caption{Accuracy and fairness measures predicting income and gender attribute ($z$) on the Adult dataset with different training approaches.}
\begin{tabular}{|l|C{2cm}C{2cm}|C{2.5cm}C{2.3cm}|}
\hline
\multicolumn{1}{|c|}{} & \multicolumn{2}{c|}{\textbf{baseline}}& \multicolumn{2}{c|}{\textbf{our approach}} \\\hline
&
{with z}&
{without z} &
{} &
{z reattached} \\ \hline\hline
\textbf{accuracy income} & 0.85 & 0.85 & 0.81 & 0.83 \\\hline
\textbf{parity gap} & 0.16 & 0.17 & 0.05 & 0.15 \\
\textbf{equality gap} & 0.069 & 0.074 & 0.0075 & 0.1\\\hline
\textbf{accuracy female} & 0.46 & 0.25 & 0.027 &1.0\\
\textbf{accuracy male} & 0.84 & 0.86 & 0.98 & 1.0\\
\hline
\end{tabular}
\label{tab:CausalityGender}
\end{table}%
\begin{figure}[tb]%
\centering
\begin{center}
\begin{subfigure}[t]{0.4\textwidth}
\includegraphics[width=1\textwidth]{figures/hists_rep1_baseline.png}
\subcaption{baseline}
\end{subfigure}
\begin{subfigure}[t]{0.4\textwidth}
\includegraphics[width=1\textwidth]{figures/hist_rep1.png}
\subcaption{our approach}
\end{subfigure}
\end{center}
\vspace{-0.6cm}
\caption{Histogram of the outputs of $r$ (see Fig.~\ref{fig:CausalityArchitect}) for male and female samples. Without enforcing any fair training the learnt distributions carry a lot of information about the sensitive attribute (a) in contrast to our approach with overlapping distributions (b), not allowing to trace back the sensitive attribute.}%
\label{fig:CausalityHists}%
\end{figure}
\paragraph{Interpretablity of the sensitive attribute.}\label{par:interpretability} The influence of the gender attribute on the classification of the input samples concerning the income is evaluated. The weight of the fair one-dim. feature representation $r$ and the weight of the input of the sensitive attribute $z$ are compared (see Fig.~\ref{fig:CausalityArchitect}). We consider the samples where the income is larger than \$50K. The weight for $z$ lays around 4, whereas the weight for $r$ is approximately 1, thus much smaller. This shows that the input of the sensitive attribute has indeed a large influence on the classification as it holds highly valuable information.
\subsection{Experiment: CelebA}
The \emph{CelebA} image dataset\footnote{\url{http://mmlab~.ie.cuhk.edu.hk/projects/CelebA.html}, 2020/07/10.} is significantly more complex than the \emph{MNIST} or \emph{Adult} dataset. This record contains a total of 202,599 images of celebrities, each with 40 attributes. 162,770 images are used for training, 19,867 for validating and the rest for testing. The annotated attributes reflect appearance of the celebrities as well as the emotional state (e.g. smiling), gender, attractiveness and age. The gender attribute is used as a binary sensitive characteristic and attractiveness as a target label for the classification of the images.
As model we use a fixed VGG19 net trained on imagenet (to speed up the training process and reduce complexity) and an additional hidden layer with 124 nodes.
Tab.~\ref{tab:fairnessCelebA} compares results with different weights $\lambda$ and shows that we can in fact debias the pretrained VGG net. The \emph{CelebA} dataset is heavily skewed; around $\sim{77\%}$ of the images showing women are labeled as attractive, compared to $\sim{23\%}$ of men. If $\lambda$ is strong enough, the influence of the skew on the fairness disappears. The downside is the decrease of accuracy to only $\sim{55\%}$ as the TNRs for female and male are getting low. Please note, the comparison with Quadrianto~\cite{Quadrianto} is not too accurate as our baseline already has a lower accuracy.
\begin{table}[tb]
\centering
\caption{Fairness of different models on the CelebA dataset. A lower weight of $\lambda$ keeps the accuracy higher, but improves fairness only to a limited amount. A higher weight decreases the accuracy significantly but makes the model fair.}
\begin{tabular}{|l|C{1.5cm}|C{1.5cm}C{1.5cm}C{1.5cm}|C{1.5cm}C{1.5cm}|}
\hline
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{\textbf{baseline}}& \multicolumn{3}{c|}{\textbf{our approach}}& \multicolumn{2}{c|}{Quadrianto'18\cite{Quadrianto}} \\
&
\textbf{}&
\textbf{$\lambda=0.05$} &
\textbf{$\lambda=0.07$} &
\textbf{$\lambda=0.1$} &
\textbf{fair} &
\textbf{baseline}\\\hline\hline
\textbf{accuracy} & 0.73 & 0.64 & 0.62& 0.55 & 0.8 & 0.8\\\hline
\textbf{parity gap} & 0.47 & 0.23 & 0.22 & 0.0038 & - & - \\
\textbf{equality gap} & 0.31 & 0.25 & 0.23 & 0.018 & 0.19 & 0.34\\
\hline
\end{tabular}
\label{tab:fairnessCelebA}
\end{table}%
\begin{table}[!tb]
\centering
\caption{Accuracies and fairness measures predicting gender attribute $z$ and income on the CelebA dataset with different training approaches.}
\begin{tabular}{|l|C{2cm}C{2cm}|C{2.5cm}C{2.3cm}|}
\hline
\multicolumn{1}{|c|}{} & \multicolumn{2}{c|}{\textbf{baseline}}& \multicolumn{2}{c|}{\textbf{our approach}} \\\hline
&
{with z}&
{without z} &
{} &
{z reattached} \\ \hline\hline
\textbf{accuracy} & 0.73 & 0.74 & 0.62 & 0.7\\\hline
\textbf{parity gap} & 0.47 & 0.49 & 0.22 & 0.63\\
\textbf{equality gap} & 0.31 & 0.32 & 0.23 & 0.59\\
\hline
\textbf{accuracy female} & 0.86 & 0.84 & 0.88 &1.0\\
\textbf{accuracy male} & 0.83 & 0.85 & 0.25 & 1.0\\
\hline
\end{tabular}
\label{tab:CausalityGenderCelebA}
\end{table}%
\begin{figure}[!tb]
\centering
\begin{center}
\begin{subfigure}[t]{0.4\textwidth}
\includegraphics[width=1\textwidth]{figures/hist_rep1_celebA_baseline.png}
\subcaption{baseline}
\end{subfigure}
\begin{subfigure}[t]{0.4\textwidth}
\includegraphics[width=1\textwidth]{figures/hist_rep1_celebA.png}
\subcaption{our approach}
\end{subfigure}
\end{center}
\vspace{-0.6cm}
\caption{Histogram of the outputs of $r$ for male and female samples for \emph{CelebA}.}%
\label{fig:CausalityHistsCelebA}%
\end{figure}
\paragraph{Fair representation \& Intepretability.} The histograms of the the fair one-dim. representation (see Fig.~\ref{fig:CausalityArchitect}) for the male and female samples in Fig.~\ref{fig:CausalityHistsCelebA} show a very similar distribution, supporting the assumption that our model contains a fair representation of the data. The influence of the gender attribute on the classification, see Tab.~\ref{tab:CausalityGenderCelebA}, is checked with the same approach as described in Sec.~\ref{sec:adultExperiment}. The similar accuracies of the baseline models show that information about the gender attribute is indeed still hidden in the input data. The reattached sensitive attribute helps the fair model to perform better in classifying faces as attractive. This can also be seen in the weight of the sensitive attribute $z$ with around 1.5, compared to the one of the fair one-dim. feature representation $r$ with around 0.7
\subsection{Experiments\label{sec:initialExp}}
We show the basic behavior of our method on an illustrative experiment based on the well known \emph{MNIST} dataset of handwritten digits\footnote{\url{http://yann.lecun.com/exdb/mnist/}, 2020/07/10.}.
Additionally we created the \emph{MNIST-I}, which contains all original \emph{MNIST} images, but inverted. Together it forms our dataset where the sensitive attribute indicates if the digit originates from the \emph{MNIST-I} or the original \emph{MNIST}. As target we still want to predict which number is depicted.
We train a simple neural network with two 128- and a single 20-width ReLU hidden layer as representation layer. For training, a batch size of 128 samples is used and the weight of the proposed affinity loss is set to $\lambda = 0.01$.
\paragraph{Embedding.}
In order to analyze our learnt representation, we perform a t-Distributed Stochastic Neighbor Embedding (t-SNE) on the representation layer. It models the higher-dim. data by a low-dim. point such that similar objects lay closer together and dissimilar ones further away.
\begin{SCfigure}
\begin{tabular}{m{1.7cm}C{3cm}C{3cm}}
&{baseline}& {our approach} \\
{target (digits)}&\parbox{3cm}{\includegraphics[width=.25\textwidth]{figures/tsne_digits_withoutloss.png}} & \parbox{3cm}{\includegraphics[width=.25\textwidth]{figures/tsne_digits_withloss.png}} \\
{sensitive attribute (MNIST/
MNIST-I)}&\parbox{3cm}{\includegraphics[width=.25\textwidth]{figures/tsne_groups_withoutloss.png}} &
\parbox{3cm}{\includegraphics[width=.25\textwidth]{figures/tsne_groups_withloss.png}}\\
\end{tabular}
\caption{2-dim. t-SNE plot showing the learnt feature representations colored based on digit (top row) and on the sensitive attribute (bottom row). By using our approach, the representation shows distinguishable digit clusters but the dataset origin cannot be traced. Best viewed in colour!}
\label{tab:t-SNE-MNIST}
\end{SCfigure}%
Fig.~\ref{tab:t-SNE-MNIST} depicts the comparison of two models, trained without (baseline) and with the affinity loss. The baseline model learns two clusters for each digit (one normal and one inverted) and the groups can easily be separated by the sensitive attribute (MNIST/MNIST-I). In contrast, adding the proposed affinity loss into the training process shows that the two groups are highly overlapping, i.e., not being distinguishable anymore, creating a fair (more in Sec.~\ref{section:fairness}) and more general (only one cluster per digit) feature representation. The digits can still be predicted very accurately as their clusters are kept very distinct from each other. The effect of generality is also used for domain adaptation in Sec.~\ref{section:domainAdap}.
\paragraph{Predictions}
After training, we fix all hidden layers and do a normal retrain of the output layer. The output layer is not only trained to predict the digits but also if the sample comes from the original MNIST or inverted MNIST-I dataset. The model trained with our approach should struggle in learning the origin of the sample (inverted/not inverted). In fact, our fair model predicts the target class with 93\% accuracy (4\% higher than the baseline), whereas the sensitive attribution is hardly predictable anymore (around 57\% accuracy). The baseline model can easily predict (nearly 100\%) the sensitive attribute.
\paragraph{Hyperparameter.}
If the weight $\lambda$ of the affinity loss is set to zero, we train the model only based on the target loss not focusing on making the model fair (see Tab.~\ref{tab:MNIST-MNISTI-Weight}, trained 5 times and averaged). For $\lambda$ equals 0.01 we get the fairest model, as the numbers are predicted well (even better than the baseline model does) and the origin dataset of the input samples can hardly be predicted. For a $\lambda$ of 0.1 the model gets as fair as possible by not learning anything at all.
If the representation layer is chosen too small (smaller than 5 nodes) the model does not perform well in predicting the digits accurately (see Tab.~\ref{tab:MNIST-MNISTI-Layers}). If the number of nodes is getting too large (more than 50 in this example) the model gets unfair again suffering from the curse of dimensionality.
\begin{table}[tb]
\centering
\caption{Influence of Hyper-parameters on fairness and accuracy.}
\begin{subtable}{\textwidth}
\caption{Tuning $\lambda$ balances accuracy vs. fairness.}
\begin{tabular}{|l|C{2cm}C{2cm}C{2cm}C{2cm}C{2cm}|}
\hline
\textbf{$\lambda$} & 0 & 0.0001 & 0.001 & 0.01 & 0.1 \\\hline\hline
\textbf{accuracy} & 0.89 & 0.9 & 0.93 & 0.93 & 0.1 \\
\textbf{digits} & $\pm$ 0.02 & $\pm$ 0.02 & $\pm$ 0.01 & $\pm$ 0.01 & $\pm$ 0.005\\\hline
\textbf{accuracy} & 0.995 & 0.99 & 0.89 & 0.57 & 0.5\\
\textbf{sensitive} & $\pm$ 0.005 & $\pm$ 0.005 & $\pm$ 0.02 & $\pm$ 0.03 & $\pm$ 0.005 \\\hline
\end{tabular}
\label{tab:MNIST-MNISTI-Weight}
\end{subtable}
\begin{subtable}{\textwidth}
\caption{Too small representations (obviously) do not allow for good classification results. A too large embedding space suffers from the curse of dimensionality.}
\begin{tabular}{|l|C{1.6cm}C{1.6cm}C{1.6cm}C{1.6cm}C{1.6cm}C{1.6cm}|}
\hline
\textbf{layer size} & 1 & 5 & 10 & 20 & 50 & 100 \\\hline\hline
\textbf{accuracy} & 0.1 & 0.78 & 0.93 & 0.93 & 0.95 & 0.96 \\
\textbf{digits} & $\pm$ 0.05 & $\pm$ 0.03 & $\pm$ 0.01 & $\pm$ 0.01 & $\pm$ 0.03 & $\pm$ 0.04\\\hline
\textbf{accuracy} & 0.5 & 0.54 & 0.57 & 0.57 & 0.74 & 0.85\\
\textbf{sensitive} & $\pm$ 0.05 & $\pm$ 0.02 & $\pm$ 0.02 & $\pm$ 0.03 & $\pm$ 0.04 & $\pm$ 0.05\\\hline
\end{tabular}
\label{tab:MNIST-MNISTI-Layers}
\end{subtable}
\end{table}%
\section{Introduction}
\label{sec:introduction}
\input{introduction}
\section{Learning Invariant Representation}
\input{invariantRepLearning}
\section{Fairness and Interpretability} \label{section:fairness}
\input{fairnessandcausality}
\section{Domain Adaptation}
\input{domainAdaptation} \label{section:domainAdap}
\section{Discussion and Conclusions}
\input{discussion}
\bibliographystyle{splncs03}
|
1,108,101,563,257 | arxiv | \section{Introduction}\label{sec:Intro}
Over the years, \emph{tensor network}~(TN) has been widely applied to various technical fields.
It enables us to resolve extremely high-dimensional problems, such as deep learning~\citep{novikov2015tensorizing,kossaifi2020tensor}, probability density estimation~\citep{glasser2019expressive,miller2021tensor,novikov2021tensor}, partial differential equations~\citep{bachmayr2016tensor,pmlr-v139-richter21a}, and quantum circuit simulation~\citep{markov2008simulating,huggins2019towards}, with acceptable computational and storage costs.
However, as an inevitable side-effect, practitioners have to face a notorious challenge when applying TNs to practical tasks: how to efficiently select the optimal TN structures from a massive quantity of candidates?
Recent works thus put effort into this challenge, in the heading of \emph{TN structure search}~(TN-SS).
Most recently, there have been studies, which focus on searching TN ranks~\cite{hashemizadeh2020adaptive,kodryan2020mars}, formats~\cite{hayashi2019exploring,li2020evolutionary}, and orders~\cite{li2020high,qiu2021memory} to achieve more compact representations.
These results also confirm numerically that the structures impact the expressive power~\citep{cohen2016expressive} of TNs in learning tasks.
In this paper, we consider a practical variant of \mbox{TN-SS}.
The goal is to improve the compactness and expressiveness of a TN while \emph{preserving} its format, such as tensor train~(TT, \citealt{oseledets2011tensor}) or tensor ring~(TR, \citealt{zhao2016tensor}).
Several existing works~\cite{zhao2016tensor,zheng2021fully} have noticed that the mapping from tensor modes onto the TN vertices, also known as core tensors, also influences the expressive power of the model.
To see this, we implement a toy experiment, in which TR is utilized to approximate a tensor of order four.
Figure~\ref{fig:cover} shows the required ranks for achieving the same approximation accuracy in the experiment.
We see that a good ``mode-vertex'' mapping (corresponding to \emph{Model 1}) would produce smaller ranks than the other two models.
It implies lower computational and storage costs and more promising generalization capability in learning tasks~\cite{khavari2021lower}.
This fact thus motivates this work for searching both the optimal TN-ranks and ``mode-vertex'' mappings.
Despite the potential benefit, searching for the optimal ``mode-vertex'' mappings is non-trivial in general.
For instance, there would be $\mathcal{O}(N!)$ different candidates for TR of order $N\geq{}3$ even though the optimal ranks are known.
It is apparently unacceptable to solve it by exhaustive search, particularly when combinatorially searching the TN-ranks is required as well.
Unlike TN-SS, it also appears new theoretical questions for the variant: how fast the scale of the mappings grows with the structures parameters, \textit{e.g.}, format, order and ranks?
Why does the growth rate change?
And what bounds the growth?
To this end, we conduct a thorough investigation of this variant, named \emph{TN permutation search}~(TN-PS), from both the theoretical and numerical aspects.
Theoretically, we answer the preceding questions by analyzing the counting property of the search space of TN-PS, proving a universal non-asymptotic bound for TNs in \emph{arbitrary} formats.
The result is helpful for fast estimation of the computational budget in searching.
We also establish the basic geometry for TN-PS with group-theoretical instruments, involving the (semi-)metric and neighborhood of the search space, such that the local searching can be applied to the task.
Numerically, we develop an efficient algorithm for TN-PS.
In contrast to the existing sampling-based methods~\cite{hayashi2019exploring,li2020evolutionary},
we draw samples in the established neighborhood to explore the `steepest-descent'' path of the landscape, thereby accelerating the searching procedure and decreasing the computational cost.
Experimental results on extensive benchmarks demonstrate that the proposed algorithm is \emph{unique} to resolving TN-PS consistently so far, and the required model evaluations are much fewer than the previous algorithm.
We summarize the main contributions of this work as follows:
\begin{itemize}
\item We propose for the first time the problem of \emph{tensor-network permutation search} (TN-PS), an important variant of TN-SS in practice;
\item We rigorously prove new theoretical properties for TN-PS, involving the counting, (semi-)metric, and neighborhood, revealing how TN structures impact these properties;
\item We develop a local-sampling-based meta-heuristic, which significantly reduces the computational cost compared to \cite{li2020evolutionary}.
\end{itemize}
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{cover_modify.pdf}
\caption{Impact of the ``mode-vertex'' relations on ranks in TR format.
The chart shows the required ranks and number of parameters ($\#$Para.) in average. See supplementary materials for details.
}\vskip -0.1in
\label{fig:cover}
\end{figure}
\subsection{Related Works}
\noindent\textbf{Searching tensor-network (TN) structures.}
Searching the optimal structures for TNs is typically thought of as an extension of the rank selection problem for tensor learning~\citep{zhao2015bayesian,yokota2016smooth,zhao2016tensor,cheng2020novel,mickelin2020algorithms,cai2021blind,hawkins2021bayesian,li2021heuristic,long2021bayesian,sedighin2021adaptive}, which is widely known to be challenging, especially when the TN formats contain cycles~\citep{landsberg2011geometry,batselier2018trouble,Ye2019Tensor}.
More recently, several studies put much effort into this problem, \textit{i.e.}, TN-SS, for exploring unknown formats~\citep{hayashi2019exploring,hashemizadeh2020adaptive,kodryan2020mars,li2020evolutionary,nie2021adaptive}.
The latest works~\cite{razin2021implicit,razin2022implicit} also study the implicit regularization over TN-ranks.
Another line of work closely related to ours is those that study the partition issue for hierarchical Tucker~(HT, \citealt{falco2020geometry,haberstich2021active}) decomposition, which aims to search for the optimal tree structures.
Compared to these works, we focus on the search over the ``mode-vertex'' mappings, which have remained unexplored until now.
\noindent\textbf{Sampling-based optimization.}
Our new algorithm is inspired by zeroth-order optimization, which
is also known as gradient-free optimization or bandit optimization.
The methods can date back to stochastic hill-climbing~\cite{russell1995artificial}, followed by numerous evolutionary programming algorithms~\cite{back1996evolutionary}, and are restudied recently by~\citet{golovin2019gradientless} and applied to various machine learning tasks~\cite{liu2020primer,liu2020min,savarese2021domain,singh2021continuum}.
Inspired by the work~\cite{golovin2019gradientless}, we refine the sampling strategy for TN-PS by taking the unique property of the neighborhood into account but maintaining its original simplicity and efficiency.
\section{Preliminaries}
We first summarize notations and elementary results used throughout the paper.
After that, definitions related to \emph{tensor networks} (TNs) are reviewed for the self-contained purpose.
Throughout the paper, we use blackboard letters, such as $\mathbb{G}$ and $\mathbb{S}$, to denote sets of subjects.
With additional structures, they are also used to represent specific algebraic subjects according to the context, such as groups, fields or linear spaces.
In particular, we use $\mathbb{S}_N$, $\mathbb{R}$, $\mathbb{Z}^+$, and $\mathbb{R}^{I_1\times{}I_2\times\cdots\times{}I_N}$ to represent the \emph{symmetric group} of order $N$, the real field, positive integers and the real linear space of dimension $I_1\times{}I_2\times\cdots\times{}I_N$, respectively.
The \emph{size} of a finite set $\mathbb{A}$ is denoted by $\vert\mathbb{A}\vert$, and the \emph{Cartesian product} of two sets $\mathbb{A}$ and $\mathbb{B}$ is denoted by $\mathbb{A}\times{}\mathbb{B}$.
We say two sets, \textit{e.g.}, $\mathbb{A,B}$, are equivalent if there exists a bijective mapping from $\mathbb{A}$ and $\mathbb{B}$, and sometimes write $\mathbb{A}=\mathbb{B}$ without explicit declaration of the mapping if it is unambiguous.
For convenience, we use $[N]\subseteq{}\mathbb{Z}^+$ to denote a set of positive integers from $1$ to $N$, where $\subseteq$ represents the subset relation.
A \emph{graph} $G=(\mathbb{V},\mathbb{E})$ consists of a \emph{vertex} set $\mathbb{V}$ and an \emph{edge} set $\mathbb{E}$.
For a graph $G$ of $N$ vertices, the set of its \emph{automorphisms}, written $Aut(G)$, is a collection of vertex permutations, under which the edges are preserved, and equivalent to a \emph{subgroup} of the symmetric group, \textit{i.e.}, $Aut(G)\leq{}\mathbb{S}_N$.
We call $H=(\mathbb{V}_H,\mathbb{E}_N)$ a \emph{(spanning) subgraph} of $G$ if $\mathbb{V}_H=\mathbb{V}$ and $\mathbb{E}_H\subseteq{}\mathbb{E}$.
Let $K_N=(\mathbb{V},\mathbb{E}_{K_N})$ be a \emph{complete graph} with $N$ vertices and $\mathbb{G}_N$ be the set containing all subgraphs of $K_N$. We then know that any simple graphs of $N$ vertices are elements of $\mathbb{G}_N$.
The \emph{minimum and maximum degree} of a graph $G$ are denoted by $\delta$ and $\Delta$, respectively.
\subsection{Tensor and Tensor Networks (TNs)}
We consider an \emph{order-$N$ tensor} as a multi-dimensional array of real numbers represented by \mbox{$\mathcal{X}_{i_1,i_2,\ldots,i_N}\in{}
\mathbb{R}^{I_1\times{}I_2\times\cdots\times{}I_N}$}, where
the indices $i_n,\,n\in{}[N]$ correspond to the $\mathbb{R}^{I_n}$-associated \emph{tensor mode}.
Sometimes we ignore the indices by representing the same tensor as $\mathcal{X}$ for notational simplicity.
\emph{Tensor contraction} roughly refers to the process of summing over a pair of repeated indices between two tensors, which is though of as a natural extension of matrix multiplication into high-order tensors.
An explicit calculation of tensor contraction used in this paper follows the definition in~\cite{cichocki2016tensor}.
We consider \emph{tensor network}~(TN) as defined by~\citet{Ye2019Tensor}.
Suppose a sequence of vector spaces $\mathbb{R}^{I_i},\,i\in{}[N]$ and an edge-labelled simple graph $(G,r)=(\mathbb{V},\mathbb{E},r)$, where
$r:\mathbb{E}\rightarrow{}\mathbb{Z}^+$ represents the function labelling edges with positive integers.
TN is thus intuitively defined as a set of tensors, whose elements are of the form of a sequence of tensor contraction of ``core tensors'' corresponding to vertices of $G$.
See \citep{Ye2019Tensor} for an explicit definition of a TN.
In the paper we refer to those core tensors as \emph{vertices}, to the unlabelled graph $G$ as TN \emph{format}, and to the function $r$ as \emph{TN-(model)-ranks}.
Being consistent with \citep{Ye2019Tensor}, we use the same mathematical expression $TNS(G,r,\mathbb{R}^{I_1},\mathbb{R}^{I_2}\ldots,\mathbb{R}^{I_N})$ to represent a TN in our analysis.
The expression is also rewritten as $TNS(G,r)$ for shorthand if $\mathbb{R}^{I_n},\,n\in{}[N]$ are unimportant in the context.
Let $\mathbb{F}_G$ be the set consisting of all possible functions of $r$'s associated to $G$.
Then note that $\mathbb{F}_G$ is equivalent to a positive cone except zero of dimension $\vert{}\mathbb{E}\vert$, \textit{i.e.}, $\mathbb{F}_G=\mathbb{Z}^{+,\vert{}\mathbb{E}\vert}$.
\subsection{TN Structure Search~(TN-SS)}
Let $\mathcal{X}\in{}\mathbb{R}^{I_1\times{}I_2\times\cdots\times{}I_N}$ be an order-$N$ tensor.
TN-SS \emph{without noise} is to solve an optimization problem as follows:
\begin{equation}
\min_{r\in{}\mathbb{F}_{K_N}}\phi\left(K_N,r\right),\quad{}s.t.\,\mathcal{X}\in{}TNS(K_N,r),\label{eq:TNSS}
\end{equation}
where $\phi:\mathbb{G}_N\times{}\mathbb{F}_{K_N}\rightarrow{}\mathbb{R}$ represents a loss function measuring the model complexity of a TN.
Note that, although in~\eqref{eq:TNSS} the first term of $\phi$ is fixed to be $K_N$ , the TN format can degenerate into any simple graphs of $N$ vertices, as the edges of labeling with ``1'', \textit{i.e.}, $\{e\in\mathbb{E}_{K_N}\vert{}r(e)=1\}$, can be harmlessly discarded from the format~\citep{Ye2019Tensor,hashemizadeh2020adaptive}.
We see that solving~\eqref{eq:TNSS} is an \emph{integer programming} problem, generally NP-complete~\cite{papadimitriou1982complexity}.
Nevertheless, thanks to the fact $\mathbb{F}_{K_N}=\mathbb{Z}^{+,\vert\mathbb{E}_{K_N}\vert}$,
some practical algorithms have been proposed~\cite{hashemizadeh2020adaptive,kodryan2020mars,li2020evolutionary}, as $\mathbb{Z}^{+,\vert\mathbb{E}_{K_N}\vert}$ is a well-defined metric space with the isotropic property.
However, we will see next that such good properties do not hold for TN-PS anymore in general.
\section{Tensor-Network Permutation Search~(TN-PS)}\label{sec:counting}
In this section, we first make precise the problem of TN-PS and then prove the properties involving counting, metric, and neighborhood, which are crucial for both understanding the problem and deriving efficient algorithms.
\subsection{Problem Setup}\label{sec:ProblemSetup}
Recall the example illustrated in Figure~\ref{fig:cover}.
Suppose a tensor $\mathcal{X}$ of order $N$ and a simple graph $G_0$, , dubbed \emph{template}, of $N$ vertices.
Apart from the TN-ranks, the primary goal of TN-PS is to find the optimal mappings in some sense from the modes of $\mathcal{X}$ onto vertices of $G_0$.
We thus easily see that the problem amounts to searching the optimal \emph{permutation} of vertices of a graph.
More precisely,
solving TN-PS is to repeatedly index the vertices of $G_0$ consecutively from $1$ to $N$, and then to seek the optimal index sequence in some sense from all possibilities.
Since the permutations are bijective to each other, the TN structures arising from these permutations naturally form an equivalence class, of which all elements preserve the same ``diagram'' as $G_0$.
Formally, such the equivalence class to the template $G_0=(\mathbb{V},\mathbb{E}_0)$ can be written as follows:
\begin{equation}
\mathbb{G}_0=\left\{G\in{}\mathbb{G}_N\vert{}G\cong{}G_0\right\},\label{eq:G0}
\end{equation}
where $\cong$ denotes the relation of \emph{graph isomorphism}, meaning that for each $G\in{}\mathbb{G}_0$ there exists a vertex permutation $g_G\in\mathbb{S}_N$ such that $G=(g_G\left(\mathbb{V}\right),\mathbb{E}_0)$ holds, or $G=g_G\cdot{}G_0$ for shorthand.
TN-PS (without noise) is thus defined by restricting the search space of~\eqref{eq:TNSS} to $\mathbb{G}_0$ as follows:
\begin{equation}
\min_{(G,r)\in{}\mathbb{G}_0\times\mathbb{F}_{G_0}}\phi\left(G,r\right),\quad{}s.t.\,\mathcal{X}\in{}TNS(G,r).\label{eq:TNPS}
\end{equation}
Compared to TN-SS, we search TN structures from a new space consisting of two ingredients: a non-trivial graph set $\mathbb{G}_0$ and $\mathbb{F}_{G_0}=\mathbb{Z}^{+,\vert\mathbb{E}_0\vert}$ that corresponds to the TN-ranks.
We see that TN-PS is no longer an integer programming problem as TN-SS due to the irregular geometry of $\mathbb{G}_0$.
Meanwhile, the size of the new search space varies with different template $G_0$.
Figure~\ref{fig:space} visualizes intuitively the ``geometrical shape'' of the search space for TN-PS associated to a template of three vertices and two edges.
We see that the search space of TN-PS is more ``collapsed'' than the original TN-SS.
One immediate consequence of collapsing is that the searching path and solutions for TN-SS would run out of the TN-PS region, thereby failing to preserve the original TN format.
Next, we will establish formal statements for these observations, and the results will help develop feasible algorithms for resolving TN-PS.
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{TNPSSpaces.pdf}
\caption{``Geometrical shape'' of search spaces of TN-SS and TN-PS, where the equivalence class $\mathbb{G}_0$ makes the ``shape'' for TN-PS as a combination of flips of low-dimensional spaces.
}\vskip -0.1in
\label{fig:space}
\end{figure}
\subsection{Counting TN Structures}
We begin by counting the size of the new search space, proving that
the graph degrees of the template $G_0$ give a universal bound for the size of the search space of TN-PS.
Suppose first a simple graph $G_0=(\mathbb{V},\mathbb{E}_0)$ of $N$ vertices as the template, by which we then construct the set $\mathbb{G}_0$ as Eq.~\eqref{eq:G0}.
As mentioned above, we have known two facts:
\emph{1)} $Aut(G_0)$ forms a subgroup of $\mathbb{S}_N$, \textit{i.e.}, $Aut(G_0)\leq{}\mathbb{S}_N$, such that $G_0=a\cdot{}G_0$ for any $a\in{}Aut(G_0)$;
and \emph{2)} for every $G\in\mathbb{G}_0$ there exists $g_G\in\mathbb{S}_N$ such that $G=g_G\cdot{}G_0$.
By these facts, $G=g^\prime\cdot{}G_0$ holds for any $g^\prime=g_G\cdot{}a$, implying that for each $G\in\mathbb{G}_0$ there exists a \emph{left coset} of $Aut(G_0)$, which is of the form \mbox{$g_G\cdot{}Aut(G_0):=\{g_G\cdot{}a\vert{}a\in{}Aut(G_0)\}$}.
According to the \emph{Lagrange's theorem} in group theory, we thus obtain the following equation with respect to the size of $\mathbb{G}_0$:
\begin{equation}
\vert{}\mathbb{S}_N\vert=\vert\mathbb{G}_0\vert\cdot\vert{}Aut(G_0)\vert.\label{eq:Lag}
\end{equation}
Table~\ref{tab:TNGraphs} lists the values of $\vert{}Aut(G_0)\vert$ associated with several commonly used TNs.
The size of $\mathbb{G}_0$ for those TNs can be therefore derived by~\eqref{eq:Lag}, shown in the last row of Table~\ref{tab:TNGraphs}.
\begin{table}[t]\small
\caption{
Illustration of several counting-related properties for commonly used TNs of order $N>3$, including \textit{tensor train} (TT, \citealt{oseledets2011tensor}), \textit{tensor tree} (TTree, \citealt{Ye2019Tensor}), TR and \textit{projected entangled pair states}~(PEPS, \citealt{verstraete2004renormalization}), where $G_0=(\mathbb{V},\mathbb{E}_0)$, and $\delta_0$ and $\Delta_0$ denote the minimum and maximum degree of $G_0$, respectively.
}
\vspace{0.3cm}
\centering
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{lcccc}\toprule
& \textbf{TT} & \textbf{TTree} & \textbf{TR} & \textbf{PEPS}\\\midrule
$G_0$ & Path & Tree & Cycle & Lattice \\
$\delta_{0}$& $1$ & $1$ & $2$ & $2$\\
$\Delta_{0}$& $2$ & $[2,N-1]$ & $2$ & $2,3,4$\\
$\vert{}\mathbb{E}_0\vert$& $N-1$ & $N-1$ & $N$ & $\leq{}N$ \\
{\scriptsize$\vert{}Aut(G_0)\vert$}& $2$ & $[2,(N-1)!]$ & $2N$ & $\leq{}N$\\
$\vert\mathbb{G}_0\vert$& $N!/2$ & $[N,N!/2]$ & {\tiny$(N-1)!/2$} & $\leq{}N!/4$\\
\bottomrule
\end{tabular}
\vskip -0.1in
\label{tab:TNGraphs}
\end{table}
However, counting the automorphisms for a general graph is difficult~\citep{chang1995finding}.
Blow we prove that the size of the search space of TN-PS is controlled by the the minimum and maximum degree of $G_0$.
For convenience, we further assume
that TN-ranks are only searched within a finite range $\mathbb{F}_{G_0,R}\subset{}\mathbb{F}_{G_0}$, meaning that the rank $r(e)\leq{}R$ holds for any $r\in{}\mathbb{F}_{G_0,R}$ and $e\in{}\mathbb{E}_0$.
We then have the following counting bounds.
\begin{theorem}\label{thm:OverallBounds}
Assume $G_0$ to be a simple and connected graph of $N$ vertices, and $\mathbb{G}_0$ is constructed as~\eqref{eq:G0}.
Let $\delta=N/d_1$ and $\Delta=N/d_2$, $d_1\geq{}d_2>1$, be the minimum and maximum degree of $G_0$, respectively.
The size of the search space of~\eqref{eq:TNPS}, written $\mathbb{L}_{G_0,R}:=\mathbb{G}_0\times{}\mathbb{F}_{G_0,R}$, is bounded as follows:
\begin{equation}
\begin{split}
R^{\frac{N^2}{2d_2}}\cdot{}N!\geq\vert{}\mathbb{L}_{G_0,R}\vert\geq{}
R^{\frac{N^2}{2d_1}}\cdot{}e^{\gamma(d_2)\cdot{}N-\frac{1}{2}\log{}d_2-1/24}
\end{split},\label{eq:SizeBound}
\end{equation}
where $\gamma(d)=\log{}d+\frac{1}{d}-1$ is a positive and monotonically increasing function for $d>1$.
\end{theorem}
Proving the above theorem requires the following lemma about an upper-bound of the size of $Aut(G_0)$, of which the proof is given in Appendix~\ref{sec:Proofs}.
\begin{lemma}\label{thm:CountingLemma}
Let $G_0$ be a simple graph of $N$ vertices, and $Aut(G_0)$ be the set containing automorphisms of $G_0$.
Assume that $G_0$ is connected and its maximum degree $\Delta$ satisfies $N/\Delta=d>1$, then the inequality
\begin{equation}
\left\vert{}Aut(G_0)\right\vert\leq{}N!\cdot{}e^{-\gamma(d)\cdot{}N+\frac{1}{2}\log{}d+1/24}\label{eq:CountingAutExplict}
\end{equation}
holds, where $\gamma(\,\cdot\,)$ is defined in Theorem~\ref{thm:OverallBounds}.
\end{lemma}
As shown in~\eqref{eq:SizeBound}, the bounds of $\vert\mathbb{L}_{G_0,R}\vert$ are determined by three factors: the number of vertices $N$, the searching range of TN-ranks $R$, and the graph degrees of $G_0$ parameterized by $d_1$ and $d_2$.
Figure~\ref{fig:Bound} shows the bounds in ~\eqref{eq:SizeBound} with varying these factors.
We see from the left panel that the upper and lower bounds go closer with increasing the value of $d$~(where we assume $d_1=d_2=d$ for brevity).
It implies that the bounds are tight for graphs with small degrees.
We also see from the middle panel that $\vert\mathbb{L}_{G_0,R}\vert$ grows fast with $N$, even though the graph degree $d$ been sufficiently small such as in TT/TR, while the growth is relatively slow with increasing $R$, the search range for TN-ranks.
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{CountingBound.pdf}
\vspace{-0.4cm}
\caption{Illustration of bounds given in Theorem~\ref{thm:OverallBounds} with varying the parameters $d$, $N$ and $R$, where ``U.b.'' and ``L.b.'' denote the upper and lower bounds, respectively, and $d_1=d_2=d$.
}\vskip -0.1in
\label{fig:Bound}
\end{figure}
\subsection{Semi-Metric and Neighborhood}
The notion of metric and neighborhood of the search space are fundamental for most \emph{steepest-descent}-based optimization methods.
We have seen that they are well-defined for TN-SS but remain unknown for TN-PS.
To address the issue, we establish below a new (semi-)metric and neighborhood for TN-PS with rigorous proofs using \emph{group-theoretic} instruments.
The application of these results to developing efficient algorithms for TN-PS will be introduced in the next section.
We begin by establishing the semi-metric, a relaxation of metric satisfying separation and symmetry except possibly for the triangle inequality, over the graph set $\mathbb{G}_0$.
Although there has been much literature in which different definitions of the graph metric or similarity are proposed, most of them are computationally hard~\cite{koutra2011algorithms}.
Unlike those works, we construct the semi-metric over $\mathbb{G}_0$ based on the equivalence property of its elements given in~\eqref{eq:G0}, so it can be built up by graph isomorphisms in a simple fashion.
Recall the symmetric group $\mathbb{S}_N$.
Let $\mathbb{T}_N\subseteq{}\mathbb{S}_N$ be the set consisting of its all adjacent transpositions, the operations of swapping adjacent two integers in $[N]$ and fixing all other integers.
We thus know from group theory that $\mathbb{T}_N$ generates $\mathbb{S}_N$.
Furthermore,
let $d_{\mathbb{T}_N}:\mathbb{S}_N\times{}\mathbb{S}_N\rightarrow{}R$ be the \emph{word metric}~\citep{luck2008survey} of $\mathbb{S}_N$ induced by $\mathbb{T}_N$.
Intuitively, the value of $d_{\mathbb{T}_N}(p_1,p_2),\,p_1,p_2\in\mathbb{S}_N$ reflects the minimum number of adjacent swapping operations required for transforming the permutation from $p_1$ to $p_2$.
Since we saw in Section~\ref{sec:ProblemSetup} that for each $G\in\mathbb{G}_0$ there is a permutation $g_G\in{}\mathbb{S}_N$ such that $G=g_G\cdot{}G_0$, we thus construct a function $d_{G_0}:\mathbb{G}_0\times{}\mathbb{G}_0\rightarrow{}\mathbb{R}$ using the word metric $d_{\mathbb{T}_N}$ as follows:
\begin{equation}
d_{G_0}\left(G_1,G_2\right)=\min_{
p_i\in{}g_i\cdot{}Aut(G_0),i=1,2
}d_{\mathbb{T}_N }(p_1,p_2),\label{eq:metric}
\end{equation}
where $G_1,G_2\in\mathbb{G}_0$ and $g_1,g_2\in\mathbb{S}_N$ are permutations satisfying $G_i=g_i\cdot{}G_0$, $i=1,2$.
The following lemma shows that \eqref{eq:metric} is in fact a semi-metric function, followed by the construction of the corresponding neighborhood in $\mathbb{G}_0$.
\begin{lemma}\label{thm:metric}
Let $G_0$ be a simple graph and $\mathbb{G}_0$ be the set defined as~\eqref{eq:G0}.
The function $d_{G_0}:\mathbb{G}_0\times{}\mathbb{G}_0\rightarrow{}\mathbb{R}$ defined by \eqref{eq:metric} is a semi-metric on $\mathbb{G}_0$.
Furthermore, let $\mathbb{I}_d\left(G\right)$ be a set constructed as follows:
\begin{equation}
\begin{split}
\mathbb{I}_d\left(G\right)=
\{G'&\in\mathbb{G}_0\vert{}G'=q\prod_{i=1}^d{}t_i\cdot{}G_0,\\
&q\in{}g\cdot{}Aut(G_0),\,t_i\in{}\mathbb{T}_N,\,i\in[d]\}
\end{split}.\label{eq:nh}
\end{equation}
Then $\mathbb{N}_D\left(G\right)=\bigcup_{d=0}^D\mathbb{I}_d(G)$ is the neighborhood of \mbox{$G=g\cdot{}G_0\in{}\mathbb{G}_0$} induced by~\eqref{eq:metric}, with the radius
$D\in{}\mathbb{Z}^+\cup\{0\}$.
\end{lemma}
\begin{algorithm}[tb]
\caption{Random sampling over $\mathbb{I}_d(G)$}
\label{alg:sampling}
\begin{algorithmic}
\STATE {\bfseries Input:}
Center: $G\in\mathbb{G}_0$ with $N$ vertices; Radius: $d$.
\STATE {\bfseries Initialize:}
$G^\prime=G$ where $G^\prime=(\mathbb{V}^\prime,\mathbb{E}^\prime)$.
\FOR{$k=1$ {\bfseries to} $d$}
\STATE Uniformly draw $i,j\in{}[N],\,i\neq{}j$ in random.
\STATE Choose $v_i,v_j\in\mathbb{V}^\prime$ and swap them.
\ENDFOR
\STATE {\bfseries Output:} $G^\prime$.
\end{algorithmic}
\end{algorithm}
We see from~\eqref{eq:nh} that $\mathbb{N}_D\left(G\right)$ consists of combinations of two sets: $Aut(G_0)$ and $\mathbb{T}_N$, followed by the permutation representative $g$ associated to the center graph $G$.
It thus suggests a straightforward sampling method over $\mathbb{N}_D\left(G\right)$, that is, combinatorially sampling over $Aut(G_0)$ and $\mathbb{T}_N$ from some distributions.
However, obtaining all elements of $Aut(G_0)$ is computationally hard~(NP-intermediate,~\citealt{goldwasser1989knowledge}) in general.
To avoid this, we prove that sampling using Alg.~\ref{alg:sampling} can cover all elements of $\mathbb{I}_d(G)$ without sampling directly over $Aut(G_0)$.
\begin{theorem}\label{thm:cover}
For every $G^\prime\in{}\mathbb{I}_d(G)$ with $G\in\mathbb{G}_0$ and $d\geq{}1$, the probability that the output of Alg.~\ref{alg:sampling} equals $G^\prime$ is positive.
\end{theorem}
The (semi-)metric and neighborhood for the overall search space of TN-PS, \textit{i.e.}, $\mathbb{G}_0\times{}\mathbb{F}_{G_0,R}$, can be thus derived by composing the Euclidean metric of $\mathbb{F}_{G_0,R}\subseteq{}\mathbb{Z}^{+,\vert\mathbb{E}_0\vert}$.
In the next section, Alg.~\ref{alg:sampling} will be applied to the new algorithm, by which the searching efficiency is significantly improved.
\section{Meta-Heuristic via Local Sampling}
We present now a new meta-heuristic algorithm for searching TN structures.
Unlike the existing methods such as~\cite{li2020evolutionary}, we exploit the information of the ``steepest-descent'' direction, estimated by sampling over a neighborhood of the search space, to accelerate the searching procedure.
Suppose a tensor $\mathcal{X}$ of order $N$.
For the practical purpose, we take the influence of noise into~\eqref{eq:TNPS}, which is given by
\begin{equation}
\begin{split}
&\min_{G,r,\mathcal{Z}}\phi\left(G,r\right)+\lambda\cdot{}RSE(\mathcal{X},\mathcal{Z})\\
s.t.&\,(G,r)\in\mathbb{G}_0\times{}\mathbb{F}_{G_0,R},\,\mbox{and }\mathcal{Z}\in{}TNS(G,r)\label{eq:PTNPS}
\end{split},
\end{equation}
where $\lambda>0$ denotes a tuning parameter associated with the noise variance, and $RSE$ is the function of \emph{relative squared error}~(RSE) for modeling the influence of Gaussian noise.
In other applications such as in generative models~\cite{liu2021tensor},
it can be replaced by KL-divergence without modifying the algorithm details.
The searching algorithm is illustrated in Alg.~\ref{alg:main}, where $N_{[R]}(a,b)$ denotes a Gaussian distribution of the mean $a$ and variance $b$, followed by a truncation operation such that the samples out of the range $[R]$ are pulled back to the closest bound, $Ber(p)$ denotes the Bernoulli distribution of the mean $p$, and $\mathbf{I}$ denotes the identity matrix of dimension $\vert\mathbb{E}_0\vert\times{}\vert\mathbb{E}_0\vert$.
In each iteration, we elaborate the algorithm into three phases: \emph{local-sampling, evaluation}, and \emph{updating}.
Suppose the starting point $(G^{\{m\}},r^{\{m\}})$ at the $m$th iteration.
In the local-sampling phase, we randomly draw samples for both the TN-ranks ($s_k$) and the ``mode-vertex'' maps ($H_k$) over the neighborhood centered at $(G^{\{m\}},r^{\{m\}})$.
The aim is to explore good descent directions within the neighborhood.
The sampling distributions, involving the rounded truncated Gaussian and the uniform distribution in Alg.~\ref{alg:sampling}, are chosen as a non-informative searching prior.
Note that for the two variance-related parameters $c_1,c_2\in[0.9,1)$, we apply the annealing trick to the shrinkage of the sampling range in each iteration.
The trick guarantees the convergence of the algorithm.
In the evaluation phase, we employ \emph{arbitrarily} proper optimization or learning methods to minimize RSE or other alternatives for the sampled structures $(H_k,s_k),\,k\in{}[\#Sample]$.
In the updating phase, we calculate the overall loss function $f_k$ for each sampled structures, and then update $(G^{\{m+1\}},r^{\{m+1\}})$ once there exists new samples, whose performance is better than $(G^{\{m\}},r^{\{m\}})$.
More precisely, let $f_0$ be the loss of~\eqref{eq:PTNPS} with respect to $(G^{\{m\}},r^{\{m\}})$ and $f_{min}$ be the minimum among all $f_k$'s.
If $f_{min}<f_0$, we update $(G^{\{m+1\}},r^{\{m+1\}})=(H_{min},s_{min})$; otherwise, we remain $(G^{\{m+1\}},r^{\{m+1\}})=(G^{\{m\}},r^{\{m\}})$.
\noindent\textbf{Discussion.}
Compared with the global-sampling methods, such as genetic algorithm~(GA)~\cite{hayashi2019exploring,li2020evolutionary}, we restrict the sampling range into the neighborhoods of the structure candidates, rather than the whole search space.
The advantages in doing so are mainly two-folds:
first, the neighborhood geometry allows to construct gradient-like directions, which result in a faster decrease of the loss function if the landscape is related smooth;
second, a smaller sampling range can mitigate the curse of dimensionality.
Otherwise, the algorithm would ``lose itself'' in searching if the TN structure is large scale.
However, it should be mentioned that the local-sampling methods would perform worse than the global-searching ones if the landscape is too ``flat'' or ``swinging''.
We conjecture with rich empirical observations that TN-PS (including TN-SS) seems more suitable for ``local-sampling'' methods.
Since a small perturbation on the structure such as ranks would not dramatically change the RSE, it implies that the landscape of~\eqref{eq:PTNPS} tends to be smooth.
Although a rigorous discussion on this issue remains open, we use extensive numerical results to verify the efficiency of Alg.~\ref{alg:main} in the next section.
\begin{algorithm}[tb]
\caption{TN-structure Local Sampling (TNLS)}
\label{alg:main}
\begin{algorithmic}
\STATE {\bfseries Input:} template graph: $G_0=(\mathbb{V},\mathbb{E}_0)$; searching range of TN-ranks: $R$; function: \mbox{$f(G,r,\mathcal{Z}):=\phi\left(G,r\right)+\lambda\cdot{}RSE(\mathcal{X},\mathcal{Z})$}, maximum iteration: $\#Iter>0$; number of sampling: $\#Sample>0$; tuning parameters: $c_1,c_2\in[0.9,1)$.
\STATE {\bfseries 0. Initialize:}\\
$(G^{\{0\}},r^{\{0\}})$ with random selection from $\mathbb{G}_0\times{}\mathbb{F}_{G_0,R}$.
\STATE Obtain $\mathcal{Z}^{\{0\}}$ by arbitrary TN approximation methods with $(G^{\{0\}},r^{\{0\}})$.
\FOR{$m=1$ {\bfseries to} $\#Iter$}
\STATE {\bfseries 1. Local sampling:}
\FOR{$k=1$ {\bfseries to} $\#Sample$}
\STATE Sample $s_k\sim{}N_{[R]}(r^{\{m\}},c_1^{m-1}\cdot\mathbf{I})$ with rounding.
\IF{\texttt{TRUE} by sampling from $Ber(c_2^{m-1})$}
\STATE Sample $H_k\in\mathbb{G}_0$ from $\mathbb{I}_1(G^{\{m\}})$ using Alg.~\ref{alg:sampling}.
\ELSE
\STATE $H_k=G^{\{m\}}$.
\ENDIF
\ENDFOR
\STATE { \textbf{2. Evaluation:} \emph{(be possible in parallel)}}
\FOR{$k=1$ {\bfseries to} $\#Sample$}
\STATE Obtain $\mathcal{Z}_k$ by arbitrary TN approximation methods with the given $(H_k,s_k)$.
\ENDFOR
\STATE {\bfseries 3. Update:}
\begin{equation}\nonumber
\begin{split}
(G&^{\{m+1\}},r^{\{m+1\}},\mathcal{Z}^{\{m+1\}})=\\
\arg&\min_{k}\{f(G,r,\mathcal{Z})\vert{}(G,r,\mathcal{Z})=(G^{\{m\}},r^{\{m\}},\mathcal{Z}^{\{m\}}),\\
&(G,r,\mathcal{Z})=(H_k,s_k,\mathcal{Z}_k)\}.
\end{split}
\end{equation}
\ENDFOR\\
{\bfseries Return:} $(G^{\{m+1\}},r^{\{m+1\}})$.
\end{algorithmic}
\end{algorithm}
\section{Experimental Results}
In this section, we numerically verify the effectiveness and efficiency of the proposed method on both the synthetic and real-world tensors.
\begin{table*}[t]
\centering
\caption{Experimental results on synthetic data in TT/TR format.
In the table, \emph{Eff.} denotes the parameter efficiency defined in~\cite{li2020evolutionary},
$\#Eva.$ denotes the number of evaluations, and ``-'' denotes that the methods fail to satisfy the condition $RSE\leq{}10^{-4}$.
}
\begin{threeparttable}\small\label{tab:TR}
\setlength{\tabcolsep}{1.7mm}{
\begin{tabular}{ccccc|cc|cc}
\toprule
\multirow{4}[0]{*}{\textbf{Trial}}&\multicolumn{4}{c}{\emph{TR methods}}&\multicolumn{2}{c}{\emph{TN-SS methods}}&\multicolumn{2}{c}{\emph{TN-PS methods}}\\
& \textbf{TR-SVD} & \textbf{TR-LM} & \textbf{TR-ALSAR} & \textbf{Bayes-TR} & \textbf{Greedy}&\textbf{TNGA}$^\star$&\textbf{TNGA+}&\textbf{TNLS}\\
\cmidrule{2-9}
{}&\multicolumn{8}{c}{\textbf{ Order 4} -- \emph{Eff.$\uparrow$~$\langle$\emph{Is TT/TR format preserved? {\color{green}Yes} or {\color{red}No}.}$\rangle$~$[$\#Eva.$\downarrow$$]$} }\\
\midrule
\textbf{A} & 1.00 & 1.00 & 0.21 & 1.00&1.00~$\langle${\color{green}Yes}$\rangle$ &1.00~$\langle${\color{green}Yes}$\rangle$~[600] & 1.00~[450]&1.00~[\textbf{361}] \\
\textbf{B} & 0.64& 1.00 & 1.00 & 0.64 &0.89$\langle${\color{red}No}$\rangle$& 1.00~$\langle${\color{green}Yes}$\rangle$~[300] &1.00~[450] &1.00~[\textbf{241}] \\
\textbf{C} & 1.17 & 1.17 &0.23 & 1.00&1.17~$\langle${\color{green}Yes}$\rangle$ &1.17~$\langle${\color{green}Yes}$\rangle$~[750] & 1.17~[450]&1.17~[\textbf{181}] \\
\textbf{D} & 0.57 & 0.57 & 0.32 & -&1.00~$\langle${\color{green}Yes}$\rangle$ &1.00~$\langle${\color{green}Yes}$\rangle$~[450] &1.00~[300]&1.00~[\textbf{301}] \\
\textbf{E} & 0.43& 0.48 & 0.40 & 0.40&1.00~$\langle${\color{green}Yes}$\rangle$ &1.00~$\langle${\color{green}Yes}$\rangle$~[1050] & 1.00~[450] &1.00~[\textbf{361}] \\
\midrule
{}&\multicolumn{8}{c}{\textbf{ Order 6} -- \emph{Eff.$\uparrow$~$\langle$\emph{Is TT/TR format preserved? {\color{green}Yes} or {\color{red}No}.}$\rangle$~$[$\#Eva.$\downarrow$$]$} }\\
\midrule
\textbf{A}&0.21&0.44&-&-&0.16~$\langle${\color{red}No}$\rangle$&0.82~$\langle${\color{red}No}$\rangle$~[1650] &\textbf{1.00}~[1500]&\textbf{{1.00}}~[\textbf{661}]\\
\textbf{B}&0.14&0.15&0.14&-&0.27$~\langle${\color{red}No}$\rangle$&- &\textbf{1.00}~[1350]&\textbf{1.00}~[\textbf{601}]\\
\textbf{C}&0.57&1.00&0.85&0.29&0.97~$\langle${\color{red}No}$\rangle$&1.00~$\langle${\color{green}Yes}$\rangle$~[3300] &1.00~[1800]&1.00~[\textbf{661}]\\
\textbf{D}&0.21&0.39&0.10&0.13&1.04~$\langle${\color{green}Yes}$\rangle$&1.04~$\langle${\color{green}Yes}$\rangle$~[2700] &\textbf{1.16}~[1500]&\textbf{1.16}~[\textbf{601}]\\
\textbf{E}&0.15&0.30&-&0.12&1.00~$\langle${\color{green}Yes}$\rangle$&1.00~$\langle${\color{green}Yes}$\rangle$~[2400] &1.00~[1050]&1.00~[\textbf{541}]\\
\midrule
{}&\multicolumn{8}{c}{\textbf{ Order 8} -- \emph{Eff.$\uparrow$~$\langle$\emph{Is TT/TR format preserved? {\color{green}Yes} or {\color{red}No}.}$\rangle$~$[$\#Eva.$\downarrow$$]$} }\\
\midrule
\textbf{A}&0.10&0.16&-&0.03&0.88~$\langle${\color{red}No}$\rangle$&0.48~$\langle${\color{red}No}$\rangle$~[2550]&\textbf{1.00}~[2850]&\textbf{1.00}~[\textbf{1021}]\\
\textbf{B}&0.09&0.43&-&-&0.61$\langle${\color{red}No}$\rangle$&-&\textbf{1.02}~[2250]&\textbf{1.02}~[\textbf{961}]\\
\textbf{C}&0.03&0.31&-&0.02&\textbf{1.16}$\langle${\color{red}No}$\rangle$&0.49~$\langle${\color{red}No}$\rangle$~[2250]&1.11~[3750]&1.11~[\textbf{1321}]\\
\textbf{D}&0.20&0.53&-&-&1.03$\langle${\color{red}No}$\rangle$&0.32~$\langle${\color{red}No}$\rangle$~[4050]&\textbf{1.06}~[1950]&\textbf{1.06}~[\textbf{781}]\\
\textbf{E}&0.33&0.33&-&-&\textbf{1.17}~$\langle${\color{green}Yes}$\rangle$&0.23~$\langle${\color{red}No}$\rangle$~[1500]&0.88~[1500]&\textbf{1.17}~[\textbf{901}]\\
\bottomrule
\end{tabular}
}
\end{threeparttable}
\end{table*}
\subsection{Synthetic Data in TT/TR Format and Beyond}
Using synthetic data, we first verify:
\textbf{(a)} TN-PS can reduce the required TN model size for the low-rank tensor approximation task, reflecting the improvement of the expressive power of TNs;
and \textbf{(b)} the proposed local-sampling method achieves more efficient searching than the existing sampling-based methods.
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{Convergence.pdf}
\vspace{-0.6cm}
\caption{Average loss with varying the number of evaluations.
}\vskip -0.1in
\label{fig:convergence}
\end{figure}
\noindent\textbf{Data generation.}
We choose TT/TR, the most commonly used TN formats in machine learning, to generate tensor data.
For each tensor order, \textit{i.e.}, the number of vertices, $N\in\{4,6,8\}$, we generate five tensors by randomly choosing ranks and values of vertices (core tensor).
In more detail, the dimensions for each tensor modes are set to equal $3$.
Here we choose a small dimension as same as the one in~\citep{li2020evolutionary} because it is typically irrelevant to the searching difficulty, as shown in Theorem~\ref{thm:OverallBounds}.
Meanwhile, we uniformly select the TN-ranks from $\{1,2,3,4\}$ in random, and \textit{i.i.d.} draw samples from Gaussian distribution $N(0,1)$ as the values of vertices.
After contracting all vertices, we finally uniformly permute the tensor modes in random.
The permutations maintain unknown for all algorithms in the experiment.
\noindent\textbf{Experiment setup.}
In our method, we set the template $G_0$ as a cycle graph, the searching range for TN-ranks $R=7$, the maximum iteration $\#Iter=30$, the number of samples $\#Sample=60$, and the tuning parameters $c_1=0.9$ and $c_2=0.9, 0.94, 0.98$ for three different tensor orders $N=4,6,8$, respectively.
In the experiment, we also implement various TR decomposition methods with adaptive rank selection for comparison, including TR-SVD and TR-ALSAR~\citep{zhao2016tensor}, Bayes-TR~\citep{tao2020bayesian}, and TR-LM~\citep{mickelin2020algorithms}.
We also compare our method with two SOTAs for TN-SS, including ``Greedy''~\cite{hashemizadeh2020adaptive} and TNGA~\citep{li2020evolutionary}.
Note that the original TNGA is forced to search only the TN formats.
For a fair comparison, we extend it into two new versions: in ``TNGA$^\star$'' we trivially allow the method to search the formats and ranks simultaneously; and in ``TNGA+'' we use the classic ``random-key'' trick~\cite{bean1994genetic} to encode $\mathbb{G}_0$ into chromosomes, such that TNGA+ is capable of solving TN-PS as well but remaining the same genetic operations as TNGA.
In these two methods and ours, we set the function $\phi$ in~\eqref{eq:PTNPS} to be the compression ratio defined in~\cite{li2020evolutionary}, the parameter $\lambda=200$, and apply the ``Adam'' optimizer~\cite{kingma2014adam} with the same parameters in the evaluation phase of Alg.~\ref{alg:main}.
We force all methods to achieve $RSE\leq{}10^{-4}$ for each tensor, a pretty small approximation error, in the experiment.
Otherwise we say the approximation fails.
For performance evaluation, we use the \emph{Eff.} index~\citep{li2020evolutionary}, the ratio of parameter number of TNs between the searched structure and the one in data generation, as the main performance measure.
A larger value of \emph{Eff.} implies using fewer parameters to achieve the close approximation error.
For the sampling-based methods, we also report the total number of evaluations required to obtain the solution, shown as $[\#Eva.]$ in Table~\ref{tab:TR}, reflecting the computational cost for those methods.
\begin{table}[t]
\centering
\caption{
Experimental results on synthetic data in various TN formats.
}
\begin{threeparttable}\small\label{tab:TN}
\setlength{\tabcolsep}{1.5mm}{
\begin{tabular}{llcccc}
\toprule
\multirow{2}[0]{*}{\textbf{TNs}}&\multirow{2}[0]{*}{\textbf{Methods}}&\multicolumn{4}{c}{\textbf{Trial} -- \emph{Eff.$\uparrow$}~$\langle$\emph{formats preserved? {\color{green}Y} or {\color{red}N}.}$\rangle$ }\\
\cmidrule{3-6}&
& \textbf{A} & \textbf{B} & \textbf{C} & \textbf{D} \\
\midrule
\multirow{3}[0]{*}{\textbf{TTree}}
&TNGA+&1.29&1.17&1.11&1.55
\\
&TNLS&1.29&1.17&1.11&1.55
\\
&Greedy&1.29~$\langle${\color{red}N}$\rangle$&1.03~$\langle${\color{red}N}$\rangle$&0.79~$\langle${\color{red}N}$\rangle$&1.27~$\langle${\color{red}N}$\rangle$
\\
\cmidrule{2-6}
\multirow{3}[0]{*}{\textbf{PEPS}}
&TNGA+&1.14&1.00&1.00&1.21
\\
&TNLS&1.14&1.00&1.00&1.21
\\
&Greedy&1.03~$\langle${\color{red}N}$\rangle$&1.00~$\langle${\color{green}Y}$\rangle$&1.13~$\langle${\color{green}Y}$\rangle$&1.21~$\langle${\color{green}Y}$\rangle$
\\
\cmidrule{2-6}
\multirow{3}[0]{*}{\textbf{HT}}
&TNGA+&1.45&1.21&1.18&1.29
\\
&TNLS&1.45&1.21&1.18&1.29
\\
&Greedy&1.81~$\langle${\color{red}N}$\rangle$&1.91~$\langle${\color{red}N}$\rangle$&1.73~$\langle${\color{red}N}$\rangle$&2.16~$\langle${\color{red}N}$\rangle$
\\
\cmidrule{2-6}
\multirow{3}[0]{*}{\textbf{MERA}}
&TNGA+&0.95&1.32&2.30&1.00~
\\
&TNLS&1.09&1.88&2.88&1.03
\\
&Greedy&-&1.65~$\langle${\color{red}N}$\rangle$&0.78~$\langle${\color{red}N}$\rangle$&0.88~$\langle${\color{red}N}$\rangle$
\\
\bottomrule
\end{tabular}
}
\end{threeparttable}
\vskip -0.1in
\end{table}
\begin{table}[t]
\caption{Average RSE$\downarrow$ for \emph{image completion} by methods TRALS~\citep{wang2017efficient}, Greedy, TNGA$+$, and TNLS (ours), where the values in square brackets are the number of evaluations required by methods.
}
\vspace{0.3cm}
\centering
\begin{threeparttable}\centering\small
\begin{tabular}{ccccc}
\toprule
& \textbf{TRALS} & \textbf{Greedy} & \textbf{TNGA+} & \textbf{TNLS} \\
\midrule
$90\%$&0.192&0.186&\textbf{0.178}~[4929]&\textbf{0.178}~[\textbf{1544}]\\
$70\%$&0.142&0.142&\textbf{0.132}~[6086]&0.136~[\textbf{1437}]\\
\bottomrule
\end{tabular}
\end{threeparttable}\label{tab:completion}
\vskip -0.1in
\end{table}
\noindent\textbf{Results.}
The experimental results are reported in Table~\ref{tab:TR}, where the symbol ``$-$'' denotes that the method fails to achieve the condition $RSE\leq{}10^{-4}$, and the ``$\langle${\color{green}Yes} or {\color{red}No}$\rangle$'' denotes whether the structures obtained by methods preserve the TT/TR format or not.
As shown in Table~\ref{tab:TR}, TNGA+ and TNLS achieve the best \emph{Eff.} values, implying that they utilize the fewest parameters to achieve the approximation accuracy similar to others.
Most TR methods with rank selection obtain poor performance in this case, and the performance of the two TN-SS methods also becomes worse with increasing the tensor order.
We also see that ``Greedy'' and TNGA$^\star$ cannot guarantee the solution preserving the TR format, even though the data are generated with TR.
Last, we observe that TNLS costs significantly fewer evaluations than TNGA+ to achieve the same \emph{Eff.}.
To see this, we generate another two tensors of order-$8,12$, respectively, and employ TNGA+ and TNLS to search the optimal TR structures.
Figure~\ref{fig:convergence} shows the loss with varying the number of evaluations, averaged from five \textit{i.i.d.} runs.
We see that TNLS obtains consistently faster loss reduction than TNGA+ with the same number of evaluations.
As well as TT/TR, we also verify the effectiveness of our method in other TN formats, including TTree (order-7), PEPS~(order-6), hierarchical Tucker~(HT, order-6,~\citealt{hackbusch2009new}) and multi-scale entanglement renormalization ansatz~(MERA, order-8,~\citealt{cincio2008multiscale,reyes2020multi}), which are widely known in both machine learning and physics.
Being consistent with the results for TT/TR,
the results in Table~\ref{tab:TN} also demonstrate the effectiveness of our method.
\begin{table*}[t]
\caption{Average compression ratio$\uparrow$ ($\log$) and RSE$\downarrow$ (in parentheses) for \emph{image compression}, where ``Reshape'' and ``VDT'', denote two different tensorization operations, and the values in square brackets are the number of evaluations required by methods.}
\vspace{0.2cm}
\centering
\begin{threeparttable}
\centering\small
\begin{tabular}{lccccc}
\toprule
&\textbf{TR-SVD}& \textbf{TR-LM} & \textbf{Greedy} & \textbf{TNGA+}&\textbf{TNLS}\\
\midrule
Reshape&0.92~(0.15)&0.90~(0.14)&0.95~(0.15)&\textbf{1.36}~(\textbf{0.13})~[5700]&1.32~(0.14)~[\textbf{1876}]\\
VDT&1.10~(0.15)&1.07~(0.15)&0.91~(0.15)&1.28~(0.15)~[5700]&\textbf{1.30}~(0.15)~[\textbf{1546}]\\
\bottomrule
\end{tabular}\label{tab:compression}
\end{threeparttable}
\vskip -0.1in
\end{table*}
\begin{table*}[t]
\caption{Number of parameters ($\times{}1000$) and mean square error (MSE, in round brackets) for \emph{TGP model compression}, where CCPP, MG and Protein are three datasets, and the values in [square brackets] show the number of evaluations required in each method.}
\vspace{0.3cm}
\centering\small
\begin{threeparttable}\centering
\begin{tabular}{lccc}
\toprule
& \textbf{CCPP} & \textbf{MG} & \textbf{Protein} \\
\midrule
\textbf{TGP}~\cite{izmailov2018scalable}&2.64~(0.06)~[N/A]&3.36~(0.33)~[N/A]&2.88~(0.74)~[N/A]\\
\textbf{TNGA+}&\textbf{2.24}~(0.06)~[1500]&\textbf{3.01}~(0.33)~[4900]&2.03~(0.74)~[3900]\\
\textbf{TNLS}&\textbf{2.24}~(0.06)~[\textbf{1051}]&\textbf{3.01}~(0.33)~[\textbf{3901}]&\textbf{1.88}~(0.74)~[\textbf{3601}]\\
\bottomrule
\end{tabular}
\end{threeparttable}\label{tab:GP}
\vskip -0.1in
\end{table*}
\subsection{Real-World Data}
We now apply the proposed method to three benchmark problems on real-world data: \emph{image completion},
\emph{image compression},
and \emph{model compression} of tensorial Gaussian process~(TGP).
In these problems, TT/TR has been widely exploited~\cite{wang2017efficient,izmailov2018scalable,yuan2019tensor}.
The goal of the experiment is to verify whether imposing TN-PS can further improve their performance.
Note that in all benchmarks TNGA$+$ and TNLS are implemented in the TR format.
See Appendix for experimental details.
\noindent
\textbf{Image completion.}
TNLS is utilized to predict the missing values from natural images.
In the experiment, seven images from USC-SIPI~\cite{weber1997usc} are chosen, resized by $256\times{}256$, and then reshaped by visual data tensorization~(VDT)~\cite{latorre2005image,bengua2017efficient,yuan2019high} into a tensor of order-9.
After that, the entries are uniformly removed at random with the missing rate $\{70\%,90\%\}$, respectively.
The average of the prediction RSE is shown in Table~\ref{tab:completion}.
\noindent
\textbf{Image compression.}
Tensor decomposition methods are utilized to compress ten natural images randomly chosen from BSD500~\cite{arbelaez2010contour}.
In the data preparation phase, the images are grayscaled, resized by $256\times{}256$, and then reshaped into order-$8$ tensors by both the VDT and the trivial reshaping operations as done in \emph{Python}.
The results are shown in Table~\ref{tab:compression}, where we demonstrate the compression ratio~(in the form of logarithm base $10$) and the corresponding RSE.
For comparison, we implement the methods, including TR-SVD, TR-LM, Greedy, TNGA+, and TNLS, in the experiment.
\noindent
\textbf{Compressing TGP models.}
In this task, we consider compressing not data but parameters of a learning model.
To be specific, we compress the well-trained high-dimensional variational mean of TGP~\cite{izmailov2018scalable} by tensor decomposition.
We evaluate the performance using three datasets for the regression task, including CCPP~\cite{tufekci2014prediction}, MG~\cite{flake2002efficient}, and Protein~\cite{Dua2019}, for which we have the targeted tensors of the order-$\{4,6,9\}$, respectively.
Table~\ref{tab:GP} shows the number of the parameters ($\times{}1000$) after decomposition and the corresponding mean square error (MSE, in the round brackets) for each dataset.
\noindent
\textbf{Results.}
We can observe from the experimental results that TN-PS can boost the performance of the TR models in all tasks.
With the search of vertex permutations, \textit{i.e.}, the ``mode-vertex'' mappings, the expressive and generalization power of the TR models can be significantly improved.
Compared with TN-SS methods like Greedy, TN-PS takes more ``inductive bias'' modeled as the template.
As a consequence, imposing suitable ``inductive bias'' accelerates the searching process and helps avoid the loss in high-dimensional landscapes.
Compared with TNGA$+$, TNLS achieves similar performance in three tasks and costs significantly less number of evaluations.
This result is expected as we mentioned that the local sampling could leverage the more efficient ``steepest-descent'' path, which is not thought of in the GA-based methods.
\section{Concluding Remarks}
The experiential results demonstrate that TN-PS, a new variant of searching TN structures, can further improve the expressive power of TNs in various tasks.
The new searching algorithm TNLS is verified as being more efficient than existing sampling-based algorithms, with fewer evaluations and faster convergence.
Our theoretical results analyze how the symmetry of TN formats determines the number of all possible ``mode-vertex'' mappings, \textit{i.e.}, the counting property, proving that a universal bound on the counting property exists if the TN formats are sufficiently sparse.
We also establish the basic geometry of the search space for TN-PS.
By the graph isomorphism relation of the TN structures, we construct a semi-metric function and prove its corresponding neighborhood for the search space.
There results are applied as the theoretical foundation to the proposed sampling algorithm.
Taken together, TN-PS explores more efficient TN structures for tensor decomposition/completion/learning tasks, preserving the TN formats in contrast to the previous TN-SS problem.
\noindent\textbf{Limitation.}
One main limitation of our method is the higher running time compared with the greedy method~\cite{hashemizadeh2020adaptive} in searching.
A rigorous analysis about the smoothness of the landscape of TN-SS/PS also remains open.
Our code is available at~\url{https://github.com/ChaoLiAtRIKEN/TNLS}.
\section*{Acknowledgements}
We appreciate the anonymous (meta-)reviewers of ICML~2022 for helpful comments.
We are indebted to Chunmei for the help in proofreading the manuscript, Cichocki and Yokota for the insightful suggestions on the motivation, and Pan and Hashemizadeh for comments that improved the manuscript.
This work was partially supported by JSPS KAKENHI (Grant No. 20K19875, 20H04249, 20H04208) and the National Natural Science Foundation of China (Grant No. 62006045, 62071132).
Part of the computation was carried out at the Riken AIp Deep learning ENvironment~(RAIDEN).
|
1,108,101,563,258 | arxiv | \section{Introduction}
In multiple-class classification problems, the relative position among classes can be very complicated within the variable-space,
the situation becomes worse for data sets with lengthy variables.
Modern machine learning methods commonly outperform the conventional statistical classification models in these situations.
These methods, such as the support vector machine (SVM), may have good prediction power;
however, the classification rules of them are difficult to be explained.
There are also some methods build multiple-class classification rules baed on the divide-and-conquer concept via a bunch of binary classifiers, and thus
rely on certain voting schemes, which make model-interpretation more difficult \citep[see also][]{tomczak2015}.
Some hybrid methods, which depend on voting-like schemes, such as random forest methods, will also suffer from model-interpretation difficulty.
These nonparametric types of methods are preferred due to their flexibility, especially for problems with complicated data, such as image, and sound, where conventional parametric models are inferior.
However, their classification rules are less informative for the follow-up applications/studies, where the further details of classification rules, or ``model interpretation'' property is essential in addition to the conventional performance measures such as accuracy, precision, and so on.
Thus, the interpretation ability of classifiers catches a lot of attention \citep{Yang2010}, and also motivates this study.
For conventional classifier construction processes, when a data set has a large number of variables, we aim to find a short set of informative variables to build ``separating'' boundaries among classes for the ease of classification rules interpretation.
This task is crucial even in binary classification problems \citep[see][]{Liu2008, mechbal2015, Zhang2017},
and it becomes more difficult to achieve in multiple-class classification problems due to the complications among classes.
Moreover, if all the conditional probabilities, given each class label, of a sample are known, we can assign this sample to the class where its largest conditional probability.
This classification rule is called the Bayes rule, which has the smallest error rate \citep[see][]{ryzin1966}.
In \citet{ryzin1966}, he also showed that if we accurately estimate each class density, then the classification error rate approaches to that of the Bayes rule.
Since then, the probability-based classification rules have been extensively discussed in the literature
\citep{
DBLP:journals/tit/DevroyeW76,
DBLP:journals/prl/GreblickiP83,
Har-Peled:2002,
John95estimatingcontinuous}.
Although there are many different probability-based methods reported, the model interpretation ability is usually out of their scopes \citep{Ancukiewicz1998, hall2005, lugosi1996, Kobos2011, 796369}.
However, to find only one variable set for all classes in a multiple-class classification problem makes the estimation of the density of each class very difficult, since all classes have their own characters, and may not be described via the same set of variables.
Thus, how to retain prediction power, while to identify proper variables of each class for improving the model-interpretation ability for multiple-class classifiers is our goal in this paper.
\cite{shao2011} and many other researchers have pointed out that a classification rule could do as bad as random guessing without considering the sparsity condition, and complex structure in high-dimensional data sets.
\cite{fan2008} state that the difficulty of high-dimensional classification problems is intrinsically caused by too many noise variables useless for reducing classification error, which offer less benefits for decision-making, and increase complexity, and confusion in model-interpretation.
Moreover, they also pointed out that the ``important variables'' for estimating the conditional densities for individual classes can be different,
and this fact makes the model-interpretation of the probability-based methods even more difficult for multiple-class classification problems.
Thus, identifying the individual high impact variable set of each class plays a crucial role in both classification performance, and model-interpretation, especially
when the follow-up research requires the details of the classification rules instead of just the predicted labels.
We study multiple-class classification problems under sparsity condition, where each sample belongs to only one class.
Our goal is to assign new sample to the most appropriate class,
and learn about ``why this sample should be assigned to a particular class'' through the identified variable sets of individual classes.
As mentioned in \cite{fan2008}, the selected variable sets for individual classes are not necessarily the same, thus we study the method that can identify variable sets for individual classes.
We propose a variable detection process that can separately identify variable sets for estimating the density of individual classes,
and then integrate such a novel variable detection procedure into a nonparametric posterior kernel density classifier approach based on a density estimation method \citep{Sugiyama10a, Sugiyama12}. {\color{black} We call the proposed classifier NPKDC-vd, which is the short for ``nonparametric posterior kernel density classifier with variable detection.'' }
In the rest of this paper, we discuss the theoretical properties of the proposed method, and show that this method can asymptotically reach the Bayes rule.
Moreover, we describe its related algorithms including construction of classification rules, variable identification, and training sample size determination.
We explain how a general probability-based-model works in Section 2.1.
We also explain how we can just use the relevant variables for this task, and how to determine the number of relevant variables in Section 2.2.
In Section 2.3, we introduce the hypotheses testing ignorer to take the estimation variance into our procedure.
Then, in Section 2.4, we explain how we can decide the training sample sizes of each class.
The asymptotic property of the proposed method is in Section 2.5, and its proof is in the Supplementary.
We then illustrate our method with numerical studies with the synthesized and real data in Section 3 and 4, respectively, and then summarize our findings in the Summary section.
\section{Nonparametric Posterior Kernel Density Classifier with Variable Detection}
\label{sec:2}
\subsection{Nonparametric classifier}
We now describe the NPKDC-vd below.
Let $\mathbf{X}\subset \mathbb{R}^d$ be a $d$-dimensional vector of variables, and $Y \in \{1, 2, \dots , c\}$ be the class label, where $c$ is the number of classes. Suppose that we do not have any prior prevalence information of each class; i.e. assume that $P(y=Y)=1/c$, for $Y \in \{1, 2, \dots , c\}$.
Assume that $\mathbf{X} \times Y$ have a joint probability density function $f_{\mathbf{X}, Y}(\mathbf{x}, y)$, and $\mathbf{x} \in \mathbf{X}$ has a marginal density $f_{\mathbf{X}}(\mathbf{x})$, then the class-posterior probability is
$P(Y=y|\mathbf{X}=\mathbf{x}) ={f_{\mathbf{X}, Y}(\mathbf{x}, y)}/{f_{\mathbf{X}}(\mathbf{x})}$.
It implies that a sample $\mathbf{x}$ belong to Class ${y}$ with probability $P(Y=y|\mathbf{X}=\mathbf{x})$,
and a probability-based classification rule will assign it to Class $\hat{y}$, if $\hat{y} := \argmax_{y \in \{1, \dots, c\}} P(Y=y|\mathbf{X}=\mathbf{x})$.
To apply this idea we need to estimate the class-posterior probabilities,
which is a challenging problem from both computational and theoretical perspectives; especially when the number of class $c$ and/or the dimension $d$ is large.
Suppose that $\{\mathbf{x}_i, y_i)$, $i=1, \dots, n\}$, are i.i.d paired samples of size $n$.
Using the idea of {\color{black} the least-squares probabilistic classifier (LSPC)} with a nonparametric kernel density estimation of conditional probabilities \citep[see][]{Sugiyama10a, Sugiyama12},
we have the class-posterior probability of Class $y$ as a summation of kernel bases $K'$:
\begin{equation}
P(Y=y|\mathbf{X}=\mathbf{x}; \bm{\alpha}) = \sum_{y'=1}^c\sum_{i=1}^n\alpha_i^{y'}K'(\mathbf{x}, \mathbf{x}_i, y, y'),
\label{LSPC0}
\end{equation}
which has $c \times n$ parameters: $\bm{\alpha} = (\alpha_1^1, \dots, \alpha_n^1, \dots, \alpha_1^c, \dots, \alpha_n^c)^T$.
Let $\{\mathbf{x}_i^y, i = 1, \dots, n_y\}$ denote the samples in Class $y$ of size $n_y$.
Then via the delta kernel for $Y$ to separate $\mathbf{X}$ and $Y$, and a symmetric Gaussian kernel $\mathcal{K}$ on samples within the target class to reduce the number of kernels, we have the class-posterior probability for Class $y$ as follows:
\begin{equation}
P(Y=y|\mathbf{X}=\mathbf{x}; \bm{\beta^y}) := \sum_{i=1}^{n_y}\beta_i^y \mathcal{K}(\mathbf{x}, \mathbf{x}_i^y),
\label{LSPC1}
\end{equation}
where $\beta^y$, $y=1, \ldots, c$, is an unknown vector to be estimated.
Assume $\beta^y$, $y=1, \ldots, c$, are known, then assign sample $\mathbf{x}$ to Class $\hat{y}$, if
\begin{align*}
\hat{y} &:= \argmax_{y \in \{1, \dots, c\}} P(Y=y|\mathbf{X}=\mathbf{x}; \bm{\beta^y})
:= \argmax_{y \in \{1, \dots, c\}} \sum_{i=1}^{n_y}\beta_i^y \mathcal{K}(\mathbf{x}, \mathbf{x}_i^y).
\end{align*}
Note that \eqref{LSPC1} suggests that we can decompose the original problem into $c$ independent problems, and use only the training samples of each class to estimate its own posterior probability. Because we do not use the whole training samples for an individual class, the number of parameters is equal to $\sum_{y=1}^c n_y$ parameters, which is smaller than that in \eqref{LSPC0}. Thus, using this decomposition can cut down the computational cost.
Moreover, via such decomposition, we can carry out this learning process with a parallel computing facility, or a machine with multi-core/multi-thread CPUs; that is, we can learn $c$ models simultaneously and separately.
\subsection{Bandwidth determination}
The conventional LSPC uses a prefixed bandwidth for its Gaussian kernels for each class.
This setting is not suitable when the density of each class has its own key variables, which
is common in multiple-class classification problems \cite[see also][]{fan2008}.
We propose a bandwidth selection process and integrate it into \eqref{LSPC1}
such that the estimates via \eqref{LSPC1} can have different smoothing parameters for individual coordinates of its kernel density estimation.
Then the class-posterior density becomes
\begin{align}
P(Y=y|\mathbf{X}=\mathbf{x}; H_y) &:= \frac{1}{n_y} \sum_{i=1}^{n_y}\frac{1}{\det(H_y)}\mathcal{K}(H_y^{-1}(\mathbf{x} - \mathbf{x}_i^y))\nonumber\\
& := \frac{1}{n_y}\sum_{i=1}^{n_y} \prod_{j=1}^d\frac{1}{h_j^y}K\Big(\frac{x_j - x_{ij}^y}{h_j^y}\Big),
\label{NPKDC-vd}
\end{align}
where $H_y = diag(h_1^y, \dots, h_d^y)$ is a diagonal matrix and its element $h_j^y$ is the bandwidth of one-dimensional Gaussian kernel $K$ in the $j$th coordinate for Class $y$.
From \eqref{NPKDC-vd}, if ${h_j^y}$ is small, then its correspondent variable will have larger contributions to the posterior density.
For a probability-based classification rule, the samples in Class $y$ should have higher class-posterior probability for Class $y$.
It follows that variables with smaller bandwidth values (i.e. $h_j^y$'s in \eqref{NPKDC-vd}) are more important to the posterior density of Class $y$.
Then the classification rule of NPKDC-vd via \eqref{NPKDC-vd} becomes
\begin{align}
\hat{y} = \argmax_{y \in \{1, \dots, c\}} \frac{1}{n_y}\sum_{i=1}^{n_y} \prod_{j=1}^d\frac{1}{h_j^y}K\Big(\frac{x_j - x_{ij}^y}{h_j^y}\Big)\label{eq:BayesianNPKDC-vd}.
\end{align}
We then treat the bandwidths as some unknown parameters to be estimated with data, and will discuss its estimation procedure below.
\subsubsection{Greedy bandwidth estimation}
\cite{scott-sain} claim that the direct estimation of the full density by kernel methods is only feasible when the dimension of variable vector is no larger than 6.
There are many criteria reported in the literature for determining bandwidth. However, as far as we know, most of them consider all variables equally and ignore the effects of the variables with no or little impact \citep{LEIVAMURILLO2012, Gu2013}, such that their methods suffer from the curse of dimensionality due to too many redundant variables included in the estimating processes \cite[see also][]{fan2008}.
The computational cost is one of the major issues for deciding the bandwidth for each dimension of data sets with lengthy variables.
Thus, identifying high impact variable sets efficiently and effectively for density estimation of each class will not only largely improve the interpretation ability and classification performance, but reduce the computational cost.
Here we combine the idea of LSPC, and the regularization of derivative expectation operator (rodeo) \citep[see][]{pmlr-v2-liu07a} together in the proposed bandwidth selection procedure such that we can detect the relevant variables with faster convergence rate and lower computational cost.
Without loss of generality and for simplicity of discussion, we rearrange the order of variables in each Class $y$, $y=1, \ldots, c$, such that the first $ r_y$ variables, $X_j: j =1 \leqslant j \leqslant r_y$, are relevant high impact variables for the given class, and $X_j: j=r_y+1 \leqslant j \leqslant d$ are the rest irrelevant ones.
(Please note that this arrangement is only for the convenience of discussion, and is not necessary in practice.)
Let $R_y = \{j: 1 \leqslant j \leqslant r_y \}$ and $R_y^c=\{j:r_y+1 \leqslant j \leqslant d\}$ be two index sets, where $r_y$ is the number of the relevant variables for Class $y$, $y=1, \ldots, c$, and this number can be different among classes.
The details about determining the largest $r_y$ for the index set $R_y$ are in Section \ref{sec:feature-identification},
and Algorithm \ref{alg2} is for this purpose.
We may omit the subscript $y$ of $R_y$ and $r_y$ from our formulas in the following discussion when there is no ambiguity.
Assume, for the moment, that $r_y=r$ is the largest number of key variables for class density estimation, and the rest variables have little or no impact,
then the posterior density function of Class $y$ can be rewritten as
\begin{align}
P\big(Y=y|\mathbf{X} = \mathbf{x}; H_y\big) & = g_{Y|\mathbf{X}_R}\big(\mathbf{x}_R; H_y^R\big)u(\mathbf{x}_{R^c})
= g_{Y|\mathbf{X}_R}(\mathbf{x}_R, H_y^R)\nonumber,
\end{align}
where $u(\cdot)$ is a uniform function, $g_{Y|\mathbf{X}_R}$ depends only on the set $\mathbf{X}_R$,
and $H_y^R = diag(h_1^y, \dots, h_r^y)$ is an $r\times r$ submatrix of $H_y$.
Then \eqref{NPKDC-vd} becomes
\begin{align}
&P\big(Y=y|\mathbf{X} = \mathbf{x}; H_y\big)
= \frac{1}{n_y}\sum_{i=1}^{n_y} \prod_{j=1}^d\frac{1}{h_j^y}K\Big(\frac{x_j - x_{ij}^y}{h_j^y}\Big) \nonumber \\
&=\frac{1}{n_y}\sum_{i=1}^{n_y} \left[\prod_{j=1}^r\frac{1}{h_j^y}K\Big(\frac{x_j - x_{ij}^y}{h_j^y}\Big)\right] \left[\prod_{j=r+1}^d\frac{1}{h_j^y}K\Big(\frac{x_j - x_{ij}^y}{h_j^y}\Big)\right] \label{uniform}\\
& \approx \frac{1}{n_y}\sum_{i=1}^{n_y} \left[\prod_{j=1}^r\frac{1}{h_j^y}K\Big(\frac{x_j - x_{ij}^y}{h_j^y}\Big)\right]. \label{approx}
\end{align}
The second term of \eqref{uniform} follows a uniform distribution by assumption.
Hence, we can use a large bandwidth value on $h_j, j = r+1, \dots, d$, as in the greedily bandwidth selection approach,
to get a smooth kernel density function to estimate such a uniform function.
Thus, we can use \eqref{approx} to approximate \eqref{uniform}.
Let $\mathbf{x}=(x_i, \dots, x_d)^T$ be a sample of Class $y$.
Using \eqref{approx}, the estimated posterior density of $\mathbf{x}$ is
\begin{equation*}
P\big(Y=y|\mathbf{X} = \mathbf{x}; H_y\big) = \frac{1}{n_y}\sum_{i=1}^{n_y} \prod_{j=1}^d\frac{1}{h_j^y}K\Big(\frac{x_j - x_{ij}^y}{h_j^y}\Big).
\end{equation*}
To determine bandwidths, we start with a bandwidth matrix $H_y = \textit{diag}(h_0, \dots, h_0)$ with a large $h_0$ via the rodeo method,
and compute derivatives, for $ 1 \leqslant j \leqslant d$,
\begin{align}
Z_j &= \frac{\partial P\big(Y=y|\mathbf{X} = \mathbf{x}; H_y\big)}{\partial h_j^y}
= \frac{1}{n_y}\sum_{i=1}^{n_y} \frac{\partial}{\partial h_j^y}\left[\prod_{k=1}^d\frac{1}{h_k^y}K\Big(\frac{x_k - x_{ik}}{h_k^y}\Big)\right]\nonumber\\
&\equiv \frac{1}{n_y}\sum_{i=1}^{n_y} Z_{ji}.
\end{align}
If $K$ is the Gaussian kernel, then $Z_j$ becomes
\begin{align}
Z_j &= \frac{1}{n_y}\sum_{i=1}^{n_y} Z_{ji}
= \frac{1}{n_y}\sum_{i=1}^{n_y}\frac{(x_j - x_{ij})^2-(h_j^y)^2}{(h_j^y)^3} \prod_{k=1}^d\frac{1}{h_k^y}K\Big(\frac{x_k - x_{ik}}{h_k^y}\Big).
\end{align}
If the derivative $|Z_j|$ is small and the corresponding value of $h_j^y$ is relatively large, it suggests that the variable $X_j$ may be irrelevant in the current density estimation.
If $|Z_j|$ is large and to change $h_j^y$ will lead to a substantial change in its corresponding estimate, then we set the bandwidth equal to a smaller number
{$\gamma \times h_j^y$}, where {$\gamma \in (0, 1)$} is a small constant.
We repeat this procedure for each $j$, $j=1, \ldots, d$, and keep shrinking its corresponding bandwidth in discrete steps, such as {$1, \gamma, \gamma^2, \dots$}, until the value of $|Z_j|$ is less than a given threshold $\lambda_j$.
To take the estimation variance into consideration, we use the sample variance of $Z_{j}$s, say $s_j^2$, to estimate $\sigma_j^2 \equiv \mbox{Var}(Z_j)$.
Following the suggestion of \cite{pmlr-v2-liu07a} and
to balance variance against bias, we set a threshold $\lambda_j = s_j\sqrt{2\log(n_y c)}$ with $c = O(\log n_y)$.
Once we have the posterior densities for all classes, we use \eqref{eq:BayesianNPKDC-vd} as a rule to assign a class label for each sample.
Algorithm~\ref{alg1} describes how to estimate the posterior density estimate with bandwidth selection for each class.
(Note that users can use other kernel functions with these algorithms via some modifications.)
\subsection{Variable identification and training sample size determination}
\label{sec:feature-identification}
We use the estimated bandwidths obtained from Algorithm~\ref{alg1} as indices of the relevant level of variables,
which depends on the estimate of posterior density of each class, and therefore can be different among classes.
This novel variable determination method for constructing multiple-class classifiers can greatly improve
the model-interpretation ability of classification rules, and provide more information applications or planning further studies.
\subsection*{Identifying high impact variables}
We have the local bandwidths $\hat h_{j,i}^y$ of each variable $X_j$, $j = 1, \dots, d$, in each Class $y$ after applying Algorithm~\ref{alg1} to the training samples of each class.
As discussed before, if $X_j$ is a relevant variable, then it should have a smaller bandwidth compared with that of irrelevant variables in this class.
Because estimates of bandwidths depend on the samples, we take the variances of bandwidth estimates into consideration via adopting statistical hypotheses testing procedures to decide whether there are significant differences among the bandwidths of variables. We state this procedure below:
\noindent{\it Step 1:}
We first want to know whether all bandwidth means are equal, so the null and alternative hypotheses are
\begin{align}
\label{anova}
H_0 : ~& \bar h_1^y = \bar h_2^y = \dots = \bar h_d^y\\
H_1 : ~& \text{not all bandwidth means are equal},\nonumber
\end{align}
where $\bar h_j^y = \sum_{i = 1}^{n_y}\hat h_{j,i}^y/n_y$, $ j = 1, \dots, d$.
In our algorithm, all bandwidths start from the same initial value $h_0$.
We use a one-way analysis of variance (ANOVA), and reject the null hypothesis if
\begin{equation}
\label{Ftest}
F \equiv MS_B/MS_w > F_{\alpha, d-1, n_y\cdot d-d},
\end{equation}
where $MS_B$ is between-class mean square, $MS_w$ is within-class mean square, and $F_{\alpha, d-1, n_y \cdot d-d}$ is the upper $100 \cdot (1-\alpha)$th percentile of the F distribution with parameters $d-1$ and $n_y \cdot d-d$ degree of freedom.
\noindent{\it Step 2.}
Rejecting the null hypothesis in \eqref{anova} indicates that all bandwidth means are not the same. We then want to make sure which means are different via a post-hoc analysis through a multiple pairwise comparison:
\begin{align}
\label{Tukey}
H_0 : ~& \bar h_m^y = \bar h_n^y, ~~ m,n = 1, ~\dots, d, m \neq n\\
H_1 : ~& \bar h_m^y \neq \bar h_n^y.\nonumber
\end{align}
Because multiple testings are performed together, we adopt Tukey's honestly significant difference (Tukey's HSD) procedure, instead of $t$-test, based on the studentized range distribution \citep{Hochberg:1987:MCP:39892,10.2307/3001913}. Thus, the null hypothesis in \eqref{Tukey} is rejected if
\begin{equation}
\label{post}
|t_{mn}| = \frac{|\bar h_m^y -\bar h_n^y|}{\sqrt{MS_w(\frac{2}{n_y})}} > \frac{1}{\sqrt{2}}q_{\alpha,d,n_y \cdot d-d},
\end{equation}
where $q_{\alpha,d,n_y \cdot d-d}$ is the upper $100(1-\alpha)$th percentile of the studentized range distribution with parameters $d$ and $n_y \cdot d - d$ degree of freedom.
To identify relevant variables, we need to know exactly which variables have the bandwidth means which are significantly smaller than others,
To accelerate the testing process, we arrange the bandwidth means in an increasing order.
Denote the sorted bandwidth means as $\bar h_{1^*}^y \leq \bar h_{2^*}^y \leq \dots \leq \bar h_{d^*}^y$, then we aim to find
\begin{equation}
\label{vs}
o = \argmax_{m = 1,\dots, d} \Big\{|t_{m^*n^*}| = \frac{|\bar h_{m^*}^y -\bar h_{n^*}^y|}{\sqrt{MSE(\frac{2}{n_y})}} > \frac{1}{\sqrt{2}}q_{\alpha,d,n_y \cdot d-d}, \forall ~n > m\Big\}.
\end{equation}
Because after sorting, bandwidths $\bar h_{m^*}^y, ~m < o$ is smaller than $\bar h_{o^*}$, they are all significantly smaller than $\bar h_{n^*}, \forall n > o$.
It implies that the corresponding $\bar h_{o^*}$ is significantly smaller than $\bar h_{n^*}, \forall n > o$.
Thus, we choose the corresponding variables $X_{1^*}, \dots, X_{o^*}$ as the relevant variables in Class $y$ as follows:
\begin{equation}
j \in
\begin{cases}
R_y,& \text{if } \ \bar h_j^y \leqslant \bar h_{o^*}^y\\
R_y^c, & \text{otherwise}.
\end{cases}
\label{Eq:xr}
\end{equation}
We summarize this variable selection procedure as Algorithm~\ref{alg2}.
For convenience, we also put the classification and this class-specific variable determination together as Algorithm \ref{alg3}.
The proposed approach emphasizes more on ``understanding how each sample is assigned to a class,'' or ``these samples belong to a particular class due to which variables.''
This is, of course, different from the conventional model interpretation of linear or other parametric models.
However, this information is useful in some research/decision-making scenarios, such as medical/pharmaceutical studies, manufacturing, and so on.
Note that these algorithms are easily implemented with multi-core/multi-thread computers, or parallel computing facilities. Thus, they
will not dramatically increase the computational cost.
\begin{algorithm}[t]\fontsize{10}{8}\selectfont
\SetAlgoLined
\caption{Variable Selection for Individual Classes}
\label{alg2}
\KwData{
\begin{itemize}
\item $\mathbf{x}_i=(x_{i1}, \dots, x_{id})^T,
i = 1, \dots, n_y$: training data set of Class $y$
\end{itemize}
}
\KwIn{
\begin{itemize}
\item $\alpha$: significant level
\end{itemize}
}
\KwOut {
\begin{itemize}
\item High impact index set $R_y$
\end{itemize}
}
\emph{\textbf{Learning}}
\begin{enumerate}
\item \For{$i = 1, \dots, n_y$}{
Find the local bandwidths $\hat H_{y,i} = \text{diag}(\hat h_{1,i}^y, \dots, \hat h_{d,i}^y)$
for data point $\mathbf{x}_i$ by Algorithm~\ref{alg1}
}
\item Calculate bandwidth means $\bar h_j^y$, $j = 1, \dots, d$.
\item one-way ANOVA test for equality of all bandwidth means by \eqref{anova} and \eqref{Ftest}
\item Stop if null hypothesis \eqref{anova} is not rejected, otherwise continue
\item Tukey's HSD for Post-Hoc Analysis by \eqref{Tukey} and \eqref{post}
\item Find the largest mean $\bar h_{o*}^y$ that has significant different than all the larger ones by \eqref{vs}
\item Decide $R^y$ by \eqref{Eq:xr}.
\end{enumerate}
\end{algorithm}
\begin{algorithm}[t]\fontsize{10}{8}\selectfont
\SetAlgoLined
\caption{Integration of Classification and Variable Detection}
\label{alg3}
\KwData{
$\{\mathbf{x}^{y'}\}=\{\mathbf{x}^{y'}_{1}, \dots, \mathbf{x}^{y'}_{n_{y'}}\}$,
${y'}= 1, \dots, c$: training data set in $y'$th class; \\
\hspace{42pt}$\{\mathbf{x}_i, y_i\}$ : $i = 1, \dots, m$ : testing data\\
}
\KwIn{$0 < \gamma < 1$: reduce rate for bandwidth, \\
\hspace{42pt}$h_0 = c_0/\log\log n$: initial bandwidth, \\
\hspace{42pt}$\tau$: threshold}
\KwOut {Estimated label: $\hat y_i$, $i = 1, \dots, m$,\\
\hspace{52pt}Selected variables: $R^{y'}$, $y' = 1, \dots, c$
}
\emph{\textbf{Learning}}\\
\For{$y'=1, \dots, c$}{
Use training data set $\{\mathbf{x}^{y'}\}$\\
\For{$i = 1, \dots, m$}{
Estimate bandwidths $\hat H_{y',i}=(\hat h^{y'}_{i1}, \dots, \hat h^{y'}_{id})$ via Algorithm \ref{alg1} on testing data $\mathbf{x}_i$.\\
Calculate the estimated density $\hat p(y=y'|\mathbf{x}_i;\hat H_{y',i})$
}
}
\emph{Classification}\\
$\hat y_i = \argmax_{y'} \frac{\hat p(y=y'|\mathbf{x}_i;\hat H_{y',i})}{\sum_{y'=1}^c \hat p(y'|\mathbf{x}_i;\hat H_{y',i})}$, $i = 1, \dots, m$ \\
Accuracy $= \{\sum_{i=1}^m I(\hat y_i = y_i)\}/m$, where $I(\cdot)$ is an indicator function.
\emph{Variable Selection}\\
\For{$y'=1, \dots, c$}{
($\hat h^{y'}_1, \dots, \hat h^{y'}_d) = \frac{\sum_{i=1}^mH_{y',i}\times I(\hat y_i = y')}{\sum_{i=1}^mI(\hat y_i = y')}$\\
Apply Variable Selection algorithm for Class $y'$ and get $R_{y'}$
}
\end{algorithm}
\subsection{Training Sample Sizes of Individual classes}
In general, when training sample sizes among classes are highly imbalanced, we will usually obtain a biased classification rule with this training set.
On the other hand, a class with more relevant variables requires larger training sample size to estimate its density in our approach.
To use an equal sample size for all classes is a simple, but inefficient, approach. Thus, different classes require different training sample sizes, and
to have appropriate training sample sizes among classes is important for multiple-class classification problems.
To this end, we propose a two-step method for estimating the ``optimal'' sample sizes of individual classes based on Theorem \ref{thm1}, when a pilot study is allowed.
We summarize the procedure as Algorithm \ref{alg4} in Supplementary, for calculating training samples sizes.
Let $K$ be a $d$-dimensional bounded symmetric kernel (see also \eqref{eq:kernel}) satisfying
\begin{equation}
\int K(u)du = 1_d, \text{ and } \int uK(u)du = 0_d,
\end{equation}
and let $\nu$ and $\kappa$ be
\begin{equation}
\nu = \int_{\mathbb{R}^d} uu^TK(u)du, \text{ and } \kappa = \Big(\int_{\mathbb{R}^d} K^2(u)du \Big)^{1/2}.
\end{equation}
Suppose that there is a set of initial samples for all individual classes, then
by Theorem \ref{thm1} (see also \eqref{eq:errorrate3}), we have
\begin{align}
L_n - L^\star &\leqslant \sum_{y' = 1}^c \Bigg( \hat p_{y'}\int \Big| f_{y'}(\mathbf{x}) - \hat f_{y'}(\mathbf{x})\Big|dx + \int f_{y'}(\mathbf{x})\Big|p_{y'} - \hat p_{y'} \Big|dx \Bigg)\nonumber\\
&=\sum_{y' = 1}^c \Bigg(\mathcal{O}\Big(n_{y'} ^{-2/(4+r_{y'})}\Big)+ \mathcal{O}\Big(\sqrt{\log(\log(n))/n}\Big)\Bigg). \nonumber
\end{align}
It follows that the convergence rate of $ L_n - L^\star$ is asymptotically dominated by
\begin{equation}\label{eq:p}
\sum_{y' = 1}^c \hat p_{y'}\int \Big| f_{y'}(\mathbf{x}) - \hat f_{y'}(\mathbf{x})\Big|dx.
\end{equation}
Thus, for a given $\epsilon >0$, we have
\begin{equation}
\sum_{y' = 1}^c \frac{n_{y'}}{n}\Bigg(\frac{\nu}{2} \int \Big|\sum_{j=1}^{r_{y'}}k_j^{y'2}f_{y'}^{(jj)}\Big| + \kappa \Big(\prod_{j=1}^{r_{y'}}k_j^{y'}\Big)^{-1/2}\int \sqrt{f_{y'}}\Bigg)n_{y'}^{-\frac{2}{4+r_{y'}}} \leqslant \epsilon.
\label{eq:samplesize1}
\end{equation}
Equation \eqref{eq:samplesize1} implies that
\begin{equation}
\sum_{y' = 1}^c n_{y'}^{\frac{2+r_{y'}}{4+r_{y'}}}\Bigg(\frac{\nu}{2} \int \Big|\sum_{j=1}^{r_{y'}}k_j^{y'2}f_{y'}^{(jj)}\Big| + \kappa \Big(\prod_{j=1}^{r_{y'}}k_j^{y'}\Big)^{-1/2}\int \sqrt{f_{y'}}\Bigg) \leqslant n\epsilon.
\label{eq:samplesize2}
\end{equation}
Let $\vec{A} = (A_1, \dots, A_c)'$, and $\vec{B} = (B_1, \dots, B_c)'$ be the vectors with components
\begin{align}
A_{y'} = & n_{y'}^{(2+r_{y'})/(4+r_{y'})}, \text{ and }
\label{eq:A1}\\
B_{y'} = & \frac{\nu}{2} \int \Big|\sum_{j=1}^{r_{y'}}k_j^{y'2}f_{y'}^{(jj)}\Big| + \kappa \Big(\prod_{j=1}^{r_{y'}}k_j^{y'}\Big)^{-1/2}\int \sqrt{f_{y'}}, \;\; y' = 1 \dots, c,
\label{eq:B1}
\end{align}
respectively. Then, $\sum_{y' = 1}^c A_{y'} B_{y'} = \vec{A}\cdot\vec{B}\leqslant n \epsilon$
suffices to guarantee that $\eqref{eq:samplesize2} \leqslant n\epsilon$.
This inner product $\vec{A}\cdot\vec{B}$ reaches its maximum, when $\vec{A}$ and $\vec{B}$ are parallel.
Hence, for a given $\epsilon >0$, if individual sample sizes satisfying $A \propto B$ with the total sample size equal to $n$,
then the difference of error rates between the proposed procedure and Bayes rule is approximately less than $\epsilon$.
Since the true density functions are unknown, we use the estimated density functions instead.
Then, via Monte-Carlo integral method, we have the components of vectors $\vec A$ and $\vec B$ as follows:
\begin{align}
\hat A_{y'} = & n_{y'}^{(2+\hat r_{y'})/(4+\hat r_{y'})}
\label{eq:A2}\\
\hat B_{y'} = & \frac{\nu}{2} \sum_{\{\mathbf{x}^{\hat y'}\}} \frac{\Big|\sum_{j=1}^{\hat r_{y'}}k_j^{y'2}\hat f_{y'}^{(jj)}(\mathbf{x})\Big|}{\hat f_{y'}(\mathbf{x})} + \kappa \Big(\prod_{j=1}^{\hat r_{y'}}k_j^{y'}\Big)^{-1/2}\sum_{\{\mathbf{x}^{\hat y'}\}} \frac{\sqrt{\hat f_{y'}(\mathbf{x})}}{\hat f_{y'}(\mathbf{x})}.
\label{eq:B2}
\end{align}
Thus, with a given error bound, a pre-decided total sample size, and the information obtained from the first stage,
we can adopt a two-stage procedure to estimate the required sample size of each class.
Algorithm \ref{alg4} in Supplementary (see Section \ref{ss})
describes a two-stage approach based on the above discussion, which includes two steps: {\it estimation and resampling}.
From the estimation step (E-step), we have the estimated bandwidths for each class, and the estimated labels of samples.
Using the estimated density and predicted label, we then use the formula above to decide whether to include
more training samples in the resampling step (R-step), for the given error bound $\epsilon$ and total sample size $n$.
Because the precision of the estimated sample sizes depends on the initial samples of individual classes, a reasonable size of initial samples for each class is required.
\subsection{Asymptotic properties and Algorithm of classification rule}
Let $\hat f_{y'}(\mathbf{x};\hat H_{y'})$ be the estimated probability density function using rodeo method with the estimated bandwidth matrix $\hat H_{y'}=diag(\hat h_1^{y'},\dots,\hat h_d^{y'})$.
Then we have the following theorem, and the proof of it is in Supplementary \ref{Proof of Theorem 1}.
\begin{thm}\label{thm1}
Assume the sample size in Class $y'$ is $n_{y'}$, and the total sample size is equal to $n = \sum_{y'\in \mathcal{Y}}n_{y'}$.
Let $\hat p_{y'} = n_{y'}/n$ be the sample proportion, and $\hat f_{y'}(\mathbf{x}; \hat H_{y'})$ be the estimates via rodeo methiod.
Then the classification rule:
\begin{equation}
\hat y = \argmax_{y' \in \mathcal{Y}}\hat p_{y'}\hat f_{y'}(\mathbf{x};\hat H_{y'})
\label{AppBayes}
\end{equation}
approximates to the Bayes rule:
\begin{equation}
y = \argmax_{y' \in \mathcal{Y}}p_{y'}f_{y'}(\mathbf{x}).
\label{Bayes}
\end{equation}
Let $L^\star$ and $L_n$ denote the probability of error rates of \eqref{Bayes} (the Bayes rule), and \eqref{AppBayes}, respectively.
Then $L_n- L^*\rightarrow 0$ with probability one, as $\min_{y'=1, \ldots, c}\{n_{y'}\}$ goes to $\infty$.
\end{thm}
Algorithm \ref{alg1} describes the steps for conducting the NPKDC-vd -- short for the {\it nonparametric posterior kernel density classifier with the proposed bandwidth selection procedure}.
\begin{algorithm}[t]\fontsize{10}{8}\selectfont
\SetAlgoLined
\caption{Nonparametric Posterior Kernel Density Classifier with Variable Detection
\label{alg1}
\KwData{
\begin{itemize}
\item $\mathbf{x}_i=(x_{i1}, \dots, x_{id})^T$,
$i = 1, \dots, n_y$: training data set of Class $y$
\item $\mathbf{x}$: a point on which we want to find the posterior density estimator
\end{itemize}
}
\KwIn{
\begin{itemize}
\item \textcolor{black}{$0 < \gamma < 1$}: reduce rate for bandwidth
\item $h_0 = c_0/\log\log n_y$: initial bandwidth for some constant $c_0$
\item $c_n = O(\log n_y)$
\end{itemize}
}
\KwOut {
\begin{itemize}
\item Bandwidths $\hat H_y = \text{diag}(\hat h_1^y, \dots, \hat h_d^y)$
\item Posterior density estimator: $\hat P\big(Y=y|\mathbf{X} = \mathbf{x}; \hat H_y\big)$
\end{itemize}
}
\bf{Initialization}\\
$h_j^y= h_0$, $j = 1, \dots, d$\\
$\mathcal{A} = \{1, 2, \dots, d\}$\\
\vspace{0.1in}
\While{$\mathcal{A}$ is nonempty}{
\For{$j \in \mathcal{A}$}{
Estimate the derivative $Z_j$ and sample variance $s_j^2$.\\
Compute the threshold $\lambda_j = s_j\sqrt{2\log(n_y c_n)}$.\\
{If $|Z_j| > \lambda_j$, set $h_{j}^y \leftarrow \gamma h_{j}^y$;} otherwise remove $j$ from $\mathcal{A}$.
}
Calculate the estimated posterior probability with the estimated $\hat H_y :\hat p (Y=y|x,\hat H_y)$
}
\end{algorithm}
\section{Numerical Results: Synthesized Data}
\label{sec:7}
We use both synthetic and real datasets for illustration, and evaluate the performance based on both prediction power and variable detection ability.
We set {\color{black} $c_0=10$, $c_n = \log n_y$, and $\gamma = 0.9$ as the default parameters for our algorithms, and also used the default parameter of Gaussian kernel ion Matlab.}
The baselines for comparison are the results obtained from classifiers without variable detection.
To have a thorough assessment, we also use different numbers of classes, allocations among classes, and different training sample sizes of each class in the numerical studies.
We also apply the method of {\color{black} support vector machine (SVM)} to the same data sets for comparison purposes.
We train a SVM based multiple-class classifier via the multiple-class error-correcting output codes (ECOC) model \cite[][]{Bagheri2012, Berger99}, which is trained with a bunch of the SVM-based binary classifiers with standardized predictors and Gaussian kernel function (with parameter equal to 1), and assign the testing data to class with the highest class posterior probabilities. To illustrate the variable selection power of the proposed method and for comparison purposes, we adopt the method of {\color{black} ``SVM with sequential variable selection'' (SVMfs),} where variables are sequentially added to train a classification rule until there is no improvement in prediction \cite[see][for further details]{Bagheri2012, Berger99}.
SVMfs will only determine a subset, and the selected subset is shared with all classes, while our method allows each class to have its own subset of variables.
\subsection{Ten-class Example}
We generate a 10-class data set with 30 variables in this synthesized data set,
and among them, each class has its own 6 relevant variables, which are not all the same among classes.
{\color{black}The relevant variables in the $y$th class are: $\{y, y+1, \dots, y+5\}$,} which are generated with distributions
\begin{equation*}
X_i^y \sim \mathcal{N}(0.5, (0.02*(i-y+1))^2),\, \text{for } i = y : y+5,\, y = 1 : 10,
\end{equation*}
and the rest irrelevant variables are generated from
\begin{equation*}
X_i^y \sim \text{Uniform}(0, 1), \,\text{for } i \in \{1:30\}\setminus \{y:y+5\},\, y = 1 : 10.
\end{equation*}
{\color{black} The relevant variables of Class 1 are Variables 1 to 6, Class 2 has Variable 2 to 7 as its relevant variables, and so on.
That is, there are only some overlapping relevant variables among nearby classes.}
The training and testing sample sizes for each class are 150 and 100, respectively.
We repeat this study 1000 times and Table \ref{Table1} shows the results based on 1000 trials.
We found that when the data set has some irrelevant variables, the proposed NPKDC-vd method has better classification accuracy ($67.52\%$) compared with that of the conventional LSPC ($21.43\%$) at a price of computational time.
The average computational time of NPKDC-vd is around 100 seconds,
SVMfs use around 140 seconds, while LSPC, and SVM just take a few seconds.
The sequential variable selection of SVMfs is based on an iterative algorithm, and that is the reason why it takes so much time.
SVMfs perform better than SVM due to its variable selection procedure.
The proposed NPKDC-vd method has the highest accuracy in this example, which is much better than that of SVMfs.
As mentioned before, these SVM based methods rely on a bunch of SVM based binary classifiers.
Thus, it is difficult to ``explain'' their classification rules.
Figure \ref{Figure1} shows the box-plots of the mean predicted bandwidths using the training samples in each class based on 1000 replications.
In this plot, we can see that the mean bandwidths of the relevant variables are much smaller than those of the irrelevant ones,
and the relevant variables for each class are separately identified.
Table \ref{Table2} summaries the probabilities of relevant variables identified via hypotheses testing described in Algorithm \ref{alg2}.
We highlight those relevant variables of each class with gray background color in this table, and
the probabilities in those gray cells are all equal to 1, and the rest are equal to 0.
Hence, the NPKDC-vd can identify the relevant variables precisely -- both false positive or false negative rates are 0.
It also confirms the predicted bandwidth is good guidance for identifying relevant variables.
Table \ref{Table2_svm} is the variable identification probability of SVMfs. We can see that the detection probability is much lower than that of NPKDC-vd. Moreover, SVMfs only selects a set of variables for all classes, while NPKDC-vd can identify the relevant variables for each class with probability 1.
Thus, the NPKDC-vd method provides more precise information about each class.
Moreover, we can see from this table that the computation of the posterior probability estimation with rodeo method via decomposition mentioned before,
the increase in computational time is feasible (see NPKDC-vd and SVMfs in Table \ref{Table1}).
In fact, it can be shortened further with a parallel/distributed computation framework, since those estimates can be conducted separately.
\begin{table}
\begin{center}
\caption{ Classification results and computation time used in Example 1}
\label{Table1}
\begin{tabular}{ccc}\hline
& \multicolumn{2}{c}{Method} \\ \cline{2-3}
& NPKDC-vd & LSPC \\ \hline
Accuracy & 0.6752( 0.0167) & 0.2143( 0.0138) \\ \hline
Time/sec & 101.8789( 1.5503) & 2.3559( 0.0540) \\ \hline
\end{tabular}
\label{Table1_svm}
\begin{tabular}{ccc}\hline
& \multicolumn{2}{c}{Method} \\ \cline{2-3}
& SVM & SVM(with fs) \\ \hline
Accuracy & 0.1361( 0.0141) & 0.4655( 0.0690) \\ \hline
Time/sec & 3.7740( 0.1224) & 138.4383( 26.0866) \\ \hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\textwidth,height=0.8\textheight]{figure1_1.pdf}
\caption{The box-plots of mean predicted bandwidths for the 10-class example. }
\label{Figure1}
\end{center}
\end{figure}
\begin{table}\fontsize{10}{8}\selectfont
\caption{ The probabilities of relevant variables identified via NPKDC-vd for Example 1 based on 1000 iterations.}
\label{Table2}
\begin{center}
\scalebox{0.65}{
\begin{tabular}{c|cccccccccc}\hline
\diagbox{group~}{variable~~}& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \hline
1$\sim$10 & 0.1100\cellcolor[gray]{0.9} & 0.2000\cellcolor[gray]{0.9} & 0.2200\cellcolor[gray]{0.9} & 0.2900\cellcolor[gray]{0.9} & 0.3900\cellcolor[gray]{0.9} & 0.5000\cellcolor[gray]{0.9} & 0.4900\cellcolor[gray]{0.9} & 0.4100\cellcolor[gray]{0.9} & 0.4400\cellcolor[gray]{0.9} & 0.3400\cellcolor[gray]{0.9} \\
\hline \hline\diagbox{group~}{variable~~} & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ \hline \hline
1$\sim$10 & 0.2700\cellcolor[gray]{0.9} & 0.1300\cellcolor[gray]{0.9} & 0.0900\cellcolor[gray]{0.9} & 0.0300\cellcolor[gray]{0.9} & 0.0100\cellcolor[gray]{0.9} & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
\hline \hline\diagbox{group~}{variable~~} & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\ \hline \hline
1$\sim$10 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
\hline \hline
\end{tabular}
}
\end{center}
\end{table}
\begin{table
\caption{ The probabilities of relevant variables identified by SVM with sequential variable selection based on 1000 iterations, for data in Example 1.}
\label{Table2_svm}
\begin{center}
\scalebox{0.65}{
\begin{tabular}{c|cccccccccc}\hline
\diagbox{group~}{variable~~}& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \hline
1$\sim$10 & 0.1100\cellcolor[gray]{0.9} & 0.2000\cellcolor[gray]{0.9} & 0.2200\cellcolor[gray]{0.9} & 0.2900\cellcolor[gray]{0.9} & 0.3900\cellcolor[gray]{0.9} & 0.5000\cellcolor[gray]{0.9} & 0.4900\cellcolor[gray]{0.9} & 0.4100\cellcolor[gray]{0.9} & 0.4400\cellcolor[gray]{0.9} & 0.3400\cellcolor[gray]{0.9} \\
\hline \hline\diagbox{group~}{variable~~} & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ \hline \hline
1$\sim$10 & 0.2700\cellcolor[gray]{0.9} & 0.1300\cellcolor[gray]{0.9} & 0.0900\cellcolor[gray]{0.9} & 0.0300\cellcolor[gray]{0.9} & 0.0100\cellcolor[gray]{0.9} & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
\hline \hline\diagbox{group~}{variable~~} & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\ \hline \hline
1$\sim$10 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
\hline \hline
\end{tabular}
}
\end{center}
\end{table}
\subsection{Asymmetric Centers of Classes}
\label{Ex2}
We now study the case with a complicated relative positions among classes.
To illustrate this situation, we generate a 5-class data set with 10 variables, and only the first two variables are relevant in all classes.
We generate 8 irrelevant variables from a uniform distribution, and the other two relevant variables from \eqref{2-relevant} below:
\begin{equation}
N\bigg(\begin{bmatrix} \mu^y_1 \\
\mu^y_2 \end{bmatrix}, \begin{bmatrix} 0.1^2 & 0 \\ 0 & 0.2^2 \end{bmatrix}\bigg), y = 1:5,
\label{2-relevant}
\end{equation}
where $\mu^y_1$ and $\mu^y_2$ are location parameters.
Figure \ref{Figure2} shows the values of the class means in the space spanned by these two relevant variables.
It shows that Class 1 (i.e. $y=1$) is surrounded by the other 4 classes in this space with different distances between their means.
\begin{figure}[h]
\begin{center}
\includegraphics[width=12cm,]{Ex2.pdf}
\caption{The means of 5 classes with asymmetric positions the first and second variables.}
\label{Figure2}
\end{center}
\end{figure}
\begin{figure} [t]
\begin{center}
\includegraphics[width=0.7\textwidth]{Ex2_acc.pdf}
\caption{The maximum and minimum accuracies of two methods based on the data with asymmetric positions among classes.}
\label{Figure3}
\end{center}
\end{figure}
We conduct studies with different combinations of these 5 classes to see how the number of classes, and their relative positions affect classification performance.
{\color{black}The training and testing data sizes of each class are 150 and 100, respectively.
Table \ref{Table3} reports the classification accuracy based on 1000 replications of each study. }
We can see that the classification performance declines for all 4 methods when the number of classes increases,
since when the number of classes increases, the relations among classes are getting complicated such that the decision boundaries among classes are vague.
From this table, we can also see that the performances are not all the same among the studies with the same number of classes, which is due to the relative positions among classes being all different.
We can find a similar situation in Table \ref{Table3_svm},
which shows that the classification accuracy of SVM and SVMfs declines for all 4 methods when the number of classes increases.
Comparing these two tables, we found that SVMfs has the highest accuracy.
NPKDC-vd has slightly lower accuracy than that of SVMfs
In general, methods with variable selection perform better than their own counterparts.
Table~\ref{Table4} shows the variable detection probabilities,
and the probabilities of correctly selecting the first two relevant are equal to 1 for both NPKDC-vd and SVMfs.
(Note that the results for other cases are similar, and therefore are omitted.)
However, the false positive probabilities of NPKDC-vd are less than $1\%$,
while SVMfs has falsely selected probabilities of irrelevant variables larger than $10\%$ in all multiple-class setups in this example.
These results show that using estimated bandwidths for variable selection is promising.
It confirms that NPKDC-vd can retain both satisfactory classification accuracy,
and reveal the information of relevant variables of individual classes, which improves the interpretation ability of the corresponding classification rules.
Figure \ref{Figure3} is a plot of the maximum and minimum accuracies of NPKDC-vd and LSPC, which shows that NPKDC-vd has higher accuracy than that of LSPC for all cases in this example.
The maximum accuracy declines as the number of classes increases for both methods .
However, the minimum accuracy of NPKDC-vd retains at the level of accuracy (around 0.77) as the number of classes increases,
while that of LSPC drops from 0.75 to 0.70.
\begin{table} [t]
\begin{center}
\caption{ Classification accuracy on different combinations of classes in example with special located between classes in Section \ref{Ex2}. The training size = 150, and testing size = 100 in each class. Results are based on 1000 repeated trials.}
\label{Table3}
\scalebox{0.63}{
\begin{tabular}{c|c|c||c|c|c}\hline
\# of classes & \multicolumn{2}{c||}{2} & \# of classes & \multicolumn{2}{c}{3} \\ \hline
\diagbox{classes~}{Method~~} & NPKDC-vd& LSPC & \diagbox{classes~}{Method~~} & NPKDC-vd & LSPC \\ \hline
1 2 & 0.7793( 0.0287) & 0.7520( 0.0291) & 1 2 3 & 0.7784( 0.0232) & 0.7225( 0.0264) \\ \hline
1 3 & 0.8813( 0.0245) & 0.8315( 0.0300) & 1 2 4 & 0.7847( 0.0229) & 0.7132( 0.0259) \\ \hline
1 4 & 0.9004( 0.0211) & 0.8876( 0.0218) & 1 2 5 & 0.7898( 0.0246) & 0.7602( 0.0244) \\ \hline
1 5 & 0.9613( 0.0150) & 0.9431( 0.0185) & 1 3 4 & 0.8232( 0.0207) & 0.7993( 0.0229) \\ \hline
2 3 & 0.9676( 0.0118) & 0.9605( 0.0133) & 1 3 5 & 0.8945( 0.0174) & 0.7804( 0.0278) \\ \hline
2 4 & 0.9877( 0.0079) & 0.9720( 0.0131) & 1 4 5 & 0.9095( 0.0166) & 0.8672( 0.0177) \\ \hline
2 5 & 0.9023( 0.0244) & 0.8817( 0.0225) & 2 3 4 & 0.9074( 0.0163) & 0.9002( 0.0158) \\ \hline
3 4 & 0.8997( 0.0209) & 0.9070( 0.0209) & 2 3 5 & 0.9130( 0.0160) & 0.8570( 0.0194) \\ \hline
3 5 & 0.9950( 0.0046) & 0.9953( 0.0043) & 2 4 5 & 0.9270( 0.0138) & 0.9120( 0.0143) \\ \hline
4 5 & 0.9932( 0.0063) & 0.9929( 0.0063) & 3 4 5 & 0.9259( 0.0138) & 0.9248( 0.0145) \\ \hline
\multicolumn{6}{c}{ } \\ \hline
\# of classes & \multicolumn{2}{c||}{4} & \# of classes & \multicolumn{2}{c}{5} \\ \hline
\diagbox{classes~}{Method~~} & NPKDC-vd & LSPC & \diagbox{classes~}{Method~~} & NPKDC-vd & LSPC \\ \hline
1 2 3 4 & 0.7636( 0.0223) & 0.7022( 0.0203) & 1 2 3 4 5 & 0.7728( 0.0211) & 0.6987( 0.0201) \\ \hline
1 2 3 5 & 0.7901( 0.0204) & 0.6924( 0.0226) & & &\\ \hline
1 2 4 5 & 0.7948( 0.0182) & 0.7314( 0.0202) & & &\\ \hline
1 3 4 5 & 0.8498( 0.0166) & 0.7636( 0.0177) & & &\\ \hline
2 3 4 5 & 0.8810( 0.0167) & 0.8486( 0.0192) & & &\\ \hline
\end{tabular}
}
\end{center}
\end{table}
\begin{table} [t]
\begin{center}
\caption{ Classification accuracy for different combinations of classes in Section \ref{Ex2}.
Results are based on 1000 replications, and training and testing sizes are equal to 150, and 100, respectively, in each class.
SVM and SVMfs denote the SVM methods with/without variable selection preprocess.}
\label{Table3_svm}
\scalebox{0.63}{
\begin{tabular}{c|c|c||c|c|c}\hline
\# of groups & \multicolumn{2}{c||}{2} & \# of groups & \multicolumn{2}{c}{3} \\ \hline
\diagbox{combinations~}{Method~~} & SVM & SVMfs& \diagbox{combinations~}{Method~~} & SVM & SVMfs \\ \hline
1 2 & 0.7981( 0.0301) & 0.8843( 0.0190) & 1 2 3 & 0.7725( 0.0264) & 0.8869( 0.0150) \\ \hline
1 3 & 0.8548( 0.0282) & 0.9470( 0.0149) & 1 2 4 & 0.8083( 0.0207) & 0.8954( 0.0159) \\ \hline
1 4 & 0.8962( 0.0223) & 0.9617( 0.0102) & 1 2 5 & 0.8036( 0.0226) & 0.8977( 0.0145) \\ \hline
1 5 & 0.9369( 0.0185) & 0.9900( 0.0051) & 1 3 4 & 0.8044( 0.0247) & 0.9178( 0.0156) \\ \hline
2 3 & 0.9347( 0.0220) & 0.9958( 0.0036) & 1 3 5 & 0.8683( 0.0168) & 0.9551( 0.0103) \\ \hline
2 4 & 0.9793( 0.0116) & 0.9974( 0.0032) & 1 4 5 & 0.8957( 0.0186) & 0.9666( 0.0095) \\ \hline
2 5 & 0.9052( 0.0254) & 0.9610( 0.0108) & 2 3 4 & 0.8822( 0.0219) & 0.9615( 0.0089) \\ \hline
3 4 & 0.8508( 0.0300) & 0.9486( 0.0141) & 2 3 5 & 0.9046( 0.0157) & 0.9702( 0.0093) \\ \hline
3 5 & 0.9735( 0.0156) & 0.9980( 0.0030) & 2 4 5 & 0.9167( 0.0156) & 0.9731( 0.0067) \\ \hline
4 5 & 0.9621( 0.0169) & 0.9982( 0.0030) & 3 4 5 & 0.9022( 0.0137) & 0.9651( 0.0082) \\ \hline
\multicolumn{6}{c}{ } \\ \hline
\# of groups & \multicolumn{2}{c||}{4} & \# of groups & \multicolumn{2}{c}{5} \\ \hline
\diagbox{combinations~}{Method~~} & SVM & SVMfs & \diagbox{combinations~}{Method~~} & SVM & SVMfs \\ \hline
1 2 3 4 & 0.7606( 0.0203) & 0.8814( 0.0172) & 1 2 3 4 5 & 0.7806( 0.0163) & 0.8871( 0.0149) \\ \hline
1 2 3 5 & 0.7881( 0.0199) & 0.8944( 0.0153) & & &\\ \hline
1 2 4 5 & 0.8086( 0.0186) & 0.9034( 0.0133) & & &\\ \hline
1 3 4 5 & 0.8295( 0.0186) & 0.9325( 0.0109) & & &\\ \hline
2 3 4 5 & 0.8720( 0.0161) & 0.9496( 0.0096) & & &\\ \hline
\end{tabular}
}
\end{center}
\end{table}
\begin{table} [t]
\caption{The probabilities of relevant variable identification, based on 1000 iterations, for the example with asymmetric positions among classes of NPKDC-vd and SVM with sequential variable selection.}
\label{Table4_all}
\begin{subtable}{0.9\linewidth}\tiny
\centering
\caption{Probabilities of the relevant variables identified via NPKDC-vd}
\begin{tabular}{cc|>{\columncolor[gray]{0.9}}c>{\columncolor[gray]{0.9}}ccccccccc}\hline
\multicolumn{2}{c|}{\diagbox{\# of groups/\\group}{variables}} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
\multirow{2}{*}{2} & 1 & 1.00 & 1.00 & 0.01 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\
& 2 & 1.00 & 1.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
\hline
\multirow{3}{*}{3} & 1 & 1.00 & 1.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\
& 2 & 1.00 & 1.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.01\\ & 3 & 1.00 & 1.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
\hline
\multirow{4}{*}{4} & 1 & 1.00 & 1.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\
& 2 & 1.00 & 1.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
& 3 & 1.00 & 1.00 & 0.01 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
& 4 & 1.00 & 1.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
\hline
\multirow{5}{*}{5} & 1 & 1.00 & 1.00 & 0.00 & 0.00 & 0.01 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\
& 2 & 1.00 & 1.00 & 0.00 & 0.00 & 0.01 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
& 3 & 1.00 & 1.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
& 4 & 1.00 & 1.00 & 0.00 & 0.00 & 0.01 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
& 5 & 1.00 & 1.00 & 0.00 & 0.01 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
\hline
\end{tabular}
\label{Table4}
\caption{Probabilities of the relevant variables identified via SVM with variable selection preprocess}
\label{Table4_svm}
\begin{tabular}{cc|>{\columncolor[gray]{0.9}}c>{\columncolor[gray]{0.9}}ccccccccc}\hline
\multicolumn{2}{c|}{\diagbox{\# of groups/\\group}{variables}} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
2 & 1$\sim$ 2 & 1.00 & 1.00 & 0.16 & 0.17 & 0.14 & 0.16 & 0.21 & 0.20 & 0.21 & 0.19 \\
\hline
3 & 1$\sim$ 3 & 1.00 & 1.00 & 0.23 & 0.21 & 0.23 & 0.22 & 0.20 & 0.25 & 0.18 & 0.09 \\
\hline
4 & 1$\sim$ 4 & 1.00 & 1.00 & 0.10 & 0.15 & 0.23 & 0.17 & 0.20 & 0.18 & 0.22 & 0.15 \\
\hline 5 & 1$\sim$ 5 & 1.00 & 1.00 & 0.21 & 0.16 & 0.27 & 0.14 & 0.18 & 0.20 & 0.22 & 0.12 \\
\hline
\end{tabular}
\end{subtable}
\end{table}
\subsection{Training samples sizes and number of classes}
To illustrate how training data sizes affect the performance of NPKDC-vd and LSPC classifiers,
we generate data sets under the same setup as in Section \ref{Ex2}, but with different sizes of training sets,
where class sizes are 50, 150, 500, and 1000, and the testing sample size is 100 for each class.
We consider classification problems with 2 to 5 classes.
Because the results are similar, when for the case with the same number of classes,
we only report the first one for each setting as their representatives.
(For example, for two-class cases, we only report the results of Class 1 versus Class 2 , and omit the other two-class cases.)
Table \ref{Table6} summarizes the accuracy of this study, and Figure \ref{Figure4} shows the corresponding plots.
The accuracy of both methods increases when the training size increases, and
the differences between the two methods become smaller when the training sample size gets larger.
When the training sample size is small, NPKDC-vd clearly outperforms LSPC.
It is worth noting that the NPKDC-vd outperforms LSPC in all cases with a number of classes larger than 2,
and maintains a stable accuracy even when the training sample sizes are small.
The LSPC can only be better in binary classification cases with more than 500 training samples.
Besides these cases, the NPKDC-vd has about $10\%$ higher accuracy than that of the LSPC.
Table \ref{Table8} lists the probabilities of detecting the two relevant variables for all different sample-size cases.
When the training sample size is no less than 150 for each class, the relevant variable detection rate of NPKDC-vd is 100\% in this example.
When there are only 50 training samples in each class, the detection rates for the second variable are less than 1 in these studies.
Please note that in this table, we simply use variables 1 and 2 to denote the first and second relevant variables of each class, and they are not the same among classes.
\begin{table} [t]
\begin{center}
\caption{Classification accuracies of NPKDC-vd and LSPC based 1000 trails with different training examples sizes.}
\label{Table6}
\scalebox{0.6}{
\begin{tabular}{c|cccc|cccc}\hline
& \multicolumn{8}{c}{Methods} \\
& \multicolumn{4}{c}{NPKDC} & \multicolumn{4}{c}{LSPC} \\ \hline \hline
\diagbox{\# of training~}{\# of groups~~} & 2 & 3 & 4 & 5 & 2 & 3 & 4 & 5 \\ \hline
\multirow{2}{*}{\bf 50} & \cellcolor[gray]{0.9} 0.7505 & \cellcolor[gray]{0.9} 0.7398 & \cellcolor[gray]{0.9} 0.7141 & \cellcolor[gray]{0.9} 0.7371 & \cellcolor[gray]{0.9} 0.6603 & \cellcolor[gray]{0.9} 0.5987 & \cellcolor[gray]{0.9} 0.5917 & \cellcolor[gray]{0.9} 0.6215 \\
& \cellcolor[gray]{0.9} ( 0.0373)& \cellcolor[gray]{0.9}( 0.0282)& \cellcolor[gray]{0.9}( 0.0274)& \cellcolor[gray]{0.9} ( 0.0240)& \cellcolor[gray]{0.9} ( 0.0421)& \cellcolor[gray]{0.9} ( 0.0354)& \cellcolor[gray]{0.9} ( 0.0323)& \cellcolor[gray]{0.9}( 0.0304)\\\hline
\multirow{2}{*}{150} & 0.7685 & 0.7601 & 0.7480 & 0.7726 & 0.7396 & 0.7143 & 0.6899 & 0.7079 \\
& ( 0.0305)& ( 0.0255)& ( 0.0212)& ( 0.0198)& ( 0.0325)& ( 0.0245)& ( 0.0203)& ( 0.0184)\\\hline
\multirow{2}{*}{500} & 0.7850 & 0.7805 & 0.7562 & 0.7854 & 0.7927 & 0.7532 & 0.7210 & 0.7342 \\
& ( 0.0296)& ( 0.0220)& ( 0.0191)& ( 0.0171)& ( 0.0286)& ( 0.0230)& ( 0.0186)& ( 0.0164)\\\hline
\multirow{2}{*}{1000} & 0.7857 & 0.7784 & 0.7596 & 0.7870 & 0.8052 & 0.7527 & 0.7299 & 0.7485 \\
& ( 0.0282)& ( 0.0233)& ( 0.0222)& ( 0.0177)& ( 0.0252)& ( 0.0228)& ( 0.0188)& ( 0.0193)\\\hline
\end{tabular}
}
\end{center}
\end{table}
\begin{table}[h]\footnotesize
\begin{center}
\caption{The relevant variable identification probabilities of the NPKDC-vd with different training sample sizes and number of groups based on 1000 iterations.}
\label{Table8}
\scalebox{0.8}{
\begin{tabular}{cc||cc|cc|cc|cc}\hline
& & \multicolumn{8}{c}{\# of training samples} \\
& & \multicolumn{2}{c}{50} & \multicolumn{2}{c}{150} & \multicolumn{2}{c}{500} & \multicolumn{2}{c}{1000} \\
\hline
\multicolumn{2}{c||}{\diagbox{\# of groups/\\group}{variables\\}} & 1 & 2 & 1 & 2 & 1 & 2 & 1 & 2 \\
\hline
\multirow{2}{*}{2} & 1 & 1.00 & 0.85 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00\\& 2 & 1.00 & 0.88 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00\\
\hline
\multirow{3}{*}{3} & 1 & 1.00 & 0.88 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00\\ & 2 & 1.00 & 0.94 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00\\ & 3 & 1.00 & 0.77 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00\\
\hline
\multirow{4}{*}{4} & 1 & 1.00 & 0.90 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00\\ & 2 & 1.00 & 0.88 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00\\ & 3 & 1.00 & 0.86 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00\\ & 4 & 1.00 & 0.87 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00\\
\hline
\multirow{5}{*}{5} & 1 & 1.00 & 0.86 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00\\ & 2 & 1.00 & 0.83 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00\\ & 3 & 1.00 & 0.79 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00\\ & 4 & 1.00 & 0.86 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00\\ & 5 & 1.00 & 0.73 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00\\
\hline
\end{tabular}
}
\end{center}
Note that the variable 1, and 2 here just denote the first and second relevant variables of each class, and they are not the same among classes.
\end{table}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\textwidth]{Ex3_acc.pdf}
\caption{The accuracy rates of two methods.}
\label{Figure4}
\end{center}
\end{figure}
\section{Real Data Examples}
\subsection{Anuran Species Classification}
Anuran calls data set was used in several classification tasks related to the challenge of anuran species recognition through their calls \citep[see][]{Dua:2017},
and was created by segmenting 60 audio records belonging to 4 different families, 8 genus, and 10 species, and each audio corresponds to one specimen.
There are 7195 syllables identified from the 60 bioacoustics signals after segmenting, and each syllable is represented by a set of variables extracted by Mel-Frequency Spectral Coefficients (MFCCs), which performs a spectral analysis based on a triangular filter-bank logarithmically spaced in the frequency domain.
Therefore, each sample in this data set is denoted by a variable set of MFCCS coefficients belonging to a special species.
We focus on classification of the main 7 out of 10 species: Leptodactylus fuscus, Adenomera andreae, Adenomera hylaedactyla, Hyla minuta, Hypsiboas cinerascens, Hypsiboas cordobae, and Ameerega trivittata.
In addition to the original dataset, we add 5 normally distributed noise attributes, with mean 0 and variance 1, as the irrelevant variables.
{\color{black}In each run, we randomly select 100 and 50 examples from each species as the training and testing (evaluation) data, respectively,
and report the results are based on 100 replications.}
Figure \ref{fig:frogs2} shows the box-plots of the mean predicted bandwidths with noise variables based on 100 trials,
where the bandwidths of the last 5 added noise variables remain large, and the other bandwidths shrink as expected.
This result indicates that the last 5 variables are irrelevant, which is consistent with our setup,
and the proposed method successfully detects the relevant variables in this example.
Table \ref{tb:frogs1} summarizes the classification performance of the proposed NPKDC-vd method
with both the original data and the data with 5 additional irrelevant variables.
The results with the original data are used as our comparison baseline.
Note that there are 3 popularly used performance indexes: accuracy, specificity, and precision, in this table.
The results of NPKDC-vd show about $3\%$ drop in accuracy and precision when there are irrelevant variables in the data set,
and specificity remains similar.
The differences in results of NPKDC-vd between these two data sets are not statistically significant.
Table \ref{SVMExAnuran} shows results of SVM and SVMfs under a similar setup.
If we are only interested in classification accuracy, then SVMfs is the best choice in this example.
However, SVMfs is built on multiple binary classifiers, and cannot have good interpretation ability.
Thus, it provides less information about each class.
We summarize the means of Z-scores of the predicted bandwidths for all variables in Table \ref{Table:Frog_Z}.
Most of the variables are identified. However, we found that besides the added irrelevant variables, some original variables such as Variable 1 and 2 in Hypsiboas Cordobae, and Ameerega Trivittata species are also regarded as irrelevant, which are new findings, and were not reported in the literature before.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.8\textwidth]{frog2.pdf}
\caption{The box-plots of mean predicted bandwidths for anuran calls.}
\label{fig:frogs2}
\end{center}
\end{figure}
\begin{table}\fontsize{8}{7}\selectfont
\caption{Classification results of anuran species.}
\label{tb:frogs1-all}
\begin{subtable}{0.9\linewidth}
\centering
\caption{Classification results of NPKDC-vd}
\label{tb:frogs1}
\begin{tabular}{l|r|r|r|r|r|r}
\hline
& \multicolumn{3} {c|}{Original dataset} & \multicolumn{3} {c}{Extended dataset}\\
& \multicolumn{3} {c|}{(without noise variables)} & \multicolumn{3} {c}{(with noise variables)}\\ \hline
& Accuracy & Precision & Specificity & Accuracy & Precision & Specificity\\ \hline
mean & 0.9155 & 0.9173 & 0.9859 & 0.8741 & 0.8807 & 0.9790\\
Std & 0.0124 & 0.0019 & 0.0021 & 0.0142 & 0.0135 & 0.0024\\ \hline
\end{tabular}
\caption{Classification results of SVM}
\label{SVMExAnuran}
\scalebox{0.9}{
\begin{tabular}{l|r|r|r|r|r|r}
\hline
& \multicolumn{3} {c|}{Original dataset} & \multicolumn{3} {c}{Extended dataset}\\
& \multicolumn{3} {c|}{(without noise features)} & \multicolumn{3} {c}{(with noise features)}\\ \hline
& Accuracy & Precision & Specificity & Accuracy & Precision & Specificity\\ \hline
mean & 0.7606 & 0.9071 & 0.9734 & 0.7488 & 0.9081 & 0.9721\\
Std & 0.0806 & 0.0118 & 0.0090 & 0.0895 & 0.0132 & 0.0099\\ \hline
\end{tabular}
}
\end{subtable}
\end{table}
\begin{table}[t]\fontsize{8}{7}\selectfont
\begin{center}
\caption{Anuran Calls: The means of Z-scores of the mean predicted bandwidths for all attributes, where the cells with gray background are {\bf relevant} variables.}
\label{Table:Frog_Z}
\begin{tabular}{l|ccccccccc}\hline
\diagbox{Species~}{variable~~}& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \hline
Leptodactylus Fuscus & -0.14\cellcolor[gray]{0.9} & -0.27\cellcolor[gray]{0.9} & -0.38\cellcolor[gray]{0.9} & -0.44\cellcolor[gray]{0.9} & -0.37\cellcolor[gray]{0.9} & -0.47\cellcolor[gray]{0.9} & -0.43\cellcolor[gray]{0.9} & -0.57\cellcolor[gray]{0.9} & -0.44\cellcolor[gray]{0.9} \\
Adenomera Andreae & 0.53 & -0.04\cellcolor[gray]{0.9} & -0.33\cellcolor[gray]{0.9} & -0.39\cellcolor[gray]{0.9} & -0.5\cellcolor[gray]{0.9}9 & -0.59\cellcolor[gray]{0.9} & -0.45\cellcolor[gray]{0.9} & -0.59\cellcolor[gray]{0.9} & -0.55\cellcolor[gray]{0.9} \\
Adenomera Hylaedactyla & -0.24\cellcolor[gray]{0.9} & -0.27\cellcolor[gray]{0.9} & -0.28\cellcolor[gray]{0.9} & -0.29\cellcolor[gray]{0.9} & -0.53\cellcolor[gray]{0.9} & -0.24\cellcolor[gray]{0.9} & -0.48\cellcolor[gray]{0.9} & -0.46\cellcolor[gray]{0.9} & -0.55\cellcolor[gray]{0.9} \\
Hyla Minuta & 0.80 & -0.10\cellcolor[gray]{0.9} & 0.01 & -0.43\cellcolor[gray]{0.9} & -0.64\cellcolor[gray]{0.9} & -0.58\cellcolor[gray]{0.9} & -0.62\cellcolor[gray]{0.9} & -0.51\cellcolor[gray]{0.9} & -0.60\cellcolor[gray]{0.9} \\
Hypsiboas Cinerascens & -0.11\cellcolor[gray]{0.9} & 0.06 & -0.31\cellcolor[gray]{0.9} & -0.40\cellcolor[gray]{0.9} & -0.42\cellcolor[gray]{0.9} & -0.55\cellcolor[gray]{0.9} & -0.23\cellcolor[gray]{0.9} & -0.57\cellcolor[gray]{0.9} & -0.40\cellcolor[gray]{0.9} \\
Hypsiboas Cordobae & 0.29 & 0.53 & -0.26\cellcolor[gray]{0.9} & -0.35\cellcolor[gray]{0.9} & -0.44\cellcolor[gray]{0.9} & -0.57\cellcolor[gray]{0.9} & -0.40\cellcolor[gray]{0.9} & -0.56\cellcolor[gray]{0.9} & -0.58\cellcolor[gray]{0.9} \\
Ameerega Trivittata & 0.08 & 0.20 & -0.28\cellcolor[gray]{0.9} & -0.40\cellcolor[gray]{0.9} & -0.50\cellcolor[gray]{0.9} & -0.22\cellcolor[gray]{0.9} & -0.55\cellcolor[gray]{0.9} & -0.37\cellcolor[gray]{0.9} & -0.36\cellcolor[gray]{0.9} \\
\hline \hline
\diagbox{Species~}{variable~~} & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 \\ \hline \hline
Leptodactylus Fuscus & -0.30\cellcolor[gray]{0.9} & -0.43\cellcolor[gray]{0.9} & -0.41\cellcolor[gray]{0.9} & -0.51\cellcolor[gray]{0.9} & -0.45\cellcolor[gray]{0.9} & -0.55\cellcolor[gray]{0.9} & -0.62\cellcolor[gray]{0.9} & -0.64\cellcolor[gray]{0.9} & -0.62\cellcolor[gray]{0.9} \\
Adenomera Andreae & -0.47\cellcolor[gray]{0.9} & -0.50\cellcolor[gray]{0.9} & -0.43\cellcolor[gray]{0.9} & -0.46\cellcolor[gray]{0.9} & -0.55\cellcolor[gray]{0.9} & -0.47\cellcolor[gray]{0.9} & -0.58\cellcolor[gray]{0.9} & -0.59\cellcolor[gray]{0.9} & -0.67\cellcolor[gray]{0.9} \\
Adenomera Hylaedactyla & -0.41\cellcolor[gray]{0.9} & -0.55\cellcolor[gray]{0.9} & -0.45\cellcolor[gray]{0.9} & -0.48\cellcolor[gray]{0.9} & -0.49\cellcolor[gray]{0.9} & -0.49\cellcolor[gray]{0.9} & -0.58\cellcolor[gray]{0.9} & -0.61\cellcolor[gray]{0.9} & -0.64\cellcolor[gray]{0.9} \\
Hyla minuta & -0.50\cellcolor[gray]{0.9} & -0.47\cellcolor[gray]{0.9} & -0.49\cellcolor[gray]{0.9} & -0.45\cellcolor[gray]{0.9} & -0.50\cellcolor[gray]{0.9} & -0.54\cellcolor[gray]{0.9} & -0.56\cellcolor[gray]{0.9} & -0.62\cellcolor[gray]{0.9} & -0.72\cellcolor[gray]{0.9} \\
Hypsiboas Cinerascens & -0.59\cellcolor[gray]{0.9} & -0.53\cellcolor[gray]{0.9} & -0.46\cellcolor[gray]{0.9} & -0.49\cellcolor[gray]{0.9} & -0.50\cellcolor[gray]{0.9} & -0.46\cellcolor[gray]{0.9} & -0.73\cellcolor[gray]{0.9} & -0.56\cellcolor[gray]{0.9} & -0.65\cellcolor[gray]{0.9} \\
Hypsiboas Cordobae & -0.55\cellcolor[gray]{0.9} & -0.49\cellcolor[gray]{0.9} & -0.47\cellcolor[gray]{0.9} & -0.51\cellcolor[gray]{0.9} & -0.48\cellcolor[gray]{0.9} & -0.55\cellcolor[gray]{0.9} & -0.68\cellcolor[gray]{0.9} & -0.68\cellcolor[gray]{0.9} & -0.71\cellcolor[gray]{0.9} \\
Ameerega Trivittata & -0.49\cellcolor[gray]{0.9} & -0.52\cellcolor[gray]{0.9} & -0.39\cellcolor[gray]{0.9} & -0.63\cellcolor[gray]{0.9} & -0.48\cellcolor[gray]{0.9} & -0.65\cellcolor[gray]{0.9} & -0.65\cellcolor[gray]{0.9} & -0.64\cellcolor[gray]{0.9} & -0.71\cellcolor[gray]{0.9} \\
\hline \hline
\diagbox{Species~}{variable~~} & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 & \\ \hline \hline
Leptodactylus Fuscus & -0.59\cellcolor[gray]{0.9} & -0.65\cellcolor[gray]{0.9} & -0.66\cellcolor[gray]{0.9} & 2.09 & 1.81 & 2.10 & 1.98 & 1.96 & \\
Adenomera Andreae & -0.72\cellcolor[gray]{0.9} & -0.63\cellcolor[gray]{0.9} & -0.65\cellcolor[gray]{0.9} & 2.03 & 1.92 & 1.93 & 1.99 & 1.85 & \\
Adenomera Hylaedactyla & -0.63\cellcolor[gray]{0.9} & -0.60\cellcolor[gray]{0.9} & -0.67\cellcolor[gray]{0.9} & 2.02 & 1.87 & 2.08 & 2.06 & 1.90 & \\
Hyla Minuta & -0.71\cellcolor[gray]{0.9} & -0.69\cellcolor[gray]{0.9} & -0.62\cellcolor[gray]{0.9} & 1.90 & 1.95 & 1.83 & 2.01 & 1.85 & \\
Hypsiboas Cinerascens & -0.60\cellcolor[gray]{0.9} & -0.70\cellcolor[gray]{0.9} & -0.61\cellcolor[gray]{0.9} & 1.90 & 2.15 & 1.92 & 1.84 & 2.04 & \\
Hypsiboas Cordobae & -0.73\cellcolor[gray]{0.9} & -0.75\cellcolor[gray]{0.9} & -0.65\cellcolor[gray]{0.9} & 1.96 & 2.06 & 1.78 & 1.92 & 1.89 & \\
Ameerega Trivittata & -0.69\cellcolor[gray]{0.9} & -0.74\cellcolor[gray]{0.9} & -0.79\cellcolor[gray]{0.9} & 1.95 & 1.97 & 2.06 & 1.93 & 1.84 & \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
\begin{table}
\begin{center}
\caption{ The probabilities of relevant variable identified by sequential variable selection, based on 1000 iterations, where the cells with gray background are variables with probability larger than 0.5}
\label{Anuranfs}
\scalebox{0.8}{
\begin{tabular}{c|cccccccccc}\hline
\diagbox{group~}{variable~~}& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \hline
1$\sim$10 & 0.1100\cellcolor[gray]{0.9} & 0.2000\cellcolor[gray]{0.9} & 0.2200\cellcolor[gray]{0.9} & 0.2900\cellcolor[gray]{0.9} & 0.3900\cellcolor[gray]{0.9} & 0.5000\cellcolor[gray]{0.9} & 0.4900\cellcolor[gray]{0.9} & 0.4100\cellcolor[gray]{0.9} & 0.4400\cellcolor[gray]{0.9} & 0.3400\cellcolor[gray]{0.9} \\
\hline \hline\diagbox{group~}{variable~~} & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ \hline \hline
1$\sim$10 & 0.2700\cellcolor[gray]{0.9} & 0.1300\cellcolor[gray]{0.9} & 0.0900\cellcolor[gray]{0.9} & 0.0300\cellcolor[gray]{0.9} & 0.0100\cellcolor[gray]{0.9} & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
\hline \hline\diagbox{group~}{variable~~} & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\ \hline \hline
1$\sim$10 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\
\hline \hline
\end{tabular}
}
\end{center}
\end{table}
\subsection{Handwritten digit dataset}
We use a subset from the MINST database of handwritten digits,
where digits have been size-normalized and centered on a fixed-size image.
We randomly select 100 gray images from this data set, for training, and another 100 images for testing from this database.
(For the details about this database, please refer to \cite{lecun2010mnist}.)
We resize each handwritten digit image to $64 = 8\times 8$ pixels, and treat each pixel as a variable.
It becomes a 10-class classification problem through 64-dimensional density estimation when applying the proposed method.
Table \ref{tb:digit1} shows the classification results of NPKDC-vd, where accuracy, precision, and specificity for all digits are greater than $95\%$,
Table \ref{tb:digit2} shows the results of both SVM and SVMfs.
From Table \ref{tb:digit1-all}, we found that NPKDC-vd largely outperforms these two SVM-based methods in all three measurements,
since the relations among classes in this example are rather complicated, and binary-classifier-based methods will suffer from complicated ``voting-like'' schemes.
Table \ref{Table_Digits} shows the means of Z-scores of the mean predicted bandwidths for all 64 attributes/pixels.
The corresponding bandwidths of the relevant pixels could drop to a very small value.
Moreover, because the background pixels of the image data have a density close to a point mass, instead of a uniform distribution,
the variables with large interquartile ranges (IQR) are irrelevant.
Figures of the box-plots for the mean of the selected bandwidths of the testing images based on 100 trials are in Supplementary,
which provides similar information as in Table \ref{Table_Digits}.
The bandwidths of variables in this situation are the pixels on the edges of the image, such as 1, 8, 9, 16, 17, 24, 25, 32, 33, 40, 41, 48, 49, 56, 57, and 64,
and their IQRs are usually large (see also Figures S.1 and S.2 in Supplementary).
Table \ref{tb:digitprobSVM} shows the probabilities of relevant variable identification by SVM with sequential variable selection, based on 1000 iterations, for handwritten digits.
We can see that the pixels/variables identified via the sequential variable selection of SVMfs scatter around all pixels, and cannot focus on some particular variables for each class (digit). Thus, in this case, SVMfs cannot provide neither stable information about each class, nor satisfactory classification performance.
\begin{table}[t]\fontsize{9}{8}\selectfont
\centering
\caption{Classification results of handwritten digit dataset.}
\label{tb:digit1-all}
\begin{subtable}{0.9\linewidth}
\centering
\caption{Classification results of NPKDC-vd.}
\label{tb:digit1}
\begin{tabular}{l|r|r|r}
\hline
& Accuracy & Precision & Specificity \\ \hline\hline
mean & 0.9698 & 0.9966 & 0.9797 \\
Std & 0.0047 & 0.0005 & 0.0030 \\ \hline
\end{tabular}
\caption{Classification results of SVM and SVMfs}
\label{tb:digit2}
\scalebox{1}{
\begin{tabular}{l|r|r|r|r|r|r}
\hline
& \multicolumn{3} {c|}{SVM} & \multicolumn{3} {c}{SVMfs (with feature selection)}\\
\hline
& Accuracy & Precision & Specificity & Accuracy & Precision & Specificity\\ \hline
mean & 0.2739 & 0.8721 & 0.9193 & 0.7606 & 0.9071 & 0.9734\\
Std & 0.0227 & 0.0303 & 0.0025 & 0.0806 & 0.0118 & 0.0090\\ \hline
\end{tabular}
}
\end{subtable}
\end{table}
\begin{table}[ht!]\fontsize{6}{5}\selectfont
\begin{center}
\caption{\small Handwritten Digits: The mean Z-scores of the mean predicted bandwidths for all attributes. The cells with {\bf gray background are irrelevant variables} for its corresponding digit.}
\label{Table_Digits}
\begin{tabular}{c|ccccccccccccc}\hline \hline
\multirow{2}{*}{Digits} & \multicolumn{13}{c}{Attribute}\\
& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline \hline
1 & -0.80\cellcolor{gray!10} & -0.77\cellcolor{gray!10} & 0.81 & 1.07 & 1.17 & 1.12 & -0.40\cellcolor{gray!10} & -0.80\cellcolor{gray!10} & -0.80\cellcolor{gray!10} & -0.33\cellcolor{gray!10} & 0.90 & 0.02 & -0.38\cellcolor{gray!10} \\
2 & -1.18\cellcolor{gray!10} & 0.04 & 1.05 & 0.71 & 0.95 & -0.82\cellcolor{gray!10} & -1.16\cellcolor{gray!10} & -1.18\cellcolor{gray!10} & -1.18\cellcolor{gray!10} & 1.12 & 0.40 & 0.77 & 0.93 \\
3 & -1.27\cellcolor{gray!10} & -0.12\cellcolor{gray!10} & 0.98 & 0.50 & 0.88 & 0.91 & -0.98\cellcolor{gray!10} & -1.25\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & 1.02 & 0.70 & 0.93 & 0.81 \\
4 & -0.94\cellcolor{gray!10} & -0.93\cellcolor{gray!10} & -0.75\cellcolor{gray!10} & 0.72 & 0.80 & 0.66 & -0.78\cellcolor{gray!10} & -0.87\cellcolor{gray!10} & -0.94\cellcolor{gray!10} & -0.87\cellcolor{gray!10} & 0.26 & 0.65 & 0.75 \\
5 & -1.22\cellcolor{gray!10} & -0.17\cellcolor{gray!10} & 0.97 & 0.91 & 0.80 & 0.92 & 0.94 & -1.17\cellcolor{gray!10} & -1.22\cellcolor{gray!10} & 0.75 & 0.18 & 0.82 & 0.69 \\
6 & -1.15\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & 0.32 & 1.02 & 1.11 & -0.34\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & -1.06\cellcolor{gray!10} & 1.14 & 0.47 & 0.95 \\
7 & -1.11\cellcolor{gray!10} & -0.91\cellcolor{gray!10} & 1.07 & 0.87 & 0.46 & 0.78 & 1.13 & -0.36\cellcolor{gray!10} & -1.11\cellcolor{gray!10} & -0.47\cellcolor{gray!10} & 1.19 & 1.09 & 1.03 \\
8 & -1.28\cellcolor{gray!10} & -1.04\cellcolor{gray!10} & 0.80 & 0.61 & 0.69 & 0.89 & -0.70\cellcolor{gray!10} & -1.28\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & 0.73 & 0.50 & 0.85 & 0.85 \\
9 & -1.26\cellcolor{gray!10} & -0.86\cellcolor{gray!10} & 0.86 & 0.81 & 0.85 & 0.69 & -0.73\cellcolor{gray!10} & -1.21\cellcolor{gray!10} & -1.26\cellcolor{gray!10} & 0.64 & 0.35 & 0.86 & 0.78 \\
10 & -1.27\cellcolor{gray!10} & -1.26\cellcolor{gray!10} & 0.91 & 0.74 & 1.11 & 0.39 & -1.25\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & 0.13 & 0.87 & 1.02 & 0.99 \\ \hline \hline
\multirow{2}{*}{Digits} & \multicolumn{13}{c}{Attribute}\\
& 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 \\ \hline \hline
1 & 1.20 & -0.22\cellcolor{gray!10} & -0.80\cellcolor{gray!10} & -0.79\cellcolor{gray!10} & -0.22\cellcolor{gray!10} & 1.23 & -0.23\cellcolor{gray!10} & -0.65\cellcolor{gray!10} & 1.15 & -0.43\cellcolor{gray!10} & -0.80\cellcolor{gray!10} & -0.78\cellcolor{gray!10} & 0.01 \\
2 & 0.27 & -1.13\cellcolor{gray!10} & -1.18\cellcolor{gray!10} & -1.18\cellcolor{gray!10} & 1.03 & 1.17 & 0.96 & 0.90 & 0.49 & -1.09\cellcolor{gray!10} & -1.18\cellcolor{gray!10} & -1.18\cellcolor{gray!10} & 0.05 \\
3 & 0.99 & -0.51\cellcolor{gray!10} & -1.26\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & 0.49 & 0.85 & 0.78 & 0.92 & 1.05 & -0.71\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & -0.98\cellcolor{gray!10} \\
4 & 0.05 & 0.01 & -0.64\cellcolor{gray!10} & -0.94\cellcolor{gray!10} & -0.56\cellcolor{gray!10} & 0.73 & 0.76 & 0.57 & 0.69 & 0.90 & -0.59\cellcolor{gray!10} & -0.94\cellcolor{gray!10} & 0.42 \\
5 & 0.90 & 0.62 & -1.19\cellcolor{gray!10} & -1.20\cellcolor{gray!10} & 0.89 & 0.33 & 0.84 & 0.53 & -0.31\cellcolor{gray!10} & -1.14\cellcolor{gray!10} & -1.22\cellcolor{gray!10} & -1.22\cellcolor{gray!10} & 0.88 \\
6 & -0.49\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & -0.14\cellcolor{gray!10} & 0.90 & 1.06 & -0.81\cellcolor{gray!10} & -1.14\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & 0.70 \\
7 & 0.66 & 0.96 & -0.55\cellcolor{gray!10} & -1.11\cellcolor{gray!10} & -0.86\cellcolor{gray!10} & -0.21\cellcolor{gray!10} & -0.56\cellcolor{gray!10} & 0.90 & 0.80 & 0.86 & -0.96\cellcolor{gray!10} & -1.11\cellcolor{gray!10} & -0.67\cellcolor{gray!10} \\
8 & 0.73 & 0.60 & -1.28\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & 0.88 & 0.70 & 0.76 & 0.88 & 0.71 & 0.54 & -1.28\cellcolor{gray!10} & -1.28\cellcolor{gray!10} & 0.33 \\
9 & 0.97 & 0.13 & -1.13\cellcolor{gray!10} & -1.26\cellcolor{gray!10} & 0.90 & 0.73 & 0.82 & 0.89 & 0.70 & 0.51 & -1.14\cellcolor{gray!10} & -1.26\cellcolor{gray!10} & 0.59 \\
10 & 1.06 & -0.81\cellcolor{gray!10} & -1.2\cellcolor{gray!10}7 & -1.27\cellcolor{gray!10} & 1.03 & 0.38 & 0.94 & 0.52 & 0.78 & 0.73 & -1.27\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & 0.86 \\ \hline \hline
\multirow{2}{*}{Digits} & \multicolumn{13}{c}{Attribute}\\
& 27 & 28 & 29 & 30 & 31 & 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 \\ \hline \hline
1 & 1.23 & -0.43\cellcolor{gray!10} & -0.61\cellcolor{gray!10} & 1.13 & -0.48\cellcolor{gray!10} & -0.80\cellcolor{gray!10} & -0.80\cellcolor{gray!10} & 0.02 & 1.26 & -0.32\cellcolor{gray!10} & -0.25\cellcolor{gray!10} & 1.26 & -0.65\cellcolor{gray!10} \\
2 & 0.49 & 0.76 & 0.94 & 0.50 & -1.17\cellcolor{gray!10} & -1.18\cellcolor{gray!10} & -1.18\cellcolor{gray!10} & -1.09\cellcolor{gray!10} & -0.62\cellcolor{gray!10} & 0.99 & 0.86 & -0.01\cellcolor{gray!10} & -1.17\cellcolor{gray!10} \\
3 & 0.44 & 1.02 & -0.01\cellcolor{gray!10} & 0.98 & -1.00\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & -1.17\cellcolor{gray!10} & 0.17 & 1.02 & 0.86 & 0.68 & 0.73 \\
4 & 0.27 & 0.93 & 0.49 & 0.70 & 0.83 & -0.83\cellcolor{gray!10} & -0.93\cellcolor{gray!10} & 0.80 & -0.32\cellcolor{gray!10} & 0.90 & 0.83 & -0.37\cellcolor{gray!10} & 0.92 \\
5 & -0.16\cellcolor{gray!10} & 0.53 & 0.58 & 0.54 & -0.58\cellcolor{gray!10} & -1.22\cellcolor{gray!10} & -1.22\cellcolor{gray!10} & 0.39 & 0.99 & 0.69 & 0.83 & 0.87 & 0.20 \\
6 & 0.55 & 1.17 & 0.31 & -0.21\cellcolor{gray!10} & -1.12\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & 0.91 & 0.30 & 0.92 & 0.93 & 1.05 & 0.34 \\
7 & 0.86 & 1.17 & 0.81 & 0.65 & 1.02 & -1.08\cellcolor{gray!10} & -1.11\cellcolor{gray!10} & 0.06 & 1.13 & 0.37 & -0.29\cellcolor{gray!10} & 0.82 & 1.15 \\
8 & 0.76 & 0.43 & 0.30 & 0.81 & -0.40\cellcolor{gray!10} & -1.28\cellcolor{gray!10} & -1.28\cellcolor{gray!10} & -0.75\cellcolor{gray!10} & 0.85 & 0.15 & 0.60 & 0.39 & -0.57\cellcolor{gray!10} \\
9 & 0.77 & 0.83 & 0.84 & 0.08 & 0.94 & -1.26\cellcolor{gray!10} & -1.26\cellcolor{gray!10} & -1.13\cellcolor{gray!10} & 0.36 & 0.77 & 0.90 & 0.70 & 0.82 \\
10 & 0.98 & 0.02 & -1.14\cellcolor{gray!10} & 1.09 & 0.49 & -1.27\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & 0.88 & 1.06 & -0.72\cellcolor{gray!10} & -1.14\cellcolor{gray!10} & 1.10 & 0.43 \\ \hline \hline
\multirow{2}{*}{Digits} & \multicolumn{13}{c}{Attribute}\\
& 40 & 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 & 49 & 50 & 51 & 52 \\ \hline \hline
1 & -0.80\cellcolor{gray!10} & -0.80\cellcolor{gray!10} & -0.30\cellcolor{gray!10} & 1.05 & -0.20\cellcolor{gray!10} & -0.28\cellcolor{gray!10} & 1.17 & -0.64\cellcolor{gray!10} & -0.80\cellcolor{gray!10} & -0.80\cellcolor{gray!10} & -0.64\cellcolor{gray!10} & 0.89 & -0.32\cellcolor{gray!10} \\
2 & -1.18\cellcolor{gray!10} & -1.18\cellcolor{gray!10} & -1.01\cellcolor{gray!10} & 0.52 & 0.87 & 1.02 & -0.22\cellcolor{gray!10} & -0.74\cellcolor{gray!10} & -1.18\cellcolor{gray!10} & -1.16\cellcolor{gray!10} & 0.11 & 0.95 & -0.31\cellcolor{gray!10} \\
3 & -1.27\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & -1.25\cellcolor{gray!10} & -0.39\cellcolor{gray!10} & -0.76\cellcolor{gray!10} & 0.22 & 0.95 & 0.84 & -1.26\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & 0.26 & 1.00 & 1.00 \\
4 & -0.94\cellcolor{gray!10} & -0.46\cellcolor{gray!10} & 0.72 & 0.44 & 0.72 & -0.23\cellcolor{gray!10} & 0.55 & 0.42 & -0.94\cellcolor{gray!10} & -0.74\cellcolor{gray!10} & -0.10\cellcolor{gray!10} & 0.19 & 0.80 \\
5 & -1.22\cellcolor{gray!10} & -1.22\cellcolor{gray!10} & -1.14\cellcolor{gray!10} & -0.87\cellcolor{gray!10} & -0.74\cellcolor{gray!10} & 0.48 & 0.79 & 0.36 & -1.21\cellcolor{gray!10} & -1.22 & 0.02 & 1.01 & 0.94 \\
6 & -1.15\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & 0.79 & 0.05 & 1.08 & 1.02 & 1.04 & 1.12 & -0.92\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & -0.55\cellcolor{gray!10} & 1.05 & 0.97 \\
7 & -1.11\cellcolor{gray!10} & -1.11\cellcolor{gray!10} & -0.42\cellcolor{gray!10} & 0.62 & 0.86 & 0.93 & 0.72 & -0.41\cellcolor{gray!10} & -1.11\cellcolor{gray!10} & -1.11\cellcolor{gray!10} & -1.07\cellcolor{gray!10} & 0.79 & 0.53 \\
8 & -1.28\cellcolor{gray!10} & -1.28\cellcolor{gray!10} & -0.12\cellcolor{gray!10} & 0.83 & 0.89 & 0.77 & 0.81 & -0.13\cellcolor{gray!10} & -1.28\cellcolor{gray!10} & -1.28\cellcolor{gray!10} & -0.25\cellcolor{gray!10} & 0.77 & 0.88 \\
9 & -1.26\cellcolor{gray!10} & -1.26\cellcolor{gray!10} & -1.16\cellcolor{gray!10} & -0.92\cellcolor{gray!10} & -0.97\cellcolor{gray!10} & -0.15\cellcolor{gray!10} & 0.96 & 0.70 & -1.25\cellcolor{gray!10} & -1.26\cellcolor{gray!10} & -0.47\cellcolor{gray!10} & 0.66 & 0.94 \\
10 & -1.27\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & 0.75 & 0.74 & -0.00\cellcolor{gray!10} & -0.26\cellcolor{gray!10} & 1.00 & 0.96 & -1.27\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & -0.24\cellcolor{gray!10} & 0.79 & 1.12 \\ \hline \hline
\multirow{2}{*}{Digits} & \multicolumn{13}{c}{Attribute}\\
& 53 & 54 & 55 & 56 & 57 & 58 & 59 & 60 & 61 & 62 & 63 & 64 & \\ \hline \hline
1 & -0.30\cellcolor{gray!10} & 1.18 & 0.03 & -0.79\cellcolor{gray!10} & -0.80\cellcolor{gray!10} & -0.78\cellcolor{gray!10} & 0.83 & 0.94 & 1.18 & 1.13 & 0.59 & -0.79\cellcolor{gray!10} \\
2 & 1.02 & 1.12 & 1.11 & -0.35\cellcolor{gray!10} & -1.17\cellcolor{gray!10} & 0.05 & 1.06 & 0.82 & 0.95 & 1.01 & 1.10 & 0.21 \\
3 & 1.05 & 0.53 & 1.03 & -1.22\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & -0.22\cellcolor{gray!10} & 1.03 & 0.70 & 0.77 & 1.04 & 0.15 & -1.27\cellcolor{gray!10} \\
4 & 0.26 & 0.93 & -0.78\cellcolor{gray!10} & -0.94\cellcolor{gray!10} & -0.94\cellcolor{gray!10} & -0.90\cellcolor{gray!10} & -0.76\cellcolor{gray!10} & 0.95 & 0.68 & 0.49 & -0.94\cellcolor{gray!10} & -0.94\cellcolor{gray!10} \\
5 & 0.75 & 0.79 & 0.17 & -1.21\cellcolor{gray!10} & -1.22\cellcolor{gray!10} & -0.19\cellcolor{gray!10} & 0.99 & 0.31 & 0.80 & 0.68 & -1.01\cellcolor{gray!10} & -1.21\cellcolor{gray!10} \\
6 & 1.14 & 1.09 & 1.06 & -0.43\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & -1.15\cellcolor{gray!10} & 0.51 & 1.02 & 0.52 & 1.00 & 1.11 & -0.86\cellcolor{gray!10} \\
7 & 0.94 & -0.99\cellcolor{gray!10} & -1.11\cellcolor{gray!10} & -1.11\cellcolor{gray!10} & -1.11\cellcolor{gray!10} & -0.94\cellcolor{gray!10} & 1.08 & 0.92 & 0.06 & -1.08\cellcolor{gray!10} & -1.11\cellcolor{gray!10} & -1.11\cellcolor{gray!10} \\
8 & 0.82 & 0.77 & 0.25 & -1.28\cellcolor{gray!10} & -1.28\cellcolor{gray!10} & -1.20\cellcolor{gray!10} & 0.82 & 0.60 & 0.71 & 0.86 & -0.53\cellcolor{gray!10} & -1.27\cellcolor{gray!10} \\
9 & 0.86 & 0.67 & 0.55 & -1.23\cellcolor{gray!10} & -1.26\cellcolor{gray!10} & -0.98\cellcolor{gray!10} & 0.72 & 0.58 & 0.76 & 0.77 & -0.09\cellcolor{gray!10} & -1.26\cellcolor{gray!10} \\
10 & 1.12 & 0.63 & 0.46 & -1.27\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & -1.27\cellcolor{gray!10} & 0.82 & 0.80 & 0.76 & 1.00 & -1.04\cellcolor{gray!10} & -1.27\cellcolor{blue!10} \\
\hline \hline
\end{tabular}
\end{center}
\end{table}
\begin{table}
\begin{center}
\caption{The probabilities of relevant variable identification of SVM with sequential variable selection for handwritten digits based on 1000 runs.}
\label{tb:digitprobSVM}
\scalebox{0.8}{
\begin{tabular}{c|ccccccccccccc}\hline \hline
Attribute& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline
Prob. & 0 & 0.16 & 0.19 & 0.19 & 0.18 & 0.34 & 0.19 & 0.18 & 0.11 & 0.10 & 0.74 & 0.21 & 0.18
\\ \hline \hline
Attribute & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 \\ \hline
Prob. & 0.42 & 0.09 & 0.16 & 0.04 & 0.09 & 0.72 & 0.66 & 0.76 & 0.96 & 0.16 & 0.11 & 0 & 0.18 \\
\hline \hline
Attribute & 27 & 28 & 29 & 30 & 31 & 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 \\ \hline
Prob. & 0.65 & 0.63 & 0.37 & 0.49 & 0.52 & 0 & 0 & 0.1 & 0.17 & 0.24 & 0.56 & 0.39 & 0.39 \\
\hline \hline
Attribute & 40 & 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 & 49 & 50 & 51 & 52 \\ \hline
Prob. & 0 & 0.01 & 0.19 & 1 & 0.73 & 0.43 & 0.32 & 0.21 & 0.03 & 0.03 & 0.14 & 0.09 & 0.33\\
\hline \hline
Attribute & 53 & 54 & 55 & 56 & 57 & 58 & 59 & 60 & 61 & 62 & 63 & 64 & \\ \hline
Prob. & 0.41 & 0.48 & 0.13 & 0.07 & 0 & 0.12 & 0.16 & 0.22 & 0.27 & 0.63 & 0.40 & 0.15 \\
\hline \hline
\end{tabular}
}
\end{center}
\end{table}
\section{Summary}
The relative position of classes in the variable-space of a multiple-class classification problem is usually complicated.
Hence, some complicated modeling procedures are preferred for their classification performance.
Using a voting scheme via building many binary classifiers is another choice.
However, this kind of approach usually aims at high prediction power, and sacrifices the ``model interpretation'' ability,
which is important from many classification application perspectives.
If we only aim our classification goal at classification performance, such as accuracy, specificity, sensitivity/precision, then finding only a set of variables for all classes, via using a complicated model, could be an easy way out.
Nevertheless, we will have less power about model interpretation when the relations among classes are complicated,
and this goal is even more difficult to be achieved when data sets have lengthy variables.
Thus, when the model-interpretation ability is indispensable to an application, deciding class-specific variable sets to clarify the classification rules of models is important in addition to the conventional classification performance measures.
In this study, we adopt NPKDC-vd for multiple-class classification, and shows that the error rate of this method approaches to that of the Bayes rule, asymptotically.
Moreover, through local bandwidths of density estimation, we integrate such a posterior density estimation procedure with a variable selection scheme such that we are able to identify the individual high impact variable set for each class, such that the proposed classification rule are easier to be understood,
and we can have a better description of the character of each class.
In the numerical examples, we see that the proposed method has a satisfactory classification performance with only a limited increase of computational burden, and the ``interpretation'' of each class is largely improved, especially for the examples with complicated relative positions among classes.
Because we can separately calculate the local bandwidths within each class, the computational efficiency can be further improved via parallel computing methods, and can be easily implemented via machines with a multi-core CPU.
Moreover, because using the bandwidth to select the relevant variables is a promising approach,
we suggest a two-step approach such that we can first filter out the irrelevant variables to reduce variable size in advance,
then apply the proposed algorithm with the rest variables.
This approach can be useful for problems with a small sample size but a large number of variables.
Since there will need different considerations for filtering out variables in the first stage,
and will go beyond the scope of the current project.
Thus, we will study this problem in our future studies, and will report its results elsewhere.
|
1,108,101,563,259 | arxiv | \section{Introduction}
\begin{figure*}[!t]
\centering
\includegraphics[scale=3.05]{figures/brat-example2.pdf}
\caption{An example standardized science exam question densely annotated with one or more fine-grained semantic categories for nearly each word. This 4-choice multiple choice question (here, under the curriculum topic \textit{``The Interdependence of Life \textgreater The Food Chain \textgreater Decomposers''}) is one of 4,239 drawn from the ARC corpus and densely annotated in this work. \label{fig:example}}
\end{figure*}
Named entity recognition (NER) \cite{grishman1996message} is a common natural language processing task that aims to abstract or categorize common classes of noun phrases in text, such as identifying ``Arthur'' as a \textit{person} or ``Montreal'' as a \textit{location}. This high-level categorization of important entities in text is a staple of most modern NLP pipelines,
and has a variety of applications for higher-level tasks including information extraction \cite{valenzuela2016odin}, knowledge base population \cite{dredze2010entity}, and question answering \cite{abujabal2017automated}.
\hspace{2em} Named entity recognition identifies common classes of entities in text, but these entity labels are generally sparse (typically occurring for between 10\% to 20\% of words in a corpus, see Section~\ref{sec:mentiondensity}), limiting utility to downstream tasks. In this work, we introduce the idea of common entity recognition (CER), which aims to tag all content words in text with an appropriate fine-grained semantic class. CER allows text to be automatically annotated with a much richer set of semantic labels, potentially providing greater utility for downstream applications. We explore CER in the context of scientific text, and present ScienceExamCER, a training corpus annotated with over 113k common entity annotations drawn from a fine-grained set of over 600 semantic categories, which include named entities, as well as verb groups, properties and quantities, thematic types, and synonyms for key terminology. We also release an off-the-shelf NER tagger modified to perform multilabel CER tagging. This BERT CER tagger achieves an accuracy of 0.85 F1 on this task, indicating that the tag ontology labels are well-defined and clearly identifiable. These two resources offer new opportunities to explore the impact of dense semantic annotation in downstream tasks.
\hspace{2em} We believe the notion of dense fine-grained semantic tagging to be potentially useful to any application domain, but explore common entity recognition here in the context of scientific text aimed at teaching and evaluating scientific knowledge.
An example of this dense semantic classification in the context of standardized science exams is shown in Figure~\ref{fig:example}. While CoreNLP \cite{manning2014stanford} does not locate any entities in the sentence \textit{``Rolanda is growing tomato plants in her garden''}, our CER annotation and system abstracts this sentence to \textit{``[Rolanda]$_\textit{Human}$ [is]$_\textit{StateOfBeing}$ [growing]$_\textit{Growth/ActionsForAgriculture}$ [tomato]$_\textit{Food}$ [plants]$_\textit{Plant}$ [in]$_\textit{RelativeLocation}$ her [garden]$_\textit{ManmadeLocation}$''}.
\hspace{2em} We detail corpus and ontology/typology construction in Section 3, including a comparison of mention density with other common corpora. Automated evaluations of CER performance are shown in Section 4, including an analyses of the training data requirements of this fine-grained classification, as well as an error analysis.
\section{Related Work}
\hspace{2em} Common sets of entity labels (or \textit{typologies}) have expanded from early experiments with a single label, \textit{organization} \cite{rau1991extracting}, to the 7 common MUC-6 types \cite{grishman1996message} typically used by NER systems, including named entities (\textit{person, organization, location}), temporal mentions (\textit{date, time}), and numeric categories (\textit{money, percent}). Subsets of the MUC-6 types have been included in the typologies of benchmark NER corpora, including CoNLL-2003 \cite{sang2003introduction}, OntoNotes \cite{weischedel2013ontonotes}, and BBN \cite{weischedel2005bbn}.
\hspace{2em} Sekine et al. \shortcite{Sekine2002ExtendedNE} proposed an extended hierarchy of MUC-6 types expanded to include 150 open-domain category labels. While most of these category labels are named entities, Sekine et al. include 10 measurement categories (e.g. \textit{weight, speed, temperature}) and 3 high-level natural object categories (\textit{animal, vegetable, mineral}) that most closely relate to the 601 fine-grained science categories in this work. A subsequent version, the Extended Named Entity (ENE) Ontology \cite{sekine-2008-extended}, expands the typology to 200 classes, including 19 fine-grained expansions of the \textit{natural\_object} type, such as \textit{bird} or \textit{reptile}, as well as adding 5 meronym categories, such as \textit{plant\_part}, that further relax the working definition of named entities from proper names to include other categories \cite{nadeau2007survey}.
\hspace{2em} While open-domain typologies are common, domain-specific typologies and corpora are also popular, occasionally making use of existing domain ontologies to reduce the burden in manually generating fine-grained typologies, such as the manual creation of the fine-grained science-domain typology in this work. An extreme example of fine-grained NER is the MedMentions corpus \cite{murty2018hierarchical}, which contains 246k mentions labelled with Universal Medical Language System (UMLS) \cite{bodenreider2004unified} categories, a fine-grained ontology of over 3.5 million medical concepts.
Similarly, large knowledge bases can be filtered to automatically produce fine-grained typologies (as in FIGER \cite{ling2012fine} and HYENA \cite{yosef2012hyena}), or used to bootstrap the entity classification process in manually-generated typologies. Magnini et al. \shortcite{magnini-etal-2002-wordnet} demonstrate combining WordNet predicates \cite{fellbaum1998wordnet} with approximately 200 handcoded rules can achieve an F1 score of 0.85 on recognizing 10 common entity types, while
Ritter et al. \shortcite{ritter2011named} use distantly supervised topic modeling over Freebase entities \cite{bollacker2008freebase} to perform named entity recognition on social media posts, achieving an F1 score of 0.59 on 10 common entity types.
With respect to larger typologies, Del Corro et al. \shortcite{del2015finet} perform super-fine grained entity typing using the 16k fine-grained WordNet types under the high-level taxonomic categories of \textit{person, organization, and location}, achieving a manually-evaluated precision of 59.9\% on the CoNLL corpus and 28.3\% on New York Times news articles. For smaller manually-generated typologies, Mai et al.~\shortcite{mai2018empirical} demonstrate a model combining LSTMs, CNNs, CRFs, and dictionary-based methods can achieve an F1 of 83.1 on an in-house corpus labeled with Sekine's \shortcite{sekine-2008-extended} 200-class ENE ontology.
\hspace{2em} NER has historically been approached using a wide variety of methods, including rules \cite{hanisch2005prominer}, feature-based machine learning systems \cite{Mayfield:2003:NER:1119176.1119205}, conditional random fields \cite{greenberg2018marginal}, contextualized embeddings \cite{peters2018deep}, and combinations thereof.
Qu et al. \shortcite{qu-etal-2016-named} demonstrate that it is possible to use a conditional random field model to transfer NER performance between datasets, at least in part. Ma et al. \shortcite{ma2016label} show embedding models can transfer performance in zero-shot settings on fine-grained named entity classification. Expanding on this, recent transformer models \cite{peters2018deep,devlin2018bert} have shown strong transfer performance on a variety of text classification tasks including named entity recognition using large pretrained contextualized embeddings that are fine-tuned on comparatively small in-domain corpora. In this work we make use of an off-the-shelf bidirectional transformer (BERT) NER system modified to support multi-label classification, and demonstrate strong performance on the fine-grained common entity recognition task.
\section{Data and Annotation}
\begin{table}
\centering
\small
\begin{tabular}{lcc}
\textbf{Label} & \textbf{Examples} & \textbf{Prop.} \\
\hline
StateOfBeing & is, are, be & 4.6\% \\
LevelOfInclusion & which, each, only & 4.0\% \\
RelativeLocation & inside, under & 3.0\% \\
Comparison & identical, difference & 1.9\% \\
RelativeDirection & forward, upward & 1.8\% \\
ProbabilityAndCertainty & likely, possible & 1.6\% \\
Cause & because, due to & 1.5\% \\
AmountComparison & most, more, less & 1.5\% \\
RelativeTime & during, after & 1.4\% \\
Creation & produce, make, form & 1.2\% \\
PhasesOfWater & steam, ice & 1.1\% \\
CardinalNumber & one, 100 & 1.0\% \\
ContainBeComposedOf & made of, contains & 1.0\% \\
IncreaseDecrease & increasing, decline & 0.9\% \\
Element & oxygen, carbon & 0.9\% \\
Plant & tree, crops, weeds & 0.8\% \\
Move & placed, motion, travel & 0.8\% \\
Use & with, apply & 0.8\% \\
AmountChangingActions & deplete, extend & 0.8\% \\
RelativeNumber & many, some, high & 0.7\% \\
Temperature & hot, warm, cold & 0.7\% \\
Energy & kinetic energy, power & 0.7\% \\
CombineAdd & add, absorb, mix & 0.7\% \\
LiquidMatter & water, oil, droplets & 0.7\% \\
Scientist & geologist, Galileo & 0.6\% \\
Quality & best, good, useful & 0.6\% \\
Size & large, thick, diameter & 0.6\% \\
AbilityAvailablity & potential, unable & 0.6\% \\
ManmadeObjects & ball, spoon, paper & 0.6\% \\
PrepDirections & on, through, along & 0.6\% \\
Human & person, astronaut & 0.6\% \\
ActionsForAnimals & eat, migrate, swim & 0.6\% \\
InnerPlanets & earth, mars, venus & 0.6\% \\
QualityComparison & advantage, benefit & 0.6\% \\
Mammal & dog, horse, bear & 0.6\% \\
Exemplar & including, such as & 0.5\% \\
PlantPart & leaves, flower, root & 0.5\% \\
PerformActivity & conduct, do & 0.5\% \\
Result & effect, impact & 0.5\% \\
Compound & carbon dioxide & 0.5\% \\
BodiesOfWater & ocean, lake, pond & 0.5\% \\
Help & benefit, support & 0.5\% \\
Require & need, must, takes & 0.5\% \\
Rock & bedrock, boulder & 0.5\% \\
TemporalProperty & first, over time & 0.5\% \\
EarthPartsGross & surface, equator & 0.5\% \\
WeatherPhenomena & wind, cloud, drought & 0.5\% \\
Communicate & explain, describe & 0.5\% \\
GeographicFormations & mountain, glacier & 0.5\% \\
ChangeInto & become, converted & 0.5\% \\
Soil & sand, ground, topsoil & 0.4\% \\
Color & green, blue, white & 0.4\% \\
Star & sun, proxima centauri & 0.4\% \\
PhaseChangingActions & melt, evaporation & 0.4\% \\
Nutrition & food, nutrient & 0.4\% \\
AnimalPart & body, organ, eye & 0.4\% \\
TimeUnit & day, second, year & 0.4\% \\
Method & process, procedure & 0.4\% \\
StateOfMatter & solid, liquid, gas & 0.4\% \\
\end{tabular}
\vspace{-4pt}
\caption{The most frequent subset of the 601 semantic classification labels used to annotate the ScienceExamCER corpus. Proportion refers to the proportion of mentions in the training set that are labeled with a given category. The full set of 601 classes is included in the supplementary material. \label{tab:examplesemanticcategories}}
\end{table}
\subsection{Corpus}
We annotate fine-grained semantic classes on standardized science exam questions drawn from the Aristo Reasoning Challenge (ARC) corpus \cite{clark2018think}, which contains 7,787 elementary and middle school ($3^{rd}$ through $9^{th}$ grade) standardized exam questions drawn from 12 US states. Each question is a 4-way multiple choice question, ranging from short direct questions to detailed multi-step problems grounded in examples. An example question is shown in Figure~\ref{fig:example}. Question text contains an average of 21 words across 1.7 sentences, while answer candidate text averages 4 words, but can be as short as a single word (as in Figure~\ref{fig:example}). In this work we draw 4,239 questions from the ARC corpus, consisting of the full training and development folds, to use for our semantic labeling and prediction tasks.
\subsection{Semantic Class Labels}
We conducted a large data-driven analysis of the 4,239 science exam questions with the aim of identifying a set of high-level semantic categories that would provide near total coverage for classifying or grouping nearly all of the 156k words found across the question and answer text in this corpus.
While named entity recognition typically focuses on proper names with specific referents \cite{nadeau2007survey}, in the end we arrived at creating 601 fine-grained categories spanning 6 classes of groups:
{\flushleft\textbf{Taxonomic Groups:}} High-level categories expressing taxonomic membership, such as that a \textit{hummingbird} is a kind of \textit{bird}. This (or stricter interpretations) is the common form of entity classification in most named entity recognition corpora.
{\flushleft\textbf{Meronym Groups:}} Categories expressing part-of relations, such as that a \textit{fin} is a part of an \textit{aquatic animal}, an \textit{x-axis} is a part of a \textit{graphical representation}, or that an \textit{individual} is a part of a \textit{group}.
{\flushleft\textbf{Action Groups:}} Collections of action words that tend to describe similar ideas. For example, \textit{decrease, increase, contract, expand, inflate, deflate, accelerate, decelerate, lower, raise} all describe a group of \textit{actions that involve increasing or decreasing quantities}.
{\flushleft\textbf{Thematic word groups:}} Groups of words that surround a particular topic. For example, \textit{observe, conduct an experiment, compare, study, consider, test, collect, record, gather, examine,} and \textit{research} are some of the words included in the \textit{performing research using the scientific method} semantic class.
{\flushleft\textbf{Properties and Values:}} Common science-domain properties of objects, such as \textit{mass, size,} or \textit{conductivity}, typically grouped with common values they might take, such as \textit{soft, brittle, or hard} in the case of \textit{hardness}.
{\flushleft\textbf{Synonyms:}} Groups of words that tend to express similar ideas in the context of science exams. For example, \textit{disease, infection, and sick} all convey the notion of \textit{illness}.
\\
\hspace{2em} To identify specific instances of these categories in the science exam domain, we first sorted questions into fine-grained curriculum topics using the 406 detailed science-domain question classification labels of Xu et al. \shortcite{Xu2019MulticlassHQ}, noting that common categories of words tended to emerge upon detailed manual inspection when questions on similar topics were examined together. We proceeded through several iterations of this process, recording candidate high-level semantic classes, as well as seed words that belonged to those categories. After assembling a large list of candidate categories, we further enumerated the seed words with encyclopedic knowledge manually through web searches. For example, while the annotators may have only observed the words \textit{Sun} and \textit{Proxima Centauri} in the corpus for the \textit{Star} category, we would manually expand this to also include other nearby stars such as \textit{Vega}, \textit{Polaris}, and \textit{Wolf 359}.
\hspace{2em} As a final step, we automatically expanded the seed word list to include lexical variations of each manually added word by first using pretrained GLoVe embeddings \cite{pennington2014glove} to compute the top-N most similar words to a given seed word using cosine similarity, then using several low-precision high-recall heuristics to identify words that had the potential to be lexical variations of an existing word on the seed list. We then generated a frequency histogram of any word present in the corpus that did not yet belong to at least one semantic category, and either placed it in an existing category, or formed a new category for that word and repeated the expansion process for seed words. This detailed manual category development process required approximately three weeks of annotator time, ultimately arriving at a list of 601 high-level semantic categories, with an extensive list of both manually and automatically populated seed words for each category. The full list of semantic categories and seed words is included in the \textit{supplementary material}.
\begin{table*}
\centering
\begin{tabular}{lcccccc}
\textbf{Measure} & \textbf{ScienceExamCER} & \textbf{OntoNotes 5}& \textbf{BBN} & \textbf{GUM} & \textbf{CoNLL 2003} \\
\hline
\textit{Entity Categories} & 601 & 18 & 64 & 11 & 4 \\
\textit{Total Mentions} & ~133k${^{*}}$ & 162k & 172k & 11k & ~35k \vspace{1.5mm}\\
\textit{Words} & 156k & 2.44M & 1.05M & 55k & 264k \\
~~\textit{Labeled Words} & 117k & 284k & 257k & 33k & 51k \\
~~\textit{Mention Density (overall)} & \textbf{75\%} & 12\% & 25\% & 59\% & ~19\% \vspace{1.5mm}\\
\textit{Content Words} & 104k & 1.39M & 677k & 34k & 190k \\
~~\textit{Labelled Content Words} & 100k & 255k & 243k & 21k & 50k \\
~~\textit{Mention Density (Content Words)} & \textbf{96\%} & 18\% & 36\% & 62\% & ~27\% \vspace{1.5mm}\\
\end{tabular}
\caption{Summary statistics including \textit{mention density} for the ScienceExamCER corpus, as well as four other common benchmark corpora. At 96\%, the ScienceExamCER is significantly more densely labeled than the next-nearest corpus. (\textit{*} denotes that approximately 16k spans have multiple labels, and as such the total mentions exceeds the total labeled words). \label{tab:summarystatistics}}
\end{table*}
\begin{table}[]
\centering
\begin{tabular}{lcc}
\textbf{Fold} & \textbf{Science Questions} & \textbf{Words} \\
\hline
Train & 2,696 & 108,396 \\
Development & 674 & 27,560 \\
Test & 869 & 35,379 \\
\end{tabular}
\caption{Summary statistics for the training, evaluation, and test sets used for evaluating semantic category classification. }
\label{tab:summarystatisticsquestionset}
\end{table}
\subsection{Annotation Procedure}
Annotating a large set of semantic classes onto more than one hundred thousand words presents challenges with annotation consistency and tractability. It would be challenging for crowdworkers to learn a detailed set of 601 fine-grained semantic categories, and extremely time consuming for research assistants to traditionally annotate a collection at this scale. To overcome these challenges, we modified the annotation task to automatically preannotate the entire corpus using the large set of bootstrapped seed words associated with each semantic class, effectively preannotating each word with a set of possible semantic category labels. These preannotated labels are effectively low-precision and high-recall, most often containing the correct label(s) for a given mention, but also containing other incorrect labels that must be manually removed by an annotator. A total of 226k preannotated mentions were generated (an average of 1.5 per word), which was reduced to 133k mentions (0.9 per word) after incorrect labels were removed by the annotator. We used the BRAT annotation tool \cite{stenetorp2012brat} for the label removal step. To ease the annotator's need for switching semantic contexts, questions were presented to the annotator sorted by curriculum topic using the question classification annotation of Xu et al. \shortcite{Xu2019MulticlassHQ}. The annotation procedure took approximately 2.5 minutes per question, for a total of 200 hours.
\hspace{2em} A clear question with this ``preannotate-then-filter'' annotation protocol is how well this procedure is able to provide both coverage and accurate labels for the words in the corpus. Our analysis in Section~\ref{sec:mentiondensity} shows that after annotation, 96\% of content words and 75\% of all words have at least one gold semantic category label, suggesting this protocol allows for near-complete coverage of content words at a fraction of the time required to make accurate 601-class annotation judgements at scale. Both our interannotator agreement (included below) and automatic classification performance are high, suggesting adequate annotated label accuracy.
{\flushleft\textbf{Label distribution:}} Named entity corpora often have many labels in their typologies, but the majority of mentions tend to cluster around a small set of possible labels \cite{choi-etal-2018-ultra}. The distribution of most frequent labels after annotation is shown in Table~\ref{tab:examplesemanticcategories}. The usage of the 601 total semantic class labels in this corpus is well distributed, with the 356 most-frequent types covering 95\% of the total mentions, while 479 types cover 99\% of mentions. At the 99\% level, categories (for example, \textit{Geometric Qualities}, such as \textit{angle, slope, or circumference}) still contain 16 mentions, highlighting the scale of the corpus.
{\flushleft\textbf{Interannotator agreement:}} Each question was annotated by a single annotator. A second annotator was trained in the annotation procedure and re-annotated 50 questions totalling 1,756 tokens. Between both annotators, a total of 1,369 mentions were annotated with semantic class labels. Total percent agreement across both annotators was 76\%.\footnote{Because the bootstrapped preannotation procedure reduces the set of possible labels for a given mention from 601 to an average of approximately 2 (the average number of preannotated labels per annotated word), Cohen's Kappa \cite{cohen1960coefficient} would either be artificially inflated (if treating the annotation as a 601 class labeling problem) or reduced (if treating annotation as a 2 class problem). As such we report raw percent agreement, which (as critiqued by Cohen) has known problems when dealing with highly skewed frequency distributions of labels, particularly when few labels are present. Here, the number of label categories is high, and (as shown in Table~\ref{tab:examplesemanticcategories}) the frequency of labels is well distributed across the label set. As such, the inflation of the percent agreement statistic is likely to be minimal. } Upon inspection, labeling multi-word sequences as either a single mention or multiple smaller mentions was a frequent source of disagreement. When these cases were removed, percent agreement rose to 83\%.
~\\
\subsection{Mention density comparison}
\label{sec:mentiondensity}
To increase the utility of our common entity corpus for downstream tasks, one of the design goals was to provide at least one high-level semantic category to nearly every word in the corpus. To measure this we define the notion of the \textit{mention density} of a corpus as the proportion of words that contain at least one entity label.\footnote{Specifically, the proportion of non-punctuation tokens in a BIO-formatted corpus that are labelled with either a B (beginning) or I (inside) tag.}
We compare the mention density of this corpus with the English subsets of the four benchmark named entity recognition corpora listed below:
{\flushleft\textbf{CoNLL}} \cite{sang2003introduction}: The CoNLL 2003 Named Entity Recognition Shared Task corpus, which includes 4 entity labels that are a subset of the MUC-6 typology: \textit{person, location, organization} and \textit{miscellaneous}.
{\flushleft\textbf{OntoNotes 5.0}} \cite{weischedel2013ontonotes}: A large multi-genre corpus of news media, blog, newsgroup, and conversational text, annotated with 18 entity labels, including the MUC-6 types.
{\flushleft\textbf{BBN}} \cite{weischedel2005bbn}: A corpus of news text annotated with 21 course entity types, including 12 named entity types (e.g. \textit{person, organization, product}) and 7 numeric types (e.g. \textit{date, percent, cardinal number}). The full set of entity labels includes 64 fine-grained types.
{\flushleft\textbf{GUM}} \cite{Zeldes2017}: An open-domain corpus annotated with a collapsed set of OntoNote entities reduced to 11 entity types, such as \textit{person, organization,} or \textit{place}. Two additional catch-all tags are added, \textit{object} and \textit{abstract}, which provide high-level but minimally informative categorical information for large noun phrases. Approximately 40.5\% of the labelled words in this corpus are labelled as either \textit{object} or \textit{abstract}.\\
\hspace{2em} The analysis of mention density is shown in Table~\ref{tab:summarystatistics}. Overall, the mention density of this science corpus is 75\%, meaning that 75\% of all words in the corpus are annotated with at least one high-level semantic category. When considering only content words (here, determined to be \textit{nouns, verbs, adjectives, adverbs, and numbers}), this proportion increases to 96\%. The mention density for the named entity corpora examined in Table~\ref{tab:summarystatistics} ranges between 12\% and 59\% for all words, and 18\% to 62\% when considering only content words. At 62\%, the GUM corpus contains the next-nearest mention density to the ScienceExamCER corpus, however a large portion of those mentions (40.5\%) of words) use the high-level \textit{object} or \textit{abstract} labels, and as such are of limited informativeness to downstream tasks. BBN, the corpus with the next-nearest mention density to GUM, has labels for only 36\% of it content words, and 25\% of all words.
\section{Experimental Results}
\subsection{Model}
Our semantic class labeling task is conceptually similar to named entity recognition or entity typing, only requiring a label for nearly every word in an input sentence. In light of this, here we use an off-the-shelf named entity recognition model, and show it also performs well on the densely-labeled common entity recognition task.
\hspace{2em} Recently, pretrained bidirectional encoder representation from transformer (BERT) models \cite{devlin2018bert} have shown state-of-the-art performance at both named entity recognition as well as a variety of other token-level classification tasks. In this work, we use an off-the-shelf implementation of a BERT-based named entity recognition system, BERT-NER\footnote{\url{https://github.com/kamalkraj/BERT-NER}}. Most approaches to named entity recognition model the task as a single-label prediction task, where each word has at most one label. We modify the BERT-NER implementation to allow for multi-label predictions using the following method.
\hspace{2em} Given a sentence $S$ consisting of $L$ tokens, such that $S=(x_1, x_2, ..., x_L)$, the original BERT-based token classification model generates $L$ respective $M$-dimensional encodings $(x_1, x_2, ..., x_L)$, one for each token. These encodings then pass through a \textit{softmax} layer and make use of a multi-classes cross entropy loss function that generates a single class prediction per token.
We adapt this system to multi-label classification by using a \textit{sigmoid} function and binary cross entropy in place of the original loss function to allow the classifications for each token to return non-zero values for more than one class. More formally, our loss function becomes:
\begin{equation}
\begin{aligned}
L_{multilabel} = {}&-\frac{1}{M}\sum_{m=1}^{M}[\tilde{y}_l^m\cdot{log\sigma(x_l^m)} + {} \\
&(1-\tilde{y}_l^m)\cdot{log(1-\sigma(x_l^m))}] \\
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\sigma(x_l^m)&=\frac{1}{1+e^{-x_l^m}}
\end{aligned}
\end{equation}
where $M$ is the number of total classes, $x_l$ is $M$-dimensional encoding for the $l$-th token in sentence, $\tilde{y}_l$ is the $l$-th token's gold label vector, and $\sigma$ is the \textit{sigmoid} activation function.
{\flushleft\textbf{Folds:}} Because of the expense associated with annotating a large corpus, only the training and development subsets of the ARC corpus were manually annotated with semantic class labels. As such we repurpose the original development set for testing, and hold out 20\% of the training corpus for development. Summary statistics on these folds are provided in Table~\ref{tab:summarystatisticsquestionset}.
{\flushleft\textbf{Hyperparameters:}} We make use of the pre-trained English \texttt{BERT-Base-cased} model\footnote{\url{https://github.com/google-research/bert}}, with a maximum sequence length of 64. The threshold for the sigmoid activation layer was tuned on the development set, with a value of 0.4 found to provide good performance. The large number of possible class labels in our task compared with typical named entity recognition datasets, combined with the modified multi-label loss function, necessitated significantly longer training times for the model to converge. We empirically found that the model tended to converge by 140 epochs, which took approximately 5 hours to train using dual RTX2080Ti GPUs. Classification of the entire test dataset is comparatively fast, providing semantic class labels at a rate of approximately 900 questions (35,000 words) per minute, enabling the pre-trained model to be run on other science-domain corpora (for example, textbooks, study guides, Simple Wikipedia, or other grade-appropriate knowledge resources) at scale.
\subsection{Evaluation}
\begin{table}
\centering
\begin{tabular}{lcccc}
\textbf{Model} & \textbf{Fold} & \textbf{Prec.} & \textbf{Recall} & \textbf{F1} \\
\hline
BERT-NER & dev & 0.84 & 0.85 & 0.84 \\
BERT-NER & test & 0.85 & 0.86 & 0.85 \\
\end{tabular}
\caption{Performance on the 601-category fine-grained semantic classification task on the development and test folds using the BERT-NER model. \label{tab:performance}}
\end{table}
The results for our semantic classification task on the ScienceExamCER corpus using the 601-class fine-grained typology are shown in Table~\ref{tab:performance}. We evaluate entity classification performance using the standard definitions of Precision, Recall, and F1. Overall classification performance is high, reaching 0.85 F1 on the held-out test set. This suggests the common entity recognition performance is sufficiently high to be useful for a variety of downstream tasks. To further characterize performance, we investigate how the availability of training data affects this fine-grained classification task, as well as common classes of prediction errors the BERT-NER model makes.
\subsection{Performance vs Training Data}
\begin{figure}[!t]
\centering
\includegraphics[scale=0.30]{figures/datadependence1.pdf}
\caption{Classification performance (F1) versus the number of training epochs when training the model with less data. Series represent training the model with the entire training set, or randomly subsampled proportions of training data summing to 75\%, 50\%, and 25\% of the original training set size. Each point represents the average of 5 randomly subsampled training sets. \label{fig:datadependence}}
\end{figure}
Manually annotating fine-grained mentions in large corpora is expensive and time consuming. To investigate how classification performance varies with availability of training data, we randomly subsampled smaller training sets from our full training corpus that were 25\%, 50\%, or 75\% as large, corresponding to spending approximately 50, 100, or 150 hours at the manual annotation task, respectively. The results are shown in Figure~\ref{fig:datadependence}. With only 25\% of training data available, F1 performance decreases dramatically from 0.85 to 0.50. 50\% of training data decreases classification performance by 7 points, while 75\% of available training data decreases classification performance by 2 points. This suggests that the scale of training data generated in this work is provides near saturated performance using the BERT-NER model, and that annotating the remainder of available standardized science exam questions in the ARC corpus would likely result in only a minimal increase on classification performance.
\subsection{Error Analysis}
\begin{table}
\centering
\begin{tabular}{lc}
\textbf{Error Class} & \textbf{Prop.} \\
\hline
\textit{Predicted label also good} & 24\% \\
\textit{Model did not generate prediction} & 24\% \\
\textit{Multiple gold labels, one found} & 21\% \\
\textit{Predicted label semantically near gold label} & 17\% \\
\textit{Gold label incorrect} & 7\% \\
\textit{Multi-word Expression} & 6\% \\
\textit{Predicted label using incorrect word sense} & 5\% \\
\end{tabular}
\caption{An analysis of common categories of model prediction errors, as a proportion of the first 100 errors on the test set. Note that a given errorful prediction may belong to more than one category, and as such the proportions do not sum to 100\%. \label{tab:erroranalysis}}
\end{table}
To better understand the sources of error in our model, we conducted an analysis of the first 100 errorful predictions on the test set, with the results shown in Table~\ref{tab:erroranalysis}. Nearly one third of errors are due to issues with the annotation, such as a mention missing an additional label that is also good (24\% of errors), or the manually annotated gold label being incorrect (7\% of errors). For a substantial portion of errors (24\%), no single semantic class rose to meet the activation threshold of the sigmoid layer and the model did not produce a prediction for that word, while, similarly, in 21\% of cases only one label of a multi-label word was produced. The remaining errors broadly cluster around technical challenges in determining the semantics of each category, including word-sense disambiguation (5\% of errors), locating multi-word expressions (6\% of errors), or predicting a label whose category is semantically similar to the gold label (17\% of errors).
\section{Conclusion}
We present ScienceExamCER, a densely annotated corpus of science exam questions for common entity recognition where nearly every word is annotated with fine-grained semantic classification labels drawn from a manually-constructed typology of 601 semantic classes. We demonstrate that BERT-NER, an off-the-shelf named entity recognition model, achieves 0.85 F1 on classifying these fine-grained semantic classes on unseen text in a multi-label setting. The data and code are released with the goal of supporting downstream tasks in question answering that are able to make use of this dense semantic category annotation.
\section{Supplementary Material}
The annotated corpora, fine-grained typology, and pretrained models for this work are available at \url{http://cognitiveai.org/explanationbank/}. A truncated version of the typology is included in the \texttt{Appendix} below.
\section{Acknowledgements}
This work was supported by the Allen Institute for Artificial Intelligence and the National Science Foundation (NSF Award \#1815948, ``Explainable Natural Language Inference'', to PJ). We thank Peter Clark for thoughtful discussions on this work and comments on an earlier draft.
\section{Bibliographical References}
\label{main:ref}
\bibliographystyle{lrec}
|
1,108,101,563,260 | arxiv | \section{Introduction}
Top-quark pair production $pp \to t(p_t) + \bar{t}(p_{\bar{t}}) + X$ is one of the most important scattering processes at the Large Hadron Collider (LHC). The experimental measurements of its differential cross sections have achieved remarkably high precisions \cite{Aaboud:2018eqg, Sirunyan:2018ucr, Sirunyan:2018wem, Sirunyan:2019lnl} in the Run 2 of the LHC with a center-of-mass energy $\sqrt{s} = \unit{13}{\TeV}$. In the meantime, the theoretical modeling of the kinematic distributions in quantum chromodynamics (QCD) has also been greatly improved with the emergence of the next-to-next-to-leading order (NNLO) predictions \cite{Czakon:2015owf, Czakon:2016dgf, Catani:2019hip} supplemented with the resummation of large logarithms to the next-to-next-to-leading logarithmic accuracy (NNLL$'$) \cite{Pecjak:2016nee, Czakon:2018nun, Pecjak:2018lif}. The complete next-to-leading order (NLO) corrections incorporating electroweak (EW) effects \cite{Bernreuther:2010ny, Pagani:2016caq} have also been combined with the higher order QCD corrections, resulting in the state-of-the-art standard model (SM) predictions for this process \cite{Czakon:2017wor, Czakon:2019txp}.
In spite of these developments, some small discrepancies between the high precision theoretical and experimental results are persistent. A notable one is the $t\bar{t}$ invariant-mass distribution near the $2m_t$ threshold measured by the CMS collaboration at the \unit{13}{\TeV} LHC \cite{Sirunyan:2018wem, Sirunyan:2018ucr}, where $m_t$ is the top-quark mass, and the invariant mass $M_{t\bar{t}}$ is defined as $M_{t\bar{t}}^2 \equiv (p_t + p_{\bar{t}})^2$.
In Figure~\ref{fig:old_results}, we depict the averaged $M_{t\bar{t}}$ distribution in the \unit{[300,380]}{\GeV} range. The CMS result \cite{Sirunyan:2018ucr} is shown as the green band, which reflects the combined statistical and systematical uncertainty. The central values of various theoretical predictions (NNLO from \cite{Czakon:2016dgf}, NNLO+EW from \cite{Czakon:2017wor}, and NNLO+NNLL$'$ from \cite{Czakon:2018nun}) are shown in comparison. One can see that there exists a clear gap between the experimental and theoretical results.
It can also be seen that the theoretical predictions in this region strongly depend on $m_t$.
As a result, this discrepancy has a strong impact on the extraction of $m_t$ from kinematic distributions \cite{Sirunyan:2019zvx}, which favors a value of $m_t$ significantly lower than the current world average.
Therefore, it is interesting to ask whether the top quark is indeed lighter than we expect, or there are missing contributions in this region which are not incorporated in the most up-to-date theoretical predictions.
\begin{figure}[t!]
\centering
\includegraphics[width=0.9\linewidth]{CMS_vs_old_results}
\vspace{-5ex}
\caption{\label{fig:old_results}The averaged $t\bar{t}$ invariant mass distribution in the \unit{[300,380]}{\GeV} range. The CMS result \cite{Sirunyan:2018ucr} is shown as the green band, while the central-values of various theoretical predictions are shown in comparison.}
\end{figure}
In this Letter, we study a class of higher order QCD corrections in the threshold region $M_{t\bar{t}} \sim 2m_t$. They originate from exchanges of Coulomb-type gluons as well as soft gluons between the top and anti-top quarks. The leading contributions take the form $(\alpha_s/\beta)^n$, where $\beta \equiv \sqrt{1-4m_t^2/M_{t\bar{t}}^2}$ is the velocity of the top and anti-top quarks in the $t\bar{t}$ rest frame. The region $M_{t\bar{t}} \sim 2m_t$ corresponds to $\beta \sim 0$, where the top and anti-top quarks are slowly moving with respect to each other. In the small $\beta$ region, the $1/\beta^n$ contributions (as well as $\ln^n\beta$ ones) are enhanced and lead to large corrections beyond the NNLO(+NNLL$'$) results. These corrections are divergent as $\beta \to 0$, signaling a breakdown of the perturbative expansion in this limit. At NNLO (order $\alpha_s^2$), this does not pose a severe problem since the integration over $M_{t\bar{t}}$ is still convergent, which gives a finite prediction for the averaged differential cross section as shown in Figure~\ref{fig:old_results}. However, at order $\alpha_s^4$ and beyond, even the integration over $M_{t\bar{t}}$ becomes divergent, and one has to perform an all-order resummation to restore the validity of theoretical predictions.
Furthermore, at fixed orders in perturbation theory, $M_{t\bar{t}} \geq 2m_t$ due to phase-space constraint. This is not physical since one knows that bound-state effects can make the mass of the $Q\bar{Q}$ system smaller than $2m_Q$, where $Q$ is a heavy quark. For charm quarks and bottom quarks, such bound-state effects are genuinely non-perturbative. For top-quark pairs, the bound-state effects can be (to a good approximation) studied in perturbation theory, as long as we integrate in a large enough range around the threshold $M_{t\bar{t}} = 2m_t$.
In the following part of this Letter, we present a factorization formula for the partonic differential cross sections valid in the region $M_{t\bar{t}} \sim 2m_t$, which schematically takes the form $d\hat{\sigma} \sim H \times J$. The hard function $H$ describes exchanges and emissions of hard gluons with typical momenta of $\mathcal{O}(M_{t\bar{t}})$, while the potential function $J$ resums exchanges of Coulomb and soft gluons between the top and anti-top quarks.
Note that we are not taking the limit $\sqrt{\hat{s}} \to M_{t\bar{t}}$ here, where $\sqrt{\hat{s}}$ is the partonic center-of-mass energy. As a result, extra hard radiations are allowed and are incorporated in our framework into the hard function $H$. Therefore, our factorization formula is intrinsically different from those in \cite{Hagiwara:2008df, Beneke:2010da, Beneke:2011mq}, where the $\sqrt{\hat{s}} \to M_{t\bar{t}}$ approximation has been employed. Our formalism is similar to that of \cite{Petrelli:1997ge, Kiyo:2008bv}, but takes into account the kinematic dependence of the renormalization and factorization scales, and differs in the treatment of subleading corrections in $\beta$.
Based on our factorization formula, we then present numeric results that are relevant to LHC phenomenology. We show that the resummation effects indeed enhance the $M_{t\bar{t}}$ distribution significantly in the threshold region, and make the theoretical prediction more compatible with the experimental data.
\section{Factorization near threshold}
To set up the stage, we begin with the $t\bar{t}$ invariant mass distribution written as a convolution of the partonic luminosity functions and partonic differential cross sections
\begin{equation}
\frac{d\sigma}{dM_{t\bar{t}}} = \sum_{i,j} \int^1_{\tau} \frac{dz}{z} \frac{\tau}{z} \int d\Theta \, \frac{d\hat{\sigma}_{ij}(z,\mu_f)}{dM_{t\bar{t}} \, d\Theta} \, f\!\!f_{ij}\Big(\frac{\tau}{z},\mu_f\Big) \, ,
\label{eq:fac1}
\end{equation}
where $i,j \in \{q,\bar{q},g\}$ denote partons within the colliding protons; $z \equiv M_{t\bar{t}}^2/\hat{s}$, $\tau \equiv M_{t\bar{t}}^2/s$ and $\mu_f$ is the factorization scale. The parton luminosity functions $f\!\!f_{ij}(y,\mu_f)$ are defined by
\begin{equation}
f\!\!f_{ij}(y,\mu_f) \equiv \int^1_{y} \frac{d\xi}{\xi} \, f_{i/p}(\xi,\mu_f) \, f_{j/p}(y/\xi,\mu_f) \, ,
\end{equation}
where $f_{i/p}$ is the parton distribution function (PDF) of the parton $i$ in the proton $p$. Note that in writing down Eq.~\eqref{eq:fac1}, we have taken into account that the factorization scale $\mu_f$ may depend on kinematic variables (which we collectively denote as $\Theta$) other than $m_t$ and $M_{t\bar{t}}$. This is necessary since we will combine our result with the NNLO result of \cite{Czakon:2016dgf}, in which the scales are correlated with the variable $H_T \equiv \sqrt{p_{T,t}^2+m_t^2} + \sqrt{p_{T,\bar{t}}^2+m_t^2}$, where $p_{T,t}$ and $p_{T,\bar{t}}$ are the transverse momenta of the top and anti-top quarks, respectively.
We are concerned with QCD corrections to the partonic differential cross sections. At the leading order (LO) in $\alpha_s$, only the $q\bar{q}$ and $gg$ channels give non-vanishing contributions
\begin{align}
\frac{d^2\hat{\sigma}_{q\bar{q}}^{(0)}}{dM_{t\bar{t}} \, d\cos\theta_t} &= \frac{2\pi\beta\alpha_s^2(\mu_r)}{M_{t\bar{t}}^3} \frac{C_FC_A}{9} \, c_{q\bar{q},8}(\cos\theta_t) \, \delta(1-z) \, , \nonumber
\\
\frac{d^2\hat{\sigma}_{gg}^{(0)}}{dM_{t\bar{t}} \, d\cos\theta_t} &= \frac{2\pi\beta\alpha_s^2(\mu_r)}{M_{t\bar{t}}^3} \bigg[ \frac{C_F}{32} \, c_{gg,1}(\cos\theta_t) \nonumber
\\
&+ \frac{(C_A^2-4)C_F}{64} \, c_{gg,8}(\cos\theta_t) \bigg] \, \delta(1-z) \, ,
\label{eq:sigma0_hat}
\end{align}
where $\mu_r$ is the renormalization scale and $\theta_t$ is the scattering angle of the top quark in the $t\bar{t}$ rest frame. The coefficients $c_{ij,\alpha}$, with $\alpha=1,8$ labelling the color configuration of the $t\bar{t}$ system, are given by
\begin{align}
c_{q\bar{q},8}(\cos\theta_t) &= \frac{1}{4} \big[ 2 - \beta^2(1-\cos^2\theta_t) \big] \, , \nonumber
\\
c_{gg,1}(\cos\theta_t) &= \frac{1}{2(1-\beta^2\cos^2\theta_t)^2} \Big[ 4 - 2(1-\beta^2)^2 \nonumber
\\
&\hspace{-2em} - 2\beta^2(1-\beta^2\cos^2\theta_t) - (1+\beta^2\cos^2\theta_t)^2 \Big] \, ,
\\
c_{gg,8}(\cos\theta_t) &= 2c_{gg,1}(\cos\theta_t) \, \bigg[ \frac{16}{5} - \frac{9}{10} (3-\beta^2\cos^2\theta_t) \bigg] \, . \nonumber
\end{align}
Note that Eq.~\eqref{eq:sigma0_hat} is proportional to $\beta$ due to the two-body phase space. This leads to the fact that the LO differential cross section approaches zero at the threshold $M_{t\bar{t}} = 2m_t$. This is no longer true at higher orders in $\alpha_s$, essentially because of the $1/\beta^n$ terms mentioned in the Introduction. More precisely, the NLO differential cross section approaches a positive constant in the $\beta \to 0$ limit, while the NNLO and higher order ones diverge in that limit. This behavior makes the perturbative expansion badly convergent in the threshold region. In the following, we analyze the factorization properties of the differential cross sections and resum the large $1/\beta^n$ and $\ln^n\beta$ corrections to all orders in $\alpha_s$. This is the only way to restore the predictive power of perturbative QCD near threshold.
In the threshold limit $M_{t\bar{t}} \to 2m_t$ or $\beta \to 0$, there exist large hierarchies among the energy scales $M_{t\bar{t}}$, $M_{t\bar{t}}\beta$ and $M_{t\bar{t}}\beta^2$. Using the method of regions, we identify the following momentum modes in the $t\bar{t}$ rest frame \cite{Beneke:1997zp}
\begin{align}
\text{hard: } \; &k^\mu \sim M_{t\bar{t}} \, , & \text{potential: } \; &k^0 \sim M_{t\bar{t}}\beta^2 \, , \; \vec{k} \sim M_{t\bar{t}}\beta \, , \nonumber
\\
\text{soft: } \; &k^\mu \sim M_{t\bar{t}}\beta \, , & \text{ultrasoft: } \; &k^\mu \sim M_{t\bar{t}}\beta^2 \, .
\label{eq:modes}
\end{align}
The top and anti-top quarks move very slowly and are non-relativistic objects whose residue momenta correspond to the potential mode. They can interact with each other through soft, ultrasoft and potential gluons which are described by the effective Lagrangian of potential non-relativistic QCD (pNRQCD) \cite{Brambilla:1999xf, Beneke:1999qg}. The hard mode describes fluctuations with typical momenta of $\mathcal{O}(M_{t\bar{t}})$, resulting in Wilson coefficients of effective operators in pNRQCD. We note that there are no collinear modes in our setup. This is related to the fact that we are not taking the limit $\sqrt{\hat{s}} \to M_{t\bar{t}}$, so that the extra emissions are not constrained. As a result, we do not need to employ the soft-collinear effective theory, as opposed to \cite{Beneke:2010da, Beneke:2011mq}.
Using the method of effective field theory (EFT), we derive a factorization formula for the partonic cross sections in the threshold region. The power expansion of pNRQCD is according to the counting $\alpha_s \sim \beta \sim 1/\ln\beta$. Up to the next-to-leading power (NLP), the factorization formula reads
\begin{multline}
\frac{d\hat{\sigma}^{\text{NLP}}_{ij}}{dM_{t\bar{t}} \, d\Theta} = \frac{16\pi^2\alpha_s^2(\mu_r)}{M_{t\bar{t}}^5} \sqrt{\frac{M_{t\bar{t}}+2m_t}{2M_{t\bar{t}}}} \sum_{\alpha} c_{ij,\alpha}(\cos\theta_t)
\\
\times H_{ij,\alpha}(z,M_{t\bar{t}},Q_T,Y,\mu_r,\mu_f) \, J^{\alpha}(E) + \mathcal{O}(\beta^3) \, ,
\label{eq:fac2}
\end{multline}
where $H_{ij,\alpha}$ are hard functions describing hard gluon exchanges and emissions, which depend on the transverse momentum $Q_T$ and the rapidity $Y$ of the $t\bar{t}$ pair; $J^\alpha$(E) with $E=M_{t\bar{t}}-2m_t$ are potential functions describing exchanges of potential, soft and ultrasoft gluons between the top and anti-top quarks.
Note that at NLP, ultrasoft gluon exchanges among initial-state partons and the $t\bar{t}$ pair do not contribute. As a result there is no soft function in our factorization formula. Such contribution could be present at higher powers in $\alpha_s$ and $\beta$, which is an interesting subject of study in the future.
The prefactors in Eq.~\eqref{eq:fac2} are chosen such that the LO cross sections in the EFT reproduce the exact ones in Eq.~\eqref{eq:sigma0_hat}. These prefactors take into account subleading corrections in $\beta$ at LO, and have significant impacts at higher orders in $\alpha_s$. This, together with the explicit kinematic dependence of the hard function, makes our factorization formula substantially different from those in \cite{Petrelli:1997ge, Kiyo:2008bv}.
The hard functions can be calculated order-by-order in $\alpha_s$ in terms of differential cross sections in the limit $p_{t}=p_{\bar{t}}=P_{t\bar{t}}/2$. For resummation at NLP, we need the hard functions up to the NLO. The LO hard functions are given by
\begin{align}
H^{(0)}_{q\bar{q},1} &= 0 \, , \quad
H^{(0)}_{q\bar{q},8} = \frac{C_A C_F}{9} \, \delta(1-z) \, \delta(Q_T) \, \delta(Y) \, , \nonumber
\\
H^{(0)}_{gg,1} &= \frac{C_F}{32} \, \delta(1-z) \, \delta(Q_T) \, \delta(Y) \, , \nonumber
\\
H^{(0)}_{gg,8} &= \frac{(C_A^2-4)C_F}{64} \, \delta(1-z) \, \delta(Q_T) \, \delta(Y) \, .
\end{align}
We have calculated the NLO corrections to the hard functions analytically, which were not known in the literature. They receive contributions from both virtual exchanges and real emissions of gluons. The virtual and real contributions are separately infrared divergent. The divergences cancel when combining the two together with the PDF counterterms. After the cancellation, we obtain finite NLO hard functions involving singular distributions of $1-z$, $Q_T$ and $Y$. The results are more complicated than those in the literature \cite{Petrelli:1997ge, Kiyo:2008bv} since we need to keep the dependencies on kinematic variables in order to use the $H_T$-based renormalization and factorization scales.
The potential function $J^\alpha(E)$ is related to the imaginary part of the pNRQCD Green function $G^\alpha(\vec{r}_1,\vec{r}_2;E)$ of the $t\bar{t}$ pair at origin. It receives contributions from potential, soft and ultrasoft modes.
The explicit expressions for the Green function up to NLP can be found in, .e.g., \cite{Beneke:2011mq, Ju:2019lwp}.
The finite width of the top quark can be taken into account by the replacement $E \to E+i\Gamma_t$ in the potential function. Such a treatment is valid at NLP, but additional considerations are required at higher powers (we count $\Gamma_t/m_t \sim \beta^2$) \cite{Hoang:1999zc, Beneke:2003xh, Beneke:2004km, Beneke:2010mp, Beneke:2017rdn}.
Combining the hard and potential functions, we can now produce NLP resummed predictions for the $M_{t\bar{t}}$ distribution near threshold. These can be further combined with fixed-order ones via the matching formula
\begin{equation}
\frac{d\sigma^{\text{(N)NLO+NLP}}}{dM_{t\bar{t}}} = \frac{d\sigma^{\text{NLP}}}{dM_{t\bar{t}}} + \frac{d\sigma^{\text{(N)NLO}}}{dM_{t\bar{t}}} - \frac{d\sigma^{\text{(n)nLO}}}{dM_{t\bar{t}}} \, ,
\label{eq:matching}
\end{equation}
where we have used ``nLO'' and ``nnLO'' to denote fixed-order expansions of the NLP resummed result \eqref{eq:fac2} up to the second and third order in $\alpha_s(\mu_r)$. Their differences with respect to the full NLO and NNLO results are higher-power terms in $\beta$ that we want to take into account through the matching procedure.
\section{Numeric results}
We now present the numeric results based on our factorization formula \eqref{eq:fac2} and the matching formula \eqref{eq:matching}. Throughout our calculation we take $m_t=\unit{172.5}{\GeV}$, $\Gamma_t=\unit{1.4}{\GeV}$ and use the NNPDF3.1 NNLO PDFs with $\alpha_s(m_Z)=0.118$. Since the small-$\beta$ resummation is valid only at low $M_{t\bar{t}}$, we restrict ourselves to $M_{t\bar{t}} \in \unit{$[300,380]$}{\GeV}$ in this work, which is the region most sensitive to $m_t$. Following \cite{Czakon:2017wor, Czakon:2018nun}, we choose the default values of the renormalization scale $\mu_r$ and the factorization scale $\mu_f$ to be $H_T/4$, the corresponding results are regarded as the central values. We then vary the scales simultaneously up and down by a factor of 2 to estimate the remaining theoretical uncertainties. Note that with such a procedure, the error bands are not necessarily symmetric around the central values.
\begin{figure}[t!]
\centering
\includegraphics[width=0.9\linewidth]{NLO_vs_approx}
\vspace{-4ex}
\caption{\label{fig:nlo_vs_approx}The exact NLO correction and its various approximations in the range \unit{[340-380]}{\GeV}.}
\end{figure}
First of all, in Fig.~\ref{fig:nlo_vs_approx} we show the exact NLO correction computed using \texttt{MCFM}~\cite{Campbell:2010ff}, in comparison with its approximations in various kinematic limits. Such a comparison is important to assess the validity of the approximations within the phase-space region of interest. Fig.~\ref{fig:nlo_vs_approx} has a few significant implications we'd like to elaborate.
\textbf{1)} The $\beta \to 0$ result corresponds to the second term in the fixed-order expansion of our NLP formula \eqref{eq:fac2}. One can see that it provides an excellent description of the full NLO correction in the range \unit{$[340,380]$}{\GeV}. This demonstrates that our small-$\beta$ approximation is indeed valid, and supports the resummation performed in this work.
\textbf{2)} The $z \to 1$ result corresponds to the ``soft'' or ``threshold'' limit $\sqrt{\hat{s}} \to M_{t\bar{t}}$ considered in, e.g., \cite{Ahrens:2010zv, Pecjak:2016nee}. It should be stressed that the concept of ``soft'' or ``threshold'' there has a completely different meaning than those used in this work. We observe that, while not perfect, the $z \to 1$ limit provides a reasonable approximation to the full result in the low $M_{t\bar{t}}$ region. It is known that this limit works better in the high $M_{t\bar{t}}$ region \cite{Ahrens:2010zv, Pecjak:2016nee}.
\textbf{3)} Finally, the double limit $\beta \to 0$ \emph{and} $z \to 1$ corresponds to the ``soft'' limit considered in \cite{Hagiwara:2008df, Kiyo:2008bv}. One can see that this approximation does \emph{not} capture the dominant contribution at NLO. This is essentially the reason why we do not consider such a ``soft'' resummation in this work.
\begin{figure}[t!]
\centering
\includegraphics[width=0.9\linewidth]{NLP_vs_expansion}
\vspace{-4ex}
\caption{\label{fig:nlp_expansion}The NLP resummed result and its fixed-order expansion.}
\end{figure}
We now turn to compare the NLP resummed result Eq.~\eqref{eq:fac2} and its fixed-order expansion. As in Eq.~\eqref{eq:matching}, we label the expansion up to the $i$-th term ($i=0,1,2,\ldots$) as n$^{i}$LO. Here the lower-case `n' implies that these are approximations to the full N$^{i}$LO results in the $\beta \to 0$ limit. Note that the n$^0$LO result equals exactly to the full LO result due to the prefactors in Eq.~\eqref{eq:fac2}. We show such a comparison in Fig.~\ref{fig:nlp_expansion}, from which we can draw several important conclusions.
\textbf{1)} The fixed-order expansion converges rather quickly when $M_{t\bar{t}}$ is much larger than $2m_t$. However, when $M_{t\bar{t}}$ approaches the threshold, the behavior becomes out-of-control. In particular, the n$^3$LO result tends to $+\infty$ while the n$^4$LO one tends to $-\infty$ in the $\beta \to 0$ limit ($M_{t\bar{t}} \to \unit{345}{\GeV}$).
\textbf{2)} The singularity at $\beta = 0$ is regularized by the resummation effects, and we obtain a finite prediction near $M_{t\bar{t}} = 2m_t$ with the NLP resummation formula. One also finds that in the resummed result the region $M_{t\bar{t}} < 2m_t$ is allowed due to bound-state effects. The shape of the NLP curve for $M_{t\bar{t}} < 2m_t$ depends crucially on the top quark width. However, we have checked that the integrated cross section in the \unit{$[300,380]$}{\GeV} range is insensitive to the value of $\Gamma_t$.
\textbf{3)} As a final implication of Fig.~\ref{fig:nlp_expansion}, we note that the NLP curve almost overlaps with the nLO and n$^2$LO curves for $M_{t\bar{t}} > \unit{360}{\GeV}$. This means that in this region, the matched (N)NLO+NLP results of Eq.~\eqref{eq:matching} are governed by the fixed-order (N)NLO calculations. Effectively, this shows that we are not applying the small-$\beta$ resummation to regions where one may worry about the break down of the EFT description. As a matter of fact, we have checked that the dominant beyond-NNLO correction comes from the region $M_{t\bar{t}} < \unit{350}{\GeV}$, where $\beta < 0.17$ and pNRQCD is perfectly applicable.
\begin{figure}[t!]
\centering
\includegraphics[width=0.9\linewidth]{CMS_vs_our_results}
\vspace{-7ex}
\caption{\label{fig:our_results}The averaged $t\bar{t}$ invariant mass distribution in the \unit{[300,380]}{\GeV} range. The CMS result \cite{Sirunyan:2018ucr} is shown as the green band. The various theoretical predictions are shown in comparison, with NNLO+NLP being our best prediction.}
\end{figure}
\begin{table}[t!]
\centering
\begin{tabular}{|c|c|}
\hline
CMS & $1.664^{+0.166}_{-0.166}$
\\ \hline
NLO & $1.208^{+0.150}_{-0.139}$
\\ \hline
NNLO & $1.319^{+0.035}_{-0.070}$
\\ \hline
NLO+NLP & $1.367^{+0.107}_{-0.124}$
\\ \hline
NNLO+NLP & $1.434^{+0.014}_{-0.060}$
\\ \hline
\end{tabular}
\caption{\label{tab:our_results}The differential cross sections (in unit of pb/GeV) entering Fig.~\ref{fig:our_results}.}
\end{table}
Finally, we match our resummed calculation to the NLO and NNLO results according to Eq.~\eqref{eq:matching}. The NLO results are computed using \texttt{MCFM} and the NNLO results are obtained from \cite{Czakon:2016dgf, Catani:2019hip, Britzger:2012bs, Czakon:2017dip}. The NLO+NLP and NNLO+NLP results are shown in Fig.~\ref{fig:our_results} and Tab.~\ref{tab:our_results} together with the NLO and NNLO ones, compared against the CMS measurement. We find that the resummation effects enhance the NNLO differential cross section by about 9\%, and make the theoretical prediction more compatible with experimental data. As discussed in the Introduction, this enhancement will have significant impacts on the extraction of the top-quark mass from kinematic distributions. Although we can't repeat the analysis of \cite{Sirunyan:2019zvx}, we can estimate the impacts by studying the $m_t$-dependence of the NLO
(used in the fit of \cite{Sirunyan:2019zvx}) and NNLO+NLP predictions for the \unit{[300,380]}{\GeV} range. For example, the central value of the NNLO+NLP differential cross section for $m_t = \unit{172.5}{\GeV}$ is given by \unit{1.434}{\picobarn\per\GeV}. To achieve the same differential cross section using the NLO calculation, one needs to lower the value of $m_t$ down to about \unit{171}{\GeV}.
This roughly explains the outcome of \cite{Sirunyan:2019zvx}, and shows that using our prediction in the fit will lead to a shift of about \unit{1.5}{\GeV} resulting in a value closer to the world average.
\section{Conclusion}
In this Letter, we studied the $t\bar{t}$ invariant-mass distribution near the $2m_t$ threshold. In this region, there exist a small gap between the most up-to-date theoretical predictions and the experimental measurements by the CMS collaboration at the LHC. This leads to a tension between the values of $m_t$ determined from direct measurements and from fitting kinematic distributions in $t\bar{t}$ production.
We show that higher-order non-relativistic effects lead to large corrections to the differential cross section in the threshold region, which were not included in the state-of-the-art theoretical predictions. We derive a factorization formula to resum such corrections to all orders in the strong coupling, and calculate necessary ingredients to perform the resummation at next-to-leading power. We combine the resummation with NLO and NNLO results, and present numeric results relevant for LHC phenomenology. We find that the resummation effect increases the differential cross section in the range $M_{t\bar{t}} \in \unit{[$300,380$]}{\GeV}$ by about 9\%. This makes the theoretical prediction more compatible with experimental data, and leads to a shift of about \unit{1.5}{\GeV} on the extracted top-quark mass from kinematic distributions, resulting in a value better consistent with the world average.
Our results can be easily combined with soft gluon resummation of \cite{Ahrens:2010zv, Pecjak:2016nee, Pecjak:2018lif, Czakon:2018nun} and with electroweak corrections of \cite{Bernreuther:2010ny, Pagani:2016caq, Czakon:2017wor, Czakon:2019txp}. This allows a precision prediction for the $t\bar{t}$ invariant-mass spectrum across the whole phase space. Our framework can be extended to study the $t\bar{t}+\text{jets}$ production process, which is also being used to extract the top quark mass \cite{Alioli:2013mxa, Aad:2019mkw}.
\begin{acknowledgments}
L.~L.~Yang would like to thank A.~Mitov for useful discussions. This work was supported in part by the National Natural Science Foundation of China under Grant No. 11975030, 11635001 and 11575004. W.-L.~Ju was supported in part by the China Postdoctoral Science Foundation under Grant No. 2017M610685.
\end{acknowledgments}
|
1,108,101,563,261 | arxiv | \section*{Introduction}
The field of integrable systems is a dynamic subject
closely related to many parts of
modern mathematics and featuring manifold applications, cf.\ e.g.\ \cite{ac, bl, blu, ca, cgs, dun, l, n, nd, olverbook, vps}
and references therein.\looseness=-1
A characteristic feature of integrable systems is that they never show up alone -- instead, they
always belong to integrable hierarchies, and the latter are typically
constructed using the recursion operators (ROs), cf.\ e.g.\ \cite{ac, bl, dun, ma, n, olver, olverbook} and references therein.
In particular, the ROs for multidimensional integrable
dispersionless systems
are now a subject of intense research, cf.\ e.g.\
\cite{ac, dun, m-s, ms3, ms-ip, m1,m2,m3, olverbook} and references therein. In fact, the overwhelming majority of integrable systems in four or more independent variable known to date, including those relevant for applications, like the (anti-)self-dual Yang--Mills equations or the (anti-)self-dual vacuum Einstein equations, is dispersionless, i.e., can be rewritten as first-order homegeneous quasilinear systems, cf.\ e.g.\ \cite{ac, dun, s} and Example 5 below for details.\looseness=-1
The study of a hierarchy associated to a given integrable system, and hence of its recursion operator,
is important for many reasons, {\em inter alia} because known exact solutions of integrable systems, like, say, multisoliton and finite-gap solutions, always happen to be invariant under a certain combination of the flows in the hierarchy, and this is instrumental in the actual construction of the solutions in questions, cf.\ e.g.\ \cite{dun, n}.\looseness=-1
Another feature making recursion operators relevant for applications is that existence of a (formal) recursion operator can in many situations be employed as an efficient integrability test or as a means of proving nonintegrability, see e.g.\ \cite{my} and references therein for general theory and \cite{sv} for a recent application.\looseness=-1
There are several methods for construction of the ROs for the
integrable dispersionless systems: the partner symmetry approach \cite{m-s}, the
method based on adjoint action of the Lax operators \cite{ms-ip},
and the approach using the Cartan equivalence method \cite{m1, m2,
m3}.
Below we present another method for construction of ROs for
integrable multidimensional dispersionless systems. Roughly
speaking, it consists in constructing a Lax-type representation for
the system under study with the following additional properties: i)
the Lax operators are linear in the spectral parameter and ii) the
solution of the associated linear system is a (nonlocal) symmetry of
the nonlinear system under study. Taking the coefficients at the
powers of spectral parameter in the operators constituting the newly
constructed Lax-type representation yields, under certain technical
conditions, the RO. A similar approach, with symmetries replaced by
cosymmetries and Proposition~\ref{Th1} by Proposition~\ref{Th2},
allows one to look for the adjoint ROs.
To the best of author's knowledge, the key idea of our approach,
i.e., that the Lax-type representation employed for the construction
of the recursion operator
should be {\em constructed} from the original Lax-type
representation but is by no means obliged to be identical with the
latter, has not yet appeared in the literature. This kind of flexibility
noticeably enhances the applicability of our method.
On the other hand,
the extraction of the RO from a Lax-type representation which is
linear in the spectral parameter was already explored to some extent
e.g.\ in \cite{p}.
The paper is organized as follows. In Section~\ref{pr} we recall
some basic facts concerning the geometric theory of PDEs. In
Section~\ref{rk} we state some general results on the construction
of recursion operators. Section~3, which is the core of the paper,
explores the construction of special Lax-type representations that
give rise to ROs through the results of Section~\ref{rk}.
Section~\ref{ex} gives a selection of examples illustrating our
approach, and Section~\ref{co} provides a brief discussion.\looseness=-1
\section{Preliminaries}\label{pr}
Let
$\mathcal{F}=0$ be a system of $m$ PDEs $F_I=0$, $I=1,\dots,m$, in
$d$ independent variables $x^i$, $i=1,\dots,d$, so we put $\vec
x=(x^1,\dots,x^d)$, for an un\-known $N$-com\-ponent vector
function $\bi{u}=(u^1,\dots,u^N)^{T}$,
where the superscript `$T$' indicates the transposed matrix.
Let \[u^\alpha_{i_1\dots i_n}=\partial^{i_1+\cdots+i_n}u^\alpha/\partial
(x^1)^{i_1}\cdots \partial (x^n)^{i_n}\] and $u^\alpha_{00\cdots0}\equiv
u^\alpha$. In what follows all functions are tacitly assumed to be meromorphic
in
all their arguments.
As e.g.\ in \cite{olverbook, kv}, $x^i$ and $u^\alpha_{i_1\dots
i_n}$ are considered here and below as independent quantities and can be
viewed as coordinates on an abstract infinite-dimensional space (a
jet space).
By a {\it local function}
we shall mean a
function of $\vec x$, $\bi u$ and of a finite number of
the derivatives of $\bi{u}$. To indicate locality of a function $f$
we shall write it as $f(\vec x,[\boldsymbol{u}])$.
We denote by
\[
{\displaystyle D_{x^j} = \frac{\partial}{\partial x^j}
} + \displaystyle\sum\limits_{\alpha=1}^{N}
\sum\limits_{i_1,\dots,i_n=0}^{\infty}{\displaystyle
u^\alpha_{i_1\dots i_{j-1} (i_j+1) i_{j+1}\dots i_n} \frac{\partial}{\partial u^\alpha_{i_1\dots i_n}}}
\]
the total derivatives with respect to $x^j$, cf.\ e.g.\ \cite{olverbook, kv}. For the sake of simplicity
the extensions of $D_{x^i}$ to nonlocal variables are again denoted by $D_{x^i}$, as e.g.\ in
\cite{ms-ip}.\looseness=-1
The system $\mathcal{F}=0$ together with all its differential consequences
\[D_{x^1}^{i_1}\cdots D_{x^n}^{i_n} \mathcal{F} = 0,\quad i_1,i_2,\dots,i_n=1,\dots,d,\quad n=1,2,\dots,
\]
defines a {\em diffiety} $\mathrm{Sol}_{\mathcal{F}}$, a submanifold of the infinite jet space,
see e.g.\ \cite{kv} and references therein for details.
This submanifold can be informally thought of as a set of all formal solutions of $\mathcal{F}=0$,
which motivates the choice of notation.
Below all equations will be required to hold only on
$\mathrm{Sol}_\mathcal{F}$, or a certain differential covering
thereof, see e.g.\ \cite{kv} for details on the latter, rather than e.g.\ on the whole jet space unless
otherwise explicitly stated.
\looseness=-1
The {\it directional derivative} (cf.\ e.g.\ \cite{bl, ms-ip} and references therein) along
$\bi{U}=(U^1,\dots,U^N)^{T}$
is the vector field on the jet space
$$
\dder{\bi{U}} = \sum_{\alpha=1}^{N}\sum_{i_1,\dots,i_n=0}^{\infty}
(D_{x^1}^{i_1}\cdots D_{x^n}^{i_n} U^\alpha)
\frac{\partial}{\partial u^\alpha_{i_1\dots i_n}}.
$$
The total derivatives as well as
the directional derivative can be applied to (possibly vector or matrix) local functions
$P$.
Recall, cf.\ e.g.\ \cite{olverbook, kv}, that
a local $N$-component vector function $\bi{U}$ is a (characteristic
of a) {\it symmetry} for the system $\mathcal{F}=0$ if $\bi{U}$
satisfies the linearized version of this system, namely,
$\ell_{\mathcal{F}}(\bi U)= 0$. We stress that by our
blanket assumption this is required to hold on
$\mathrm{Sol}_\mathcal{F}$ only rather than on the whole jet space.
Informally this definition means that the flow
$\bi{u}_\tau=\bi{U}$ leaves $\mathrm{Sol}_\mathcal{F}$ invariant.
Here
\[
\ell_f=\sum_{\alpha=1}^{N}\sum_{i_1,\dots,i_n=0}^{\infty}
\frac{\partial f}{\partial u^\alpha_{i_1\dots i_n}} D_{x^1}^{i_1}\cdots D_{x^n}^{i_n}
\]
is the operator of linearization and
$\ell_{\mathcal{F}}=(\ell_{F_1},\dots,\ell_{F_m})^T$.
Notice an
important identity
$\ell_{f}(\boldsymbol{U})=\partial_{\boldsymbol{U}}(f)$
which holds for any local $f$ and $\boldsymbol{U}$, see e.g.\ \cite{kv}.
Finally, recall that
if we have an operator in total derivatives
\[
Q=Q^0+\sum\limits_{k=1}^q \sum\limits_{j_1=1}^d\cdots
\sum\limits_{j_k=1}^d Q^{j_1\dots j_k} D_{x^{j_1}}\cdots
D_{x^{j_k}},
\]
where $Q^0$ and $Q^{j_1\dots j_k}$ are $s\times s'$-matrix-valued
local functions, its formal adjoint reads (cf.\ e.g.\ \cite{olverbook})
\[
Q^\dagger=(Q^0)^T+\sum\limits_{k=1}^q \sum\limits_{j_1=1}^d\cdots
\sum\limits_{j_k=1}^d (-1)^k D_{x^{j_1}}\cdots D_{x^{j_k}}\circ
(Q^{j_1\dots j_k})^T.
\]
For integrable systems in more than two independent variables their
symmetries usually depend, in addition to local variables, described
above, also on nonlocal variables. In other words, we are forced to
consider solutions $\boldsymbol{U}$ of
$\ell_{\mathcal{F}}(\boldsymbol{U})=0$ involving nonlocal variables.
A number of authors refers to such objects as to the shadows of nonlocal
symmetries, see e.g.\ \cite{kv} and references therein for more
details on this terminology; cf.\ also \cite{blu} and references therein. However, as in the present paper we shall not deal with full nonlocal symmetries
in the sense of \cite{kv}, for the sake of simplicity in
what follows we shall refer to solutions $\boldsymbol{U}$ of
$\ell_{\mathcal{F}}(\boldsymbol{U})=0$ involving nonlocal variables
just as to {\em nonlocal symmetries}.
A {\em cosymmetry}, also known as adjoint symmetry, see e.g.\ \cite{blu},
$\boldsymbol{\gamma}$ is a quantity which is dual to a symmetry: it has $m$ rather than $N$ components and
satisfies (cf.\ e.g.\ \cite{bl,kv} for details)
the
system
\[
\ell_{\mathcal{F}}^{\dagger}(\boldsymbol{\gamma})=0.
\]
Note that cosymmetries, just like symmetries, may depend on nonlocal variables in addition to the local ones.
A {\em recursion operator} (RO) for our system $\mathcal{F}=0$ is then, roughly speaking, an operator that sends any symmetry of $\mathcal{F}=0$ into
another symmetry thereof \cite{olver, olverbook}. However, this definition works well only when the RO contains no nonlocal terms, as often is the case for linear and linearizable equations, cf.\ e.g.\ \cite{bp, olverbook} and references therein.\looseness=-1
In order to properly handle nonlocal terms in ROs it is more appropriate
to view the RO as a B\"acklund auto-transformation for
$\ell_{\mathcal{F}}(\boldsymbol{G})=0$, see \cite{ms-ip, p, ph} and references therein for details, and below we adhere to this point of view.
Likewise, an {\em adjoint recursion operator} is from this perspective a B\"acklund
auto-transformation for $\ell_{\mathcal{F}}^{\dagger}(\boldsymbol{\gamma})=0$, cf.\ e.g.\ \cite{kv}.
As a closing remark, recall that a partial differential system is
called {\em dispersionless} (cf.\ e.g.\ \cite{ms3} and references
therein) if it can be written in the form of a quasilinear
homogeneous first-order system
\begin{equation}
\sum_{j=1}^d \sum_{\alpha=1}^n A_{I\alpha}^j(\bi{u})\frac{\partial u^\alpha}{\partial x^j}=0,\qquad
\label{hds}
\end{equation}
where $I=1,\dots,m$, $m\geq N$, and $\bi{u}=(u^1,\dots,u^N)^T$.
This class of systems is quite rich: for instance, quasilinear scalar second-order PDEs which do not explicitly involve the dependent variable $u$,
$$
\sum_{i=1}^d \sum_{j=i}^d f^{ij}(\vec x,\partial u/\partial x^1,
\dots, \partial u/\partial x^d)\partial^2 u/\partial x^i\partial
x^j=0,$$ can be brought into the form (\ref{hds}) by putting
$\bi{u}=(\partial u/\partial x^1, \dots, \partial u/\partial x^d)^T$.
\section{Some remarks on recursion operators}\label{rk}
Consider the following differential operators in total derivatives
\begin{equation}\label{ablm}
\begin{array}{l}
A_i=A_i^0+\sum\limits_{j=1}^d
A_{i}^{j} D_{x^{j}},\quad B_i=B_i^0+\sum\limits_{j=1}^d B_{i}^{j} D_{x^{j}},\quad i=1,2,\\
L=L^0+\sum\limits_{k=1}^d L^{k} D_{x^{k}},\quad
M=M^0+\sum\limits_{k=1}^d M^{k} D_{x^{k}},
\end{array}
\end{equation}
where
$A_i^j=A_i^j(\vec x, [\boldsymbol{u}])$ and $B_i^j=B_i^j(\vec x, [\boldsymbol{u}])$ for $i=1,2$ and $j=1,\dots,d$ are scalar functions, $A_i^0=A_i^0(\vec x, [\boldsymbol{u}])$ and $B_i^0=B_i^0(\vec x, [\boldsymbol{u}])$
are $N\times N$ matrices,
$L^0=L^0(\vec x, [\boldsymbol{u}])$ and
$L^{k}=L^{k} (\vec x, [\boldsymbol{u}])$ for $k=1,\dots,d$ are $N\times m$ matrices, while
$M^0=M^0(\vec x, [\boldsymbol{u}])$ and $M^{k}=M^{k} (\vec x, [\boldsymbol{u}])$ for $k=1,\dots,d$
are $m\times N$ matrices.
\begin{prop}\label{Th1}Let the operators $A_i,B_i,L,M$ of the form (\ref{ablm}) be such that
\begin{eqnarray}\label{com}
\mbox{i)}\ [A_1,A_2]=0,\hspace{149mm}\\
\label{LABa}
\mbox{ii)}\
(A_1 B_2-A_2 B_1)= L\circ \ell_{\mathcal{F}},\hspace{126mm}\\
\label{LAB}
\mbox{iii)}\ \ell_{\mathcal{F}}=
M \circ (B_1 A_2-B_2 A_1),\hspace{125mm}
\end{eqnarray}
iv)
there exist two distinct indices $p,q\in\{1,\dots,d\}$
such that we
can express $D_{x^p} \tilde{\boldsymbol{U}}$ and $D_{x^q}
\tilde{\boldsymbol{U}}$ from the relation
\begin{equation}\label{R}
A_i (\tilde{\boldsymbol{U}})=B_i (\boldsymbol{U}),
\ i=1,2.
\end{equation}
Then relations (\ref{R}) define a recursion operator for $\mathcal{F}=0$, i.e.,
whenever $\boldsymbol{U}=(U^1,\dots,U^N)^T$ is a (possibly nonlocal)
symmetry
for $\mathcal{F}=0$, so is
$\tilde{\boldsymbol{U}}=(\tilde{U}^1,\dots,\tilde{U}^N)^T$ defined by
(\ref{R})
\end{prop}
{\em Proof.} First of all, if $\boldsymbol{U}=(U^1,\dots,U^N)^T$ is a
symmetry for $\mathcal{F}=0$, then the system (\ref{R}) for $\tilde{\boldsymbol{U}}$ is compatible
by virtue of i)--iii).
We now have $\ell_{\mathcal{F}}(\tilde{\boldsymbol{U}})=0$ by virtue of (\ref{LAB}),
and hence $\tilde{\boldsymbol{U}}$ is a shadow of symmetry for our system. $\Box$
As an aside
note that the condition (\ref{LAB}) in a slightly
different form
has appeared
in \cite{dun-09, fkr, gs}, and was given there a certain geometric
interpretation.\looseness=-1
In complete analogy with the above we can also readily prove the counterpart of
Proposition~\ref{Th1} for adjoint recursion operators.
\begin{prop}\label{Th2}Suppose that $A_i,B_i,L,M$ are as before, but with
$A_i^0$ and $B_i^0$ being $m \times m$ matrices.
Further assume that these operators are such that
\begin{eqnarray}
\label{comARO}
\mbox{i)}\ [A_1,A_2]=0,\hspace{145mm}\\
\label{LABARO3a}
\mbox{ii)}\ \ell_{\mathcal{F}}^{\dagger}=
L\circ (B_1 A_2-B_2 A_1),\hspace{122mm}\\
\label{LABN2a}
\mbox{iii)}\
(A_1 B_2-A_2 B_1)=M\circ \ell_{\mathcal{F}}^{\dagger},\hspace{120mm}
\end{eqnarray}
\hspace*{1mm}
iv)
there exist two distinct indices $p,q\in\{1,\dots,d\}$
such that we
can express $D_{x^p} \tilde{\boldsymbol{\gamma}}$
and $D_{x^q} \tilde{\boldsymbol{\gamma}}$ from the
syste
\begin{equation}\label{ARO}
A_i (\tilde{\boldsymbol{\gamma}})=B_i (\boldsymbol{\gamma}),\quad i=1,2
\end{equation}
Then (\ref{ARO}) defines an adjoint recursion operator for $\mathcal{F}=0$, i.e.,
whenever $\boldsymbol{\gamma}=(\gamma^1,\dots,\gamma^m)^T$
is a cosymmetry for $\mathcal{F}=0$, then so is $\tilde{\boldsymbol{\gamma}}$
defined by (\ref{ARO}).\looseness=-
\end{prop}
\section{Recursion operators from Lax-type representations}\label{roltr}
We start with the following observation which is readily checked by
straightforward computation.
\begin{prop}\label{Th3} Suppose that the operators $A_i$ and $B_i$
define a RO (resp.\ an adjoint RO) as in Proposition~\ref{Th1}
(resp.\ as in Proposition~\ref{Th2}). Then the operators
$\mathcal{L}_i=\lambda A_i- B_i$, $i=1,2$, where $\lambda$ is a
spectral parameter, satisfy $[\mathcal{L}_1,\mathcal{L}_2]=0$, i.e.,
they constitute a Lax-type representation for $\mathcal{F}=0$.
\end{prop}
Thus, if our system admits an (adjoint) RO, it also admits a
Lax-type representation which is linear in the spectral parameter.
Hence, a natural source
of $A_i,B_i$ and $L, M$ satisfying the conditions
of Proposition~\ref{Th1}
is provided by the
Lax-type representations for $\mathcal{F}=0$ of the form\looseness=-1
\begin{equation}\label{Lax0}
\mathcal{L}_i\psi=0,\quad i=1,2,
\end{equation}
with $\mathcal{L}_i$ linear in $\lambda$ such that $\psi$ is a
nonlocal symmetry of $\mathcal{F}=0$, i.e., we have
$\ell_\mathcal{F}(\psi)=0$, cf.\ e.g.\ \cite{p}. Then putting
$\mathcal{L}_i=\lambda B_i- A_i$ or $\mathcal{L}_i=\lambda A_i- B_i$
gives us natural candidates for $A_i$ and $B_i$ which then should be
checked against the conditions of Proposition~\ref{Th1}, and, if the
latter hold, yield a recursion operator for $\mathcal{F}=0$.\looseness=-1
Even if it is impossible to construct the recursion operator from (\ref{Lax0}),
one still can construct infinite series of nonlocal symmetries for $\mathcal{F}=0$ expanding $\psi$ into
formal Taylor or Laurent series in $\lambda$, cf.\ e.g.\ \cite{p, as}.\looseness=-1
It is important to stress that the Lax representation with the
operators $\mathcal{L}_{i}$ employed in the above construction does
not have to be the {\em original} Lax representation of our system.
In general, we should {\em custom tailor} the operators
$\mathcal{L}_{1,2}$ constituting the Lax pair for the construction
in question, so that the solutions of the associated linear problem
(\ref{Lax0}) are symmetries, i.e., satisfy the linearized version of
our system.\looseness=-1
The natural building blocks for these $\mathcal{L}_i$ are the
original Lax operators $\mathscr{X}_i$, their formal adjoints
$\mathscr{X}_i^\dagger$, and the operators
$\mathrm{ad}_{\mathscr{X}_i}=[\mathscr{X}_i,\cdot]$, but in general
one has to twist them, cf.\ e.g.\ Example 4
below.
More explicitly, suppose that the system under study, i.e.,
$\mathcal{F}=0$, admits a Lax representation of the form
\begin{equation}\label{lax-x}
\mathscr{X}_i\psi=0,\quad i=1,2,
\end{equation}
where $\mathscr{X}_i$ are linear
in the spectral parameter $\lambda$
but such $\psi$ does {\em not} satisfy $\ell_{\mathcal{F}}(\psi)=0$,
i.e., $\psi$ is not a (nonlocal) symmetry.
Then we can seek for a nonlocal symmetry of $\mathcal{F}=0$ of the
form
\begin{equation}\label{phi}
\Phi=\Phi(\vec x, [\boldsymbol{u},\psi,\chi,\vec{\zeta}]),
\end{equation}
i.e., for a vector function of $\vec x$, and of $\boldsymbol{u}$,
$\psi$, $\chi$, and $\vec{\zeta}$ and a finite number of the
derivatives of $\boldsymbol{u}$, $\psi$, $\chi$, and $\vec\zeta$.
Here $\chi$ satisfies the system \[ \mathscr{X}^\dagger_i\chi=0,\ i=1,2,\]
$\mathscr{Z}=\sum\limits_{j=1}^d \zeta^j\partial/\partial x^j$
satisfies
\[
[\mathscr{X}_i,\mathscr{Z}]=0,\ i=1,2,
\]
and $\vec{\zeta}=(\zeta^1,\dots,\zeta^d)^T$.
Moreover, we should also require that there exist the operators
$\mathscr{L}_i$ which are linear in $\lambda$ and such that
\[\mathscr{L}_i\Phi=0,\quad i=1,2.\]
Then one should extract $A_i$ and $B_i$ for
Proposition~\ref{Th1} from these $\mathscr{L}_i$ rather than
from the original $\mathscr{X}_i$.
The study of specific examples strongly suggests that
the nonlocal variables $\psi$, $\chi$ and $\vec{\zeta}$ usually do not mix and enter $\Phi$ linearly,
so it typically suffices to look for nonlocal symmetries $\Phi$ of $\mathcal{F}=0$ in
one of the following forms,\looseness=-1
\begin{eqnarray}
\Phi^\alpha&=&\sum\limits_{s=1}^r a^\alpha_s \psi^s+
\sum\limits_{s=1}^r\sum\limits_{k=1}^d
a^{k}_s D_{x^{k}} (\psi^s),
\label{psi1}\\
\Phi^\alpha&=&\sum\limits_{s=1}^r b^{\alpha, s} \chi_s+
\sum\limits_{s=1}^r\sum\limits_{k=1}^d
b^{\alpha,s,k} D_{x^{k}}(\chi_s),
\label{psi2}\\
\Phi^\alpha&=&\sum\limits_{j=1}^d c^\alpha_j \zeta^j+
\sum\limits_{k,l=1}^d
c^{\alpha,k}_l D_{x^{k}} (\zeta^l),
\label{psi3}
\end{eqnarray}
where $\alpha=1,\dots,N$, $p\in\mathbb{N}$,
$a^\alpha_s$, $a^{\alpha,k}_s$, $b^{\alpha, s}$, $b^{\alpha,s,k}$,
$c^\alpha_j$ and $c^{\alpha,k}_l$ are local functions,
and $r$ is the number of components of $\psi$,
instead of general form (\ref{phi}). Moreover, in many cases it is possible to restrict
oneself to considering only zero-order terms in the nonlocal variables, i.e., use the simpler Ans\"atze
$\Phi^\alpha=\sum\limits_{s=1}^r a^\alpha_s \psi^s$,
$\Phi^\alpha=\sum\limits_{s=1}^r b^{\alpha, s} \chi_s$, and
$\Phi^\alpha=\sum\limits_{j=1}^d c^\alpha_j \zeta^j$ instead of (\ref{psi1}), (\ref{psi2}) and (\ref{psi3}).\looseness=-1
A similar approach, with symmetries replaced by cosymmetries and
Proposition~\ref{Th1} by Proposition~\ref{Th2} can, of course, be
applied to the construction of adjoint recursion operators.
Finally, let us point out that the assumption of
linear dependence of the Lax operators on $\lambda$ is not as restrictive as it seems:
a great many of known today multidimensional dispersionless
hierarchies include systems with this property and, moreover, by
Proposition~\ref{Th3} if a system admits an RO of the form described
in Proposition~\ref{Th1} then it necessarily possesses a Lax-type
representation whose operators are linear in $\lambda$.\looseness=-1
Moreover, in some cases the nonlinearity in $\lambda$ can be removed
upon a proper rewriting of the Lax pair.
Consider for example the Pavlov equation \cite{mvp}
\begin{equation}\label{pe}
u_{yy}+u_{xt}+u_x u_{xy}-u_y u_{xx}=0,
\end{equation}
which possesses a Lax pair of the form \cite{mvp} (cf.\ also \cite{ms})
\begin{equation}\label{lax-pe}
\psi_y=(-u_x-\lambda) \psi_x,\quad \psi_t=(-\lambda^2-\lambda u_x+u_y) \psi_x
\end{equation}
which is quadratic in $\lambda$.
However, rewriting the second equation of (\ref{lax-pe}) as
\[
\psi_t=\lambda(-\lambda- u_x)\psi_x+u_y\psi_x
\]
and substituting $\psi_y$ for $(-\lambda- u_x)\psi_x$, we obtain
\[
\psi_t=\lambda\psi_y+u_y\psi_x,
\]
and thus (\ref{lax-pe}) can be rewritten in the form linear in $\lambda$:
\begin{equation}\label{lax-pe1}
\psi_y=(-u_x-\lambda) \psi_x,\quad \psi_t=-\lambda\psi_y+u_y\psi_x.
\end{equation}
\section{Examples}\label{ex}
\subsection*{Example 1} It is readily checked that if $\psi$ satisfies (\ref{lax-pe1})
then $\Phi=1/\psi_x$ is a nonlocal symmetry for
(\ref{pe}), i.e., $U=1/\psi_x$ satisfies the
linearized version of (\ref{pe}),
\begin{equation}\label{lpe}
U_{yy}+U_{xt}+u_x U_{xy}-u_y U_{xx}+ u_{xy} U_x- u_{xx}U_y=0.
\end{equation}
Now by virtue of (\ref{lax-pe1}) $\Phi=1/\psi_x$ also satisfies a pair of linear equations of the form $\mathscr{L}_i\Phi=0$, $i=1,2$, where
\[
\mathscr{L}_1=-D_y+(\lambda-u_x)D_x+u_{xx},\quad \mathscr{L}_2=D_t+\lambda D_y-u_y D_x+u_{xy}.
\]
Now, as $\Phi=1/\psi_x$ satisfies (\ref{lpe})
we can identify $\mathscr{L}_i$ with $\mathcal{L}_i$
from (\ref{Lax0}) and put $\mathscr{L}_i=\lambda A_i- B_i$, where
\[
A_1=D_x, \quad A_2=D_y, \quad B_1=D_y+u_x D_x-u_{xx},\quad
B_2=-D_t+u_y D_x-u_{xy}.\]
Then it is easily seen that these operators satisfy
the conditions of Proposition~\ref{Th1} for suitably chosen $L$ and
$M$, so we arrive at the recursion operator for the Pavlov equation
given by the formulas
\[
\tilde U_x=U_y+u_x U_x-u_{xx} U,\quad \tilde U_y=-U_t+u_y U_x-u_{xy} U,
\]
which maps a (possibly nonlocal) symmetry $U$ to a new nonlocal symmetry
$\tilde U$. This is nothing but the recursion operator found in
\cite{ms} rewritten as a B\"acklund auto-transformation for (\ref{lpe}).
\subsection*{Example 2} Consider the general heavenly equation
\begin{equation}\label{ghe}
a u_{xy}u_{zt} + b u_{xz}u_{yt}+c u_{xt}u_{yz} = 0,\qquad a+b+c=0,
\end{equation}
where $a,b,c$ are constants. Here $d=3$, $N=1$, $x^1=x$, $x^2=y$,
$x^3=z$, $x^4=t$, $u^1=u$. Note that this equation describes {\em
inter alia} a class of anti-self-dual solutions of the Einstein
field equations as shown in \cite{mas}.
By definition, a (nonlocal) symmetry $U$ of (\ref{ghe}) satisfies
the linearized version of (\ref{ghe}), that is,
\begin{equation}\label{linghe}
a u_{xy}U_{zt} +a u_{zt}U_{xy} + b u_{xz}U_{yt}+
b u_{yt}U_{xz}+c u_{xt}U_{yz} +c u_{yz}U_{xt}= 0.
\end{equation}
Eq.(\ref{ghe}) admits \cite{df,mas}
a Lax-type representation with the operators
\begin{equation}\label{lax}
\begin{array}{l}
L_1=(1+c\lambda)D_x-\displaystyle\frac{u_{xz}}{u_{zt}}D_t-c\lambda \displaystyle\frac{u_{xt}}{u_{zt}}D_z,\\[5mm]
L_2=(1-b\lambda)D_y-\displaystyle\frac{u_{yz}}{u_{zt}}D_t+b\lambda \displaystyle\frac{u_{yt}}{u_{zt}}D_z.
\end{array}
\end{equation}
It is readily seen that $\psi$ that satisfies $L_i\psi=0$, $i=1,2$, also satisfies (\ref{linghe}), i.e., it is a nonlocal symmetry for (\ref{ghe}), so in spirit of Proposition~\ref{Th3}
let
\[
\begin{array}{l}
A_1=c D_x-c \displaystyle\frac{u_{xt}}{u_{zt}}D_z,\quad
A_2=-b D_y+b \displaystyle\frac{u_{yt}}{u_{zt}}D_z,\\[5mm]
B_1= -D_x+\displaystyle\frac{u_{xz}}{u_{zt}}D_t,\quad
B_2= -D_y+\displaystyle\frac{u_{yz}}{u_{zt}}D_t.
\end{array}
\]
Then all conditions of Proposition~\ref{Th1} are readily seen to be
satisfied, e.g.\ we have $B_1 A_2-B_2 A_1=(1/u_{zt})\ell_F$, where
$F=a u_{xy}u_{zt} + b u_{xz}u_{yt}+c u_{xt}u_{yz}$.
Hence
the relations
\begin{equation}\label{ghero}
\displaystyle\tilde{U}_x=\frac{u_{xz}U_t+c u_{xt}\tilde{U}_z
-u_{zt} U_x}{cu_{zt}},\
\displaystyle\tilde{U}_y=-\frac{u_{yz}U_t-b u_{yt}\tilde{U}_z-u_{zt} U_y}{bu_{zt}},
\end{equation}
where $U$ is a (possibly nonlocal) symmetry of (\ref{ghe}), define a new symmetry $\tilde U$
and thus a novel recursion operator (\ref{ghero}) for (\ref{ghe}), i.e., a B\"acklund auto-transformation for (\ref{linghe}).
This operator has first appeared in the very first version of the present paper (arXiv:1501.01955v1) and was later rediscovered, in a slightly different form, in \cite{msy}.
\subsection*{Example 3} Consider the Mart\'\i{}nez Alonso--Shabat \cite{m-a-s} equation
\begin{equation}\label{ase}
u_{yt}=u_z u_{xy}-u_y u_{xz}
\end{equation}
where $d=4$, $N=1$, $x^1=x$, $x^2=y$, $x^3=z$, $x^4=t$, $u^1=u$.
Eq.(\ref{ghe}) admits \cite{m3} a Lax-type representation with the
operators\vspace{-2mm}
\begin{equation}\label{lax-ase}
L_1=D_y-\lambda u_y D_x,\quad
L_2=D_z-\lambda u_z D_x+\lambda D_t.
\vspace{-2mm}
\end{equation}
Consider the action of operators
\[
\mathcal{L}_i=\mathrm{ad}_{L_i}=[L_i,\cdot]
\]
on operators of the form $w D_x$ and spell it out as
\[
\mathrm{ad}_{L_i}(w D_x)\equiv((\lambda B_i-A_i)w)D_x,
\]
so we have
\[
A_1= D_y,\
A_2= D_z,\quad
B_1=u_y D_x-u_{xy},\
B_2=u_z D_x-D_t-u_{xz}
\]
All conditions of Proposition~\ref{Th1} are again satisfied, so the relations
\begin{equation}\label{asero}
\begin{array}{l}
\displaystyle\tilde{U}_y=u_y U_x-u_{xy}U,
\\[3mm]
\displaystyle\tilde{U}_z=u_z U_x-u_{xz}U-U_t,
\end{array}
\end{equation}
where $U$ is any (possibly nonlocal) symmetry of (\ref{ase}), define
a new symmetry $\tilde U$, i.e., Eq.(\ref{asero}) provides a recursion
operator for (\ref{ase}). This operator was already found in
\cite{m3}.
Applying (\ref{asero}) to the simplest symmetry $U=u_x$ we obtain (modulo an arbitrary function of $x$ and $t$ resulting from the integration) a nonlocal symmetry of the form $\tilde{U}=w-u_x^2/2$, where the nonlocal variable $w$ is defined by the relations
\[
\begin{array}{l}
\displaystyle w_y=u_y u_{xx},\qquad
\displaystyle w_z=u_z u_{xx}-u_{xt}.
\end{array}
\]
This example suggests that Eq.(\ref{asero}) can be further simplified: a new symmetry $\tilde{U}$ can be constructed by putting
\[\tilde{U}=W-u_x U,
\]
where $W$ is defined by the relations
\[
\begin{array}{l}
\displaystyle W_y=u_y U_x+u_{x}U_y,
\\[3mm]
\displaystyle W_z=u_z U_x+u_{x}U_z-U_t.
\end{array}
\]
Thus, the recursion operators produced within our approach are not
necessarily in the simplest possible form.
\subsection*{Example 4} Consider a system \cite{fk, fkk}
\begin{equation}
m_{t}=n_{x}+nm_{r}-mn_{r},\qquad n_{z}=m_{y}+m n_{s}-n m_{s},
\label{fks}
\end{equation}
where $x,y,z,r,s,t$ are independent and $m,n$ are dependent variables.
Eq.(\ref{fks}) can be written as a condition of commutativity for the following pair of vector fields,
\begin{equation}\label{cc0}
[D_{z}-m D_{s}-\lambda D_{x}+\lambda m D_{r},
D_{y}-n D_{s}-\lambda D_{t}+\lambda n D_{r}]=0.
\end{equation}
where $\lambda$ is a spectral parameter, and is therefore integrable.
By virtue of (\ref{fks}) there exists a potential $u$ such that
\begin{equation}\label{sub}
m=u_z/u_s,\quad n=u_y/u_s.
\end{equation}
Substituting this into (\ref{fks}) gives a single second-order equation for $u$,
\begin{equation}\label{equ}
u_s u_{zt}-u_z u_{st}- u_s u_{xy}+u_y u_{sx}-u_y u_{rz}+u_z u_{ry}=0,
\end{equation}
which can be written as a commutativity condition $[L_1,L_2]=0$ for
\[
L_1=\displaystyle D_{z}-\frac{u_z}{u_s}D_{s}-\lambda D_{x}+\lambda \frac{u_z}{u_s} D_{r},\
L_2=\displaystyle D_{y}-\frac{u_y}{u_s}D_{s}-\lambda D_{t}+\lambda \frac{u_y}{u_s}D_{r}.
\]
Let $\chi$ satisfy $L_i^\dag \chi=0$, $i=1,2$.
Then $\zeta=u_s/\chi$ is a nonlocal symmetry for (\ref{equ})
and we can readily obtain a linear system for $\zeta$ of the form
$\mathcal{L}_1\zeta=0, \mathcal{L}_2\zeta=0$ from that for $\chi$.
Upon
spelling out $\mathcal{L}_i$ as $\mathcal{L}_i=A_i-\lambda B_i$
and checking that the conditions of Theorem~\ref{Th1} are satisfied
for suitable $L$ and $M$ we arrive at
the following novel RO
for (\ref{equ}):
\begin{equation}\label{invro1}
\begin{array}{l}
\tilde{U}_y =\displaystyle \frac{u_y}{u_s} \tilde{U}_s-\frac{u_y}{u_s} U_r+ U_t-\frac{(u_{st}-u_{ry})}{u_s} U,\\[7mm]
\tilde{U}_z =\displaystyle \frac{u_z}{u_s} \tilde{U}_s-\frac{u_z}{u_s} U_r+ U_x+\frac{(u_{rz}-u_{sx})}{u_s} U.
\end{array}
\end{equation}
Here again $U$ is a (in general nonlocal) symmetry of (\ref{equ}), and $\tilde U$
is a new nonlocal symmetry for (\ref{equ})
\subsection*{Example 5} Consider (see e.g.\ \cite{ac, atcmp, Mason} and references therein)
the following Lax operators:
\begin{equation}\label{kmlax}
L_1=D_y+\partial_{\tilde x} K-\lambda D_{\tilde x},\quad
L_2=D_x+\partial_{\tilde y} K-\lambda D_{\tilde y}.
\end{equation}
The commutativity condition
$[L_1,L_2]=0$
yields the system
\begin{equation}\label{kme}
\partial_x \partial_{\tilde x} K-\partial_y \partial_{\tilde y} K
-[\partial_{\tilde x} K,\partial_{\tilde y} K]=0.
\end{equation}
Here $K$ takes values in a (matrix) Lie algebra $\mathfrak{g}$ and is
known as the Yang $K$-matrix.
Note that (\ref{kme}) can be rewritten in the dispersionless form of the type (\ref{hds})
upon introducing (cf.\ e.g.\ \cite{Mason, ph}) an additional dependent variable $J$
taking values in $G=\exp(\mathfrak{g})$.
Namely, the dispersionless form of (\ref{kme}) reads
\begin{equation}\label{jk}
\partial_x J= J \partial_{\tilde y} K,\quad \partial_y J= J \partial_{\tilde x} K.
\end{equation}
The compatibility condition $\partial_x\partial_y J=\partial_y\partial_x J$ yields an equation
\begin{equation}\label{jme}
\partial_{\tilde x} (J^{-1}\partial_x J) = \partial_{\tilde y} (J^{-1}\partial_y J),
\end{equation}
while (\ref{kme}) is identically satisfied by virtue of (\ref{jk}).
In other words, (\ref{jk}) defines a B\"acklund transformation relating (\ref{kme}) and (\ref{jme}).
If we define the gauge potentials by
setting $A_x=\partial_{\tilde y} K$, $A_y=\partial_{\tilde x} K$, $A_{\tilde x} =0$, $A_{\tilde y}=0$,
then they satisfy the anti-self-dual Yang--Mills equations on $\mathbb{R}^4$ with Euclidean metric with coordinates $X^i$
such that $\sqrt{2}x=X^1+ \mathrm{i} X^2$, $\sqrt{2}\tilde x=X^1- \mathrm{i} X^2$, $\sqrt{2}y=X^3- \mathrm{i} X^4$,
$\sqrt{2}\tilde y=X^3+ \mathrm{i} X^4$, where $\mathrm{i}=\sqrt{-1}$, cf.\ e.g.
\cite{Mason}. Conversely, any sufficiently smooth solution of
the anti-self-dual Yang--Mills equations can, up to a suitable gauge transformation, be obtained in this way \cite{Mason}. In other words, the anti-self-dual Yang--Mills equations, which play an important role in modern physics, cf.\ e.g.\ \cite{Mason} and references therein, can, in a suitable gauge, be written in dispersionless form, exactly as claimed in Introduction.\looseness=-1
It is natural to assume that in the Lax pair equations $L_i\psi=0$ we
have $\psi\in G=\exp(\mathfrak{g})$. However, such $\psi$ cannot be a symmetry
for (\ref{kme}), since a symmetry must live in $\mathfrak{g}$ just like $K$.
However, we can get a Lax representation
for $\Phi\in\mathfrak{g}$ using the adjoint action: $[L_1,\Phi]=[L_2, \Phi]=0$, i.e., we use the Lax operators $\mathcal{L}_i=\mathrm{ad}\ L_i$.
Then all conditions of Proposition~\ref{Th1} are satisfied for $A_i$ and $B_i$ obtained
by spelling out $\mathcal{L}_i$ as
$\mathcal{L}_i=\lambda A_i -B_i$,
which yields the following RO for (\ref{kme}):
\begin{equation}\label{kmero}
V_{\tilde x}=U_y+[\partial_{\tilde x} K,U],\quad V_{\tilde y}=U_x+[\partial_{\tilde y} K,U],
\end{equation}
where $U$ and $V$ are symmetries for (\ref{kme}). This RO is
known from the literature, see \cite{blr, ph, atcmp}.
If in the above construction we take $\mathfrak{g}=\mathfrak{diff}(\mathbb{R}^N)$, i.e., we put
\[
K=\sum\limits_{i=1}^N u_i
\partial/\partial z^i,\]
where
$u_i=u_i(x,y,\tilde x,\tilde y,z^1,\dots,z^N)$ are scalar functions,
then (\ref{kme}) becomes \cite{ms3} the Manakov--Santini system \cite{ms-pla},
an integrable system in $(N+4)$ independent variables, and
(\ref{kmero}) provides a recursion operator for this system which was
found earlier in \cite{ms-ip} (see also \cite{ms3}) by a different
method.
Further examples of recursion operators obtained using our approach
can be found in \cite{km}.
\section{Concluding remarks}\label{co}
We presented above a method of construction for recursion operators
and adjoint recursion operators for a broad class of
multidimensional integrable systems that can be written as
commutativity conditions for a pair of vector fields, or, even more
broadly, of linear combinations of vector fields with zero-order
matrix operators, linear in the spectral parameter and free of
derivatives in the latter. This method is based on constructing a
special Lax-type representation for the system under study from the
original Lax-type representation, see Section~\ref{roltr} for
details. Note that if a system under study admits several
essentially distinct Lax representations then our approach applied
to these Lax representations in principle can give rise to several
essentially distinct recursion operators.
To the best of author's knowledge, the method in question
works for all known examples of multidimensional dispersionless integrable
systems admitting recursion operators in the form of B\"acklund
auto-transformation for linearized systems, including e.g.\ the ABC
equation \cite{z}
\
a u_x u_{yt} + b u_y u_{xt} + c u_t u_{xy} = 0,\quad a+b+c=0
\]
the simplest (2+1)-dimensional equation of the so-called universal hierarchy
of Mart\'\i{}nez Alonso and Shabat \cite{m-a-s}
\[
u_{yy}-u_y u_{tx}+u_x u_{ty}=0
\]
the complex Monge--Amp\`ere equation (see \cite{nsky} for its
recursion operator), first, second and modified heavenly equations,
etc. In particular, this method enabled us to find hitherto unknown
recursion operators for the general heavenly equation of Doub\-rov
and Ferapontov (\ref{ghe})
and for the
equation (\ref{equ}).
Note that our approach, when applicable, is in general computationally less demanding
than the methods of \cite{ms-ip} and \cite{m2}.
While all examples to which we have applied our approach so far are dispersionless,
Propositions~\ref{Th1} and \ref{Th2} and the method described in Section~\ref{roltr}
do not explicitly require this to be the case, so
it would be interesting to find out whether it is possible to extend
our approach to the construction of recursion operators
to other classes of integrable systems
\subsection*{Acknowledgments}
This research was supported in part by the Ministry of Education, Youth and Sports of the Czech Republic (M\v{S}MT \v{C}R) under RVO funding for I\v{C}47813059, and by the Grant Agency of the Czech Republic (GA \v{C}R) under grant P201/12/G028.
The author is pleased to thank O.I. Morozov and R.O. Popovych for stimulating discussions.
\vspace{-2mm}
|
1,108,101,563,262 | arxiv | \section{Introduction}
It is well-known that QCD with light fermions has a chiral symmetry
restoration phase transition at the temperature $T\simeq 170$ MeV.
This phase transition has been studied on the
lattice and with the help of various
QCD-motivated models (see, for example \cite{Kogut,Fleming}, \cite{Berges}), however, not much is
known about its driving mechanism.
When the temperature is high the effective coupling constant becomes
small and the theory can be studied semiclassically. From the other
side, at zero temperature it is very likely that the instanton --
anti-instanton ensemble (the instanton liquid) is responsible for
the spontaneous chiral symmetry breaking
\cite{DP2a,DP2b,DP3,Dobzor}. Therefore, in order to get an insight
into the mechanism of chiral symmetry breaking and restoration at
non-zero temperatures, it is natural to consider an ensemble of the
(anti) self-dual fields at finite temperature. A generalization of
the usual Belavin--Polyakov--Schwartz--Tyupkin (BPST)
instantons~\cite{BPST} for arbitrary temperatures is the Kraan--van
Baal--Lee--Lu (KvBLL) caloron with non-trivial
holonomy~\cite{KvB,LL}. These configuration were extensively
studied on the lattice in ~\cite{Brower,IMMPSV,Gatt} for $SU(2)$ and
$SU(3)$ gauge groups. To construct an ensemble of these
configurations theoretically, one should first compute the quantum weight of the
single KvBLL caloron, or the probability with which this
configuration occurs in the partition function of the theory. In
Ref.~\cite{DGPS} this problem was solved for the pure Yang--Mills
theory. To extend that result to the theory with fermions one has to
calculate the fermionic determinant in the background of the KvBLL
caloron. This calculation is the aim of the present paper.
Speaking of finite temperature one implies that the Euclidean
space-time is compactified in the `time' direction whose inverse
circumference is the temperature $T$, with the usual periodic
boundary conditions for boson fields and anti--periodic conditions
for the fermion fields. In particular, it means that the gauge field
is periodic in time, and the theory is no longer invariant under
arbitrary gauge transformations, but only under gauge
transformations that are periodical in time. As the space topology
becomes nontrivial the number of gauge invariants increases. The new
invariant is the holonomy or the eigenvalues of the Polyakov line
that winds along the compact 'time' direction~\cite{Polyakov}
\begin{equation} L=
\left.{\rm P}\,\exp\left(\int_0^{1/T}\!dt\,A_4\right)\right|_{|\vec
x|\to\infty}.
\label{Pol1}
\end{equation}\\
This invariant together with the topological charge and the magnetic
charge can be used for the classification of the field
configurations \cite{GPY} its zero vacuum average is
one of the common criteria of confinement.
The general expression for the self-dual electrically neutral
configuration with topological charge $1$ and arbitrary holonomy was
constructed a few years ago by Kraan and van Baal \cite{KvB} and Lee
and Lu \cite{LL}; it has been named the KvBLL caloron. In the
limiting case, when the KvBLL caloron is characterized by trivial
holonomy (meaning that the Polyakov line (\ref{Pol1}) assumes values
belonging to the group center $Z(N)$ for the $SU(N)$ gauge group),
it is reduced to the periodic Harrington-Shepard \cite{HS} caloron
known before. In this limit the quantum weights were studied in
detail by Gross, Pisarski and Yaffe \cite{GPY}.
Besides the neutral self-dual configurations there are charged
configurations. The ``elementary" soliton with unit electric and
magnetic charges is the dyon, also called the
Bogomolnyi--Prasad--Sommerfeld (BPS) monopole~\cite{Bog,PS}. It is a
self-dual solution of the Yang--Mills equations of motion with
static ({\it i.e.} time-independent) action density, which carries
both the magnetic and electric fields of the Coulomb type at
infinity, decaying as $1/r^2$. In the $SU(2)$ gauge theory there are
in fact two types of self-dual dyons~\cite{LY} $M$ and $L$ with
(electric, magnetic) charges $(+,+)$ and $(-,-)$, and two types of
anti-self-dual dyons $\overline{M}$ and $\overline{L}$ with charges
$(+,-)$ and $(-,+)$, respectively. Formally, $L$ and $M$ dyons are
related by a non-periodical gauge transformation.
\begin{figure}[t]
\centerline{ \epsfxsize=0.6\textwidth
\epsfbox{action_density_plot1.eps}} \caption{The action density of
the KvBLL caloron as function of $z,t$ at fixed $x=y=0$, with the
asymptotic value of $A_4$ at spatial infinity $\v=0.9\pi T,\;
\overline{{\rm v}}=1.1\pi T$. It is periodic in $t$ direction. At large dyon
separation the density becomes static (left, $r_{12}=1.5/T$). As the
separation decreases the action density becomes more like a $4d$
lump (right, $r_{12}=0.6/T$). In both plots the L,M dyons are centered
at $z_{\rm L}=-\v\,r_{12}/2\pi T,\;z_{\rm M}=\overline{{\rm v}}\,r_{12}/2\pi T,\;x_{\rm
L,M}=y_{\rm L,M}=0$. The axes are in units of temperature
T.\label{fig:adp1}}
\end{figure}
The KvBLL caloron of the $SU(2)$ gauge group (to which we restrict
ourselves in this paper) is ``made of" one $L$ and one $M$ dyon,
with total zero electric and magnetic charges. It means that for
large $r_{12}$ the KvBLL caloron field becomes a sum of L and M dyons
fields ($r_{12}$ is a parameter of the KvBLL caloron field, having a
natural meaning of the dyons' separation; it is associated with the
usual instanton size by $r_{12}=\pi\rho^2 T$). One can consider the
KvBLL caloron as a two-monopole solution. Although the action
density of the isolated $L$ and $M$ dyons does not depend on time,
their combination in the KvBLL solution is generally non-static.
When the distance $r_{12}$ becomes small compared to $1/T$ the KvBLL
caloron in its core domain $\sqrt{x_\mu x_\mu}\sim \rho$ reduces
to the usual BPST instanton (for explicit formulas
see \cite{KvB,DG}). The holonomy remains nontrivial as outside the
small core domain $A_4$ tends to a constant value.
The gluonic quantum weight computed in Ref.~\cite{DGPS} in the limit
when the separation between dyons $r_{12}$ is much larger than their
core sizes $1/\v$ and $1/\overline{{\rm v}}$ (they are constrained by $\overline{{\rm v}}+\v=2\pi
T$) has the form: \beqa \nonumber {\cal Z}_{\rm KvBLL}&= &\int d^3z_1\,
d^3z_2\, T^6\, (2\pi)^{\frac{8}{3}}\,C\, \left(\frac{8 \pi^2}{g^2}
\right)^4 \left(\frac{\Lambda e^{\gamma_E}}{4 \pi
T}\right)^{\frac{22}{3}} \left(\frac{\v}{2\pi T}\right)^{\frac{4
\v}{3\pi T}}
\left(\frac{\overline{{\rm v}}}{2\pi T}\right)^{\frac{4 \overline{{\rm v}}}{3\pi T}}\\
& \times &
\exp\left[-2\pi\,r_{12}\,P''(\v)\right]\;\exp\left[-V^{(3)}P(\v)\right],
\eeqa where $P(\v)=\frac{\v^2\overline{{\rm v}}^2}{12\pi^2 T}$ and the overall
factor $C$ is a combination of universal constants; numerically
$C=1.0314$. $\Lambda$ is the scale parameter in the Pauli--Villars
regularization scheme; the factor $g^{-8}$ is not renormalized at
the one-loop level.
To account for fermions we have to multiply the partition function
by $\prod\limits_{j=1}^{N_f} {\rm Det} (i \nabla\hspace{-.65em}/\hspace{.3em} +i m_j)$, where
$\nabla\hspace{-.65em}/\hspace{.3em}$ is the spin-1/2 isospin-1/2 covariant derivative in
background considered, and $N_f$ is the number of light flavors. We
consider only the case of massless fermions here. The operator
$i\nabla\hspace{-.65em}/\hspace{.3em}$ has zero modes \cite{Cherndb} therefore a meaningful object is ${\rm Det}'(i
\nabla\hspace{-.65em}/\hspace{.3em})$ --- a normalized and regularized product of non-zero
modes. In the self-dual background it is equal to $\left
({\rm Det}(-\nabla^2) \right)^2$, where $\nabla$ is the spin-0
isospin-1/2 covariant derivative~\cite{BC}.
In this paper we calculate $\log{\rm Det}(-\nabla^2)$ exactly, find a
compact analytical expression for the large $r_{12}$ asymptotic, find
$1/r_{12}^n$ corrections up to the $5^{\rm th}$ order, and the small
$r_{12}$ asymptotic. These results are analytical and give an almost
exact answer for all $r_{12}$ except $r_{12}\sim 1/T$. In section
\ref{sec_num} we present our results of numerical evaluation of the
determinant for arbitrary $r_{12}$ and $\v$, which are consistent with
both asymptotics.
In the forthcoming publication~\cite{GSN} we are going to generalize our
results to arbitrary $SU(N)$. The first step in this direction has
been made in Ref.~\cite{DG} where the gluonic Jacobian over
zero-modes, or the volume form on the moduli space, has been
computed for the general $SU(N)$ KvBLL caloron (the metrics of the
caloron moduli space was found earlier by Thomas Kraan \cite{Kraan}). We believe that our
results will be useful to construct the ensemble of KvBLL calorons
at any temperatures and to obtain a better understanding of the
chiral symmetry restoration mechanism at the phase transition
temperature.
\section{The KvBLL caloron solution}
In this section we remind some basic facts about the caloron with
nontrivial holonomy just to establish the notations, see
Refs.~\cite{KvB,LL,DGPS} for a detailed discussion. We use here the
gauge convention and the formalism of Kraan and van Baal (KvB)
\cite{KvB}, the notations are taken from \cite{DGPS}.
The key quantity characterizing the KvBLL solution for the $SU(2)$
gauge group to which we restrict ourselves in this paper, is the
holonomy ${\rm tr}\,L$, \eq{Pol1}. In the gauge where $A_4$ is static and
diagonal at spatial infinity, i.e. $\left.A_4\right|_{{\vec
x}\to\infty}= i\v\frac{\tau_3}{2}$, it is this asymptotic value $\v$
which characterizes the caloron solution in the first place. We
shall also use the complementary quantity $\overline{{\rm v}} \equiv 2\pi T-\v$.
Their relation to parameters $\omega, \bar\omega$ introduced by
KvB~\cite{KvB} is $\omega= \frac{\v}{4\pi T},\; \bar\omega=
\frac{\overline{{\rm v}}}{4\pi T}= \frac{1}{2}-\omega$. Both $\v$ and $\overline{{\rm v}}$ vary from 0
to $2\pi T$. At $\v= 0,2\pi T$ the holonomy is said to be `trivial',
and the KvBLL caloron reduces to that of Harrington and
Shepard~\cite{HS}. Note that the fields in the fundamental
representation feel the sign of ${\rm tr} L$, therefore the cases $\v=0$
and $\v=2 \pi T$ differ when we account for fermions.
We shall parameterize the solution in terms of the coordinates of
the dyons' `centers' (we call the constituent dyons L and M
according to the classification in Ref.~\cite{DPSUSY}): \beqa \nonumber
L\;{\rm dyon}:\quad &&\vec z_1= -\frac{2\omega{\overrightarrow r_{12}}}{T},\\
\nonumber
M\;{\rm dyon}:\quad &&\vec z_2= \frac{2\bar\omega{\overrightarrow r_{12}}}{T},\\
\nonumber {\rm dyon\; separation}: &&\vec z_2-\vec z_1= {\overrightarrow
r_{12}},\quad \quad |r_{12} |= \pi T\,\rho^2, \eeqa where $\rho$ is the
parameter used by Kraan and van Baal; it becomes the size of the
caloron at $\v\to 0$. We introduce the distances from the
`observation point' $\vec x$ to the dyon centers, \beqa \nonumber
&&\vec r = \vec x-\vec z_1=\vec x + 2\omega\overrightarrow{r_{12}},\qquad r= |\vec r|,\\
&&\vec s = \vec x-\vec z_2=\vec x -
2\bar\omega\overrightarrow{r_{12}},\qquad s= |\vec s|. \eeqa Henceforth
we measure all dimensional quantities in units of temperature for
brevity and restore $T$ explicitly only in the final results.
The KvBLL caloron field in the fundamental representation
is~\cite{KvB} (we choose the separation between dyons to be in the
third spatial direction, ${\overrightarrow r_{12}}= r_{12}\vec e_3$): \begin{equation}
A^{\rm KvB}_\mu=\frac{i}{2}\bar\eta^3_{\mu\nu}\tau_3\d_\nu\log\Phi
+\frac{i}{2}\Phi\;{\rm
Re}\left[(\bar\eta^1_{\mu\nu}-i\bar\eta^2_{\mu\nu})(\tau_1+i\tau_2)\d_\nu\chi\right]
\label{APvB}\end{equation} where $\tau_i$ are Pauli matrices,
$\bar\eta^a_{\mu\nu}$ are 't~Hooft's symbols~\cite{tHooft} with
$\bar\eta^a_{ij}= \epsilon_{aij}$ and $\bar\eta^a_{4\nu}=
-\bar\eta^a_{\nu 4}= \delta_{a\nu}$. ``Re'' means hermitian part of
the matrix, and the functions used are \beqa \nonumber &&\chi=
\frac{r_{12}}{\psi}\left(e^{-4\pi\bar\omega
ix_4}\frac{\mathrm{sh}\,}{s}+e^{4\pi\omega ix_4}\frac{\overline{\mathrm{sh}}\,}{r}\right),
\qquad\Phi= \frac{\psi}{\hat\psi}\,
\label{eq:psihat},\\
&&\hat\psi= -\cos(2\pi x_4)+\overline{\mathrm{ch}}\,\mathrm{ch}\,+\frac{\vec r\vec
s}{2rs}\overline{\mathrm{sh}}\,\mathrm{sh}\,,\qquad \psi=
\hat\psi+\frac{r_{12}^2}{rs}\overline{\mathrm{sh}}\,\mathrm{sh}\,+\frac{r_{12}}{s}\mathrm{sh}\,\overline{\mathrm{ch}}\,+\frac{r_{12}}{
r}\overline{\mathrm{sh}}\,\mathrm{ch}\,.\eeqa We have introduced the short-hand notations for
hyperbolic functions: \begin{equation} \mathrm{sh}\,\equiv\sinh(s\v),\qquad
\mathrm{ch}\,\equiv\cosh(s\v),\qquad \overline{\mathrm{sh}}\,\equiv\sinh(r\overline{{\rm v}}),\qquad
\overline{\mathrm{ch}}\,\equiv\cosh(r\overline{{\rm v}}) \,. \end{equation} One can note that the field
(\ref{APvB}) is not periodical. However, one can formally make a
(time dependent) gauge transformation so that the resulting field
\textit{is} periodical. It turns out that there are two inequivalent
possibilities to make a time dependent gauge transformation:
$g_P=e^{-2\pi i x_4\omega\tau_3}$ and $g_A=e^{2\pi i
x_4\bar\omega\tau_3}$. Correspondingly, we have two self-dual
periodical fields \beqa A^P_\mu(\omega,\vec x,x_4)=\delta_{\mu
4}2\pi i
\omega\tau_3+\frac{i}{2}\bar\eta^3_{\mu\nu}\tau_3\d_\nu\log\Phi
+\frac{i}{2}\Phi\;{\rm Re}\left[(\bar\eta^1_{\mu\nu}
-i\bar\eta^2_{\mu\nu})(\tau_1+i\tau_2)e^{-4\pi i x_4\omega}\d_\nu\chi\right]\label{APvBP}\\
A^A_\mu(\omega,\vec x,x_4)=-\delta_{\mu 4}2\pi i
\bar\omega\tau_3+\frac{i}{2}\bar\eta^3_{\mu\nu}\tau_3\d_\nu\log\Phi
+\frac{i}{2}\Phi\;{\rm
Re}\left[(\bar\eta^1_{\mu\nu}-i\bar\eta^2_{\mu\nu})(\tau_1+i\tau_2)e^{4\pi
i x_4\bar\omega}\d_\nu\chi\right] \label{APvBA}\eeqa Note that they
are related by an anti-periodical gauge transformation
$g_Pg_A^\dag=e^{-\pi i x_4\tau_3}$. In general, an anti-periodical
gauge transformation may affect different quantities. For example,
the determinant of the ghost operator in the fundamental
representation ${\rm Det}(-\nabla^2)$ is not invariant under this
transformation, whereas the determinant in the adjoint
representation ${\rm Det}(-D^2)$ is invariant. We have to calculate
${\rm Det}(-\nabla^2)$ for both fields (\ref{APvBP}) and (\ref{APvBA}).
Fortunately, in the case of one caloron we can limit ourselves to
one of the fields by the observation that these fields differ only
by the exchange of constituent dyons and by the substitution
$\omega\rightarrow\bar\omega$. More precisely, \begin{equation}
A^P_\mu(\omega,x_1,x_2,x_3,x_4)={\rm s_{\mu\nu}}\;
e^{-i\pi\frac{\tau_1}{2}}A^A_\nu(\bar\omega,x_1,-x_2,2[\omega-\bar\omega]r_{12}-x_3,x_4)e^{i\pi\frac{\tau_1}{2}}
\end{equation} where $s$ is a diagonal matrix such that $\rm s_{11}=s_{44}=+1$
and $\rm s_{22}=s_{33}=-1$. We make a rotation around $\vec e_1$,
which exchanges the two dyons, and a global gauge transformation.
Obviously space rotation and gauge transformation of a background do
not change the determinant and we can conclude \begin{equation}
{\rm Det}\!\left(-\nabla^2[A^A_\mu(\omega)]\right)={\rm Det}\!\left(-\nabla^2[A^P_\mu(\bar
\omega)]\right) \end{equation} This allows us to consider only $A^P_\mu$ in
what follows. We shall systematically drop the $P$ subscript.
The first term in \ur{APvBP} corresponds to a constant $A_4$
component at spatial infinity ($ A_4 \approx i\v\frac{\tau_3}{2} $)
and gives rise to the non-trivial holonomy.
In the situation when the separation between dyons $r_{12}$ is large
compared to both their core sizes $\frac{1}{\v}$ (M) and
$\frac{1}{\overline{{\rm v}}}$ (L), the caloron field can be approximated by the
sum of individual BPS dyons, see Figs.~1 (left). We give below the
field inside the cores and far away from both cores.
\subsection{Inside dyon cores}
In the vicinity of the M dyon and far away from the L dyon ($r \overline{{\rm v}}
\gg1$) the field becomes that of the M dyon. It is static up to the
corrections of the order of $e^{-2\pi\bar\omega r}$ as can be seen
directly from \eq{APvBP} as $e^{-4\pi i x_4\omega}\chi$ become
static in this domain. We write it in spherical coordinates centered
at $\vec z_2$:
\[
A_4^M =\frac{i\tau_3}{2}\left(\v\coth(\v s)-\frac{1}{s}\right),\quad
A_\theta^M=\v\frac{\sin\!\phi\;\tau_1-\cos\!\phi\;\tau_2}{2i\sinh(\v
s)}\,,
\]
\begin{equation} A_r^M =0,\quad
A_\phi^M=\v\frac{\cos\!\phi\;\tau_1+\sin\!\phi\;\tau_2}{2i\sinh(\v
s)} +i\tau_3\frac{\tan(\theta/2)}{2s}, \label{Mdyon}\end{equation}
In the vicinity of the L dyon the field is substantially time
dependent. However, this time-dependence can be removed by an
\textit{anti-periodical} gauge transformation. It is instructive to
write the L dyon field in spherical coordinates centered at $\vec
z_1$: \beqa
\nonumber A_4^L&=&\frac{i\tau_3}{2}\left(\frac{1}{r}+2\pi-\overline{{\rm v}}\coth(\overline{{\rm v}} r)\right),\;\;\;A^L_r=0,\\
\nonumber A^L_\theta &=& i\overline{{\rm v}}\frac{-\sin(2\pi x_4-\phi)\;\tau_1
+\cos(2\pi x_4-\phi)\tau_2}{2\sinh(\overline{{\rm v}} r)},\\
A^L_\phi &=& i\overline{{\rm v}}\frac{\cos(2\pi x_4-\phi)\;\tau_1+\sin(2\pi
x_4-\phi)\tau_2}{2\sinh(\overline{{\rm v}} r)}-i\tau_3\frac{\tan(\theta/2)}{2r}\;.
\label{Ldyon}\eeqa We shall use the fact that the L dyon field can be
obtained from the M dyon field by interchanging $\v$ with $\overline{{\rm v}}$ and
by making an appropriate anti-periodical gauge transformation.
We see that in both cases the L,M fields become Abelian at large
distances, corresponding to the (electric, magnetic) charges $(-,-)$
and $(+,+)$, respectively. The corrections to the fields of M and L
dyons are hence of the order of $1/r_{12}$, arising from the presence of
the other dyon.
\subsection{Far away from dyon cores}
Far away from both dyon cores ($r \overline{{\rm v}}\gg 1,\;s\v \gg 1$; note that
it does not necessarily imply large separations -- the dyons may
even be overlapping) one can neglect both types of exponentially
small terms, ${\cal O}\left(e^{-r\overline{{\rm v}} }\right)$ and ${\cal
O}\left(e^{-s\v}\right)$. With the exponential accuracy the function
$\chi$ in \eq{eq:psihat} is zero, and the KvBLL field \ur{APvB}
becomes Abelian~\cite{KvB}: \beqa \label{A4as}
&&A_4^{\rm as}= \frac{i\tau_3}{2}\left(\v+\frac{1}{r}-\frac{1}{s}\right),\\
\label{Aphias} &&A_\varphi^{\rm as}=
-\frac{i\tau_3}{2}\left(\frac{1}{r}+\frac{1}{s}\right)
\sqrt{\frac{(r_{12}-r+s)(r_{12}+r-s)}{(r_{12}+r+s)(r+s-r_{12})}}\;. \eeqa In
particular, far away from both dyons, $A_4$ is the Coulomb field of
two opposite charges.
\section{The scheme for computing Det${\bf (-\nabla ^2)}$}
As explained in section I, to find the quantum weight of the KvBLL
caloron, one needs to calculate the small oscillation determinant,
${\rm Det}(-\nabla^2)$, where $\nabla_\mu= \d_\mu+A_\mu$ and $A_\mu$
is the caloron field \cite{KvB} in the fundamental representation.
We employ the same method as in \cite{DGPS,Zar}. Instead of
computing the determinant directly, we first evaluate its derivative
with respect to the holonomy $\v$, and then integrate the derivative
using the known determinant at $\v=0$ or $\v=2\pi$~\cite{GPY} as a
boundary condition.
If the background field $A_\mu$ depends on some parameter ${\cal
P}$, a general formula for the derivative of the determinant with
respect to such parameter is \begin{equation} \frac{\partial\,\log {\rm
Det}(-\nabla^2[A])}{\partial {\cal P}} = \!-\!\int
d^4x\,{\rm Tr}\left(\partial_{\cal P} A_\mu\, J_\mu\right),
\label{dvDet}\end{equation} where $J_\mu$ is the vacuum current in the
external background, determined by the Green function: \begin{equation}
J^{ab}_\mu\!\equiv\!\left.(\delta^a_c\delta^b_d\d_x\!
-\!\delta^a_c\delta^b_d\d_y\!+\!A^{ac}\delta^b_d\!
+\!A^{db}\delta^a_c) {\cal G}^{cd}(x,y)\right|_{y= x}\qquad {\rm
or\;simply}\quad J_\mu\equiv \overrightarrow{\nabla}_\mu {\cal G}
+{\cal G} \overleftarrow{\nabla}_\mu. \label{defJ}\end{equation} Here $\cal G$
is the Green function or the propagator of spin-0, isospin-1/2
particle in the given background $A_\mu$ defined by \begin{equation}
-\nabla^2_xG(x,y)= \delta^{(4)}(x-y) \label{Gdef}\end{equation} The anti-periodic
propagator can be easily obtained from it by a standard procedure:
\begin{equation}\label{greenP}
{\cal G}(x,y)= \sum_{n= -\infty}^{+\infty} (-1)^n G(x_4,{\vec
x};y_4+n,{\vec y}).
\end{equation}
\Eq{dvDet} can be easily verified by differentiating the identity
$\log{\rm Det}(-D^2)= {\rm Tr}\log(-D^2)$. The background field $A_\mu$
in \eq{dvDet} is taken in the fundamental representation, as is the
trace. Hence, if the anti--periodic propagator ${\cal G}$ is known,
\eq{dvDet} becomes a powerful tool for computing quantum
determinants. Specifically, we take ${\cal P}= \v$ as the parameter
for differentiating the determinant, and there is no problem in
finding $\partial_{\v}A_\mu(r,s,r_{12},x_0,\v)$ for the caloron field
\ur{APvB}. We assume {\it positions of dyons fixed} for convenience,
it differs simply by a global translation from $\partial_{\v}A_\mu$
with the fixed center of mass position.
The Green functions in the self-dual backgrounds are generally
known~\cite{CWS,Nahm80} and are built in terms of the
Atiah--Drinfeld--Hitchin--Manin (ADHM) construction~\cite{ADHM} for
the given self-dual field. A subtlety appearing at nonzero
temperatures is that the Green function is defined by \eq{Gdef} in
the Euclidean $\relax{\rm I\kern-.18em R}^4$ space, where the topological charge is
infinite because of the infinite number of repeated stripes in the
compactified time direction, whereas one actually needs an
explicitly anti--periodic propagator \ur{greenP}. To overcome this
nuisance, Nahm~\cite{Nahm80} suggested to pass on to the Fourier
transforms of the infinite-range subscripts in the ADHM
construction. We review this program in Appendix A of \cite{DGPS},
first for the single dyon field and then for the KvBLL caloron. This
way, we get the finite-dimensional ADHM construction both for the
dyon and the caloron, with very simple periodicity properties.
The isospin-1/2 propagator in $\mathbb{R}^4$ is known to be
\begin{equation}
\label{green12} G(x,y)= \frac{\langle v(x)|v(y)\rangle}
{4\pi^2(x-y)^2}\;.
\end{equation}
In what follows it will be convenient to split it into two parts:
\beqa \nonumber
{\cal G}(x,y)&=& {\cal G}^{\rm r}(x,y)+{\cal G}^{\rm s}(x,y)\\
\label{green3} {\cal G}^{\rm s}(x,y)\equiv G(x,y),&&\qquad{\cal G}^{\rm r}(x,y)
\equiv\sum\limits_{n\neq 0} (\mp1)^n G(x_4,{\vec x};y_4+n,{\vec
y})\;. \eeqa The vacuum current \ur{defJ} will be also split into
two parts: ``singular'' and ``regular'' , in accordance to which
part of the periodic propagator \ur{green3} is used to calculate it:
\begin{equation} J_\mu= J^{\rm r}_\mu+J^{\rm s}_\mu. \end{equation}
\begin{figure}[t]
\centerline{ \epsfxsize=0.5\textwidth \epsfbox{LMdyons.eps}}
\caption{\label{fig:LMdyons} Three regions of integration for well
separated dyons.}
\end{figure}
Although there is no principle difficulty in doing all calculations
exactly for the whole caloron moduli space, at the stage of spatial
integration of the determinant density we loose the capacity of
performing analytical calculations for the simple reason that the
expressions become too long, and so far we have not been able to put
them into a manageable form in the general case. Therefore, we have
to adopt a more subtle attitude. First of all we restrict ourselves
to the part of the moduli space corresponding to large separations
between dyons ($r_{12}\gg 1$). Physically, it seems to be the most
interesting case. Furthermore, at the first stage we take
$r_{12}\v,r_{12}\overline{{\rm v}}\gg 1$, meaning that the dyons are well separated and do
not overlap since the separation is then much bigger than the core
sizes, see Fig.~1 (left). In this case, the vacuum current $J_\mu$
\ur{defJ} becomes that of single dyons inside the spheres of some
radius $R$ surrounding the dyon centers, such that
$\frac{1}{\v},\frac{1}{\overline{{\rm v}}}\ll R \ll r_{12}$, and outside these spheres
it can be computed analytically with the exponential precision, in
accordance with subsection II.B, see Fig.~2. Adding up the
contributions of the regions near two dyons and of the far-away
region, we get $d{\rm Det}(-D^2)/d\v$ for well-separated dyons.
Integrating it over $\v$ we obtain the fermion determinant itself up
to a constant and possible $1/r_{12}$ terms. Finally, we compute the
$1/r_{12}$ corrections up to the $1/r_{12}^5$ term in the ${\rm Det}(-D^2)$,
which turn out to be quite non-trivial.
This is already an interesting result by itself, however, we would
like to compute the constant, which can be done by matching our
calculation with that for the trivial caloron at $\v=0$ and $\v=2
\pi T$. It means that we have to extend the domain of applicability
to $r_{12}\v=\O(1)$ (or $r_{12}\overline{{\rm v}}=\O(1)$) implying overlapping dyons,
presented in Fig.~1 (right).
\section{Near each of the dyons } \label{nearcores}
In this section we calculate the right hand side of \eq{dvDet} in
the core domains of the L and M dyons. For M dyon this task is
nearly solved in \cite{DGPS}. In Appendix A the derivation of the
vacuum current in the background is given and we have to multiply it
by an expression for the gauge field (\ref{Mdyon}) and integrate
over M-dyon core domain (see Fig. 2).
The result is
\begin{equation}
{\rm tr}[\d_\v A^M_\mu J^M_\mu]=\frac{
{\left(1-s\v \coth (s\v)\right) }^3
\left( \sinh (2 s \v)-6 s \v \right) }{96 {\pi }^2 s^3 \sinh^2(s\v)} -
\frac{
\left( 1-s\v \coth (s\v)\right)
\left( \sinh (2 s\v)-2 s\v\right) }{24 s\sinh^2(s\v)}
\label{otvA} \end{equation} and integrating over the ball of radius $R$
surrounding the dyon we obtain
\begin{equation} \int_0^R {\rm tr}[\d_\v
A^M_\mu J^M_\mu]d^4x=
\frac{\pi}{432}+\frac{\gamma_E}{12\pi}+\frac{31}{144\pi}+
\frac{\log(\v/\pi)}{12\pi} -\int^R \d_\v
P\left[\frac{1}{2}\left(\overline{{\rm v}}+\frac{1}{s}\right)\right]4\pi s^2 ds\;,
\label{corA} \end{equation} where we denote
$P(\v)\equiv\frac{\v^2\overline{{\rm v}}^2}{12\pi^2}$ (in fact this
function is a one-loop perturbative effective potential \cite{GPY,NW})
and thus \begin{equation} -\int^R \d_\v
P\left[\frac{1}{2}\left(\overline{{\rm v}}+\frac{1}{s}\right)\right]s^2 ds=
\frac{R^3}{6}P'\left(\frac{\overline{{\rm v}}}{2}\right)+\frac{R^2}{8}P''\left(\frac{\overline{{\rm v}}}{2}\right)
+\frac{R}{16}P'''\left(\frac{\overline{{\rm v}}}{2}\right)+\frac{\log
R}{96}P^{IV}\left(\frac{\overline{{\rm v}}}{2}\right) \end{equation} The calculation in the
L-dyon background is similar but slightly more difficult. The
strategy is to reduce this problem to the calculation in the M-dyon
background, which is simpler. For this purpose we note that the
L-dyon gauge field is a gauge transformation of the M-dyon gauge
field with the parameter $\v$ taken to be equal to $\overline{{\rm v}}=2\pi-\v$. We
should note, however, that this transformation is anti-periodical in
time and the anti-periodical Green function becomes a periodical one
under this gauge transformation. It means that we have to the use
the periodical vacuum current in the M-dyon background computed in
the Appendix A and substitute $\v$ with $\overline{{\rm v}}$ to get the right
answer: \beqa {\rm tr}[\d_\v A^L_\mu J^L_\mu]&=&
\frac{(r\overline{{\rm v}}\coth(r\overline{{\rm v}})-1)(\sinh(2r\overline{{\rm v}})-2r\overline{{\rm v}})}{12 r\sinh^2(r\overline{{\rm v}})}
+\frac{(r\overline{{\rm v}}\coth(r\overline{{\rm v}})-1)^3(\sinh(2r\overline{{\rm v}})-6r\overline{{\rm v}})}{96\pi^2 r^3\sinh^2(r\overline{{\rm v}})}\\
\nonumber&-&\frac{r\overline{{\rm v}}\coth(r\overline{{\rm v}})-1}{\coth\left(\frac{r\overline{{\rm v}}}{2}\right)}\;
\frac{r\overline{{\rm v}}[2\cosh(r\overline{{\rm v}})+\cosh(2r\overline{{\rm v}})-4r\overline{{\rm v}}
\coth\left(\frac{r\overline{{\rm v}}}{2}\right)+7]-\sinh(2{\rm r}\overline{{\rm v}})}{32\pi
r^2\sinh^2(r\overline{{\rm v}})}. \eeqa Integration leads to the following result
\begin{equation} \int_0^R {\rm tr}[\d_\v A^L_\mu J^L_\mu]d^4x=
-\frac{\pi}{432}-\frac{\gamma_E}{12\pi}-\frac{31}{144\pi}-
\frac{\log(\overline{{\rm v}}/\pi)}{12\pi}-\int^R \d_\v
P\left[\frac{1}{2}\left(\overline{{\rm v}}-\frac{1}{r}\right)\right]4\pi r^2 dr.
\label{corP} \end{equation}
Finally, substituting (\ref{corA}) and (\ref{corP}) into
(\ref{dvDet}) we arrive at the contribution from the cores \begin{equation}
\left.\d_\v\log{\rm Det}(-\nabla^2)\right|_{\rm
cores}=\frac{\log(\overline{{\rm v}}/\v)}{12\pi} -\frac{4\pi
R^3}{3}P'\left(\frac{\overline{{\rm v}}}{2}\right)-\frac{\pi
R}{2}P'''\left(\frac{\overline{{\rm v}}}{2}\right)\,.
\label{twocores} \end{equation}
\section{Contribution from the far region}\label{FarCurrent}
In this region we can drop the exponentially small terms $e^{-\v s}$
and $e^{-\overline{{\rm v}} r}$ in the r.h.s. of \eq{dvDet}. It turns out to be a
great simplification. The calculations are similarly to the calculations for
isospin-1 \cite{DGPS}. We are not repeating them here because the
result is simple and natural. The full vacuum current can be easily
expressed through the perturbative potential energy
$P(\v)\equiv\frac{\v^2\overline{{\rm v}}^2}{12\pi^2}$. The only non-zero component
is \begin{equation}
J_4=-\frac{i \tau_3}{2} \left.P'(q)\right|_{q=\frac{1}{2}\left ( \overline{{\rm v}}+\frac{1}{s}-\frac{1}{r}\right)}\,.
\label{J4asP}\end{equation} Making use of \eq{A4as} we come to the contribution
from the far region \begin{equation} \left.\d_\v\log{\rm Det}(-\nabla^2)\right|_{\rm
far}=\int \d_{\v} P\left[\frac{1}{2}\left (
\overline{{\rm v}}+\frac{1}{s}-\frac{1}{r}\right)\right] d^4 x \end{equation} The integral is
taken over the whole space with two holes (see Fig.
\ref{fig:LMdyons}). We can use the symmetry between dyons to write
\begin{equation} \int \d_{\v} P\left[\frac{1}{2}\left (
\overline{{\rm v}}+\frac{1}{s}-\frac{1}{r}\right)\right] d^4 x
=-\frac{1}{2}P'\left(\frac{\overline{{\rm v}}}{2}\right)\left(V-2\frac{4\pi}{3}R^3\right)
-\frac{1}{16}P'''\left(\frac{\overline{{\rm v}}}{2}\right)(4\pi r_{12}-8\pi
R)+\O\left(\frac{\log R}{R}\right) \label{far1} \end{equation} where $V$ is
the 3D volume of the system.
\section{Combining all three regions}
Adding (\ref{far1}) to (\ref{twocores}) we get the expression for
the $\v$ derivative of the determinant for large distances $r_{12}$
between constituents with the $1/r_{12}$ precision: \beqa
\d_\v\log{\rm Det}(-\nabla^2)
=-\frac{\log(\v/\overline{{\rm v}})}{12\pi}-\frac{1}{2}P'\left(\frac{\overline{{\rm v}}}{2}\right)V-
\frac{1}{16}P'''\left(\frac{\overline{{\rm v}}}{2}\right)4\pi r_{12} \eeqa
This equation can be easily integrated over $\v$ up to a constant,
which in fact can be a function of the separation $r_{12}$: \begin{equation}
\left.\log{\rm Det}(-\nabla^2)\right|_{\rm far}
=\left[P\left(\frac{\overline{{\rm v}}}{2}\right)-\frac{\pi^2}{12}\right]V+
P''\left(\frac{\overline{{\rm v}}}{2}\right)\frac{\pir_{12}}{2}-\frac{\overline{{\rm v}}\log{\overline{{\rm v}}}}{12\pi}-\frac{\v\log{\v}}{12\pi}+
f(r_{12}) \label{result1} \end{equation} This result is valid with ${\cal
O}(\exp(-\v r_{12}),\exp(-\overline{{\rm v}} r_{12}),1/r_{12})$ precision, and we have
separated a constant $\frac{\pi^2}{12} = P(\pi),$ which accounts
for normalization to the $\v=0$ determinant. Since in the above
calculation of the determinant for well-separated dyons we have
neglected the Coulomb field of one dyon inside the core region of
the other, we expect that the unknown function
$f(r_{12})=\O(1/r_{12})+c_{\frac{1}{2}}$ where $c_{\frac{1}{2}}$ is the true integration
constant. One should consider the $r_{12}$-derivative of the determinant
to see this explicitly. We perform it in Section \ref{sec1/r}. Our
next aim will be to find the constant $c_{\frac{1}{2}}$. The $\O(1/r_{12})$
corrections will be found later.
\section{Expression for the constant in the determinant}
To find the integration constant, one needs to know the value of the
determinant at $\v\!=\!0$, where the KvBLL caloron with non-trivial
holonomy reduces to the Harrington--Shepard caloron with trivial
holonomy, and for which the determinant has been computed by Gross,
Pisarski and Yaffe (GPY)~\cite{GPY}. Before we match our determinant
at $\v\!\neq \!0$ with that at $\v\!= \!0$, let us recall the GPY
result for the isospin-1/2 determinant.
\subsection{Det${\bf (-\nabla^2)}$ at $\v=0$}
The $\v\!=\!0,\;2\pi$, periodic instanton is traditionally
parameterized by the instanton size $\rho$. It is
known~\cite{Rossi,GPY} that the periodic instanton can be viewed as
composed of two BPS monopoles one of which has an infinite size. It
becomes especially clear in the KvBLL construction~\cite{KvB,LL},
where the size of the M (L) dyon becomes infinite as $\v\to
0\;(2\pi)$, see section II. Despite one dyon being infinitely large,
one can still continue to parameterize the caloron by the distance
$r_{12}$ between dyon centers, with $\rho= \sqrt{r_{12}/\pi}$. Since the
determinant \ur{result1} is given in terms of $r_{12}$ we have first of
all to rewrite the GPY determinant in terms of $r_{12}$, too. Actually,
GPY have interpolated the determinant in the whole range of $\rho$
(hence $r_{12}$) but we shall be interested only in the limit $r_{12}\gg 1$.
In this range the GPY result reads:
\begin{equation}
\log{\rm Det}(-\nabla^2)=\left.\log{\rm Det}(-\nabla^2)\right|_{\v,T=0}+\eta
\frac{\pir_{12}}{3}-\frac{1}{12}\logr_{12}+\frac{c_0}{16}+\O\left(\frac{1}{r_{12}}\right)
\end{equation}
where $\eta=-1/2$ for anti-periodic boundary conditions (i.e.
$\v=0$, and M-dyon is infinitely large), $\eta=1$ for periodic
conditions (i.e. $\v=2\pi$, and L-dyon is large). One can verify
that the term linear in $d$ in this expression is exactly equal to
the term linear in $d$ in \eq{result1} when $\v=0$ for anti-periodic
and when $\v=2\pi$ for periodic boundary conditions. In
Ref.~\cite{DGPS} we have managed to find a constant $c_0$
analytically:
$$ c_0= \frac{8}{9}-\frac{8\,\gamma_E}{3}-\frac{2\,\pi^2}{27}+\frac{4\,\log \pi}{3} $$
The zero-temperature instanton determinant is~\cite{tHooft}:
\beqa
\label{Dinst} \left.\log {\rm Det}(-\nabla^2)\right|_{\v, T= 0}&= & \frac{1}{6}
\log \mu + \frac{1}{12} \log\left(\frac{r_{12}}{\pi}\right) +\alpha(1/2)
\;,
\eeqa where it is implied that the determinant is regularized by
the Pauli--Villars method and $\mu$ is the Pauli--Villars mass. Thus
the isospin-1/2 result for the case of trivial holonomy is \begin{equation}
\log{\rm Det}(-\nabla^2)=\eta\frac{\pir_{12}}{3}+\frac{c_0}{16}+\alpha(1/2)-\frac{1}{12}\log\pi+\frac{1}{6}\log\mu\;,
\label{GPY2} \end{equation} where \begin{equation}
\alpha(1/2)=-\frac{17}{72}+\frac{\gamma_E}{6}+\frac{\log\pi}{6}-\frac{\zeta'(2)}{\pi^2}
\end{equation}
We have to match our \eq{result1} with \eq{GPY2} at $\v=0$ or
$\v=2\pi$, but \eq{result1} has been derived assuming $r_{12}\gg
\frac{1}{\v},\frac{1}{\overline{{\rm v}}}$ and one cannot formally take the limit
$\v\to 0$ in that expression without taking simultaneously
$r_{12}\to\infty$. In the next subsection we will show that one
\textit{can} actually take this limit and get a correct constant. We
will rely on exact results to show that \eq{result1} is valid for
arbitrary values of $\vr_{12}$ and $\bvr_{12}$ (but large $r_{12}\gg 1$).
\subsection{Extending the result to arbitrary values of $\vr_{12}$}
Let us take a fixed but large value of the dyon separation $r_{12} \gg
1$ such that both \eq{result1} and \eq{GPY2} are valid. Actually,
our aim is to integrate the exact expression for the derivative of
the determinant \begin{equation} \label{aim}
\partial_\v \log {\rm Det}(-\nabla^2)= \int \wp(x)d^4x ,\qquad
\wp(x)\equiv -{\rm Tr}\left[\partial_\v A_\mu J^\mu\right], \end{equation} from
$\v=0$, where the determinant is given by \eq{GPY2}, to some small
value of $\v\ll 1$ (but such that $\vr_{12}\gg 1$), where \eq{result1}
becomes valid. We shall parameterize this $\v$ as $\v= k/r_{12}\ll 1$
with $k \gg 1$. The result of the integration over $\v$ must be
equal to the difference between the right hand sides of
\eqs{result1}{GPY2}. We want to show that it is ${\cal O}(k/r_{12})$; it
means that there are no large corrections to \eq{result1} at small
$\v$ that can alter the constant term in the determinant.
Denoting by $\bar{\wp}$ our $\wp$ with subtracted asymptotic terms
we have to show that \begin{equation} \int_0^{\frac{k}{r_{12}}} d\v
\int\!d^4x\,\bar\wp(x)= \O\left(\frac{1}{r_{12}}\right)\;.
\label{intwp2}\end{equation}
In this integration we are in the domain $1/\v\gg 1$ and $r_{12}\gg 1$
and we can simplify the integrand dropping terms which are small in
this domain. At this point it will be convenient to restore
temporarily the temperature dependence. With $\beta\equiv 1/T$ our
domain of interest is $1/\v\gg \beta$ and $r_{12}\gg \beta$. Therefore
we are in the small-$\beta$ range and can expand $\bar\wp$ in series
with respect to $\beta$: \begin{equation}
\bar\wp=\frac{1}{\beta^2}\wp_0+\frac{1}{\beta}\wp_1+\O(\beta^0)\;.
\label{beta_exp}\end{equation} As we shall see in a moment, only the first two
terms can be not small in this range and we need to know only them
to compute $c_2$. It is a great simplification because $\wp_{0,1}$
does not contain terms proportional to $e^{-\overline{{\rm v}} r}$ since $\overline{{\rm v}}= 2\pi
T-\v \to \infty$ at $\beta \to 0$, and what is left is time
independent. Moreover, what is left after we neglect exponentially
small terms are homogeneous functions of $r,s,r_{12},\v$ and we can turn
to the dimensionless variables: \beqa &&\wp_0(r,s,r_{12},\v)=
\frac{1}{r_{12}}\widetilde{\wp}_0\left(\frac{r}{r_{12}},
\frac{s}{r_{12}},\vr_{12}\right) \;. \eeqa From the exact expressions we see
that $\wp_1 \equiv 0$. The reason is that there is no $1/\beta$ term
in \eq{Asum}. Actually the same phenomenon can be seen for the
single dyon in \eq{otvA}. We rewrite the l.h.s. of \eq{intwp2} in
terms of the new variables: \beqa
\int\!d^4x\,\bar\wp(x)=\frac{r_{12}^2}{\beta}\int\! d^3\tilde
x\,\widetilde{\wp}_0 +\O(\beta) =\frac{r_{12}^2}{\beta} F_0(\v r_{12}) +
\O(\beta) \;, \label{intwp01}\eeqa where $\tilde x = x/r_{12}$ is
dimensionless. We see that it is indeed sufficient to take just the
first two terms in the expansion \ur{beta_exp} at $\beta\to 0$.
Let us try to integrate \eq{intwp01} over $\v$ from 0 to $k/r_{12}$ even
without explicit knowledge of $F_{0}$. We get \begin{equation}
\int \limits_0^{k/r_{12}}\frac{r_{12}^2}{\beta} F_0(\v r_{12})
d \v = \frac{r_{12}}{\beta} \int \limits_0^{k} F_0(\vr_{12}) d(\v r_{12})
\end{equation} We know that the approximate result \eq{result1} is correct
with at least ${\cal O}(\exp(-k) , 1/(\vr_{12}))$ precision, so at large
$k$ the correction to \eq{result1} must be a constant, plus a term
of the order of $1/(\vr_{12}) = 1/k$. Therefore, $F_0$ must be zero. We
have checked numerically that it is indeed so (actually this
function $F_0$ is the same as in \cite{DGPS}).
Thus, we have proved \eq{intwp2}. We {\it can} now compare
\eq{result1} with \eq{GPY2} and get the constant: \begin{equation}
c_\frac{1}{2}=\frac{c_0}{16}+\alpha(1/2)+\frac{1}{12}\log 4
\pi+\frac{1}{6}\log\mu=\frac{1}{6}\log\mu+
\frac{36\log(2\pi^2)-39-\pi^2}{216}-\frac{\zeta'(2)}{\pi^2}=\frac{1}{6}\log\mu+0.36584648
\label{c_half} \end{equation}
It means that \eq{result1} is right in whole range $0\leq \v\leq
2\pi$ with the accuracy $1/r_{12}$.
\section{$1/r_{12}$ corrections} \label{sec1/r}
In the previous section we have seen that \Eq{result1} is valid to
all orders in $1/(\vr_{12}),1/(\bvr_{12})$; however, there are other $1/r_{12}$
corrections which are not accompanied by the $1/\v,\;1/\overline{{\rm v}}$ factors:
the aim of this subsection is to find them using the exact vacuum
current.
To this end, we again consider the case
$r_{12}\gg\frac{1}{\v},\,\frac{1}{\overline{{\rm v}}}$ such that one can split the
integration over $3d$ space into three domains shown in Fig.~2. In
the far-away domain one can use the same vacuum current as we had in
Section \ref{FarCurrent}, as it has an exponential precision with
respect to the distances to both dyons. In the core regions,
however, it is now insufficient to neglect completely the field of
the other dyon, as we did in Section IV looking for the leading
order. Since we are now after the $1/r_{12}$ corrections, we have to use
the exact field and the exact vacuum current of the caloron but we
can neglect the exponentially small terms of order $e^{-r_{12}}$.
Another modification with respect to section IV is that we find it
more useful this time to choose $r_{12}$ as the parameter $\cal P$ in
\eq{dvDet}. We shall compute the $1/r_{12}^2$ and higher terms in
$\d{\rm Det}(\!-\!D^2)/\dr_{12}$ and then restore the determinant itself
since the limit of $r_{12}\to\infty$ is already known. Let us define how
the KvBLL field depends on $r_{12}$. As is seen from \eq{APvBP} the
KvBLL field is a function of $r,\;s,\;\v,\;r_{12}$ only. We define the
change in the separation $r_{12}\to r_{12}+dr_{12}$ as the symmetric
displacement of each monopole center by $\pm dr_{12}/2$. It corresponds
to \begin{equation} \frac{\d r}{\d r_{12}}=\frac{r_{12}^2+r^2-s^2}{4r_{12} r},\qquad
\frac{\d s}{\d r_{12}}=\frac{r_{12}^2+s^2-r^2}{4r_{12} s}. \label{drsd}\end{equation} We
shall use the definition \ur{drsd} to compute the derivative of the
caloron field \ur{APvB} with respect to $r_{12}$.
Let us start from the $M$-monopole core region. To get the $1/r_{12}$
correction to the determinant we need to compute its derivative in
the $1/r_{12}^2$ order and expand correspondingly the caloron field and
the vacuum current to this order. Wherever the distance $r$ from the
far-away L dyon appears in the equations, we replace it by
$r=(r_{12}^2+2sr_{12}\cos\theta+s^2)^{1/2}$, where $s$ is the distance from
the M-dyon and $\theta$ is the polar angle seen from the M-dyon
center. Expanding in inverse powers of $r_{12}$ we get the coefficients
that are functions of $s,\cos\theta$. One can easily integrate over
$\theta$ as the integration measure in spherical coordinates is
$2\pi s^2 ds\;d\cos\theta$. Then we integrate over $s$, the distance
to the core of the M-monopole. After that we have to add
contributions from the L and M dyon cores and from the far-away
domain. It turns out that contributions from dyons differ by terms
dependent on $R$ only: \beqa \d_{r_{12}}\log{\rm Det}_{\rm core}&=&
\frac{1}{r_{12}^2}\left(\frac{23}{72\pi}+\frac{\gamma}{6\pi}+\frac{\pi}{216}+\frac{\log(R^2
\v \overline{{\rm v}}/\pi^2)}{12\pi}\right)
+\frac{1}{12r_{12}^3\pi}\left(\frac{1}{\v}+\frac{1}{\overline{{\rm v}}}\right)\\
\nonumber&-&\frac{1}{24r_{12}^4\pi
}\left(\frac{1}{\v^2}+\frac{1}{\overline{{\rm v}}^2}\right)
+\frac{60-\pi^4}{2160r_{12}^5\pi}\left(\frac{1}{\v^3}+\frac{1}{\overline{{\rm v}}^3}\right)+\O\left(\frac{1}{r_{12}^6}\right)
\eeqa (we drop here powers of $R$ as they cancel with the
contribution from the far domain)
Now let us turn to the far-away domain. Recalling \eq{J4asP} we
realize that the contribution of this region is determined by the
potential energy: \beqa \left.\frac{\d \log
{\rm Det}(\!-\!\nabla^2)}{\dr_{12}}\right|_{\rm far}
& = & \int\!d^3x\, \d_{r_{12}}P\left(\frac{1}{2} \left[\overline{{\rm v}}-\frac{1}{r}+\frac{1}{s}\right] \right) \\
\nonumber& = & \frac{1}{8}P''(\overline{{\rm v}}/2)\int\!d^3x
\,\d_{r_{12}}\left(\frac{1}{r}-\frac{1}{s}\right)^2 +\frac{1}{384}P^{\rm
IV}(\overline{{\rm v}}/2)\int\!d^3x
\,\d_{r_{12}}\left(\frac{1}{r}-\frac{1}{s}\right)^4. \label{smpfn} \eeqa
The integration range is the $3d$ volume with two balls of radius
$R$ removed. We use \beqa \int
\d_{r_{12}}\left(\frac{1}{r}-\frac{1}{s}\right)^2d^3x&=&4\pi-\frac{16\pi
R^2}{3 r_{12}^2}
+\sum_{n=1}^\infty\frac{8 n \pi}{(2n-1)(2n+1)}\left(\frac{R}{r_{12}}\right)^{2n+1}\\
\int \d_{r_{12}}\left(\frac{1}{r}-\frac{1}{s}\right)^4d^3x&=&
\nonumber\frac{32\pi\log(r_{12}/R)}{r_{12}^2}+\frac{2\pi(8-9\pi^2)}{3r_{12}^2}+\frac{80\pi
R}{r_{12}^3} -\frac{32\pi R^2}{r_{12}^4}+\frac{304\pi R^3}{15
r_{12}^5}+\O(1/r_{12}^{6})
\eeqa
Adding up all three contributions we see that the region
separation radius $R$ gets canceled (as it should), and we come to:
\beqa
\d_{r_{12}}\log{\rm Det}(-\nabla^2)&=&\frac{\pi}{2}P''\left(\frac{\overline{{\rm v}}}{2}\right)
+\frac{1}{12r_{12}^2\pi}\left(\log(\v\bvr_{12}^2/\pi^2)-\frac{23\pi^2}{72}+2\gamma+\frac{25}{6}\right)\\
&+&\frac{1}{12r_{12}^3\pi}\left(\frac{1}{\v}+\frac{1}{\overline{{\rm v}}}\right)
-\frac{1}{24r_{12}^4\pi }\left(\frac{1}{\v^2}+\frac{1}{\overline{{\rm v}}^2}\right)
\nonumber+\frac{60-\pi^4}{2160r_{12}^5\pi}\left(\frac{1}{\v^3}+\frac{1}{\overline{{\rm v}}^3}\right)+\O\left(\frac{1}{r_{12}^6}\right)
\;. \eeqa Integrating over $r_{12}$ gives \beqa
\label{r12corr}\log{\rm Det}(-\nabla^2)&=&\frac{\pi}{2} P''\left
(\frac{\overline{{\rm v}}}{2}\right) r_{12} -
\frac{1}{12r_{12}\pi}\left(\log(\v\bvr_{12}^2/\pi^2)-\frac{23\pi^2}{72}+2\gamma+\frac{37}{6}\right)
-\frac{1}{24r_{12}^2\pi}\left(\frac{1}{\v}+\frac{1}{\overline{{\rm v}}}\right)\\
\nonumber&+&\frac{1}{72r_{12}^3\pi
}\left(\frac{1}{\v^2}+\frac{1}{\overline{{\rm v}}^2}\right)
-\frac{60-\pi^4}{8640r_{12}^4\pi}\left(\frac{1}{\v^3}+\frac{1}{\overline{{\rm v}}^3}\right)
+\frac{12-\pi^4}{2880r_{12}^5\pi}\left(\frac{1}{\v^4}+\frac{1}{\overline{{\rm v}}^4}\right)
+\bar c+\O\left(\frac{1}{r_{12}^6}\right)\;, \eeqa where $\bar c$ is the
integration constant that does not depend on $r_{12}$. Comparing
\eq{r12corr} with \eq{result1} at $r_{12}\to\infty$ we conclude that
\begin{equation} \bar c =
V\,\left[P\left(\frac{\overline{{\rm v}}}{2}\right)-\frac{\pi^2}{12}\right]
-\frac{\overline{{\rm v}}\log{\overline{{\rm v}}}}{12\pi}-\frac{\v\log{\v}}{12\pi}+c_\frac{1}{2} \end{equation}
and $c_\frac{1}{2}$ is given in \eq{c_half}. We note that the leading
correction, $\logr_{12}/r_{12}$, arises from the far-away range and is
related to the potential energy, similar to the leading $r_{12}$ term.
This series in $1/r_{12}$ is asymptotic as the coefficients
are rapidly increasing.
Interesting, we have revealed that the contribution from the regular
part of the current $J^{\rm r}$ has no corrections $1/r_{12}^n$ (we
verified that up to the $10^{\rm th}$ order) and is determined
completely by the far asymptote \begin{equation} \!-\!\int
d^4x\,{\rm Tr}\left(\partial_{r_{12}} A_\mu\, J^{\rm r}_\mu\right)\simeq
P''\left(\frac{\overline{{\rm v}}}{2}\right)\frac{\pi}{2}\;. \label{regcontr1} \end{equation} It
seems that ${\rm Tr}\left(\partial_{r_{12}} A_\mu\, J^{\rm r}_\mu\right)$ is
a full derivative of a simple expression.
\section{Asymptote of small separation }
In this domain of the moduli space of the KvBLL caloron it is
important to realize that KvBLL caloron is a chain of the usual BPST
instantons equally separated in time direction with a fixed relative
gauge orientation between the neighbors.
When the size of instantons is large compared to their separation
$1/T$, they overlap and cannot be thought of as individual
pseudoparticles. It is much simpler to consider the KvBLL caloron as
a bound state of two BPS dyons. Small separations correspond to the
small instanton size $\rho$ since $\rho=\sqrt{r_{12}/\pi}$. It means
that instantons do not overlap, and the KvBLL caloron reduces to the
usual BPST caloron.
Formally, one switches to dimensionless variables $r=\bar
r\rho,\;s=\bar s\rho$ and then takes the limit $\rho\rightarrow 0$.
In this limit the gauge field of the KvBLL caloron becomes that of
the ordinary instanton of vanishing size, \textit{plus} a constant
field. It is then clear that the determinant becomes that of the
ordinary instanton, plus a contribution from a constant field: \begin{equation}
\log {\rm Det}(-\nabla^2)=\left[P\left(\frac{\overline{{\rm v}}}{2}\right)
-\frac{\pi^2}{12}\right]V+\frac{1}{6} \log \mu + \frac{1}{12}
\log\left(\frac{r_{12}}{\pi}\right) +\alpha(1/2)+\O(\rho)
\;,\;\;\;\;\;\alpha(1/2)=0.145873 \end{equation} One can expand in powers of
$\rho$ further, but it requires knowledge of the exact expression
for the vacuum current (\ref{defJ}). Here we give only the final
result. The contribution from the regular part of the current
$J^{\rm r}$ for small $r_{12}$ is \begin{equation} \!-\!\int
d^4x\,{\rm Tr}\left(\partial_{r_{12}} A_\mu\, J^{\rm r}_\mu\right)
=P''\left(\frac{\overline{{\rm v}}}{2}\right)\frac{\pi}{2}+\O(\rho)
\label{regcontr}\end{equation} and from the singular part
\begin{equation}
\!-\!\int d^4x\,{\rm Tr}\left(\partial_{r_{12}} A_\mu\, J^{\rm s}_\mu\right)=\frac{1}{12\pi\rho^2}+(3\v\overline{{\rm v}}-2\pi^2)\frac{1}{72\pi}+\O(\rho)
\label{singcontr}
\end{equation}
taking into account these contributions we have
\begin{equation}
\label{small_rho}
\log {\rm Det}(-\nabla^2)=\left[P\left(\frac{\overline{{\rm v}}}{2}\right)-\frac{\pi^2}{12}\right]V+
\alpha(1/2)+\frac{1}{6} \log(\mu\rho) -(7\pi^2-3\pi\v-3\v^2)\frac{\rho^2}{36}+\O(\rho^3) \;,
\end{equation}
Note that the contribution from the regular part of the vacuum current
for small values of $r_{12}$ is exactly the same as it has been for
large $r_{12}$ in \eq{regcontr1}. It suggests that \eq{regcontr1} is
exact. Unfortunately, we have not been able to prove it
analytically, but we have verified it numerically. We have found out
that \eq{regcontr1} is correct for all $r_{12}$ and $\v$ with the
precision of at least $10^{-3}$.
\section{Numerical evaluation} \label{sec_num}
\begin{figure}[t]
\label{fig_2sht}
\centerline{\epsfxsize=0.5\textwidth \epsfbox{aaa.eps}
\epsfxsize=0.5\textwidth \epsfbox{daaa.eps}}
\caption{Result of the
numerical evaluation of $A(\v,r_{12})$. The dashed line corresponds to
the trivial holonomy (left). Fist, we have computed $\d_{r_{12}}
A(\v,r_{12})$ (right), and then integrated it from $r_{12}=\infty$ to
$r_{12}=0$. All the curves on the left plot go to zero as $r_{12}\rightarrow
0$ with an accuracy $10^{-4}$; that proves the consistency of our
small $r_{12}$ and large $r_{12}$ asymptotics. }
\end{figure}
One can hardly expect a compact analytical expression for the
determinant at arbitrary separations (or instanton sizes) and
holonomies as even for the case of the trivial holonomy, studied by
Gross, Pisarski and Yaffe~\cite{GPY}, it is not known. In this
section we present our results of the numerical evaluation of the
determinant for arbitrary separations and holonomies.
We start from \eq{dvDet} and calculate the exact analytical
expression for the vacuum current (\ref{defJ}) using (\ref{green12})
and the exact expression from \cite{DGPS} for the ADHM construction.
The resulting expression is too lengthy for printing but can be
provided by request. We calculate the trace in the integrand in the
r.h.s. of \eq{dvDet} with a particular choice of
the parameter $\cal P$.
We have taken ${\cal P}=r_{12}$. Using the axial symmetry we can reduce
the number of integrations to 3. We have managed to perform the
integration numerically with the precision $10^{-4}$, despite the
complexity of the expression. The numerical data are shown in
Fig.~3. One can see that it is consistent with all our analytical
results. Following \cite{GPY} we denote
\begin{equation}
\log{\rm Det}(-\nabla^2)=\left.\log{\rm Det}(-\nabla^2)\right|_{\v,T=0}
+A(\v,r_{12})+\left[P\left(\frac{\overline{{\rm v}}}{2}\right)-\frac{\pi^2}{12}\right]V+P''\left(\frac{\overline{{\rm v}}}{2}\right)\frac{\pir_{12}}{2}\,,
\end{equation}
where $\left. \log{\rm Det}(-\nabla^2)\right|_{\v,T=0}$ is given in \eq{Dinst}.
Note that if \eq{regcontr1} is exact, only the singular part of
the current $J^{\rm s}$ contributes to $A(\v,r_{12})$, thus $A(\v,r_{12})$
must be symmetric under the exchange $\v\leftrightarrow\overline{{\rm v}}$. We have
verified this symmetry numerically. From
(\ref{r12corr},\ref{singcontr}) we see that $A(\v,r_{12})$ has the
following asymptotics (valid for all $\v$)
\begin{equation}
A(\v,r_{12})=\frac{\log(2\pi)}{6}-\frac{\v\log\v}{12\pi}-\frac{\overline{{\rm v}}\log\overline{{\rm v}}}{12\pi}+\frac{1}{18}
-\frac{\gamma}{6}-\frac{\pi^2}{216}-\frac{\log
(r_{12}/\pi)}{12}+\O\left(\frac{1}{r_{12}}\right)
=\frac{(3\v\overline{{\rm v}}-2\pi^2)r_{12}}{72\pi}+\O\left(r_{12}^{3/2}\right)
\label{Aasy}\end{equation}
We find that $A(\v,r_{12})$ can be fitted by the following
expression
\begin{equation}
A(\v,r_{12})\simeq-\frac{1}{12}\log\left(1+\frac{\pir_{12}}{3}\right)
-\frac{r_{12}\alpha}{216\pi(1+r_{12})} +\frac{0.00302\;
r_{12}(\alpha+9 \v\overline{{\rm v}})}{2.0488+r_{12}^2}
\label{fit}\end{equation}
where $\alpha=18\v\log\v+18\overline{{\rm v}}\log\overline{{\rm v}}-216.611$. This expression has a maximum
absolute error $5\times 10^{-3}$.
\section{Summary
}
In this paper we have considered finite temperature QCD with light fundamental fermions and
calculated the fermionic contribution to the 1-loop quantum weight of the instanton with
non-trivial value of holonomy (or Polyakov line) at spatial infinity.
Finite temperature theory is formulated in terms of the Euclidean QCD partition function
with (anti-)periodic boundary conditions in time with the period $\beta=1/T$, where $T$ is
the temperature.
The instanton with non-trivial holonomy (or the KvBLL caloron~\cite{KvB,LL} for brevity)
is the most general self-dual solution of the YM equations of motion with a unit topological
charge and the above-mentioned periodicity conditions imposed. The solution is parameterized
by the holonomy $\v$ at spatial infinity, the temperature $T=1/\beta$, the size $r_{12}$,
center position, and color orientation. When $r_{12} \ll \beta$ the solution reduces to the ordinary
BPST instanton with size $\rho=\sqrt{r_{12} T/\pi}$ in the domain $x_\mu^2\sim \rho^2$ which
gives the main contribution to the action density. In the opposite limit $r_{12} \gg \beta$
it can be described as a superposition of two (for $N_c=2$) properly gauge-combed
BPS monopoles with the separation $r_{12}$ between them.
One can assume that the effective gauge coupling is reasonably small at the deconfinement and chiral
symmetry restoration temperatures and that therefore the semiclassical ensemble of the KvBLL calorons
makes sense and can be relevant for describing those phase transitions. It is therefore important to calculate
the quantum weight of an individual KvBLL caloron as the first step in the study of the ensemble of calorons.
The gluonic contribution to the quantum weight has been calculated in Ref.~\cite{DGPS}. In this paper we have
extended the methods and the result of that work to the theory with light fermions -- by computing the
fermionic determinant over non-zero modes ${\rm Det}' (i \nabla\hspace{-.65em}/\hspace{.3em})$. The zero fermion modes
\cite{Cherndb} should be taken separately when constructing the ensemble,
in the spirit of Ref.~\cite{DP2a,DP2b}; it will be considered elsewhere.
The main result of the paper is the determinant in the fundamental representation in the background
field of the KvBLL caloron (\ref{APvBP}):
\begin{equation}
\frac{1}{2} \log {\rm Det}'(i \nabla\hspace{-.65em}/\hspace{.3em}(A^P)) = \left.\log{\rm Det}(-\nabla^2)\right|_{\v,T=0}+A(\v/T,r_{12} T)
+\left[P\left(\frac{\overline{{\rm v}}}{2 }\right)-\frac{\pi^2}{12}T^3\right]V
+ P''\left(\frac{\overline{{\rm v}}}{2}\right)\frac{\pir_{12}}{2} \,,
\label{resu1}\end{equation}
where $V$ is a $3d$ volume and $\overline{{\rm v}}=2 \pi T - \v$.
Since the fundamental representation is sensitive to the sign of the asymptotic holonomy $e^{i \v T \tau_3/2}$,
the fermionic determinant is $4 \pi T$ periodic in $\v$, in contrast to the period of $2 \pi T$ for the gluon
and ghost determinants. We have also considered the background field (\ref{APvBA}) for which one has to exchange
$\v$ with $\overline{{\rm v}}$ in \eq{resu1}.
The numerical values of the quantity $A(\v,r_{12})$ are tabulated in Appendix B; they can be fitted by \eq{fit}
consistent with the asymptotic expansion of this quantity at large dyons' separations,
\beqa
\label{r12corrA}
A(\v,r_{12})&=&\frac{\log (2\pi)}{6}-\frac{\v\log\v}{12\pi}-\frac{\overline{{\rm v}}\log\overline{{\rm v}}}{12\pi}+\frac{1}{18}
-\frac{\gamma}{6}-\frac{\pi^2}{216}-\frac{\log (r_{12}/\pi)}{12} \nonumber \\
&&- \frac{1}{12r_{12}\pi}\left(\log(\v\bvr_{12}^2/\pi^2)-\frac{23\pi^2}{72}+2\gamma+\frac{37}{6}\right)
-\frac{1}{24r_{12}^2\pi}\left(\frac{1}{\v}+\frac{1}{\overline{{\rm v}}}\right)\\
\nonumber&+&\frac{1}{72r_{12}^3\pi }\left(\frac{1}{\v^2}+\frac{1}{\overline{{\rm v}}^2}\right)
-\frac{60-\pi^4}{8640r_{12}^4\pi}\left(\frac{1}{\v^3}+\frac{1}{\overline{{\rm v}}^3}\right)
+\frac{12-\pi^4}{2880r_{12}^5\pi}\left(\frac{1}{\v^4}+\frac{1}{\overline{{\rm v}}^4}\right)
+\O\left(\frac{1}{r_{12}^6}\right)\;,
\eeqa
and with the expansion at small $r_{12}$ (see \eq{small_rho}),
\begin{equation}
A(\v,r_{12})=\frac{r_{12} \left(3 \v\overline{{\rm v}}-2\pi^2\right) }{72\pi } + {\cal O}(r_{12}^{3/2})
\end{equation}
The generalization of these results to the case of larger groups, $N_c>2$, will be published elsewhere~\cite{GSN}.
\section*{Acknowledgements}
We thank Dmitry Diakonov and Victor Petrov for useful and
inspirating discussions, and for careful and critical reading
of the manuscript.
Both authors also thank the foundation of non-commercial programs
`Dynasty' for partial support. This work was partially supported by
RSGSS-1124.2003.2.
|
1,108,101,563,263 | arxiv | \section{ Introduction}
It is well-known nowadays that quantum field theory allows violations of
all the known pointwise energy conditions in classical general relativity,
such as the weak energy
condition and null energy condition \cite{HE}.
Specific examples are the Casimir effect \cite{C,BM}, squeezed states
of light
\cite{WKHW} and
black hole evaporation \cite{H75} which involves negative energy densities
and fluxes in a crucial way. On the other hand, if the laws
of quantum field theory place no restrictions on negative energy, then it
might be
possible to produce gross macroscopic effects such as: violation of the second
law of thermodynamics \cite{F78,D82} or of cosmic censorship \cite{FR90,FR92},
traversable wormholes \cite{MT,MTY}, ``warp drive''\cite{WARP},
and possibly time machines \cite{MTY,AEVERETT}. As a result, much effort has
been recently directed toward determining the extent to which
quantum field theory permits violations of the local energy conditions.
One approach involves so-called
``averaged energy conditions'' (see, for example, \cite{T}-\cite{VISSER}),
i.e., averaging the local
energy conditions over timelike or null geodesics. Another method employs
``quantum inequalities'' (QI's) \cite{F78,FR95,F91}, which are
constraints on the magnitude and duration of negative energy fluxes and
densities. In a recent paper Ford and Roman \cite{FR96} have investigated the
averaged weak(AWEC)
and averaged null (ANEC) energy conditions, together with quantum inequalities
on negative energy, in the context
of evaporating black hole backgrounds in both two and four dimensions. They
find that, in all cases where the averaged energy
conditions fail, there exist quantum inequalities which bound the magnitude and
extent of the negative energy, and hence
the degree of the violation. In the four dimensional case, they first wrote
down a QI-type bound by analogy with the 2D case, and
then made use of the limited range of available numerical data outside the
event horizon \cite{ELSTER}-\cite{JMO} to check whether the inequality
is satisfied.
However, generally, the energy integrals are to be evaluated over an infinite
range to test the AWEC and ANEC\footnote{
and furthermore there seems to be an error in one set of the numerical data
\cite{JMO} they used, as pointed out by Visser\cite{VIS97}}. Therefore it
remains interesting to see whether
the qualitative
features of the Ford-Roman results are still true if we can have analytical
expressions for the energy-momentum tensor and employ them to evaluate the
energy integrals over the whole range. Fortunately, Visser \cite{VIS97} has
recently constructed a semi-analytical model for the stress-energy tensor
corresponding to the conformally coupled massless scalar field that nicely
fits all the known numerical data \cite{ELSTER}-\cite{JMO}, and satisfies all
the known analytic features of the stress-energy tensor \cite{CF} in the Unruh
vacuum describing a black hole evaporating into empty space. It is worthwhile to point
out that the original Ford-Roman quantum inequality was proven for the minimally coupled,
rather than the conformally coupled scalar field.
In this paper, we will use Visser's semi-analytical results to explicitly
evaluate the relevant integrals to
examine the energy conditions both outside and inside\footnote{ Although Visser's result
is obtained using data from outside the event horizon there is nothing to prevent us from
applying it inside.\cite{VIS97})} the black hole event horizon and test quantum
inequalities whenever the averaged energy conditions fail. It is worth noting that
recently Pfenning and Ford have generalized the Ford-Roman quantum
inequality-type restrictions on the stress-energy tensor for the minimally coupled
scalar field in Minkowski spacetime to some static curved spacetimes.\cite{PFORD}
\section{ Averaged energy conditions and quantum inequalities}
Let us write the four-dimensional Schwarzschild metric in the form
\begin{equation}
ds^2=-C dt^2+C^{-1}dr^2+ r^2d\Omega^2
\end{equation}
where $ C=1-2M/r$. Define a new variable $z=2M/r$. Then according to Visser's
semi-analytical model, the non-zero stress tensor components $T_{\mu\nu}$
(in Schwarzschild coordinates) are given by
\begin{eqnarray}
T_{tt} &=& p_\infty
\Bigg[ \left({\xi\over20} - {k_5\over24} - {k_6\over20}\right) +
k_4 z^2 +
\left(-k_4 + {4 k_5\over3}\right) z^3
\nonumber\\
&&\qquad
+ \left({-\xi} -{5 k_5\over4} + {3 k_6\over2} \right)z^4
+ \left( {9 \xi\over10} -{7 k_6\over5} \right)z^5 \Bigg],
\\
\nonumber\\
T_{rr} &=& -{z^2\over(1-z)^2} p_\infty
\Bigg[ - \left({\xi\over20} - {k_5\over24} - {k_6\over20}\right)
+ k_4 z^2 +
\left(-k_4 + {2 k_5\over3}\right) z^3
\nonumber\\
&&\qquad
+ \left(-{3 k_5\over4} + {k_6\over2} \right)z^4
+ \left( { \xi\over10} -{3 k_6\over5} \right)z^5 \Bigg],
\\
\nonumber\\
T_{\theta\theta} &=& T_{\phi\phi}= p_\infty \; z^4 \; ( k_4 + k_5 z + k_6 z^2 ),
\\
\nonumber\\
T_{rt} &=& T_{tr}= -\left( {\xi\over20} - {k_5\over24} - {k_6\over20}\right) \; p_\infty \;
{z^2\over1-z},
\end{eqnarray}
where
\begin{eqnarray}
p_\infty =\frac{1}{90(16\pi)^2(2M)^4}\nonumber\\
k_4=26.5652,\quad k_5=-59.0214,\quad k_6=38.2068.\\
\end{eqnarray}
and $ T_{\mu\nu} $ is understood to denote the quantum expectation value in the Unruh vacuum
state. In this paper, we only
consider the Unruh vacuum, as this is the vacuum which describes physically realistic
collapse spacetimes in which the black hole forms at a finite time in the past.
We shall evaluate AWEC and ANEC integrals along half-complete timelike and
null geodesics to determine where the averaged energy conditions fail and whether
there exists any bound on the extent of violation. Usually the ANEC is defined using an integral over $(-\infty,
\infty)$, rather than over semi-infinite domains (i.e. over geodesics with past endpoints), as we shall consider here.
However, it is of some interest to study whether ANEC is satisfied in the spacetime of an object collapsing to form a
black hole. The answer to this question might determine, for example, whether Penrose's singularity theorem \cite{P65}
will still
hold in the presence of local violations of the energy conditions, such as the Hawking evaporation process. In this case,
one would want ANEC to hold over half-complete null geodesics.
\paragraph*{1. Outgoing radial timelike observers}
The four-velocity of an outgoing radial timelike observer in 4D Schwarzschild spacetime is given by
\begin{equation}
u^{\mu}=(u^t,u^r)=\biggl({dt\over d\tau}, {dr\over d\tau},{d\theta \over d\tau},{ d\phi\over d\tau}
\biggr)=\biggl(\frac{k}{C},\sqrt{k^2-C},0,0\biggr)
\end{equation}
where $k$ is the energy per unit rest mass.
The energy density in this observer's frame is
\begin{equation}
T_{\mu\nu}u^{\mu}u^{\nu}=\frac{k^2}{C^2}(T_{tt}+C^2T_{rr})-CT_{rr}+\frac{2k\sqrt{k^2-C}}{C}\;T_{rt}
\end{equation}
We consider, for simplicity, the case in which an observer is shot outward initially very close to the
horizon at very large velocity, i.e., we are interested in the limits
\begin{equation}
\epsilon<<M, \qquad k>>1,
\end{equation}
where $r=2M+\epsilon $ is the observer's initial position.
Examine AWEC along the observer's trajectory to obtain
\begin{equation}
\int^\infty_{2m+\epsilon}\;T_{\mu\nu}u^{\mu}u^{\nu}\;d\tau=\int^\delta_0\;
\frac{2MT_{\mu\nu}u^{\mu}u^{\nu}}{z^2\sqrt{k^2-(1-z)}}
\;dz
\end{equation}
where $\delta= 1+\epsilon/2M. $ If we expand the integrand in inverse powers of $k$, then perform
the integration using Maple and expand the result in powers of $\epsilon$, we get
\begin{eqnarray}
\int^\infty_{2M+\epsilon}\;T_{\mu\nu}u^{\mu}u^{\nu}\;d\tau & \sim &
-\frac{k}{\eta M^2}\biggl( {2a_1\over15}\frac{1}{\epsilon}
-{a_2\over5}+O(\epsilon^0)\biggr) \nonumber\\
&&-\frac{a_3}{30\eta M^3}\frac{1}{k}+O(k^{-3})
\end{eqnarray}
where we have defined new constants
\begin{eqnarray}
\eta=90\times(16)^3\pi^2, \quad a_1=6\xi-5k_5-6k_6=641.8662,
\nonumber\\
a_2 =6\xi-5k_5-6k_6=471.5354, \quad a_3=6\xi-20k_4-12k_6-15k_5=128.37324.
\end{eqnarray}
Plugging in the numerical values and keeping only the leading order, we arrive at
\begin{equation}
\int^\infty_{2M+\epsilon}\;T_{\mu\nu}u^{\mu}u^{\nu}\;d\tau
\sim -2.352\times10^{-5}{1\over M^2}{k\over\epsilon} \gprox
-\frac{1}{M^2 \delta \tau}
\end{equation}
where
\begin{equation}
\delta\tau={\epsilon\over k}
\end{equation}
It is easy to show that near the event horizon,
\begin{equation}
T_{\mu\nu}u^{\mu}u^{\nu}\sim -{k^2\over{M^2(r-2M)^2}}
\end{equation}
By the same argument as in Ref.\cite{FR96}, we see that $\delta\tau$ is the characteristic
time scale for the changes in the energy density observed by the outgoing radial timelike
observers. Clearly, AWEC is violated , but we still have a Ford-Roman type quantum inequality
Eq.~(14) bounding the degree of
violation seen by timelike geodesic observers who start out very close to the horizon at very high
speed. In this limit, the longer the timescale over which the observer sees a significant change in
energy density, the smaller the magnitude of the integrated negative density seen by that observer.
Since the negative energy density drops off rapidly with increasing $r$,
to remain in the negative energy density the observer must stay close
to the horizon. The closer the observer is to the horizon, the larger
is the magnitude of the negative energy density.
However, in order to remain close to the horizon for a long time as seen
by a distant observer, the observer's trajectory must be nearly lightlike.
Therefore, although the observer spends a long time in the negative energy
region as seen by the distant observer, the {\it proper} time spent
in the region of appreciable negative energy (as measured by
$\delta \tau$) decreases with the observer's proximity to the horizon.
\paragraph*{2. Radial Null Geodesic}
ANEC is satisfied for ingoing radial null geodesics \cite{VIS97}, therefore we shall only examine
the outgoing case. Outgoing radial null geodesics are characterized by
\begin{equation}
K^{\mu}=E(C^{-1},1,0,0),
\end{equation}
where $E$ is an arbitrary positive constant, whose value fixes the scale of the affine parameter
$\lambda. $
Thus, we have
\begin{eqnarray}
T_{\mu\nu}K^{\mu}K^{\nu}&=&{E^2\over(1-z)}\biggl[C^{-1}T_{tt}+CT_{rr}+2T_{tr}\biggl]\nonumber\\
&&= p_\infty {z^5\over(1-z)^2}
\Bigg[
{2k_5\over3} - \left(\xi+{k_5\over2}-k_6\right) z +
\left({4\xi\over5} - {4k_6\over5}\right)z^2
\Bigg]
\nonumber\\
&&= p_\infty {z^5\over(1-z)^2}
\Bigg[
-\alpha-\beta z+\gamma z^2
\Bigg],
\nonumber\\
\end{eqnarray}
where
\begin{equation}
\alpha={2k_5\over3}=39.3476, \quad \beta= \left(\xi+{k_5\over2}-k_6\right)=28.2825,
\quad \gamma=\left({4\xi\over5} - {4k_6\over5}\right)=46.23456.
\end{equation}
Note that there is a discontinuity at the event horizon in the above expression, so we shall
address the null rays on the horizon separately. It is easy to show that if
\begin{equation}
r> \frac{4\gamma}{\beta +\sqrt{\beta^2+4\alpha\gamma}} M \sim 1.56523376 M
\end{equation}
then $T_{\mu\nu}K^{\mu}K^{\nu}$ is negative, therefore the local
energy condition is also violated along the null geodesics slightly inside the horizon.
Consider first an outgoing null geodesic starting at $2M+\epsilon,$ with $ \epsilon<<M $,
and examine the ANEC along this ray to get
\begin{equation}
I_1=\int^{\infty}_0\;T_{\mu\nu}K^{\mu}K^{\nu}d\lambda
\sim -{2a_1\over15\eta M^2}{E\over\epsilon}+O(\epsilon^0).
\end{equation}
let us now define
\begin{equation}
\delta\lambda={\epsilon\over E}
\end{equation}
and note that $\delta\lambda$
is the characteristic affine parameter distance over which the negative energy is
decreasing along the outgoing geodesics. Thus ANEC is violated along this ray, but we can rewrite Eq.~(21 )
in the form of a QI
\begin{equation}
\int^{\infty}_0\;T_{\mu\nu}K^{\mu}K^{\nu}d\lambda
\sim -2.3522\times 10^{-5}{1\over M^2{\delta\lambda}}
\gprox -{1\over M^2{\delta\lambda}}
\end{equation}
This is the null version of Eq.~(14).
For outgoing rays inside the horizon starting at $r=2M-\epsilon$, we have
\begin{equation}
I_2=\int^{\lambda_f}_0\;T_{\mu\nu}K^{\mu}K^{\nu}d\lambda=
{1\over E}\int^{2M-\epsilon }_{r_{min}}\;T_{\mu\nu}K^{\mu}K^{\nu}dr\;,
\end{equation}
where $\lambda_f$ is the value of the affine parameter at $r=r_{min}$,
the minimum value of $r$ attained by the null rays.
If we perform the integration and expand the result in the small $r_{min}$ limit,
with $\epsilon$ fixed, we get
\begin{equation}
I_2\sim {32(\xi-k_6)\over{5\eta}}\frac{ME}{r^4_{min}}+O(r^{-3}_{min})>0
\end{equation}
However, if we take the limit of $ \epsilon\rightarrow 0$ with $r_{min}$ fixed, then
\begin{equation}
I_2\sim -{2a_1\over{15\eta M^2}}{E\over\epsilon}+O(\epsilon^0)<0
\end{equation}
Thus we see that ANEC is violated for outgoing null rays just outside the
horizon, but it is satisfied for outgoing null rays inside the horizon
in the limit $r_{min} \rightarrow 0$.
(The divergence in Eq.~(24) as $\epsilon \rightarrow 0$
may be circumvented by an appropriate choice of scaling of the affine
parameter, i.e., by an appropriate choice of $E$ \cite{FR96}.
Outgoing rays which
originate in the region $1.56523376M<r<2M$ initially encounter negative energy,
and later encounter positive energy near the singularity, $r=0$. These 4D results would
seem to suggest that the semiclassical effects of negative energy, in processes such as black
hole evaporation, might not invalidate the Penrose singularity theorem \cite{P65} before quantum
gravity effects become significant. Our four-dimensional
results here are essentially in accordance with the 2D results obtained by Ford and Roman except
for the numerical constants and factors of $M^{-2}$. This may
indicate that results in the 2D are good examples of what is going on in the 4D.
\paragraph*{3. Null geodesics on the horizon}
The Schwarzschild coordinates for the outgoing null geodesics outside the horizon are singular
on the horizon. To avoid this singularity, let us transform to the Kruskal null coordinates
\begin{equation}
U=-e^{-\kappa u}, \qquad V=e^{\kappa v}
\end{equation}
where $ u=t-r^*, v=t+r^*$, with $r^*$ the usual tortoise coordinate and $\kappa=1/(4M)$
the surface gravity. In thses coordinates the null vectors are
\begin{equation}
K^{\bar\mu}=\biggl( {dU\over d\lambda},{dV\over d\lambda}\biggr)=
\biggl( 0, {2E\over C}\kappa e^{\kappa v}\biggr)
\end{equation}
However, this is still not an affinely parameterized null tangent vector on the horizon,
although the singularity can be removed by a proper choice of the scaling of $ E$ \cite {FR96}.
The affinely parametrized null tangent vector on the horizon is \cite{WALDGR}
\begin{equation}
k^{\bar\mu}=(0,\kappa)
\end{equation}
To examine the ANEC on the horizon, let us consider the integral
$\int_{}^{}\, T_{\bar \mu \bar \nu} \, k^{\bar \mu}\,k^{\bar \nu}\, dV$
along a portion of the future event horizon. Here $T_{\bar \mu \bar \nu}$
is the vacuum expectation value of the stress-tensor in the Unruh vacuum state
expressed in Kruskal null coordinates. The Kruskal
advanced time coordinate $V$, is
an affine parameter on the horizon ( \cite{WALDGR}, \cite{WALDQFT}). Using
Eq.~(29), and a straightforward coordinate transformation of
Eqs.~(2)-~(5), we obtain
\begin{equation}
\int_{V_0}^{\infty}\, T_{\bar \mu \bar \nu} \, k^{\bar \mu}\,k^{\bar \nu}\, dV
\,\sim \, - { a_4 \over {24\,\eta\,M^4\,V_0} }
\sim -3.2493\times 10^{-7} {1\over M^4 V_0}\,. \label{eq:Kint}
\end{equation}
where
\begin{equation}
a_4={6\xi\over 5}-k_5-{6k_6\over5}=28.37324.
\end{equation}
It is worth pointing out that for the Unruh vacuum describing a realistic collapse situation,
$V_0$ will not be zero.
There is a smallest allowed value of $V_0=V_{min}$, which is the value of the $V$=const. line at which the
worldline of the surface of the collapsing star intersects the future event horizon.
So, the ANEC integral along a portion of the future event horizon is negative but finite.
To see if there is a Ford-Roman QI-type bound in this case, let us rescale the coordinates as follows
\begin{equation}
\hat U = {\kappa}^{-1}\,U\,, \hat V = {\kappa}^{-1}V\;.
\end{equation}
Then we find
\begin{equation}
T_{\hat \mu \hat \nu} \, k^{\hat \mu}\,k^{\hat\nu}\,= - { 2a_4 \over {3\,\eta\,M^2 \hat V^2} } \,
\end{equation}
Thus
\begin{eqnarray}
\int_{\hat {V_0}}^{\infty}\, T_{\hat \mu \hat \nu} \, k^{\hat \mu}\,k^{\hat
\nu}
\, d{\hat V} \,& = & \, -
\frac{2a_4}{3\eta M^2} \int_{\hat {V_0}}^{\infty}\, \Biggl( {1 \over { {\hat V}^2} }
\Biggr)\, d{\hat V} \nonumber \\
& = & \, - { 2a_4 \over {3 \eta M^2 \, \hat {V_0} } }
\sim - 2.5994\times 10^{-6} { 1 \over { M^2 \, \hat {V_0} } } \,.
\end{eqnarray}
Obviously, $\hat V_0$ can be interpreted as the characteristic affine parameter distance over
which the integrand falls off in the sense that if we increase $\hat V$
from $\hat V = \hat {V_0}$ to $\hat V = 2 \hat {V_0}$,
$T_{\hat \mu \hat \nu} \, k^{\hat \mu}\,k^{\hat \nu}$ falls off to $1/4$ of
its initial value. Thus the interpretation of the QI for null geodesics
on the horizon is the same as that for Eq.~(23).
\section{Conclusions}
In this paper, we have examined, by explicitly evaluating the relevant integrals, AWEC and
ANEC in four-dimensional evaporating Schwarzschild black hole spacetimes using Visser's
semi-analytical model for the stress-energy tensor for the conformally coupled
massless scalar field. We found that AWEC is violated for outgoing radial timelike geodesics
which start just outside the horizon and reach infinity. Similarly, ANEC is violated for
half-complete outgoing radial null geodesics which start just outside the horizon, and
in these two cases, there exist QI-type bounds on the degree of violation. These confirm
Ford-Roman's results. With the semi-analytical expressions for the stress-energy tensor
components, we were able to further examine the ANEC on the horizon and ANEC for outgoing null
rays inside the horizon. It was discovered that ANEC is satisfied along the outgoing null geodesics
inside the horizon because of a large positive contribution to the integral near the singularity.
This suggests that the semiclassical effects of negative energy, in processes
such as black hole evaporation, might not invalidate the Penrose singularity \cite{P65} before
quantum gravity effects become significant. It is interesting to notice that
our 4D results are essentially in accordance with the 2D results obtained by
Ford and Roman, and this would seem to suggest that 2D results may be a very
good indicator of what is happening in the 4D in some cases. However, it was
found that ANEC is violated for
radial null geodesics on the horizon and that there exists a similar Ford-Roman
QI-type bound on
the degree of violation. It is worth pointing out that in our discussions it is assumed that
the background metric is static, thus ignoring the backreaction of the Hawking radiation.
This approach may well be valid during the early stages of black hole evaporation. However,
since the Hawking evaporation is a highly dynamic process, sooner or later
a point will be reached in which the backreaction can no longer be ignored. We then need
to use an evolving background metric, such as the Vaidya metric, to model realistic evaporating
black holes. Therefore it would be interesting to do a similar analysis using an evolving metric
to examine the energy conditions and test quantum inequalities proposed by Ford
and Roman. Hopefully, we shall address this issue in the future.
\begin{acknowledgments}
I wish to thank Prof. Larry Ford and Prof. Tom Roman for their very helpful discussions, comments and
their critical readings of the manuscript. I also wish to thank Prof. Tom Roman for suggesting
this topic to me.
\end{acknowledgments}
|
1,108,101,563,264 | arxiv | \section{Introduction}
Accompanying with the rapid technological progress in the last decades the topic "space weather" is becoming extremely important for life in our society. Space weather disturbances in terms of geomagnetic storms, aurorae or geomagnetically induced currents (GIC) can affect space-borne and ground based technological systems. The strong effect of turbulent magnetic field from interplanetary coronal mass ejection (ICMEs) on the Earth's magnetosphere, driving strong geomagnetic storms and GICs \citep[see][]{Huttunen2008,Guo2011,Lugaz2016}. In a previous study~\citep{Krauss2015} the thermospheric and geomagnetic response to 104 ICMEs observed by the Advanced Composition Explorer (ACE;~\cite{Stone1998}) and Gravity Recovery and Climate Experiment (GRACE;~\citet{Tapley2004}) satellites during the period July 2003 to August 2010 has been analyzed. It was found that the strength of the $B_{\rm z}$ component, either in the shock-sheath or in the magnetic structure~\citep{Burlaga1982} of the ICME, is highly correlated with the neutral density enhancement. An increase in thermospheric neutral densities affects Earth-orbiting satellites in such a way that the drag force on the spacecraft is enhanced, which may lead to severe consequences for the satellite's altitude. For example, during the ICME event on October 29, 2003 the orbit decay for the International Space Station (ISS) for one day increased by a factor of 4~\citep{Bean2007}. Therefore, enhanced statistics are needed in order to better estimate the response of the thermospheric density to impacting solar phenomena. In that respect, not only ICMEs but also co-rotating interaction regions (CIRs) are of interest~\citep{Chen2014}. The interaction between slow and fast solar wind streams, emanating from coronal holes, results in compression regions. Those regions are related to a very turbulent and strong magnetic field and usually last longer compared to the shock-sheath region of a ICME. However, the negative peak in the $B_{\rm z}$ component of the magnetic field is much larger. Within a CIR the plasma pressure peaks close to the stream interface region, which is accompanied by a large shear flow~\citep{Gosling1999}. Especially during low solar activity phases of the solar cycle, CIR induced storms occur very frequently and quasi-periodically~\citep{Temmer2007,Lei2008}. Even though, the compression region of CIRs might be less energetic compared to those of ICMEs, it lasts longer and it is a recurrent phenomenon. Following~\citet{Turner2009}, magnetic storms driven by CIRs seem to deposit more energy in the ionosphere and ring current and thus appear to be more geoefficient than ICMEs.
In this paper, we rely on an extended dataset of thermospheric density measurements compared to our predecessor study. At that time, the investigation period was rather limited due to missing calibration parameters for the GRACE accelerometer data beyond 2010. However, through the development of a new calibration method~\citep{Klinger2016}, the analysis period for the current study could be nearly doubled (2003--2015).
\begin{figure*}[htb]
\begin{center}
\includegraphics[width=8cm]{figure1.png}
\end{center}
\caption{Temporal distribution of the analyzed ICME events (blue bars) between 2003 and 2015. The red bars indicate the level of geomagnetic activity in terms of the geomagnetic disturbance storm time index (Dst)~\citep{Sugiura1964,SugiuraKamei1991}.}
\label{fig:1}
\end{figure*}
Against this background Figure~\ref{fig:1} illustrates an overview of all ICME events analyzed in this study at annual intervals. Even though, very strong geomagnetic storms during solar cycle 24 were missing, due to the lower solar activity level~\citep{Guo2011}, Figure~\ref{fig:1} clearly reveals that, through the temporal extension, nearly 100 additional ICME events could be included in the analysis. In total, the present study comprises measurements of about 400 solar events including the complementary data set of CIRs.
\section{Data and Analysis}
For the current study we analyzed geomagnetic storms induced by 196 ICME and 195 CIR events during the 13-years time period from 2003 to 2015. The basis for the selection of the ICME events forms the catalog of near-Earth interplanetary ICMEs maintained by~\citet{Richardson2010} (R\&C). Arrival times for CIR events are provided through reference lists by S. Vennerstrom and \citet{Jian2011}. Technically speaking, these lists cover stream interaction regions (SIR); the only difference between a SIR and a CIR is that the latter recurs for two or more solar rotations. The effects of these solar storms on the near-Earth environment are thoroughly analyzed by using in-situ magnetic field measurements performed by the ACE spacecraft and in addition by accelerometer measurements on board the low Earth orbiting satellites GRACE and Challenging Mini Payload Satellite (CHAMP;~\citet{Reigber2002}).
\subsection{Magnetic field measurements}
In general, the evaluation of the ICME kinematics is similar to those described in~\citet{Krauss2015}. We process in-situ measurements (level 2 data) by the ACE spacecraft, which is located at the Lagrange point L1, in terms of solar wind flow speed (proton bulk speed) and the magnetic field component $B_{\rm z}$ (in Geocentric Solar Ecliptic, GSE, coordinates) with a time resolution of four minutes (OMNI database;~\citet{King2004}). The presumed starting times of the disturbance are taken from the R\&C catalog and act as a starting point for a 36h time window, from which the negative $B_{z}$ peak was extracted. The analysis of the 195 CIR events was performed by applying the same methods. In contrast to ICMEs and their rather smooth long-term negative $B_{\rm z}$ component in the magnetic structure, geomagnetic storms induced by CIRs are characterized by lower energy input but show long-persistent variations of the $B_{\rm z}$ component in the compression region. To illustrate this behavior Figure~\ref{fig:2} shows the $B_z$ and speed component of the solar wind, as well as the total perpendicular pressure ($P_t$) for a ICME and CIR event.
\begin{figure*}[htb]
\begin{center}
\includegraphics[width=8cm]{figure2.png}
\end{center}
\caption{CIR (top) and ICME (bottom) magnetic field observation, total perpendicular pressure (sum of magnetic pressure and perpendicular plasma thermal pressure) and bulk speed measured by ACE. Green shaded area marks the compression region. Yellow shaded area marks the magnetic structure.}
\label{fig:2}
\end{figure*}
\subsection{Thermospheric neutral density}
As it is generally known, the Earth's atmosphere density decreases exponentially with increasing height. The further away from the surface, the less weight is in the air column above. To examine the response of the Earth's thermosphere at different satellite altitudes we calculated neutral densities using accelerometer measurements from two satellite missions, namely CHAMP and GRACE. The former spacecraft represents the first dedicated Earth gravity field mission and was launched in July 2000 in a near-circular and near-polar orbit (eccentricity $\approx~0.004$, inclination $\approx 87^{\circ}$). During the ten-years mission duration the satellite altitude decreased steadily from the initially 454\,km to below 300\,km.
The second mission which we examined in our analysis is the U.S.-German project GRACE, which was launched in 2002. The mission consisted of two identical spacecraft following each other in the same orbit, separated by about 220\,km. The orbit characteristics (eccentricity $\approx~0.001$, inclination $\approx 89^{\circ}$) are similar to those from CHAMP. Regarding the satellite altitude, Fig.~\ref{fig:3} illustrates the course of both satellite missions during the investigation period 2003--2015.
\begin{figure*}[htb]
\begin{center}
\includegraphics[width=8cm]{figure3.png}
\end{center}
\caption{Course of the satellite altitudes of the CHAMP and GRACE spacecraft during the period of investigation (2003--2015).}
\label{fig:3}
\end{figure*}
Most of the time the GRACE mission was orbiting the Earth at around 490\,km. However, in the last few years of the mission duration (decommission was in November 2017) the altitude decreases more and more rapidly. Comparing the temporal course of the satellite altitude's we can recognize that the two missions are approximately separated by 80--200\,km height difference. Subsequently, the estimated neutral densities ($\rho$) for the lower satellite mission CHAMP must be significantly higher than those from GRACE (cf. Fig.~\ref{fig:7} [c, d]). In either case the density determinations are based on accelerometer measurements ($a$) aboard the satellites, that can be calculated as
\begin{equation}
\vec{a} = -\frac{1}{2} \rho \left(\sum\limits_{i=1}^{plates} C_{D,i} \frac{A_i}{m} (\vec{v} \cdot \vec{n}_i)\cdot \vec{v} \right).
\label{eq:rho}
\end{equation}
Therein, $m$ indicates the current satellite mass, $C_{D,i}$ the variable drag coefficient, $A_i$ the effective cross sectional area, $v$ the satellite velocity relative to the co-rotating atmosphere and $n_i$ the unit vector of the individual satellite plate $i$. These satellite specific information were taken from a satellite 9- and 15-plate macro-models for GRACE and CHAMP, respectively~\citep{Bruinsma2003,Bettadpur2007}. For a more detailed description of the neutral density estimation procedure we refer the reader to~\citet{Sutton2008,Doornbos2009,Krauss2013}.
It has to be noted that, due to the ellipticity of the satellite orbits and the long analysis period the neutral densities are normalized to an average altitude of 400\,km ($\rho_{\rm 400}$) and 490\,km ($\rho_{\rm 490}$) for CHAMP and GRACE, respectively. To accomplish this task we used the empirical thermosphere model Jacchia-Bowman 2008~\citep{Bowman2008}.
In~\citet{Krauss2015}, the bias and scale parameters for the proper calibration of the raw GRACE accelerometer measurements were kindly provided by the Centre National d'\`{E}tudes Spatiales (CNES). The obtained calibration parameters were calculated till December 2010. To save energy and to extend the satellite mission the active thermal control system of the GRACE satellites was switched off in April 2011 - resulting in the necessity to revise the processing routine. The method used for this study was elaborated in the course of the joint project SPICE (Environmental space geodesy: detection of changes in glacier mass time-variable gravity) and since then already applied for the latest Earth gravity field release ITSG-Grace2016. It already takes into account, the temperature dependency of the accelerometer (bias and scale factor) and should therefore be suitable to analyze GRACE accelerations during the extended investigation period between 2010 and 2015. For further information about the developed calibration routine the reader is referred to~\citet{Klinger2016} and~\citet{Krauss2018}.
Since the results of the present study are solely based on the newly formulated calibration method, we initially validated the method by comparing the results with those published in~\citet{Krauss2015}. For this purpose, the data of the former investigation period (July 2003 till December 2010) were reprocessed by using accelerations calibrated with the new routine. Figure~\ref{fig:4} presents the outcome of these computations, in terms of correlation coefficients between the estimated normalized neutral density increase and various analyzed parameters. This includes the magnetic field component $B_{\rm z}$ as well as different geomagnetic indices and corresponding ICME parameters (proxy for the convective electric field [$E$] and for the energy input via the Poynting flux into the magnetosphere [$S$]).
\begin{figure*}[htb]
\begin{center}
\includegraphics[width=12cm]{figure4.png}
\end{center}
\caption{Validation of the new acceleration calibration method [current study] by comparing it with the results from the former~\citep{Krauss2015} study in terms of correlation coefficients with the neutral density. Basis therefore is the former event database from August 2003 to December 2010.}
\label{fig:4}
\end{figure*}
Compared to the predecessor study (red bar), Fig.~\ref{fig:4} reveals that, the correlation coefficients are slightly lower when applying the new calibration method. This can be attributed to the fact, that the new neutral density estimates show a fairly higher noise and scatter behavior. As the differences are within reasonable limits we propose that it is feasible to apply the routine elaborated by \citet{Klinger2016} for the entire period of investigation.
Since~\citet{Krauss2015} has revealed that the correlations between the neutral density enhancement and the $B_{\rm z}$ component of the magnetic field are higher than those with the convective electric field [$E$] or the energy input via the Poynting flux into the magnetosphere [$S$]) we will solely focus on the former one for the further analysis of the in-situ measurements from the ACE spacecraft.
\section{Results}
\subsection{Influence of different solar activity levels on the evaluation results}
When incorporating the new data from solar cycle 24 we have to keep in mind that, the solar activity levels were clearly different in the time range for 2003--2010 (previous study) and 2011--2015 (new extended period). While 2003--2010 covers the maximum and decaying phase of cycle 23, 2011--2015 comprises the rising and maximum phase of the weak solar cycle 24~\citep{Gopalswamy2015}. Hence, ICME speeds and impacts are expected to be much lower for the extended study than for the results given in~\citet{Krauss2015}. For this reason, we initially carried out a comparison between results obtained for the two separate time periods. Figure~\ref{fig:5} shows scatter plots between the estimated neutral density increase based on GRACE data, the magnetic field component $B_{\rm z}$ measured by ACE as well as the Sym-H index as a representative for the geomagnetic activity level.
\begin{figure*}[htb]
\begin{center}
\includegraphics[width=14cm]{figure5.png}
\end{center}
\caption{Scatter plots comprising 196 analyzed ICME events. Red circles cover the period of the predecessor study from 2003 till 2010, blue squares the extended analysis period from 2011 to 2015. In the left panel the magnetic field component $B_{\rm z}$ versus GRACE neutral densities are shown; in the right panel $B_{\rm z}$ versus the geomagnetic index Sym-H. For a better visibility, extrapolated regression lines are illustrated in dashed lines, but should be interpreted with care.}
\label{fig:5}
\end{figure*}
For each of the three parameters the expected lower solar activity level during the time period 2011--2015 is apparent. The maximal negative $B_{\rm z}$ magnetic field measured for an ICME event during the extended period is $-31.6$\,nT for an event on October 02, 2013. Due to the absence of extreme ICME events we obtain slightly different results in the regression line trends for the different time ranges. Thus, it has to be taken care when extrapolating data from a limited subset (e.g. time range 2011--2015) of measurements, which might lead to erroneous assumptions (cf. Fig.~\ref{fig:5} dashed lines). However, if we constrain our analysis to smaller CME events (see Fig.~\ref{fig:6}b) we can state that the data of the extended period fits pretty well in the overall picture.\\
In Tab.~\ref{tab:correlation1} an overview of all determined correlation coefficients regarding the increase in the neutral density from GRACE as well as the magnetic field component $B_{\rm z}$ for the complete analysis period is given.
\begin{table}[ht]
\caption{Correlation coefficient between the various parameters and the increase in neutral density $\Delta \rho_{\rm490}$ (column 2) as well as the $B_{\rm z}$ component measured by ACE (column 3) - time period 2003--2015.}
\centering
\begin{tabular}{l c c}
\hline
{} & {\bf{$\Delta \rho_{\rm 490}$}} & {\bf{$B_{\rm z}$}} \\
\hline
$\Delta \rho_{\rm 490}$ & 1.00 & -0.89 \\
$B_{\rm z}$ & -0.89 & 1.00 \\
$v_{\rm max}$ & 0.65 & -0.60 \\
$E$ & -0.85 & 0.86 \\
$S$ & 0.78 & 0.77 \\
Dst & -0.89 & 0.85 \\
AE & 0.71 & -0.73 \\
a\textsubscript{a} & 0.89 & -0.87 \\
a\textsubscript{m} & 0.89 & -0.88 \\
a\textsubscript{p} & 0.84 & -0.83 \\
k\textsubscript{p} & 0.67 & -0.75 \\
PCN & 0.80 & -0.80 \\
PCS & 0.76 & -0.78 \\
SYM--H & -0.88 & 0.89 \\
\hline
\end{tabular}
\label{tab:correlation1}
\end{table}
The resulting correlations are slightly lower than those obtained for the period 2003--2010 (cf. Fig.~\ref{fig:4}), due to the already mentioned fact that the extended period missed extreme ICME events, which are important in a statistical manner. However, overall the results from both time periods fit together well. Based on that, we safely can extend our statistical analysis from the time period 2003--2010 (cf. Fig.~\ref{fig:4}) to the entire investigation period January 2003 to December 2015.\\
We note that Tab.~\ref{tab:correlation1} only refers to density increases computed for the GRACE satellites. This is because the analyzed CHAMP data lead to similar results due to a nearly 1:1 correlation between GRACE and CHAMP densities (cc=0.97).
\subsection{Investigating the impact of CIR induced geomagnetic storms on the Earth thermosphere}
In a second step of the study we examined, how CIR induced geomagnetic storms affect the thermospheric density increase. The illustration in Fig.~\ref{fig:6}a represents a scatter plot including all analyzed ICME and CIR events in terms of increasing neutral densities and the maximal negative $B_{\rm z}$ component measured by GRACE and ACE, respectively. CIR induced geomagnetic storms have weaker effects on the thermosphere density than those caused by ICME events. This can be attributed to their overall smaller $B_{\rm z}$ magnetic field as well as to the fact that the relevant energy input into the thermosphere comes from the compression region of the CIR.
\begin{figure*}[htb]
\begin{center}
\includegraphics[width=14cm]{figure6.png}
\end{center}
\caption{(left to right) scatter plots of the minimum $B_{\rm z}$ component measured by ACE with the increase in neutral density and a magnified version which includes the outcome of the ICME analysis including only ICME events which triggered a density enhancement $\Delta \rho < 1.6\times10^{-12}$~kg\,m$^{-3}$ (yellow line). For a better visibility, extrapolated regression lines are illustrated in dashed lines, but should be interpreted with care.}
\label{fig:6}
\end{figure*}
During the analysis period, CIR events cause variations in the magnetic field component $B_{\rm z}$ only up to about -22~{nT} and density increases at GRACE altitudes in the order of approximately $~1.4\times10^{-12}$~kg\,m$^{-3}$. This means that, the influence of CIR events is in the same range of magnitude as the majority of the ICMEs (186 out of 196) and thus can be seen as an extended distribution of the density enhancements related to weak ICMEs. This can be further illustrated by excluding the ten most extreme ICME events, with a density enhancement greater than $~1.6\times10^{-12}$~kg\,m$^{-3}$, in the analysis. This yields a significant better agreement between the ICME and CIR analysis and indicates that the effect of negative $B_{\rm z}$ is not a linear one. The findings are given in Fig.~\ref{fig:6}b, which reveals a magnified view of Fig.~\ref{fig:6}a, and additionally includes the derived regression line from the reduced ICME analysis.\\
CIR events are generally characterized through a turbulent and strong magnetic field due to the compression between the slow and fast solar wind regime. Against this background, the effect on the Earth's thermosphere should agree better with those from ICMEs where the negative $B_{\rm z}$ component is located in the shock-sheath region~(cf.~\citet{Krauss2015}).
\begin{figure*}[htb]
\begin{center}
\includegraphics[width=8cm]{figure7.png}
\end{center}
\caption{Scatter plot of the minimum $B_{\rm z}$ component measured by ACE with the geomagnetic index Sym-H. Illustrated are events over the entire investigation period 2003--2015: 195 CIR (green squares) and 196 ICME (red/blue circles).}
\label{fig:7}
\end{figure*}
Figure~\ref{fig:7} shows the relationship between the $B_{\rm z}$ measured by ACE and the geomagnetic activity level on the example of the Sym-H index for all analyzed CIR and ICME events (separated in shock-sheath and magnetic structure). It can be clearly seen that, the impact of CIR induced geomagnetic storms on the Sym-H index is very similar to the effects from shock-sheath regions of an ICME that carry a lower $B_{\rm z}$ than the magnetic structure.
\subsection{Height depended impact of geomagnetic storms induced by ICME and CIR events}
The relative density increase, due to ICME and CIR induced geomagnetic storms, is found to be very similar for GRACE or CHAMP. Therefore, all results so far referred solely to the neutral density estimate for the GRACE spacecraft (see Tab.~\ref{tab:correlation1}). However, due to their different orbit heights (cf. Fig.\ref{fig:3}), the satellites are exposed to different absolute atmospheric densities. Examining the subplots Fig.~\ref{fig:8}~[c, d], we see that the estimated densities differ by an order of magnitude. Hence, density increases due to an extreme solar event will have different effects on orbit decays for GRACE or CHAMP.
\begin{figure*}[ht]
\begin{center}
\includegraphics[width=14cm]{figure8.png}
\end{center}
\caption{Overview of various analyzed parameters during an ICME event in August 25, 2005. Including the ACE measurements of the $B_{\rm z}$ component (a), the Dst-index (b), the estimated neutral densities for CHAMP and GRACE (c,d) and finally the determined orbit decay for both satellite missions (e,f).}
\label{fig:8}
\end{figure*}
Following ~\citet{Chen2012} the orbit decay can be calculated in terms of changes in the mean semi--major axis. Figures~\ref{fig:8} summarize various analyzed parameters during an ICME event in August 2005. It can be seen that, due to the large differences of both satellite missions regarding their absolute neutral atmospheric densities, tantamount to their orbital altitude, the orbit decay for GRACE and CHAMP is significantly different, even though it is triggered from the same solar event.
\begin{figure*}[ht]
\begin{center}
\includegraphics[width=14cm]{figure9.png}
\end{center}
\caption{Overview of the same parameters as shown in Fig.~\ref{fig:8} but for a CIR event in July 03, 2006.}
\label{fig:9}
\end{figure*}
Regarding the CHAMP orbit decay rates our results confirm the findings from~\citet{Chen2014}. The authors compared the effects of CIR- and CME-induced geomagnetic storms using data from this specific satellite mission and found that both types of solar events might trigger comparable orbit decay rates. Figure~\ref{fig:9} illustrates various analyzed parameters during such an CIR event in July 2006.\\
However, as noted before, the decay rate depends significantly on the actual satellite altitude. For the GRACE orbit, which was located about 150\,km higher than CHAMP, the expected decay is much lower due to the less denser atmosphere. When analyzing the effects of solar events on the orbital altitude of satellites over a 13-years investigation period different changing conditions must be kept in mind: Firstly, due to the varying solar conditions (cf. solar cycle 23, 24), solar events have different impact intensities. Secondly, the actual altitude (cf.~\ref{fig:3}) of a satellite represents a significant factor in the expected orbital decay. Figure~\ref{fig:10} summarizes the impact of nearly 200 ICME events for the CHAMP (upper panel) and the GRACE satellite mission (lower panel) exemplified by the estimated orbital decay, the prevailing satellite altitude and the intensity of the specific solar event in terms of the in-situ measurements of the magnetic field component $B_{\rm z}$ - the larger the bubble the stronger the negative IMF $B_{\rm z}$ event.
\begin{figure*}[ht]
\begin{center}
\includegraphics[width=10cm]{figure10.png}
\end{center}
\caption{Orbit decays versus orbital altitude and event strength in terms of $B_{\rm z}$ [nT] measurements for the CHAMP (top) and GRACE spacecraft (bottom). To obtain a better visibility, two average events which triggered large orbit decays due to an extreme low satellite orbital altitude are neglected for the present illustration.}
\label{fig:10}
\end{figure*}
Therein, Fig.~\ref{fig:10} reveals that, a large number of ICME events which occurred during the 13-years investigation period had minor effects on the GRACE satellite orbit (around 450--490\,km) but already perceptible implications for the CHAMP mission. Extreme ICME-induced geomagnetic storms may lead to decay rates of 40--50\,m and 90--120\,m for GRACE and CHAMP, respectively. Furthermore, it demonstrates, that during periods of lower orbital altitudes even smaller solar events are sufficient to trigger GRACE orbit decay rates in the order of 20--50\,m. For CHAMP such effects are nearly not existent due to low solar activity at the end of the satellite mission (decommission in 2010). Concerning the time delay between the ICME/CIR arrival and the orbit decay onset an average value of $5 \rm{hrs}\pm 2 \rm{hrs}$ was estimated.
\clearpage
\section{Discussion}
This paper deals with the analysis of the impact of 196 Earth-directed ICME and 195 CIR events on the Earth upper atmosphere and the direct implications for satellite orbits. The study comprises solar events which occurred in the time period between 2003 and 2015 using in-situ measurements recorded by the ACE spacecraft in the Lagrange point L1 and the two low-Earth orbiting missions CHAMP and GRACE. As a result, of the additional analyzed CIR events, the extended investigation period (beyond 2010) and the added satellite mission CHAMP, the current study represents an extended analysis of the predecessor study~\citet{Krauss2015}. For the time-based delimitation of the various solar events we rely on the ICME catalog maintained by R\&C and CIR-lists provided by~\citet{Jian2011} and S. Vennerstrom. To process accelerometer measurements beyond 2010 we applied a new calibration routine, whose results were compared with our previous results and calibration method. The analysis has revealed, that the new developed routine delivers reasonable results and is suitable to analyze solar events after the shutdown of the thermal control system in April 2011. Based on these findings we thoroughly examined the effects of solar events between 2003 and 2015 with regard to the occurred thermospheric and geomagnetic response. The derived linear regression parameters confirm our previous findings, which showed a high correlation between the majority geomagnetic indices and ICME parameters with the variations in the atmospheric neutral density. Furthermore, the increase in the neutral densities during an event yield a nearly 1:1 correlation (cc=0.97) between estimates for CHAMP and GRACE satellites.\\
A subsequent investigation on CIR events showed that the impacts on the near-Earth environment are quite similar to those from Earth-directed ICME events - about 90\% of ICMEs are in the same order of magnitude. A direct comparison of ICME and CIR induced geomagnetic storms indicated that both show the same behavior when the main disturbances (minimum in $B_{\rm z}$) are triggered at the time of the shock-sheath region and not in the magnetic structure part.\\
Finally, as proposed in the discussion section of~\citet{Krauss2015}, the current study now addressed an important field of application of space weather, namely the induced satellite orbit decay.
For this purpose, the effects of all ICME and CIR between 2003 and 2015 on the CHAMP and GRACE satellite altitude were estimated. In doing so, we can confirm findings by~\citet{Chen2014} who demonstrated that CIR events have similar effects on the total orbital decay than ICME induced geomagnetic storms. However, when comparing the two satellite missions CHAMP and GRACE it became obvious that the actual orbital height is a crucial factor. One and the same solar events triggers substantial different decay rates (up to 70\,m) at different altitudes.
\acknowledgments
We acknowledge the use of the satellite data from ACE, CHAMP and GRACE as well as the world data center (WDC) for Geomagnetism, Kyoto. ACE data used in this study were obtained through the OMNI database (omniweb.gsfc.nasa.gov). The access to the accelerometer measurements from the CHAMP and GRACE satellites was provided by the Information System and Data Center (ISDC) in Potsdam (isdc.gfz-potsdam.de). The authors additionally acknowledge the use of the ICME list of Richardson and Cane as well as the CIR lists of L.K. Jian and S. Vennerstrom.
Furthermore, we would like to thank B. Klinger (University of Technology Graz) for her support concerning the GRACE accelerometer calibration. Martina Edl contributed to the CIR related data preparation. S.K.\,acknowledges the support by the FFG/ASAP Program under grant no. 847971 (SPICE) and M.T. the FFG/ASAP Program under grant no. 859729 (SWAMI).
|
1,108,101,563,265 | arxiv | \section{Introduction}
Recent years have seen increasing interest in various spin models that are defined on regular lattices and nevertheless exhibit ``fractal'' properties.
These include gapped spin liquid models in which immobile topological excitations are created at the corners of operators with fractal support, or
spin models with symmetries that act on a fractal subsystem.
One example of the former is Haah's code~\cite{Haah}, a canonical model of type-II~\cite{vijay2016fracton} fracton topological order~\cite{nandkishorereview,pretkoreview,chamon,bravyi2011topological,haah2011local,kim20123d,yoshida2013exotic,vijayhaahfu2015,vijay2016fracton,williamsongauge,Dua2019a}.
Such 3D phases are characterized by topological quasiparticle excitations that are strictly immobile.
As quantum codes, they lack string-like logical operators, and instead have logical operators supported on a fractal subset of sites.
The fractal nature of these codes leads to promise as quantum memories~\cite{PhysRevLett.107.150504,bravyi2013quantum,brown2014quantum}.
More generally, fracton phases have received tremendous attention in a wide variety of contexts~\cite{PhysRevB.95.155133,shriyaprx,gromovprr, PhysRevLett.123.136401,PhysRevB.101.165145,PhysRevB.99.155126, PhysRevLett.120.195301,gromov2017fractional,PhysRevB.100.134113,Gromov2019,PhysRevLett.119.257202,prem2018pinch,Doshi2020,PhysRevD.96.024051,Brown2019,vijay2017generalization,Williamson2020,Bulmash2019,Prem2019,Stephen2020,Pai2019,haah2014bifurcation,shirley2017fracton,Dua2019b,Slagle2018,Slagle2018b,Wen2020,Aasen2020,wang2020nonliquid,bulmash2018generalized,Gromov2018,PretkoGauge2018,Williamson2018,Seiberg_2020,Tantivasadakarn_2020,Shirley2020,Tantivasadakarn_JW,Devakul2020,PhysRevB.97.041110}.
An example of the latter type of fractal models are the fractal Ising models~\cite{fractalSSPT}.
These are classical spin models on the square lattice with symmetries that flip the Ising spins on a fractal subset of sites.
These have been studied as classical codes~\cite{CAcode,yoshidaclassical,Nixon2020} for their information storage capacity and also as translation invariant models of glassiness~\cite{newmanmoore}.
In these codes, classical information is stored in the spontaneous symmetry-broken ground states of the model.
With the same set of fractal symmetries, there may also be non-trivial phases without spontaneous symmetry breaking, known as fractal symmetry-protected topological (SPT) phases~\cite{fractalSSPT,kubica2018ungauging,Devakul2018}.
An example is the cluster model~\cite{cluster} on the honeycomb lattice, which realizes a non-trivial SPT phase protected by fractal subsystem symmetries~\cite{fractalSSPT}.
In 2D, such phases have received attention as they have been proven to be useful resources for universal measurement-based quantum computation~\cite{Raussendorf_2019,devakul2018universal,stephen2019subsystem,Daniel2019}.
In this paper, we unify all these fractal spin models via a process called ``fractalization''.
Fractalization maps certain operators defined on a $D$ dimensional lattice to one on a $D+m$ dimensional lattice, taking as input a set of $m$-dimensional linear cellular automaton (LCA) rules.
In particular, the various fractal models above may all be understood as fractalized versions of simple lower-dimensional models.
A class of type-II fracton topological phases~\cite{yoshida2013exotic} in 3D, including Haah's cubic code, may be understood as fractalized 2D toric codes~\cite{kitaev}; the fractal Ising models in 2D are simply fractalized 1D Ising models; and the 2D cluster models realizing fractal SPT phases are fractalized 1D cluster models.
We also present an interpretation of the fractalization map as a three-step process which directly relates many properties of the fractalized model to that of the original.
For instance, we show that the ground state manifold of a fractalized commuting Hamiltonian is (non-local) unitarily related to that of stacks of the original model.
We discuss the nature of the excitations in fractalized models in relation to those of the original model, taking as examples the toric code in various dimensions.
Point-like excitations of the original model lead to immobile point-like fracton excitations in the fractalized model, whose immobility is guaranteed by the lack of string-like logical operators.
When the original model has loop-like excitations, on the other hand,
we determine a criterion under which the fractalized model lacks any loop or string-like excitations whatsoever.
The excitations of such models are instead ``fractalized loop'' excitations: extended deformable fractal excitations with energy cost scaling faster than linearly with the linear size.
When applied to quantum codes, fractalization results in a higher dimensional code with information storage capabilities similar to that of decoupled stacks of the original, but with improved code distance.
In particular, for a locally generated $[n,k,d]$ code in $D$ dimensions, where $n$ is the number of physical qubits, $k$ is the number of logical qubits, and $d$ is the code distance, the $D\rightarrow D+m$ fractalized code is a locally generated $[L^m n, L^m k, d^\prime]$ code in $D+m$ dimensions, where $d^\prime\geq d$ depending on the LCA rules and $L$ is the linear size of the system.
Fractalization therefore has a useful application in deriving new fractal codes from known lower-dimensional codes.
By fractalizing the 4D toric code, for instance, we construct a 5D model which lacks any membrane logical operators, a direct generalization of Type-II fracton phases in 3D whose hallmark is the absence of string logical operators~\cite{vijay2016fracton}.
We also discuss in detail fractalization applied to the 2D Bacon-Shor subsystem code~\cite{bacon,poulin}, which results in the 3D fractal Bacon-Shor code.
By incorporating ideas from Bravyi~\cite{bravyi} and Yoshida~\cite{yoshidaclassical}, we are able to improve the information storage capabilities of the 3D fractal Bacon-Shor code to $[n,k,d]\sim [L^3,L^2,L^{\eta}]$, where we conjecture $\eta\rightarrow 2$ in a limit of large physical qudit dimension.
To the best of our knowledge, if our conjecture proves true this would be the first code to saturate the information storage tradeoff bound for locally generated subsystem codes in 3D~\cite{bravyi,flammia}, $k\sqrt{d}\leq \mathcal{O}(n)$.
In Sec~\ref{sec:prelim}, we review how LCA rules generically lead to fractal structures along with other necessary background.
In Sec~\ref{sec:fractalization}, we define and discuss the fractalization procedure and its interpretation in terms of a three-step process.
In Sec~\ref{sec:examples}, we go through each of the mentioned examples in detail, and discuss connections to other recent works.
In Sec~\ref{sec:fractalizingloops}, we discuss fractalized loop excitations in detail through examples, including higher dimensional toric codes.
In Sec~\ref{sec:fsc}, we present the fractalized Bacon-Shor code and discuss its memory storage capabilities.
Finally, concluding remarks and future directions are laid out in Sec~\ref{sec:conclusion}.
\section{Preliminaries}\label{sec:prelim}
\subsection{Fractals and linear cellular automata}\label{sec:review}
We begin by reviewing linear cellular automata (LCA) and their connection to fractals.
Let $c_{\mathbf{r},t}\in\{0,1\}$ represent the state of cell $\mathbf{r}=(r_1,\dots, r_D)\in \mathbb{Z}^D$ of a cellular automaton at time $t$.
The LCA update rule determines how the state is determined as a linear function (modulo 2) of the state at previous time.
Throughout this paper, we are mostly concerned with first-order, local, and translation invariant LCA update rules.
That is,
\begin{equation}
c_{\mathbf{r},t} = \sum_{\mathbf{r}^\prime} F_{\mathbf{r},\mathbf{r}^\prime} c_{\mathbf{r}^\prime,t-1} \equiv \mathbf{F}\mathbf{c}_{t-1}
\label{eq:updatevec}
\end{equation}
where the binary matrix $\mathbf{F}$ specifies the LCA rules, and all arithmetic is implied modulo 2.
First-order means that $\mathbf{c}_{t}$ only depends on the states $\mathbf{c}_{t-1}$ at time $t-1$, locality implies that $\mathbf{F}$ is only non-zero for small $|\mathbf{r}-\mathbf{r}^\prime|$, and translation invariance means that $F_{\mathbf{r},\mathbf{r}^\prime}=f_{\mathbf{r}-\mathbf{r}^\prime}$ only depends on the difference $\mathbf{r}-\mathbf{r}^\prime$.
Starting at time $t=0$ with the state $\mathbf{c}_{0}$, the space-time trajectory at all future times is completely determined by
\begin{equation}
\mathbf{c}_{t} = \mathbf{F}^t \mathbf{c}_{0}
\label{eq:updatevect0}
\end{equation}
The set of cells with $c_{\mathbf{r},t}=1$ will generically form a fractal subset of the $(D+1)$ dimensional space-time lattice.
This is most naturally seen by adopting a polynomial representation for LCAs.
In the polynomial representation, the state $\mathbf{c}_t$ is represented by a $D$-variate polynomial with $\mathbb{F}_2$ coefficients,
\begin{equation}
c_t(\mathrm{x}) = \sum_\mathbf{r} c_{\mathbf{r},t} \mathrm{x}^{\mathbf{r}}\in\mathbb{F}_2[\mathrm{x}],
\end{equation}
where $\mathrm{x}=\{x_i\}_{i\in[1\dots D]}$, and $\mathrm{x}^{\mathbf{r}} \equiv \prod_i x_i^{r_i}$ is a monomial.
Similarly, the update rule is represented by
\begin{equation}
f(\mathrm{x}) = \sum_{\mathbf{r}} f_\mathbf{r} \mathrm{x}^\mathbf{r}
\end{equation}
in terms of which Eqs.~\ref{eq:updatevec} and \ref{eq:updatevect0} take the particularly simple form
\begin{equation}
c_t(\mathrm{x}) = f(\mathrm{x}) c_{t-1}(\mathrm{x}) = f(\mathrm{x})^t c_0(\mathrm{x}).
\end{equation}
We will utilize both the vectoral representation and the polynomial representation, depending on whichever one is most convenient.
We always assume $f(\mathrm{x})$ is chosen to have a non-zero constant term and only positive powers of $x_i$, and that $f(\mathrm{x})\neq 1$ is non-trivial.
To see that this generates fractal structures, notice that for any $t=2^n$, we have, due to the properties of $\mathbb{F}_2$ polynomials, $f(\mathrm{x})^{2^n} = f(\mathrm{x}^{2^n})=f(\{x_i^{2^n}\})$.
Assuming the initial state $c_0(\mathrm{x})$ only contained finitely high powers of any $x_i$, then at exponentially long times $c_t(\mathrm{x})=f(\mathrm{x}^{2^n})c_0(\mathrm{x})$ describes copies of the initial state each shifted by $2^n$ for each term in $f(\mathrm{x})$.
We most often consider LCA defined on finite $(L_1,\dots,L_d)$ systems with periodic boundary conditions, which is enforced by the identification $x_i^{L_i}=1$.
An important property is the reversibility of an LCA rule for a given system size, which corresponds to the existence of $f(\mathrm{x})^{-1}$ satisfying $f(\mathrm{x})^{-1} f(\mathrm{x}) = 1$, after taking into account periodic boundary conditions.
A simple family of reversible LCA is given by any polynomial $f(\mathrm{x})$ satisfying $f(1)=1$ on a system of size $L_i=2^n$.
This follows from the fact that $f(\mathrm{x})^{2^n} = f(\mathrm{x}^{2^n}) = f(1)=1$, and so the inverse is given by $f(\mathrm{x})^{-1}\equiv f(\mathrm{x})^{2^n-1}$.
As an example, consider $d=1$ spatial dimensions, $f(x)=1+x$ (corresponding to $F_{\mathbf{r},\mathbf{r}^\prime}=\delta_{\mathbf{r},\mathbf{r}^\prime} + \delta_{\mathbf{r},\mathbf{r}^\prime+1}$) and the initial state $c_0(x)=1$ (corresponding to $c_{\mathbf{r},0} = \delta_{\mathbf{r}, 0}$).
Then $c_t(x)=f(x)^t$, and listing out terms for the powers of $f(x)$ we find
\begin{equation}
\begin{matrix}
f(x)^0 & = & 1 & & & & &\\
f(x)^1 & = & 1 &x& & & &\\
f(x)^2 & = & 1 & &x^2& & &\\
f(x)^3 & = & 1 &x&x^2&x^3 &&\\
f(x)^4 & = & 1 & & & &x^4&\\
f(x)^5 & = & 1 &x & & &x^4&x^5\\
f(x)^6 & = & 1 & &x^2 & &x^4&&x^6\\
f(x)^7 & = & 1 &x &x^2&x^3&x^4&x^5&x^6&x^7\\
\end{matrix}
\label{eq:sierpinski}
\end{equation}
and so on.
Zooming out, the fractal generated is the Sierpinski triangle (Pascal's triangle mod 2).
We remark that $f(x)=1+x$ is irreversible on any system size.
A simple example of a reversible LCA is ${f(x)=1+x+x^2}$ on sizes $L=2^n$.
\subsection{Pauli operators}
In this paper, we present a procedure by which a set of LCA rules may be used to extend a quantum CSS code into a higher dimensional fractal code.
To this end, it is useful to use a vectoral (and polynomial) representation of Pauli operators~\cite{nla.cat-vn2487773,haah2013commuting,yoshida2013exotic}.
Let us consider qubits on the sites $\mathbf{r}$ of a hypercubic lattice.
Acting on these qubits, we have Pauli operators $X_{\mathbf{r}}$ and $Z_{\mathbf{r}}$.
Define a map $\sigma_X$ which maps a binary vector $a_{\mathbf{r}}$ to a tensor product of $X$ Pauli operator as
\begin{equation}
\sigma_X[\mathbf{a}] = \prod_\mathbf{r} X_\mathbf{r}^{a_\mathbf{r}},
\end{equation}
and similarly $\sigma_Z$,
\begin{equation}
\sigma_Z[\mathbf{b}] = \prod_\mathbf{r} Z_\mathbf{r}^{b_\mathbf{r}}.
\end{equation}
The commutation relation between two Pauli operators is straightforward to compute,
\begin{equation}
[[\sigma_X[\mathbf{a}] ,\sigma_Z[\mathbf{b}] ]]
=
(-1)^{\mathbf{b}^T \mathbf{a}}
\end{equation}
where $[[A,B]] = A^{-1} B^{-1} A B$ is the group commutator.
Using the correspondence between vectors and polynomials, $\mathbf{a} \leftrightarrow \sum_\mathbf{r} a_\mathbf{r} \mathrm{x}^\mathbf{r}$, $\sigma_{X/Z}$ may also be interpreted as a map from $\mathbb{F}_2$ polynomials to Pauli operators.
Translations are nicely expressed in this language as multiplication by a monomial:
$\sigma_X[a(\mathrm{x})]$ translated by $\mathbf{r}$ is $\sigma_X[\mathrm{x}^\mathbf{r} a(\mathrm{x})]$.
Note that we use the same symbol $(\sigma_{X/Z})$ in both the vectoral and polynomial representation.
The commutation relation for translations of two operators can be neatly summed up by a single commutation polynomial $c(\mathrm{x})$,
\begin{equation}
[[\sigma_X[\mathrm{x}^\mathbf{r} a(\mathrm{x})], \sigma_Z[\mathrm{x}^{\mathbf{r}^\prime}b(\mathrm{x})] ]]
=
(-1)^{c_{\mathbf{r}^\prime-\mathbf{r}}}
\end{equation}
where
\begin{equation}
c(\mathrm{x}) = \sum_{\mathbf{r}} \mathrm{x}^\mathbf{r} c_\mathbf{r} = a(\mathrm{x}) b(\bar{\mathrm{x}})
\end{equation}
and $\bar{\mathrm{x}}\equiv \mathrm{x}^{-1}$.
In particular, if $c(\mathrm{x})=0$, then all translations commute.
Throughout this paper, we typically consider more than one physical qubit per lattice site.
The generalization to $N$ qubits per site, with operators $X_\mathbf{r}^{(n)}$ and $Z_{\mathbf{r}}^{(n)}$ for $n=1\dots N$, is straightforward:
in the vectoral representation, $\sigma_{X/Z}$ now takes as input $N$ vectors (or a tensor), $a_{\mathbf{r},n}$, one for each of the qubits.
Similarly, in the polynomial representation, $\sigma_{X/Z}$ takes as input a vector of $N$ polynomials, $\mathbf{a}(\mathrm{x})=a_n(\mathrm{x})$.
The commutation polynomial is then given by $c(\mathrm{x}) = \mathbf{a}^T(\mathrm{x}) \mathbf{b}(\bar{\mathrm{x}})$.
\section{Fractalization}
\label{sec:fractalization}
\subsection{Definition}\label{sec:fracdef}
We are now ready to discuss fractalization.
In the most general case, fractalization maps a $D$-dimensional $X$ or $Z$ Pauli operator to one in $D+m$ dimensions.
We consider a hypercubic lattice with $N$ qubits per site.
As input, we take a set of $D$, $m$-dimensional LCA rules, ${\mathrm{f}(\mathrm{y})\equiv\{f_i(\mathrm{y})\}_{i\in[1\dots d]}}$, where $\mathrm{y}=\{y_j\}_{j\in[1\dots m]}$.
Let $A$ be a local $X$ Pauli operator in $D$ dimensions, in the polynomial representation,
\begin{equation}
A = \sigma_X\left[\mathrm{x}^{\mathbf{r}_0}\mathbf{a}(\mathrm{x})\right],
\end{equation}
where $\mathrm{x}=\{x_1,\dots,x_d\}$ and we have chosen an ``anchor point'' $\mathbf{r}_0$.
We uniquely specify the anchor point for this operator by maximizing each $(\mathbf{r}_0)_i$ subject to the requirement that $\mathbf{a}(\mathrm{x})$ has only positive powers of $x_i$.
Locality means that $\mathbf{a}(\mathrm{x})$ contains only finitely high powers of $x_i$.
Going to $D+m$ dimension, let $\mathbf{s}$ denote the position vector along the final $m$ dimensions, such that the full position of a site is $(\mathbf{r},\mathbf{s})$.
Similarly, in the polynomial notation, let $\mathrm{y}$ denote the variables corresponding to the final $m$ dimensions.
For each $\mathbf{s}_0$, the fractalized operator $A^{\mathrm{frac}}_{\mathbf{s}_0}$ is then defined as
\begin{equation}
A^{\mathrm{frac}}_{\mathbf{s}_0} = \sigma_X \left[\mathrm{x}^{\mathbf{r}_0} \mathrm{y}^{\mathbf{s}_0} \mathbf{a}(\mathrm{f}(\mathrm{y})\circ \mathrm{x})\right]\label{eq:afracpoly}
\end{equation}
where
\begin{equation}
\label{eq:fdotx}
\mathrm{f}(\mathrm{y})\circ \mathrm{x}\equiv \left\{f_i(\mathrm{y}) x_i\right\}_{i\in[1\dots D]}.
\end{equation}
Thus, fractalization maps a $D$-dimensional operator to a set of $D+m$-dimensional operators, $A\rightarrow \{A^{\mathrm{frac}}_{\mathbf{s}_0}\}$, one for each shift $\mathbf{s}_0$ in the new dimensions.
Most of the examples in this paper will involve only $m=1$ new dimension.
Similarly, suppose we start out with the $Z$ Pauli operator,
\begin{equation}
B = \sigma_Z\left[\mathrm{x}^{\mathbf{r}_0}\mathbf{b}(\bar{\mathrm{x}})\right],
\end{equation}
where the anchor point $\mathbf{r}_0$ is chosen similarly as before but spatially inverted: each $(\mathbf{r}_0)_i$ is minimized subject to the constraint that $\mathbf{b}(\bar{\mathrm{x}})$ contains only positive powers of $\bar{\mathrm{x}}$.
Then, the fractalized operator is defined as
\begin{equation}
B^{\mathrm{frac}}_{\mathbf{s}_0} = \sigma_Z \left[\mathrm{x}^{\mathbf{r}_0} \mathrm{y}^{\mathbf{s}_0} \mathbf{b}(\mathrm{f}(\bar{\mathrm{y}})\circ \bar{\mathrm{x}})\right]
\label{eq:bfracpoly}
\end{equation}
For completeness, let us also express fractalization using the vectoral representation.
The input LCAs are given by matrices $\{\mathbf{F}_i=F_{\mathbf{s},\mathbf{s}^\prime}^{(i)}\}_{i\in[1\dots D]}$.
Starting with an $X$ operator
\begin{equation}
A = \sigma_X\left[a_{\mathbf{r},n}\right],
\end{equation}
we find the anchor point $\mathbf{r}_0$ by maximizing each $(\mathbf{r}_0)_i$ subject to the constraint that $a_{\mathbf{r},n}=0$ for any $(\mathbf{r} - \mathbf{r}_0)_i<0$.
Then, the $D+m$ dimensional fractalized operator is
\begin{equation}
\begin{split}
A^{\mathrm{frac}}_{\mathbf{s}_0} &=
\sigma_X\left[a^{\mathrm{frac}}_{(\mathbf{r},\mathbf{s}),n}\right]\\
&=
\sigma_X\left[\left(\prod_{i}\mathbf{F}^{(\mathbf{r}-\mathbf{r}_0)_i}_i\right)_{\mathbf{s},\mathbf{s}_0}a_{\mathbf{r},n}\right].
\end{split}\label{eq:Afrac}
\end{equation}
where the ordering of each $\mathbf{F}_i$ in the product is arbitrary, as all $\mathbf{F}_i$ commute due to translation invariance (this is obvious from the polynomial representation, $f_i(x)f_j(x)=f_j(x)f_i(x)$).
Note that since $a_{\mathbf{r},n}=0$ if $(\mathbf{r}-\mathbf{r}_0)_i<0$, $\mathbf{F}_i$ is always raised to a non-negative power.
Similarly, for a $Z$ operator,
\begin{equation}
B = \sigma_Z\left[b_{\mathbf{r},n}\right],
\end{equation}
we must choose the anchor $\mathbf{r}_0$ such that each $(\mathbf{r}_0)_i$ is minimized subject to the constraint that $b_{\mathbf{r},n}=0$ for any $(\mathbf{r}_0-\mathbf{r})_i<0$.
Then,
\begin{equation}
B^{\mathrm{frac}}_{\mathbf{s}_0} =
\sigma_Z\left[\left(\prod_{i}\mathbf{F}^{(\mathbf{r}_0-\mathbf{r})_i}_i\right)_{\mathbf{s}_0,\mathbf{s}}
b_{\mathbf{r},n}\right].
\end{equation}
Given a CSS stabilizer group $\mathcal{S}=\langle \{\mathcal{O}_l\}\rangle$, or Hamiltonian
\begin{equation}
H = -\sum_l \mathcal{O}_l,
\end{equation}
where $\mathcal{O}_l$ are local $X$ or $Z$ generators of the stabilizer group,
we may define the fractalized stabilizer group $\mathcal{S}^\mathrm{frac}=\langle \{\mathcal{O}^\mathrm{frac}_{\mathbf{s},l}\}\rangle$, or Hamiltonian
\begin{equation}
H^{\mathrm{frac}} = -\sum_{\mathbf{s},l} \mathcal{O}^{\mathrm{frac}}_{\mathbf{s},l}
\end{equation}
which, as we will show, inherits many of the properties of $H$.
Note that $H^{\mathrm{frac}}$ is always translation invariant along the final $m$ dimensions by construction.
Finally, fractalization may also be applied to non-local operators, provided certain conditions are satisfied.
If the original system has open boundary conditions, fractalization can be applied straightforwardly.
In the case of periodic boundary conditions, however, it may not be possible to choose an anchor point consistently for a non-local operator.
Suppose an operator is non-local in the $i$th direction.
If $f_i(y)^{L_i}=1$, then the fractal is commensurate with the system size and the anchor $r_{0,i}$ can be chosen arbitrarily.
If this is not the case, then we must find a set of polynomials $Q_i=\{ q(\mathrm{y}) : q(\mathrm{y})f_i(\mathrm{y})^{L_i} = q(\mathrm{y})\}$, and make a choice of basis polynomials $\{q_j(\mathrm{y})\}_j$ for $Q_i$.
Then, an operator $A$ or $B$ maps on to a set of fractalized operators, one for each $q_j(\mathrm{y})$.
Eq~\ref{eq:afracpoly} and \ref{eq:bfracpoly} generalize to
\begin{equation}
\begin{split}
A^{\mathrm{frac}}_j &= \sigma_X\left[\mathrm{x}^{\mathbf{r}_0} q_j(\mathrm{y}) \mathbf{a}(\mathrm{f}(\mathrm{y})\circ\mathrm{x})\right]\\
B^{\mathrm{frac}}_j &= \sigma_Z \left[\mathrm{x}^{\mathbf{r}_0} q_j(\bar{\mathrm{y}}) \mathbf{b}(\mathrm{f}(\bar{\mathrm{y}})\circ \bar{\mathrm{x}})\right].
\end{split}
\end{equation}
where the choice of $r_{0,i}$ can be arbitrary.
Thus, each operator only maps on to $\log_2|Q_i|$ fractalized operators.
If an operator is non-local in multiple directions $i\in I$, then $q_j(\mathrm{y})$ must be chosen from $\cap_{i\in I} Q_i$ such that the fractal is commensurate with all directions.
\subsection{Properties}
\subsubsection{Locality}
We first point out that fractalization is locality preserving.
As long as $A$ has support only on a local patch of the $d$-dimensional lattice, $A^{\mathrm{frac}}_{\mathbf{s}_0}$ will also have support on a local patch of the $D+m$-dimensional lattice near $\mathbf{s}_0$.
This follows from the locality of the LCA rules: $F^{(i)}_{\mathbf{s},\mathbf{s}_0}$ is only non-zero for small $|\mathbf{s}-\mathbf{s}_0|$.
Non-local operators map on to non-local, potentially fractal, operators.
To see what this means, consider the $D=1$ chain with length $L$ and open boundary conditions, and the $1$-dimensional LCA rule $f(y)=1+y$.
The non-local operator
\begin{equation}
S = \prod_{r} X_r =\sigma_X\left[\sum_{r} x^r\right]
\end{equation}
maps on to the fractal operators
\begin{equation}
S^{\mathrm{frac}}_{s} = \sigma_X\left[y^s \sum_{r} f(y)^r x^r\right].
\end{equation}
in two dimensions.
Recall that $f(y)^n$ generates the $n$th row of the Sierpinski triangle fractal (Eq~\ref{eq:sierpinski}).
Hence, $S^{\mathrm{frac}}_s$ is an operator that acts as $X$ on sites along a Sierpinski triangular fractal subsystem, and $s$ just denotes an overall shift in the $y$ direction.
This operator has support on a fraction of sites scaling as $L^{d_f}$, where $d_f\approx 1.58$ is the Hausdorff dimension of the Sierpinski triangle.
\subsubsection{Commutativity}
The next property of fractalization is commutativity preservation.
Suppose we have two operators ${A=\sigma_X[\mathrm{x}^{\mathbf{r}_0}\mathbf{a}(\mathrm{x})]}$ and ${B=\sigma_Z[\mathrm{x}^{\mathbf{r}_1}\mathbf{b}(\bar{\mathrm{x}})]}$ satisfying $[A,B]=0$.
Then, $[A^{\mathrm{frac}}_{\mathbf{s}_0},B^\mathrm{frac}_{\mathbf{s}_1}]=0$ for all $\mathbf{s}_0,\mathbf{s}_1$ as well.
This can be seen by computing the commutation polynomial.
The commutation polynomial for $A$ and $B$ is
\begin{equation}
c(\mathrm{x}) = \mathrm{x}^{\mathbf{r}_0-\mathbf{r}_1}\mathbf{a}^T(\mathrm{x}) \mathbf{b}(\mathrm{x}),
\end{equation}
which has a zero constant term iff $[A,B]=0$.
The commutation polynomial for $A^{\mathrm{frac}}_{\mathbf{s}_0}$ and $B^{\mathrm{frac}}_{\mathbf{s}_1}$ is
\begin{equation}
\begin{split}
c^{\mathrm{frac}}(\mathrm{x},\mathrm{y}) &= \mathrm{x}^{\mathbf{r}_0-\mathbf{r}_1}\mathrm{y}^{\mathbf{s}_0-\mathbf{s}_1}\mathbf{a}^T(\mathrm{f}(\mathrm{y})\circ\mathrm{x})\mathbf{b}(\mathrm{f}(\mathrm{y})\circ \mathrm{x})\\
&=\mathrm{y}^{\mathbf{s}_0-\mathbf{s}_1} c(\mathrm{f}(\mathrm{y})\circ \mathrm{x})
\end{split}
\end{equation}
Therefore, if $c(x)$ has a zero constant term, $c^\mathrm{frac}(\mathrm{x},\mathrm{y})$ also has a zero constant term regardless of $\mathbf{s}_0,\mathbf{s}_1$.
It is also instructive to work in the vectoral representation.
Let $AB = (-1)^c B A$,
then, commutativity of $A=\sigma_X[(a_{\mathbf{r},n})]$ and $B=\sigma_Z[(b_{\mathbf{r},n})]$ implies that
\begin{equation}
c = \sum_{\mathbf{r},n} b_{\mathbf{r},n}a_{\mathbf{r},n}
\end{equation}
is zero.
The fractalized operators satisfy $[[A^\mathrm{frac}_{\mathbf{s}_0},B^\mathrm{frac}_{\mathbf{s}_1}]] =(-1)^{c_{\mathbf{s}_0,\mathbf{s}_1}^\mathrm{frac}} $
where
\begin{equation}
c^\mathrm{frac}_{\mathbf{s}_0,\mathbf{s}_1} = \left(\prod_{i} \mathbf{F}_i^{(\mathbf{r}_1-\mathbf{r}_0)_i}\right)_{\mathbf{s}_1,\mathbf{s}_0} c
\label{eq:fraccomm}
\end{equation}
which is zero if $c=0$.
If instead $A$ and $B$ anticommute, then $c=1$ and $c^\mathrm{frac}_{\mathbf{s}_0,\mathbf{s}_1}$ may be zero or non-zero depending on their separation $\mathbf{s}_1-\mathbf{s}_0$.
\subsection{Three-step process}\label{sec:interp}
We now present a three-step process which is equivalent to fractalization, but offers insight into the relation between the original and fractalized operators.
The three steps are $1)$ constructing a layered system, $2)$ unitary transformation, and $3)$ choosing a different set of generators.
Each of these three steps is explained in detail below.
While fractalization may be applied to systems of arbitrary size and LCA rules, the three-step process is only applicable under certain conditions:
the boundary conditions of the $D$ original directions must be either open or periodic with $\mathbf{F}_i^{L_i}=\mathbb{1}$,
$\mathbf{F}_i$ must all be invertible, and
the boundary conditions along the $m$ new directions must be periodic.
Let us start with an $X$ or $Z$ operator in the vectoral representation, $A=\sigma_X[a_{\mathbf{r},n}]$ or $B=\sigma_Z[b_{\mathbf{r},n}]$, defined in $D$ dimensions, and choose an anchor point $\mathbf{r}_0$ in the same way as before.
The first step is to extend these operators to a layered system, with each layer labeled by its coordinate $\mathbf{s}$ along the perpendicular direction.
The operator $A$ or $B$ now maps on to a set of operators on each layer separately,
\begin{equation}
A_{\mathbf{s}_0} = \sigma_X[a_{\mathbf{r},n}\delta_{\mathbf{s},\mathbf{s}_0}],\;\;\;\;
B_{\mathbf{s}_0} = \sigma_Z[b_{\mathbf{r},n}\delta_{\mathbf{s},\mathbf{s}_0}].
\end{equation}
on the $D+m$ dimensional lattice.
The next step involves a non-local (along the $\mathbf{s}$ directions) unitary transformation.
The unitary transformation involves a series of controlled-$X$ (CX) gates.
In particular, for any invertible binary matrix $\mathbf{M}$ with $M_{ii}=1$, there is a corresponding CX circuit which implements the transformation
\begin{equation}
\sigma_X[\mathbf{v}]\rightarrow \sigma_X[\mathbf{M}\mathbf{v}], \;\;\;\;
\sigma_Z[\mathbf{w}]\rightarrow \sigma_Z[\mathbf{M}^{-1,T}\mathbf{w}].
\end{equation}
In Appendix~\ref{app:cx}, we give a general algorithm for determining a CX circuit for any such $\mathbf{M}$.
Here, we choose the matrix
\begin{equation}
M_{(\mathbf{r},\mathbf{s},n),(\mathbf{r}^\prime,\mathbf{s}^\prime,n^\prime)} = \delta_{\mathbf{r},\mathbf{r}^\prime}\delta_{n,n^\prime} \left(\prod_i \mathbf{F}_i^{r_i}\right)_{\mathbf{s},\mathbf{s}^\prime}
\end{equation}
which maps $A_{\mathbf{s}_0}\rightarrow\tilde{A}_{\mathbf{s}_0},~B_{\mathbf{s}_0}\rightarrow\tilde{B}_{\mathbf{s}_0}$,
\begin{equation}
\begin{split}
\tilde{A}_{\mathbf{s}_0} = \sigma_X\left[\left(\prod_{i} \mathbf{F}_i^{r_i}\right)_{\mathbf{s},\mathbf{s}_0} a_{\mathbf{r},n}\right],\\
\tilde{B}_{\mathbf{s}_0} = \sigma_Z\left[\left(\prod_{i} \mathbf{F}_i^{-r_i}\right)_{\mathbf{s}_0,\mathbf{s}} b_{\mathbf{r},n}\right].
\end{split}
\end{equation}
which are highly non-local operators along the $\mathbf{s}$ directions as they involve $\mathbf{F}_i$ being raised to potentially high powers $r_i$.
The third and final step involves choosing a different set of generators for the groups $\langle \{\tilde{A}_{\mathbf{s}_0}\}\rangle$ and $\langle \{\tilde{B}_{\mathbf{s}_0}\}\rangle$.
For $X$ operators we choose
\begin{eqnarray}
A^{\mathrm{frac}}_{\mathbf{s}_0} &=& \prod_{\mathbf{s}^\prime} \tilde{A}_{\mathbf{s}^\prime}^{\left(\prod_i \mathbf{F}^{-r_{0,i}}_i\right)_{\mathbf{s}^\prime,\mathbf{s}_0}}
\label{eq:genchange}
\\
&=& \sigma_X \left[\sum_{\mathbf{s}^\prime} \left(\prod_i \mathbf{F}_i^{-r_{0,i}}\right)_{\mathbf{s}^\prime,\mathbf{s}_0} \left(\prod_i\mathbf{F}_i^{r_i}\right)_{\mathbf{s},\mathbf{s}^\prime} a_{\mathbf{r},n}\right]\nonumber\\
&=& \sigma_X \left[\left(\prod_i \mathbf{F}_i^{(\mathbf{r}-\mathbf{r}_0)_i}\right)_{\mathbf{s},\mathbf{s}_0} a_{\mathbf{r},n}\right]
\end{eqnarray}
which matches Eq~\ref{eq:Afrac}.
Similarly, for $Z$ operators,
\begin{equation}
\begin{split}
B^{\mathrm{frac}}_{\mathbf{s}_0} &= \prod_{\mathbf{s}^\prime} \tilde{B}_{\mathbf{s}^\prime}^{\left(\prod_i \mathbf{F}^{r_{0,i}}_i\right)_{\mathbf{s}_0,\mathbf{s}^\prime}}
\end{split}
\end{equation}
results in the fractalized operator.
Since $\mathbf{F}_i$ are all invertible, this just corresponds to a different choice of generators for their generated groups.
Although $\tilde{A}_{\mathbf{s}_0}$ were highly non-local, $A^\mathrm{frac}_{\mathbf{s}_0}$ are all local operators.
Thus, locality is restored.
This three-step process has important implications for fractalized quantum codes.
Consider the CSS stabilizer Hamiltonian
\begin{equation}
H = -\sum_l \mathcal{O}_l
\end{equation}
with the $2^k$-dimensional ground state manifold obtained as the simultaneous $+1$ eigenstate of all $\mathcal{O}_l$.
Let us choose LCA rules and the system sizes such that the three-step process can be applied.
After the first step, $H\rightarrow \sum_{\mathbf{s}} H_{\mathbf{s}}$ simply describes a layered system with ground state degeneracy $2^{L k}$.
The second step is a unitary transformation, $H\rightarrow U H U^\dagger$, which does not affect the ground state degeneracy.
The ground states themselves are related to the original ground states of the layered system by the non-local unitary transformation $\ket{\psi}\rightarrow U\ket{\psi}$.
Finally, since all ground states are $+1$ eigenstates of each term, choosing a different set of generators for the stabilizer group does not affect the ground state manifold (although higher excited eigenstates are shifted).
Thus, starting with a locally generated CSS stabilizer Hamiltonian with ground state degeneracy $2^k$, the fractalized Hamiltonian will have a $2^{L k}$ degenerate ground state manifold which is unitarily related to those of the layered system by the non-local unitary transformation $U$.
In particular, this implies that upon compactifying the additional $m$ dimensions the fractalized model is in the same $D$ dimensional gapped quantum phase of matter as the layered system.
\subsection{Generalizations}
\subsubsection{Beyond qubits}
Everything discussed above straightforwardly generalizes from qubit degrees of freedom to $\mathbb{Z}_p$ qudits of prime dimension $p$.
The $X$ and $Z$ Pauli operators are replaced by $\mathbb{Z}_p$ clock operators, satisfying the relation
$XZ = e^{\frac{2\pi i}{p}} ZX$.
Fractalization now takes as input $p$-state LCA rules, specified by polynomials $f(\mathrm{x})$ with $\mathbb{F}_p$ coefficients.
Such LCA still generate fractal structures due to the property $f(\mathrm{x})^{p^n}=f(\mathrm{x}^{p^n})$, where self-similarity occurs at scales $p^n$.
Our previous discussions readily applies, with arithmetic now done modulo $p$.
\subsubsection{Higher order LCA}\label{sec:higherorder}
The fractalization procedure generalizes to higher order LCA.
In an $n$th order LCA, the state at time $t$, $c_t$, may be determined by the states at times back to $t-n$,
\begin{equation}
c_t(\mathrm{y}) = \sum_{j=1\dots n} f^{(j)}(\mathrm{y}) c_{t-j} (\mathrm{y})
\end{equation}
where the LCA update rule is now specified by a list of $n$ polynomials $\{f^{(j)}(\mathrm{y})\}$.
Fractalization from $D$ to $D+m$ dimensions now takes as input $D$, $m$-dimensional, $n$-th order LCA rules, $\{f^{(j)}_i(\mathrm{y})\}$.
One possible generalization for
Eq.~\ref{eq:Afrac} is
\begin{equation}
\begin{split}
A^{\mathrm{frac}}_{\mathbf{s}_0} =& \sigma_X\left[
\mathrm{x}^{\mathbf{r}_0} \mathrm{y}^{\mathbf{s}_0} \mathbf{a}\left(\left\{\sum_{j} {f}^{(j)}_i(\mathrm{y})x_i^j\right\}_{i}\right)
\right]
\end{split}\label{eq:hoAfrac}
\end{equation}
and
\begin{equation}
\begin{split}
B^{\mathrm{frac}}_{\mathbf{s}_0} =& \sigma_Z\left[
\mathrm{x}^{\mathbf{r}_0} \mathrm{y}^{\mathbf{s}_0} \mathbf{b}\left(\left\{\sum_{j} {f}^{(j)}_i(\bar{\mathrm{y}})\bar{x}_i^j\right\}_i\right)
\right]
\end{split}
\end{equation}
This generalization is chosen such that the fractalized Ising model (see Sec~\ref{sec:ising}) has as its ground state the set of valid space-time LCA trajectories.
However, this higher order generalization no longer has as many of the nice properties as before.
For one, $[A,B]=0$ is no longer a sufficient criterion for the commutativity of the fractalized operators.
Instead, commutativity is only guaranteed if all translations commute.
Let $A_{\mathbf{r}}$ and $B_{\mathbf{r}}$ be translations of $A$ and $B$, such that their anchor point is at $\mathbf{r}$.
Then, $[A_{\mathbf{r}_0},B_{\mathbf{r}_1}]=0$ for all $\mathbf{r}_{0/1}$, implies that $[A^{\mathrm{frac}}_{\mathbf{r}_0,\mathbf{s}_0},B^{\mathrm{frac}}_{\mathbf{r}_1,\mathbf{s}_1}]=0$ for all $\mathbf{r}_{0/1},\mathbf{s}_{0/1}$.
This follows from the fact that, for operators whose translations all commute,
\begin{equation}
\mathbf{a}^T(\mathrm{x})\mathbf{b}(\mathrm{x})=\mathbf{a}^T(\mathrm{f}(\mathrm{y})\circ \mathrm{x})\mathbf{b}(\mathrm{f}(\mathrm{y})\circ \mathrm{x})=0 \, .
\end{equation}
Thus, fractalization with higher order LCA should only be applied for translation invariant commuting codes.
We also do not have a generalization of the three-step process for fractalizing higher order LCA, even for translation invariant codes.
Thus, the precise connection between the original and fractalized code for a general higher order LCA remains an open question.
\section{Examples and Discussion}\label{sec:examples}
In this section, we go through a number of examples of fractalization applied to well-understood models, connecting them with more exotic fractal models in the literature.
We start with the 1D quantum Ising model, which we show fractalizes to a spin model studied by Newman and Moore~\cite{newmanmoore} as a translation invariant model of glassiness, and also as a classical code~\cite{CAcode,yoshidaclassical,Nixon2020}.
We then move on to the cluster model~\cite{cluster} and find that it fractalizes into cluster states on higher dimensional lattices, known examples of fractal subsystem SPT phases~\cite{fractalSSPT} which have been shown to be useful for universal measurement-based quantum computation~\cite{Raussendorf_2019,devakul2018universal}.
We then fractalize Kitaev's 2D toric code model~\cite{kitaev}, which results in Yoshida's fractal spin liquids~\cite{yoshida2013exotic}, a family of models of type-II fracton topological order.
\subsection{1D Ising model}\label{sec:ising}
The 1D Ising model is defined on a chain with one qubit per site $r$.
The Hamiltonian simply consists of pairwise nearest neighbor Ising interactions
\begin{equation}
H_\mathrm{1DIsing} = -\sum_{r} Z_{r-1} Z_r = -\sum_r \sigma_Z[x^r (1+\bar{x})]
\end{equation}
which, after fractalization with a $1$ dimensional LCA $f(y)$, becomes
\begin{equation}
H_\mathrm{1DIsing}^\mathrm{frac} = -\sum_{r,s} \sigma_Z[x^ry^s (1+f(\bar{y})\bar{x})].
\end{equation}
Using the choice $f(y)=1+y$, we arrive at the Newman-Moore model~\cite{newmanmoore},
\begin{equation}
H_\mathrm{NM} = -\sum_{r,s} Z_{(r,s)} Z_{(r-1,s)} Z_{(r-1,s-1)}.
\end{equation}
where $(r,s)$ labels the $x,y$ coordinates on the square lattice.
The global $\mathbb{Z}_2$ symmetry of the Ising model maps on to a set of fractal symmetries,
\begin{equation}
S=\sigma_X\left[\sum_r x^r\right] \rightarrow S^{\mathrm{frac}}_{s} =\sigma_X\left[y^s \sum_r f(y)^r x^r\right]
\end{equation}
which act on a subsystem of sites corresponding to the Sierpinski triangle fractal, shifted by $s$ along the $y$ direction.
The ground states of $H_{\mathrm{1DIsing}}^\mathrm{frac}$ are spin configurations that are valid space-time LCA trajectories.
Let us denote a product state in the $Z_{(r,s)}$ basis by $|\{z_{(r,s)}\}\rangle$, where $Z_\mathbf{r}|\{z_{(r,s)}\}\rangle=(-1)^{z_{(r,s)}}$.
Then, $z_{(r,s)}$ forms a valid LCA trajectory, with $r$ playing the role of time, and $s$ playing the role of space.
This remains true in higher dimensions, as well as for higher-order LCA.
For the $D$-dimensional Ising model, fractalized with $m$-dimensional LCA, the resulting $D+m$-dimensional $H_{\mathrm{1DIsing}}^\mathrm{frac}$ has as its ground state valid space-time LCA trajectories,
where $\mathbf{r}$ are time dimensions, and $\mathbf{s}$ are space dimensions.
Multiple time dimensions, each governed by their own update rule, are possible since translation-invariant LCA rules commute.
The Ising chain may also be viewed as the classical repetition code, whereby the logical bit is represented by the two symmetry-broken ground states.
Similarly, the fractalized Ising chain, in $1+m$ dimensions, is also useful as a classical code~\cite{yoshidaclassical,CAcode,Nixon2020}, in which logical bits are again represented by the symmetry-broken ground states, of which there are now $2^{\mathcal{O}(L^m)}$.
The fractal symmetry operators $S^{\mathrm{frac}}_\mathbf{s}$ are logical operators that flip between the symmetry-broken ground states.
Upon generalizing to $\mathbb{Z}_p$ degrees of freedom, this code saturates an information storage tradeoff bound~\cite{BravyiPoulinTerhal} in the limit $p\rightarrow\infty$~\cite{yoshidaclassical}.
The nature of the excitations of the fractalized Ising model can be deduced from the excitations of the Ising model.
The Ising model in 1D has point-like domain-wall excitations, which are created from a ground state $\ket{\psi}$ by flipping a segment of spins, $W\ket{\psi}\equiv \prod_{r\in[\ell_1\dots \ell_2-1]}X_r \ket{\psi}$.
Let $B_r=Z_{r-1}Z_r$ be a term in $H_{\mathrm{1DIsing}}$, which satisfies the commutation relation
\begin{equation}
[[W,B_r]] = (-1)^{\delta_{r,\ell_1} + \delta_{r,\ell_2}} \, ,
\end{equation}
making it clear that $W$ only creates two excitations at $\ell_1$ and $\ell_2$ when acting on the ground state.
From Eq.~\ref{eq:fraccomm}, the fractalized versions of these operators satisfy
$[[W^\mathrm{frac},B^{\mathrm{frac}}_{r,s}]] = (-1)^c$ with
\begin{equation}
c=\delta_{r,\ell_1}\delta_{s,s_1} + \delta_{r,\ell_2}(\mathbf{F}^{\ell_2-\ell_1})_{s,s_1} \, .
\end{equation}
Therefore, $B^{\mathrm{frac}}_{s_1}$ creates a single excitation at $(\ell_1,s_1)$,
while creating potentially many excitations along $r=\ell_2$, depending on $\mathbf{F}^{\ell_2-\ell_1}$.
However, if $\ell_2-\ell_1=2^n$, ${f(y)^{2^n} = f(y^{2^n})}$ and so $(\mathbf{F}^{2^n})_{s_1,s}$ is only non-zero for a few $s$; for $f(y)=1+y$, only two further excitations are created at $(\ell_1+2^n,s_1)$ and $(\ell_1+2^n,s_1+2^n)$.
The excitations of $H^{\mathrm{frac}}_{\mathrm{1DIsing}}$ are therefore point-like, and may be created in special configurations separated by distances $2^n$ via fractal operators.
They are an example of (non-topological) fractons.
This can be generalized to higher dimensions.
The 2D Ising model, for example, has loop-like domain wall excitations and is treated in section~\ref{sec:fractalizingloops}.
\subsection{1D Cluster model}\label{sec:clusterstate}
The 1D cluster model, in CSS form, has two qubits per site and is described by the Hamiltonian
\begin{equation}
\begin{split}
H_{\mathrm{clus}} &= -\sum_{r} X_r^{(1)} X_r^{(2)} X_{r+1}^{(1)} - \sum_r Z_{r-1}^{(2)} Z_r^{(1)} Z_r^{(2)}\\
&= -\sum_{r}\sigma_X\left[x^r\begin{pmatrix}1+x\\1\end{pmatrix}\right]
-\sum_{r}\sigma_Z\left[x^r\begin{pmatrix}1\\1+\bar{x}\end{pmatrix}\right] .
\end{split}
\end{equation}
After fractalization with a $1$-dimensional LCA $f(y)$,
\begin{equation}
\begin{split}
H_{\mathrm{clus}}^\mathrm{frac} =
& -\sum_{r,s}\sigma_X\left[x^r y^s\begin{pmatrix}1+f(y)x\\1\end{pmatrix}\right]\\
&-\sum_{r,s}\sigma_Z\left[x^r y^s\begin{pmatrix}1\\1+f(\bar{y})\bar{x}\end{pmatrix}\right] ,
\end{split}
\end{equation}
which describes another cluster state on a 2D bipartite lattice.
With the choice $f(y)=1+y$, for example, $H^{\mathrm{frac}}_{\mathrm{clus}}$ describes the cluster state on the honeycomb lattice~\cite{fractalSSPT}.
Higher dimensional models may be created by using $m$-dimensional LCA rules,
for example, using the $2$-dimensional LCA rule $f(\mathrm{y})=1+y_1+y_2$, one arrives at a 3-dimensional cluster state, with Sierpinski tetrahedron fractal subsystem symmetries.
The cluster model describes a non-trivial symmetry-protected topological (SPT) phase, protected by the ${\mathbb{Z}_2\times\mathbb{Z}_2}$ symmetry that is generated by $S_1=\prod_r Z_r^{(1)}$ and $S_2=\prod_r X_r^{(2)}$.
The fractalized cluster model also describes an SPT phase, protected by the set of fractal subsystem symmetries generated by $S^{\mathrm{frac}}_{\alpha,s}$, $\alpha\in\{1,2\}$.
The non-triviality of $H^{\mathrm{frac}}_{\mathrm{clus}}$ as a fractal subsystem SPT follows from that of $H_{\mathrm{clus}}$.
The cluster chain $H_{\mathrm{clus}}$ realizes a non-trivial SPT phase, as can be observed from the fact that, at a boundary, the $G=\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry acts as a non-trivial projective representation~\cite{PhysRevB.84.235128}.
We may apply fractalization directly to the open cluster chain and its symmetry operators, following the three-step process.
In the first step, the system is composed of decoupled cluster chains along $x$, each with their own symmetry $S_{\alpha,s}$ realizing a projective representation on the left edge.
The second step is a unitary transformation, local along $x$, which does not affect the projective representation.
The third step is a different choice of generators for the symmetry group $\langle \{ S_{\alpha,s}\}\rangle$.
We can simply choose the origin such that the anchor point for $S_{\alpha,s}$ is $r_0=0$, in which case the change of generators (Eq~\ref{eq:genchange}) is trivial.
Thus, $S^{\mathrm{frac}}_{\alpha,s}$ realizes the same projective representation in $H^{\mathrm{frac}}_{\mathrm{clus}}$ as $S_{\alpha,s}$ from the layered $H_{\mathrm{clus}}$.
If we take periodic boundary conditions along the $x$ direction as well, along with an irreversible LCA, then the three-step process cannot be applied.
Following the discussion in Sec.~\ref{sec:fracdef}, the symmetry group of $H^{\mathrm{frac}}_{\mathrm{clus}}$ is generated by $\{S_{\alpha,j}^{\mathrm{frac}}\}$, which is smaller than the symmetry group of the layered system (or may even be trivial in the most extreme case), depending on the number of solutions to $q(y)f(y)=q(y)$.
These are pseudo-SPT phases, as defined in Ref.~\cite{Devakul2018}.
For a given set of fractal symmetries, there are an infinite number of non-trivial SPT phases~\cite{Devakul2018}.
The fractal subsystem SPT phase obtained from fractalizing the cluster state is only the simplest non-trivial phase.
The remaining phases are generically not strictly translation invariant (they are translation invariant with a larger period $2^n$), and therefore cannot be realized by fractalization.
\subsection{2D toric code model}\label{sec:toriccode}
The 2D toric code~\cite{kitaev} is defined on the square lattice with a qubit on every bond.
Associating with each site the two qubits on bonds straddling it in the $+\mathbf{x}_1$ and $+\mathbf{x}_2$ direction, the Hamiltonian may be written as
\begin{align}
H_{\mathrm{TC}} =&
-\sum_{\mathbf{r}} \sigma_X\left[\mathrm{x}^\mathbf{r}\begin{pmatrix}1+x_2\\1+x_1\end{pmatrix}\right]
-\sum_{\mathbf{r}} \sigma_Z\left[\mathrm{x}^\mathbf{r}\begin{pmatrix}1+\bar{x}_1\\1+\bar{x}_2\end{pmatrix}\right]
\nonumber \\
=& -\sum_\mathbf{r} A_\mathbf{r} - \sum_\mathbf{r} B_\mathbf{r} \, .
\end{align}
Applying fractalization with two 1-dimensional LCA rules, $f_1(y),f_2(y)$, we get
\begin{equation}
\begin{split}
H^{\mathrm{frac}}_{\mathrm{TC}} = &
-\sum_{\mathbf{r},s} \sigma_X\left[\mathrm{x}^\mathbf{r} y^s\begin{pmatrix}1+f_2(y)x_2\\1+f_1(y)x_1\end{pmatrix}\right] \\
&-\sum_{\mathbf{r},s} \sigma_Z\left[\mathrm{x}^\mathbf{r} y^s \begin{pmatrix}1+f_1(\bar{y})\bar{x}_1\\1+f_2(\bar{y})\bar{x}_2\end{pmatrix}\right].
\end{split}
\label{eq:fractc}
\end{equation}
which is Yoshida's fractal spin liquid model~\cite{yoshida2013exotic}.
We take $H_{\mathrm{TC}}$ on an $L_1\times L_2$ torus, and $H^{\mathrm{frac}}_{\mathrm{TC}}$ on an ${L_1\times L_2\times L_3}$ $3$-torus.
The logical operators of the toric code,
\begin{equation}
\begin{split}
\ell^X_{1} = \sigma_X\left[\sum_{r_1}\begin{pmatrix} x_1^{r_1} \\ 0\end{pmatrix}\right]\;\;\;
\ell^Z_{1} = \sigma_Z\left[\sum_{r_2}\begin{pmatrix} \bar{x}_2^{r_2} \\ 0\end{pmatrix}\right]\\
\ell^X_{2} = \sigma_X\left[\sum_{r_2}\begin{pmatrix} 0\\x_2^{r_2}\end{pmatrix}\right]\;\;\;
\ell^Z_{2} = \sigma_Z\left[\sum_{r_1}\begin{pmatrix} 0 \\ \bar{x}_1^{r_1}\end{pmatrix}\right]
\end{split}
\end{equation}
generate the $2$ qubit Pauli algebra on the ground state manifold of $H_{\mathrm{TC}}$.
If $f_1(y)^{L_1}=f_2(y)^{L_2}=1$, then they can be straightforwardly fractalized,
\begin{equation}
\begin{split}
\ell^{X,\mathrm{frac}}_{1,s} = \sigma_X\left[y^s\sum_{r_1}\begin{pmatrix} f_1(y)^{r_1}x_1^{r_1} \\ 0\end{pmatrix}\right]\\
\ell^{Z,\mathrm{frac}}_{1,s} = \sigma_Z\left[y^s\sum_{r_2}\begin{pmatrix} f_2(\bar{y})^{r_2}\bar{x}_2^{r_2} \\ 0\end{pmatrix}\right]
\end{split}
\end{equation}
and similarly for $\ell^{X/Z,\mathrm{frac}}_{2,s}$.
It can be easily checked that $[[\ell^{X,\mathrm{frac}}_{\alpha,s_0},\ell^{Z,\mathrm{frac}}_{\beta,s_1}]]=(-1)^{\delta_{\alpha \beta}\delta_{s_0 s_1}}$,
thus generating a $2 L_3$ qubit Pauli algebra on the ground state manifold.
If $f_i(y)^{L_i}\neq 1$, then some smaller subgroup is realized instead.
The toric code describes a 2D topologically ordered phase.
The fractalized toric codes describe a family of (type-II) fracton topological ordered phases.
Fracton topological order is characterized by a subextensive ground state degeneracy (scaling with an envelope $2^{\mathcal{O}(L)}$) and quasiparticle excitations with restricted mobility.
When the three-step process for fractalization holds, the ground state degeneracy of $H^{\mathrm{frac}}_{\mathrm{TC}}$ is $2^{2 L_3}$, but will be less for generic system sizes where $f_i(y)^{L_i}\neq 1$.
The excitations of the toric code are created at the ends of string operators.
Fractalizing these string operators and following a similar analysis as in Sec.~\ref{sec:ising}, one finds that the excitations of the fractalized toric code are also point-like and must be created in particular configurations.
As pointed out in Ref.~\cite{yoshida2013exotic}, these quasiparticle excitations are only strictly immobile if $f_1(y)$ and $f_2(y)$ are algebraically unrelated.
Two polynomials are algebraically related if there is a non-trivial solution to
\begin{equation}
f_1(y)^{n_1} = c f_2(y)^{n_2}
\label{eq:algebraicrelation}
\end{equation}
for any finite constants $n_i$,$c$, without periodic boundary conditions.
Haah's cubic code~\cite{Haah} can also be viewed as a higher order fractal spin liquid~\cite{yoshida2013exotic}.
Using the generalization of fractalization to higher order LCA, with
\begin{equation}
\begin{split}
f_1^{(1)}(y)&=1+y+y^2\\
f_1^{(2)}(y)&=0\\
f_2^{(1)}(y)&=1+y\\
f_2^{(2)}(y)&=1+y+y^2
\end{split}
\end{equation}
one arrives at a model that is locality preserving unitarily related~\cite{yoshida2013exotic} to Haah's cubic code, up to a redefinition of the lattice vectors.
The toric code may also be regarded as a limit of the Ising gauge theory (IGT), which is obtained by gauging the global $\mathbb{Z}_2$ symmetry of the 2D quantum Ising model $H_{\mathrm{2D Ising}}$.
After gauge fixing, the IGT takes the form of a perturbed toric code,
\begin{equation}
H_{\mathrm{IGT}} =-h\sum_\mathbf{r} A_\mathbf{r} -K\sum_\mathbf{r} B_\mathbf{r} - J\sum_{\mathbf{r},n}Z_\mathbf{r}^{(n)} - \Gamma\sum_{\mathbf{r},n}X_{\mathbf{r}}^{(n)}
\end{equation}
where $n=1,2$ labels the qubits on each site.
$H_{\mathrm{IGT}}$ is in the topologically ordered phase when $J/h, \Gamma/K \ll 1$.
The fractalized IGT, $H^{\mathrm{frac}}_{\mathrm{IGT}}$, can either be obtained by fractalizing $H_{\mathrm{IGT}}$ or alternatively by applying a generalized gauging procedure~\cite{williamsongauge,vijay2016fracton,shirley2018FoliatedFracton} to the fractal subsystem symmetries of the fractalized 2D Ising model.
Thus, the generalized gauging procedure naturally arises in this context as the ``fractalized'' version of the usual gauging procedure.
In Refs.~\cite{Dua2019,Williamson2018b}, the effect of dimensional compactification of 3D fracton models was investigated.
Compactifying a 3D translation invariant type-II fracton code along one dimension leads to a 2D code which is equivalent to copies of toric code and trivial states by a locality-preserving unitary transformation~\cite{haah2016algebraic,Haah2018a,bombin2014structure}.
Take the $L_1\times L_2\times L_3$ fractalized toric code models, and consider compactifying along the third dimension.
In cases where the three-step process for fractalization can be applied, we can easily show that the resulting compactified model is local unitarily equivalent to $L_3$ copies of the toric code:
The first step creates the layered toric codes, the second step is a unitary transformation, and the third step is an irrelevant change of generators for the stabilizer group.
The unitary transformation, the most important step, is non-local only in the third direction.
Thus, once the third dimension is compactified, the second step is a local unitary transformation which transforms the ground state into $L_3$ copies of the toric code ground state.
In a recent manuscript~\cite{coupledlayers} by the present authors, a coupled cluster state construction was presented for a number of models, including the toric code and many type-II fracton phases.
The Hamiltonian takes the form $H=H_{\mathrm{clus}} + \lambda H_{\mathrm{coup}}$, where $H_{\mathrm{clus}}$ is the cluster Hamiltonian on decoupled chains or layers, and $H_{\mathrm{coup}}$ couples them together.
By choosing the the cluster states and couplings appropriately, as discussed in Ref.~\cite{coupledlayers},
the low energy effective Hamiltonian in the limit $\lambda\rightarrow\infty$ coincides with the desired fracton or toric code model.
In particular, a construction of the toric code in terms of coupled wires, and Yoshida's type-II fracton phases~\cite{yoshida2013exotic} in terms of coupled layers was obtained.
This coupled layer construction of Yoshida's models follows directly from fractalization applied to the coupled wire construction of the toric code.
Specifically, the Hamiltonian $H$ describing the coupled wire construction of the toric code can be written in terms of only $X$ or $Z$ operators.
Applying fractalization, $H\rightarrow H^\mathrm{frac}$, the cluster state wires map on to fractal cluster state layers, and in the limit $\lambda\rightarrow\infty$, $H^\mathrm{frac}$ realizes the fractalized toric code, which is exactly Yoshida's models.
\section{Fractalized loop excitations}
\label{sec:fractalizingloops}
In this section we describe several examples that involve fractalizing stabilizer models with loop-like excitations.
We find that the excitations in the fractalized models are no longer generically loop or string-like, but form more general extended ``fractalized loop'' excitations.
We determine the criterion for such models to lack any string-like excitations whatsoever.
This implies the lack of any membrane logical operators and can lead to a boost in the energy barrier for applying a logical operator from $\mathcal{O}(L)$ in the original model to $\mathcal{O}(L^\alpha)$, for some $1<\alpha\leq 2$ determined by the choice of fractalization, where $L$ is the linear size of the system.
For this section, we make use of a more compact notation for the sum of $K$ Hamiltonian terms,
\begin{equation}
\sum_{k=1\dots K}\sigma_X[\mathbf{a}_k(\mathrm{x})] \equiv \sigma_X[\mathbf{A}(\mathrm{x})]
\end{equation}
where
\begin{equation}
\mathbf{A}(\mathrm{x}) = \begin{pmatrix} \mathbf{a}_1(\mathrm{x}) & \dots & \mathbf{a}_K(\mathrm{x})\end{pmatrix}
\end{equation}
is an $N\times K$ matrix whose columns correspond to each of the Hamiltonian terms (and rows correspond to the physical qubits on each site as before).
A similar notation is also used for $\sigma_Z[\mathbf{B}(\mathrm{x})]$.
This representation is useful for translation invariant models with multiple types of $X$ or $Z$ terms, such as the higher dimensional toric code models.
\subsection{2D quantum Ising model} \label{sec:fracloop2d}
As a warmup we fractalize the 2D quantum Ising model which supports looplike domain wall excitations.
The Hamiltonian is given by
\begin{align}
H_{\text{2DIsing}} = - \sum_{\textbf{r}} \sigma_Z[\mathrm{x}^{\mathbf{r}} \mathbf{B}(\bar{\mathrm{x}})]
\end{align}
with
\begin{equation}
\mathbf{B}(\bar{\mathrm{x}})
= \begin{pmatrix}
1+\bar{x}_1 & 1+\bar{x}_2
\end{pmatrix}
\, ,
\label{eq:2DQIBr}
\end{equation}
where, as explained above, the two columns correspond to the Ising interactions along the $x_1$ and $x_2$ directions.
The logical operators for the code on periodic boundary conditions are given by
\begin{align}
\label{eq:2DQIL}
\ell^X =
\sigma_X \left[
\sum_{\textbf{r}}
\mathrm{x}^{\mathbf{r}}
\right] \, ,
&&
\ell^Z =
\sigma_Z \left[ 1
\right] \, ,
\end{align}
which represent a global $X$ spin flip and single site $Z$ operator, respectively.
Hence, the Ising model is not topologically ordered as the ground space degeneracy can be split locally.
A rectangular loop excitation can be created with a truncated membrane operator,
\begin{equation}
W = \sigma_X\left[\sum_{\mathbf{r} \in \mathcal{R}} \mathrm{x}^\mathbf{r}\right] \equiv \sigma_X[\mathbf{w}(\mathrm{x})]\, ,
\end{equation}
where $\mathcal{R}=\mathcal{R}_1\times\mathcal{R}_2$ is a set of sites in some rectangular region $\mathcal{R}_i = \{0\dots l_i-1\}$.
The excitations created in response to $W$ acting on the ground state is described by the polynomial vector
\begin{equation}
\mathbf{E}_{W}(\mathrm{x}) = \mathbf{B}(\mathrm{x})^T \mathbf{w}(\mathrm{x}) = \begin{pmatrix}
(1+x_1^{l_1})\sum_{r_2\in\mathcal{R}_2} x_2^{r_2}\\
(1+x_2^{l_2})\sum_{r_1\in\mathcal{R}_1} x_1^{r_1}
\end{pmatrix},
\end{equation}
whose elements denote the locations of excited $\sigma_Z$ terms, and the two rows correspond to the two types of $\sigma_Z$ terms.
In this case, this means that a line of $\sigma_Z[\mathrm{x}^\mathbf{r}(1+\bar{x}_1)]$ terms are excited along the edges of the rectangle at $r_1=0,l_1$, and a line of $\sigma_Z[\mathrm{x}^\mathbf{r}(1+\bar{x}_2)]$ excitations are created along the other edges at $r_2=0,l_2$.
This is simply the domain wall excitation of the Ising model.
The 3D fractalized 2D Ising model is described by the Hamiltonian
\begin{align}
H_{\text{2DIsing}}^{\text{frac}} = - \sum_{\textbf{r},s}
\sigma_Z[\mathrm{x}^{\mathbf{r}}y^s \textbf{B}(\mathrm{f}(\bar{y}) \circ \bar{\mathrm{x}})]
\end{align}
for some choice of $\mathbb{F}_2$ polynomials $f_1,f_2$.
For periodic boundary conditions $L_1,L_2,L_3$ satisfying $f_1(y)^{L_1}=f_2(y)^{L_2}=1$ there are $L_3$ logical qubits encoded within the ground state manifold with pairs of independent logical operators obtained by fractalizing the logical operators in Eq.~\eqref{eq:2DQIL},
\begin{align}
\ell_{s}^X =
\sigma_X \left[ y^s
\sum_{\textbf{r}}
f_1(\bar{y})^{r_1} f_2(\bar{y})^{r_2} \bar{\mathrm{x}}^{\textbf{r}}
\right] \, ,
&&
\ell_s^Z =
\sigma_Z \left[ y^s
\right] .
\end{align}
We see that the fractalized model is also not topologically ordered as the ground space degeneracy is split by local operators.
Suppose we now act on the ground state with a truncated $\ell^X_s$ logical operator, i.e. a fractalized $W$ operator,
$W^{\mathrm{frac}} \equiv \sigma_X [\mathbf{w}(\mathrm{f}(\mathrm{y})\circ \mathrm{x})]$.
This creates excitations located at
\begin{equation}
\mathbf{E}_{W}^\mathrm{frac}(\mathrm{x}) = \begin{pmatrix}
(1+f_1(y)^{l_1} x_1^{l_1})\sum_{r_2\in\mathcal{R}_2} f_2(y)^{r_2}x_2^{r_2}\\
(1+f_2(y)^{l_2} x_2^{l_2})\sum_{r_1\in\mathcal{R}_1} f_1(y)^{r_1} x_1^{r_1}
\end{pmatrix},
\end{equation}
which we refer to as a fractalized loop excitation.
It is clear that such excitations have an energy cost scaling faster than linearly with the size $l_1,l_2$.
Thus, these are not loop-like excitations.
They appear loop-like when projected on to the $(x_1,x_2)$ plane, but extend non-trivially into fractals in the $y$ direction.
A natural question is whether there exists \emph{any} extended topological excitations which are string-like.
By topological, we mean that the excitation is not simply a string of locally-created excitation clusters (recall that $H^{\mathrm{frac}}_{\mathrm{2DIsing}}$ is not topologically ordered).
To answer this question, observe that the Hamiltonian terms obey a \emph{local relation} on each cube, where a local relation is defined to be a product of Hamiltonian terms that result in unity.
The local relation is represented by a polynomial vector $\mathbf{l}(\bar{\mathrm{x}})$
which satisfies $\mathbf{B}(\bar{\mathrm{x}})\mathbf{l}(\bar{\mathrm{x}})=0$.
Going back to the original 2D Ising model, the local relation
\begin{equation}
\mathbf{l}(\bar{\mathrm{x}}) = \begin{pmatrix}1+\bar{x}_2 \\ 1+\bar{x}_1\end{pmatrix} \, ,
\end{equation}
where the two columns correspond to the two Hamiltonian terms,
tells us that $\sigma_Z[\mathrm{x}^\mathbf{r} \mathbf{B}(\bar{\mathrm{x}}) \mathbf{l}(\bar{\mathrm{x}})]=1$: a product of four Ising interaction terms around any square plaquette is trivial.
This simply means that excitations (which are domain walls) must come in pairs around every square plaquette, and therefore form closed $\mathbb{Z}_2$ loops.
For the fractalized Ising model, we have the fractalized local relation
$\mathbf{l}^\mathrm{frac}(\bar{\mathrm{x}},\bar{y}) = \mathbf{l}(\mathrm{f}(\bar{y})\circ\bar{\mathrm{x}})$ which implies that excitations of Hamiltonian terms are only allowed in certain configurations.
Meanwhile, a local cluster of excitations can always be created by acting with a single $X$ on a ground state, described by
\begin{equation}
\mathbf{E}_{X}(\mathrm{x},y) = \mathbf{B}^T(\mathrm{f}(y)\circ\mathrm{x})=
\begin{pmatrix}
1+f_1(y)x_1\\
1+f_2(y)x_2
\end{pmatrix} \, .
\end{equation}
A topological string-like excitation, $\mathbf{E}_{\mathrm{string}}(\mathrm{x},y)$, would have to be a valid excitation satisfying $\mathbf{l}^\mathrm{frac}(\mathrm{x},y)^T \mathbf{E}_{\mathrm{string}}(\mathrm{x},y)=0$, and also not be made up of a product of $\mathbf{E}_{X}(\mathrm{x},y)$.
This is equivalent to the problem of finding an $X$-type string logical operator $\ell^X = \sigma_X[\mathbf{E}_{\mathrm{string}}(\mathrm{x},y)]$ for the ``excitation Hamiltonian'' which we define in terms of the local relation and local excitation cluster,
\begin{equation}
H_{\mathrm{exc}} = -\sum_{\mathbf{r},s} \sigma_Z[\mathrm{x}^\mathbf{r} y^s \mathbf{l}^{\mathrm{frac}}(\bar{\mathrm{x}},\bar{y})] + \sigma_X[\mathrm{x}^\mathbf{r} y^s\mathbf{E}_{X}(\mathrm{x},y)]\\
\end{equation}
and is equivalent to the fractalized toric code $H^{\mathrm{frac}}_{TC}$ ( Eq.~\ref{eq:fractc}).
As shown in Ref.~\cite{yoshida2013exotic}, this code lacks string-like logical operators iff $f_1$ and $f_2$ are algebraically unrelated, as defined in Eq.~\ref{eq:algebraicrelation}.
Hence, for algebraically unrelated $f_{1,2}$, $H^{\mathrm{frac}}_{2D\mathrm{Ising}}$ lacks any topological string or loop-like excitations, otherwise there exists string-like excitations (which will only be allowed to lie along a particular direction).
\subsection{3D toric code}
Next we consider fractalizing the 3D toric code, a topologically ordered model with both loop and particle excitations.
It is described by the Hamiltonian
\begin{align}
\label{eq:3DTCH}
H_{\text{3DTC}} = - \sum_{\textbf{r}}
\sigma_X[\mathrm{x}^{\textbf{r}} \textbf{A}(\mathrm{x})] + \sigma_Z[\mathrm{x}^{\textbf{r}} \textbf{B}(\bar \mathrm{x})] \, ,
\end{align}
where
\begin{align}
\textbf{A}(\mathrm{x}) =
\begin{pmatrix}
0 & 1+x_3 & 1+x_2 \\
1+x_3 & 0 & 1+x_1 \\
1+x_2 & 1+x_1 & 0
\end{pmatrix} , &&
\textbf{B}(\bar \mathrm{x}) =
\begin{pmatrix}
1+\bar{x}_1 \\
1+\bar{x}_2 \\
1+\bar{x}_3
\end{pmatrix} .
\label{eq:3DTCAB}
\end{align}
The logical operators for the code on periodic boundary conditions correspond to conjugate string-membrane pairs given by
\begin{align}
\ell_1^X =
\sigma_X \left[
\sum_{r_1}
x_1^{r_1} \mathbf{e}_1
\right]
&&
\ell_1^Z =
\sigma_Z \left[
\sum_{r_2,r_3}
\bar{x}_2^{r_2}\bar{x}_3^{r_3} \mathbf{e}_1
\right] \, ,
\end{align}
where $\mathbf{e}_1=(1,0,0)^T$ is a unit vector.
$\ell^{X/Z}_{2/3}$ are defined similarly by cyclic permutations of $1,2,3$.
Since this model is topologically ordered, no local operators can split the ground space degeneracy.
Truncating the string logical operator $\ell^X_i$ leads to topological quasiparticle excitations at its ends, and truncating the membrane logical operator $\ell_i^Z$ leads to a topological loop excitation along its boundary, when acting on the ground state manifold.
The 4D fractalized 3D toric code model is described by the Hamiltonian
\begin{align}
H_{\text{3DTC}}^{\mathrm{frac}} = - \sum_{\textbf{r},s}
\sigma_X[\mathrm{x}^{\textbf{r}} y^s \textbf{A}(\mathrm{f}( y) \circ\mathrm{x})] + \sigma_Z[\mathrm{x}^{\textbf{r}}y^s \textbf{B}(\mathrm{f}(\bar y) \circ \bar \mathrm{x})] \, ,
\end{align}
for the $\textbf{A}$ and $\textbf{B}$ matrices introduced in Eq.~\eqref{eq:3DTCAB} and a choice of $\mathbb{F}_2$ polynomials $\mathrm{f}( y)=\{f_1(y),f_2(y),f_3(y)\}$.
Excitations of the $X$ term are point-like, while excitations of the $Z$ term are fractalized loops.
Similar to before, these fractalized loops resemble loops when projected to $(x_1,x_2,x_3)$, but extend non-trivially in the $y$ direction.
Following Ref.~\cite{yoshida2013exotic}, a necessary and sufficient condition for the model to have no string logical operators is that the triple $f_1,f_2,f_3$ is not algebraically related.
A triple $f_1,f_2,f_3$ is said to be algebraically related iff an equation of the form
\begin{align}
f_1(y)^{n_1}= c~f_2(y)^{n_2}f_3(y)^{n_3} \, ,
\label{eq:3algebraicrelated}
\end{align}
holds for some finite constants $n_i,c,$ where $n_i$ are non-negative integers and not all $n_i=0$, or for some other permutation of $1,2,3$, without periodic boundary conditions.
We may also ask whether the excitations of the $Z$ terms can be string-like.
Repeating the same analysis as before,
we may reduce the problem to that of finding string-like logical operators for an excitation Hamiltonian, on which the arguments of Ref.~\cite{yoshida2013exotic} can again be applied.
The condition for the existence of string-like excitations is once again Eq.~\ref{eq:3algebraicrelated}.
Hence, algebraic independence of $f_1,f_2,f_3$ imply both that there are no string-like logical operators and also no string-like $X$ excitations.
The lack of string-like $X$ excitations further implies the non-existence of a membrane $Z$-type logical operator (or some fractal structure than can be embedded onto a membrane), since if one existed then its truncation would produce a string excitation around its perimeter.
This model still has $X$-type logical operators which are fractals that can be embedded on to a single $(x_i,y)$ plane, however.
By fractalizing the 4D toric code in the next section, we construct a 5D code which lacks any membrane logical operators altogether.
For periodic boundary conditions $L_1,L_2,L_3,L_4$ satisfying $f_i(y)^{L_i}=1$ there are $3L_4$ logical qubits encoded within the ground space and a basis of logical operators is given by
$\ell^{X/Z,\mathrm{frac}}_{i,s}$.
To construct an explicit example we take
\begin{equation}
\begin{split}
f_1(y)&= 1+y+y^2 \, , \\
f_2(y)&=1+y+y^3 \, , \\
f_3(y)&=1+y^2+y^3 \, ,
\end{split}
\end{equation}
which satisfy the algebraic independence condition, as the chosen polynomials are prime, and the boundary conditions when $L_i=2^l$.
\subsection{4D toric code}
Next we consider fractalizing the 4D toric code which only supports loop-like excitations.
The model is described by the Hamiltonian
\begin{align}
H_{\text{4DTC}} = - \sum_{\textbf{r}}
\sigma_X[\mathrm{x}^{\textbf{r}} \textbf{A}(\mathrm{x})] + \sigma_Z[\mathrm{x}^{\textbf{r}} \textbf{B}(\bar \mathrm{x})] \, ,
\end{align}
with
\begin{align}
\label{eq:4DTCa}
\textbf{A}(\mathrm{x}) &=
\begin{pmatrix}
1+x_2 & 1+x_1 & 0 & 0 \\
1+x_4 & 0 & 0 & 1+x_1 \\
1+x_3 & 0 & 1+x_1 & 0 \\
0 & 1+x_3 & 1+x_2 & 0 \\
0 & 1+x_4 & 0 & 1+x_2 \\
0 & 0 & 1+x_4 & 1+x_3
\end{pmatrix}
\, ,
\\
\label{eq:4DTCb}
\textbf{B}(\bar \mathrm{x}) &=
\begin{pmatrix}
0 & 0 & 1+\bar{x}_4 & 1+\bar{x}_3 \\
0 & 1+\bar{x}_3 & 1+\bar{x}_2 & 0 \\
0 & 1+\bar{x}_4 & 0 & 1+\bar{x}_2 \\
1+\bar{x}_4 & 0 & 0 & 1+\bar{x}_1 \\
1+\bar{x}_3 & 0 & 1+\bar{x}_1 & 0 \\
1+\bar{x}_2 & 1+\bar{x}_1 & 0 & 0
\end{pmatrix}
\, ,
\end{align}
where the six rows correspond to qubits living on an $(x_i,x_j)$-plaquette, with $ij=12,13,14,23,24,34$, in that order.
On periodic boundary conditions the ground space encodes six logical qubits with logical operators
\begin{align}
\label{eq:4DTCl}
\ell_{12}^X =
\sigma_X \left[
\sum_{r_3,r_4}
x_3^{r_3} x_4^{r_4}\mathbf{e}_{12}
\right],
&&
\ell_{12}^Z =
\sigma_Z \left[
\sum_{r_1,r_2}
\bar{x}_1^{r_1} \bar{x}_2^{r_2}\mathbf{e}_{12}
\right],
\end{align}
where $\mathbf{e}_{12}$ is the unit vector corresponding to the qubit on the $(x_1,x_2)$ plaquette.
The logical operators $\ell_{ij}^{X/Z}$ for the remaining $ij$ are defined similarly by permuting $1,2,3,4$.
$\ell^{X}_{ij}$ is a membrane operator acting in the plane orthogonal to $ij$, while $\ell^Z_{ij}$ acts in the plane $ij$.
This model, in fact, extends to a nontrivial finite temperature quantum phase of matter and forms a self-correcting quantum memory at finite temperature, in part due to the linear scaling of the energy barrier that needs to be overcome to implement any logical operator~\cite{Dennis2001}.
The 5D fractalized 4D toric code is described by the following Hamiltonian
\begin{align}
H_{\text{4DTC}}^{\mathrm{frac}} = - \sum_{\textbf{r},s}
\sigma_X[\mathrm{x}^{\textbf{r}} y^s \textbf{A}(\mathrm{f}( y) \circ\mathrm{x})] + \sigma_Z[\mathrm{x}^{\textbf{r}}y^s \textbf{B}(\mathrm{f}(\bar y) \circ \bar \mathrm{x})] \, ,
\end{align}
for the $\textbf{A}$ and $\textbf{B}$ matrices defined in Eqs.~\eqref{eq:4DTCa}~\&~\eqref{eq:4DTCb}, and some $\mathbb{F}_2$-polynomials $\mathrm{f}( y)=\{f_1(y),f_2(y),f_3(y),f_4(y)\}.$
On periodic boundary conditions $L_i,$ for $i=1 \dots 5$, such that $f_i(y)^{L_i}=1,$ for $i=1 \dots 4$, there are $6 L_5$ encoded qubits.
The logical operator pairs $\ell^{X/Z,\mathrm{frac}}_{ij,s}$ are given by fractalizing Eq.~\eqref{eq:4DTCl}.
Acting on the ground state with a truncated version of these operators, as in earlier examples, results in the creation of a fractalized loop excitation which appears loop-like when projected to a particular $(x_i,x_j)$ plane but extends non-trivially into the $y$ direction.
Following the same argument, we find that the requirement for non-existence of string-like excitations is that $f_{1\dots 4}$ satisfy a generalized algebraic relation where
either
\begin{equation}
f_1(y)^{n_1} = c~f_2(y)^{n_2} f_3(y)^{n_3} f_4(y)^{n_4}
\end{equation}
or
\begin{equation}
f_1(y)^{n_1} f_2(y)^{n_2} = c~f_3(y)^{n_3} f_4(y)^{n_4}
\end{equation}
can be satisfied for a set of finite constants $n_i,c$ (where not all $n_i=0$) or for any
permutation of $1,2,3,4$, without periodic boundary conditions.
Hence, for four algebraically unrelated polynomials $f_i$, $H_{\mathrm{4DTC}}$ lacks any string-like excitations.
This also implies the non-existence of any membrane logical operators, including those with dimension less than 2 which can be embedded within a membrane.
This is therefore a 5D code which lacks membrane logical operators,
a direct generalization of Type-II fracton models in 3D which are characterized by the non-existence of string-like logical operators.
We note that a 5D model without string-like excitations has appeared before~\cite{Haah5d}.
To find an explicit example we can take prime polynomials $f_i$.
For instance
\begin{align}
f_1(y)= 1+y+y^2 \, , \ \, &&
f_2(y)=1+y+y^3 \, ,\\
f_3(y) =1+y^2+y^3 \, , &&
f_4(y)=1+y+y^4 \, .
\end{align}
To achieve a more local model we can change our choice by the modification ${f_4(y)=1+y}$, at the cost of encoding fewer logical qubits for a given system size.
\subsection{Fractalized membrane excitations and beyond}
There is another toric code in 4D given by a direct generalization of the 3D toric code to include a further variable $x_4$. This model supports point-like and membrane-like excitations. Similar to the above examples it can be fractalized, leading to a model that supports ``fractalized membrane'' excitations.
In 6D there is a toric code supporting only membrane-like excitations, which again can be fractalized.
For an algebraically unrelated choice of fractalizing polynomials this produces a 7D model with no 3D (or lower) operators.
This follows from a dimensional reduction scheme generalizing our proof for lack of membrane logical operators.
The presence of a 3D logical operator in $H$ implies the existence of a 2D (membrane) excitation, which implies the existence of a 2D logical operator in its excitation Hamiltonian $H_{\mathrm{exc}}$ (as defined in Sec~\ref{sec:fracloop2d}).
This in turn implies the existence of 1D (string-like) excitations in $H_{\mathrm{exc}}$, which implies the existence of a 1D logical operator in \emph{its} excitation Hamiltonian $H_{\mathrm{exc}^2}$.
At this point, the proof of Ref~\cite{yoshida2013exotic} can be applied to show that string logical operators exist for $H_{\mathrm{exc}^2}$ iff $f_i$ are algebraically related.
The contrapositive of this chain of implications then means that for an algebraically unrelated choice of $f_i$, $H$ does not have any 3D logical operators.
More generally higher dimensional toric codes exist that can support excitations of any dimensionality which can then be fractalized leading to higher dimensional extended excitation generalizations of fracton particles.
In particular in $(2n+1)$D there is a fractalized toric code that is a generalized type-II fracton model in the sense that it supports no $2n$, or lower, dimensional logical operators.
Fractalizing higher dimensional models requires larger sets of algebraically unrelated polynomials to construct type-II models (where the algebraic independence condition is further generalized following the form of the above generalizations). These are easy to find by simply taking all the $f_i$ to be prime polynomials of low degree, as the uniqueness of polynomial factorization over a field guarantees algebraic independence.
\section{Application: Fractal Bacon-Shor subsystem code}
\label{sec:fsc}
\subsection{Fractal Bacon-Shor code}
In this section we focus on a subsystem code obtained by fractalizing the Bacon-Shor code~\cite{bacon} and Bravyi's generalization thereof~\cite{bravyi}. We investigate the code's quantum information storage capacity which we conjecture asymptotically saturates an information storage tradeoff bound~\cite{bravyi,flammia}.
In a subsystem code~\cite{poulin,bacon}, the full Hilbert space is structured as $\mathcal{H}=(\mathcal{H}_L\otimes \mathcal{H}_G)\oplus \mathcal{H}_E$, where $\mathcal{C}=\mathcal{H}_L\otimes\mathcal{H}_G$ is the codespace, composed of the logical and gauge subsystems, and $\mathcal{H}_E$ is an error subspace.
The information is stored in the state of the logical subsystem $\mathcal{H}_L$, while the state of the gauge subsystem can be arbitrary and is not protected.
A subsystem code is completely specified by its gauge group $\mathcal{G}$.
From $\mathcal{G}$, the stabilizer group $\mathcal{S}$ and logical operators follow.
In this section, we explain this structure for the Bacon-Shor (BS) code~\cite{bacon} in parallel with the fractalized Bacon-Shor (FBS) code obtained by applying fractalization to the BS gauge group generators.
We take the 2D BS code defined on an $L_1\times L_2$ lattice with one qubit per site and open boundary conditions.
The gauge group $\mathcal{G}$ is generated by products of two adjacent $X$s in the same row, or $Z$s in the same column,
\begin{align}
\mathcal{G} =& \langle \{X_{\mathbf{r}} X_{\mathbf{r}+\mathbf{x}}, Z_{\mathbf{r}} Z_{\mathbf{r}-\mathbf{y}}\}\rangle
\\
=&
\left\langle
\{\sigma_X[\mathrm{x}^\mathbf{r} (1-x_1)],\sigma_Z[\mathrm{x}^\mathbf{r}(1-\bar{x}_2)]\}
\right\rangle
\equiv \langle \{A_{\mathbf{r}}, B_{\mathbf{r}}\}\rangle
\, .
\nonumber
\end{align}
The Hamiltonian model based on these generators is known as the quantum compass model~\cite{Dorier2005}.
The minus signs, although irrelevant for qubits, are included as we later generalize to $\mathbb{Z}_p$ qudits.
We apply fractalization to the gauge group generators with a pair of $1$-dimensional LCA rules, $f_1(y),f_2(y)$.
The gauge group of the FBS code is
$\mathcal{G}^{\mathrm{frac}}=\langle \{{A}_{\mathbf{r}}^{\mathrm{frac}},{B}_{\mathbf{r}}^{\mathrm{frac}}\}\rangle$
where
\begin{equation}
\begin{split}
{A}_{\mathbf{r},s}^{\mathrm{frac}} =& \sigma_X\left[\mathrm{x}^\mathbf{r} y^s (1-f_1(y) x_1) \right]\, ,\\
{B}_{\mathbf{r},s}^{\mathrm{frac}} =& \sigma_Z\left[\mathrm{x}^\mathbf{r} y^s (1-f_2(\bar{y}) \bar{x}_2)\right] \, ,
\end{split}
\end{equation}
on an $L_1\times L_2\times L_3$ system, with periodic boundary conditions only along $L_3$. In the above,
$\mathbf{r}=(r_1,r_2)$ where $r_i\in R_i\equiv \{0\dots L_i-1\}$, and $s\in R_3$ with $s=L_3$ and $s=0$ identified.
The above local gauge generators define a family of Hamiltonians parameterized by the relative coupling strengths of the $X$ and $Z$ type generators, similar to the quantum compass model. Unlike stabilizer codes, such gauge code Hamiltonians may become gapless for certain values of the relative coupling strengths.
We remark that when the three-step process of fractalization holds exactly, then the code generated by $\mathcal{G}^{\mathrm{frac}}$ is unitarily related to the stack of $L_3$ BS codes.
Code properties that are unaffected by unitary transformations, such as the number of encoded qubits, are therefore equivalent to those of $L_3$ BS codes.
Other properties, such as the code distance which is defined below, are affected by the unitary transformation.
The stabilizer group $\mathcal{S}$ is obtained as the center of $\mathcal{G}$, $\mathcal{S}=C(\mathcal{G})\cap \mathcal{G}$, where $C(\cdot)$ is the centralizer in the Pauli group.
The codespace $\mathcal{C}$ is the simultaneous $+1$ eigenspace of all operators in $\mathcal{S}$.
For the BS code, $\mathcal{S}$ is generated by the the product of $X$ along two adjacent columns, or $Z$ along two adjacent rows,
\begin{equation}
\begin{split}
S^X_{r_1} =& \sigma_X\left[x_1^{r_1} (1-x_1) \sum_{r_2} x_2^{r_2}\right]\\
S^Z_{r_2} =& \sigma_Z\left[x_1^{L_1-1}x_2^{r_2} (1-\bar{x}_2) \sum_{r_1} \bar{x}_1^{r_1}\right]
\end{split}\,,
\end{equation}
where $S^{X}_{r_1}$ is defined for $r_1\in R_1\setminus \{L_1-1\}$, and $S^{Z}_{r_2}$ is defined for $r_2\in R_2\setminus \{0\}$ (due to the boundaries).
The generators of the stabilizer group of the FBS code, $\mathcal{S}_{\mathrm{frac}}$, is then
\begin{equation}
\begin{split}
S^{X,\mathrm{frac}}_{r_1,s} =& \sigma_X\left[ x_1^{r_1} y^s (1-f_1(y) x_1)\sum_{r_2}f_2(y)^{r_2} x_2^{r_2}\right]\, ,\\
S^{Z,\mathrm{frac}}_{r_2,s} =& \sigma_Z\left[x_1^{L_1-1}x_2^{r_2} y^{s}(1-f_2(\bar{y}) \bar{x_2}) \sum_{r_1}f_1(\bar{y})^{r_1} {\bar{x}_1}^{r_1}\right]\, .
\end{split}\label{eq:Sfrac}
\end{equation}
for all $s\in R_3$.
These operators act on two adjacent rows or columns when projected on to $(x_1,x_2)$, like those of the BS code, but spread out non-locally into a fractal along the $y$ direction.
The codespace is further decomposed into the logical and gauge subsystems, $\mathcal{C}=\mathcal{H}_L\otimes\mathcal{H}_G$.
The gauge operators act on the codespace as $\mathbb{1}_L\otimes G_G$.
Logical operators are those that act non-trivially in $\mathcal{H}_L$, and come in two types: bare and dressed logical operators.
Bare logical operators act as $ \ell_L\otimes\mathbb{1}_{G}$.
Dressed logical operators are products of bare logical operators and gauge operators, and therefore act on the codespace as $\ell_L\otimes G_G$.
The set of bare logical operators is defined as $L_{\mathrm{bare}} = {C}(\mathcal{S}) \setminus \mathcal{G}$.
Bare logical operators of the BS code are products of $X$ along an odd number of columns and/or $Z$ along an odd number of rows.
Let us focus on the simplest ones, consisting of products along a single row/column, which we choose to be
\begin{equation}
\begin{split}
\ell^X = \sigma_X\left[ x_1^{L_1-1} \sum_{r_2} x_2^{r_2}\right] \, ,\,
\ell^Z = \sigma_Z\left[ x_1^{L_1-1} \sum_{r_1} {\bar{x}_1}^{r_1} \right]\, ,
\end{split}
\end{equation}
and $\ell^Y=i\ell^X\ell^Z$,
from which all other logical operators can be obtained by multiplying with $\mathcal{S}$.
Analogously, the corresponding bare logical operators for the FBS code are
\begin{equation}
\begin{split}
\ell^{X,\mathrm{frac}}_s = \sigma_X\left[x_1^{L_1-1} y^s\sum_{r_2}f_2(y)^{r_2} x_2^{r_2}\right] \, ,\\
\ell^{Z,\mathrm{frac}}_s = \sigma_Z\left[x_1^{L_1-1} y^s\sum_{r_1}f_1(\bar{y})^{r_1} {\bar{x}_1}^{r_1} \right]\, ,
\end{split}\label{eq:fraclog}
\end{equation}
of which there are now $L_3$ independent $X$ and $Z$ type bare logical operators.
These satisfy the commutation relation $[[\ell^{X,\mathrm{frac}}_{s_0} ,\ell^{Z,\mathrm{frac}}_{s_1}]] = (-1)^{\delta_{s_0,s_1}}$, and thus generate an $L_3$-dimensional Pauli algebra on $\mathcal{H}_L$.
This code therefore protects $L_3$ logical qubits, as expected from the three-step interpretation of fractalization.
Dressed logical operators are then obtained as products of bare logical operators and gauge operators.
The code distance $d$ is defined to be the minimum support of a non-trivial dressed logical operator.
Let us take all $L_i\sim L$.
The minimum weight of bare logical operators of the FBS code scale as $d\sim L^{\mathrm{min}(d_1,d_2)}$ where $d_{1,2}$ are the Hausdorff dimensions of the fractal generated by $f_{1,2}$~\cite{yoshidaclassical}.
However, estimating the minimum weight of dressed logical operators is a difficult optimization problem in general.
This is because it is generally possible to reduce the support of a bare logical operator by multiplying it with gauge operators.
Let us assume that $d\sim L^{\eta}$, where $\eta$ is the scaling dimension of the minimum weight dressed operator.
If $f_{1,2}$ are algebraically related, then it is straightforward to find a dressed logical operator scaling with $\eta=1$.
However, if they are algebraically unrelated, we expect that $1<\eta\leq\mathrm{min}(d_1,d_2)\leq 2$.
The simplest example of two algebraically unrelated polynomials is $f_1(y)=1+y$ and $f_2(y)=1+y+y^2$.
We have taken periodic boundary conditions along the third dimension, which may be difficult to implement in systems with locality constraints.
This can be easily circumvented.
With open boundaries, we choose to keep all the the local gauge generators $A^\mathrm{frac}_{\mathbf{r},s}$ and $B^\mathrm{frac}_{\mathbf{r},s}$ with $s\in R_3$, truncating them at the edge.
With this choice of gauge generators, the logical operators are given by Eq.~\ref{eq:fraclog} with $s\in R_3$, also truncated at the edge.
As a result, logical operators near the boundary have reduced weight:
$\ell^{Z,\mathrm{frac}}_{0}= \prod_{r_1} Z_{(r_1,0)}$ is the minimal weight logical operator with support on only $L_1$ sites, and so $d = L_1$.
To avoid this, we can increase the system size
$L_3\rightarrow L_3+ 2\delta L$
with
\begin{equation}
\delta L = L_1 \mathrm{deg}(f_1) + L_2 \mathrm{deg}(f_2)
\end{equation}
where $\mathrm{deg}$ is the degree function, and use only the central $L_3$ logical qubits, $\{\ell^{X/Z,\mathrm{frac}}_s\}_{\delta L \leq s < L_3+\delta L}$ to store information.
The logical operators for qubits within this range are unaffected by the boundaries, and the overall scaling of the code parameters with $L$ remains the same.
\subsection{Information storage capacity and the fractal Bravyi-Bacon-Shor code}
We now discuss the information storage capacity of the fractal subsystem code.
The 2D BS code with ${n=L^2}$ physical qubits can encode $k=1$ qubits of quantum information with a code distance $d= L$.
The $[n,k,d]$ parameters of subsystem codes with a locally generated gauge group must satisfy the information storage tradeoff bound~\cite{bravyi,flammia}
\begin{equation}
k d^{\frac{1}{D-1}} \leq O(n)\label{eq:subbound}
\end{equation}
(Ref.~\cite{bravyi} only derived the bound in $D=2$, but the proof can be extended to higher dimensions as written down in Ref.~\cite{flammia}).
Bravyi's generalization~\cite{bravyi} of the BS code, henceforth referred to as the Bravyi-Bacon-Shor (BBS) code, saturates this bound in 2D.
The 2D BBS code is parameterized by a binary matrix $K_{\mathbf{r}}$, which specifies a set of sites $K_\mathbf{r}=0$ to be removed from the 2D lattice.
The gauge generators are modified accordingly to skip over the removed sites, i.e. $X_{\mathbf{r}} X_{\mathbf{r}+n\mathbf{x}}$, where $\mathbf{r}$,$\mathbf{r}+n\mathbf{x}$ are two nearest unremoved sites along that row.
Although this may induce long range couplings, additional auxiliary qubits may be introduced to ensure all generators are local~\cite{bravyi}.
The end result is that logical operators from different rows or columns can no longer generically be related to one another by $\mathcal{S}$, and the number of logical qubits is given by $k=\mathrm{rank}(K)$.
Furthermore, it is possible~\cite{bravyi} to find a family of matrices $K_\mathbf{r}$ such that
$k\sim L$, while maintaining $n\sim L^2$ and $d\sim L$.
Thus, $[n,k,d]\sim[L^2,L,L]$ saturates the 2D bound in Eq.~\ref{eq:subbound}, $k d \sim L^2 \leq \mathcal{O}(L^2)$.
In 3D, however, the current best known subsystem code does not quite saturate the scaling in the tradeoff bound~\cite{baconflammia}.
The FBS code has $n=L^3$, $k=L$, and $d\sim L^{\eta}$ for some $1\leq \eta \leq 2$ as discussed previously.
This is still far from saturating the 3D tradeoff bound, $k\sqrt{d}\sim L^{1+\eta}\leq \mathcal{O}(L^3)$.
To further improve the quantum information storage capacity, we may apply fractalization to the BBS code, resulting in the fractal BBS (FBBS) code.
Removing a single site $K_\mathbf{r}=0$ from the original model corresponds to removal of a whole line of sites $\{(\mathbf{r},s)\}_{s}$ from the fractal model.
The number of logical qubits is improved to $k=L \,\mathrm{rank}(K)$.
Again, there is a choice of binary matrices such that the parameters scale as $[n,k,d]\sim [L^3,L^2,L^{\eta}]$ (for a potentially different $\eta$).
Thus, the FBBS code nearly saturates the tradeoff bound depending on the value of $\eta$,
$k\sqrt{d}\sim L^{2+\eta}\leq O(L^3)$.
Next, we may further generalize beyond qubits to qudits of prime dimension $p$.
In the limit of large $p$, the Hausdorff dimensions of the generated fractals approach $d_{1,2}\rightarrow 2$ (this was used by Yoshida~\cite{yoshidaclassical} to asymptotically saturate the information storage bound in a classical spin system).
We conjecture that in the limit $p\rightarrow\infty$, for $f_1,f_2$ algebraically unrelated polynomials, the code distance scaling parameter also approaches $\eta\rightarrow 2$.
The FBBS code is therefore conjectured to asymptotically saturate the 3D information storage bound $k\sqrt{d}\sim n$ in the limit of large $p$.
We may also generalize to higher dimension.
Suppose we wish to apply fractalization to a code from ${D\rightarrow D+m}$ dimensions.
Let us start with an $[n,k,d]$ subsystem code on an $n=L^D$ system, and apply fractalization via the three-step process as described in Sec~\ref{sec:interp}.
The first step is to construct a layered system, in which the code now has parameters $[L^m n, L^m k, d]$.
The second step is a different choice of gauge generators, which does not affect any parameters of the code.
Finally, the third step is a non-local unitary transformation.
This does not affect $n$ or $k$, but does affect the distance, which now scales as $d\sim L^{\eta}$, where $1\leq \eta \leq m+1$.
Thus, starting with the 2D BBS code, the $2+m$ dimensional FBBS code has parameters $[n,k,d]\sim [L^{m+2},L^{m+1},L^{1\leq \eta \leq m+1}]$.
The analogue of our conjecture in higher dimensions would be that, for certain choices of LCA rules and $\mathbb{Z}_p$ qudits, $\eta\rightarrow m+1$ in the limit $p\rightarrow\infty$.
In this case, the tradeoff bound
\begin{equation}
k d^{\frac{1}{m+1}}\sim L^{m+1+\eta/(m+1)}\xrightarrow[\eta\rightarrow m+1]{p\rightarrow\infty} L^{m+2} \leq O(L^{m+2})
\end{equation}
is saturated in the large $p$ limit in any dimension $m+2$.
\section{Conclusions}
\label{sec:conclusion}
We have introduced the fractalization procedure, by which certain spin models can be extended into higher dimensional fractal models.
Many exotic fractal models, such as type-II fracton models or fractal SPTs, may be understood as fractalized versions of familiar spin models, such as the toric code or the cluster model, as we have discussed in detail with many examples.
We provided an interpretation of fractalization in terms of a three-step process, which elucidates how the properties of the fractal model are inherited from those of the original model.
Fractalizing models with loop-like or higher dimensional excitations, for example, leads to exotic extended fractal excitations.
We have introduced the fractal Bacon-Shor subsystem code which was, after applying Bravyi's generalization~\cite{bravyi} and in the limit of large on-site Hilbert space dimension~\cite{yoshidaclassical}, conjectured to saturate the information storage tradeoff bound in 3D.
A more detailed analysis of the code properties of the fractal code is important, but beyond the scope of this current paper.
This would include a numerical estimation of the code distance, especially in the limit of large $p$, as well as finding and implementing error correction algorithms. Due to the presence of non-local stabilizers, similar to the BS code, we expect that the fractalized BS code does not give rise to a positive threshold for local noise, without further modifications.
There are still many open directions regarding the fractalization procedure.
Currently, fractalization can only be applied to purely $X$ or $Z$ operators while preserving commutativity;
it would be interesting if a generalization beyond such operators is possible.
An intriguing potential application would then be the fractalization of Kitaev's honeycomb model~\cite{kitaev2006anyons} in the non-Abelian Ising phase to possibly obtain a non-Abelian type-II fracton model.
Other possibilities would be generalizing beyond translation-invariant LCA, perhaps to quantum cellular automata.
These would allow fractalization to be applied to more complicated spin models.
Unfortunately, the present formalism in this paper does not readily admit such generalizations.
Another question is whether there exists an interpretation, similar to the three-step interpretation, of fractalization with higher-order LCA.
Such an interpretation would greatly improve our understanding of type-II fracton phases described by higher-order LCA, such as Haah's cubic code.
There are also many more CSS codes to which fractalization can be applied, particularly in higher dimensions.
In this work we considered fractalizing 3D and 4D toric codes, the latter lead to 5D models with no membrane operators that are, in some sense, a generalization of the type-II fracton models that exist in 3D.
Beyond this, each other $D$ dimensional toric code or type-I fracton model defines a family of $D+m$ dimensional fracton models with fractal logical operators and excitations beyond the examples covered in the current work.
\acknowledgements
TD acknowledges support from the Charlotte Elizabeth Procter Fellowship at Princeton University.
DW acknowledges support from the Simons foundation.
|
1,108,101,563,266 | arxiv | \section{Introduction}
Active galactic nuclei (AGN) are believed to be powered by accretion of gas
onto the black hole located at the centre of galaxies.
The AGN bolometric luminosity (L$_{b}$) depends on the mass accretion rate and on the efficiency for the conversion of gravitational energy into radiation.
In this scenario, the luminosity produced by a black hole has a physical
limit, the Eddington limit
(L$_{Edd}$$\simeq$1.3$\times$10$^{38}$(M$_{BH}$/M$_{\odot}$) erg/s), at which
the radiation pressure due to the accretion of the infalling matter balances
the gravitational force of the black hole.
The ratio between the bolometric and Eddington luminosity, $\lambda$=
L$_{b}$/L$_{Edd}$, is referred to as the Eddington ratio.
The knowledge of the Eddington ratio distribution and its evolution is very
important to constrain the predictions of theoretical models which link the evolution of the galaxies in the hierarchical clustering
scenario with the quasar evolution (e.g. Menci et al. 2003, 2004). In fact,
all predictions regarding AGN, such as the luminosity function or the black hole mass
function of AGN relics, depends on the assumptions about this quantity. The
estimate of the Eddington ratio
requires measurements of the black hole
mass, M$_{BH}$, and of the bolometric luminosity, L$_{b}$.
According to the unification model (Antonucci 1993), in the AGN classified as
type 1 the optical/UV continuum source and the surrounding broad emission line
region (BLR) are viewed without any substantial obscuration, while in type 2 AGN these regions suffer obscuration along the
observer's line of sight by intervening dusty material, most of which is
likely associated with a circumnuclear structure often referred to as the
torus. Even if some exceptions have been found (e.g. Bianchi et al. 2008),
the unification model can be safely assumed to provide a useful reference
frame for the vast majority of sources.
Progress in reverberation mapping of type 1 AGN (Peterson et al. 2004, 2005)
provided rather good evidence that the BLR size depends on the optical
continuum luminosity (Kaspi et al. 2000, 2005; Bentz et al. 2006).
This size-luminosity relation, combined with the gas
velocity derived from the width of the emission lines, under the assumption
that the BLR clouds are in Keplerian motion around the black hole, has been
widely used to estimate the black hole masses, in large samples of type 1 AGN
up to z$\simeq$4 (e.g. Woo \& Urry 2002; McLure \& Dunlop 2004; Warner et
al. 2004; Kollmeier et al. 2006; Vestergaard 2002, 2004; Netzer \&
Trakhtenbrot 2007). The Eddington ratio is then obtained by applying a bolometric
correction to the optical luminosity. These authors seemed to agree on a
practically constant value of
$\lambda$ ($\lambda$$\simeq$0.25 Kollmeier et al. 2006) at all z, irrespective of luminosity and black hole mass, after selection
effects had been properly taken into account (but see Netzer \&
Trakhtenbrot 2007 for a recent result indicating a dependence of $\lambda$ on both M$_{BH}$ and $z$ up to $z$ $\simeq$0.75).\\
In type 2 AGN the BLR is not visible and the M$_{BH}$ estimate is usually obtained through the empirical
relationships between the black hole mass and the properties of the spheroidal
component of the host galaxy (e.g. Magorrian et al. 1998; Gebhardt et
al. 2000a; Ferrarese $\&$ Merritt 2000; Tremaine et al. 2002). This method has
been applied to a very large sample of type 2 AGN in the local Universe
(Kauffmann et al. 2003; Heckman et al. 2004 here-in-after H04; Kauffmann \& Heckman 2008). At higher $z$, however this
method becomes more difficult to apply, especially in spiral galaxies where it
is necessary to disentangle the contribution of the bulge from that of the
disk. A step forward in this direction has been recently made by Bian et
al. (2006, here-in-after B06), which were able to estimate the black hole masses for a
subsample of SDSS type 2 AGN at 0.3$<$ $z$ $<$0.83 from the Zakamska et
al. (2003) sample. The black hole mass estimate was obtained from the stellar
velocity dispersion measurements ($\sigma_{\ast}$) and the M$_{BH}$-$\sigma_{\ast}$ relation of Tremaine et
al. (2002).
In type 2 AGN, since the primary optical/UV radiation is much affected
by dust extinction, the bolometric luminosity is generally inferred from the [OIII] emission line luminosity (L$_{OIII}$).
The [OIII] emission line arises in the much larger narrow line region (NLR),
where the gas is photoionized by the continuum radiation escaping within the opening angle of the torus. The observed flux is therefore
weakly affected by the viewing angle relative to the torus and its luminosity
provides an indication of the nuclear luminosity. The key issue, in order to
estimate $\lambda$ in these AGN, is therefore to have a
bolometric correction factor in hand to be applied to L$_{OIII}$.\\
H04 computed a mean bolometric correction factor to the
observed [OIII] luminosity, and used it to estimate the Eddington ratio
distribution of local (z$\simeq$0.1) type 2 AGN. This bolometric correction factor was also used by B06 to estimate the
$\lambda$-distribution of their sample. B06 found a mean
value of this distribution $\langle \lambda \rangle$ $\simeq$1, which would reveal a population
of high Eddington ratio sources which differs from that observed in the local
Universe and in type 1 AGN. If confirmed, this result would have enormous
impact on the way that massive black holes grow along cosmic time, as most of
this growth as expected to occur at significant redshift and in obscured objects.\\
However, L$_{OIII}$ is only an indirect estimator of the nuclear luminosity,
which depends on the geometry of the system, on the amount of gas, and on any
possible shielding effect which may affect the ionizing radiation seen by the NLR.
This implies an intrinsic scatter in the L$_{OIII}$-L$_{b}$ relation and therefore an uncertainty in the $\lambda$ estimate.
A further complication arises from the fact that the [OIII] emission line is affected by extinction, likely due to dust within the NLR.
In some cases the reddening due to the NLR can be evaluated, and the extinction-corrected [OIII] luminosity (L$^{c}_{OIII}$) is a more direct estimator
of the nuclear luminosity than L$_{OIII}$. We estimated the [OIII] bolometric
correction following a similar method adopted by H04 , but we used the extinction-corrected [OIII] luminosity instead of
the observed one.
Then we
used this bolometric correction to calculate the Eddington ratio of the sources from the B06 sample for which the extinction-corrections are available.
Moreover, we have observed with XMM-{\it Newton} 8 sources of the B06 sample,
in order to have an independent estimate
of the bolometric luminosity through X-ray luminosity.
The paper is organised as follows.
In Sect. 2 the bolometric correction estimate is described. Sect. 3 is devoted
to the estimate of the Eddington ratio distribution. Discussion and
conclusions follow in Sect. 4.
\section{The L$_{X}$-L$^{c}_{OIII}$ luminosity relation}
H04 estimated the bolometric
correction (BC) to the observed [OIII] luminosity, L$_{OIII}$, in a two-step process. First they estimated the mean ratio between the monochromatic
continuum luminosity at 5000 $\AA$ and the [OIII]
luminosity, L$_{5000}$/ L$_{OIII}$, in a sample of type 1 AGN. Then they calculated the mean ratio
between the bolometric luminosity and L$_{5000}$ using the average type 1 AGN
intrinsic spectral energy distribution (SED) of Marconi et al. (2004).
They found L$_{b}$/L$_{OIII}$$\simeq$3500. As discussed in H04,
they did not correct L$_{OIII}$ for dust extinction in the NLR,
because the extinction correction is usually made by measuring the observed Balmer
decrement (H$\alpha$/H$\beta$), and in type 1 AGN this procedure requires deblending the narrow
components from the broad ones.
In Seyfert galaxies optically classified as type 1.5, 1.8, 1.9 and 2, in
which the narrow components of the H$\alpha$ and
H$\beta$ lines dominate the line profiles (Osterbrock \& Ferland 2006), the estimate of the
Balmer decrement can instead be considered reliable.
So we used these Seyfert types to estimate the BC to convert the
extinction-corrected [OIII] luminosity, L$^{c}_{OIII}$, to the bolometric
luminosity.
In these Seyfert types the bolometric luminosity
cannot be easily determined because the primary optical/UV radiation is highly obscured.
In X-rays, however, the nuclear emission could be directly estimated,
provided that the absorbing matter is Compton-thin (i.e. $N_H$ $<$
$\sigma_T^{-1}=1.5\times 10^{24}$ cm$^{-2}$; if the matter is Compton-thick,
no nuclear radiation is visible below 10 keV, with the Compton reflection
component usually dominating the spectrum).
Assuming that the X-ray bolometric correction is the same in type 1 and in
type 2 AGN, we used the relation between L$^{c}_{OIII}$ and the absorption
corrected X-ray luminosity and the X-ray
bolometric correction of type 1 and Compton-thin type 2
AGN to estimate the [OIII] bolometric correction (C$_{OIII}$).\\
We have collected a sample from the literature for which reliable estimates of the Balmer decrement, L$_{OIII}$ and L$_{X}$ were available. We have discarded Seyfert galaxies with optical classification $\le$ 1.2 and X-ray Compton-thick sources.
The final sample consists of 61 sources: 5 from Mulchaey et al. (1994)\footnote{We have used L$_{OIII}$ and Balmer decrement from Mulchaey et al. (1994) and the X-ray
luminosities from Bianchi et al. 2009a.}, 12 from Panessa et al. (2006, P06
hereafter), and 44 from Bassani et al. (1999, B99 hereafter, 8 of these 44 sources
are also included in the P06 sample).
We used the B99 relation to derive L$^{c}_{OIII}$ from the Balmer decrement:\begin{equation}\label{bassani}
L^{c}_{OIII}=L_{OIII}\left(\frac{(H\alpha/H\beta)_{obs}}{3.0}\right)^{2.94}.
\end{equation}
which assumes an intrinsic Balmer decrement equal to 3.0 as expected in the NLR
(see Osterbrock \& Ferland 2006).\\
In Fig. \ref{oiiix} (top panel) the distribution of the ratio between the
(2-10) keV luminosity and the extinction-corrected [OIII] luminosity for our sample is shown. The mean of this distribution is
\begin{equation} \label{pan}
\langle \log \left( \frac{L_{X}}{L^{c}_{OIII}} \right)\rangle=1.09
\end{equation}
and the dispersion is 0.63 dex.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=8 cm]{fig1a.ps}
\includegraphics[width=8 cm]{fig1b.ps}
\caption{Distribution of log(L$_{X}$/L$^c_{OIII}$) (top) and log L$_{X}$-log
L$^{c}_{OIII}$ relation (bottom). The solid line is
the best fit relation Eq. (\ref{fit}) for the 61 sources from Mulchaey et
al. (1994) (crosses) , B99 (squares) and P06 (triangles), while the dotted
line is the best fit relation obtained by P06.}
\label{oiiix}
\end{center}
\end{figure}
The bottom panel of Fig. \ref{oiiix} shows L$_{X}$ versus L$^c_{OIII}$. The
correlation is highly significant\footnote{We consider statistically
significant the correlations whose null hypothesis probability, P, is lower
than 1$\times$10$^{-3}$, which correspond to a confidence level of
99.9$\%$.} (Spearman rank correlation coefficient $\rho$=0.86,
P=1.6$\times$10$^{-18}$). In order to quantify any distance-driven
effects which may be present in a luminosity-luminosity correlation, we also
performed a ``scrambling test'' (see e.g. Bregman 2005; Merloni et al. 2006;
Bianchi et al. 2009b). The correlation appears not to be affected by distance effects:
not even a data-set out of the 100000 simulated ones has a correlation
coefficient as large as the one measured in the real data-set.
An ordinary least-square bisector procedure gives
the following best fit:
\begin{equation} \label{fit}
\log\left(\frac{L_{X}}{10^{42}erg/s}\right)=(1.11\pm0.10)+(1.02\pm0.06)\log\left(\frac{L^{c}_{OIII}}{10^{42}erg/s}\right)
\end{equation}
i.e. we found a linear correlation between L$_{X}$ and L$^{c}_{OIII}$ .\\
P06 investigated the same correlation finding that L$_{X}$$\propto$L$^c_{OIII}$$^{(1.22\pm0.06)}$.
They included in their analysis type 1 AGN, type 2 AGN and
Compton-thick sources (in the latter case they assumed that the intrinsic X-ray
luminosity is 60 times larger than the observed one) and covered about five orders
of magnitude in both [OIII] luminosity and X-ray luminosity.
It should be recalled that the P06 sample is optically
selected while our sample, mainly based on the B99 catalogue, includes both optically selected and
X-ray selected AGN. As discussed in several papers, most recently in Heckman et
al. 2005 and Netzer et al. (2006, N06 hereafter), X-ray selected samples are likely to be biased towards X-ray bright
sources and hence produce a larger X/[OIII] ratio compared to
optically selected ones. The 44 sources which we have selected from the B99
catalogue have all log L$^{c}_{OIII}$ $>$39.3. The presence of X-ray selected
objects among these sources should therefore tend to make the
L$_{X}$-L$^{c}_{OIII}$ relationship steeper than found by P06, which, as it
can also be seen in Fig. \ref{oiiix}, is exactly the opposite of what we found.\\
On the other hand, the high luminosity ranges in P06 are populated nearly completely by type 1 AGN, for
which we consider unreliable the estimate of L$^c_{OIII}$. The steeper slope
of the L$_{X}$-L$^{c}_{OIII}$ relation found by P06 could indeed
be due to an underestimate of the reddening in these AGN types.\\
N06 studied the relation between L$_{OIII}$
and L$_{X}$ in samples of type 1 and type 2 AGN. They found that the ratio
between L$_{OIII}$ and L$_{X}$ decreases with L$_{X}$, and for type 2 AGN the
same result was obtained also using the extinction-corrected [OIII]
luminosity. In the latter case, they used the B99 catalogue excluding sources
with optical classification $\le$ 1.5 and Compton-thick sources. They found
log(L$^{c}_{OIII}$/L$_{X}$)=(15.0$\pm$4.0)-(0.38$\pm$0.09)log L$_{X}$, which
implies L$_{X}$ $\propto$ L$^c_{OIII}$$^{1.61}$, i.e. a correlation with an even steeper slope
than that obtained by P06.
Our analysis differs from that of N06 only for the inclusion of the AGN
optically classified as type 1.5. Excluding these sources from our sample, we found
for the remaining 52 sources:
$\log(L_{X}/10^{42}erg/s)=(1.08\pm0.11)+(1.04\pm0.06)\log(L^{c}_{OIII}/10^{42}erg/s)$ ($\rho$=0.85,
p=2.0$\times$10$^{-15}$ ), i.e. a significant linear correlation also in this
case. The discrepancy between the results may be due to the smaller range of
luminosities covered by N06, which do not include sources with L$_{X}$ $\le$ 10$^{40}$erg/s. Unfortunately we
cannot make a clear comparison with N06 because they did not perform a simple
L$_{X}$-L$^{c}_{OIII}$ fit and an extrapolation of the resulting slope from their fit may be
misleading. Moreover N06 did not explicitly report the method they used to compute the
linear regression and the significance of the correlation they found.\\
The relation between L$_{X}$ and L$^c_{OIII}$ can be combined with
the X-ray bolometric correction (XBC) to obtain the [OIII] bolometric correction factor,
C$_{OIII}$. Adopting the luminosity-dependent XBC of Marconi et al. (2004), we found a
mean value of C$_{OIII}$ in the luminosity ranges log L$_{OIII}$=38-40,
40-42, and 42-44 of 87, 142 and 454 respectively.
As a cautionary remark, it must be noted that the XBC may suffer from large
uncertainties.
Marconi et al. (2004) and Hopkins et al. (2007) accounted for variations
of AGN SEDs with luminosity using the anti-correlation between the optical-to-X-ray spectral index
($\alpha_{OX}$) and the luminosity at 2500 $\AA$ (L$_{2500}$) (e.g. Vignali
et al. 2003a), but
Hopkins et al. (2007) found that intrinsic spread in AGN SEDs could give
rise to variation of a factor of $\sim$ 2 in the XBC, even when luminosity
dependence is taken into account.\\
Moreover, the very dependence on luminosity of the X-ray bolometric correction
has been questioned by Vasudevan \& Fabian (2007), who estimate it from the observed SEDs of 54 AGNs.
They found evidence of a large scatter in the XBC,
with no simple dependence on luminosity, and suggested instead the Eddington ratio
as the physical quantity the bolometric correction depends on. \\
Very recently, Kauffmann \& Heckman (2008) anticipated an estimate of C$_{OIII}$.
They found C$_{OIII}$ in the range $\sim$ 500 to 800. Because the details of their
analysis have not been published yet, for the time being we are not able
to discuss the origin of this discrepancy.
\section{The Eddington ratio distribution of type 2 SDSS AGN at 0.3$<z<$0.4}
B06 measured the stellar velocity dispersion, $\sigma_{\ast}$, for 30 sources
of the 0.3$<z<$0.8 SDSS type 2 AGN sample of Zakamska et al. (2003). The
stellar velocity dispersion was obtained by fitting the profile of heavy
element absorption lines which are mainly associated with the old stellar
population in the bulge. Then, they estimated the black hole masses from
$\sigma_{\ast}$, adopting the Tremaine et al. (2002) relation.\\
The B06 subsample is representative of the total
sample of Zakamska et al. (2003) with respect to L$_{OIII}$. Indeed, the observation of significant stellar absorption features, which is the key point to accurately measure $\sigma_{\ast}$, does not appear to depend on the nuclear luminosity of the sources, as confirmed by a t-test (see B06).
For 15 sources B06 calculated
the [OIII] line luminosity corrected for extinction using the observed Balmer
decrement and Eq. (\ref{bassani}). Therefore all these sources have $z$ $\leq$ 0.4, because of the limited
wavelength coverage of the SDSS spectroscopy ($\simeq$3800-9200 $\AA$).
The ratio between L$^{c}_{OIII}$ and L$_{OIII}$, is between 0.6 and 17.8. In three objects the observed Balmer decrement
is lower than the theoretical value, most likely due to an imperfect
subtraction of the starlight.
We assigned zero extinction to these objects.
Figure 4 in B06 shows the distribution of the Eddington ratios of the 15
sources where we have a reliable estimate of L$^{c}_{OIII}$. This distribution is surprising
for its mean value, $\langle \lambda \rangle$ $\simeq$ 1, because it would reveal a population of
high Eddington ratio sources which is not observed in the local Universe, such
Eddington ratios are neither observed in type 1 AGN at higher redshift.
However, it is important to note that B06 obtained this distribution estimating the bolometric luminosity from L$^{c}_{OIII}$ and the BC given by H04. This bolometric correction applies to L$_{OIII}$ and not to L$^{c}_{OIII}$, therefore B06 overestimated in a systematic manner L$_{b}$ and hence $\lambda$.\\
In this section we used C$_{OIII}$, as computed in Sect. 2, to calculate the Eddington ratio distribution of the
type 2 SDSS AGN of the B06 sample.
First of all we have tested
the reliability of deriving $\lambda$ from the [OIII] line luminosity in an
optically selected sample, adopting the C$_{OIII}$ we derived in a collection
of X-ray selected and optically selected sources.
To this aim we used the P06 subsample used in Sect. 2.
In the top panel of Fig. \ref{lambda} the Eddington ratio estimates obtained from L$^{c}_{OIII}$
and C$_{OIII}$ ($\lambda_{OIII}$) are compared with those obtained from L$_{X}$ and the XBC of Marconi et
al. 2004 ($\lambda_{X}$) (the black hole masses are taken from Table 2 of
P06). We found a good agreement between the X-ray and the [OIII] estimates.\\
For the 8 SDSS AGN of the B06 sample for which we have
XMM-{\it Newton} observations we can repeat the above check.
The bottom panel of Fig. \ref{lambda} shows $\lambda_{X}$ versus $\lambda_{OIII}$ obtained from L$^{c}_{OIII}$
and C$_{OIII}$ (crosses) and versus $\lambda_{OIII}$ obtained by B06 (squares).
As expected, the $\lambda_{OIII}$ obtained by B06 are systematically
larger than those derived from the X-ray luminosity, while this effect
disappears when the proper bolometric correction is used.
The black hole mass, the [OIII] line luminosity, the extinction-corrected [OIII] line luminosity, the (2-10) keV observed flux (F$^{obs}_{X}$) and the intrinsic (i.e. de-absorbed) (2-10) keV luminosity (L$^{int}_{X}$) of the X-ray observed sources are listed in Table \ref{SDSS_source}. The X-ray data reduction and the estimate of L$^{int}_{X}$ either from X-ray spectral fitting or extrapolated from the observed X-ray count rates (or upper limits) are reported in the Appendix.\\
\begin{figure}[h!]
\begin{center}
\includegraphics[width=6.5 cm]{fig2a.ps}
\includegraphics[width=6.5 cm]{fig2b.ps}
\caption{Top: $\lambda_{X}$ vs. $\lambda_{OIII}$ for the sources in the P06
subsample. Bottom: $\lambda_{X}$ vs. $\lambda_{OIII}$ obtained by B06 (squares) and from
L$^{c}_{OIII}$ and C$_{OIII}$ (crosses), for the B06 sources
with the XMM-{\it Newton} observations.}
\label{lambda}
\end{center}
\end{figure}
Figure \ref{histo_Bian} shows the $\lambda$-distribution of the B06 sample
obtained using L$^{c}_{OIII}$ and C$_{OIII}$ to estimate the bolometric
luminosity. This distribution is obviously shifted to lower $\lambda$ values
with respect to that obtained by B06: the mean of this distribution is
$\langle \lambda \rangle$$\simeq$0.1, a value which is similar to that
estimated in other AGN samples.\\
We have checked whether the width of the observed $\lambda$-distribution
indicates a physical spread in the Eddington ratios or it can be due
to the dispersion in the L$^{c}_{OIII}$- L$_{b}$ relation. In practice,
we checked if the observed distribution is consistent with a log-normal
distribution centered at log$\lambda$=-1 and with width equal to the
dispersion in the L$_{X}$-L$^{c}_{OIII}$ relation (0.63 dex). A
Kolmogorov-Smirnov test gives D=0.27, which
corresponds to a probability of 0.18 for the null hypothesis that the two
distributions are drawn from the same parent population.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=7 cm]{fig3.ps}
\caption{Eddington ratio distribution of the B06 sample which is obtained estimating $\lambda$ from L$^{c}_{OIII}$ and C$_{OIII}$.}
\label{histo_Bian}
\end{center}
\end{figure}
\begin{table*}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Source & M$_{BH}$ &L$_{OIII}$ &L$^{c}_{OIII}$ & F$^{obs}_{X}$& L$^{int}_{X}$ \\
(1)&(2)&(3)&(4)&(5)&(6)\\
\hline
SDSSJ075920.21+351903.4& 8.36 & 7.59&8.84&1.3$\times 10^{-14}$ &43.92$^{\ast}$ \\
SDSSJ083945.98+384319.0& 8.16 & 8.60$^{\dagger}$&9.71$^{\dagger}$ &2.3$\times 10^{-13}$ & 44.26 \\
SDSSJ084041.05+383819.9& 7.71 & 8.62& 9.48&$<$4.4$\times 10^{-14}$&$<$44.07 \\
SDSSJ092318.04+010144.7&8.53&8.94&9.31&$<$4.0$\times 10^{-14}$&$<$44.24\\
SDSSJ094820.38+582526.4& 7.01 & 7.89&9.08&$<$4.2$\times 10^{-14}$&$<$44.17 \\
SDSSJ133735.01-012815.6& 8.75 & 8.72&9.56&$<$6.0$\times 10^{-14}$&$<$44.25 \\
SDSSJ143027.66-005614.8& 7.61 & 8.36&9.23&$<$8.6$\times 10^{-14}$&$<$44.38 \\
SDSSJ143928.23+001538.0& 8.01 & 8.08&9.22&6.6$\times 10^{-14}$& 43.88\\
\hline
\end{tabular}
\caption{(1) SDSS name. (2) log of the black hole mass in units of M$_{\odot}$ from B06. (3) log of the [OIII] line luminosity in units of L$_{\odot}$ from Zakamska et al (2003). (4) log of the extinction-corrected [OIII] line luminosity in units of L$_{\odot}$ from B06. (5) Observed X-ray flux in units of erg s$^{-1}$ cm$^{-2}$(see the Appendix). (6) log of the de-absorbed X-ray luminosity in units of erg s$^{-1}$ (see the Appendix).
($\ast$) Compton-thick source: L$^{int}_{X}$ is obtained assuming that the
ratio between the reflected and the primary component is 0.05.
($^{\dagger}$)
this source has $z$=0.424 so the SDSS spectrum does not
cover the spectral range necessary to measure the H$\alpha$ line. The [OIII] line luminosity and the Balmer decrement measurements are obtained from observation collected with the 2.2 m telescope at the Centro Astron\'{o}mico Hispano Alem\'{a}n (CAHA) at
Calar Alto. }
\label{SDSS_source}
\end{table*}
\section{Conclusions}
In this paper we estimate the bolometric correction
factor, C$_{OIII}$, needed to convert the extinction-corrected [OIII] line luminosity to
bolometric luminosity.
We determined C$_{OIII}$ in a two-step process. First we studied the
L$_{X}$-L$^{c}_{OIII}$ relation in a sample of 61 Seyfert galaxies with
reliable measurement of both L$^{c}_{OIII}$ and L$_{X}$, i.e. we selected
galaxies with reliable measurement of the Balmer decrement
(Seyfert galaxies with optical classification $\ge$1.5) and X-ray Compton-thin
sources. The bolometric correction C$_{OIII}$ was then obtained by combining the mean
ratio between L$_{X}$ and L$^{c}_{OIII}$ with the luminosity-dependent X-ray bolometric
correction of Marconi et al. (2004). We found a
mean value of C$_{OIII}$ in the luminosity ranges log L$_{OIII}$=38-40,
40-42, and 42-44 of 87, 142 and 454 respectively.\\
Similarly, H04 estimated the [OIII] bolometric correction combining
the mean ratio between L$_{5000}$ and L$_{OIII}$ in a sample of
type 1 AGN with the 5000 Angstrom continuum to L$_{bol}$ correction from Marconi et
al. (2004). H04 did not correct L$_{OIII}$ for the extinction in the NLR,
therefore their bolometric correction applies to the ``observed'' [OIII] luminosities.
Compared to H04 our estimate of the [OIII] bolometric correction is not
subject to the scatter due to the extinction in the NLR, which we estimated
to be about 0.6 dex from our sample.\\
The [OIII] line luminosity is an indirect estimator of the nuclear
luminosity, depending on the geometry of the system, and on any
possible shielding effect which may affect the ionizing radiation seen by the NLR.
The amount of gas which is exposed to the ionizing radiation depends on the
torus opening angle. Bianchi et al. (2007, and references therein) confirmed the
Iwasawa-Taniguchi (IT) effect, i.e. the anti-correlation between the iron line equivalent
width and the X-ray luminosity. The simplest explanation for the IT effect is in terms
of a decrease of the torus covering factor (and hence an increase of its opening angle)
with luminosity, which may also explain the observed decrease with
luminosity of the ratio between mid-IR and bolometric luminosities (Maiolino et al.
2007; Treister et al. 2008). If this explanation is correct, we would expect a less
than linear relation between L$_{X}$ and L$^{c}_{OIII}$, differently from what we
found (the situation is even worse with the more than linear relations found by
P06 and N06). This problem will be addressed in a forthcoming paper.
We used C$_{OIII}$ to estimate the Eddington ratio distribution of
type 2 SDSS AGN at 0.3$<z<$0.4 from the B06 sample. B06 found that the mean of
this distribution
is $\langle \lambda \rangle$ $\simeq$ 1, thus revealing a population of high Eddington ratio
sources which is absent in the local Universe and in type 1 AGN at
higher redshift (e.g. Woo \& Urry 2002; McLure \& Dunlop 2004; Warner et
al. 2004; Kollmeier et al. 2006; Vestergaard 2002, 2004; Netzer \&
Trakhtenbrot 2007).
However, we have demonstrated, with the help of the X-ray
luminosities, that B06 have overestimated in a
systematic manner the bolometric luminosities and hence the Eddington
ratios, which we found to be
$\langle \lambda \rangle$ $\simeq$0.1, similar to what found in other AGN samples.
\section*{Acknowledgements}
This work benefited from an Italy-Spain Integrated Actions, reference HI-0079.
AL, GM, and SB acknowledge financial support from ASI (grant I/088/06/0).
XB and FJC acknowledge financial support by the Spanish Ministerio de Ciencia e Innovaci\'on under project ESP2006-13608-C02-01
. The authors thank Cristian Vignali for useful discussions, and the anonymous
referee for his constructive comments.
Based on observations obtained with {\it XMM-Newton}, an ESA science mission
with instruments and contributions directly funded by ESA Member States and
the USA (NASA). Based on observation collected with the
2.2 m telescope at the Centro Astron\'{o}mico Hispano Alem\'{a}n (CAHA) at
Calar Alto, operated jointly by the Max-Planck Institut f\"ur Astronomie and the Instituto de Astrof\'{i}sica de Andaluc\'{i}a (CSIC).
|
1,108,101,563,267 | arxiv | \section{Introduction}
Population {\sc III} stars are the first potential producers of UV photons that can
contribute to the reionization process and are the first producers of the
metals required for the formation of population {\sc II} stars.
Metals produced by the first stars will also be injected into some fraction of the ionized volume,
and so will be present in gas falling into new or existing protogalactic halos.
The question then arises as to how this low level of metal enrichment affects the ability of the
gas to cool and collapse.
In \cite{BRO01}, the collapse of cold, metal-enriched gas in a top-hat potential was simulated.
These simulations included the effects of atomic fine structure cooling, but did not
include cooling from ${\rm H_{2}}$. In the absence of molecular cooling,
fragmentation suggestive of a modern IMF was found to occur only for
metallicities above a threshold value of $Z \simeq 10^{-3.5} Z_\odot$. However,
the authors noted that the neglect of molecular cooling could be significant.
In \cite{OMU05}, Omukai et~al.\ argued, based on the results of their detailed one-zone
models, that molecular cooling would indeed dominate the cooling over many orders of
magnitude in density. Also, by starting with cold gas, the authors of \cite{BRO01}
implicitly assume that no extra entropy or energy has been added to the gas during its
enrichment, although this is unlikely to be the case \cite{OH03}.
It is therefore important to consider whether a metallicity threshold
appears in simulations with different initial conditions. In this
paper, we examine a different, still idealized, set of initial
conditions, using a more detailed treatment of the cooling and chemistry of the gas,
and find no threshold. We argue that the transition from
the primordial to the modern IMF therefore depends on the conditions under
which stars form as much as on the metal abundances present.
\section{Numerical Approach}
To help us to assess the influence of metals on the fragmentation properties of
gas in small protogalactic halos, we have performed a number of numerical
simulations. Here we present a portion of our results. Detailed discussion of
the full set of simulations can be found in \cite{jmgk07} (hereafter paper V).
Our study requires us to follow the collapse of the gas over many orders of
magnitude in density, and therefore the problem is well-suited for the
use of a Lagrangian numerical method. We use
SPH, a Lagrangian method described in \cite{BEN90}, \cite{MON92}
and \cite{MON05}, in which the fluid is followed with an ensemble of particles.
Fluid properties at each point are computed by averaging over neighboring particles.
The calculations presented in this paper use version 1 of the parallel SPH
code GADGET \cite{SPR01}.
In order to achieve a higher mass resolution, we refine the mass of the gas particles using
a method known as particle splitting \cite{KIT02}. In \cite{jgkm07}, hereafter paper II,
we presented low-resolution simulations that show that the density increases most rapidly close to the center of the dark matter halo. On-the-fly splitting with two levels of refinement at two
different radii provides us with the highest mass resolution at the region of interest. In the region of collapse we can resolve Jeans masses down to 1.5~$M_{\odot}$, as compared to $M_{\mathrm{res}} = 200~M_{\odot}$ in the low resolution simulations.
We discuss the details of the implementation of the particle splitting algorithm in paper V.
As we want to represent gas that has collapsed beyond the
resolution limit of the simulation in a numerically robust manner, we
have modified the code to allow it to create sink particles --
massive, non-gaseous particles, designed to represent dense cores,
that can accrete gas from their surroundings but otherwise interact
only via gravity \citep{BAT95}.
Sink particles are created once the density rises above $10^4\,\mathrm{cm^{-3}}$
and are endowed with an accretion radius of $0.3\,\mathrm{pc}$. On every time step,
any gas within this accretion radius that is gravitationally bound to the sink particle
is accreted by it. The design and implementation of our sink particle algorithm is
discussed in more detail in \cite{JAP05}.
\section{Chemistry and Cooling}
We have further modified GADGET to allow us to follow the non-equilibrium
chemistry of the major coolants in both primordial and low-metallicity gas.
Our chemical model is presented in \cite{gj07} and \cite{glo07} (hereafter
papers I and III). Provided that carbon and oxygen are amongst
the most abundant metals, the major coolants will be largely the same as in
local atomic and molecular gas, namely $\rm{H}_{2}$, HD, C, $\rm{C^{+}}$, O, Si, $\rm{Si^{+}}$, CO,
OH and ${\rm H_{2}O}$ \citep{OMU05}. We therefore follow the abundances of
these ten species, together with an additional 29 species that play important
roles in determining the abundances of one or more of these coolants.
Our chemical network contains a total of 189 collisional
gas-phase reactions between these 39 species, as well as 7 grain surface
reactions, 43 reactions involving the photoionization or photodissociation of
chemical species by ultraviolet radiation, and 8 reactions involving cosmic
rays. For simplicity, in the simulations presented in this paper we do not include
the effects of dust, UV radiation or cosmic rays, and so use a simplified
version of the model that contains only the collisional reactions.
The thermal evolution of the gas in our simulations is modelled using a
cooling function that includes the effects of atomic fine structure cooling
from $\rm{C}$, $\rm{C^{+}}$, O, Si and $\rm{Si^{+}}$, rotational and vibrational cooling
from $\rm{H}_{2}$, HD, CO and ${\rm H_{2}O}$, Lyman-$\alpha$ cooling,
Compton cooling, and $\rm{H}^{+}$ recombination cooling, as well as a number
of other processes of lesser importance.
For additional details of our chemical networks and cooling function, please consult
papers I and III; further details of their implementation within Gadget are also given
in paper II.
\section{Initial Conditions}
Our initial conditions are based on those used in paper II,
although collapse is followed to much higher density in the current work.
We study protogalaxies forming from fully ionized gas with initial
temperature $T_g = 10^4$~K, representing a fossil H~{\sc ii} region
\citep{OH03} polluted by supernovae from the ionizing object.
We model one such halo by using a fixed background potential with a spherically symmetric density profile \cite{NAV97}. We study the evolution both of zero-metallicity, primordial gas and of
metal-enriched gas with a metallicity ${Z} = 10^{-3} {Z_{\odot}}$. To simplify the discussion of our simulation
results, we take as a fiducial example a halo with a dark matter mass
$M_{\rm dm} = 7.8 \times 10^{5} \: {M_{\odot}}$, an initial redshift
$z_{i} = 25$, and a spin parameter $\lambda = 0.05$. For this example halo, the virial temperature
$T_{\rm vir} = 1900 \, \mathrm{K}$, the virial radius $r_{\rm vir} = 0.1 \, {\rm kpc}$,
and the truncation radius $r_{{t}} = 0.49\:\mathrm{kpc}$, where both
radii are given in physical units. The scale radius $r_{{s}}$ of this
example halo is $29\,\mathrm{pc}$, and the full computational volume is a box
of side length $1\,\mathrm{kpc}$. The gas mass is $M_{{g}}=0.19\,M_{\mathrm{dm}}$.
\section{Results and Discussion}
In the simulations presented here we focus on gas with a metallicity
${\rm Z} = 10^{-3} \: {Z_{\odot}}$. This metallicity
is at the upper limit of the range of values proposed
for the so-called critical metallicity ${Z_{\rm crit}}$,
the value of the metallicity at which efficient fragmentation
and low-mass star formation is hypothesized to first occur
\citep{BRO01,OMU05,BL03,SCH02}. It is also comparable to the
globally averaged metallicity produced by the sources responsible
for reionization \citep[see e.g.][]{RIC04}.
In Figure~\ref{fig1} we compare the temperature-density evolution of the run with primordial gas and the run with enriched gas. The two runs differ by less than 10\% up to densities of $n = 10^4 \: {\rm cm^{-3}}$.
This result demonstrates that the cooling of the gas is barely influenced
by the presence of metals: fine-structure cooling contributes
only marginally to the total cooling rate. $\rm{H}_{2}$ is the dominant coolant,
rather than metal fine structure lines.
At first sight, the fact that fine structure cooling from metals has little impact on
the thermal or dynamical evolution of the gas at metallicities below
$0.1 \: {Z_{\odot}}$ is somewhat surprising, given that it was previously
found that gas with a metallicity of only ${Z} = 10^{-3} \: {Z_{\odot}}$ could cool
rapidly and fragment even in the complete absence of molecular hydrogen
\cite{BRO01}. However, this difference in conclusions appears to be a
consequence of the different initial conditions used in the two sets of simulations,
and the difference in the physics included. The simulations presented in \cite{BRO01}
neglected molecular cooling, and started with cold ($T_{\mathrm{initial}}= 200\,\mathrm{K}$),
low ionization ($x_{e}=10^{-4}$) gas in a flat, top-hat dark matter potential lacking a central condensation, although with local density perturbations. In these simulations,
the gas cools, collapses and fragments only if the metallicity is high enough to
render the cooling time shorter the the local dynamical timescale. With our arguably more
realistic initial conditions, and, crucially, with our inclusion of ${\rm H_{2}}$, we find no
metallicity threshold -- ${\rm H_{2}}$ cooling alone is sufficient to allow the gas
to collapse, but the collapsing gas does not fragment (see Figure~\ref{fig2} and \cite{jkgm07}).
In \cite{jmgk07}, we present a set of additional runs with metallicities up to
${Z} = 10^{-1} {Z_{\odot}}$ that confirm this result. We conclude that the transition
from the primordial to the modern-day IMF therefore depends on the conditions under
which stars form as much as on the metal abundances present. The actual conditions
that are appropriate for second generation star formation still need to be determined in
detail by observation and modeling of galaxy formation.
\begin{figure}
\includegraphics[width=.47\textwidth]{fig1.eps}
\caption{Gas temperature vs. number density for the runs with metallicities 0.0 ({\it solid line}), $10^{-3}$ ({\it dotted line}), and $10^{-1} \;Z_{\odot}$ ({\it dashed line}). The spin parameter of the runs is 0.05. The time of the plot is half a Hubble time. The thin lines show the 1-$\sigma$ deviation.}
\label{fig1}
\end{figure}
\begin{figure}
\includegraphics[width=.23\textwidth]{fig2.eps}
\includegraphics[width=.245\textwidth]{fig3.eps}
\caption{Cut at $z = 0.5 \,\mathrm{kpc}$ through the gas number density for the run with zero metallicity ({\it left panel}) and the run with a metallicity of $ {Z} = 10^{-3} {Z_{\odot}}$ ({\it right panel}). We show a box with a size of 20 pc.}
\label{fig2}
\end{figure}
\begin{theacknowledgments}
RSK and AKJ acknowledge support from the Emmy Noether Program of the DFG (grant KL1385/1). M-MML and SCOG were supported in part by AST03-07793 and M-MML was supported in part by a DAAD stipend. The simulations described here were performed on the PC cluster "sanssouci" at the AIP and the cluster "McKenzie" at CITA.
\end{theacknowledgments}
\bibliographystyle{aipproc}
|
1,108,101,563,268 | arxiv | \section{Introduction}
Visual Simultaneous Localization and Mapping (vSLAM) is the process of creating a map based on the visual information without a global reference, being used in multiple applications where GPS signal is not available or precise enough. In the context of vSLAM based on features, this is done by using the relative pose obtained by detecting interest points, generating a descriptor for them and matching the extracted keypoint in the previous frame to the current one. Those matches are then processed using algorithms similar to the normalized eight-point algorithm \cite{8point_alg}. On top of that, loop closing and global optimization processes are normally implemented to improve the estimated trajectory, removing drift error.
All of that is dependent on the correct association of keypoints. Most feature extraction methods generate descriptors based solely on photometric information like ORB \cite{ORB}, SURF \cite{SURF} and SIFT \cite{SIFT}, with only the first one being both fast and reliable. Although they achieve excellent results in SLAM systems, handmade descriptors are highly susceptible to illumination changes, which decreases the quality of the pose estimation. This problem is important to be considered since most SLAM applications are made to run in an uncontrolled environment.
Convolutional neural networks are normally used in computer vision tasks to increase robustness, and most often obtain better results than handmade methods \cite{conv_better_hand1, conv_better_hand2}. Similarly, methods such as SuperPoint \cite{superpoint} and LIFT \cite{LIFT} were developed to improve the coherency of feature extraction algorithms. Both are superior to ORB and present comparable or better results than SIFT while keeping a real-time performance.
In the context of deep learning, models that learn multiple related tasks can have a better result if trained with a shared encoder due to the inductive bias, as seen in the related literature \cite{MTL_kendall, MTL_ang}, \cite{MTL_sener}. In comparison to using multiple models and soft parameter sharing, the computational cost of using a shared encoder for multi-task is relatively low.
This article proposes adding semantic information intrinsically to the keypoint and descriptor, which would help to detect meaningful pixels and create a better descriptor. In this way, the pose estimation method can consider this semantic information as an additional criterion in the matching process of the extracted features. To accomplish that, multi-task learning methods are essential to effectively train the model. A diagram of the proposed model can be seen in Figure~\ref{fig:ssp}.
\begin{figure}[t]
\centering
\includegraphics[width=.4\textwidth]{imgs/SP/SSp-cropped.pdf}
\caption{\textbf{Semantic SuperPoint Model}. The dashed line represents that the semantic decoder was only used in training and C is the number of classes.}
\label{fig:ssp}
\end{figure}
In this work, we also evaluate the models trained with and without the semantic decoder on the HPatches \cite{hpatches_2017_cvpr} according to the metrics defined by DeTone et al. \cite{superpoint} and on the KITTI dataset \cite{kitti_seq} at the SLAM task using the ORB-SLAM2 \cite{orb-slam2} as the base framework. We show that the proposed Semantic SuperPoint (SSp) model improves the matching score metric and estimates better trajectories. For a 480×640 image, our model runs at \textbf{208 FPS} or \textbf{4.8 ms} per frame on an NVIDIA GTX 3090.
In summary, the key contributions of this paper are:
\begin{itemize}
\item Evaluation of multi-task learning methods applied to SuperPoint-like models.
\item A novel training paradigm for deep feature extraction.
\item A slight improvement in the Matching Score metric.
\end{itemize}
\section{RELATED WORK}
\textbf{Semantic SLAM}. Adding high-level semantic information to SLAM-related applications was previously reported in the literature on multiple occasions to improve methods based only on the intensities of the pixels. Examples of those projects are the use of high-level semantic information for loop closure detection with LiDAR 3D points to avoid local minimums \cite{semantic_scan}, detection of dynamic objects to calculate their estimated speed and take that into account for a more accurate pose estimation process \cite{dyna_SLAM}, segmentation of objects for better description considering a database of selected artifacts \cite{sSLAM4AUV}, and for a handmade semantic descriptor combined with a photometric one \cite{sem_vslam_health}.
All of those highly depend on the quality of the network prediction, therefore are sensible to reflections and unknown objects. Even though the same can be said about our model, the semantic information only helps to extract better features and is not used in the inference, so it is more robust to misclassification.
\textbf{Multi-task Learning}. The simplest way to address multiple tasks is to do a uniform sum of each loss, as presented in Equation \eqref{eq:mtl_sum}, in the specific case of $w_t = 1$ for every task $t$. This usually obtains the worst result because of the different nature of the tasks and scales of the loss functions used to minimize them. Consequently, the model can converge prioritizing one task over another.
One way to avoid this is to compute a weighted sum of each loss according to a positive scalar $w_t$. This has a better result than the uniform sum. However, it does not achieve the same result when compared to multi-task learning methods, requiring a grid search for the weighting parameters, which is time-consuming and computationally costly. The mentioned approach tries to minimize the following loss objective:
\begin{equation}
L_{total} = \sum_{t=1}^N w_t L_t \text{,}
\label{eq:mtl_sum}
\end{equation}
\noindent where $w_t$ is the weight that scales the $t$-th task loss $L_t$, and $N$ is the number of tasks.
To avoid this undesired convergence behavior and the grid search, this work implements two multi-task learning methods proposed in \cite{MTL_kendall, MTL_ang}.
The first method \cite{MTL_kendall} is based on the uncertainty modeling to weigh losses and derive a multi-task loss function by maximizing a Gaussian likelihood with homoscedastic uncertainty to classification and regression problems.
Let $f^\mathbf{W}(\mathbf{x})$ be the output of a neural network with $\mathbf{W}$ weights and input $\mathbf{x}$ and $\mathbf{y}$ for the respective label. For regression problems, considering a normal distribution, the log-likelihood associated is:
\begin{equation}
\text{log} \, p(\mathbf{y}|f^\mathbf{W}(\mathbf{x})) \propto -\frac{1}{2\sigma^2}||y - f^\mathbf{W}(\mathbf{x})||^2 - \text{log}\, \sigma \text{,}
\label{eq:mtl_kendall_reg}
\end{equation}
\noindent where $\sigma$ is an observation noise scalar. In the case of a classification problem:
\begin{equation}
\text{log} \, (\mathbf{y}=c|f^\mathbf{W}(\mathbf{x}), \sigma) \propto -\frac{1}{\sigma^2}f^\mathbf{W}(\mathbf{x}) -\text{log}\, \sigma \text{,}
\label{eq:mtl_kendall_class}
\end{equation}
\noindent where $c$ is the predicted class.
The second approach \cite{MTL_ang} consists in finding a common descent direction for the shared encoder. This is done by computing the gradient related to each task, normalizing them, and
searching for weighting coefficients that minimize the norm of the convex combination, obtaining the central direction.
To consider the gradient history, a period of $T$ iterations of the gradients’ norm of each task are tracked and, in case of some task diverging, the central dir. is pulled toward it as the tensioner's idea.
Nakamura et al. \cite{MTL_ang} shows that by doing this the shared encoder parameters are adjusted to equally benefit all tasks, besides decreasing the convergence time significantly.
For a more detailed explanation, please refer to the original article of each method.
\textbf{SuperPoint}. SuperPoint \cite{superpoint} is a deep feature extractor trained by self-supervision that can perform in real-time while executing both keypoint detection and descriptor decoders.
For the detection decoder, a base detector MagicPoint is trained in the Synthetic Shapes dataset, witch renders simple 2D objects, such as quadrilaterals, triangles, lines, and ellipses. In the dataset, the keypoint label is created by marking certain junctions positions, end of segments, and ellipses center.
The MagicPoint model is then used to extract the interest points in the selected dataset by performing a number $N_h$ of homographic adaptations and combining its output to generate a final heatmap, filtered by Non-Maximum Suppression (NMS) \cite{NMS}.
The next step is joint training with both detector and descriptor decoders. A set with the original image and its warped version is passed through the model, outputting the detected interest points and the descriptors. Since the warp transformation is known, the same goes for the position relation between each pixel. Thus, pairs of matches and non-matches can be created and compared to the ground truth, illustrated in Figure \ref{fig:sp_pipeline}.
\begin{figure*}[htpb]
\setbox9=\hbox{\centering\includegraphics[width=.32\linewidth]{imgs/SP/SSP_joint_training-cropped.pdf}
\subcaptionbox{Training in Synthetic Shapes dataset}
{\raisebox{\dimexpr\ht9/2-\height/2}{\includegraphics[width=.32\linewidth]{imgs/SP/magicpoint_train-cropped.pdf}}}
\unskip\ \vrule\
\hfill
\subcaptionbox{Keypoint extraction}
{\raisebox{\dimexpr\ht9/2-\height/2}{\includegraphics[width=.32\linewidth]{imgs/SP/self-label-cropped.pdf}}}
\unskip\ \vrule\
\hfill
\subcaptionbox{Joint training}{\includegraphics[width=.32\linewidth]{imgs/SP/SSP_joint_training-cropped.pdf}}
\caption{\textbf{Semantic SuperPoint training overview}. A base betector is trained in the Synthetic Shapes dataset and used to extract pseudo-ground truth interest points in unseen real images using $N_h$ homographic transformations. The joint training step uses the generated labels with the semantic label to train the SSp model. Adapted from DeTone et al. \cite{superpoint}.}
\label{fig:sp_pipeline}
\end{figure*}
\section{SEMANTIC SUPERPOINT MODEL}
SSp is a fully convolutional neural network that receives a grayscale image as input and outputs a heatmap, descriptor, and semantic segmentation prediction. The architecture uses a single shared encoder and its output (intermediary features) passes through each decoder.
Due to the aggressive data augmentation process and the large variety of classes in the MS-COCO dataset \cite{MSCOCODATA}, used for training, the semantic prediction is not usable in practical application. Nonetheless, adding a semantic decoder for training improved the quality of the deep feature extractor. It can therefore be said that rather than being a model, SSp is more like a training method.
\subsection{Shared Encoder}
According to the open source implementation of SuperPoint (Sp) \cite{SP_open_git}, the encoder is based on the U-Net \cite{unet}, using \texttt{double\_conv} blocks, which consists of $2\times$(conv2d-BN-ReLu) and three layers of max pooling to reduce the dimension between them. Therefore the output has a shape of $\frac{H}{8}\times \frac{W}{8}\times C_{enc}$. We adopt $C_{enc} = 256$ as the encoder output dimension. The same encoder model is used for Sp and SSp architectures.
\subsection{Detector Decoder}
Both the detector and descriptor decoders are the same as SuperPoint's original model. The detector has a shape of $\frac{H}{8}\times \frac{W}{8}\times 65$ to avoid upsampling operations and has one dummy dimension 'no keypoints'.
The loss function used was the binary cross entropy (BCE):
\begin{multline}
L_{d} = \sum_{n=1}^{N} [y_n\log(\text{Softmax}(x_n))\\
+(1-y_n)\log(1-\text{Softmax}(x_n))] \text{,}
\label{eq:det_loss}
\end{multline}
\noindent where N is the batch size.
\subsection{Descriptor Decoder}
For the descriptor decoder, also to avoid adding computational cost related to the upsampling, the output is shaped as $\frac{H}{8}\times \frac{W}{8} \times D$ and to extract the descriptor, it passes through a Bicubic Interpolation and is normalized.
To train the descriptor, it is necessary to relate two images and evaluate the matches and non-matches while balancing both influences. This was done by dividing the loss value by the number of positive ($n_p$) and negative ($n_n$) matches in the Hinge loss:
\begin{multline}
L_{desc} = \underbrace{\frac{s}{n_p}\max(0,m_p-d^Td^{'})}_\text{$L_p$} \\
+ \underbrace{\frac{(1-s)}{n_n} \max(0,d^Td^{'} - m_n)}_\text{$L_n$}
\text{,}
\label{eq:desc_loss}
\end{multline}
\noindent where $m_p = 1$, $m_n = 0.2$, $d$ and $d^{'}$ are the descriptor decoder outputs for the normal and warped image, respectively.
\subsection{Semantic Decoder}
The semantic decoder consists of a conv3x3-BN-ReLU block followed by a conv1x1. The cross-entropy (CE) loss was used for training this decoder:
\begin{equation}
L_{s} = -\sum_{n=1}^N \sum_{c=1}^{C} w_clog (\text{Softmax}(x_c))y_{n,c}.
\label{eq:sem_loss}
\end{equation}
\section{MULTI-TASK LOSS} \label{sec:mtl}
In this section, we discuss the losses applied in the training process. In all cases, the semantic loss will be included to avoid rewriting the equations, but the non-semantic loss can be obtained by removing the $L_{s_i}$ terms.
\subsection{Uniform loss}
For comparison purposes, the model was also trained with the uniform sum loss function as follows:
\begin{equation}
L_{total} = L_{d_1} + L_{d_2} + \lambda L_{desc} + L_{s_1} + L_{s_2}.
\label{eq:uni_loss_sp}
\end{equation}
The losses $L_{d_i}$ and $L_{s_i}$ are obtained from the interest point detector and the semantic decoder, respectively. The subscribed index 1 is for the normal image and 2 for the warped version.
In the original SuperPoint article \cite{superpoint}, they use the term $\lambda$ to weight the descriptor loss. In our case, the code was build on the unofficial implementation of Jau et al. \cite{SP_open_git}, and they use $\lambda=1$. Besides, the model was able to learn both tasks without the need for tuning the value $\lambda$.
\subsection{Uncertainty loss}
In the case of the uncertainty loss proposed by Kendall et al. \cite{MTL_kendall}, the detection loss can be directly related to the classification case, Equation \eqref{eq:mtl_kendall_class}, since the BCE loss behaves the same way as a CE loss for 2 classes. The same goes for the semantic decoder as it also uses CE.
The descriptor loss is more complex. To jointly calculate the terms of the hinge loss, the assumption that the losses are not correlated must be true. Even though this is not the case for most multi-task models, the approximation remains valid.
The problem with the hinge loss is that the positive and negative terms are inversely proportional. The more the positive loss improves, the negative loss will not get better at the same rate as the model would create a bias towards positive matches and vice versa.
The descriptor loss was considered as two independent regression problems, each one with a related normal distribution. Considering the loss given by Equation \eqref{eq:desc_loss}, it is expected that the output $D_{out}=f^{\mathbf{W_{sh, desc}}}(\mathbf{x}_1)^Tf^{\mathbf{W_{sh, desc}}}(\mathbf{x}_2)$ is going to be grouped toward $m_p$ or $m_n$ for the positive matches ($s=1$) and the negative matches ($s=0$), respectively. So we can describe the distribution as:
\begin{equation}
p(\mathbf{y}|D_{out}, s) = s \mathcal{N}_p (D_{out}, \sigma_p^2) + (1-s)\mathcal{N}_n (D_{out}, \sigma_n^2) \text{,}
\label{eq:mtl_norm_desc}
\end{equation}
\noindent where $\mathcal{N}_i$ and $\sigma_i$ are the normal distribution and variance associated with each match type.
Expanding the normal distributions and approximating the loss terms, we obtain:
\begin{equation}
\text{log} \, p(\mathbf{y}|D_{out}) \propto -\frac{1}{2\sigma^2} \left(L_p + L_n\right) - \text{log} \, \sigma.
\label{eq:mtl_norm_desc_vf}
\end{equation}
See Appendix \ref{ap:mtl_norm_desc} for more detail. The normal and warped loss terms are summed because they represent the same objective on the same weights, so the joint probability can be modeled as:
\begin{multline}
p(\mathbf{y_{d_1},y_{d_2}, y_{desc}, y_{s_1}, y_{s_2}} | f^{\mathbf{W}}(\mathbf{x_1}), f^{\mathbf{W}}(\mathbf{x_2})) = \\ p(y_{d_1}, y_{d_2}|f^{\mathbf{W_{sh, d}}}(\mathbf{x_1}), f^{\mathbf{W}_{sh, d}}(\mathbf{x_2})) p(y_{desc} | D_{out})\\
p(\mathbf{y}_{s_1} = c_1, \mathbf{y}_{s_2} = c_2|f^{\mathbf{W_{sh, s}}}(\mathbf{x_1}), f^{\mathbf{W_{sh, s}}}(\mathbf{x_2})).
\label{eq:loss_prob_t_v2}
\end{multline}
\noindent So the final equation for the uncertainty loss is given by:
\begin{multline}
-log \, p(\mathbf{y_{d_1},y_{d_2}, y_{desc}, y_{s_1}, y_{s_2}} | f^{\mathbf{W}}(\mathbf{x_1}), f^{\mathbf{W}}(\mathbf{x_2})) \propto \\
(L_{d_1}+L_{d_2}) \text{exp}(\eta_{d}) +\frac{L_{desc}}{2} \text{exp}(\eta_{desc})\\
+(L_{s_1}+L_{s_2}) \text{exp}(\eta_{s}) + \eta_d + \frac{\eta_{desc}}{2} + \eta_s = L_{total}.
\label{eq:mtl_uncertanty}
\end{multline}
In order to avoid numerical instability during training, we do the variable change: $\eta_i = 2\text{log} \, \sigma_i$. The initial values for $\eta_d, \, \eta_{desc},\, \eta_s$ were $1.0, \, 2.0, \, 1.0$ , respectively.
\subsection{Central dir. + tensor}
To apply the Central dir. + tensor method the normal and warped loss were summed to simplify the common gradient descent calculation and the descriptor loss was treated as a single task.
The $\alpha$ term, which is used to regulate the tension factor’s sensitivity, was set to $0.3$, as indicated by the original authors \cite{MTL_ang}.
\section{DATA PREPARATION}
The MS-COCO 2017 dataset consists of 123k 640×480 RGB images split into 118k images for training and 5k for validation, with 133 segmented classes. This large amount and variability of data can provide the model with the necessary generalization capability to perform in several conditions while being similar to the 2014 one, used to train the original SuperPoint model.
\subsection{Keypoint extraction}
In order to create the pseudo-ground truth labels for the keypoint extractor, a pretrained MagicPoint model in the Synthetic Shapes dataset from Jau et al. \cite{SP_open_git} was applied to the MS-COCO dataset. It was used $N_h = 100$ as the number of homograph transformations, resized to a resolution of $240\times320$ to decrease training time.
\section{TRAINING}
All training and evaluation processes were done in a machine with an AMD Ryzen 9 5950X 16-Core Processor and an NVIDIA GTX 3090 24GB using PyTorch \cite{PyTorch}.
For data augmentation, multiple transformations, such as Gaussian noise, scale, rotation, translation, and others were applied to improve the network's robustness to illumination and viewpoint changes. The descriptor dimension is $D=256$ for all experiments.
\subsection{Sp \& SSp training}
The Sp and SSp models were trained using the Adam optimizer \cite{Adam} for 200k iterations with batch size 16. We tested with a fixed learning rate as in the original article, and with a starting value of $0.0025$ with polynomial decay and an end value of $0.001$. The first option worked better for the non-semantic version and the second one was better for the SSp model.
We also varied the multi-task loss used for each model, as described in Section \ref{sec:mtl}. Empirically, we found that the central dir. $+$ tensor method took too long to converge. So we used the model saved at the 100K$^{th}$ iteration trained by the uncertainty loss as the starting point, and trained for more 100K iterations using the central dir. $+$ tensor optimization method with the same learning rate parameters.
In every variation, the models were saved every interval of 5K iterations and evaluated as described in Section \ref{sec:eval}.
To rank each model, we adopt the matching score as the main metric, since it is the percentage of inliers. Hence, it represents an overall performance of the keypoint detection and descriptor generation. Furthermore, it is the most relatable to SLAM methods.
\section{EVALUATION} \label{sec:eval}
\subsection{HPatches}
The HPatches dataset consists of 116 scenes each one with six different images. The first 57 scenes are subject to illumination changes and in the other 59 scenes, viewpoint changes.
The models were evaluated according to the metrics: repeatability (Rep.), localization error (MLE), homography estimation ($e=\{1, 3 , 5\}$), nearest neighbor mean average precision (NN mAP), and matching score (M.S.). Each metric is described in detail by DeTone et al. \cite{superpoint}.
Considering the abbreviations \texttt{unc} for the uncertainty loss, \texttt{ct} for central dir. $+$ tensor and \texttt{uni} for the uniform loss, the results of each model are presented in Table \ref{tab:hpatches_metric}.
\begin{table}[ht]
\caption{Evaluation on HPatches dataset}
\label{tab:hpatches_metric}
\begin{center}
\resizebox{\linewidth}{!}{
{\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{1}{|c|}{\multirow{2}{3cm}{\centering Feature extractor}} & \multicolumn{3}{c|}{Homography Estimation} & \multicolumn{2}{c|}{Detector metrics} & \multicolumn{2}{c|}{Descriptor metrics}\\ \cline{2-8}
& $\epsilon = 1$ & $\epsilon = 3$ & $\epsilon = 5$ & Rep. & MLE & NN mAP. & M.S.\\
\hline
ORB$^{\mathrm{a}}$ & .150 & .395 & .538 & \textbf{.641} & 1.157 & .735 & .266 \\
SIFT$^{\mathrm{a}}$ & .424 & .676 & .759 & .495 & \textbf{0.833} & .694 & .313 \\
LIFT$^{\mathrm{a}}$ & .284 & .598 & .717 & .449 & 1.102 & .664 & .315 \\
\hline
Sp + uni (baseline) & .476 & .748 & .817 & .599 & 1.017 & \textbf{.864} & .519 \\
Sp + unc & .460 & .745 & .812 & .599 & 1.010 & \textbf{.864} & .520\\
Sp + ct & .493 & .753 & .812 & .602 & 1.001 & .862 & .519\\
\hline
SSp + uni (\textbf{ours}) & .398 & .710 & .797 & .584 & 1.052 & .843 & .506\\
SSp + unc (\textbf{ours}) & .450 & .745 & .816 & .598 & 1.005 & \textbf{.864} & .522\\
SSp + ct (\textbf{ours}) & .466 & .762 & .805 & .598 & 0.999 & .858 & .519\\ \hline
Pretrained model \cite{superpoint} & \textbf{.497} & \textbf{.766} & \textbf{.840} & .610 & 1.086 & .843 & \textbf{.540} \\
\hline
\multicolumn{7}{l}{$^{\mathrm{a}}$Original results reported by DeTone et al. (2017).}
\end{tabular}}}
\caption*{All metrics are higher is better, except MLE.}
\end{center}
\end{table}
Analyzing the results of the SSp model, the uniform loss had the worst result
compared to the others SuperPoint models. The uncertainty and central dir. methods improved significantly the initial result obtained using the uniform loss for the SSp model and had a very similar performance in the case of the Sp model.
Considering the matching score, the SSp model had the best result with a slight improvement of $\mathbf{+}$\textbf{0.2\%} over the baseline Sp. As for the Homography Estimation it had a better result than the Sp + unc for $\epsilon = 5$ while balancing the detector metrics.
Thus, adding semantic information can help deep feature extraction methods, but still needs some improvement to increase all metrics.
It was not possible to replicate the result obtained from the Magic Leaps pretrained model \cite{superpoint} since the official implementation is not public available as well its specifics. For this reason, the results were not directly compared with the pretrained model, instead we considered the Sp + uni as baseline for comparison.
\subsection{KITTI}
The best version of each model was evaluated in the KITTI dataset according to the implementation SuperPoint\_SLAM \cite{ORB_SLAM2_SP}. Each sequence was executed 10 times and the Absolute Pose Error (APE) and Relative Pose Error (RPE) metrics were extracted with the open source lib \texttt{evo} \cite{evo}. The mean values and the standard variation together with the p-values are exhibited in Table \ref{tab:KITTI_metrics}.
\begin{table*}[ht]
\sisetup{
table-align-uncertainty=true,
separate-uncertainty=true,
}
\renewrobustcmd{\bfseries}{\fontseries{b}\selectfont}
\renewrobustcmd{\boldmath}{}
\centering
\caption{Mean APE and RPE in the KITTI dataset for each model}
\begin{tabular}{|c||S[table-format=3.3,detect-weight,mode=text] @{${}\pm{}$} S[table-format=3.3,detect-weight,mode=text]|S[table-format=3.3,detect-weight,mode=text] @{${}\pm{}$} S[table-format=3.3,detect-weight,mode=text]|c||S[table-format=1.3,detect-weight,mode=text] @{${}\pm{}$} S[table-format=1.3,detect-weight,mode=text]|S[table-format=1.3,detect-weight,mode=text] @{${}\pm{}$} S[table-format=1.3,detect-weight,mode=text]|c|}
\hline
\multicolumn{1}{|c||}{\multirow{2}{3cm}{\centering Sequence}} & \multicolumn{5}{c||}{\centering ATE} & \multicolumn{5}{c|}{RPE} \\ \cline{2-11}
& \multicolumn{2}{c|}{\centering Sp} & \multicolumn{2}{c|}{\centering SSp} & p-value & \multicolumn{2}{c|}{\centering Sp} & \multicolumn{2}{c|}{\centering SSp} & p-value\\
\hline
00 & 6.767 & 1.112 & \bfseries 6.651 & \bfseries 0.614 & 0.705 & \bfseries 0.124 & \bfseries 0.028 & 0.130 & 0.032 & 0.804 \\
01 & 286.709 & 140.875 & \bfseries 209.985 & \bfseries 130.585 & 0.131 & 4.779 & 1.910 & \bfseries 3.991 & \bfseries 2.183 & 0.174 \\
02 & 22.459 & 2.428 & \bfseries 22.314 & \bfseries 2.988 & 0.821 & \bfseries 0.098 & \bfseries 0.008 & \bfseries 0.098 & \bfseries 0.009 & 0.705 \\
03 & \bfseries 1.319 & \bfseries 0.142 & 1.630 & 0.262 & 0.003 & \bfseries 0.040 & \bfseries 0.002 & 0.046 & 0.004 & 0.007 \\
04 & \bfseries 0.840 & \bfseries 0.135 & 0.905 & 0.261 & 0.940 & \bfseries 0.034 & \bfseries 0.004 & 0.039 & 0.008 & 0.406 \\
05 & \bfseries 5.803 & \bfseries 1.233 & 6.315 & 2.931 & 0.821 & \bfseries 0.172 & \bfseries 0.046 & 0.199 & 0.101 & 1.000 \\
06 & 11.953 & 0.939 & \bfseries 11.833 & \bfseries 0.415 & 0.406 & \bfseries 0.240 & \bfseries 0.048 & 0.267 & 0.075 & 0.326 \\
07 & 3.388 & 2.947 & \bfseries 2.124 & \bfseries 0.516 & 1.000 & 0.108 & 0.071 & \bfseries 0.084 & \bfseries 0.010 & 0.884 \\
08 & 31.324 & 1.736 & \bfseries 26.700 & \bfseries 1.013 & 0.000 & 0.273 & 0.012 & \bfseries 0.235 & \bfseries 0.009 & 0.000 \\
09 & 35.945 & 2.140 & \bfseries 31.788 & \bfseries 1.595 & 0.001 & 0.172 & 0.013 & \bfseries 0.164 & \bfseries 0.010 & 0.226 \\
10 & 5.515 & 0.437 & \bfseries 4.953 & \bfseries 0.316 & 0.023 & 0.064 & 0.003 & \bfseries 0.060 & \bfseries 0.003 & 0.019 \\
\hline
\end{tabular}
\label{tab:KITTI_metrics}
\end{table*}
Note that the SLAM application used here is only for comparison purposes as the matching time adds to much computational cost, being responsible for most of it \cite{SP_time}. To deal with these issues, approaches similar to GCNv2 \cite{SP_time} can be used.
\section{CONCLUSIONS}
We have shown that adding semantic information in a shared encoder based architecture in combination with multi-task learning methods can improve deep feature extraction methods and, in this case, without increasing computational cost in the inference step.
Future works can tune the intensity of the data augmentation process and the complexity of the training dataset to balance generalization capability and semantic learning or train the feature extractor using another approach that does not rely on intensive data augmentation.
Additionally, if achieved a satisfying semantic prediction, it is possible to combine the high-level semantic information into the already existing semantic SLAM methods for further improvements.
\addtolength{\textheight}{-2cm}
\section*{APPENDIX}
\subsection{Descriptor distribution} \label{ap:mtl_norm_desc}
The descriptor distribution is initially defined as two normal distributions as in Equation \eqref{eq:mtl_norm_desc}, copied bellow for convenience:
\begin{equation}
p(\mathbf{y}_{desc}|D_{out}, s) = s \mathcal{N}_p (D_{out}, \sigma_p^2) + (1-s)\mathcal{N}_n (D_{out}, \sigma_n^2).
\label{eq:mtl_norm_desc_copy}
\end{equation}
\noindent Expanding the norm distribution we have:
\begin{multline}
p(\mathbf{y}_{desc}|D_{out}, s) = \frac{s}{\sigma_p \sqrt{2\pi}} \exp{\left[\frac{-1}{2} \left(\frac{\mathbf{y}_{desc}-D_{out}}{\sigma_p} \right)^2 \right]} \\
+ \frac{1-s}{\sigma_n \sqrt{2\pi}} \exp{\left[\frac{-1}{2} \left(\frac{\mathbf{y}_{desc}-D_{out}}{\sigma_n} \right)^2 \right]}.
\label{eq:mtl_norm_desc_extended}
\end{multline}
Considering the same variance for both losses and substituting the term $(y-D)^2$ for the respective loss we have:\cite{orb-slam2}
\begin{equation}
p(\mathbf{y}_{desc}|D_{out}) = \frac{1}{\sigma\sqrt{2\pi}} \left[ \exp\left( { \frac{-L_p}{2\sigma^2} }\right)
+ \exp{\left(\frac{-L_n}{2\sigma^2} \right)}\right].
\end{equation}
The $s$ term is no longer needed because each loss is related to each match case. Since we are not balancing the losses between each other, we define the likelihood as:
\begin{equation}
p(\mathbf{y}_{desc}|D_{out}) \propto \frac{1}{\sigma \sqrt{2\pi}} \exp\left( { \frac{-(L_p + L_n)}{2\sigma^2} }\right).
\end{equation}
\noindent So the log likelihood is:
\begin{equation}
\text{log} \, p(\mathbf{y}_{desc}|D_{out}) \propto -\frac{1}{2\sigma^2} \left(L_p + L_n\right) - \text{log} \, \sigma.
\label{eq:mtl_norm_desc_vf_ap}
\end{equation}
\bibliographystyle{plain}
|
1,108,101,563,269 | arxiv | \section{Introduction\label{sec:intro}}
Understanding the\ microscopic structure\
of matter,\ how daily goods are formed,\ and
how they might be developed\ is always beneficial.\
Low-temperature\ polymer electrolyte membrane\
fuel cells\ (PEMFCs),\ also known as proton exchange\
membrane fuel cells,\ have stimulated interest in\
recent years as\ a feasible power source,\
particularly in\ automotive applications~\cite{ham2011pd}.\
PEM fuel cells\ generate electricity by combining\
hydrogen\ (or hydrogen-rich fuel) and oxygen\
(from the air),\ resulting in low pollution\
and high fuel economy~\cite{vielstich2009handbook}.\
The development of\ highly active and low-cost catalysts\
is a major challenge\ before proton exchange membrane\
fuel cells (PEMFCs)\ can find large-scale practical\
applications~\cite{wieckowski2003catalysis,weber2018carbon,
antolini2003formation, shao2007understanding,yu2007recent,
chen2009shape, peng2009designer, liu2011tuning}.
Platinum has\ been widely chosen\ for a catalytic\
role in\ PEMFC since a\ decades ago~\cite{HL69}.\
However,\ there is a need to find\ replacement\
materials\ because Pt has\ been used as a\
catalyst for fuel\ cells for\ a long time\ and\
might face\ being\ a short\ supply in the future.\
Even currently, one\ of the\ major barriers\
for a\ cheap\ commercialization\ of\ fuel cell\
engines\ is the electrodes'\ high cost,\
which is\ mostly\ due to\ the\ use of\
expensive\ platinum~(Pt)-containing\
electro-catalysts.\ Furthermore,\ the oxygen\
reduction\ reaction~(ORR)\ at the cathode is\
the PEM fuel cell's\ rate-limiting phase,\
in which molecular\ oxygen is\ destroyed and\
mixed with protons\ and\ electrons\ supplied by\
the anode through\ the membrane and\ external\
circuit\ to form water\ as\ by-product.\
As a result,\ platinum-based\ catalysts can\
be poisoned by\ carbon monoxide~(CO)\ and other\
pollutants~\cite{shao2006palladium}.\
Because of these\ characteristics,\ pursuing for\
alternative\ catalyst materials\ have gotten a\
lot of research\ attention.\
A\ recent\ study\ within\ our\ research\ group~\cite{Nigussa_2019}\
has\ investigated\ palladium\ as\ a possible\
candidate\ for\ replacement.\ This\ corresponds\ to\
$\textit{\rm{x}}=0$,\ in\ the\ alloy\ composition.\
However,\ still\ because\ of\ the\ rare\ availability\
of\ Pd,\ consideration\ of\ further\ other\
alloys\ as\ a\ replacement\ candidate\ is\ of\
interest\ in\ this\ work.\
As a result,\ one of our primary\ motives is to\
investigate the properties\ of the Pd$\rm_{1-x}$Ag$\rm_{{x}}$\
($\textit{\rm{x}}=0.25-0.75$)\ alloy as\ components\
of fuel\ cell,\ as electrode.\ This\ could\ also\
represent\ a\ study\ on\ a\ part\ of\ a\ grand\ alloy\
matrix\ Pt$\rm_{1-x}$Ag$\rm_{x}$Pd$\rm_{1-x}$Ag$\rm_{x}$,\
making\ the\ fuel\ cell\ electrode,\ where\ investigating\
how\ certain\ region\ of\ the\ alloy\ dominated\ by\
Pd$\rm_{1-x}$Ag$\rm_{x}$\ behaves\ is\ the\ central\
interest\ point\ of\ this\ work.\ Studying\ efficiency\
of\ candidate\ replacement\ catalysts\ goes\ through\
investigating\ interactions\ of\ molecules\
of\ fuel\ with\ components\ of\ the\ fuel cell\
as\ well\ as\ also\ transport\ of\ the\ molecules\
of\ fuel\ within\ the\ fuel\ cell.\
The interaction of molecules\ of fuel with the fuel cell\
electrode\ is studied by Density\ Functional Theory~(DFT),\
while\ desorption of molecules\ of the\ fuel from\ the fuel\
cell electrode\ is studied\ by Kinetic Monte Carlo~(KMC)\
simulation.\ A\ Simplified diagram of the PEM fuel cell\
operating principles\ is shown in Fig.~\ref{fig:fig1}.\
\begin{figure}[htp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
\includegraphics[height=6cm,width = 9cm]{Figures/Fig1.eps}
\end{adjustbox}
\caption{Simplified diagram of the PEM fuel cell\
operating process,\ taken\
from~\cite{salameh2014dynamic}.\
\label{fig:fig1}}
\end{figure}
In this paper,\ we present the results\
adsorption properties based\ on first\
principles DFT calculation,\ as well as\
a\ monte carlo study\ of desorption\ process\
from\ Pd$\rm_{1-x}$Ag$\rm_{x}$~($\rm{x}=$~0.25,\
0.50,\ 0.75) surface alloy where the\ DFT\
outputs\ are used as empirical~parameters.\
Our\ presentation\ focuses\ on\
Pd$\rm_{3}$Ag~(111),~specifically\
due to its stability\ compared to\
PdAg and PdAg$\rm_{3}$\ surfaces.\
The paper is organized as follows.~In the\
next\ section (sec.~\ref{sec:comp}),\ a\
detail account\ of the computational\ method\
is presented.\ Results\ and discussion are\
presented in\ section~\ref{sec:res},\ with\
the conclusion\ being presented in\ section~\ref{sec:conc}.
\section{Computational Methods\label{sec:comp}}
\subsection{Density Functional Theory~(DFT)}
An ab-initio simulations within gpaw code~\cite{55enkovaara2010electronic}\
is\ used\ to examine the\ electronic structure\
properties of the\ Palladium-Silver alloy.\
The electron wavefunction\ is approximated in gpaw\
using\ a projector augmented wave~(PAW) modality\
$\&$ expanded over\ a planewave basis set~\cite{56mortensen2005real,
57blochl1994improved}.\ The PAW technique\ is designed\
to successfully deal\ with the so-called pseudopotential\
idea,\ which is used\ in other DFT codes.\
The exchange-correlation\ energies are\ treated\
using\ PBE~\cite{24perdew1996generalized}.\
The PAW data set takes care of the electron-ion\
interactions.\ The\ k-points\ of\ the\ Brillouin zone~(BZ)\ are\
generated from the input ${\bf{k}}$-mesh\ using\
the\ Monkhorst-Pack scheme~\cite{59monkhorst1976special}.\
The\ number\ of valence electrons\ considered\ for\
each element\ within the\ paw\ data\ sets\ is\ Pd:16,\
Ag:17,\ H:1, C:4,\ and O:6.\ Geometry\
relaxations are\ carried out using BFGS\ minimizer~\cite{BS82},\
where optimizations\ of\ the atomic\ coordinates\
and the unit\ cell degrees of freedom\ is done\
within\ the concept\ of the Hellmann-Feynman forces\
and\ stresses~\cite{PRF39, NM85}\ as\ calculated on\
the Born-Oppenheimer~(BO) surface~\cite{WM91}.\
The convergence\ criteria for the forces\ were set\
at 0.05 eV/{\AA}.
Surfaces\ are\ modelled\ by\ a\ slab\ containing\
four\ atomic layers of\ the\ desired\ $hkl$\ indices.\
The\ top two atomic\ layers are\ allowed\ to relax\
upon\ geometry\ optimizations\ while\ the\ bottom\
two\ layers\ are\ frozen\ to\ a\ bulk\ geometry.\
A\ vacuum\ layer\ of\ 10~{\AA}\ is\ added\ to\
the\ slabs\ making\ a\ supercell\ which\ avoids\
lateral\ interaction\ between\ slabs,\ while\
also\ mimicks\ a\ three\ dimensional cell.\
Free\ adsorbate\ molecules/atoms\ prior\ to
adsorption are\ relaxed\ in\ a\ cubic\ unit cell\
of\ side length\ 10~{\AA}.\ Adsorptions\ are\ done\
by\ putting\ adsorbate\ molecules / atoms\ on\ the\
top\ most\ atomic\ layers\ of\ the\ slabs\ $\&$\
then\ allowing\ geometry\ relaxations\ to\ take\
place.\ Adsorption\ energies\ are\ calculated\
according\ to:\
\begin{equation}
E_{ads} = -(E_{SM} - E_S - E_M) \label{eads}
\end{equation}
Here,\ $E_S$ denotes total\ energy of a clean surface,\
$E_M$\ denotes total energy\ of an isolated molecule,\
and $E_{SM}$ denotes\ total energy of a relaxed geometry\
containing both\ substrate and molecule.\
All energies are per supercell.\ Accordingly to\
Eq.~\eqref{eads},\ thus,\ positive values\
in energies indicate\ exothermic reactions\
while negative\ values denote endothermic.\
A\ cubic\ unit cell\ with 3~Pd and 1~Ag\ atoms\
has been\ considered for Pd$\rm_{3}$Ag bulk\
structure.~For\ PdAg structure, the\ cubic\ unit\
cell with 2~Pd and 2~Ag\ atoms\ has been\ used.\
For\ PdAg$\rm_{3}$\ structure,\ the cubic unit\
cell with 1~Pd and 3~Ag\ atoms\ has\ been\ used.\
Since\ plane wave basis set\ is used\ for the\
expansion\ of electron\ wavefunctions,\ the\
k-points $\&$\ cut-off\ energies\ chosen\ would\
impact\ on\ total\ energy\ of\ the\ systems.\
Our\ calculations\ show\ that\
{\bf{k}}-mesh of 8{$\times$}8{$\times$}8\ can\
be\ optimum\ in\ the\ bulk\ calculations.\
With a fixed {\bf{k}}-mesh of $8{\times}8{\times}8$,\
a\ convergence test\ of the total energy\ with\
respect to\ energy cutoff\ is performed.\
As a result,\ we have chosen\ a\ cut-off\
energy~(ecut)~of\ 600 eV\ as\ an\ optimum\
in\ the\ rest\ of\ our\ calculations.\ With\
surface\ calculations,\ correspondingly,\
a\ {\bf{k}}-mesh of\ 8{$\times$}8{$\times$}1\
and an ecut of 600~eV\ is\ used.\
\subsection{Kinetic Monte Carlo~(kMC)}
In\ experiments,\ thermodynamic\ and kinetic\
parameters\ of\ desorption\ processes or\
decomposition\ reactions\ can be\ determined\
using\ temperature programmed desorption~(TPD).\
In simulations,\ there are\
numerous approaches\ for simulating\ TPD\
spectra,\ with Monte Carlo\ simulation\
being one of\ the most useful\ and crucial.\
This approach\ has the distinct\ advantage\
of requiring\ as fewest assumptions as\
possible.\ The probability\ for a particle at\
site $i$ to\ desorb in\ the interval~between\
$t$,~$\&$\ ($t$ + $\Delta$t)~may be\ calculated\
directly\ using Eq.~\eqref{rd1},\
\begin{equation}
r\rm_{d} = \nu \theta^n exp\left(-\frac{E\rm_{\text{ads}}}{RT}\right)
\label{rd1}
\end{equation}
Where $\nu = 10^{13} s^{-1}$,\ is the vibrational\
pre-factor\ or pre-exponential\ factor.\ R and T\
are\ universal\ gas constant and absolute\ temperature,\
respectively.\ The\ temperature\ in the simulation\
is updated\ by \ Eq.~\eqref{beta}.\
\begin{equation}
T = T\rm_{0} + {\beta}\hspace{0.2mm}t
\label{beta}
\end{equation}
where\ $t$\ is\ time for a\ desorption event and
$\beta$ is the heating rate.\
The probability\ for a particle at\ site $i$ to\
desorb in\ the interval~between\ $t$,~$\&$\
($t$ + $\Delta$t) when lateral\ interaction is\
considered\ is given by Eq.~\eqref{rd},\
\begin{equation}
r_d = -\frac{d\theta}{dt} = \nu \theta^n
exp\left(-\frac{E\rm_{\text{ads}}
+E\rm_{\text{int}}(\theta)}{RT}\right)
\label{rd}
\end{equation}
$\nu$ is\ the so-called\ pre-exponential factor,\
$n$ is\ the reaction order\ or kinetic desorption\
order~(with values~0, 1, and 2),\ $E\rm_{\text{ads}}$\
is\ the activation energy\ for desorption (for this\
work taken from DFT),\ $E\rm_{\text{int}}$ is\
the interaction\ energies between\ the adsorbate\
molecules,\ and $R$ and T\ are\ the universal\
gas\ constant and sample\ surface\ temperature,\
respectively.\ The\ interaction\ term\
of\ Eq.~\eqref{rd}\ is\ approximated\ according\
to
\begin{equation}
E\rm_{int}={\sum\limits_{k}}{N\rm_{kn}}E\rm_{kn}
\label{eqint2}
\end{equation}
with\ $E\rm_{kn}$~(pair-interaction\ energy)\
values\ are\ taken\ from\ literature~\cite{sraaen}.\
$N\rm_{kn}$~denotes\ number\ of\ $k\rm^{th}$\
nearest\ neighbor.\ For\ the\ Markovian\
process,\ the\ metropolis\ algorithm\ within\
the\ concept\ of\ Meng~and~Weinberg~\cite{BW94}\
is\ applied.\
\section{Results and discussion \label{sec:res}}
\subsection{Equation of state~(EOS) and cohesive energy}
We determine\ the equilibrium lattice\ constant\
and\ bulk modulus~(material's\ resistance to\
external\ pressure),\ using the equation\ of\
state curve-fitting\ approach\ for\ different\
palladium-silver\ bulk\ structures.\ The\ EOS\
is\ obtained\ by\ a\ curve\ fits\ within\
Stabilized Jellium model~(sjeos)~\cite{APBAF2003}\
to\ an\ energy versus volume\ calculation\ data,\
which\ also\ additionally\ outputs\
other\ important\ bulk structural\ quantities,\
such\ as minimum volume\ and\ minimum energy.\
This fit approach\ seems to yield a\ bulk modulus
values which\ is more\ closer to\ experiments\
than\ with using\ other fit methods\
such as Birch-Murnaghan~\cite{FDM44,FB78}~approach.\
However,\ the predicted values of the optimum\
lattice\ constants by using all\ the fit methods\
have\ appeared to\ be identical.\
The\ output\ parameters\ obtained\ within\ the\
sjeos\ fit\ approach,\ along\ with\ comparison\
with\ experiment\ results,\ is\
presented\ in\ Table~\ref{tab1}.\ The experimental\
results are\ taken from\ literatures~\cite{lovvik2002density,
coles1956lattice, kittel1996introduction,chan1969thermodynamic}.\
The cohesive\ energy~[eV/atom]~is\ defined\ as\ the\
energy\ required\ to\ dissociate\ the\ alloy\
compound into~free~(neutral)\ atoms.\ It\ is\
calculated\ according\ to\
\begin{equation}
E\rm_{coh} = {\sum\limits_{\rm_{atom}}}{E\rm_{atom}}-
\frac{1}{N}{E\rm_{bulk}}
\end{equation}
where $E\rm_{coh}$ is the\ cohesive energy,\
$N$\ is\ the number\ of atoms\ in\ the\ unit\
cell\ of\ the\ palladium-silver\ alloy.\
$E\rm_{atom}$\ is the atomic\ energy of a\
free\ palladium and silver\ atoms.\
Accordingly,\ as\ also\ shown\ in\ Table~\ref{tab1},\
the\ cohesive energy of Pd$\rm_{3}$Ag $>$ PdAg $>$ PdAg$\rm_{3}$.\
That\ means\ Pd$\rm_{3}$Ag\ is\ more\ stable.\\
\begin{table*}[htp!]
\addtolength{\tabcolsep}{4.8mm}
\renewcommand{\arraystretch}{2.2}
\centering
\caption{The lattice constant $a$~[{\AA}],\ the bulk\
modulus B~[GPa],\ formation energy~$E\rm_{f}$~[eV/atom],\
and\ the cohesive energies~$E\rm_{coh}$~[eV/atom]~of\
selected\ metals\ $\&$\ alloy\ systems\ of\ this\
work.\label{tab1}}
\begin{tabular}{lcccccc}
\hline
\multirow{2}*{Quantity} & \multirow{2}*{Source} & \multicolumn{5}{c}{System}\\
\cline{3-7}
{} & {} & Pd & Pd$\rm_{3}$Ag & PdAg & PdAg$\rm_{3}$ & Ag\\
\hline
\multirow{2}*{$a$} & {This work} & 3.94 & 3.98 & 4.02 & 4.07 & 4.07\\
& {Exp.} & 3.88 & 3.92 & 3.97 & 4.02 & 4.07\\
\hline
\multirow{2}*{B} & {This work} & 188 & 139.5 & 127 & 103.7 & 109\\
& {Exp.} & 180.8 & 139.4 & 116 & 103.1 & 100\\
\hline
\multirow{2}*{$E\rm_{f}$} & {This work} & - & 3.08 & 3.31 & 3.61 & -\\
& {Exp.} & - & - & - & - & -\\
\hline
\multirow{2}*{$E\rm_{coh}$} & {This work} & - & 3.31 & 2.89 & 2.60 & -\\
& {Exp.} & - & 3.60 & 3.37 & 3.16 & -\\
\hline
\end{tabular}
\end{table*}
The energy\ necessary to\ form the\
alloy compound\ from\ its separate\ elemental
bulk structures\ is represented by\
formation\ energy,~$E\rm_{f}$~[eV/atom],~which\
is\ given\ by\
\begin{equation}
E\rm_{f}=\frac{1}{N}{E\rm_{bulk}}
- {\sum\limits_{i}}\frac{1}{N\rm_{i}}{E\rm_{b}^{i}}
\end{equation}
where\ $N$~is\ number\ of\ atoms\ in\ the\
unit\ cell\ of\ the\ alloy\ compound,\
$E\rm_{bulk}$\ is\ total\ energy\ of\
the\ bulk\ alloy\ compound,\ $E\rm_{b}^{i}$\ is\
total energy\ of\ bulk\ of\ element\ $i$,\
$\&$~$N\rm_{i}$\ is\ number\ of\ atoms\
in\ the\ unit\ cell\ of\ the\ elemental\
bulk\ forms.\ In terms of\ formation energy,\
thus,\ $E\rm_{f}$\ of\ Pd$\rm_{3}$Ag $<$ PdAg $<$ PdAg$\rm_{3}$\
(see Table~\ref{tab1}).\ This\ means\ Pd$\rm_{3}$Ag\
alloy\ is\ relatively\ a\ more\ favorable\
to\ be\ formed.\
From\ the\ EOS\ output,\ it is~investigated\
that\ the total energy per atom\ values\
are\ about\ -3.6~eV,~-3.4~eV,~-3.1~eV,\ respectively,\
for Pd$\rm_3$Ag,~PdAg,~$\&$~PdAg$\rm_{3}$~structures.\
Furthermore,~the\ bulk modulus~(B)~value\ for Pd$\rm_{3}$Ag\
$>$ PdAg $>$ PdAg$\rm_{3}$\ means\ that\ Pd$\rm_{3}$Ag\
has\ relatively\ better\ operating~$\&$~endurance\
capacity\ under\ extreme\ pressures.\ The\ fact\
that\ the\ optimal\ geometry\ structure\ properties\
presented\ in\ Table~\ref{tab1}~compare\ reasonably\
well\ with\ experiments\ also\ gives\ the\ authors\
a\ full\ confidence\ on\ the\ validity\ of\ the\
chosen\ model\ alloy\ compound~$\&$~surfaces\ of\
this\ study.\
\subsection{Surface energy}
Surface energy is used to characterize\
the surface of a substrate.\ Its values\ range\
from high to low.\ Adhesion between\ unlike materials\
is determined by\ the molecular force\ of attraction\
between them.\ The strength of\ the attraction is\
determined by\ the substrate's surface energy.\
A strong molecule\ attraction is\ indicated by\
a high surface energy,\ whereas a weak molecular\
attraction is\ indicated by\ a low surface energy.\
A high surface energy\ would\ also indicate\
increased\ reactivity\ with\ adsorbates,\ while\
low\ surface\ energy\ would\ indicate\ increased\
stability.\ Surface characteristics\ are\ extremely\
important,\ especially in catalysis.\ Generally,\
the surface\ energy~($\sigma$~[eV/{\AA}$\rm^{2}$])\
is defined\ as\
\begin{equation}
\sigma = \frac{1}{2A}\left(E\rm_{slab} -
\frac{N\rm_{slab}}{{N\rm_{bulk}}} E\rm_{bulk}\right)
\end{equation}
where~$E\rm_{slab}$ denotes the total\ energy of a slab\
unit cell,\ $N\rm_{slab}$ means number\ of atoms in the\
slab unit cell,\ $N\rm_{bulk}$ means number\ of atoms\
in the bulk\ unit\ cell,\ and $E\rm_{bulk}$\ is the\
total energy of\ a bulk unit~cell.\ $A$ is the\ surface\
area of\ a slab\ unit cell,\ and the\ factor\ half\
accounts\ for the top\ and bottom surfaces.\\
\begin{table*}[htp!]
\addtolength{\tabcolsep}{12mm}
\renewcommand{\arraystretch}{2.2}
\centering
\caption{Surface energies of Pd$\rm_{1-x}$Ag$\rm_{{x}}$~surfaces,\
$\text{x}$~goes from 0.25~to~0.75.~Values~corresponding\
to\ 100,~110,~$\&$~111 facets\ are\ given.\label{tab2}}
\begin{tabular}{lccc}
\hline
\multirow{2}*{System}{} & \multicolumn{3}{c}{{Surface energy~($\sigma$)~[eV/{\AA}$\rm^{2}$]}}\\
\cline{2-4}
{} & {100} & {110} & {111}\\
\hline
Pd$\rm_3$Ag & 0.085 & 0.089 & 0.072\\
PdAg & 0.076 & 0.078 & 0.062\\
PdAg$\rm_3$ & 0.056 & 0.060 & 0.049\\
\hline
\end{tabular}
\end{table*}
As\ shown\ in\ Table~\ref{tab2},\ the surface\
energies\ for the surface facets increase according\
to\ $(111)<(100)<(110)$\ for\ all\ the\ structures.\
This\ means\ (111)~surfaces\ offer\ a\ relatively\
more\ stable\ geometries,\ while\ (110)\ facets\
would\ likely\ be\ expected\ to\ be\ more\ reactive\
to\ impurities / adsorbates.\ However,\ the\
surface\ energies\ values,\ when\ compared\
across\ the\ different\ structures,\ are\ closely\
related,\ albeit\ the\ small\ differences\ can\ be\
taken\ as\ being\ a\ marginal\ errors.\ Studies of\
interactions of\ these surfaces\ with\
adsorbates:~hydrogen~(H),\ and carbon monoxide~(CO)\
are discussed in\ the next~subsections.\
The adsorption site\ terminologies of\ top~(T),\
bridge~(B),\ and hollow~(H)~refer to the\
placement\ of adsorbates\ at the geometries\
depicted in Fig.~\ref{fig2}.\
The corresponding\ adsorption results are\
shown in\ Tables~\ref{tab3}~$\&$~\ref{tab4}.\
\begin{figure*}[ht!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
(\centering a){{\includegraphics[scale=0.1]{Figures/Fig2a.eps}}}
\qquad
(\centering b) {{\includegraphics[scale=0.12]{Figures/Fig2b.eps}}}
\qquad
(\centering c){{\includegraphics[width=6cm]{Figures/Fig2c.eps}}}
\qquad
(\centering d) {{\includegraphics[width=6cm]{Figures/Fig2d.eps}}}
\end{adjustbox}
\caption{(a) Pd$\rm_{3}$Ag~(111) surface and\
(b) Pd$\rm_{3}$Ag~(111)~top~(T)~site view.
(c) Pd$\rm_{3}$Ag~(111)~bridge~(B)~site view\ and\
(d) Pd$\rm_{3}$Ag~(111)~hollow~(H)~site view.
Color online. Colors:~Pd-light~grey,~H-cyan,\
Ag-dark~cyan.\label{fig2}}
\end{figure*}
\subsection{Adsorption~energy~of~Hydrogen~(H)~and~Carbon~monoxide~(CO)}
Upon\ the\ adsorptions,~a CO molecule\ or H is placed\
on\ the surfaces on\ one side of\ the slab at\ either the\
on-top~(T) site,\ on bridge~(B),\ or fcc hollow~(H)~site.\
The adsorption energies\ are calculated as\
\begin{eqnarray}
E\rm_{ads} &=& -\left(E\rm_{slab + CO} - E\rm_{slab}
- E\rm_{CO}\right) \hspace{4mm} \text{for~CO}\\
E\rm_{ads}& = & -\left(E\rm_{slab + H} - E\rm_{slab}
- 0.5\times E\rm_{H\rm_2}\right) \hspace{4mm} \text{for~H}
\end{eqnarray}
where $E\rm_{CO}$ is\ the energy of an isolated\
(free)~$\rm{CO}$ molecule,\ and $E\rm_{H\rm_{2}}$ is\
the energy of\ an isolated~H$\rm_{2}$\ molecule.\
The results are\ given in Tables~\ref{tab3}~$\&$~\ref{tab4}.\
\begin{table*}[ht!]
\addtolength{\tabcolsep}{9.0mm}
\renewcommand{\arraystretch}{1.3}
\centering
\caption{Adsorption energies\ of hydrogen~(H)\ on different\
surface faces~(i.e.,\ 100,\ 110,\ $\&$\ 111)~and change\
in charge of\ adsorbate for selected\
system~(-~sign indicates a loss) to palladium-silver\
alloy~system~(Pd$\rm_{1-x}$Ag$\rm_{x}$,\
$\text{x}$~goes~from~0.25~to~0.75).\
The\ adsorptions\ correspond\ to\ adsorbate\
coverage of 1 adsorbate species per slab\
unit cell~(1~ML).\label{tab3}}
\begin{tabular}{lcccc}
\hline
{System} & {Surface} & {Site} & E$\rm_{ads}$~[eV] & ${\Delta}Q\rm_{H}~[e]$\\
\hline
\multirow{9}*{Pd$\rm_{3}$Ag} & \multirow{3}*{100} & T & -0.81 & {-}\\
{} & {} & B & 0.15 & -0.06\\
{} & {} & H & 0.32 & {-}\\
\cline{3-5}
{} & \multirow{3}*{110} & T & -0.34 & {-}\\
{} & {} & B & -0.15 & -0.06\\
{} & {} & H & -0.52 & {-}\\
\cline{3-5}
{} & \multirow{3}*{111} & T & -0.34 & {-}\\
{} & {} & B & 0.34 & -0.16\\
{} & {} & H & 0.59 & -0.10\\
\cline{2-5}
\multirow{9}*{PdAg} & \multirow{3}*{100} & T & -0.84 & {-}\\
{} & {} & B & -0.34 & -0.11\\
{} & {} & H & -0.29 & {-}\\
\cline{3-5}
{} & \multirow{3}*{110} & T & -0.40 & {-}\\
{} & {} & B & 0.14 & -0.14\\
{} & {} & H & -0.27 & {}\\
\cline{3-5}
{} & \multirow{3}*{111} & T & -0.40 & -0.09\\
{} & {} & B & 0.02 & -0.14\\
{} & {} & H & 0.34 & -0.06\\
\cline{2-5}
\multirow{9}*{PdAg$\rm_{3}$} & \multirow{3}*{100} & T & -0.88 & {-}\\
{} & {} & B & -0.39 & -0.09\\
{} & {} & H & -0.57 & {}\\
\cline{3-5}
{} & \multirow{3}*{110} & T & -0.97 & {-}\\
{} & {} & B & -0.37 & {-}\\
{} & {} & H & -0.31 & {-}\\
\cline{3-5}
{} & \multirow{3}*{111} & T & -0.34 & -0.09\\
{} & {} & B & 0.04 & -0.16\\
{} & {} & H & 0.08 & -0.19\\
\hline
\end{tabular}
\end{table*}
\begin{table*}[ht!]
\addtolength{\tabcolsep}{9.0mm}
\renewcommand{\arraystretch}{1.3}
\centering
\caption{Adsorption energies\ of carbon monoxide~(CO)\ on different\
surface faces~(i.e.,\ 100,\ 110,\ $\&$\ 111)~and change\
in charge of\ adsorbate for selected\
system~(+~sign indicates a gain) from palladium-silver\
alloy~system~(Pd$\rm_{1-x}$Ag$\rm_{x}$,\
$\text{x}$~goes~from~0.25~to~0.75).\
The\ adsorptions\ correspond\ to\ adsorbate\
coverage of 1 adsorbate species per slab\
unit cell~(1~ML).\label{tab4}}
\begin{tabular}{lcccc}
\hline
{System} & {Surface} & {Site} & E$\rm_{ads}$~[eV] & ${\Delta}Q\rm_{CO}~[e]$\\
\hline
\multirow{9}*{Pd$\rm_{3}$Ag} & \multirow{3}*{100} & T & 0.09 & {}\\
{} & {} & B & 0.67 & +0.16\\
{} & {} & H & -1.62 & {-}\\
\cline{3-5}
{} & \multirow{3}*{110} & T & 1.66 & {-}\\
{} & {} & B & 1.68 & +0.15\\
{} & {} & H & -0.95 & {-}\\
\cline{3-5}
{} & \multirow{3}*{111} & T & 1.35 & +0.16\\
{} & {} & B & 1.26 & +0.15\\
{} & {} & H & 0.83 & {}\\
\cline{2-5}
\multirow{9}*{PdAg} & \multirow{3}*{100} & T & 0.06 & {-}\\
{} & {} & B & -0.78 & +0.26\\
{} & {} & H & -2.89 & {-}\\
\cline{3-5}
{} & \multirow{3}*{110} & T & 1.54 & {-}\\
{} & {} & B & 0.55 & +0.18\\
{} & {} & H & -2.18 & {}\\
\cline{3-5}
{} & \multirow{3}*{111} & T & 1.37 & +0.18\\
{} & {} & B & 0.15 & +0.25\\
{} & {} & H & -0.01 & {-}\\
\cline{2-5}
\multirow{9}*{PdAg$\rm_{3}$} & \multirow{3}*{100} & T & -0.20 & {-}\\
{} & {} & B & -0.80 & +0.15\\
{} & {} & H & -3.06 & {}\\
\cline{3-5}
{} & \multirow{3}*{110} & T & -0.12 & {-}\\
{} & {} & B & -1.07 & {-}\\
{} & {} & H & -4.25 & {-}\\
\cline{3-5}
{} & \multirow{3}*{111} & T & 1.33 & +0.19\\
{} & {} & B & 0.10 & +0.30\\
{} & {} & H & -0.92 & {-}\\
\hline
\end{tabular}
\end{table*}
The\ results\ in\ the\ tables\ show\ that\
Pd$\rm_{3}$Ag~(111)~shows\ modest\ reactivity\
towards\ both\ adsorbates,~while~Pd$\rm_{3}$Ag~(110)\
looks\ to\ be\ more\ reactive\ with\ CO~(albeit\
it\ may\ result\ in a\ poisoning\ effect).\
However,\ Pd$\rm_{3}$Ag~(111)\ is\ also\
a\ more\ favored\ surface for consideration,\
since\ it\ shows\ a\ relatively\ more\ stability.\
The\ charge\ differences\ ${\Delta}Q$\ are\
computed\ according\ to\
\begin{equation}
{\Delta}Q = e{\int}{\Delta}n(r){d\rm^{3}}r
\label{eqdelq}
\end{equation}
where\ ${\Delta}n(r)$\ is\ electron\ density\
differences obtained\ from the\ ground state\
calculations.\
The\ charge\ differences\ are\ calculated\ according\
to\ the\ concept\ in\ Eq.~\eqref{eqdelq}~$\&$~using\
Bader\ charge\ formulation~\cite{RB90}\
as implemented\ by\ literature~\cite{TSH2009}.
Hydrogen\ seems\ to\ release\ charges\
to\ the\ surfaces,\ while\ CO\ gains.\
These values~(see~Table~\ref{tab3})\ show\ a\
donation\ of\ 0.16$e$~per\ one\ hydrogen\ atom\
to\ the\ system.~CO\ gains\ nearly\ about\ the\
same\ from\ the\ system~(see~Table~\ref{tab4}).\
\subsubsection{Adsorption\ energies\ on\ Pd$\rm_{3}$Ag~surfaces}
On~(100)~surface,\ an adsorption energy\ of up to\
0.32~eV~is\ investigated\ upon\ a\ dissociated\
adsorption\ as hydrogen atoms\ at the H site.\ At\
such\ a\ higher\ adsorbate\ coverages\ as 1~ML,\
the\ energetically\ favorable adsorption\ sites\
are H~$\&$~B,\ with\ a\ relative\ favorability\
order\ being~H~$>$~B.\ In\ the\ case\ of\ CO\
adsorption,\ the\ adsorption\ energy is\
up\ to\ 0.67~eV,~$\&$~the\ favored\ sites\
are\ T~$\&$~B,~with\ the\ order\ being B~$>$~T.\\
On~(110)~surface,~adsorption\ of\ hydrogen\ atoms\
seems\ to\ be\ endothermic\ at\ all\ sites,\
while\ adsorption\ of\ CO\ can have\ adsorption\
energy\ of\ up to\ 1.68~eV,\ with\ favorable\
sites\ being\ T~$\&$~B,\ and\ the\ order\
of\ favorability\ being\ B~$>$~T.\
On~(111)~surface,\ a dissociated adsorption\
as\ hydrogen atoms\ is\ preferred\ with\
adsorption\ energies\ of\ up to 0.59~eV,\
as shown in the\ Table~\ref{tab3}.\ The\
favorable\ sites\ are\ H $\&$ B,\ with\ the\
order\ of\ relative\ favorability\ being\
H~$>$~B.\ With\ adsorption\ of\ CO,\ all\
sites\ are\ favorable\ with\ the\ adsorption\
energy of up to 1.35~eV,~$\&$~the\ relative\
favorability\ order\ of\ the\ sites\
seems\ as~T~$>$~B~$>$~H.\ The\ desorption\ energy\
values on\ Pd$\rm_{3}$Ag~(111)\ is\ comparable\
to\ but\ slightly\ smaller\ than\ a\ counterpart\
on\ Pt~(111)~\cite{KN2020},~means\ an\ improved\
performance\ of\ the\ model\ for\ the\
desired\ application.~Smaller\ adsorption\
energy\ of hydrogen\ would\ be\ good\ to\
a\ fuel\ cell\ operation\ at\ lower\
temperatures,\ while\ smaller\ adsorption\
energy\ of\ CO means\ good\ chance\ for\
reducing\ CO\ poisoning.\
\subsubsection{Adsorption\ energies\ on\ PdAg~surfaces}
With\ adsorption\ of\ hydrogen\ at\ a\ coverage\
of\ 1~ML,\ the\ B\ site\ of\ (110)~and\
the\ B~$\&$~H~sites~of\ (111)\ surface\
are\ favorable\ adsorption\ sites.~The\ adsorption\
energies\ of\ up to~0.34~eV,\ which occurs\ at\
H~site\ of\ (111)\ is\ investigated~(Table~\ref{tab3}).\\
With\ adsorption\ of\ CO,\ T~site\ of\ (100),\
T~$\&$~B\ sites\ of\ (110),\ and T~$\&$~B\ sites\
of\ (111)\ are\ the\ favored\ adsorption\
sites,\ with\ adsorption\ energies\ of\ up to\
1.54~eV,\ which\ occurs\ at\ T\ site\ of\
(110)~surface.\
\subsubsection{Adsorption\ energies\ on\ PdAg$\rm_{3}$~surfaces}
At\ such\ a\ higher\ adsorbate\ coverage\ of\
1~ML,~hydrogen\ adsorbs\ on\ (111)\ surface\
at\ a\ favorable\ sites\ of\ B~$\&$~H,\
with\ adsorption\ energy\ of\ up to\ 0.1~eV.\
Adsorption\ of\ CO\ happens\ at\ a\ favorable\
sites\ of\ T~$\&$~B,\ with\ adsorption\
energies\ of\ up to\ 1.3~eV~at\ the\ T~site.\
\subsection{Electronic Properties}
\subsubsection{Density of States~(DOS) and Band structures}
The\ DOS~$\&$~PDOS\ curves
are\ presented\ in~Fig.~\ref{fig34}.\
\begin{figure*}[htp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
(\centering a)~{{\includegraphics[scale=0.9]{Figures/Fig3.eps}}}
\qquad
(\centering b)~{{\includegraphics[scale=0.9]{Figures/Fig4.eps}}}
\end{adjustbox}
\caption{(a)~Density of states of bulk~Pd$\rm_{3}$Ag.~(b)~Projected
density of states of bulk~Pd$\rm_{3}$Ag\label{fig34}}
\end{figure*}
\begin{figure*}[htbp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
\includegraphics[scale=1.0]{Figures/Fig5.eps}
\end{adjustbox}
\caption{Band structure of bulk~Pd$\rm_{3}$Ag.\label{fig5}}
\end{figure*}
The\ dos\ peaks\ of\ the valence\ states\
show-up\ with\ significant\ amount\ in\
the\ energy\ region\ of\ E-E$\rm_{F}$~{$\in$}~[-6,0]~eV,\
see~Fig.~\ref{fig34}.~The\ $d$,~$p$,~$\&$~$s$\
orbitals~(Fig~\ref{fig34})\ contribute\ to\
the\ occupation\ states\ of\ dos.~The\
$d$-orbital\ has\ most\ states\ for\ occupation\
followed\ by\ $p$-orbital,~which in turn is\
followed by\ $s$-orbital.\ The\ $d$\ states\
dominate\ states\ near\ the\ Fermi level,\
while\ the\ $p$\ states\ contribute\ in\ the\
middle\ energy\ level,\ and\ the\ $s$-states\
contribute\ in\ the\ deeper\ energy\ levels.\
The\ band\ structure~(Fig.~\ref{fig5})~shows\
no\ energy\ gap,~$\&$~that\ the\ system\
has\ a\ typical\ metallic\ character.The\ dos\
plots\ of\ hydrogen,~$\&$~CO~adsorbed\
on\ Pd$\rm_{3}$Ag~(111)~is\ presented~in~Fig.~\ref{fig6}.\
\begin{figure*}[htbp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
(\centering a)~{{\includegraphics[scale=0.9]{Figures/Fig6a.eps}}}
\qquad
(\centering b)~{{\includegraphics[scale=0.9]{Figures/Fig6b.eps}}}
\end{adjustbox}
\caption{(a)~Density of states of Pd$\rm_{3}$Ag~(111)~before\
$\&$~after adsorption\ of~H.~(b)~Density of\
states of Pd$\rm_{3}$Ag~(111)~before~$\&$~after~adsorption\
of~CO.\label{fig6}}
\end{figure*}
A~slightly\ decreased\ states\ is\ seen\
upon\ adsorption,~as shown\ in\ the\ figures,\
due\ to\ partial\ occupation\ of\ the\ empty\
orbital\ states\ by\ the\ adsorptions.\ No\
major\ energy\ level\ shift\ is\ seen\ between\
dos curves\ of\ before\ $\&$~after\ adsorptions.\
However,\ the\ nature\ of\ curves\ at\ energy\
levels\ E-E$\rm_{F}$~of~-5~eV~$\&$-1~eV\
(see\ Fig.~\ref{fig6}~left\ side)\ means\
a\ sign\ of\ $d$-$s$\ hybridization\ between\
$d$ states\ of\ surface\ atoms~$\&$~$s$ states\
of\ the\ adsorbate\ hydrogen.~Similarly,\ from\
Fig.~\ref{fig6}~(right\ side),~we\ can\
see\ a\ $d$-$p$~hybridization~with\ the\
adsorption\ of\ CO.\
\begin{figure*}[htbp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
(\centering a)~{{\includegraphics[scale=0.5]{Figures/Fig7a.eps}}}
\qquad
(\centering b)~{{\includegraphics[scale=0.5]{Figures/Fig7b.eps}}}
\end{adjustbox}
\caption{(a)~Projected density of states of Pd$\rm_{3}$Ag~(111)~before\
$\&$~after\ adsorption of~hydrogen.~(b)~Projected\ density\
of states of\ Pd$\rm_{3}$Ag~(111)~before~$\&$~after adsorption\
of~CO.~The\ left\ sides\ within\ each\ graph~(a,~b)~is before\
adsorptions,\ while\ the\ right\ sides\ is\ after\
adsorption.\label{fig7}}
\end{figure*}
From\ Fig.~\ref{fig7},\ the\ $d$-$s$,~$\&$\
$d$-$p$ orbitals hybridization\ can be\ seen\
from\ the\ coinciding\ curve\ peaks~(see the\
figure\ on\ left,~$\&$~right\ side,~respectively).\\
By\ the\ adsorptions,\ the\ band\ structure\
plots~(Fig.~\ref{fig8})~show\ that\ no\
typical\ difference\ to\ the\ electrical\
properties\ of\ the\ surfaces\ happens,~except\
that\ the\ energy\ curves\ at\ the\ special\
k-points\ become\ more\ dense\ due\ to\ an\
added\ energy\ eigenvalues\ from\ the\
adsorbate\ states.\
\begin{figure*}[htbp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
(\centering a)~\includegraphics[scale=0.9]{Figures/Fig8a.eps}
(\centering b)~\includegraphics[scale=0.9]{Figures/Fig8b.eps}
\end{adjustbox}
\caption{a)~Band structure of clean~Pd$\rm_{3}$Ag~(111)~(left~side).\
b)~Band structure of Pd$\rm_{3}$Ag~(111)~after~adsorption\
of hydrogen~(right~side).\label{fig8}}
\end{figure*}
\subsection{KMC\ output\ of\ desorption\ process}
In\ this\ section,~a\ Monte\ Carlo\ simulation\
of\ the\ desorption\ process\ of\ adsorbates\
from\ the\ alloy\ surfaces\ is\ presented.\
In the simulation,~two cases are\ considered,\
with\ adsorbate coverage being~$\theta$ = 1 ML~(i.e.,\
1~molecule/atom\ of adsorbate per a ($1\times$1)\
surface unit cell).\ The first case\ is by\
considering a\ lateral interaction on the\
adsorbate which is given by~$15.8~(\theta - 0.25)^{1.3}$~eV,\
and the second case is by neglecting lateral\
interaction.\
\subsubsection{Simulation of desorption with\ a\ lateral\ interaction}
First,~we present\ the\ desorption\ process\ of\
hydrogen adsorbate~as\ follows.\
The adsorption energy\ of hydrogen at bridge\
site\ of\ Pd$\rm_{3}$Ag~(111)~is 0.34 eV~(Table~\ref{tab3}).\
Figure~\ref{fig9} shows the simulation\ output of\
first\ order desorption process by including\ the\
effect\ of\ lateral\ interaction for different\
heating\ rates of\ $\beta$ = 2 K/s, $\beta$ = 5 K/s\
and $\beta$ = 10 K/s.\
\begin{figure}[htbp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
\includegraphics[scale=0.5]{Figures/Fig9.eps}
\end{adjustbox}
\caption{Simulation output\ of desorption of hydrogen
from Pd$\rm_{3}$Ag~(111),~within first order\
for $\beta$ = 2 K/s~(dashed line,~blue color),\
$\beta$ = 5 K/s~(dash dot line,~red color),\
and~$\beta$ = 10 K/s~(solid line,~green color),\
with lateral\ interaction~included.\label{fig9}}
\end{figure}
The desorption peaks\ occur\ at temperatures\ of $T\rm_{m}$ = 280 K\
for~$\beta$ = 2 K/s,~ $T\rm_{m}$ = 335 K~for $\beta$ = 5 K/s,\
and $T\rm_{m}$ = 390 K~for~$\theta$ = 10 K/s.~The peaks\ shift\
to the right~(higher temperature) as\ the heating rate\
increases.\ For\ a simulation\ within\ second order,\ the\
output\ is\ presented\ in~Fig.~\ref{fig10}.\
\begin{figure}[htbp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
\includegraphics[scale=0.5]{Figures/Fig10.eps}
\end{adjustbox}
\caption{Simulation output\ of desorption of hydrogen
from Pd$\rm_{3}$Ag~(111),~within second order\
for $\beta$ = 2 K/s~(dashed line,~blue color),\
$\beta$ = 5 K/s~(dash dot line,~red color),\
and~$\beta$ = 10 K/s~(solid line,~green color),\
with lateral\ interaction~included.\label{fig10}}
\end{figure}
From the graph,\ the\ desorption\ peaks\ occur\ at\
temperatures\ of\ $T_{m}$ = 287 K~for~$\beta$ = 2 K/s),\
$T\rm_{m}$ = 345 K~for $\beta$ = 5 K/s,~and $T\rm_{m}$ = 400 K\
for $\theta$ = 10 K/s.~The peaks\ shift\ to the right~(higher\
temperature) as\ the heating rate\ increases.~The peaks\
occur at a\ relatively higher temperature,~by\ up to 10~K,\
compared\ to the case\ of\ first\ order.\\
Secondly,\ the simulation\ of desorption rate for\
the adsorbate\ CO\ is given\ as\ follows.\
The adsorption energy of CO at bridge\ site\ of\
Pd$\rm_{3}$Ag~(111)~is 1.26 eV~(Table~\ref{tab4}).\
The\ outcomes\ of\ simulation\ within\ first\
$\&$~second\ order\ rates\ is\ presented\ in\
Figs.~\ref{fig11}~$\&$~\ref{fig12}.
\begin{figure}[htbp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
\includegraphics[scale=0.5]{Figures/Fig11.eps}
\end{adjustbox}
\caption{Simulation output\ of desorption of CO\
from Pd$\rm_{3}$Ag~(111),~within first order\
for $\beta$ = 2 K/s~(dashed line,~blue color),\
$\beta$ = 5 K/s~(dash dot line,~red color),\
and~$\beta$ = 10 K/s~(solid line,~green color),\
with lateral\ interaction~included.\label{fig11}}
\end{figure}
\begin{figure}[htbp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
\includegraphics[scale=0.5]{Figures/Fig12.eps}
\end{adjustbox}
\caption{Simulation output\ of desorption of CO\
from Pd$\rm_{3}$Ag~(111),~within second order\
for $\beta$ = 2 K/s~(dashed line,~blue color),\
$\beta$ = 5 K/s~(dash dot line,~red color),\
and~$\beta$ = 10 K/s~(solid line,~green color),\
with lateral\ interaction~included.\label{fig12}}
\end{figure}
With\ lateral interaction included,~$\&$~within\
first\ order\ rate\ concept,\ the\ desorption\
peaks\ occur\ at\ temperatures\ of $T\rm_{m}$ = 300 K~for\
$\beta$ = 2 K/s),~$T\rm_{m}$ = 360 K~for $\beta$ = 5 K/s,\
and $T\rm_{m}$ = 420 K~for $\theta$ = 10 K/s~(see~Fig.~\ref{fig11}).\
The peaks\ shift\ to the right~(higher temperature)~as\
the heating\ rate\ increases.\ Within\ the second order\
rate\ concept,~$\&$~the lateral interaction\ included,\
the desorption peaks occur at temperatures of $T\rm_{m}$ = 308 K\
for~$\beta$ = 2 K/s,~$T\rm_{m}$ = 370 K~for $\beta$ = 5 K/s,\
and\ $T\rm_{m}$ = 430 K~for $\theta$ = 10 K/s~(see~Fig.~\ref{fig12}).\
The peaks\ shift to the right~(higher temperature) as the\
heating rate\ increases.~Compared\ to\ the\ peaks\ from\
first order rate concept,\ the peaks\ within the second\
order rate concept occur at a\ relatively higher~temperatures,\
by\ up to 10~K.~The width\ of\ desorption curves\ increases\
for second order\ process rate compared\ to\ that\ of\ the\
first\ order. The\ area under the curves\ is\ related the\
amount\ of desorbed\ particles.\
\subsubsection{Simulation of desorption\ without\ a\ lateral\ interaction}
Wherever\ it\ applies,~1$\rm^{st}$~$\&$~2$\rm^{nd}$\
order\ rate\ concepts\ is described\ in\ Eq.~\eqref{rd},\
while\ the\ concept\ of\ lateral\ interaction\ is\
given\ in\ Eq.~\eqref{eqint2}.\
In\ this\ case,\ the\ same adsorption energy\ of\
hydrogen on\ Pd$\rm_{3}$Ag~(111),\ 0.34~eV,~is\ used,\
but\ no\ lateral\ interaction\ is\ used.\ The\ output\
of\ the\ corresponding\ simulation\ is\ given in\
Figs.~\ref{fig13}~$\&$~\ref{fig14}.\
For a first order\ desorption rate\ concept,\ the\
desorption peaks\ occur at temperatures of\
$T\rm_{m}$ = 21 K~for\ $\beta$ = 2 K/s,\
$T\rm_{m}$ = 26 K~for $\beta$ = 5 K/s,~and $T\rm_{m}$ = 31 K\
for $\beta$ = 10 K/s~(see Fig.~\ref{fig13}).\
The peaks\ shift to the right~(higher\ temperature) as\
the heating\ rate increases.\ With\ the\ second\ order\
rate\ concept,\ the\ desorption peaks\ occur\
temperatures of $T\rm_{m}$ = 23 K\ for $\beta$ = 2 K/s\
$T\rm_{m}$ = 31 K~for $\beta$ = 5 K/s,\
and $T\rm_{m}$ = 41 K~for $\beta$ = 10 K/s~(see Fig.~\ref{fig14}).\
The peaks\ and temperature of the\ maximum desorption\
rate shift to the right~(higher temperature)\ as the\
heating rate increases.~Similar\ phenomena\ of\ such\
shifts\ with\ increased\ heating\ rate\ is\ also\
investigated\ for CO desorption from Cu~\cite{zakeri2004monte}.\
\begin{figure}[htbp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
\includegraphics[scale=0.5]{Figures/Fig13.eps}
\end{adjustbox}
\caption{Simulation output\ of desorption of hydrogen\
from Pd$\rm_{3}$Ag~(111),~within first order\
for $\beta$ = 2 K/s~(dashed line,~blue color),\
$\beta$ = 5 K/s~(dash dot line,~red color),\
and~$\beta$ = 10 K/s~(solid line,~green color),\
without including a lateral interaction~effect.\label{fig13}}
\end{figure}
\begin{figure}[htbp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
\includegraphics[scale=0.5]{Figures/Fig14.eps}
\end{adjustbox}
\caption{Simulation output\ of desorption of hydrogen\
from Pd$\rm_{3}$Ag~(111),~within second order\
for $\beta$ = 2 K/s~(dashed line,~blue color),\
$\beta$ = 5 K/s~(dash dot line,~red color),\
and~$\beta$ = 10 K/s~(solid line,~green color),\
without including a lateral interaction~effect.\label{fig14}}
\end{figure}
Furthermore,~the width\ of\ desorption curves\ increases\
for second order\ process rate compared\ to\ that\ of\ the\
first\ order.
\subsubsection{Coverage dependence of desorption\ with\ a\ lateral\ interaction}
In the this section,\ we discuss how\ the peak temperature\
depends on\ adsorbate coverage~$\theta$~in the presence\
of\ the lateral\ interaction~$\&$~compare it with
literature.\
With\ hydrogen adsorbate,\ the adsorption energy\
on\ Pd$\rm_{3}$Ag~(111)\ is 0.34~eV\ at adsorbate\
coverage of\ 1~ML~(1 molecule of\ adsorbate per a\
($1\times$1)~surface unit cell).\
At\ adsorbate\ coverage\ of\ 0.5~ML~(i.e.,\
corresponding\ to 1 molecule of adsorbate\ per a\
($2\times$1)~surface unit cell),\ a\ lateral\
interaction of\ -0.14 eV (taken from paper \cite{sraaen}, similar\
work~(adsorption of CO on Pt(111))) is used.~The\ outcome\
of\ the\ simulation\ is\ presented\ in\
Fig.~\ref{fig15}.\
\begin{figure}[htbp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
\includegraphics[scale=0.5]{Figures/Fig15.eps}
\end{adjustbox}
\caption{The first order simulation of\ desorption\
hydrogen from Pd$\rm_{3}$Ag~(111) for $\theta$ = 0.5 and
$\theta$ = 1 with lateral interaction.
Color online. Colors:~{$\theta$}=1~(dashed
line~$\&$~blue\ color),~{$\theta$}=0.5~(dash-dot
line~$\&$~red\ color).\label{fig15}}
\end{figure}
From\ the\ Fig.~\ref{fig15},\ it\ can\ be\ seen\ that\
the desorption\ peak occurs\ at a\ temperatures of
$T\rm_{m}$ = 37 K~for~$\theta$ = 0.5 ML, and\
$T\rm_{m}$ = 15 K~for $\theta$ = 1 ML.\ The\ temperature\
of the peak\ of\ the\ desorption rate increases when\
the coverage change from 1~ML to 0.5~ML.\\
For\ a second\ order\ desorption\ rate\ concept,\
the\ outcome\ is\ presented\ in\ Fig.~\ref{fig16}.\
It looks that the\ desorption\ peak\ occurs\ at\
temperature of $T\rm_{m}$ = 40 K~for $\theta$ = 0.5 ML,\
and $T\rm_{m}$ = 13 K~for $\theta$ = 1 ML.\ The\
temperature of the desorption peak increases\
when the coverage\ changes from\ 1 ML to 0.5 ML.\\
\begin{figure}[htbp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
\includegraphics[scale=0.5]{Figures/Fig16.eps}
\end{adjustbox}
\caption{The second order simulation of\ desorption\
hydrogen from Pd$\rm_{3}$Ag~(111) for $\theta$ = 0.5 and
$\theta$ = 1 with lateral interaction.
Color online. Colors:~{$\theta$}=1~(dashed
line~$\&$~blue\ color),~{$\theta$}=0.5~(dash-dot
line~$\&$~red\ color).\label{fig16}}
\end{figure}
With CO adsorbate,~the\ adsorption\ energy of CO\
from Pd$_{3}$Ag~(111)~ at 1~ML\ is\ 1.26 eV.\
Within first order desorption~rate,~$\&$~using\
a\ lateral interaction of -0.14 eV~gives\
a\ simulation\ output\ presented\ in\ Fig.~\ref{fig17}.\
\begin{figure}[htbp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
\includegraphics[scale=0.5]{Figures/Fig17.eps}
\end{adjustbox}
\caption{The first order simulation of\ desorption\
CO from Pd$\rm_{3}$Ag~(111) for $\theta$ = 0.5 and
$\theta$ = 1 with lateral interaction.
Color online. Colors:~{$\theta$}=1~(dashed
line~$\&$~blue\ color),~{$\theta$}=0.5~(dash-dot
line~$\&$~red\ color).\label{fig17}}
\end{figure}
The\ desorption\ peak\ occurs\ at a\ temperature\
of\ $T\rm_{m}$ = 72 K~for $\theta$ = 0.5 ML,\
and $T\rm_{m}$ = 55 K~for $\theta$ = 1 ML.~The\
temperature of the desorption\ peak increases\
when the coverage changes from 1 ML to 0.5 ML.\
Within\ second order desorption~rate,~the simulation\
output\ is presented\ in\ Fig.~\ref{fig18}.\
The\ desorption\ peak\ occurs\ at\ a\ temperature of\
$T\rm_{m}$ = 77 K~for $\theta$ = 0.5 ML,\ and\
$T\rm_{m}$ = 67 K~for $\theta$ = 1 ML.\ The temperature\
of the maximum\ desorption peak increases\ when the\
coverage change from 1 ML to 0.5 ML.\ There\ is\
a confidence\ in such shifts of\ desorption\
peaks as coverage decreases,~since\ TPD~experiments\
report similar pattern, albeit on different\
system~\cite{sraaen}.\ However,~the\ temperature\
values\ could\ be\ possibly\ improved\ further\
by\ computing for exact value of adsorption\
energies at coverage of 0.5~ML.~It's value\
would\ be\ expected to\ be\ an\ increased\
one\ compared\ to the~1.26~eV~value~used\
in\ this\ work.\
\begin{figure}[htbp!]
\centering
\begin{adjustbox}{max size ={\textwidth}{\textheight}}
\includegraphics[scale=0.5]{Figures/Fig18.eps}
\end{adjustbox}
\caption{The second order simulation of\ desorption\
CO from Pd$\rm_{3}$Ag~(111)~for~$\theta$ = 0.5 and
$\theta$ = 1 with lateral interaction.\
Color online. Colors:~{$\theta$}=1~(dashed
line~$\&$~blue\ color),~{$\theta$}=0.5~(dash-dot
line~$\&$~red\ color).\label{fig18}}
\end{figure}
\section{Conclusion\label{sec:conc}}
The\ most\ favorable\ adsorptions\
have\ adsorption\ energies\
in\ the\ range~[0.02, 0.59]\ eV\
for\ a\ hydrogen\ adsorbate\ at\
1~ML~adsorbate\ coverage.~With\
the\ adsorption\ of\ carbonmonoxide,\
these values are\ in the range\
of~[0.10, 1.68]~eV.~The relatively\
lower adsorption energies of hydrogen
compared to CO could indicate that the
respective ions have better mobility
within the electrolyte.~Furthermore,\
the\ adsorption\ energies\ investigated\
for\ CO\ on Pd$\rm_{3}$Ag~surfaces\ are\
desorbable\ within\ the\ PEMFC operating\
temperature without\ causing\ a\ poisoning\
effect.\ Pd$\rm_{3}$Ag\ seems\ to\ be\
a\ preferred\ structure,\ compared\ to\
PdAg,\ $\&$\ PdAg$\rm_{3}$.\ Furthermore,\
Pd$\rm_{3}$Ag~(111)~is a more\ favorable\
as well as reactive surface with the\
adsorbates\ at higher fuel supplies,\
thus, it better represents a\ typical\
surface of Pd$\rm_{3}$Ag alloy.\
A\ charge\ of\ about\ 0.16$e$ per hydrogen\
atom\ appears\ to\ be\ released\ to\
the\ system\ for\ electricity\ contribution.\\
The\ only\ possible\ draw\ back of the model\
is\ its\ having\ a slightly\ reduced opportunity\
of\ resisting\ extreme external conditions\ due\
to pressure\ fluctuations,\ as\ compared\ to\
the\ Pt-only\ or\ Pd-only electrode.\ The\
adsorbate species\ considered\ in this\ study\
can be seen to be\ part of hydrogen\ fuel in\
which case 100{$\%$}\ of the hydrogen\ fuel\
contains hydrogen species\ or other fuels\
such as kerosene,~diesel\ oil,~gasoline,~benzene,\
etc,\ which can\ contain up to 10-20$\%$\ carbon\
and hydrogen\ components.\ These\ realities\
justify\ the\ importance\ of\ dealing\ with\
hydrogen and CO\ adsorption,\ and\ desorption\
processes.\\
The\ montecarlo simulation\ of\ desorption\
process\ suggests that desorption\ peak\
of\ carbonmonoxide\ occurs\ at\
relatively higher\ temperature~(ca.~400~K)\
compared\ to\ desorption peak\ of\
hydrogen which occurs\ at 207~K.\
This indicates\ a\ relatively\
better operation\ efficiency\ at low\
temperatures\ for a hydrogen fuel as\
compared to hydrocarbon containing\
fuels.~With\ the\ 2$\rm^{nd}$\ order\
rate\ concept,\ the\ peak\ of\ the\
desorption\ rates\ shifts\ to\ a\ higher\
temperature\ compared\ to\ that\ of\ the\
1$\rm^{st}$\ order\ rate.\ Furthermore,\
within\ a\ given\ rate\ concept,\
the\ temperature\ of\ the\ desorption\
peak\ shifts\ to\ a\ higher\ temperature\
as\ the\ adsorbate\ coverage\ decreases\
from\ {$\theta$}~=~1~ML~to\ {$\theta$}~=~0.5~ML.\
The\ temperature\ of\ the\ desorption\
peaks\ does\ also\ seem\ to\ increase\
when\ heating\ rate\ is\ increased.\
Such\ phenomena\ of\ increases\ in\
desorption\ temperatures\ when\
a\ heating\ rate\ is\ increased is\
also\ investigated\ for CO desorption\
from Cu~\cite{zakeri2004monte}.
Furthermore,\ while\ assuming\ 2$\rm^{nd}$\
order\ rate\ concept,\ the\ width\ of\
the\ desorption\ curves\ seem\ to\ be\
broadened\ compared\ to\ the\ width\
within\ the\ 1$\rm^{st}$\ order\ rate\
concept.\ The\ width\ of\ the\ curves\
is\ in turn\ directly\ proportional\
to\ the\ amount\ of\ desorbed\ particles.\\
In\ recommendation\ for\ a\ need\ area\
of\ further\ studies,\ the\ authors\
commend\ for\ a\ continuous\ study\
on\ this\ subject\ which\ focuses\
on\ dealing\ with\ transport\
processes\ of\ desorbed\ ions\
within\ the\ electrolyte\ of\
the\ PEMFC.\
\section*{Disclosure\ statement}
The authors declare\ that there is no conflict\
of interest.
\section*{Acknowledgments}
We are grateful to the Ministry of Education\
of\ Ethiopia for financial support.\ The\
authors also acknowledge\ the\ Department of\
Physics at\ Addis Ababa University and Haramaya\
University.\
\newpage
\section*{References}
\bibliographystyle{elsarticle-num}
|
1,108,101,563,270 | arxiv | \section{Introduction}
A quantum computer harnesses unique features of quantum theory, such as superposition and entanglement, to tackle classically challenging tasks. To perform faithful computation, quantum information must be protected against errors due to decoherence mechanisms and operational imperfections. While these errors are relatively insignificant individually, they can quickly accumulate to completely scramble the information.
To protect quantum information from scrambling, the theoretical frameworks of quantum error correction (QEC) ~\cite{shor1995_scheme, steane1996_error} and fault-tolerant quantum computation~\cite{preskill1998_reliable} were developed in the early days of quantum computing. Essentially, these frameworks devise encodings which map a collection of physical elements onto a single `logical' bit of quantum information. Such a logical qubit is endowed with cleverly chosen symmetry properties that allow us to extract error syndromes and enact error correction without disturbing the encoded information.
An important metric for evaluating the effectiveness of QEC implementations is the break-even point, which is achieved when the lifetime of a logical qubit exceeds that of the best single physical element in the system. Achieving the break-even point entails that additional physical elements and operations introduced to a QEC process do not cause more degradation than the protection they afford. Hence, reaching the break-even point is a critical pre-requisite for implementing fault-tolerant gates and eventually performing robust quantum information processing on a large scale.
In the conventional approach to QEC, the physical elements are realized by discrete two-level systems. In this approach, even a simple QEC scheme designed to correct single errors, such as the Steane code~\cite{steane1996_error}, requires tens of two-level systems, ancillary qubits, and measurement elements. Constructing physical devices that contain these many interconnected elements can be a significant engineering challenge. More crucially, having many interconnected elements often degrades the device performance and introduces new uncorrectable errors such as cross-talk due to undesired couplings between the elements. Over the last decade, many proof-of-principle demonstrations of QEC schemes have been realized with encoding schemes based on two-level systems~\cite{chiaverini2004_realization, reed2012_realization, taminiau2014_universal, waldherr2014_quantum, linke2017_faulttolerant, andersen2020_repeated, erhard2021_entangling}. However, given the practical challenges described above, these demonstrations have not deterministically extended the performance of the logical qubits beyond that of the best available physical qubit in the system.
A promising alternative with the potential to realize robust universal quantum computing with effective QEC beyond the break-even point involves encoding logical qubits in continuous variables~\cite{braunstein1998_quantum, bartlett2002_universal, bartlett2002_quantum}. In particular, superconducting microwave cavities coupled to one or more anharmonic elements in the circuit quantum electrodynamics (cQED) architecture provide a valuable resource for the hardware-efficient encoding of logical qubits~\cite{devoret2013_superconducting, blais2020_circuit}. These cavities have a large Hilbert space for encoding information in multi-photon states compactly and thus form a logical qubit in a single piece of hardware. The anharmonic element, typically in the form of a transmon and henceforth referred to as the `ancilla', provides the necessary non-linearity to control and measure cavity states. This strategy of using multi-photon states of superconducting cavities to encode logical information is also known as bosonic codes. Implementations of bosonic codes in cQED have thus far not only demonstrated QEC at the break-even point~\cite{ofek2016_extending}, but also robust operations~\cite{heeres2017_implementing, chou2018_deterministic, gao2019_entanglement, xu2020_demonstration, reinhold2020_error-corrected} and fault-tolerant measurement of error syndromes~\cite{rosenblum2018_fault-tolerant}, thus making rapid progress in recent years.
In this article, we review the recent developments in bosonic codes in the cQED setting. In particular, we highlight the progress made in demonstrating effective QEC and information processing with logical elements implemented using bosonic codes. These recent works provide compelling evidence for the vast potential of bosonic codes in cQED as the fundamental building blocks for robust universal quantum computing.
\subsection{Organization of the article}
In Sec.~\ref{sec:concepts}, we begin by outlining the basic principles of QEC as well as bosonic codes. Here, we highlight their nature by comparing a bosonic code with a multi-qubit code in the presence of similar errors. In Sec.~\ref{sec:codes}, we present various bosonic encoding schemes proposed in literature and compare their respective strengths and limitations. In particular, we wish to emphasize crucial considerations for constructing these codes and evaluating their performance in the presence of naturally-occurring errors.
In Sec.~\ref{sec:cavities}, we introduce the key hardware building blocks required for cQED implementations of bosonic encoding schemes. In this section, we also consolidate the progress made in improving the intrinsic quality factors of superconducting microwave cavities over the last decade. Subsequently, in Sec.~\ref{section: realization}, we explore the latest developments in implementing robust universal control on bosonic qubits encoded in superconducting cavities, both in terms of single-mode gates as well as novel two-mode operations. We then describe the different strategies for detecting and correcting quantum errors on bosonic logical qubits encoded in superconducting cavities.
In Sec.~\ref{sec:FT}, we discuss the concept of fault-tolerance and how that might be realized with protected bosonic qubits. Here, we also feature some novel schemes that concatenate bosonic codes with other QEC codes to protect against quantum errors more comprehensively. Finally, in Sec.~\ref{sec:future}, we provide some perspectives for achieving QEC on a larger scale. We conclude by remarking on the appeal of the modular architecture, which offers a promising path for practical and robust quantum information processing with individually protected bosonic logical elements.
\section{Concepts of Bosonic Quantum Error Correction}\label{sec:concepts}
The general principle of QEC is to encode logical quantum information redundantly in a large Hilbert space with certain symmetry properties, which can be used to detect errors. In particular, logical code states are designed such that they can be mapped onto orthogonal subspaces under distinct errors. Crucially, the logical information can be recovered faithfully only if the mapping between the logical and the error states does not distort the code words.
Mathematically, these requirements can be succinctly described by the Knill-Laflamme condition~\cite{knill1997_theory}, which states that an error-correcting code $\mathcal{C}$ can correct any error operators in the span of an error set \mbox{$\mathcal{E} \equiv \lbrace \hat{E}_{1},\cdots,\hat{E}_{|\mathcal{E}|} \rbrace$} if and only if it satisfies:
\begin{linenomath}
\begin{gather}
\hat{P}_{\mathcal{C}} \hat{E}_{\ell}^{\dagger}\hat{E}_{\ell'} \hat{P}_{\mathcal{C}} = \alpha_{\ell\ell'}\hat{P}_{\mathcal{C}}
\end{gather}
\end{linenomath}
for all $\ell,\ell'\in \lbrace 1,\cdots,|\mathcal{E}| \rbrace$, where $|\mathcal{E}|$ is the size of the error set, $\hat{P}_{\mathcal{C}}$ is the projection operator to the code space $\mathcal{C}$, and $\alpha_{\ell\ell'}$ are matrix elements of a Hermitian and positive semi-definite matrix. A derivation of the Knill-Laflamme condition is given in Ref.~\cite{nielsen2000_quantum}.
Bosonic codes achieve these requirements by cleverly configuring the excitations in a harmonic oscillator mode. For instance, in the cQED architecture, information is encoded in multi-photon states of superconducting microwave cavities. We will illustrate how bosonic QEC codes work by describing the simplest binomial code~\cite{michael2016_new}, also known as the `kitten' code. We will then compare this code with its multi-qubit cousin, the `4-qubit code' (or the $[[4,1,2]]$ code or the distance-$2$ surface code)~\cite{leung1997_approximate}, in the presence of similar errors in the cQED architecture. This comparison, adapted from Ref.~\cite{girvin2019_quantum}, emphasizes the hardware-efficient nature of the bosonic QEC approach.
\begin{figure*}[t!]
\centering
\includegraphics[scale=1]{figs/hardware_efficiency.pdf}
\caption[width=\textwidth]{\textbf{Hardware comparison between the kitten code and the 4-qubit code.} (a) Encoding a single logical qubit using multiple energy levels of a single harmonic oscillator, where the only dominant error channel is photon loss. (b) The 4-qubit scheme based on a collection of two-level systems. Individually, each two-level system can experience errors due to spontaneous absorption and emission. Moreover, additional errors can be introduced due to undesired couplings between these systems. (c) The typical hardware for implementing a bosonic logical qubit. The logical information is encoded in a single superconducting cavity (orange) in the kitten code, which can be fully controlled by a single non-linear ancilla (green) and read out via another cavity (gray). (d) Example of a system designed to implement the 4-qubit code, where the logical information is spread out across four data qubits (orange) checked by three ancillary ones (green). The interactions between each two-level system are mediated by coupling cavities (teal), and their respective states are read out via seven planar cavities (gray).
} \label{fig:hardware}
\end{figure*}
The kitten code is designed to correct only single photon loss events, or $\hat{a}$, which are the dominant error channel in superconducting cavities. This scheme encodes logical information in the even photon number parity subspace of a harmonic oscillator (Fig.~\ref{fig:hardware}a):
\begin{linenomath}
\begin{align}
|0_{L}\rangle = \frac{1}{\sqrt{2}}( |0\rangle + |4\rangle ) , \quad |1_{L}\rangle = |2\rangle. \label{eq:bin N1S1}
\end{align}
\end{linenomath}
With these code words, a single photon loss event maps an even parity logical state $|\psi_{L}\rangle = \alpha|0_{L}\rangle + \beta|1_{L}\rangle$ to an odd parity error state $\hat{a}|\psi_{L}\rangle = \sqrt{2}(\alpha|3\rangle+\beta|1\rangle)$. Because of the parity difference, the code space and the error space are mutually orthogonal. Therefore, single photon loss events can be detected by measuring the photon number parity operator $\hat{\Pi}_{2} = (-1)^{\hat{n}}$. Importantly, since the error states $\hat{a}|0_{L}\rangle$ and $\hat{a}|1_{L}\rangle$ have the same normalization constant,
\begin{linenomath}
\begin{align}
\langle 0_{L}| \hat{a}^{\dagger}\hat{a}|0_{L}\rangle = \langle 1_{L}| \hat{a}^{\dagger}\hat{a}|1_{L}\rangle = 2,
\end{align}
\end{linenomath}
or equivalently, the logical states $|0_{L}\rangle$ and $|1_{L}\rangle$ have the same average photon number, a single photon loss event does not distort the encoded information. In other words, by mapping the normalized error states $|3\rangle$ and $|1\rangle$ back to the original code states $|0_{L}\rangle$ and $|1_{L}\rangle$, we can recover the input logical state up to an overall normalization constant:
\begin{linenomath}
\begin{align}
\hat{a}|\psi_{L}\rangle &= \sqrt{2}(\alpha |3\rangle + \beta |1\rangle) \nonumber \\ &\xrightarrow{\textrm{recovery}} \sqrt{2}(\alpha |0_{L}\rangle + \beta |1_{L}\rangle ) \propto |\psi_{L}\rangle.
\end{align}
\end{linenomath}
Note that we cannot faithfully recover logical information if the logical states have different average photon numbers. For instance, with $|0_{L}\rangle = \frac{1}{2}|0\rangle + \frac{\sqrt{3}}{2}|4\rangle$, which has $3$ photons on average, as opposed to $2$ photons for $|1_{L}\rangle = |2\rangle$, a single photon loss and recovery event yields a state which is not proportional to $|\psi_{L}\rangle$:
\begin{linenomath}
\begin{align}
\hat{a}|\psi_{L}\rangle &= \sqrt{3}\alpha|3\rangle + \sqrt{2}\beta|1\rangle \nonumber \\ &\xrightarrow{\textrm{recovery}} \sqrt{3}\alpha|0_{L}\rangle + \sqrt{2}\beta|1_{L}\rangle. \label{eq:adjusted weight}
\end{align}
\end{linenomath}
Specifically, the error and recovery process distorts the relative phase between $|0_{L}\rangle$ and $|1_{L}\rangle$. An intuitive way to understand such a phase distortion is by considering the role of the environment. When $|0_{L}\rangle$ and $|1_{L}\rangle$ have different average photon numbers, the environment gains partial information on whether $|\psi_{L}\rangle$ was in $|0_{L}\rangle$ or in $|1_{L}\rangle$. In the example above, $|0_{L}\rangle$ has more photons than $|1_{L}\rangle$ and hence has a higher probability of losing a photon. When one photon is lost, the environment knows that the encoded state was more likely to be $|0_{L}\rangle$ than $|1_{L}\rangle$ and the weights in Eq.\ \eqref{eq:adjusted weight} are therefore adjusted accordingly. Alternatively, we can say that the environment performs a weak measurement on the logical state in the $|0_{L}/1_{L}\rangle$ basis, thus leading to the dephasing of $|\psi_{L}\rangle$ in the $|0_{L}/1_{L}\rangle$ basis.
Hence, while the even parity structure allows the detection of single photon loss events, it does not guarantee the recoverability of the logical information without any distortion. Faithful recovery is ensured by selecting $|0_{L}\rangle$ and $|1_{L}\rangle$ to have the same average photon number. In the case of the kitten code, average photon numbers are matched by choosing equal coefficients for the vacuum and the four photon state components in the logical zero state.
More generally, a code $\mathcal{C}$ can correct single photon loss events if it satisfies the Knill-Laflamme condition for the error set $\lbrace \hat{I},\hat{a} \rbrace$. That is, the projection operator to the code space should satisfy $\hat{P}_{\mathcal{C}} \hat{x} \hat{P}_{\mathcal{C}} \propto \hat{P}_{\mathcal{C}}$ for all \mbox{$\hat{x}\in \lbrace \hat{I},\hat{a},\hat{a}^{\dagger},\hat{a}^{\dagger}\hat{a} \rbrace$}. The first condition with $\hat{x}=\hat{I}$ is trivially satisfied for any code. For even parity codes, which are composed of logical states of even photon number parity, the second and the third conditions with $\hat{x} = \hat{a} $ and $\hat{x} = \hat{a}^{\dagger}$ are satisfied due to the parity structure. This implies that even parity codes are capable of detecting, but not necessarily correcting, any single photon loss and gain events. Single photon loss events can be corrected if the fourth condition is met, that is, $\hat{P}_{\mathcal{C}} \hat{a}^{\dagger}\hat{a} \hat{P}_{\mathcal{C}} \propto \hat{P}_{\mathcal{C}}$, or equivalently, if all logical states have the same average photon number.
Now, using an example inspired by Ref.~\cite{girvin2019_quantum}, we compare the kitten code with its multi-qubit cousin, the 4-qubit code, whose logical states consist of 4 distinct two-level systems (Fig.~\ref{fig:hardware}b):
\begin{linenomath}
\begin{align}
|0_{L}\rangle = \frac{1}{\sqrt{2}} ( |0000\rangle + |1111\rangle ), \nonumber \\
|1_{L}\rangle = \frac{1}{\sqrt{2}} ( |0101\rangle + |1010\rangle ).
\end{align}
\end{linenomath}
The $4$-qubit code is stabilized by three stabilizers: \mbox{$\hat{S}_{1} = \hat{Z}_{1}\hat{Z}_{3}$}, \mbox{$\hat{S}_{2} = \hat{Z}_{2}\hat{Z}_{4}$}, and \mbox{$\hat{S}_{3} = \hat{X}_{1}\hat{X}_{2}\hat{X}_{3}\hat{X}_{4}$}. This scheme is capable of detecting any arbitrary single-qubit errors. Moreover, it can correct single excitation loss errors, $\lbrace\hat{\sigma}^{-}_{1},\hat{\sigma}^{-}_{2},\hat{\sigma}^{-}_{3},\hat{\sigma}^{-}_{4}\rbrace$, via approximate QEC~\cite{leung1997_approximate}, which happens when the Knill-Laflamme condition is approximately satisfied only to a certain low order in the error parameters. This capability is comparable to the protection afforded by the kitten code against single photon loss errors.
Despite their comparable error-correcting capability, the $4$-qubit code and the kitten code incur significantly different hardware overheads. A cQED implementation of the kitten code (Fig.~\ref{fig:hardware}c) requires a single bosonic mode to store logical information, a single ancilla (typically a transmon) to measure and control the cavity state, and a single readout cavity mode to measure the ancilla state. In contrast, a cQED realization of the $4$-qubit code (Fig.~\ref{fig:hardware}d) uses $4$ data qubits to encode logical information, $3$ ancillae to measure the three stabilizers, and additional cavity modes to connect and measure all $7$ physical qubits. Apart from the pure complexity of realizing such a device, the presence of additional elements introduces other error channels such as cross-talk arising from spurious couplings between the physical qubits. While these effects can be calibrated and mitigated on a small scale with clever techniques~\cite{sheldon2016_procedure, rol2019_fast}, they can quickly become intractable for more complex devices. This comparison thus illustrates the advantage of using the multiple levels of a single bosonic mode over using multiple two-level systems as a redundant resource for QEC.
While we focus on bosonic codes that encode a qubit in a single bosonic mode, there are proposals for encoding a qubit in many bosonic modes, namely permutation-invariant codes \cite{ouyang2014_permutationinvariant, ouyang2016_permutationinvariant, ouyang2017_permutationinvariant, ouyang2020_permutationinvariant}. These codes are tailored for excitation loss errors and generalize the simple 4-qubit code. Other codes of a similar nature have also been studied in Refs.~\cite{wasilewski2007_protecting, kapit2016_hardwareefficient, bergmann2016_quantum, kapit2018_errortransparent}.
\section{Performance of Bosonic Codes for Loss and Dephasing Errors}\label{sec:codes}
In general, bosonic modes in cQED systems typically undergo both photon loss and dephasing errors. Photon loss is considered to be the dominant error-channel. The rate of photon loss is determined by the internal quality factor ($Q_{\mathrm{int}}$) of the superconducting cavity. Intrinsic dephasing is usually insignificant for such cavities~\cite{reagor2013_reaching}.
However, as the cavity is dispersively coupled to a non-linear ancilla, if the ancilla experiences undesired absorption ($\Gamma_{\uparrow}$) or emission ($\Gamma_{\downarrow}$) of excitations due to stray radiation, then the encoded logical information undergoes induced dephasing. The rate of this dephasing depends on $\Gamma_{\uparrow}$, $\Gamma_{\downarrow}$, and the coupling strength $\chi$. In general, a transition of the ancilla state leads to a rotation on the logical qubit by an angle $\sim \chi t$, where $t$ is the time the ancilla spends in the resulting state after the transition. In the limit where this rotation is small, we can apply the Markovian approximation and use the following Lindblad master equation to describe the evolution of the encoded qubit:
\begin{align}
\frac{d\hat{\rho}(t)}{dt} = \big{(} \kappa \mathcal{D}[\hat{a}] + \kappa_{\phi}\mathcal{D}[\hat{a}^{\dagger}\hat{a}] \big{)} \hat{\rho}(t),
\end{align}
where $\kappa$ and $\kappa_{\phi}$ are the photon loss and dephasing rates respectively, and \mbox{$\mathcal{D}[\hat{A}](\hat{\rho})\equiv \hat{A}\hat{\rho}\hat{A}^{\dagger}-\frac{1}{2}\lbrace \hat{A}^{\dagger}\hat{A},\hat{\rho} \rbrace$} is the dissipation superoperator. Note that loss and dephasing errors are generated by the jump operators $\hat{a}$ and $\hat{n} = \hat{a}^{\dagger}\hat{a}$ respectively. We say that a system is loss dominated if $\kappa\gg \kappa_{\phi}$ and dephasing dominated if $\kappa \ll \kappa_{\phi}$.
\begin{figure*}[t!]
\centering
\includegraphics[scale=1]{figs/codes.png}
\caption[width=\textwidth]{\textbf{Visualization and performance of different bosonic codes.} (a) - (d) Wigner functions of the code words for the four-component cat, binomial kitten, GKP square lattice, and two-component cat code, respectively. (e) A qualitative description of the robustness of various classes of bosonic codes against dephasing and photon loss errors. Note that for the four-component cat code and the binomial kitten code, the logical states in the $|0/1_{L}\rangle$ basis are invariant under a $90\degree$ rotation, up to a global phase. However, a general code state, that is, an arbitrary superposition of $|0_{L}\rangle$ and $|1_{L}\rangle$), is invariant only under the $180\degree$ rotation.} \label{fig:codes}
\end{figure*}
Importantly, in typical experimental regimes, dephasing errors induced by transitions of the ancilla state are generally non-Markovian. This limits the performance of some current implementations~\cite{lescanne2020_exponential} of bosonic QEC codes.
In this section, we review the various single-mode bosonic codes, namely, cat (Fig.~\ref{fig:codes}a) and binomial (Fig.~\ref{fig:codes}b) codes (rotation-symmetric), Gottesman-Kitaev-Preskill (GKP) codes (translation-symmetric) (Fig.~\ref{fig:codes}c), and two-component cat codes (biased-noise bosonic qubits) (Fig.~\ref{fig:codes}d), and discuss their error-correcting capability against both photon loss and dephasing errors.
\subsection{Rotation-symmetric codes: Binomial and cat codes}\label{subsection:rotation symmetric codes}
The class of rotation-symmetric bosonic codes~\cite{grimsmo2020_quantum} refers to encodings that remain invariant under a set of discrete rotations in phase space. Encoding schemes with rotation-symmetry, such as the cat and binomial codes, are stabilized by a photon number super-parity operator \mbox{$\hat{\Pi}_{N} = e^{i(2\pi/N)\hat{n}}$ ($N \in \lbrace 2,3,\cdots \rbrace$)}, which is equivalent to the $360/N\degree$ rotation operator. As such, these codes are, by design, capable of detecting $N-1$ photon loss events.
Furthermore, these rotation-symmetric codes can also be made robust against dephasing errors in addition to photon loss errors~\cite{ouyang2020_tradeoffs}. For instance, the kitten code may be modified to have the following logical states:
\begin{linenomath}
\begin{align}
|0_{L}\rangle &= \frac{1}{2}( |0\rangle + \sqrt{3}|4\rangle ), \nonumber \\
|1_{L}\rangle &= \frac{1}{2}( \sqrt{3}|2\rangle + |6\rangle ). \label{eq:bin N2S1}
\end{align}
\end{linenomath}
This version of the binomial code is also stabilized by the parity operator and is invariant under the $180\degree$ rotation, thus making it robust against single photon loss errors. However, the modified code is now also higher in energy as its logical states have $3$ photons on average, as opposed to $2$ in the case of the kitten code. This additional redundancy makes the modified binomial code robust against single dephasing events as well. In other words, the logical states of the modified binomial code satisfy the Knill-Laflamme condition for an extended error set \mbox{$\lbrace \hat{I},\hat{a},\hat{a}^{\dagger}\hat{a} \rbrace$ where $\hat{a}^{\dagger}\hat{a}$} describes single dephasing errors in the cavity. This means that the two logical code words also have the same second moment of the photon number probability distribution besides having the same average photon number. Note that when more moments of the photon number probability distribution are the same for the two code words, distinguishing them becomes more difficult for the environment, which enhances the error-correcting capability of the code.
Another way to generalize rotation-symmetric codes is by considering different rotation angles. For instance, we can design codes that are invariant under a $120\degree$ rotation instead of a $180\degree$ rotation and are thus stabilized by the super-parity modulo $3$ operator \mbox{$\hat{\Pi}_{3} = e^{ i (2\pi /3) \hat{n} }$}. By taking advantage of a larger spacing in the photon number basis, $120\degree$-rotation-symmetric codes can detect two-photon loss events as well as single photon loss events, thus providing enhanced protection to the logical information. For instance, a variant of the binomial code with the logical states
\begin{linenomath}
\begin{align}
|0_{L}\rangle &= \frac{1}{2} ( |0\rangle + \sqrt{3} |6\rangle ), \nonumber \\
|1_{L}\rangle &= \frac{1}{2}(\sqrt{3}|3\rangle + |9\rangle), \label{eq:bin N2S2}
\end{align}
\end{linenomath}
is invariant under the $120\degree$ rotation, or equivalently, has photon numbers that are integer multiples of $3$. Moreover, the particular coefficients associated with each photon number state in the code words, which are derived from the binomial coefficients (hence the name `binomial' code), ensure that the code satisfies the Knill-Laflamme condition for an error set \mbox{$\lbrace \hat{I},\hat{a},\hat{a}^{2},\hat{a}^{\dagger}\hat{a} \rbrace$}. This indicates that the code is robust against both single and two-photon loss events, as well as single dephasing errors. The binomial code can be further generalized to protect against higher-order effects of loss and dephasing errors~\cite{michael2016_new} as well as used for autonomous QEC~\cite{lihm2018_implementation}.
Cat codes are another important example of rotation-symmetric bosonic codes. The four-component cat codes (or 4-cat codes), which are composed of four coherent states $|\pm\alpha\rangle$, $|\pm i\alpha\rangle$, are the simplest variants of the cat codes that are robust against photon loss errors~\cite{leghtas2013_hardware-efficient, mirrahimi2014_dynamically}. In this encoding, the logical states are defined by superpositions of coherent states:
\begin{linenomath}
\begin{align}
|0_{L}\rangle &\propto |\alpha\rangle + |i\alpha\rangle + |-\alpha\rangle + |-i\alpha\rangle
\nonumber \\
&\propto \sum_{n=0}^{\infty}\frac{\alpha^{4n}}{\sqrt{ (4n)! }}|4n\rangle,
\nonumber \\
|1_{L}\rangle &\propto |\alpha\rangle - |i\alpha\rangle + |-\alpha\rangle - |-i\alpha\rangle
\nonumber\\
&\propto \sum_{n=0}^{\infty}\frac{\alpha^{4n+2}}{\sqrt{ (4n+2)! }}|4n+2\rangle .
\end{align}
\end{linenomath}
Note that the amplitude of the coherent states $|\alpha|$ determines the size of the cat code. Similar to the kitten code (Eq.~\eqref{eq:bin N1S1}), the logical states of the 4-cat code have an even number of photons. Thus, the 4-cat code is invariant under the $180\degree$ rotation and is able to detect single photon loss events. Here, we reinforce that the parity (or rotation-symmetry) alone does not ensure the recoverability of logical information against single photon loss errors, which is only guaranteed when the Knill-Laflamme condition is satisfied for the error set \mbox{$\lbrace \hat{I},\hat{a} \rbrace$}. For even parity codes, as explained in Sec~\ref{sec:concepts}, recoverability requires the two logical states to have the same number of photons on average.
For large cat codes with $|\alpha|\gg 1$, the average photon number is approximately given by \mbox{$\bar{n} \simeq |\alpha|^{2}$} for both logical states, and the Knill-Laflamme condition is approximately fulfilled. More importantly, the average photon numbers of the two logical states are exactly the same for certain values of $|\alpha|$ (also known as `sweet spots'~\cite{li2017_cat}) such that:
\begin{linenomath}
\begin{align}
\tan|\alpha|^{2} = - \tanh |\alpha|^{2}.
\end{align}
\end{linenomath}
The smallest such $|\alpha|$ is given by $|\alpha| = 1.538$, which corresponds to the average photon number $\bar{n} = 2.324$. By increasing the size of the cat codes, we can construct logical states that are robust against both single photon loss and dephasing errors. In particular, cat codes with a large average photon number $|\alpha|^{2}$ satisfy the Knill-Laflamme condition for the dephasing error set $\lbrace \hat{I}, \hat{n}, \hat{n}^{2}, \cdots \rbrace$ approximately modulo an inaccuracy that scales as $e^{-2|\alpha|^{2}}$ \cite{albert2018_performance}. However, an extra error-correcting mechanism, such as a multi-photon engineered dissipation \cite{mirrahimi2014_dynamically}, is required to exploit the intrinsic error-correcting capability of the cat codes against dephasing errors. Moreover, increasing the number of coherent state components in the cat code introduces further protection~\cite{albert2016_holonomic,bergmann2016_quantum}. For instance, with six coherent state components, the code words become $120\degree$-rotation-symmetric and can thus detect up to two-photon loss events. Generalizations of cat codes and analyses of their performance can be found in Ref.~\cite{li2017_cat}, while a multi-mode generalization is available in Ref.~\cite{albert2019_pair-cat}.
\subsection{Translation-symmetric codes: GKP codes}\label{subsection:translation symmetric codes}
Another class of bosonic codes are translation-symmetric, with the Gottesman-Kitaev-Preskill (GKP) codes~\cite{gottesman2001_encoding} being a prominent example. The simplest variant of the GKP codes is the square lattice GKP code, which encodes a logical qubit in the phase space of a harmonic oscillator stabilized by two commuting displacement operators:
\begin{linenomath}
\begin{align}
\hat{S}_{q} &\equiv \exp[i2\sqrt{\pi}\hat{q}] = \hat{D}(i\sqrt{2\pi}), \nonumber \\
\hat{S}_{p} &\equiv \exp[-i2\sqrt{\pi}\hat{p}] = \hat{D}(\sqrt{2\pi}),
\label{eq:square gkp stabilizers}
\end{align}
\end{linenomath}
where $\hat{q}$ and $\hat{p}$ are the position and momentum operators respectively.
A key motivation for choosing these stabilizers is to circumvent the Heisenberg uncertainty principle, which dictates that the position and momentum operators cannot be measured simultaneously as they do not commute. Since the two displacement operators in Eq.~\eqref{eq:square gkp stabilizers} commute with each other, we can measure them simultaneously. Measuring the displacement operators $\exp[i2\sqrt{\pi}\hat{q}]$ and $\exp[-i2\sqrt{\pi}\hat{p}]$ is equivalent to measuring their phases $2\sqrt{\pi}\hat{q}$ and $-2\sqrt{\pi}\hat{p}$ modulo $2\pi$, which is in turn the same as simultaneously measuring $\hat{q}$ and $\hat{p}$ modulo $\sqrt{\pi}$. Therefore, the square lattice GKP code is, by design, capable of addressing two non-commuting quadrature operators by simultaneously measuring them within a unit cell of a square lattice. The uncertainty now lies in the fact that we do not know which unit cell the state is in.
The two logical states of the square lattice GKP code are explicitly given by:
\begin{linenomath}
\begin{align}
|0_{L}\rangle &\propto \sum_{n \in \mathbb{Z}}
|\hat{q} = (2n) \sqrt{\pi} \rangle \propto \sum_{n\in\mathbb{Z}} |\hat{p} = n\sqrt{\pi} \rangle ,
\nonumber \\
|1_{L}\rangle &\propto \sum_{n\in\mathbb{Z}} |\hat{q} = (2n+1)\sqrt{\pi} \rangle \propto \sum_{n\in\mathbb{Z}} (-1)^{n}|\hat{p} = n\sqrt{\pi} \rangle,
\end{align}
\end{linenomath}
and satisfy \mbox{$\hat{q} = \hat{p} = 0$} modulo $\sqrt{\pi}$, thus clearly illustrating that a periodic simultaneous quadrature measurement is indeed possible if the spacing is chosen appropriately. Additionally, the code states are invariant under discrete translations of length $2\sqrt{\pi}$ in both the position and momentum directions, which makes the code symmetric under translations.
Conceptually, ideal GKP code states consist of infinitely many infinitely squeezed states where each component is described by a Dirac Delta function. However, precisely implementing these ideal states is not feasible in realistic quantum systems. In practice, only approximate GKP states can be realized where each position or momentum eigenstate is replaced by a finitely squeezed state and large position and momentum components are suppressed by a Gaussian envelope~\cite{gottesman2001_encoding, terhal2016_encoding, matsuura2019_equivalence}. Many proposals to realize approximate GKP states in various physical platforms~\cite{gottesman2001_encoding, travaglione2002_preparing, pirandola2004_constructing, pirandola2006_generating, vasconcelos2010_alloptical, terhal2016_encoding, motes2017_encoding, weigand2018_generating, arrazola2019_machine, shi2019_fault-tolerant, su2019_conversion, eaton2019_nongaussian, hastrup2019_measurementfree, weigand2020_realizing, hastrup2020_improved, royer2020_stabilization} and ways to simulate approximate states efficiently~\cite{wan2019_memoryassisted, tzitrin2020_progress, terhal2020_scalable, pantaleoni2020_modular, walshe2020_continuousvariable, mensen2020_phasespace} have been explored in the field. The quality of such states can be characterized by the degree of squeezing in both the position and momentum quadratures. As the squeezing, and hence the average photon number, increases, the approximate state will converge towards an ideal code state. Recently, approximate states of squeezing \mbox{$5.5$ -- $9.5$ dB} have been realized in trapped ion~\cite{fluhmann2019_encoding, deNeeve2020_error} and cQED \cite{campagne-ibarcq2019_quantum} systems. The cQED realization is discussed further in Sec.~\ref{section: realization}.
With a discrete translation-symmetry, GKP codes are naturally robust against random displacement errors as long as the size of the displacement is small compared to the separation between distinct logical states. For instance, the square lattice GKP code is robust against any displacements of size less than $\sqrt{\pi}/2$ as they can be identified via the quadrature measurements modulo $\sqrt{\pi}$ and then countered accordingly. For moderately squeezed approximate GKP states that contain a small number of photons, photon loss errors can be decomposed as small shift errors and therefore can be effectively addressed by the code~\cite{gottesman2001_encoding,terhal2016_encoding}. On the other hand, for large approximate GKP states that are highly squeezed, even a tiny fraction of photon loss results in large shift errors which cannot be corrected by the code. Thus, naively using the standard GKP error correction protocol to decode does not work for large GKP states under photon loss errors. Nevertheless, studies have observed that if an optimal decoding scheme is adopted, excellent performance against photon loss errors can be achieved even with large GKP states~\cite{albert2018_performance, noh2019_quantum}. This improvement happens because photon loss errors can be converted into random displacement errors via amplification. This implies that for highly squeezed large GKP codes, a suitable decoding strategy is to first amplify the contracted states and then correct the resulting random shift errors by measuring the quadrature operators modulo $\sqrt{\pi}$~\cite{gottesman2003_secure}. We note that at present, no analogous techniques are known for dephasing errors. Thus, highly squeezed GKP codes are not robust against dephasing errors since even a small random rotation can result in large shift errors~\cite{terhal2016_encoding}.
While the preparation of GKP states is challenging, implementing logical operations on GKP states is relatively straightforward. Any logical Pauli or Clifford operation on GKP states can be realized by a displacement or Gaussian operation (via a linear drive or a bilinear coupling). Moreover, magic states~\cite{bravyi2005_universal} encoded in the GKP code, which are necessary for implementing non-Clifford operations, can be prepared with only Gaussian operations and GKP states~\cite{terhal2016_encoding, baragiola2019_allgaussian}. Thus, the preparation of code words is the only required non-Gaussian operation for performing universal quantum computation with the GKP code. Non-Clifford operations can be directly enacted on GKP qubits~\cite{gottesman2001_encoding}, although the cubic phase gate suggested in the original proposal has been recently shown to perform poorly for this purpose~\cite{hastrup2020_cubic}.
Furthermore, GKP states can be defined over lattices other than the square lattice. For instance, hexagonal GKP codes can correct any shift errors of size less than \mbox{$(2/\sqrt{3})^{1/2} \sqrt{\pi}/2 \simeq 1.07\sqrt{\pi}/2$}, which is larger than the size of shifts that are correctable by the square lattice GKP code~\cite{gottesman2001_encoding, noh2019_quantum}. A recent work has also explored the rectangular GKP code~\cite{hanggli2020_enhanced}. More generally, a multi-mode GKP code can be defined over any symplectic lattice~\cite{harrington2001_achievable}. Lastly, the GKP code may also be used for building robust quantum repeaters for long-distance quantum communication~\cite{rozpedek2020_quantum, fukui2020_alloptical}.
\subsection{Biased-noise bosonic qubits: Two-component cat codes}\label{subsection:biased noise qubits}
Recently, there has been growing interest in biased-noise bosonic qubits, where a quantum system is engineered to have one type of error occur with a much higher probability than other types of errors. This noise bias can simplify the next layer of error correction~\cite{aliferis2008_faulttolerant}. For instance, we can design a biased-noise code that suppresses bit-flips and then correct the dominant phase-flip errors by using repetition codes~\cite{guillaud2019_repetition, guillaud2020_error}. Alternatively, we can also tailor the surface code to leverage the advantages of biased-noise models to increase fault-tolerance thresholds and reduce resource overheads~\cite{tuckett2018-ultrahigh, tuckett2019_tailoring, tuckett2020_faulttolerant, chamberland2020_building}.
A promising candidate for biased-noise bosonic qubits is the two-component cat code~\cite{cochrane1999_macroscopically, jeong2002_efficient, ralph2003_quantum, glancy2004_transmission, lund2008_faulttolerant}, or 2-cat code, whose logical code words are given by:
\begin{linenomath}
\begin{align}
|+_{L}\rangle &\propto |\alpha\rangle + |-\alpha\rangle \propto \sum_{n=0}^{\infty} \frac{\alpha^{2n}}{\sqrt{ (2n)! }}|2n\rangle ,
\nonumber \\
|-_{L}\rangle &\propto |\alpha\rangle - |-\alpha\rangle \propto \sum_{n=0}^{\infty} \frac{\alpha^{2n+1}}{\sqrt{ (2n+1)! }}|2n+1\rangle .
\end{align}
\end{linenomath}
If $\alpha$ is large enough, 2-cat codes are capable of correcting dephasing errors. One way to implement 2-cat codes is by autonomously stabilizing them via an engineered dissipation of the form \mbox{$\kappa_{2}\mathcal{D}[\hat{a}^{2}-\alpha^{2}]$}~\cite{mirrahimi2014_dynamically}. This engineered two-photon dissipation can exponentially suppress the logical bit-flip error in the code states due to dephasing in superconducting cavities \mbox{($\kappa_{\phi}\mathcal{D}[\hat{a}^{\dagger}\hat{a}]$)}, i.e.,
\begin{linenomath}
\begin{align}
\gamma_{\textrm{bit-flip}} \simeq 2\kappa_{\phi}|\alpha|^{2}e^{-2|\alpha|^{2}} \,\,\, \textrm{if}\,\,\, \kappa_{\phi}\ll \kappa_{2},
\end{align}
\end{linenomath}
where $|\alpha|^{2}$ is the size of the cat code, $\kappa_{2}$ is the two-photon dissipation rate, and $\kappa_{\phi}$ is the dephasing rate.
However, the dominant error source in superconducting cavities is photon loss. While dephasing of the cavity does not change the photon number parity, a single photon loss can directly flip the photon number parity of the state. In other words, a single photon loss maps an even parity state $|+_{L}\rangle$ to an odd parity state $|-_{L}\rangle$, thus causing a phase-flip error. As such, phase-flip errors due to single photon loss cannot be mitigated by the 2-cat code. Nevertheless, bit-flip errors due to single photon loss can be suppressed exponentially in the photon number $\alpha^{2}$ provided that the two-photon dissipation is strong enough. Once bit-flip errors are countered, the next layer of QEC will only need to tackle the phase-flip errors, which can be induced by spurious transitions of the ancilla states. In particular, in the stabilized 2-cat codes, thermal excitation of the ancilla during idle times can result in a significant rotation of the cavity state and complete dephasing of the encoded qubit. Currently, such events cannot be corrected by the code and are observed to be a limiting factor for the logical lifetime of the encoded qubit~\cite{lescanne2020_exponential}.
Biased-noise 2-cat qubits can also be realized by using an engineered Hamiltonian \mbox{$\hat{H} = -K(\hat{a}^{\dagger}-(\alpha^{*})^{2})( \hat{a}^{2}-\alpha^{2} )$}, which has two coherent states $|\pm\alpha\rangle$ as degenerate eigenstates~\cite{puri2017_engineering, puri2019_stabilized, puri2020_bias-preserving}. A crucial general consideration for these biased-noise bosonic codes is that the asymmetry in the noise must be preserved during the implementation of gates. The theoretical framework to achieve a universal gate set on the 2-cat codes in a bias-preserving manner has been proposed recently~\cite{guillaud2019_repetition, puri2020_bias-preserving}.
\subsection{Comparison of various bosonic codes for loss and dephasing errors}\label{subsection:code comparision}
In summary, rotation-symmetric codes (e.g., 4-cat codes and binomial codes) are robust against both photon loss and dephasing errors. Translation-symmetric codes such as the GKP codes can be made highly robust against photon loss errors but are susceptible to dephasing errors. Thus, translation-symmetric codes are suited for loss dominated systems. In contrast, 2-cat codes can correct dephasing errors well if they are stabilized by an engineered two-photon dissipation, but are not capable of correcting photon loss errors. Hence, 2-cat codes are naturally suited for dephasing dominated systems. However, as discussed in Sec.~\ref{subsection:biased noise qubits}, 2-cat codes can also be useful in the loss dominated regime as their large noise bias can simplify any higher-level error correction schemes. In Fig.~\ref{fig:codes}(e), we provide a qualitative schematic that represents different regimes of photon loss and dephasing where each code is designed to perform well. Note that if the loss and dephasing error probabilities are too high, the quantum capacity~\cite{lloyd1997_capacity, devetak2005_private} of the corresponding quantum channel will vanish. When this happens, encoding logical quantum information in a reliable way becomes impossible even with an optimal QEC code~\cite{holevo2001_evaluating, wolf2007_quantum, wilde2012_quantum, wilde2018_energyconstrained, arqand2020_quantum}.
While photon loss is the dominant error channel in typical cQED implementations, excitation gain events do also occur, although at a much lower rate. In general, codes robust against photon loss errors tend to be robust against photon gain as well. For rotation-symmetric codes, parity (or super-parity) measurements can detect both photon loss and photon gain events. In the case of translation-symmetric GKP codes, photon gain can be converted into a random shift error by applying a suitable photon loss channel and hence be monitored with the modular quadrature measurement. On the other hand, since the 2-cat codes are not robust against photon loss, they are consequently susceptible to photon gain too, which results in phase-flip errors. Nevertheless, in the presence of engineered dissipation, the stabilized 2-cat codes can still preserve their noise bias under both photon loss and gain errors.
Until now, we have focused only on the intrinsic error-correcting capability of various bosonic codes without considering practical imperfections. In the following sections, we will discuss the implementation of bosonic QEC in cQED systems and examine errors that occur in practical situations.
\section{The {cQED} Hardware for Bosonic Codes}
\label{sec:cavities}
In the preceding sections, we have discussed the concepts and merits of various bosonic QEC codes. These ideas are brought to reality by developing robust quantum hardware with both coherent harmonic modes for encoding information as well as non-linear ancillae for effective control and tomography of the encoded information. Such systems have been realized with the motional degree of freedom in trapped ions~\cite{fluhmann2019_encoding, deNeeve2020_error}, electromagnetic fields of microwave~\cite{deleglise2008_reconstruction} and optical cavities~\cite{hacker2019_deterministic}, Rydberg atom arrays~\cite{omran2019_generation}, and flying photons~\cite{ourjoumtsev2006_generating}, and there are proposals that use other physical systems~\cite{hou2016_generation, bulutay2017_cat-state} as well. Among the various platforms, the cQED architecture consisting of superconducting microwave cavities and Josephson junction-based non-linear ancillae have enabled many prominent experimental milestones towards realizing QEC using bosonic codes. In this section, we will introduce the key hardware building blocks necessary for the successful realization of bosonic codes in cQED.
\subsection{Components of the cQED architecture}
Circuit quantum electrodynamics (cQED) explores light-matter interactions by confining quantized electromagnetic fields in precisely engineered compositions of superconducting inductors and capacitors~\cite{blais2020_circuit}. These superconducting circuit elements can be tailor-made by conventional fabrication techniques~\cite{frunzio2005_fabrication} and controlled by commercially available microwave electronics and dilution refrigerators~\cite{schoelkopf2008_wiring} to access strong coupling regimes~\cite{wallraff2004_strong, schuster2007_resolving, niemczyk2010_circuit}. These characteristics make cQED a compelling platform for universal quantum computation, as noted by several review articles~\cite{girvin2009_circuit, ladd2010_quantum, devoret2013_superconducting, wendin2017_quantum, krantz2019_quantum, kjaergaard2019_superconducting, blais2020_quantum, cai2020_bosonic}.
Quantum devices in the cQED framework are typically built by coupling two components, a linear oscillator mode and an anharmonic mode, in different configurations~\cite{blais2020_circuit}. The anharmonic modes are discrete few-level systems that can be implemented using Josephson junctions. They can be designed to interact with one or more harmonic modes, akin to atoms in an optical field in the cavity QED framework~\cite{haroche2006_exploring}. Unlike naturally occurring atoms, the parameters of these anharmonic oscillators, or artificial atoms, can be precisely engineered. For instance, they may be made to have a fixed resonance frequency in devices with a single Josephson junction or have tunable frequencies by integrating multiple junctions in the presence of an external magnetic flux. The lowest two or three energy levels of these artificial atoms, typically in the form of transmons~\cite{koch2007_charge-insensitive, schreier2008_suppressing}, can be used to effectively encode and process quantum information, as demonstrated by successful realizations of various NISQ era~\cite{preskill2018_quantum} processors~\cite{xiang2017_experimental, otterbach2017_unsupervised, kandala2017_hardware, arute2019_quantum, jurcevic2020_demonstration, google2020_hartree-fock, lacroix2020_improving}.
Linear oscillators are typically realized in cQED by superconducting microwave cavities. Commonly used architectures include coplanar waveguide (CPW)~\cite{wallraff2004_strong}, three-dimensional (3D) rectangular~\cite{paik2011_observation} and cylindrical co-axial~\cite{reagor2016_quantum}, and micromachined~\cite{brecht2015_demonstration} cavities. These quantum harmonic oscillators have well-defined but degenerate energy transitions. Therefore, to selectively address their transitions, we must introduce some non-linearities in them. In cQED, non-linearities are introduced by coupling superconducting cavities to artificial atoms in either the resonant or dispersive regimes.
Under resonant coupling, the transition frequencies of the artificial atom and the cavity coincide, which allows the direct exchange of energy from one mode to the other. In this configuration, cavities can act as an on-demand single photon source~\cite{houck2007_generating} or a quantum bus that mediates operations between two isolated artificial atoms by sequentially interacting with each of them~\cite{majer2007_coupling, sillanpaa2007_coherent, dicarlo2009_demonstration}.
Dispersive coupling is achieved by detuning the frequencies of the cavity, $\omega_{a}$, and the artificial atom, $\omega_{b}$, such that the detuning is much larger than the direct interaction strength between them. In this regime, there is no resonant energy exchange between the modes. Instead, the coupling translates into a state-dependent frequency shift, which can be described by the following Hamiltonian:
\begin{linenomath}
\begin{align}
\hat{H} &= \omega_{a}\hat{a}^{\dagger}\hat{a} + \omega_{b}\hat{b}^{\dagger}\hat{b} - \chi_{ab}\hat{b}^{\dagger}\hat{b}\hat{a}^{\dagger}\hat{a} - \frac{\alpha}{2}\hat{b}^{\dagger2}\hat{b}^2 - \frac{K}{2}\hat{a}^{\dagger2}\hat{a}^2,
\label{eq:dispersive_hamiltonian}
\end{align}
\end{linenomath}
where $\hat{a},\hat{b}$ are the annihilation operators associated with cavity and artificial atom respectively, $\chi_{ab}$ is the dispersive coupling strength between them, $\alpha$ is the anharmonicity of the artificial atom, and $K$ is the non-linearity of the cavity inherited from the atom. Eq.~\eqref{eq:dispersive_hamiltonian} further illustrates that the coupling between the two modes is symmetric. In other words, the frequency of the cavity shifts conditioned on the state of the artificial atom, and vice versa.
Superconducting cavities dispersively coupled to an artificial atom constitute a highly versatile tool that can be employed to fulfill many different roles for quantum information processing. For instance, cavities that are strongly coupled to a transmission line are useful for the efficient readout of the quantum state of the artificial atom~\cite{wallraff2004_strong, wallraff2005_approaching, johansson2006_vacuum}. Conversely, cavities weakly coupled to the environment can be used as quantum memories for storing information coherently~\cite{reagor2016_quantum}. Moreover, when the linewidth of the cavity is narrow compared to the dispersive shift $\chi_{ab}$, we can selectively address the individual energy levels of the cavity via the artificial atom~\cite{schuster2007_resolving}. In this case, the artificial atoms act as non-linear ancillae whose role is to enable conditional operations and perform efficient tomography of the cavity state~\cite{blais2004_cavity}. This configuration where multi-photon states of the cavity encode logical information and the ancilla affords universal control~\cite{krastanov2015_universal} has become an increasingly prevalent choice for implementing bosonic QEC.
\subsection{Coherence of superconducting cavities}\label{subsection:oscillators}
Superconducting microwave cavities may be realized in several geometries, with each having their respective advantages. Typically, cavities are constructed in two main architectures, which are 2-dimensional (2D) and 3-dimensional (3D), based on the dimensionality of the electric field distribution. A third possibility combines the advantages of both the 2D and 3D designs to realize a compact and highly coherent `2.5D' cavity structure, for instance, using micromachining techniques~\cite{brecht2015_demonstration, brecht2017_micromachined}.
In the 2D architecture, such as the coplanar waveguide (CPW), the cavity is defined by gaps between circuit elements printed on a substrate which is typically made from silicon or sapphire (Fig.~\ref{fig:cavities}a). In such planar structures, the energy is mostly stored in the substrate, surfaces, and interfaces, all of which suffer from losses due to spurious two-level systems in the resonator dielectrics~\cite{martinis2005_decoherence, oconnell2008_microwave, gao2008_experimental, muller2019_towards}. Hence, the $Q_{\mathrm{int}}$ of 2D cavities is currently limited to \mbox{$\sim {10^5 - 10^6}$} but can potentially be improved with more sophisticated cavity design~\cite{sage2011_study, vissers2012_reduced, geerlings2012_improving, calusine2018_analysis}, materials selection~\cite{wang2009_improving, barends2010_minimal, vissers2010_low, place2020_new}, and surface treatment~\cite{megrant2012_planar, sandberg2012_etch, bruno2015_reducing, melville2020_comparison}. Despite their limited coherence properties, 2D cavities are widely featured in cQED, and especially in NISQ processors, as they have a small footprint and a straightforward fabrication process. Furthermore, in these processors, these cavities are typically used as readout or bus modes, which do not require long coherence times.
In contrast, 3D cavities (Fig.~\ref{fig:cavities}c) achieve a higher $Q_{\mathrm{int}}$ of \mbox{$\sim {10^7 - 10^8}$}, at the cost of a much larger footprint than their 2D counterparts, by storing energy in the vacuum between the walls of a superconducting box. Among 3D designs, 3D co-axial cavities machined out of high-purity aluminum \mbox{($\simeq {99.999}\%$)} have shown a $Q_{\mathrm{int}}$ as high as \mbox{$1.1 \times 10^8$~\cite{kudra2020_high}}. This is achieved by significantly suppressing the dissipation due to current flowing along the seams of the superconducting walls and imperfections on the inner surfaces of the structure. Additionally, this design is particularly effective as a platform for implementing bosonic codes as one or more ancillae and readout modes can be conveniently integrated into the platform~\cite{axline2016_architecture}.
One strategy to take advantage of both the long lifetimes of 3D designs and the small footprint and scalable fabrication of 2D geometries is to construct compact 2.5D cavities~\cite{minev2013_planar, brecht2015_demonstration, brecht2016_multilayer, minev2016_planar, axline2016_architecture, brecht2017_micromachined, zoepfl2017_characterization}. In particular, the micromachining technique provides a promising method to fabricate lithographically defined 2.5D cavities etched out of silicon wafers~\cite{brecht2015_demonstration}. In this configuration, the energy can be stored primarily in the vacuum but the depth of the cavity is much smaller compared to the other dimensions (Fig.~\ref{fig:cavities}b). Here, the key challenge is to realize a high-quality contact between the top wall and the etched region. With carefully optimized indium bump-bonding methods, internal quality factors surpassing 300 million have recently been achieved in the 2.5D architecture~\cite{lei2020_high}. Moreover, the integration of a transmon ancilla in this design has also been demonstrated~\cite{brecht2017_micromachined}, thus making these 2.5D cavities promising candidates for realizing large-scale quantum devices based on bosonic modes.
\begin{figure*}[t!]
\centering
\includegraphics[scale=1]{figs/cavities.pdf}
\caption[width=\textwidth]{\textbf{Designs of superconducting cavities and their coherence} (a)-(c) Illustrations of 2D CPW, 2.5D micromachined, and 3D co-axial cavities respectively. (d) A selected set of $Q_{\mathrm{int}}$ for four commonly used cavity designs in cQED extracted from the literature. Overall, the coherence properties of superconducting cavities have been steadily increasing over the last decade, with the 3D co-axial cavities currently being the main architecture for realizing bosonic QEC. The references for each cavity design, in chronological order, are: 2D cavities: Refs. \cite{hofheinz2009_synthesizing}, \cite{leek2010_cavity}, \cite{megrant2012_planar}, \cite{axline2016_architecture}, \cite{calusine2018_analysis}; 3D rectangular cavities (3DR): Refs. \cite{paik2011_observation}, \cite{vlastakis2013_deterministically}, \cite{ofek2016_extending}, \cite{ hu2019_quantum}; 3D cylindrical co-axial cavities (3DC): Refs. \cite{reagor2016_quantum}, \cite{reinhold2020_error-corrected}, \cite{ma2020_error-transparent}, \cite{kudra2020_high}; 2.5D cavities: Refs.~\cite{minev2016_planar}, \cite{brecht2017_micromachined}, \cite{zoepfl2017_characterization}, \cite{lei2020_high}. We have highlighted studies that successfully integrated one or more non-linear ancillae with an asterisk (*).}
\label{fig:cavities}
\end{figure*}
Regardless of the architecture, the effective implementation of bosonic QEC schemes requires a careful balance between the need for isolation and coherence as well as the ability to effectively manipulate and characterize the cavity. The various loss channels~\cite{oliver2013_materials, mcrae2020_materials} as well as lossy interfaces~\cite{wenner2011_surface, wang2015_surface, woods2019_determining} of superconducting microwave cavities have been extensively studied and their intrinsic coherence properties have been improving significantly over the last 15 years~\cite{mcrae2020_materials}. In Fig.~\ref{fig:cavities}d, we compile a non-exhaustive summary of the internal quality factors of cavities demonstrated in various geometries over the last decade. Besides enhancing the $Q_{\mathrm{int}}$ of the cavities, integrating them with ancillary mode(s) is crucial for realizing bosonic logical qubits. However, note that introducing an ancilla results in the degradation of the $Q_{\mathrm{int}}$, as the best ancilla coherence times \mbox{($\sim {50 - 100} {\mu}$s)} are typically about $10 - 20$ times lower than those of the state-of-the-art superconducting cavities. Hence, while comparing the performance of the cavities in Fig.~\ref{fig:cavities}d, we have only included demonstrations that are compatible with being coupled to non-linear ancillae in the cQED architecture. From the figure, the 3D co-axial cavities emerge as the leading design currently used to realize bosonic qubits.
\section{Realization of Bosonic Logical Qubits}\label{section: realization}
The remarkable improvements in the performance of cQED hardware components highlighted in Sec.~\ref{sec:cavities} have made realizing protected logical qubits using bosonic codes a realistic goal. Studies that encode a single logical mode and protect it against dominant error channels have been reported for the four-component cat~\cite{ofek2016_extending}, binomial kitten~\cite{hu2019_quantum}, and square and hexagonal GKP~\cite{campagne-ibarcq2019_quantum} codes. Moreover, robust operations on both single~\cite{heeres2017_implementing, ma2020_error-transparent, reinhold2020_error-corrected} and two bosonic modes~\cite{gao2018_programmable, gao2019_entanglement, xu2020_demonstration} have also been explored, thus paving the way towards building a fault-tolerant universal quantum computer based on bosonic logical qubits. In this section, we highlight recent efforts to implement universal control of as well as error correction protocols with bosonic modes.
\subsection{Operations on single bosonic modes}\label{subsection:operations_single}
For processing quantum information encoded in multi-photon states of superconducting cavities, we must be able to perform effective operations on and characterization of the cavity states. The only operation available for a standalone cavity mode is a displacement, \mbox{$\hat{D}(\alpha) = e^{\alpha \hat{a}^{\dagger} - \alpha^{*}\hat{a}}$}, which displaces the position and/or momentum of the harmonic oscillator depending on the value of the complex number $\alpha$. Real values of $\alpha$ correspond to pure position displacements, while imaginary values of $\alpha$ correspond to pure momentum ones~\cite{glauber1963_coherent}. Displacements can only result in the generation of coherent states from vacuum without the possibility to selectively address individual photon number states in the cavity. Therefore, non-trivial operations on these bosonic logical qubits are implemented by dispersively coupling the bosonic mode to a non-linear ancilla in combination with simple displacements.
A key capability enabled by this natural dispersive coupling is a controlled phase shift (CPS). CPS is a unitary operation that imparts a well-defined ancilla state-dependent phase on arbitrary cavity states, and is governed by \mbox{$\hat{U}_{\mathrm{CPS}}(t) = |g\rangle\langle g|\otimes\hat{I} + |e\rangle\langle e|\otimes e^{i\hat{n}\chi t}$}, where $\hat{n}$ is the cavity photon number operator, $\chi$ is the dispersive coupling strength, and $t$ is the evolution time. With this unitary, we can efficiently implement conditional phase operations by simply adjusting the evolution time. In particular, when \mbox{$t = \pi/\chi$}, all the odd photon number states acquire an overall $\pi$-phase while the even states get none. This allows us to effectively map the photon number parity of the bosonic mode onto the state of the ancilla (Fig.~\ref{fig:operations}a).
Note that the time required for the parity-mapping operation scales inversely with the dispersive coupling strength, $\chi$. Naively, one might want to minimize the operation time by engineering a large $\chi$. However, increasing $\chi$ can also result in a stronger inherited non-linearity in the cavity (also known as the Kerr effect), which distorts the encoded information. Therefore, the coupling strength between the cavity and its ancilla, as well as the type of the non-linear ancillary mode, such as the transmon or the SNAIL design~\cite{frattini2017_3-wave}, are usually carefully optimized for each system to produce the desired Hamiltonian configuration.
\begin{figure*}[t!]
\centering
\includegraphics[scale=1]{figs/gates.pdf}
\caption[width=\textwidth]{\textbf{Operations on single bosonic modes.} (a) The sequence for mapping the photon number parity of a bosonic mode to the states of its ancilla. (b) Creation of arbitrary quantum states via repeat SNAP operations on the bosonic mode. (c) Implementing universal control on a single bosonic mode by concurrently driving the cavity and its ancilla with numerically optimized pulses.
} \label{fig:operations}
\end{figure*}
With $\hat{U}_{\mathrm{CPS}}$, we can deterministically create complex bosonic encodings using analytically designed protocols such as the qcMAP operation, which maps an arbitrary qubit state onto a superposition of coherent states in the cavity~\cite{leghtas2013_deterministic}. This scheme is useful for preparing 4-cat states in both single~\cite{vlastakis2013_deterministically} and multiple~\cite{wang2016_schrodinger} cavities. Furthermore, \mbox{$\hat{U}_{\mathrm{CPS}}(\pi/\chi)$} is also employed as the parity-mapping operation which is crucial for the characterization and tomography of the encoded bosonic qubits and the engineered gates on these qubits.
In general, the quantum states encoded in a cavity can be fully characterized by probing the cavity's quasi-probability distributions, which is commonly achieved by performing Wigner tomography. The Wigner function can be defined as the expectation value of the displaced photon number parity operator, \mbox{$W(\beta) = \frac{2}{\pi} \mathrm{Tr}[\hat{D}(\beta)^\dagger \rho \hat{D}(\beta) \hat{P}]$}. In cQED, the Wigner functions of arbitrary quantum states can be measured precisely with a well-defined sequence that uses only the cavity displacement, ancilla rotation, and controlled phase shift operations~\cite{blais2020_circuit}. From the results of the Wigner tomography, we can reconstruct the full density matrix and characterize the action enacted on the cavity states in either the Pauli Transfer Matrix~\cite{chow2012_universal} or the process matrix~\cite{nielsen2000_quantum}.
Another crucial operation arising from the natural dispersive coupling is the non-linear selective number-dependent arbitrary phase (SNAP) gate~\cite{krastanov2015_universal, heeres2015_cavity}. A SNAP gate (Fig.~\ref{fig:operations}b), defined as \mbox{$\hat{S}_{n}(\theta_n)= e^{i\theta_n(|n\rangle\langle n|)}$}, selectively imparts a phase $\theta_n$ to the number state $|n\rangle$. Due to the energy-preserving nature of this operation, we can simultaneously perform $\hat{S}_{n}(\theta_n)$ on multiple number states. By numerically optimizing the linear displacement gates and the phases applied to each photon number state in the cavity, we can effectively cancel out the undesired Fock components via destructive interference to obtain the intended target state.
While schemes like qcMAP and SNAP are sufficient to realize universal control on the cavity state, they quickly become impractical for handling more complex bosonic states. For instance, an operation on $n$ photons requires $O(n^2)$ gates using the SNAP protocol. To address this challenge, a fully numerical approach using optimal control theory (OCT) has been developed and widely adopted in recent years. The OCT framework provides an efficient general-purpose tool to implement arbitrary operations. In particular, the Gradient Ascent Pulse Engineering (GRAPE) method~\cite{khaneja2005_optimal, defouquieres2011_second} has been successfully deployed in other physical systems~\cite{dolde2014_high-fidelity, anderson2015_accurate} to implement robust quantum control. By constructing an accurate model of the time-dependent Hamiltonian of the system in the presence of arbitrary control fields, we can apply this technique to cQED systems to realize high-fidelity universal gate sets on any bosonic qubit encoded in cavities, as demonstrated in Ref.~\cite{heeres2017_implementing}. A typical example of a set of pulses obtained through the
gradient-based OCT framework is shown in (Fig.~\ref{fig:operations}c).
More recently, various concepts from classical machine learning, such as automatic differentiation~\cite{abdelhafez2019_gradient-based} and reinforcement learning~\cite{niu2019_universal, zhang2019_reinforcement}, have been applied to enhance the efficiency of the numerical optimization. Crucially, the success of these techniques does not only rely on the robustness of the algorithms, but also depends on the choice of boundary conditions. Knowledge of these boundary conditions requires comprehensive investigations of the physics of the quantum system and the practical constraints of the control and measurement apparatus.
\subsection{Operations on multiple bosonic modes}\label{subsection:operations_multiple}
Apart from robust single-mode operations, universal quantum computation using bosonic qubits also requires at least one entangling gate between two modes. Realizing such an operation can be challenging due to the lack of a natural coupling between cavities. Moreover, the individual coherence of each bosonic qubit must be maintained while maximizing the rate of interactions between them. One promising strategy to tackle this issue is to use the non-linear frequency conversion capability of the Josephson junction to provide a driven coupling between two otherwise isolated cavities~\cite{bergeal2010_phase-preserving}. Such operations are fully activated by external microwave drives which can be tuned on and off on-demand without modifying the hardware. This arrangement ensures that the individual bosonic modes remain well-isolated during idle times and undergo the engineered interaction only when an operation is enacted.
Using this strategy, a CNOT gate was the first logical gate enacted on two bosonic qubits~\cite{rosenblum2018_cnot}. This gate is facilitated by a parametrically-driven sideband transition between the ancilla and the control mode together with a carefully chosen conditional phase gate between the ancilla and the target cavity. By achieving a gate fidelity above 98\%, this study showcases the potential of such engineered quantum gates between bosonic modes. More recently, a controlled-phase gate has been demonstrated between two binomial logical qubits in Ref.~\cite{xu2020_demonstration}. Here, the microwave drives are tailored to induce a geometric phase that depends on the joint state of the two bosonic modes. However, these two types of operations are both customized for a selective set of code words and do not yet generalize readily to other bosonic encoding schemes.
A code-independent coupling mechanism between two otherwise isolated bosonic modes is a crucial ingredient for realizing universal control on these logical qubits. The isolated cavities must be sufficiently detuned from each other to ensure the coherence of each mode and the absence of undesired cross-talk. Ref.~\cite{gao2018_programmable} demonstrated how the four-wave mixing process in a Josephson junction can provide a frequency converting bilinear coupling of the form \mbox{$H_{\mathrm{int}} (t)/\hbar = g(t) (e^{i\varphi}\hat{a}\hat{b}^{\dagger} + e^{-i\varphi}\hat{a}^{\dagger}\hat{b})$}. Here, $\hat{a}, \hat{b}$ are the annihilation operations associated with each of the cavities, the time-dependent coefficient $g(t)$ is the coupling strength, and $\varphi$ is the relative phase between the two microwave drives. The coefficient $g(t)$ depends on the effective amplitudes of the drives, which satisfy the frequency matching condition \mbox{$|\omega_2 - \omega_1| = |\omega_a - \omega_b|$}. Most notably, this coupling can be programmed to implement an identity, a 50:50 beamsplitter, or a full SWAP operation between the stationary microwave fields in the cavities by simply adjusting the duration of the evolution. Such an engineered coupling provides a powerful tool for implementing programmable interferometry between cavity states~\cite{gao2018_programmable}, which is a key building block for realizing various continuous-variable information processing tasks such as boson sampling~\cite{aaronson2011_computational}, simulation of vibrational quantum dynamics of molecules \cite{huh2015_boson, sparrow2018_simulating, clements2018_approximating, wang2020_efficient}, and distributed quantum sensing~\cite{zhuang2018_distributed, zhuang2020_distributed, xia2020_demonstration, noh2020_encoding}.
Moreover, this bilinear coupling is also a valuable resource for enacting gates on two logical elements encoded in GKP states~\cite{gottesman2001_encoding}. As mentioned in Sec.~\ref{sec:codes}, GKP encodings rely on the non-linearity of the code words and only require linear or bilinear operations for universal control \cite{baragiola2019_allgaussian}. Therefore, this engineered bilinear coupling provides a simple and effective strategy for implementing a deterministic entangling operation for the GKP code.
For other bosonic codes, this bilinear interaction is not alone sufficient to generate a universal gate set, which requires at least one entangling gate. In this case, the exponential SWAP (eSWAP) operation can be designed to provide deterministic and code-independent entanglement~\cite{lau2016_universal}. The eSWAP operation, akin to an exchange operation between spins, implements a programmable unitary of the form:
\begin{linenomath}
\begin{equation}\label{eq:eswap}
\hat{U}(\theta) = \cos{(\theta)} \hat{I} + \sin{(\theta)}\mathrm{SWAP},
\end{equation}
\end{linenomath}
where $\theta$ is the rotation angle on the ancilla and $\hat{I}$ is the identity operation. Intuitively, this unitary implements a weighted superposition of the identity and SWAP operations between two bosonic modes regardless of their specific encodings. The eSWAP unitary has been realized in Ref.~\cite{gao2019_entanglement} between two bosonic modes housed in 3D co-axial cavities bridged by an ancilla. In this demonstration, an additional ancilla is introduced to one of the cavity modes and the resultant dispersive coupling is used to enact a $\hat{U}_{\mathrm{CPS}}$ operation to provide the tunable rotation necessary for the eSWAP unitary. The eSWAP unitary is then enacted on several encoding schemes in the Fock, coherent, and binomial basis. The availability of such a deterministic and code-independent entangling operation is a crucial step towards universal quantum computation using bosonic logical qubits. A recent study has shown that universal control and operations on tens of bosonic qubits can be achieved in a novel architecture comprising a single transmon coupled simultaneously to a multi-mode superconducting cavity~\cite{chakram2020_seamless}.
\subsection{Implementations of QEC}\label{subsection:qec}
In general, implementations of QEC with bosonic codes suffer from an initial rise in error rates due to the presence of and need to control multi-photon states. Hence, the main experimental challenge is to achieve an enhancement in the lifetime of the encoded qubit despite this initial penalty. In the context of bosonic codes, the break-even point is defined relative to the $|0, 1\rangle$ Fock states, which are the longest-lived physical elements in the cavity without error correction. Beyond the break-even point, we can be confident that the chosen QEC protocol does not introduce more errors into the system, thus attaining an improvement in the lifetime of the logical qubit. Till date, three studies have approached~\cite{campagne-ibarcq2019_quantum, hu2019_quantum} or achieved~\cite{ofek2016_extending} the `break-even point' for QEC with bosonic codes in cQED devices without any post-selection.
Broadly, QEC schemes fall into three categories based on how they afford protection to the logical qubit. In active QEC, error syndromes are repeatedly measured during the state evolution of the logical qubit and any errors detected are subsequently corrected based on the measurement outcomes. In autonomous QEC, errors are removed by tailored dissipation or by coupling to an auxiliary system, without repeatedly probing the logical qubit. In passive QEC, the logical information is intrinsically protected from decoherence because of specifically designed physical symmetries~\cite{terhal2015_quantum}. In this discussion, we have opted to distinguish between autonomous and passive QEC to highlight potential differences in system design. In the autonomous approach, the environment is intentionally engineered to suppress or mitigate errors. Whereas in the passive case, the system Hamiltonian itself is tailored to be robust against certain errors, which often involves constructing a unique physical element in the hardware~\cite{gyenis2019_experimental, grimm2020_stabilization, smith2020_magnifying}.
Typically, active QEC requires robust measurements of the error syndrome and real-time feedback. In superconducting cavities, the dominant source of error is single photon loss. For encoding schemes with rotational symmetries, such as the cat and binomial codes, single photon loss results in a flip in the parity of the code words. Therefore, measuring the parity operator tells us whether a photon jump has taken place, thus allowing us to detect the error syndrome of these logical qubits~\cite{sun2014_tracking}. In Ref.~\cite{ofek2016_extending}, which corrected a four-component cat state under photon loss (Fig.~\ref{fig:catcycle}), the correct logical state was recovered by making appropriate adjustments in the decoding step based on the number of parity flips detected with real-time feedback. The corrected logical qubit showed an improved lifetime compared to both the uncorrected state and the $|0, 1\rangle$ Fock state encoding, thereby achieving the break-even point. However, this error correction strategy is not suitable for protecting bosonic qubits for timescales exceeding the intrinsic cavity lifetime as the loss of energy from the system is accounted for but not physically rectified. Thus, such energy attenuation should be physically compensated to significantly surpass the break-even point.
\begin{figure}[h!]
\centering
\includegraphics[scale=1]{figs/cat_cycle.pdf}
\caption{\textbf{Four-component cat code under photon loss.} Every photon loss event ($\hat{a}$) changes not only the parity of the basis states, but also changes the phase relationship between them. The encoded state cycles between the even (logical) and odd (error) parity subspaces, while also rotating about the Z-axis by $\pi/2$. The decoding sequence must take both these effects into account to correctly recover the logical information.}
\label{fig:catcycle}
\end{figure}
In contrast, Ref.~\cite{hu2019_quantum} admits a photon pumping operation to achieve QEC on a logical qubit encoded in the binomial code. The errors are detected by photon number parity measurements, as in Ref.~\cite{ofek2016_extending}. The errors occurring on the logical state are then corrected by an appropriate recovery operation as soon as they are detected. An approximated recovery operation is still required in case no errors are detected as the system evolves under the no-parity-jump operator~\cite{michael2016_new}. In this experiment, the lifetime of the logical qubit was greater than that of the uncorrected binomial code state, but was marginally below that of the $|0, 1\rangle$ Fock state.
In addition to the cat and binomial encodings, active QEC has also been recently demonstrated on a high-quality GKP state stored in a superconducting cavity~\cite{campagne-ibarcq2019_quantum}. As explained in Sec.~\ref{subsection:translation symmetric codes}, errors occurring on a moderately squeezed GKP state simply manifest as displacements of the cavity state, and are revealed and mitigated by measuring the displacement stabilizers. While mitigating errors is relatively straightforward, experimental challenges in implementing the GKP code lie in preparing finitely squeezed approximate GKP states and performing modulo quadrature measurement of stabilizers. The QEC protocol used in Ref.~\cite{campagne-ibarcq2019_quantum} suppresses all logical errors for a GKP state by alternating between two peak-sharpening and envelope-trimming rounds, each consisting of different conditional displacements on the cavity. The magnitude of these conditional displacements is dependent on the measured state of the ancilla onto which displacement stabilizers are mapped. This sharpen-and-trim technique can be generalized based on the second-order Trotter formula~\cite{royer2020_stabilization}.
An alternative strategy for QEC that does not require real-time feedback based on measurement outcomes involves engineering the dynamics of the system to correct errors autonomously. In practice, autonomous QEC can be achieved by introducing a tailored dissipation~\cite{leghtas2015_confining} such that the logical subspace is stabilized via controlled interactions with the environment. In Refs.~\cite{leghtas2015_confining, touzard2018_coherent}, the desired dissipation process subtracts photons in pairs from the bosonic mode via four-wave mixing in the Josephson junction of the ancilla. This process defines a stabilized manifold of steady states spanned by superpositions of coherent states (two-component cat states). The relative strength between the engineered two-photon dissipation and the intrinsic single photon loss of the cavity defines the separation of the two components of the cat state, which determines the extent of protection against bit-flip errors. In a more recent study, significant improvements in the suppression of bit-flip errors were demonstrated by using a tailored multi-junction ancilla to enact the two-photon dissipation process~\cite{lescanne2020_exponential}.
Similarly, an engineered four-photon dissipation achieved by an eight-wave mixing process, can protect four-component cat codes against dephasing~\cite{mundhada2019_experimental}. However, this strategy only accounts for dephasing errors and does not recover logical errors caused by photon loss. In contrast, in another autonomous QEC implementation with a truncated 4-cat state~\cite{gertler2020_protecting}, a photon was added to the cavity upon detecting a photon loss using a synthetic dissipation operator. This work effectively corrects photon loss events but does not account for dephasing errors. Similarly, another study performed autonomous QEC against photon loss with the kitten code by triggering an engineered jump operation~\cite{ma2020_error-transparent}. This operation recovered the code states whenever the system entered an error state without the need for any external probing. Although these current state-of-the-art demonstrations of autonomous QEC techniques only address one type of errors, they provide convincing evidence for the viability of the respective bosonic encoding schemes, thus paving the way for higher-order QEC protocols that can protect logical information against both photon loss and dephasing errors.
Finally, QEC with bosonic codes can also be realized using a passive approach, for instance, by designing logical qubits with highly biased-noise channels to provide intrinsic protection without the need for probing any error syndromes. Very recently, two studies~\cite{lescanne2020_exponential, grimm2020_stabilization} verified that the bias between phase and bit-flip errors increases exponentially with $\alpha$, the size of the coherent state components of the 2-cat code. This strong asymmetry between the different types of errors can be exploited to significantly reduce the hardware overhead for fault-tolerant quantum computation~\cite{tuckett2018-ultrahigh,tuckett2019_tailoring,tuckett2020_faulttolerant}.
\section{Towards Fault-tolerant Quantum Computation with Bosonic Modes}\label{sec:FT}
A crucial consideration for a successful QEC implementation is that even the attempts to correct errors can themselves be erroneous. Thus, a key requirement for realizing `fault-tolerant' quantum computation is to carefully design error correction circuits such that faults that occur during the execution of a QEC protocol are also tolerated~\cite{gottesman2009_introduction}. A simple example of fault-tolerant circuit design is the transversal implementation of logical operations in the conventional paradigm of quantum error correction with multiple two-level systems. For instance, by implementing logical gates transversally for distance-$3$ codes that can correct any single qubit errors, one can ensure that a single fault in the circuit induces at most a single qubit error in any logical code block which remains correctable by the code~\cite{aliferis2006_quantum}.
One key challenge in achieving fault-tolerant operations on bosonic qubits in the cat and binomial encodings is the propagation of uncorrectable errors due to ancilla decoherence~\cite{ofek2016_extending, hu2019_quantum}. The ancilla, often in the form of a transmon, is employed during state preparation, gates, measurements, and error correction on the encoded qubit via the natural dispersive coupling. Spontaneous emission of the ancilla during these processes induces dephasing on the logical qubit, thus irreversibly corrupting the encoded information. Furthermore, two-photon loss and Kerr effects also limit the effectiveness of these QEC implementations. One technique used to mitigate these imperfections is to optimize the cadence of the parity measurements~\cite{ofek2016_extending}. Frequent parity measurements allow more accurate tracking of the error syndrome but increase the likelihood of the ancilla relaxing during measurement and thereby dephasing the cavity. On the other hand, a larger interval between two measurements not only increases the probability of two-photon loss, but also results in a higher phase accumulation due to self-Kerr interactions. As the self-Kerr operator does not commute with the annihilation operator, photon jumps cause cavity dephasing. In Ref.~\cite{ofek2016_extending}, the phase accumulated due to the self-Kerr effect was first estimated and then accounted for by adjusting the phase of subsequent drive pulses.
Several approaches have been devised to reduce the propagation of uncorrectable errors from the ancilla to the encoded information. In Ref.~\cite{rosenblum2018_fault-tolerant}, the third energy level of the ancilla is employed in the parity-mapping process to suppress the dephasing of the bosonic mode. A similar strategy can also be applied to general cavity operations, as shown in Ref.~\cite{reinhold2020_error-corrected}. In these schemes, additional levels of the ancilla as well as carefully engineered cavity-ancilla interactions are introduced to prevent incoherent evolution of the bosonic mode in the event of ancilla decay~\cite{ma2020_pathindependent}. Alternatively, error-transparent gates have been realized for binomial encodings~\cite{ma2020_error-transparent} by cleverly engineering the evolution of the bosonic qubit in both the code and error space.
For GKP codes, early discussions on fault-tolerance were focused on whether a small shift error occurring in the middle of an error correction circuit remains small throughout the entire circuit~\cite{gottesman2001_encoding,glancy2006_error}. Recently, numerous studies have approached fault-tolerant quantum computation with the GKP code by combining the GKP code with conventional error correction schemes based on multiple two-level systems that are already shown to be fault-tolerant. In performing such a concatenation of codes, additional analog information acquired during the error correction protocol can significantly boost the performance of a next-level QEC scheme based on multiple error-corrected GKP qubits~\cite{fukui2017_analog, fukui2018_tracking}. For instance, studies have shown that the fault-tolerance thresholds of the circuit-based toric code and the surface code can be increased by using the additional analog information~\cite{vuillot2019_quantum, noh2020_faulttolerant}. Another way to realize fault-tolerant quantum computing with the GKP code is to combine the GKP code with the idea of fault-tolerant measurement-based quantum computation using cluster states, for which the additional analog information also plays an important role~\cite{menicucci2014_faulttolerant, fukui2018_highthreshold, walshe2019_robust, fukui2019_highthreshold, yamasaki2020_polylog, bourassa2020_blueprint, larsen2021_faulttolerant}. From the circuit-based computing studies, we know that fault-tolerant quantum computing with the GKP code is possible if the squeezing of the GKP states is larger than $11$dB. Whereas from the measurement-based computing scenarios, a more favorable threshold value of $8$dB can be obtained by taking advantage of post-selection. However, as the measurement-based schemes use post-selection, they require a higher resource overhead. Recently, Ref. \cite{shi2019_fault-tolerant} made the phase estimation protocol to prepare GKP states in Ref.~\cite{terhal2016_encoding} fault-tolerant by using the concept of flag qubits~\cite{chao2018_quantum, chamberland2018_flag}.
There have also been general studies on the fault-tolerance of biased-noise qubits~\cite{aliferis2008_faulttolerant, tuckett2018-ultrahigh, tuckett2019_tailoring, tuckett2020_faulttolerant}. These studies highlight the potential of achieving higher fault-tolerance thresholds and lower resource overheads by taking advantage of the noise bias. Refs.~\cite{guillaud2019_repetition,guillaud2020_error} suggested pieceable fault-tolerant~\cite{yoder2016_universal} Toffoli circuits while Ref.~\cite{puri2020_bias-preserving} offered a magic state distillation scheme tailored to biased-noise models~\cite{webster2015_reducing} as a means of achieving fault-tolerant universal quantum computation with biased-noise 2-cat qubits. A detailed roadmap for realizing biased-noise 2-cat qubits using acoustic nanomechanical resonators as well as alternative fault-tolerant universal quantum computation schemes were recently proposed~\cite{chamberland2020_building}.
\section{Outlook and Perspectives}\label{sec:future}
Continuous variables have long been recognized as promising candidates for the efficient encoding and processing of quantum information~\cite{lloyd1999_quantum}. Bosonic codes leverage the advantages afforded by continuous variable quantum devices to realize hardware-efficient QEC and pave the way towards the eventual realization of a fault-tolerant quantum computer. In this review, we have presented some of the most compelling results that attest to the viability of the cQED platform. In particular, many milestones, from reaching the break-even point to implementing logical operations between bosonic logical qubits, have thus far been demonstrated using 3D cQED devices. An important next milestone would be to demonstrate the break-even point for logical operations, i.e., to achieve gate fidelities with error-correctable bosonic qubits that are higher than the highest gate fidelity attainable with the best physical element without error correction. Achieving the break-even point in both QEC and single or two-mode logical operations forms a critical foundation upon which fault-tolerant gates and algorithms can be constructed.
We further emphasize that the techniques and methodologies developed in these 3D cQED implementations of bosonic codes are agnostic to the hardware architecture. They can therefore be readily adapted to more compact designs, such as planar or 2.5D devices, as their performance improves with further developments in materials engineering and new fabrication techniques~\cite{place2020_new}.
Crucially, encoding logical information in bosonic codes is naturally compatible with a modular approach to scalability. In the modular architecture, each logical element can be encoded, protected, and optimized individually, and then connected to other elements via on-demand communication channels~\cite{jiang2007_distributed, kimble2008_quantum, monroe2014_large-scale}. In contrast to directly wiring up an increasing number of physical elements, modularity has the benefit of reduced cross-talk, robustness against local failure modes, and enhanced re-configurability. Encoding and manipulating individual logical qubits encoded in cavities is a crucial primitive for eventually realizing a large-scale modular quantum device. In addition, recent experiments have also demonstrated other key elements of a modular architecture, including programmable quantum communication and entanglement generation between two distant bosonic modes~\cite{axline2018_on-demand, campagne-ibarcq2018_deterministic, burkhart2020_error-detected} as well as a teleported CNOT gate~\cite{chou2018_deterministic}.
As we continue to improve the performance and scale of these bosonic quantum systems on the hardware level, we must also expand our technological repertoire to effectively control and characterize these more complex devices. This expansion may involve, for instance, the development of more sophisticated measurement electronics and protocols as well as new theoretical frameworks to capture the signatures of multipartite entanglement. The small-scale bosonic devices developed so far provide an indispensable platform to test and refine these crucial elements.
In recent years, cQED devices have become a workhorse for the implementations of quantum error correction and quantum information processing. Alongside the developments in cQED, we are also witnessing remarkable progress with other platforms based on trapped-ions~\cite{landsman2019_two-qubit, egan2020_fault-tolerant}, silicon spins~\cite{veldhorst2015_two-qubit, watson2018_programmable}, neutral atoms~\cite{levine2019_parallel}, etc, from both academic and industrial initiatives. While the ultimate hardware to achieve fault-tolerance may not consist of millions of 3D cavities, the insights gained from and the technical achievements demonstrated by the results summarized in this article will serve as crucial building blocks for the eventual realization of a scalable and robust universal quantum computer.
\let\oldaddcontentsline\addcontentslin
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\section*{Acknowledgments}\label{sec:acknowledgments}
The authors thank S. M. Girvin and L. Sun for fruitful discussions and comments on the manuscript. Yvonne Y. Gao acknowledges funding from the Singapore National Research Foundation Fellowship (Class of 2020). This work is also supported by the Ministry of Education in Singapore.
\let\addcontentsline\oldaddcontentslin
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1,108,101,563,271 | arxiv | \section{Introduction}
Spectroscopy of positronium (Ps) provides a sensitive test of bound
state theory based on the Quantum Electrodynamics (QED). Because of
the small mass of electron and positron, the effects of strong and
weak interactions are negligible compared with the accuracy of present
experiments. For this reason positronium represents a unique system
which can, in principle, be described with very high precision by
means of the QED only. Tests of the QED predictions are made possible
by the very high experimental accuracy of positronium spectroscopy
\cite{Mills88}.
The gross spectrum of positronium is well described by the
Schr\"odinger equation with the Coulomb potential. Energy levels
are
\begin{equation}
E(n) = -\frac {m\alpha^2}{4n^2},
\label{ELO}
\end{equation}
where $n$ is the principal quantum number.
For the purpose of interpreting modern experiments
the precision of Eq.~(\ref{ELO}) is insufficient. Corrections to the
energy levels can in part be described by the
Quantum Mechanics; however, for a complete description one has to
resort to the Quantum Field Theory (QFT). Unfortunately, an
application of the QFT to the
bound states is difficult and special methods have to be devised
\cite{Barbieri:1978mf,FGross,Lepage:1977gd,Caswell:1986ui}.
Various approaches to bound state calculations have been reviewed
e.g.~in \cite{SapYen}. Here we focus on a method close to the
so--called Non-Relativistic Quantum Electrodynamics (NRQED)
\cite{Caswell:1986ui}, which is an effective field theory based on
the QED, for small energies and momenta. Eq.~(\ref{ELO}) implies that
the characteristic velocity of the electron and positron in
positronium is of the order of the fine structure constant $\alpha \ll
1$. It is appropriate to apply a non-relativistic approximation to
this system.
Recently much progress has been achieved in the framework of
non-relativistic effective theories, mainly by employing dimensional
regularization. It has been shown \cite{Pineda:1997bj} that this
regularization procedure permits an exact separation of effects
arising at various characteristic energy scales. Using that method,
which we will call dimensionally regularized NRQED (NRQED$_\epsilon$)
the complete energy spectrum of Ps has been reproduced to order
$m\alpha^5$ \cite{Pineda:1998kn}. More recently, we have computed
$m\alpha^6$ corrections to the hyperfine splitting (HFS) of the Ps
ground state \cite{Czarnecki:1998zv}, confirming one of previously
obtained numerical results \cite{Ph}. In the present paper we
generalize that result to all $S$ states, confirming \cite{PhPRL}, and
compute also their spin independent shift at ${\cal O}(m\alpha^6)$
(obtained numerically in \cite{PhPRL}).
It is convenient to describe the energy of an $nS$ state of Ps by
dividing it up into the spin--averaged part and a part dependent on
the total Ps spin (hyperfine splitting):
\begin{equation}
E(J,n) = E_{\rm aver}(n) +\mbox{$\vec{s}$}_{+}\mbox{$\vec{s}$}_{-}E_{\rm hfs}(n),
\label{param}
\end{equation}
where $J$ is the total spin value of the Ps
and $\mbox{$\vec{s}$} _{\pm}$ are the spins of the electron and positron, respectively.
One finds:
\begin{eqnarray}
J&=&1 \quad {\rm (triplet\; state):} \qquad
\mbox{$\vec{s}$}_{+}\mbox{$\vec{s}$}_{-}= +\frac{1}{4},
\nonumber \\
J&=&0 \quad {\rm (singlet\; state):} \qquad \mbox{$\vec{s}$}_{+}\mbox{$\vec{s}$}_{-}= -\frac{3}{4}.
\end{eqnarray}
Both the spin--averaged energy and the hyperfine splitting can be
represented by series in powers and logarithms of the fine structure
constant. In the lowest order $E_{\rm aver}(n)=E(n)$ is given by
Eq.~(\ref{ELO}), and $E_{\rm hfs}(n) = {\cal O}(m\alpha^4)$.
To order $m\alpha^5$ the results for $E_{\rm aver}$ and $E_{\rm hfs}$
were found in \cite{KK,Fulton,Gupta}. Those corrections have several
sources: electron and positron charge radii and anomalous magnetic
moments, vacuum polarization, two-photon exchange, two-photon
annihilation and one-loop correction to the single-photon
annihilation.
Current accuracy of high precision experiments requires a complete
calculation of the ${\cal O}(m\alpha^6)$ corrections $\Delta E_{\rm
aver}$ and $\Delta E_{\rm hfs}$.
The most precisely measured property of positronium is the
ground state HFS, i.e.~the energy difference between
the two lowest states with total spin $1$ and $0$.
Two best experimental values are
\begin{equation}
\Delta \nu \equiv E(1^3S_1)-E(1^1S_0)
=203\;387.5(1.6)\;{\rm MHz},
\label{Mills}
\end{equation}
found in \cite{Mills1,Mills2} and
\begin{equation}
\Delta \nu =203\;389.10(0.74)\;{\rm MHz},
\label{Hughes}
\end{equation}
obtained in \cite{Ritter}.
Another quantity of the experimental interest is the
energy difference of $2^3S_1$ and $1^3S_1$ states
\cite{Fee}:
\begin{equation}\label{Fee}
E(2^3S_1)-E(1^1S_1) = 1\;233\;607\;216.4(3.2)\; {\rm MHz}.
\end{equation}
The absolute accuracy of this measurement is clearly less impressive
than that of the hyperfine splitting. However, since $m\alpha^6 =
18.658\; {\rm MHz}$, a complete calculation of the energy levels at
this order is warranted.
At order $m\alpha^6$ both $\Delta E_{\rm
aver}$ and $\Delta E_{\rm hfs}$ can be written as
\begin{equation}
\Delta E = \Delta E_{\rm rad}+\Delta E_{\rm annih}
+\Delta E_{\rm rad\;rec}+\Delta E_{\rm rec}.
\end{equation}
The logarithmic contributions at this order, ${\cal O}(m\alpha^6
\ln\alpha)$, present in the annihilation $\Delta E_{\rm annih}$ and
recoil $\Delta E_{\rm rec}$ corrections, were found first
\cite{Bodwin:1978ut,Caswell:1979vz}. $\Delta E_{\rm rad}$ arises from
the radiative corrections to the Breit potential at ${\cal
O}(\alpha,\alpha^2)$ \cite{Brodsky:1966vn,BMR1}.
The three, two, and one-photon annihilation
contributions giving $\Delta E_{\rm annih}$ were found in
\cite{Adkins:1988nd}, \cite{AAB}, and \cite{Adkins97,Hoang:1997ki},
respectively. The non-annihilation radiative recoil contributions
$\Delta E_{\rm rad\; rec}$ were calculated in \cite{STY,PhK}, while pure
recoil corrections $\Delta E_{\rm rec}$ were obtained in
\cite{Ph,Caswell:1986ui,AS} for the HFS and in \cite{PhPRL} for
$E_{\rm aver}$.
In this paper we present an analytic calculation of the recoil and
radiative recoil corrections, $\Delta E_{\rm rad}$ and $\Delta E_{\rm
rad\; rec}$, to energy levels of arbitrary $nS$ positronium states.
The rest of this paper is organized as follows: in Section
\ref{sec:frame} we discuss our method in general terms. Section
\ref{sec:hfs} is devoted to the calculation of the HFS. Many technical
details of this calculation are discussed there. In Section
\ref{sec:lev} we present a calculation of the average energy $E_{\rm
aver}$. It is very similar to HFS, except that some additional
operators contribute. Also the ${\cal O}(m\alpha^6)$ radiative
recoil corrections are discussed. Our results are summarized in
Section \ref{sec:sum}, where also an overview of the theoretical and
experimental situation is given and a complete analytic formula for
the $nS$ energy levels to order $m\alpha^6$ is presented.
\section{Framework of the calculation}
\label{sec:frame}
Before getting into details, let us describe the general framework of
our calculation of the ${\cal O}(m\alpha^6)$ corrections to energy
levels.
First, we calculate an on--shell scattering amplitude for
non-relativistic $(v \ll 1)$ particles to the needed order (the fact
that $v \sim \alpha$ in Ps serves as a counting rule for contributions
of various operators). In addition to the leading, single Coulomb
exchange, this includes the relative ${\cal O}(v^2)$ Breit corrections
and also higher order ${\cal O}(v^4,\alpha v^3)$ terms. This
non-relativistic amplitude is gauge invariant, and taken with a minus
sign provides the potential for non--relativistic particles.
Next, we use the ordinary quantum mechanical perturbation theory to
find the corrections due to that potential; as unperturbed states we
use the solutions of the Schr\"odinger equation with the Coulomb
potential. We get the ${\cal O}(m\alpha^6)$ correction to energy
levels as the sum of the first order correction due to ${\cal
O}(v^4,\alpha v^3)$ perturbation and of the second order correction
due to the Breit Hamiltonian. Previously, this scheme was used for
the calculation of the ${\cal O}(m\alpha^6 \ln\alpha)$ corrections to
the levels of $S$-states \cite{KMYlog} and of the ${\cal
O}(m\alpha^6)$ corrections to the levels of $P$-states
\cite{KMYp,KMYp2}.
In the present calculation the result of the non-relativistic
calculation is divergent. This is because also the short-distance
(``hard'') corrections contribute. They arise from virtual momenta
regions of the order of electron mass and cannot be obtained from the
non--relativistic expansion.
Our calculation is performed in the spirit of NRQED. We apply
dimensional regularization, which offers technical advantages over
more common techniques, based on the introduction of an intermediate
cut-off to separate the relativistic and non--relativistic momentum
regions. Dimensional regularization makes the matching of the
low-scale effective theory and the complete QED extremely simple. We
find that in the sum of the short and long--distance contributions the
singularities in the parameter\footnote{Throughout the paper, we use
the following notations: $D=4-2\epsilon$ and $d=3-2\epsilon$.} $\epsilon$
disappear and one arrives at a finite result.
The spinor algebra in dimensional regularization requires some
comments. In order to obtain the energy shift due to an operator
${\cal O}_i$ one has to calculate the trace of the form $\mbox{Tr}
\left [ \Psi^\dagger {\cal O}_i \Psi \right ]$, where $\Psi$ is an
appropriate wave function. The spinor parts of the relevant wave
functions are
$$
\Psi_P = \frac{1+\gamma_0}{2\sqrt{2}} \gamma_5,\qquad
\Psi_O = \frac{1+\gamma_0}{2\sqrt{2}} \mbox{$\vec{\gamma}$} \mbox{$\vec{\xi}$} ,
$$
for para and orthopositronium states, respectively. In the latter
case, $\vec \xi$ is the polarization vector (we average over its
directions). The traces are calculated in a standard way
in the $D$-dimensional space. One encounters only even numbers of
$\gamma_5$ matrices, and we treat them as anticommuting.
Since the matrix elements involve the positronium
wave function, it is easiest to calculate for the ground state ($n=1$).
However, once the corrections to the ground state have been found,
there is a convenient way of finding them for excited states, with an
arbitrary value of the principal quantum number $n$. Only the
non-relativistic contributions have a non-trivial dependence on $n$.
Their computation in dimensional regularization would be difficult.
However, this task is simplified using other regularizations.
Finally, we eliminate the cut-off dependence by requiring that for
$n=1$ the result matches the formula we found for $n=1$. The freedom
of choosing the regularization scheme simplifies considerably this
part of the calculation.
\section{HFS of the positronium ground state}
\label{sec:hfs}
In this Section we present a calculation of the recoil corrections to
the Ps ground state, $\Delta_{\rm rec}E_{\rm hfs}$. It is given as a
sum of soft (non-relativistic)
Eq.~(\ref{HFSnonrel}) and hard Eq.~(\ref{hard}) scales:
\begin{equation}
\Delta_{\rm rec} E_{\rm hfs}
= \Delta_{\rm nonrel} E_{\rm hfs} +\Delta_{\rm hard} E_{\rm hfs}
= m\alpha^6
\left( - \frac{ 1 }{ 6 } \ln \alpha + \frac{331}{432}
- \frac{ \ln 2 }{ 4 }
- \frac{17\zeta(3)}{8\pi^2} +
\frac{5}{12\pi^2} \right).
\label{Ehfsrecground}
\end{equation}
Those two groups of contributions are computed, respectively, in
Sections \ref{sec:soft} and \ref{sec:hard}. Further, in Section
\ref{sec:nhfs}, we find a
generalization of this result for radially excited states (arbitrary
$n$):
\begin{equation}
\Delta_{\rm rec} E_{\rm hfs}(n) = \frac {m \alpha^6}{n^3} \left [
-\frac {1}{6} \left ( \ln \frac {\alpha}{n} +\Psi (n) + \gamma_E \right )
+ \frac {7}{12 n} - \frac {1}{2n^2 }
+ \frac{295}{432}
- \frac{ \ln 2 }{ 4 }
- \frac{17\zeta(3)}{8\pi^2} +
\frac{5}{12\pi^2}
\right ],
\label{Ehfsrecn}
\end{equation}
where $\Psi(n)$ is the logarithmic derivative of the
$\Gamma$-function and $\gamma_E\simeq 0.577216$ is the Euler constant.
The $n$ dependence of this result and its numerical
value at $n=1$ are in agreement with \cite{Ph}.
\subsection{Soft scale contributions}
\label{sec:soft}
We divide up the non-relativistic contributions to HFS into 6 parts:
tree level Coulomb and magnetic photon exchanges, retardation,
one-loop operators, and second iteration of Breit Hamiltonian which
includes intermediate S and D wave states:
\begin{equation}
\Delta_{\rm nonrel} E_{\rm hfs}
= \Delta_{\rm C} E_{\rm hfs} +\Delta_{\rm M} E_{\rm hfs}
+\Delta_{\rm ret} E_{\rm hfs}
+\Delta_{\rm 1-loop} E_{\rm hfs} +\Delta_{\rm S} E_{\rm hfs}
+\Delta_{\rm D} E_{\rm hfs}.
\end{equation}
These partial results, given in
Eqs.~(\ref{DeltaC}, \ref{DeltaM}, \ref{Deltaret},
\ref{Delta1}, \ref{Swavehfs}, \ref{DeltaD2}) add up to
\cite{Czarnecki:1998zv}
\begin{equation}
\Delta_{\rm nonrel} E_{\rm hfs} = \frac{ \pi\alpha^3 }{ 3 m^2 } \psi^2(0)
\left( \frac{ 1 }{\epsilon } - 4 \ln(m\alpha) + \frac{331}{18} \right).
\label{HFSnonrel}
\end{equation}
In the remainder of this Section we discuss in detail how these
contributions are calculated.
According to standard procedure \cite{GuptaQED} we identify the
on--shell scattering amplitude, taken with the minus sign, with the
matrix element of an interaction operator in the momentum
representation. The soft scale contributions are calculated using the
time-independent ``old-fashioned'' perturbation theory and the Coulomb
gauge. Since this technique is not very common, let us recall its
basic ingredients.
Exchange of a Coulomb or magnetic photon is described, respectively, by
$-4\pi\alpha/\mbox{$\vec{q}$}^2$ or $-4\pi\alpha\alpha_i
\otimes \alpha_j (\delta_{ij}- q_i q_j/\mbox{$\vec{q}$}^2)/2|\mbox{$\vec{q}$}|$. In the latter
case, the denominator $2|\mbox{$\vec{q}$}|$ arises from the magnetic photon's phase
space element.
An intermediate state
introduces the factor $(E-E_{\rm int}+i0)^{-1}$, where $E_{\rm int}$
is the energy of the intermediate state and $E$ is the total energy of
the process.
Dirac spinors are
\begin{equation}
\label{u}
u(\mbox{$\vec{p}$}) =
\sqrt{\frac{ 2\omega_{p} }{ \omega_{p}+m }} \Lambda_{+ }(\mbox{$\vec{p}$}) w,
\end{equation}
where $w$ denotes the four-spinor of a particle at rest;
projectors on the positive and negative electron energy states are
given by
$$
\Lambda_{\pm }(\mbox{$\vec{p}$})=\frac{1}{2} \left( 1 \pm
\frac{ \vec{\alpha}\mbox{$\vec{p}$}+\beta m }{ \omega_{p} } \right),
\qquad \omega_{p} = \sqrt{\mbox{$\vec{p}$}^2+m^2}.
$$
In an expression for the potential the projector
$\Lambda_{-}$ contributes an additional minus sign.
We begin with the contributions of the tree level effective operators,
describing an exchange of the Coulomb or magnetic quanta. The tree
level operators, relevant for the ${\cal O}(m\alpha^6)$ calculation of
the HFS, arise as ${\cal O}(v^2)$ corrections to the Breit potential.
\subsubsection{Tree-level Coulomb photon exchange}
For the HFS we need
the spin-dependent part of the ${\cal O}(v^4)$ correction to the
Coulomb exchange (see Eq.~(\ref{eqa:coul})):
\begin{equation}\label{VC}
V_C(\mbox{$\vec{p}$}',\mbox{$\vec{p}$}) = -\frac{\pi\alpha}{ 16 m^4}
\frac{ [\mbox{$\vec{\sigma}$}\mbox{$\vec{p}$},\mbox{$\vec{\sigma}$}\mbox{$\vec{p}$}'][\mbox{$\vec{\sigma}$}'\mbox{$\vec{p}$},\mbox{$\vec{\sigma}$}'\mbox{$\vec{p}$}'] }{ \mbox{$\vec{q}$}^2 }.
\end{equation}
To calculate the spin part of the matrix element, we take the
trace with $d$-dimensional sigma--matrices and find (the factor $1/d$
in Eq.~(\ref{14}) arises from the average over directions of the o-Ps
polarization vector)
\begin{eqnarray}
\frac{1}{ 2d } {\rm Tr} \left( \sigma_i [\mbox{$\vec{\sigma}$}\mbox{$\vec{p}$},\mbox{$\vec{\sigma}$}\mbox{$\vec{p}$}'] \sigma_i
[\mbox{$\vec{\sigma}$}'\mbox{$\vec{p}$}',\mbox{$\vec{\sigma}$}'\mbox{$\vec{p}$}] \right)
&=&4\frac{ d-4 }{ d }
\left[ \mbox{$\vec{p}$}'^2 \mbox{$\vec{p}$}^2 - (\mbox{$\vec{p}$}'\mbox{$\vec{p}$})^2 \right],
\label{14}
\\
\frac{1}{ 2 } {\rm Tr} \left( [\mbox{$\vec{\sigma}$}\mbox{$\vec{p}$},\mbox{$\vec{\sigma}$}\mbox{$\vec{p}$}'] [\mbox{$\vec{\sigma}$}'\mbox{$\vec{p}$}',\mbox{$\vec{\sigma}$}'\mbox{$\vec{p}$}] \right)
&=& 4 \left[ \mbox{$\vec{p}$}'^2 \mbox{$\vec{p}$}^2 - (\mbox{$\vec{p}$}'\mbox{$\vec{p}$})^2 \right],
\end{eqnarray}
respectively for ortho and parapositronium.
Using
\begin{equation}
\mbox{$\vec{p}$}'^2 \mbox{$\vec{p}$}^2 - (\mbox{$\vec{p}$}'\mbox{$\vec{p}$})^2 = (\mbox{$\vec{p}$}'\mbox{$\vec{p}$}) \; \mbox{$\vec{q}$}^2 - (\mbox{$\vec{p}$}'\mbox{$\vec{q}$})(\mbox{$\vec{q}$}\mbox{$\vec{p}$}),
\end{equation}
and noting that the average value of $\mbox{$\vec{p}$}'\mbox{$\vec{p}$}$ in an $S$--state
vanishes, we obtain the contribution of $V_C(\mbox{$\vec{p}$},\mbox{$\vec{p}$}')$ to the ground
state HFS:
\begin{equation}
\Delta_C E_{\rm hfs} = \left\langle V_C(\mbox{$\vec{p}$}',\mbox{$\vec{p}$}) \right\rangle \Big|_{S=0}^{S=1}
= - \frac{\pi\alpha}{dm^4}
\left\langle \frac{ (\mbox{$\vec{p}$}'\mbox{$\vec{q}$})(\mbox{$\vec{q}$}\mbox{$\vec{p}$}) }{ \mbox{$\vec{q}$}^2 } \right\rangle.
\label{int}
\end{equation}
In Eq.~(\ref {int}) the matrix element is to be calculated over the ground
state wave function in $d$ dimensions:
$$
\left\langle f(\mbox{$\vec{p}$},\mbox{$\vec{p}$}') \right\rangle \equiv
\int {{\rm d}^d p \over (2\pi)^d}{{\rm d}^d p' \over (2\pi)^d}
\phi(p)\phi(p') f(\mbox{$\vec{p}$},\mbox{$\vec{p}$}').
$$
Let us briefly explain how the integral in Eq.~(\ref {int}) is
calculated. Although the integrand does not look complicated, the
difficulty is that the exact form of the wave function $\psi(r)$ in
$d$ dimensions is not known. Fortunately, it turns out to be
unnecessary.
There are two alternative ways to calculate this integral. One is to
transform it to the coordinate space. A divergence arises at $r=0$
and in the final result is proportional to the $d$-dimensional
$\psi(0)$; the remaining, finite part can be easily calculated in
$d=3$.\footnote{In general, also the derivative of the wave function
at the origin, ${\rm d}\psi(r)/{\rm d} r$ at $r=0$, can appear in the
divergent part of the integral. However, the Schr\"odinger equation
relates it to $\psi(0)$.}
In the alternative approach we use the fact that the wave function
in Eq.~(\ref {int}) satisfies the $d$-dimensional Schr\"odinger
equation, which in the momentum space reads\footnote{This equation
corresponds to a summation of an infinite number of ladder diagrams in
the Coulomb gauge. For consistency it is essential to use here
dimensional regularization in the same way as in the
other loop integrations.}
\begin{equation}
\phi(p) = \frac {4\pi \alpha m}{\mbox{$\vec{p}$}^2-mE}
\int \frac {{\rm d}^dk}{(2\pi)^d}
\frac {\phi(k)}{(\mbox{$\vec{p}$}-\mbox{$\vec{k}$})^2}.
\end{equation}
Using this equation we rewrite the integral in Eq.~(\ref {int}) as
\begin{equation}
\left\langle \frac{ (\mbox{$\vec{p}$}'\mbox{$\vec{q}$})(\mbox{$\vec{q}$}\mbox{$\vec{p}$}) }{ \mbox{$\vec{q}$}^2 } \right\rangle _{\mbox{$\vec{p}$},\mbox{$\vec{p}$}'} =
\left\langle \frac{(4\pi\alpha m)^2(\mbox{$\vec{p}$}'\mbox{$\vec{q}$})(\mbox{$\vec{q}$}\mbox{$\vec{p}$}) }
{(\mbox{$\vec{p}$}^2-mE)(\mbox{$\vec{p}$}-\mbox{$\vec{k}$})^2 \mbox{$\vec{q}$}^2 (\mbox{$\vec{p}$}'^2-mE)(\mbox{$\vec{p}$}'-\mbox{$\vec{k}$}')^2}
\right\rangle _{\mbox{$\vec{k}$},\mbox{$\vec{k}$}'},
\label {int1}
\end{equation}
where the integration over $\mbox{$\vec{p}$}$, $\mbox{$\vec{p}$}'$, as well as $\mbox{$\vec{k}$}$, $\mbox{$\vec{k}$}'$, in the
last expression is
understood. The integral over $\mbox{$\vec{p}$}$ and $\mbox{$\vec{p}$}'$ receives a divergent
contribution only from the region where $\mbox{$\vec{p}$}$ and $\mbox{$\vec{p}$}'$ simultaneously
become infinite. Therefore, a single subtraction is sufficient to make
this integral finite. It is convenient to subtract from (\ref{int1})
the following expression:
\begin{equation}
\left\langle \frac{(4\pi \alpha m)^2 (\mbox{$\vec{p}$}'\mbox{$\vec{q}$})(\mbox{$\vec{q}$}\mbox{$\vec{p}$}) }{(\mbox{$\vec{p}$}^2-mE)^2 \mbox{$\vec{q}$}^2 (\mbox{$\vec{p}$}'^2-mE)^2}
\right\rangle _{\mbox{$\vec{k}$},\mbox{$\vec{k}$}'}.
\label {int2}
\end{equation}
After the subtraction is done, two nice features emerge. In Eq.~(\ref
{int2}) the integration over $\mbox{$\vec{k}$},\mbox{$\vec{k}$}'$ factorizes and leads to
$\psi^2(0)$ times a two-loop integral, which can be easily calculated
for arbitrary $d$. On the other hand, the difference between the last
integral in Eq.~(\ref {int1}) and the integral in Eq.~(\ref {int2}) is
finite and can be calculated for $d=3$ using the explicit form of the
wave function,
$$
\phi(p) = \sqrt{\pi\alpha m\over 2}{2m^2\alpha^2\over (\mbox{$\vec{p}$}^2-mE)^2},
\qquad
E=-{m\alpha^2\over 4}.
$$
We note that the counterterm (\ref {int2}) is
constructed in such a way that the above mentioned difference vanishes
for the ground state. This can be easily seen by integrating over
$\mbox{$\vec{k}$},\mbox{$\vec{k}$}'$ in Eq.~(\ref {int2}) and using the fact that the
$\mbox{$\vec{p}$},\mbox{$\vec{p}$}'$-dependent terms in the denominator of Eq.~(\ref {int2})
coincide (up to a normalization factor) with the three-dimensional
ground state wave functions in the momentum representation.
Both methods described above lead to the same result. For $d = 3 -
2\epsilon$ we obtain:\footnote{We neglect factors
$\Gamma^2(1+\epsilon)$ and $(4\pi\mu^2)^{2\epsilon}$ which do not
contribute to the final, finite result.}
\begin{equation}
\Delta_C E_{\rm hfs} = \frac{ \pi\alpha^3 }{ 24 m^2 } \psi^2(0)
\left( \frac{ 1 }{\epsilon } - 4 \ln(m\alpha) - \frac{1}{3} \right),
\label{DeltaC}
\end{equation}
where $\psi(0)$ is the value of the $d$-dimensional ground state wave
function at the origin.
\subsubsection{Tree-level exchange of a magnetic photon}
We now consider the correction caused by the tree level exchange of a
magnetic photon, Fig.~\ref{fig:soft}(b). We neglect the energy
dependence in the photon propagator; it will be restored in the
following Section, where we discuss retardation effects. The relevant
potential is obtained from Eq.~(\ref{eqa:magn}):
\begin{equation}
\label{VM}
V_M(\mbox{$\vec{p}$}',\mbox{$\vec{p}$})
= \frac{ \pi\alpha }{ 16m^4 }
\frac{[\mbox{$\vec{\sigma}$}'\mbox{$\vec{q}$},\sigma'^i]}{ \mbox{$\vec{q}$}^2 } \left\{
\left[ \mbox{$\vec{\sigma}$}\frac{ \mbox{$\vec{p}$}'+\mbox{$\vec{p}$} }{ 2 }, \sigma^i \right]
\left(\mbox{$\vec{p}$}'^2-\mbox{$\vec{p}$}^2\right)
+ [\mbox{$\vec{\sigma}$}\mbox{$\vec{q}$},\sigma^i]\left(\mbox{$\vec{p}$}^2+\mbox{$\vec{p}$}'^2\right) \right\}
+ (\mbox{$\vec{\sigma}$} \leftrightarrow \mbox{$\vec{\sigma}$}').
\end{equation}
Contribution of this interaction to the ground state HFS is
\begin{equation}\label{EM}
\Delta_M E_{\rm hfs} = \left\langle V_M(\mbox{$\vec{p}$}',\mbox{$\vec{p}$}) \right\rangle \Big|_{S=0}^{S=1}
= -2 \frac{d-1}{d} \frac{ \pi\alpha }{ m^4 }
\left\langle \mbox{$\vec{p}$}^2 + \mbox{$\vec{p}$}'^2 + \frac{ \left(\mbox{$\vec{p}$}^2-\mbox{$\vec{p}$}'^2\right)^2 }{ 2\mbox{$\vec{q}$}^2 } \right\rangle.
\end{equation}
In $d=3$ this matrix element is linearly divergent.
To demonstrate how we treat linear divergences let us consider the
$\mbox{$\vec{p}$}^2$ term on the RHS of the above equation:
\begin{equation}
\left\langle \mbox{$\vec{p}$}^2 \right\rangle = \psi(0)
\int \frac{{\rm d}^d p}{(2\pi)^d} \mbox{$\vec{p}$}^2 \phi(p)
= m \psi(0)
\int \frac{{\rm d}^d p}{(2\pi)^d} \left( E \phi(p)
+
\int \frac{{\rm d}^d k}{(2\pi)^d} \frac{ 4\pi\alpha }{ (\mbox{$\vec{p}$}-\mbox{$\vec{k}$})^2 }
\phi(k) \right).
\label {tadpole}
\end{equation}
Shifting the integration variable $\mbox{$\vec{p}$} \to \mbox{$\vec{p}$}+\mbox{$\vec{k}$}$ we find that the
$\mbox{$\vec{p}$}$-integral in the last term is scale-less. In dimensional
regularization such integrals vanish. The first term in Eq.~(\ref
{tadpole}) is finite in three dimensions. We obtain
\begin{equation}
\left\langle \mbox{$\vec{p}$}^2 \right\rangle = mE \psi^2(0).
\end{equation}
Applying a similar procedure to the last term in Eq.~(\ref{EM})
we find the contribution of $V_M(\mbox{$\vec{p}$}',\mbox{$\vec{p}$})$ to the ground state HFS:
\begin{equation}
\Delta_M E_{\rm hfs} = \frac{ \pi\alpha }{ m^4 }
\left[ m^2\alpha^2 \psi^2(0)
- 4\frac{d-1}{d} \left\langle \frac{ (\mbox{$\vec{p}$}'\mbox{$\vec{q}$})(\mbox{$\vec{q}$}\mbox{$\vec{p}$}) }{ \mbox{$\vec{q}$}^2 } \right\rangle
\right]
= \frac{ \pi\alpha^3 }{ 3 m^2 } \psi^2(0)
\left( \frac{ 1 }{\epsilon } - 4 \ln(m\alpha) + \frac{5}{3} \right).
\label{DeltaM}
\end{equation}
\subsubsection{Retardation effects}
Let us now consider the retardation effects, which mean that the
magnetic photon emitted by the electron propagates for a finite amount
of time before being absorbed by the positron. During this time, the
electron and positron can interact by several Coulomb exchanges
(Fig.~\ref{fig:soft}(c,d,e)). To calculate the influence of these
effects on the HFS, it is sufficient to take the spin-dependent parts
of the current $\vec{j}(\mbox{$\vec{p}'$}, \mbox{$\vec{p}$}) = u^+(\mbox{$\vec{p}'$})\vec{\alpha} u(\mbox{$\vec{p}$})$
in the leading nonrelativistic approximation:
\begin{equation}
\vec{j}(\mbox{$\vec{p}'$}, \mbox{$\vec{p}$}) \to \frac{ [\mbox{$\vec{\sigma}$}\mbox{$\vec{q}$},\mbox{$\vec{\sigma}$}] }{ 4m }.
\end{equation}
The scattering operator describing the retardation effects
is nonlocal both in space and time:
\begin{equation}\label{Aret}
- A_{\rm ret} = - \alpha
\int \frac{{\rm d}^d k}{(2\pi)^d}
\exp \left( -i\mbox{$\vec{k}$}\mbox{$\vec{r}$}_p \right) \frac{ [\mbox{$\vec{\sigma}$}'\mbox{$\vec{k}$},\sigma'_i] }{ 4m }
\frac{ 4\pi }{ 2k }\frac{ \delta_{ij} - \frac{k_i k_j}{\mbox{$\vec{k}$}^2} }{ k+H-E }
\frac{ [\mbox{$\vec{\sigma}$}\mbox{$\vec{k}$},\sigma_j] }{ 4m }
\exp \left( i\mbox{$\vec{k}$}\mbox{$\vec{r}$}_e \right) + {\rm H.c.}
\label{retar}
\end{equation}
Here we assume that the magnetic photon with the momentum $\mbox{$\vec{k}$}$ is emitted
by the electron at a point $\mbox{$\vec{r}$}_e$
and absorbed by the positron at a point $\mbox{$\vec{r}$}_p$.
Between those
moments, the evolution of the system ``positronium + photon'' is governed by
the propagator $4\pi(\delta_{ij}-k_i k_j/k^2)/(2k)(k+H-E)^{-1}$, $H$ being
the Hamiltonian of the nonrelativistic positronium slowly moving due to recoil.
In the region of interest $k \gg E$ and one can expand the amplitude
(\ref{Aret}) over the powers of $(H-E)/k\sim \alpha$. The zeroth term
of this expansion is the spin--dependent part of the Breit
potential,
\begin{equation}\label{A0ret}
- A_{\rm ret}^{(0)}(\mbox{$\vec{q}$}) = - \frac{ \pi\alpha }{ 4m^2 }
\frac{ [\mbox{$\vec{\sigma}$}'\mbox{$\vec{q}$},\sigma'^i][\mbox{$\vec{\sigma}$}\mbox{$\vec{q}$},\sigma^i] }{ \mbox{$\vec{q}$}^2 }.
\end{equation}
We need the second order term:
\begin{equation}\label{Vretpos}
V_{\rm ret} = \alpha
\int \frac{{\rm d}^d k}{(2\pi)^d} \frac{ 4\pi }{ 2\mbox{$\vec{k}$}^4 }
\frac{ [\mbox{$\vec{\sigma}$}'\mbox{$\vec{k}$},\sigma'^i] }{ 4m }[H,\exp \left( -i\mbox{$\vec{k}$}\mbox{$\vec{r}$}_p \right)]
[H,\exp \left( i\mbox{$\vec{k}$}\mbox{$\vec{r}$}_e \right)]\frac{ [\mbox{$\vec{\sigma}$}\mbox{$\vec{k}$},\sigma^i] }{ 4m }
+ {\rm H.c.}
\end{equation}
Only kinetic part of the Hamiltonian,
\begin{equation}
H_{\rm kin} = \frac{ \mbox{$\vec{p}$}_e^2 }{ 2m } + \frac{ \mbox{$\vec{p}$}_p^2 }{ 2m },
\end{equation}
has to be retained in the commutators.
We find
\begin{equation}
V_{\rm ret} = - \alpha
\int \frac{{\rm d}^d k}{(2\pi)^d} \frac{ 4\pi }{ 2\mbox{$\vec{k}$}^4 }
\frac{ [\mbox{$\vec{\sigma}$}'\mbox{$\vec{k}$},\sigma'^i] }{ 8m^2 }(\mbox{$\vec{k}$}^2+2\mbox{$\vec{k}$}\mbox{$\vec{p}$}_p)\exp \left
( i\mbox{$\vec{k}$}(\mbox{$\vec{r}$}_e-\mbox{$\vec{r}$}_p) \right)
(\mbox{$\vec{k}$}^2+2\mbox{$\vec{k}$}\mbox{$\vec{p}$}_e)\frac{ [\mbox{$\vec{\sigma}$}\mbox{$\vec{k}$},\sigma^i] }{ 8m^2 }
+ {\rm H.c.}
\end{equation}
Transforming back to the relative coordinate $\mbox{$\vec{r}$}=\mbox{$\vec{r}$}_e-\mbox{$\vec{r}$}_p$ and the relative
momentum $\mbox{$\vec{p}$}=\mbox{$\vec{p}$}_e=-\mbox{$\vec{p}$}_p$, we get for
the ground state HFS:
\begin{equation}
\Delta_{\rm ret} E_{\rm hfs} = \frac{ \pi\alpha }{ m^4 }
\left( \frac{m^2\alpha^2}{3} \psi^2(0)
- 4\frac{d-1}{d} \left\langle \frac{
(\mbox{$\vec{p}$}'\mbox{$\vec{q}$})(\mbox{$\vec{q}$}\mbox{$\vec{p}$}) }{ \mbox{$\vec{q}$}^2 } \right\rangle \right)
=
\frac{ \pi\alpha^3 }{ 3 m^2 } \psi^2(0)
\left( \frac{ 1 }{\epsilon } - 4 \ln(m\alpha) - \frac{1}{3} \right).
\label{Deltaret}
\end{equation}
\subsubsection{One-loop operators}
Now we turn to the operators generated by
one-loop diagrams. For the HFS the only contribution comes from the
graph in Fig.~\ref{fig:soft}(f), which describes the mixed
Coulomb-magnetic exchange with a transition of one of the particles
to a negative energy state. In other words, this corresponds to a
creation of an additional electron-positron pair by the electric or
magnetic field of the electron or positron.
Using Feynman rules for the time-independent perturbation
theory, given at the beginning of this Section, we derive
the corresponding potential:
\begin{equation}
V_{\rm 1-loop}(\mbox{$\vec{q}$}) = \frac{ 2\pi^2\alpha^2 }{ m^3 }
\int \frac{ d^d k }{ (2\pi)^d }
\frac{ [\mbox{$\vec{\sigma}$}( \mbox{$\vec{q}$}-\mbox{$\vec{k}$} ),\sigma^i] }{ (\mbox{$\vec{q}$}-\mbox{$\vec{k}$})^2 } \frac{ [\mbox{$\vec{\sigma}$}'\mbox{$\vec{k}$},\sigma'^i] }{\mbox{$\vec{k}$}^2 }.
\end{equation}
It induces the following correction to the ground state HFS
($d$-dimensional integration over $\mbox{$\vec{k}$}$ is implicitly assumed below)
\begin{eqnarray}\label{E1loop}
\Delta_{\rm 1-loop} E_{\rm hfs} &=& - 2\frac{ d-1 }{ d }
\frac{ \alpha^2 }{ m^3 }
\left\langle \frac{ 4\pi( \mbox{$\vec{p}$}'-\mbox{$\vec{k}$} ) }{ (\mbox{$\vec{p}$}'-\mbox{$\vec{k}$})^2 }
\frac{ 4\pi( \mbox{$\vec{k}$} -\mbox{$\vec{p}$}) }{ (\mbox{$\vec{k}$}-\mbox{$\vec{p}$})^2 } \right\rangle \nonumber \\
&=& - 2\frac{ d-1 }{ d }\frac{ \alpha^2 }{ m^3 }
\left\langle \frac{ \mbox{$\vec{p}$}'^2 + \mbox{$\vec{p}$}^2 }{2} \frac{ 4\pi }{ (\mbox{$\vec{p}$}'-\mbox{$\vec{k}$})^2 }
\frac{ 4\pi }{ (\mbox{$\vec{k}$}-\mbox{$\vec{p}$})^2 }
- \frac{ 4\pi p'^i }{ (\mbox{$\vec{p}$}'-\mbox{$\vec{k}$})^2 }
\frac{ 4\pi p^i }{ (\mbox{$\vec{k}$}-\mbox{$\vec{p}$})^2 } \right\rangle \nonumber \\
&=& \frac{ \pi\alpha }{ m^4 } \left\{
\frac{m^2\alpha^2}{3} \psi^2(0)
- 8\frac{ d-1 }{ d }\left\langle \frac{ \mbox{$\vec{p}$}'^2 \mbox{$\vec{p}$}^2 }{ \mbox{$\vec{q}$}^2 }
\right\rangle \right\}
\nonumber \\
&=&
-\frac{ 4\pi\alpha^3 }{ 3 m^2 } \psi^2(0)
\left( \frac{ 1 }{\epsilon } - 4 \ln(m\alpha) - \frac{1}{3} \right).
\label{Delta1}
\end{eqnarray}
\subsubsection{Breit Hamiltonian}
To complete the calculation of the soft scale
contributions to the HFS we have to consider
the second iteration of the Breit Hamiltonian.
It is obtained by including the effects of tree level Coulomb and
magnetic photon exchanges, as well as a correction to the kinetic
energy.
Using Eqs.~(\ref{eqa:coul}) and (\ref{eqa:magn}) we find
\begin{equation}\label{HBp}
U(\mbox{$\vec{p}$}',\mbox{$\vec{p}$}) = - \frac{ \mbox{$\vec{p}$}^4 }{ 4m^3 } (2\pi)^d\delta(\mbox{$\vec{p}$}'-\mbox{$\vec{p}$})
+ \frac{ \pi\alpha }{ m^2 }
+ \frac{ 4\pi\alpha }{ m^2 }\frac{(\mbox{$\vec{p}$}'\mbox{$\vec{q}$})(\mbox{$\vec{q}$}\mbox{$\vec{p}$})-(\mbox{$\vec{p}$}'\mbox{$\vec{p}$}) \mbox{$\vec{q}$}^2}{\mbox{$\vec{q}$}^4}
- \frac{ \pi\alpha }{ 4m^2 }
\frac{[\mbox{$\vec{\sigma}$}\mbox{$\vec{q}$},\sigma^i][\mbox{$\vec{\sigma}$}'\mbox{$\vec{q}$},\sigma'^i]}{\mbox{$\vec{q}$}^2}.
\end{equation}
In the position representation this Hamiltonian becomes
\begin{equation}\label{HBr}
U(\mbox{$\vec{r}$},\mbox{$\vec{p}$}) = - \frac{ \mbox{$\vec{p}$}^4 }{ 4m^3 }
+ \frac{ d-1 }{ 4m } \left\{ \frac{ \mbox{$\vec{p}$}^2 }{ m },C(r) \right\}
+ \frac{ d\pi\alpha }{ m^2 }\delta(\mbox{$\vec{r}$})
- \frac{ 1 }{ 16m^2 }
\left[ [\mbox{$\vec{\sigma}$}\nabla,\sigma^i][\mbox{$\vec{\sigma}$}'\nabla,\sigma'^i], C(r)\right],
\end{equation}
where
\begin{equation}
C(r) \equiv - \frac{ \alpha\Gamma(d/2-1) }{ \pi^{d/2-1} r^{d-2} }
\end{equation}
is the $d$--dimensional Coulomb potential.
\subsubsection{Second iteration of the Breit Hamiltonian: $S$-wave}
We consider first the contribution of the intermediate
$S$--states. The $S$--wave part of the Breit Hamiltonian
(\ref{HBr}) reads
\begin{equation}\label{BS}
U_S(\mbox{$\vec{r}$},\mbox{$\vec{p}$}) = - \frac{ \mbox{$\vec{p}$}^4 }{ 4m^3 }
+ \frac{ d-1 }{ 4m } \left\{ \frac{ \mbox{$\vec{p}$}^2 }{ m },C(r) \right\}
+ \frac{ d\pi\alpha }{ m^2 }\delta(\mbox{$\vec{r}$})
- \frac{ \pi\alpha }{ 4dm^2 }
[\sigma_i,\sigma_j][\sigma'_i,\sigma'_j] \delta(\mbox{$\vec{r}$}).
\end{equation}
It is convenient to divide up the calculation of the $U_S$
contribution to the HFS into two parts and consider the first and the
last two terms in Eq.~(\ref{BS}) separately. We begin with the
latter, which we denote by $\Delta_{S1} E_{\rm hfs}$:
\begin{eqnarray}
\label{DES1}
\Delta_{S1} E_{\rm hfs} &=& 8\frac{(d-1)(3d-2)}{d^2}
\left( \frac{ \pi\alpha }{ m^2 } \right)^2
\left\langle \delta(\mbox{$\vec{r}$}') \sum_m{\displaystyle'} \frac{
|m(\mbox{$\vec{r}$}') \rangle\langle m(\mbox{$\vec{r}$}) | }{ E-E_m } \delta(\mbox{$\vec{r}$}) \right\rangle \nonumber \\
&=& 8\frac{(d-1)(3d-2)}{d^2}
\left( \frac{ \pi\alpha }{ m^2 } \right)^2 \psi^2(0)
\sum_m{\displaystyle'} \frac{ |m(0)|^2 }{ E-E_m }.
\end{eqnarray}
In three dimensions the last sum is ill-defined due to ultraviolet
divergences in the zeroth and first terms of its expansion in
$\alpha$. We denote these singular terms by $G_0(0,0)$ and
$G_1(0,0)$, respectively, and obtain
\begin{equation}
\Delta_{S1} E_{\rm hfs} = 8\frac{(d-1)(3d-2)}{d^2}
\left( \frac{ \pi\alpha }{ m^2 } \right)^2
\psi^2(0)
\left[ - \frac{ 3m^2\alpha }{ 8\pi }
+ G_0(0,0) + G_1(0,0) \right].
\end{equation}
$G_0(0,0)$ and $G_1(0,0)$ are calculated in $d$ dimensions,
\begin{eqnarray}
G_0(0,0) &=& - \int \frac{ {\rm d}^d p }{ (2\pi)^d
} \frac{ m }{\mbox{$\vec{p}$}^2 -m E } =
\frac{m^2\alpha}{8\pi},
\nonumber \\
G_1(0,0) &=& - \int \frac{
{\rm d}^d p' }{ (2\pi)^d } \int \frac{ {\rm d}^d p }
{ (2\pi)^d } \frac{ m }{\mbox{$\vec{p}$}'^2 - mE }
\frac{ 4\pi\alpha }{ \mbox{$\vec{q}$}^2 } \frac{ m }{\mbox{$\vec{p}$}^2-m E }
=- \frac{m^2\alpha}{16\pi}
\left( \frac{ 1 }{\epsilon } - 4 \ln(m\alpha) + 2 \right),
\end{eqnarray}
and one finds
\begin{equation}
\Delta_{S1} E_{\rm hfs} = - \frac{ 7\pi\alpha^3 }{ 9 m^2 } \psi^2(0)
\left( \frac{ 1 }{\epsilon } - 4 \ln(m\alpha) + \frac{115}{21} \right).
\label{DeltaS1}
\end{equation}
The contribution of the first two terms in Eq.~(\ref{BS}) is
calculated in the following way. We first write them as
\begin{equation}\label{S2}
-\frac{ \mbox{$\vec{p}$}^4 }{ 4m^3 } + \frac{ d-1 }{ 4m } \left\{ \frac{ \mbox{$\vec{p}$}^2 }{ m },
C(r) \right\} = - \frac{1}{ 4m }
\left[ H^2 - d\left\{ H, C(r) \right\} + (2d-1) C^2(r) \right],
\end{equation}
where $H = \mbox{$\vec{p}$}^2/m + C(r)$ is the leading order Hamiltonian.
Correction to the HFS induced by Eq.~(\ref{S2}) reads
\begin{equation}\label{DES2}
\Delta_{S2} E_{\rm hfs} = \frac{d-1}{d} \frac{ \pi\alpha }{ m^3 }
\left\langle
\left[ d\left\{ H, C(r) \right\} -(2d-1) C^2(r) \right]
G(\mbox{$\vec{r}$},\mbox{$\vec{r}$}') \delta(\mbox{$\vec{r}$}') + {\rm H.c.}
\right\rangle.
\end{equation}
We introduced here the reduced Green function
\begin{equation}
G(\mbox{$\vec{r}$},\mbox{$\vec{r}$}') = \sum_m{\displaystyle'} \frac{ |m(\mbox{$\vec{r}$}) \rangle\langle m(\mbox{$\vec{r}$}') | }{
E-E_m },
\end{equation}
which satisfies the equation
$
(H - E)G(\mbox{$\vec{r}$},\mbox{$\vec{r}$}') = \psi(r)\psi(r') - \delta(\mbox{$\vec{r}$}-\mbox{$\vec{r}$}').
$
Using obvious short-hand notations one can rewrite Eq.~(\ref{DES2}) as
follows:
\begin{eqnarray}
\Delta_{S2} E_{\rm hfs} &=& - \frac{ 2\pi\alpha }{ 3m^3 }
\left\langle \left( 6 \alpha E \frac{1}{r} G
+ 3 \frac{ \alpha }{r}\psi(r)\psi(r')
+ 5 \frac{ \alpha^2 }{ r^2 } ( G - G_0 )
\right.\right. \nonumber
\\
&& \qquad \qquad\qquad\qquad \left.\left.
+ \frac{3}{2} \frac{(d-1)(2d-1)}{d} C^2(r) G_0 \right) \delta(\mbox{$\vec{r}$}')
+ {\rm H.c.} \right\rangle,
\label{57}
\end{eqnarray}
We dropped massless tadpoles and separated the
contribution of $G_0$, which is the only one we have to calculate
keeping $d \neq 3$. We find
\begin{eqnarray}
\left\langle C^2 G_0 \delta(\mbox{$\vec{r}$}') + {\rm H.c.} \right\rangle &=&
\frac{ 8\pi\alpha }{ m } \psi^2(0) G_1(0,0),
\nonumber \\
\left\langle \frac{\alpha}{r} G \delta(\mbox{$\vec{r}$}') + {\rm H.c.} \right\rangle &=&
-\alpha \partial_{\alpha} \left\langle \delta(\mbox{$\vec{r}$}) \right\rangle =
- 3\psi^2(0),
\nonumber \\
\left\langle \frac{ \alpha }{ r } \right\rangle &=& -2E,
\nonumber \\
\left\langle \frac{ 1 }{ r^2 } ( G - G_0 ) \delta(\mbox{$\vec{r}$}') + {\rm H.c.} \right\rangle &=& -4m\psi^2(0).
\end{eqnarray}
To obtain the last line we used the following equation:
\begin{equation}
G(r,0) - G_0(r,0) = \frac{m^2\alpha}{4\pi} e^{-\gamma r}
\left( \ln (2\gamma r) + \gamma_E - \frac{5}{2} + \gamma r \right),
\end{equation}
where $\gamma=m\alpha/2$.
>From Eq.~(\ref{57}) we now find
\begin{equation}
\Delta_{S2} E_{\rm hfs} = \frac{ 5\pi\alpha^3 }{ 3 m^2 } \psi^2(0)
\left( \frac{ 1 }{\epsilon } - 4 \ln(m\alpha) + \frac{88}{15} \right).
\label{DeltaS2}
\end{equation}
The sum of $\Delta_{S1}E_{\rm hfs}$ and $\Delta_{S2}E_{\rm hfs}$ gives
the final result for the correction to the ground state HFS induced by
the second iteration of the $S$-wave Breit Hamiltonian:
\begin{equation}
\Delta_{S}E_{\rm hfs} = \frac{ 8\pi\alpha^3 }{ 9 m^2 } \psi^2(0)
\left( \frac{ 1 }{\epsilon } - 4 \ln(m\alpha) + \frac{149}{24} \right).
\label{Swavehfs}
\end{equation}
\subsubsection{Second iteration of the Breit Hamiltonian: $D$-wave}
Because of the last term in Eq.~(\ref{HBp}) Breit Hamiltonian has
non-vanishing matrix elements with $|\Delta L| = 2$. In our case this
causes virtual transitions from the triplet $S$--state into
$D$--states (transitions from the singlet state are forbidden by the
total angular momentum conservation). Again, power counting shows that
only the zeroth and the first order terms in the Green function
expansion in $\alpha$ diverge in three dimensions. We first compute
the remaining, higher order terms, which are finite for $d=3$.
The sum of those higher order terms can be written as
\begin{equation}\label{D}
\Delta_D^{\rm h-o} E_{\rm hfs} = \left\langle U_D G_0 C G C G_0 U_D
\right\rangle,
\end{equation}
where
\begin{equation}
U_D = \frac{\alpha}{4m^2}
\frac{ 3(\vec{\sigma}\vec{n})(\vec{\sigma}'\vec{n}) -
\mbox{$\vec{\sigma}\vec{\sigma}'$} }{ r^3 }
\end{equation}
is the $|\Delta L| = 2$ part of the Breit Hamiltonian in three
dimensions, $G$ and $G_0$ are defined in the previous Section,
and $C=-\alpha/r$ is the Coulomb potential.
The correction to the ground--state wave function,
\begin{equation}
\delta_0\psi(r) = G_0 U_D \psi(r),
\end{equation}
which appears in Eq.~(\ref{D}), satisfies an inhomogeneous
Schr\"odinger equation:
\begin{equation}
\left( E - \frac{\mbox{$\vec{p}$}^2}{m} \right) \delta_0\psi(r) = U_D \psi(r).
\end{equation}
Solving this equation for $\delta_0\psi(r)$ we obtain
\begin{equation}
\Delta_D^{\rm h-o} E_{\rm hfs} = 8 \delta_{S1} \left
( \frac{\alpha^2}{24} \right)^2
\left\langle \frac{1}{mr^2} G_D(r,r_1)
\frac{1}{mr_1^2} \right\rangle,
\label{72}
\end{equation}
where $G_D(r,r_1)$ is the $D$-wave part of the Green function $G$, and
the factor $8\delta_{S1}$ arises from
\begin{equation}
\left\langle
\left
[
\mbox{$\vec{\sigma}\vec{\sigma}'$}-3(\vec{\sigma}\vec{n})(\vec{\sigma}'\vec{n})
\right]^2
\right\rangle = \left\langle
3+4\mbox{$\vec{\sigma}\vec{\sigma}'$}+(\mbox{$\vec{\sigma}\vec{\sigma}'$})^2
\right\rangle = 8\delta_{S1}.
\end{equation}
To calculate the matrix element in Eq.~(\ref{72}) we note that
\begin{equation}
\frac{1}{mr^2} = \frac{1}{6} \left( H_D - H \right),
\end{equation}
where $H_D(H)$ is the radial Hamiltonian for $D(S)$-states.
Using equations of motion for both the Green function and
the wave function in Eq.~(\ref{72}) one finds
\begin{equation}\label{deltaD}
\Delta_D^{\rm h-o} E_{\rm hfs} = 8 \left( \frac{\alpha^2}{24} \right)^2
\left\langle - \frac{1}{6mr^2} \right\rangle
= - \frac{ \pi\alpha^3\psi^2(0) }{ 108m^2 },
\end{equation}
in agreement with \cite{Bodwin:1978ut}.
To complete the calculation of the $D$-wave contribution
we have to consider the
zeroth and first order terms in the $\alpha$
expansion of the Green function,
\begin{eqnarray}
\Delta_{D0} E_{\rm hfs} &=& \left\langle U_D G_0 U_D \right\rangle,
\label{D0r}
\\
\Delta_{D1} E_{\rm hfs} &=& \left\langle U_D G_1 U_D \right\rangle. \label{D1r}
\end{eqnarray}
The perturbation $U_D(\mbox{$\vec{p}$}',\mbox{$\vec{p}$})$ is extracted from Breit Hamiltonian,
Eq.~(\ref{HBp}), and reads
\begin{equation}
U_D(\mbox{$\vec{p}$}',\mbox{$\vec{p}$}) = \frac{ \pi\alpha }{ 4m^2 }
\left( \frac{ [\sigma_i,\sigma_j][\sigma'_i,\sigma'_j] }{ d }
- \frac{[\mbox{$\vec{\sigma}$}\mbox{$\vec{q}$},\sigma_i][\mbox{$\vec{\sigma}$}'\mbox{$\vec{q}$},\sigma'_i]}{\mbox{$\vec{q}$}^2} \right).
\end{equation}
The average is taken over the $d$--dimensional wave function.
Calculating the trace using the triplet wave function we obtain
\begin{eqnarray}
&&\left\langle \left( \frac{ [\sigma_i,\sigma_j][\sigma'_i,\sigma'_j] }{ d }
- \frac{[\mbox{$\vec{\sigma}$}\mbox{$\vec{q}$}',\sigma^i][\mbox{$\vec{\sigma}$}'\mbox{$\vec{q}$}',\sigma'^i]}{\mbox{$\vec{q}$}'^2} \right)
\left( \frac{ [\sigma_k,\sigma_l][\sigma'_k,\sigma'_l] }{ d }
- \frac{[\mbox{$\vec{\sigma}$}\mbox{$\vec{q}$},\sigma^k][\mbox{$\vec{\sigma}$}'\mbox{$\vec{q}$},\sigma'^k]}{\mbox{$\vec{q}$}^2} \right)
\right\rangle \nonumber \\
&&\qquad\qquad\qquad
= \frac{4(d-2)^2}{d}\left\langle B_{ij}(\mbox{$\vec{q}$}') B_{ij}(\mbox{$\vec{q}$}) \right\rangle, \qquad
B_{ij}(\mbox{$\vec{q}$}) \equiv 4\pi \left( \frac{ q_i q_j }{ \mbox{$\vec{q}$}^2 }
- \frac{\delta_{ij}}{ d } \right).
\end{eqnarray}
Therefore
\begin{eqnarray}
\Delta_{D0} E_{\rm hfs} &=& - \frac{ \alpha(d-2)^2 }{ 4m^4d }
\left\langle B_{ij}(\mbox{$\vec{p}$}'-\mbox{$\vec{k}$})g(k)B_{ij}(\mbox{$\vec{k}$}-\mbox{$\vec{p}$}) \right\rangle, \label{D02}
\\
\Delta_{D1} E_{\rm hfs} &=& - \frac{ \alpha(d-2)^2 }{ 4m^4d }
\left\langle B_{ij}(\mbox{$\vec{p}$}'-\mbox{$\vec{k}$}')g(k')
\frac{4\pi}{(\mbox{$\vec{k}$}'-\mbox{$\vec{k}$})^2}g(k)B_{ij}(\mbox{$\vec{k}$}-\mbox{$\vec{p}$}) \right\rangle, \label{D12}
\end{eqnarray}
where
\begin{equation}
g(k) = \frac{ 2\gamma }{ \mbox{$\vec{k}$}^2 + \gamma^2 },
\end{equation}
and $d$--dimensional integrations over $\mbox{$\vec{k}$}$ in (\ref{D02}) and over
$\mbox{$\vec{k}$},\mbox{$\vec{k}$}'$ in (\ref{D12}) are understood. Some details of the
integrations in Eqs.~(\ref{D02},\ref{D12}) are given
in Appendix \ref{app:int}.
Adding the higher-order
effects found in Eq.~(\ref{deltaD}) we obtain the complete
$D$-wave contributions to HFS
\begin{eqnarray}
\Delta_{D} E_{\rm hfs} &=&
\Delta_{D0} E_{\rm hfs} + \Delta_{D1} E_{\rm hfs}
+\Delta_D^{\rm h-o} E_{\rm hfs}
\nonumber \\
&=&
\frac{ 5\pi\alpha^3 }{ 72 m^2 } \psi^2(0)
\left( \frac{ 1 }{\epsilon } - 4 \ln(m\alpha) - \frac{19}{5} \right).
\label{DeltaD2}
\end{eqnarray}
\subsection{Hard scale contribution}
\label{sec:hard}
Another contribution to the HFS arises from virtual momenta scales of
the order of the electron mass. It can be calculated by considering
the on--shell $e^+e^-$ scattering amplitude with an exchange of three
photons in the $t$-channel (see Fig.~\ref{fig1}) exactly at the
threshold, i.e.~for zero relative velocity of the incoming electron
and positron, in dimensional regularization. The use of the
dimensional regularization brings in essential simplifications,
since almost any other
regularization would bring in power-like divergences and hence require
additional subtractions. This so-called hard scale contribution gives
rise to four-fermion operators in the low-scale Lagrangian or,
equivalently, to the $\delta(r)$ terms in the effective quantum
mechanical Hamiltonian.
Technically, this calculation is similar to the derivation of
the matching coefficient of the vector quark-antiquark current in QCD
and its NRQCD counterpart, described e.g. in \cite{threshold,BSS}.
Here we outline the main steps of this calculation.
An arbitrary Feynman integral which contributes to the hard scale part of
the calculation can be written as
\begin{equation}
I(a_1,...a_9) = \int \frac {{\rm d}^D k_1}{(2\pi)^D} \frac {{\rm d}^D k_2}{(2\pi)^D}
{1\over S_1^{a_1}S_2^{a_2}S_3^{a_3}S_4^{a_4}S_5^{a_5}S_6^{a_6}S_7^{a_7}
S_8^{a_8} S_9^{a_9}},
\end{equation}
where
\begin{eqnarray}
&&
S_1 = {k_1^2},\qquad
S_2={k_2^2},\qquad
S_3={(k_1-k_2)^2}, \qquad
S_4 = {k_1^2+2pk_1},
\nonumber \\
&&
S_5= {k_2^2+2pk_2},\qquad
S_6 = {k_1^2-2pk_1},\qquad
S_7 = {k_2^2-2pk_2},
\nonumber \\
&&
S_8 = {(k_1-k_2)^2+2p(k_1-k_2)},\qquad
S_9 = {(k_1-k_2)^2-2p(k_1-k_2)},
\end{eqnarray}
and $a_1,\ldots,a_9$ are integers. In practice we encounter diagrams
with
only at most 6 different propagators, so that at least 3 exponents
$a_i$ are zero. Applying the integration by parts technique
\cite{che81} to an integral $I(\{a_i\})$, one obtains a set of relations
among integrals with various values of indices $\{a_i\}$. Using these
relations one can express any $I(\{a_i\})$ in terms of a few master
integrals. This is most easily done using symbolic manipulation
programs.
The result for the hard scale recoil corrections (Fig.~\ref{fig1})
to the HFS reads \cite{Czarnecki:1998zv}
\begin{equation}
\Delta_{\rm hard} E_{\rm hfs} = \frac{ \pi\alpha^3 }{ 3 m^2 } \psi^2(0)
\left( - \frac{ 1 }{\epsilon } + 4 \ln m
- \frac{51\zeta(3)}{\pi^2} +
\frac{10}{\pi^2} - 6\ln 2 \right).
\label{hard}
\end{equation}
\subsection{HFS for excited \boldmath $S$ states}
\label{sec:nhfs}
The result for the HFS of the ground state can be used to obtain the
HFS for an arbitrary excited state. The non--trivial dependence on
the principal quantum number
$n$ arises only from the soft scale contributions. Therefore, one has
to repeat the quantum mechanical calculation of the non-relativistic
part using any convenient regularization (we use a cut-off at $1/m \ll
r_0\ll
1/m\alpha$) and compare the result with
the known formula for $n=1$, Eq.~(\ref {Ehfsrecground}). One finds
\begin{equation}
\Delta_{\rm rec} E_{\rm hfs}(n) = \frac {m \alpha^6}{n^3} \left [ [div]
-\frac {1}{6} \left ( \ln \frac {\alpha}{n} +\Psi (n) + \gamma_E \right )
+ \frac {7}{12 n} - \frac {1}{2n^2}
\right ],
\end{equation}
The quantity $[div]$ in the above equation stands
for the unknown and $n$-independent constant,
easily
determined by requiring that for $n$=1 Eq.~(\ref {Ehfsrecground})
is reproduced. We then obtain the final result for the
recoil corrections to the HFS splitting
for an arbitrary $nS$ state, Eq.~(\ref{Ehfsrecn}).
\section{Spin-averaged energy levels}
\label{sec:lev}
To obtain ${\cal O}(m\alpha^6)$ corrections to the triplet and singlet
energy levels separately, we have to calculate $E_{\rm aver}(n)$
(cf. Eq.~(\ref{param})). An appropriate formula for this calculation
is
$$
E_{\rm aver} = \frac {3\, E_{\rm triplet} + E_{\rm
singlet}}{4} \to \frac {d\; E_{d\rm -plet} + E_{\rm singlet}}{d+1}.
$$
It is known \cite{Fell1,Fell2,KMYlog} that the recoil corrections
$E_{\rm aver}$ do not
contain $\ln(\alpha)$ at the order $m\alpha^6$. In dimensional
regularization this means that the hard-scale and soft-scale
contributions are separately finite.\footnote{Because of power-like
singularities, this is not necessarily the case in other
regularization schemes.}
Conceptually, determination of $E_{\rm aver}$ is very similar to the
calculation of the HFS discussed above in detail. The only difference
is that several new operators appear, which contribute to
$E_{\rm aver}$ but not to the HFS.
\subsection{The ground state average energy shift}
We begin with the correction to $E_{\rm aver}$ induced by the
relativistic corrections to the dispersion law, $\omega_{p} =
\sqrt{\mbox{$\vec{p}$}^2+m^2}$ (Fig.~\ref{fig:soft}(g)). Expanding $\omega_p$ in
$|\mbox{$\vec{p}$}|/m$, we obtain:
\begin{equation}
\omega_{p}=m+\frac {\mbox{$\vec{p}$}^2}{2m}-\frac {\mbox{$\vec{p}$}^4}{8m^3}+\frac {\mbox{$\vec{p}$}^6}{16m^5}+\ldots.
\end{equation}
The last term induces a correction of the appropriate order:
\begin{equation}
\Delta_{\rm disp} E_{\rm aver} =
\frac {1}{8m^5} \left\langle \mbox{$\vec{p}$}^6 (2\pi)^d \delta^{(d)} (\mbox{$\vec{p}$} - \mbox{$\vec{p}$} ') \right\rangle =
-\frac{3}{64}\frac{\pi\alpha^3}{m^2}\psi^2(0).
\label{averdips}
\end{equation}
The ${\cal O}(v^4)$ spin-independent part of the
tree level Coulomb exchange amplitude (cf.~Eq.~(\ref{AC}) and
Fig.~\ref{fig:soft}(a); we neglect terms odd in $\mbox{$\vec{p}$}$, whose average
vanishes in an $S$-state),
\begin{equation}
\overline V_C(\mbox{$\vec{p}$}',\mbox{$\vec{p}$}) = - \frac{\pi\alpha}{16m^4} \left(
7\left(\mbox{$\vec{p}$}^2 + \mbox{$\vec{p}$}'^2\right)
+ \frac{5(\mbox{$\vec{p}$}^2 - \mbox{$\vec{p}$}'^2)^2}{\mbox{$\vec{q}$}^2} \right),
\end{equation}
gives rise to the following correction:
\begin{equation}
\Delta_{\rm C} E_{\rm aver} = \frac {5\pi \alpha^3 }{32 m^2}
\psi^2(0)
\left( \frac {1}{\epsilon} - 4 \ln(m\alpha)+\frac {7}{5} \right).
\end{equation}
Virtual transitions to negative energy states induced by the Coulomb
exchanges, Fig.~\ref{fig:soft}(h), generate an effective
spin-independent operator
\begin{equation}
V_{C-}(\mbox{$\vec{r}$}) = - \frac{1}{4m^3} [\mbox{$\vec{p}$},C(r)]^2.
\end{equation}
This operator describes the energy shift due to a creation
of an additional $e^+ e^-$ pair by the Coulomb field of either electron
or positron. The resulting energy shift is
\begin{equation}
\Delta_{C-} E_{\rm aver} = - \frac {\pi \alpha^3 }{4m^2}
\psi^2(0)
\left (
\frac {1}{\epsilon} - 4 \ln(m\alpha) \right ).
\end{equation}
The spin-independent part of the tree level magnetic exchange,
Fig.~\ref{fig:soft}(b), induces the following shift in the energy
levels:
\begin{equation}
\Delta_{\rm M} E_{\rm aver} = \frac{2\pi\alpha}{m^4}
\left\langle (\mbox{$\vec{p}$}'^2+\mbox{$\vec{p}$}^2)
\left( {\mbox{$\vec{p}$}\p'\over \mbox{$\vec{q}$}^2}- {(\mbox{$\vec{p}$}\mbox{$\vec{q}$})(\mbox{$\vec{q}$}\mbox{$\vec{p}$}')\over \mbox{$\vec{q}$}^4}\right)
\right\rangle
= \frac {\pi \alpha^3}{2 m^2}\psi^2(0)
\left ( \frac {1}{\epsilon} - 4 \ln(m\alpha)+\frac {1}{2} \right ).
\end{equation}
To account for the retardation in the magnetic photon propagation,
Fig.~\ref{fig:soft}(c,d,e), we use the approach described in the HFS
case. Our starting point is similar to Eq.~(\ref{retar}), except that
now the full expression for the currents must be used, rather than
just their spin-dependent part. We obtain
\begin{equation}
\Delta_{\rm ret} E_{\rm aver} = \frac {\pi \alpha^3}{8m^2}
\psi^2(0)
\left (
\frac {1}{\epsilon} - 4 \ln(m\alpha) -8 \right ).
\end{equation}
The next contribution comes from the exchange of
two magnetic photons with creation of an additional
$e^+ e^-$ pair in the intermediate state, Fig.~\ref{fig:soft}(i).
We find
\begin{equation}
\Delta_{\rm MM-} E_{\rm aver} = -\frac {\pi \alpha^3}{2 m^2}
\psi^2(0)
\left (
\frac {1}{\epsilon} - 4 \ln(m\alpha) -3 \right ).
\end{equation}
We proceed further with the correction to $E_{\rm aver}$,
induced by the second iteration of the $S$-wave
Breit Hamiltonian. The calculation closely follows
the HFS case. We arrive at the following result:
\begin{equation}
\Delta_{\rm S}^{(2)} E_{\rm aver} = - \frac {\pi \alpha^3}{12 m^2}
|\psi(0)|^2
\left (
\frac {1}{\epsilon} - 4 \ln(m\alpha) + \frac {433}{24} \right ).
\end{equation}
The iteration of the $D$-wave part of the Breit Hamiltonian only
influences the energy levels of the triplet state because of the total
angular momentum ($\vec{L}+\vec{S}$) conservation. For this reason, to
obtain the required correction to $E_{\rm aver}$ it is sufficient to
multiply Eq.~(\ref{DeltaD2}) by the factor $d/(d+1)$. We find
\begin{equation}
\Delta_D^{(2)} E_{\rm aver} = \frac {5 \pi \alpha^3}{96 m^2}
\psi^2(0)
\left (
\frac {1}{\epsilon} - 4 \ln(m\alpha) - \frac {119}{30} \right ).
\end{equation}
It is easy to see that in 3 dimensions the spin-dependent operators do
not contribute to $E_{\rm aver}$. However, since we work with
divergent integrals and use dimensional regularization, this is no
longer valid for $d \ne 3$. In this case an ``anomalous'' situation
arises: spin-dependent operators provide contributions of the form
$(d-3)/\epsilon$ to $E_{\rm aver}$, which are finite as $\epsilon \to 0$. Part
of these contributions has already been accounted for in the corrections
induced by the Breit Hamiltonian. The remaining contributions give
\begin{equation}
\Delta_{\rm anom} E_{\rm aver} =
- \frac {15}{64} \frac {\pi \alpha^3 }{m^2}\psi^2(0).
\end{equation}
The hard scale contribution, Fig.~\ref{fig1}, is calculated in the
same way as for the HFS. One finds:
\begin{equation}
\Delta_{\rm hard} E_{\rm aver} = -\frac {\pi \alpha^3 }{3m^2}
\psi^2(0)
\left ( \frac {13}{8} + \frac {9 \zeta (3)}{\pi^2}
+\frac {33}{2\pi ^2} \right ).
\end{equation}
The sum of all contributions presented above provides the
${\cal O}(m\alpha^6)$ pure recoil correction to the ground state energy:
\begin{equation}\label{n=1shift}
\Delta_{\rm rec} E_{\rm aver} = -\frac {m\alpha^6}{8}
\left ( \frac {901}{576} +\frac {11}{2\pi^2}
+ \frac {3 \zeta (3)}{\pi ^2} \right ) = -\frac
{m\alpha^6}{8} (2.48688\ldots),
\end{equation}
in very good agreement with the numerical result of Eq.~(20) in
\cite{PhPRL}, $-\frac {m\alpha^6}{8} (2.484(5))$.
\subsection{Energy levels for arbitrary $n$}
To generalize the result Eq.~(\ref{n=1shift}) for arbitrary $n$, we
proceed according to the program outlined in Section \ref{sec:nhfs}. We
repeat the calculation of the soft-scale contributions to $E_{\rm
aver}$ for arbitrary $n$ using a different regularization scheme.
Namely, we set $d=3$ and cut off the divergent integrals over $r$ from
below at some $r_0 \ll 1/(m\alpha)$. The transition to three
dimensions simplifies the calculation. We find
\begin{equation}
\Delta_{\rm rec} E_{\rm aver} (n) =
-\frac {m \alpha^6}{8n^3} \left ( [div] +\frac {69}{64n^3}
-\frac {8}{3n^2}+\frac {2}{n} \right ).
\label{Eprel}
\end{equation}
The $n$-independent term in the above equation is regularization
dependent. It cannot be determined by considering the soft-scale
contributions alone. Nevertheless, by matching Eq.~(\ref{Eprel}) to
the shift for the ground state Eq.~(\ref{n=1shift}), the ``value'' of
the divergent constant $[div]$ is completely determined. We obtain
\begin{equation}
\Delta_{\rm rec} E_{\rm aver} (n) = -\frac {m \alpha^6}{8n^3} \left (
\frac {83}{72} + \frac {11}{2\pi^2} + \frac {3 \zeta (3)}{\pi ^2}
+\frac {69}{64n^3} -\frac {8}{3n^2}+\frac {2}{n}
\right ).
\label{Eaverfin}
\end{equation}
This is our main result for the recoil corrections to the energy
levels of positronium. It agrees with the partially numerical result
derived in \cite{PhPRL}.
\section{Radiative recoil corrections}
So far in this paper we have been considering pure recoil effects.
Another class of the ${\cal O}(m\alpha^6)$ corrections
to positronium energy levels and their HFS are the so-called
radiative recoil corrections, where one of the three exchanged
photons is created and absorbed by the same particle
(see Fig.~\ref{fig2}).
Our technique is very convenient for the calculation of these
corrections. The key point is that at ${\cal O}(m \alpha^6)$ the
radiative recoil corrections do not receive any contribution from
the non-relativistic scales. Thus it is sufficient to calculate the
diagrams shown in Fig.~\ref{fig2} (supplemented by the electric
charge, electron wave function and mass renormalization) exactly at
the threshold. For the same reason, the
$n$-dependence of the radiative recoil corrections comes only from the
$1/n^3$ behavior of the $nS$-wave function at the origin.
Some details of this calculation are described in Section
\ref{sec:hard} in the context of the HFS. We obtain
\begin{eqnarray}
\Delta_{\rm rad\; rec} E_{\rm hfs}
&=& \frac {m\alpha^6}{n^3} \left [ \frac {\zeta(3)}{2\pi^2}
-\frac{79}{48}+\frac{41}{36\pi^2} +\frac {4}{3}\ln 2 \right ],
\label{radrechfs}
\\
\Delta_{\rm rad\; rec} E_{\rm aver} &=& \frac {m\alpha^6}{n^3}
\left[
\frac {9\zeta (3)}{8\pi^2} + \frac {97}{144}-\frac {1025}{432\pi^2}
\right],
\label{radreclevels}
\end{eqnarray}
respectively for corrections to the HFS and to the average energy, in
full agreement with the analytic results of Ref.~\cite{PhK}.
For completeness, we give here separately the contributions of
electron vacuum polarization effects to radiative recoil corrections
\cite{Eides95,PhK,Sapirstein:1984xr}
(they are included in (\ref{radrechfs},\ref{radreclevels})):
\begin{eqnarray}
\Delta_{\rm rad\; rec}^{\rm vac\;pol} E_{\rm hfs}
=\frac {m\alpha^6}{n^3} {5\over 9\pi^2},
\qquad
\Delta_{\rm rad\; rec}^{\rm vac\;pol} E_{\rm aver} = \frac {m\alpha^6}{n^3}
\left( {1\over 36}-{5\over 27\pi^2}\right).
\end{eqnarray}
\section{Summary and Conclusions}
\label{sec:sum}
The main new results of the present paper are the analytic formulas
(\ref{Ehfsrecn}) and (\ref{Eaverfin}) for the pure recoil ${\cal
O}(m\alpha^6)$ corrections to the HFS and spin--averaged energy levels
of positronium $nS$ states. These recoil effects provide the
last pieces needed to present complete analytical formulas for the
total corrections to $E_{\rm aver}$ and $E_{\rm hfs}$.
We use the parameterization introduced in
Eq.~(\ref{param}),
\begin{equation}
E(J,n) = E_{\rm aver}(n) +\left( {1\over 4} - \delta_{J0} \right)
E_{\rm hfs}(n),
\end{equation}
and find
\begin{eqnarray}
E_{\rm aver}(n)&=& -\frac {m\alpha^2}{4n^2}
+\frac {m\alpha^4}{16n^3} \left (\frac {11}{4n} - 1 \right )
\nonumber \\
&&+\frac {m\alpha^5}{8\pi n^3}
\left[
-6\ln \alpha
-\frac {16}{3} \ln k_0(n,0)
+ \frac {14}{3} \left( \ln \frac {4}{n} + \Psi(n) + \gamma_E \right)
-\frac {37}{45} - 3 \ln 2 + \frac{7}{3n}
\right ]
\nonumber \\
&&+\frac{m\alpha^6}{32n^3} \left[
-\ln\frac{\alpha}{n} - \Psi(n) - \gamma_E
+ \frac{141}{4}\frac{\zeta(3)}{\pi^2}
+ \left( \frac{137}{6} - \frac{68}{\pi^2} \right) \ln 2
+ \frac{1421}{27\pi^2} \right. \nonumber \\
&&\qquad \qquad \left. - \frac{2435}{432} - \frac{7}{n}
+ \frac{17}{12n^2} - \frac{69}{16n^3} \right],
\label{aver:final}
\end{eqnarray}
and
\begin{eqnarray}
E_{\rm hfs}(n)&=&
{7\over 12}\frac {m\alpha^4}{n^3}
-\frac {m\alpha^5}{\pi n^3} \left( \frac {8}{9} + \frac {1}{2} \ln 2 \right)
+\frac {m\alpha^6}{n^3}
\left [
-\frac {5}{24} \left ( \ln \frac {\alpha}{n} + \Psi(n) + \gamma_E \right )
+\frac {1367}{648\pi^2}
\right.
\nonumber \\ &&
\left.
-\frac {4297}{3456} +
\left (\frac {221}{144} + \frac {1}{2\pi^2 } \right ) \ln 2
-\frac {53}{32\pi^2 } \zeta (3)
+ \frac {5}{8n}-\frac {85}{96n^2}
\right].
\label{hfs:final}
\end{eqnarray}
We have also recalculated the radiative recoil corrections,
Eqs.~(\ref{radrechfs},\ref{radreclevels}), confirming recent result of
Ref.~\cite{PhK}. Let us make a technical remark. In dimensional
regularization, used in this paper, the calculation of the radiative
recoil corrections is particularly simple. Since there are no
low-scale contributions to the radiative recoil corrections, it
suffices to calculate corresponding Feynman graphs exactly at the
threshold. No matching or subtractions are required.
Formulas (\ref{aver:final},\ref{hfs:final}), together with $P$ state
energy levels given in Appendix \ref{app:P}, can be used to compute
quantities which can be directly confronted with experimental data.
We use the following values for the Rydberg \cite{Hansch} and fine
structure \cite{Czarnecki:1998nd} constants:
\begin{equation}
R_\infty =\frac{m\alpha^2}{2}=3\,289\,841\,960.394(27)\;{\rm MHz},
\;\;\;\;\; \alpha=1/137.035\,999\,59(51).
\end{equation}
In addition to the full corrections ${\cal O}(m\alpha^6)$ we include
the leading logarithmic terms ${\cal O}(m\alpha^7\ln^2\alpha)$ found
in \cite{DL} for HFS, and in \cite{KirSash} for the spin-averaged energy
levels:
\begin{equation}
\Delta_{LL} E(J,n) = - \left( \frac{499}{15}+7\left(1-4\delta_{J0}\right) \right)
\frac{m\alpha^7 \ln^2 \alpha}{32\pi n^3} \delta_{l0}.
\label{eq:llogs}
\end{equation}
For the most precisely measured quantity, the Ps ground state HFS, we
find
\begin{equation}
\label{HFSth}
\Delta \nu = 203\,392.01(46) \; {\rm MHz}.
\end{equation}
The ${\cal O}(m\alpha^6)$ recoil corrections to this observable have
been subject of some debate. In the literature three different
results have been reported \cite{Caswell:1986ui,Ph,AS}.\footnote{After
our HFS calculation was completed, we were informed about an
independent numerical calculation of the recoil corrections \cite{AB}.
Although that study is still in progress, its preliminary results seem
to agree with Ref.~\cite{Ph} and the present paper.} Our result for
this correction, Eq.~(\ref{Ehfsrecground}), evaluates numerically to $
m\alpha^6 \left ( - \frac{ 1 }{ 6 } \ln \alpha + 0.37632 \right )$.
This is in excellent agreement with Ref.~\cite{Ph}, where for the
non-logarithmic part of the correction a number 0.3767(17) was
obtained. The framework of our calculation is similar to
Ref.~\cite{Ph}. However, in that study a different regularization
method was used. The agreement of the results gives us confidence in
their correctness.
Comparing Eq.~(\ref{HFSth}) with the experimental results,
Eqs.~(\ref{Mills},\ref{Hughes}), we observe a significant deviation of
the order of $3-4$ experimental errors. It is not very likely that
the uncalculated higher order effects alone can account for this
discrepancy. The size of the ${\cal O}(m\alpha^6)$ corrections gives
no indication of bad behavior of the perturbative expansion. On the
other hand, the leading logarithmic term ${\cal O}
(m\alpha^7\ln^2\alpha)$ is sizable. A calculation of the subleading
terms at this order remains an important theoretical challenge.
For another experimentally interesting quantity, the energy interval
of the $1S-2S$ transition, we get
\begin{equation}\label{Edifth}
E(2^3S_1)-E(1^3S_1) =1233\,607\,222.18(58) \; {\rm MHz}.
\end{equation}
in fair agreement with the experimental result, Eq.~(\ref{Fee}).
Other quantities, for which high precision measurements have been made
or are being planned, have recently been reviewed in \cite{PhK}. In
Table \ref{tab:one} we update the theory predictions for those
observables. Our predictions are in good agreement with \cite{PhK}.
We have been able to decrease the error bars by including the
analytical results (\ref{aver:final},\ref{hfs:final}) and the value of the
leading quadratic logarithms (\ref{eq:llogs}).
Finally we would like to comment on our error estimates. The errors
due to uncertainties in the fine structure constant and the electron
mass are well below $0.1$ MHz level. The dominant theoretical error
source is the uncalculated remainder of the perturbation expansion.
Although formally $m\alpha^7 \sim 0.1\; {\rm MHz}$, the leading ${\cal
O}(m\alpha^7 \ln^2 \alpha)$ terms contribute $-0.92\; {\rm MHz}$ to
the HFS \cite{DL}. It remains very important to calculate the
remaining, non-leading terms in ${\cal O} (m\alpha^7)$.
For the present analysis we assume that the leading logs ${\cal
O}(m\alpha^7 \ln^2 \alpha)$ dominate the higher order contributions
and take half their size as the theoretical error estimate.
The spectrum of the $nS$ and $nP$ positronium energy levels is now
known analytically, including effects ${\cal O}(m\alpha^6)$. Our
calculation for the $nS$ levels was made possible by new theoretical
tools which have their roots in the recent perturbative calculations
in high-energy physics. We hope that these methods will find further
applications.
The agreement between theoretical predictions and experimental results
in Ps spectroscopy is impressive with a few exceptions. One can only
hope to find something new and unexpected by trying to put these
exceptions in line with the overall picture. We look forward to
future improved measurements of positronium energy levels and their
confrontation with QED.
\subsection*{Acknowledgments}
We are grateful to A.~Burichenko for informing us about his results
prior to publication. We thank S. Karshenboim, K. Pachucki and
E. Remiddi for reading the manuscript and helpful comments.
K.M.~would like to thank the High Energy Theory Group at Brookhaven
National Laboratory for hospitality during the final stage of this
project. This research was supported in part by the U.S.~Department
of Energy under grant number DE-AC02-98CH10886, by BMBF under grant
number BMBF-057KA92P, by Gra\-duier\-ten\-kolleg
``Teil\-chen\-phy\-sik'' at the University of Karlsruhe and by the
Russian Foundation for Basic Research under grant number 99-02-17135.
|
1,108,101,563,272 | arxiv | \section{Introduction}
\bigskip
\bigskip
Let $\Omega \subset \mathbb{R}^{d}$, $d\geq 1$, be a bounded domain, with a
C^{2}$ boundary $\partial \Omega $. Let $V$ be a function in $L^{\infty
}\left( \Omega \right) $ with its norm $\left\Vert \cdot \right\Vert
_{\infty }$. Define
\begin{equation*}
A:=\Delta -V\text{, with }D\left( A\right) =H^{2}\left( \Omega \right) \cap
H_{0}^{1}\left( \Omega \right) \text{ .}
\end{equation*
Write $\left\{ e^{tA},t\geq 0\right\} $ for the semigroup generated by $A$
on $L^{2}\left( \Omega \right) $. It is well-known that when $V=0$, the
semigroup $\left\{ e^{tA},t\geq 0\right\} $ has the exponential decay with
the rate $\alpha _{1}$, which is the first eigenvalue of $-\Delta $ with the
homogeneous Dirichlet boundary condition. The aim of this study is to build
up, for each $\gamma >0$, an output feedback law $\mathcal{F}_{\gamma }$ so
that any solution to the closed-loop controlled heat equation, associated
with $A$ and $\mathcal{F}_{\gamma }$, has an exponential decay with the rate
$\gamma $. Further, two constraints are imposed: We only have accesses to
the system at time $\frac{1}{2}T+\overline{\mathbb{N}}T$ on an open
subdomain $\omega _{1}\subset \Omega $; We only can control at time $T
\overline{\mathbb{N}}T$ on another open subset $\omega _{2}\subset \Omega $.
Here, $T$ is an arbitrarily fixed positive number and $\overline{\mathbb{N}
:=\mathbb{N}\cup \left\{ 0\right\} $, with $\mathbb{N}$ the set of all
positive natural numbers. Therefore, the closed-loop controlled equation
under consideration reads
\begin{equation}
\left\{
\begin{array}{ll}
y^{\prime }\left( t\right) -Ay\left( t\right) =0\text{ ,} & t\in \left(
0,\infty \right) \backslash \mathbb{N}T\ \text{,} \\
y\left( 0\right) \in L^{2}\left( \Omega \right) \text{ ,} & \\
y\left( nT\right) =y\left( nT_{-}\right) +1_{\omega _{2}}\mathcal{F}_{\gamma
}\left( 1_{\omega _{1}}^{\ast }y\left( \left( n-\frac{1}{2}\right) T\right)
\right) \text{ ,} & \forall n\in \mathbb{N}\text{ .
\end{array
\right. \tag{1.1} \label{1.1}
\end{equation
Here, $y\left( nT_{-}\right) $ denotes the left limit of the function:
t\mapsto y\left( t\right) $ (from $\left[ 0,\infty \right) $ to $L^{2}\left(
\Omega \right) $) at time $nT$; $1_{\omega _{2}}$ denotes the zero-extension
operator from $L^{2}\left( \omega _{2}\right) $ to $L^{2}\left( \Omega
\right) $ (i.e., for each $f\in L^{2}\left( \omega _{2}\right) $, $1_{\omega
_{2}}\left( f\right) $ is defined to be the zero-extension of $f$ over
\Omega $); $1_{\omega _{1}}^{\ast }$ stands for the adjoint operator of
1_{\omega _{1}}$; $\mathcal{F}_{\gamma }$\ is a linear and bounded operator
from $L^{2}\left( \omega _{1}\right) $ to $L^{2}\left( \omega _{2}\right) $.
The operator $\mathcal{F}_{\gamma }$ is what we will build up. The evolution
distributed system (\ref{1.1}) is well-posed and can be understood as the
coupling of a sequence of heat equations:
\begin{equation*}
y\left( t\right) :=\left\vert
\begin{array}{ll}
y^{0}\left( t\right) \text{ ,} & \text{if }t\in \left[ 0,T\right) \\
y^{n}\left( t\right) \text{ ,} & \text{if }t\in \left[ nT,\left( n+1\right)
T\right
\end{array
\right.
\end{equation*
for any $n\in \mathbb{N}$, where
\begin{equation*}
\left\{
\begin{array}{ll}
\begin{array}{ll}
\partial _{t}y^{0}-\Delta y^{0}+Vy^{0}=0\text{ ,} & \text{\ in }\Omega
\times \left( 0,T\right) \ \text{,} \\
y^{0}=0\text{ ,} & \text{\ on }\partial \Omega \times \left( 0,T\right) \
\text{,
\end{array}
& \\
\text{~}y^{0}\left( 0\right) =y\left( 0\right) \in L^{2}\left( \Omega
\right) \text{ ,} &
\end{array
\right.
\end{equation*
an
\begin{equation*}
\left\{
\begin{array}{ll}
\begin{array}{ll}
\partial _{t}y^{n}-\Delta y^{n}+Vy^{n}=0\text{ ,} & \text{\ in }\Omega
\times \left( nT,\left( n+1\right) T\right) \ \text{,} \\
y^{n}=0\text{ ,} & \text{\ on }\partial \Omega \times \left( nT,\left(
n+1\right) T\right) \ \text{,
\end{array}
& \\
\text{~}y^{n}\left( nT\right) =y^{n-1}\left( nT\right) +1_{\omega _{2}
\mathcal{F}_{\gamma }\left( 1_{\omega _{1}}^{\ast }y^{n-1}\left( \left( n
\frac{1}{2}\right) T\right) \right) \in L^{2}\left( \Omega \right) \text{ .}
&
\end{array
\right.
\end{equation*}
\bigskip
Throughout this paper, we denote by $\left\Vert \cdot \right\Vert $ and
\left\langle \cdot ,\cdot \right\rangle $ the norm and the inner product of
L^{2}\left( \Omega \right) $ respectively. The notation $\left\Vert \cdot
\right\Vert _{\omega _{1}}$ and $\left\langle \cdot ,\cdot \right\rangle
_{\omega _{1}}$ will mean the norm and the inner product of $L^{2}\left(
\omega _{1}\right) $ respectively. We write $\mathcal{L}\left(
H_{1},H_{2}\right) $ for the space consisting of all bounded linear
operators from one Hilbert space $H_{1}$ to another Hilbert space $H_{2}$.
Lastly, we introduce the set $\left\{ \lambda _{j}\right\} _{j=1}^{\infty }$
for the family of all eigenvalues of $-A$ so tha
\begin{equation}
\lambda _{1}\leq \lambda _{2}\leq \cdot \cdot \leq \lambda _{m}\leq
0<\lambda _{m+1}\leq \cdot \cdot \cdot \text{ and }\underset{j\rightarrow
\infty }{\text{lim}}\lambda _{j}=\infty \text{ ,} \tag{1.2} \label{1.2}
\end{equation
and let $\left\{ \xi _{j}\right\} _{j=1}^{\infty }$ be the family of the
corresponding orthogonal normalized eigenfunctions.
\bigskip
The main theorem of this paper will be precisely presented in section 5. It
can be simply stated as follows: For each $\gamma >0$, there is $\mathcal{F
_{\gamma }\in \mathcal{L}\left( L^{2}\left( \omega _{1}\right) ,L^{2}\left(
\omega _{2}\right) \right) $ and a positive constant $C_{\gamma }$
(depending on $\gamma $ but independent of $t$) so that each solution $y$ to
the equation (\ref{1.1}) satisfies the inequality
\begin{equation}
\left\Vert y\left( t\right) \right\Vert \leq C_{\gamma }e^{-\gamma
t}\left\Vert y\left( 0\right) \right\Vert \quad \text{\ for any }t\geq
\text{ .} \tag{1.3} \label{1.3}
\end{equation
We now give two comments on this result. First, the aforementioned $\mathcal
F}_{\gamma }$ has the form
\begin{equation}
\mathcal{F}_{\gamma }\left( p\right) =-\sum_{j=1}^{K_{\gamma }}e^{\lambda
_{j}T/2}\langle g_{j},p\rangle _{\omega _{1}}f_{j}\left( x\right) \quad
\text{\ for any }p\in L^{2}\left( \omega _{1}\right) \text{ .} \tag{1.4}
\label{1.4}
\end{equation
Here, $K_{\gamma }\in \mathbb{N}$ is the number of all eigenvalues $\lambda
_{j}$ which are less that $\gamma +\frac{\text{ln}2}{T}$; $g_{j}$ and $f_{j}$
are vectors in $L^{2}\left( \omega _{1}\right) $ and $L^{2}\left( \omega
_{2}\right) $ respectively. These vectors are minimal norm controls for a
kind of minimal norm problems which can be given by constructive methods.
Second, the operator norm of $\mathcal{F}_{\gamma }$ is bounded by
C_{1}e^{C_{2}\gamma }$, with $C_{1}>0$ and $C_{2}>0$ independent of $\gamma
.
\bigskip
We next explain our strategy and key points to prove the above-mentioned
results. First, we realize that if a solution $y$ to the equation (\ref{1.1
) satisfies the inequality
\begin{equation*}
\left\Vert y\left( \left( \tfrac{5}{4}+n\right) T\right) \right\Vert \leq
e^{-\gamma T}\left\Vert y\left( \left( \tfrac{1}{4}+n\right) T\right)
\right\Vert \quad \text{\ for all }n\in \overline{\mathbb{N}}\text{ ,}
\end{equation*
then (\ref{1.3}) holds. Next, by the time translation invariance of the
equation, we can focus our study on the interval $\left[ \left( \frac{1}{4
+n\right) T,\left( \frac{5}{4}+n\right) T\right) $. And the problem of
stabilization is transferred into the following approximate controllability
problem (of the system (\ref{1.1}) over the above interval): Find a control
in a feedback form driving the system from each initial datum $y\left(
\left( \frac{1}{4}+n\right) T\right) $ to $y\left( \left( \frac{5}{4
+n\right) T\right) $ with the above estimate. When building up the feedback
law, we propose a new method to reconstruct approximatively the initial data
$y\left( \left( \frac{1}{4}+n\right) T\right) $ from the knowledge of
1_{\omega _{1}}^{\ast }y\left( \left( \frac{1}{2}+n\right) T\right) $. All
along such process, we need to take care of the cost of $\mathcal{F}_{\gamma
}$. Naturally, we may expect that smaller is $e^{-\gamma T}$, larger is the
cost. It is worth mentioning, that by projecting the initial data $y\left(
\left( \frac{1}{4}+n\right) T\right) $ into subspaces span$\left\{ \xi
_{1},\cdot \cdot \cdot ,\xi _{K_{\gamma }}\right\} $ and span$\left\{ \xi
_{K_{\gamma }+1},\xi _{K_{\gamma }+2},\cdot \cdot \cdot \right\} $
respectively, we have
\begin{equation*}
\left\Vert P_{K_{\gamma }}y\left( \left( n+1\right) T\right) \right\Vert
\leq e^{-\lambda _{K_{\gamma }+1}3T/4}\left\Vert y\left( \left( \tfrac{1}{4
+n\right) T\right) \right\Vert
\end{equation*
where $P_{K_{\gamma }}$ denotes the orthogonal projection of $L^{2}\left(
\Omega \right) $ onto the second subspace. Therefore, there is no interest
to control the initial data $P_{K_{\gamma }}y\left( \left( \frac{1}{4
+n\right) T\right) $ when $\lambda _{K_{\gamma }+1}\gg \gamma $. This
suggests the form of our feedback law.
In our analysis, a precise estimate is established and we will build the
output stabilization law via some minimal norm impulse control problems. It
requires to link the approximate impulse control and a quantitative unique
continuation estimate called observation at one time. Our program follows
the orientation described in \cite{Li} where stabilization, optimal control
and exact controllability for hyperbolic systems are closely linked. Here,
exact controllability for the wave equation is replaced by approximate
impulse control for the heat equations.
\bigskip
\bigskip
Several notes are given in order.
\begin{enumerate}
\item Impulse control belongs to a class of important control and has wide
applications. In many cases impulse control is an interesting alternative to
deal with systems that cannot be acted on by means of continuous control
inputs, for instance, relevant control for acting on a population of
bacteria should be impulsive, so that the density of the bactericide may
change instantaneously; indeed continuous control would be enhance drug
resistance of bacteria (see \cite{TWZ}). There are many studies on optimal
control and controllability for impulse controlled equations (see, for
instance, \cite{BL1}, \cite{R}, \cite{BL2}, \cite{LM}, \cite{Z}, \cite{LY},
\cite{BC}, \cite{DS}, \cite{MR}, \cite{OS}, \cite{Be} and references
therein). However, we have not found any published paper on stabilization
for impulse controlled equations. From this perspective, the problem studied
in the current paper is new.
In the system (\ref{1.1}), we do not need to follow the rule: Make
observation at each time and then add simultaneously control with a feedback
form. (When systems have continuous feedback control inputs, one has to
follow such a rule.) Instead of this, we only need to observe $1_{\omega
_{1}}^{\ast }y$ at time points $\left( n-\frac{1}{2}\right) T$ (for all
n\in \mathbb{N}$) and then add the controls at time points $nT$. In this
way, we can not only save observation and control time, but also allow the
control time $nT$ having a delay with respect to the observation time
\left( n-\frac{1}{2}\right) T$. Sometime, such time delay could be important
in practical application.
\item Stabilization is one of the most important subject in control theory.
In most studies of this subject, the aim of stabilization is to ask for a
feedback law so that the closed loop equation decays exponentially. The
current work aims to find, for each decay rate $\gamma $, a feedback law so
that the closed loop equation has an exponential decay with the rate $\gamma
$. Such kind of stabilization is called the rapid stabilization. About this
subject, we would like to mention the works \cite{K}, \cite{U}, \cite{CCr},
\cite{V}, \cite{CCo}, \cite{CL2}, and \cite{CL1}.
\item When observation region is not the whole $\Omega $, the corresponding
stabilization is a kind of output stabilization. Such stabilization is very
useful in applications. Unfortunately, there is no systematic study on this
subject, even for the simplest case when the controlled system is
time-invariant linear ODE (see \cite{Br}). Most of publications on this
subject focus on how to construct an output feedback law for some special
equations (see, for instance, \cite{I}, \cite{Cu}, \cite{YY}, \cite{Co},
\cite{NS} and references therein). Our study also only provides an output
feedback law for a special equation.
\item In this paper, we present a new way to build up the feedback law. In
particular, the structure of our feedback law is not based on LQ theory or
Lyapunov functions (see, for instance, \cite{Ba} and \cite{C}).
\item In the current study, one of the keys to build up our feedback law is
the use of the unique continuation estimate at one time, established in \cit
{PWZ} (see also \cite{PW} and \cite{PW1}). Some new observations are made on
it in this paper (see Theorem \ref{theorem2.1}, Remark \ref{remark2.2} and
Remark \ref{remark2.3}).
\item The following extensions of the current work should be interesting:
The first case is that $V$ depends on both $x$ and $t$ variables; The second
case is that the equation is a semi-linear heat equation; The third case is
that the equation is other types of PDEs.
\end{enumerate}
\bigskip
\bigskip
The rest of this paper is organized as follows. Section 2 provides several
inequalities which are equivalent to the unique continuation estimate at one
time. Such observation estimates are used in Section 3 in which we deal with
the impulse control problem. In Section 4, we link the impulse control
problem with an approximate inverse source problem. Finally, Section 5
presents the main result, as well as its proof.
\bigskip
\bigskip
\section{Observation at one time}
\bigskip
In this section, we present several equivalent inequalities. One of them is
the unique continuation estimate at one time built up in \cite{PWZ} (see
also \cite{PW} and \cite{PW1}).
\bigskip
\begin{thm}
\label{theorem2.1} Let $\omega $ be an open and nonempty subset of $\Omega
. Then the following propositions are equivalent and are true:
\begin{description}
\item[$\left( i\right) $] There are two constants $C_{1}>0$ and $\beta \in
\left( 0,1\right) $, which depend only on $\Omega $ and $\omega $, so that
for all $t>0$ and $\Phi \in L^{2}\left( \Omega \right) $
\begin{equation*}
\left\Vert e^{tA}\Phi \right\Vert \leq e^{C_{1}\left( 1+\frac{1}{t
+t\left\Vert V\right\Vert _{\infty }+\left\Vert V\right\Vert _{\infty
}^{2/3}\right) }\left\Vert \Phi \right\Vert ^{\beta }\left\Vert 1_{\omega
}^{\ast }e^{tA}\Phi \right\Vert _{\omega }^{1-\beta }\text{ .}
\end{equation*}
\item[$\left( ii\right) $] There is a positive constant $C_{2}$, depending
only on $\Omega $ and $\omega $, so that for each $\lambda \geq 0$ and each
sequence of real numbers $\left\{ a_{j}\right\} \subset \mathbb{R}$,
\begin{equation*}
\sum_{\lambda _{j}<\lambda }\left\vert a_{j}\right\vert ^{2}\leq
e^{C_{2}\left( 1+\left\Vert V\right\Vert _{\infty }^{2/3}+\sqrt{\lambda
\right) }\int_{\omega }\left\vert \sum_{\lambda _{j}<\lambda }a_{j}\xi
_{j}\right\vert ^{2}dx\text{ .}
\end{equation*}
\item[$\left( iii\right) $] There is a positive constant $C_{3}$, depending
only on $\Omega $ and $\omega $, so that for all $\theta \in \left(
0,1\right) $, $t>0$ and $\Phi \in L^{2}\left( \Omega \right) $
\begin{equation*}
\left\Vert e^{tA}\Phi \right\Vert \leq e^{C_{3}\left( 1+\frac{1}{\theta t
+t\left\Vert V\right\Vert _{\infty }+\left\Vert V\right\Vert _{\infty
}^{2/3}\right) }\left\Vert \Phi \right\Vert ^{\theta }\left\Vert 1_{\omega
}^{\ast }e^{tA}\Phi \right\Vert _{\omega }^{1-\theta }\text{ .}
\end{equation*}
\item[$\left( iv\right) $] There is a positive constant $C_{3}$, depending
only on $\Omega $ and $\omega $, so that for all $\varepsilon ,\beta >0$,
t>0$ and $\Phi \in L^{2}\left( \Omega \right) $
\begin{equation*}
\left\Vert e^{tA}\Phi \right\Vert \leq \frac{1}{\varepsilon ^{\beta }
e^{C_{3}\left( 1+\beta \right) \left( 1+\frac{1+\beta }{\beta t}+t\left\Vert
V\right\Vert _{\infty }+\left\Vert V\right\Vert _{\infty }^{2/3}\right)
}\left\Vert 1_{\omega }^{\ast }e^{tA}\Phi \right\Vert _{\omega }+\varepsilon
\left\Vert \Phi \right\Vert \text{ .}
\end{equation*}
\item[$\left( v\right) $] There is a positive constant $c$, depending only
on $\Omega $ and $\omega $, so that for all $\varepsilon >0$, $t>0$ and
\Phi \in L^{2}\left( \Omega \right) $
\begin{equation*}
\left\Vert e^{tA}\Phi \right\Vert \leq e^{c\left( 1+\frac{1}{t}+t\left\Vert
V\right\Vert _{\infty }+\left\Vert V\right\Vert _{\infty }^{2/3}\right)
\normalfont{\text{exp}}\left( \sqrt{\frac{c}{t}\normalfont{\text{ln}}^{+
\frac{1}{\varepsilon }}\right) \left\Vert 1_{\omega }^{\ast }e^{tA}\Phi
\right\Vert _{\omega }+\varepsilon \left\Vert \Phi \right\Vert \text{ .}
\end{equation*}
\end{description}
Here $\normalfont{\text{ln}}^{+}\frac{1}{\varepsilon }:=\normalfont
\text{max}}\{\normalfont{\text{ln}}\frac{1}{\varepsilon },0\}$. Moreover,
constants $C_{3}$ in $\left( iii\right) $ and $\left( iv\right) $ can be
chosen as the same number, and $c=4C_{3}$ in $\left( v\right) $.
\end{thm}
\bigskip
\begin{rem}
\label{remark2.2} We would like to give several notes on Theorem~\re
{theorem2.1}.
\begin{enumerate}
\item The inequality in $(i)$ of Theorem \ref{theorem2.1} implies a
quantitative version of the unique continuation property of heat equations
built up in \cite{L} (see also \cite{EFV}). Indeed, if $\left\Vert 1_{\omega
}^{\ast }e^{tA}\Phi \right\Vert _{\omega }=0$ for some $\Phi \in
L^{2}(\Omega )$, then $\left\Vert e^{tA}\Phi \right\Vert =0$, which,
together with the backward uniqueness for heat equations, implies that $\Phi
=0$. Moreover, such inequality is equivalent to a kind of impulse control,
with a bound on its cost, which will be explained in the next section.
Further, such controllability is the base for our study on the stabilization.
\item In the following studies of this paper, we will essentially use the
inequality in $(iii)$ of Theorem \ref{theorem2.1}. However, other
inequalities seem to be interesting independently. For instance, when $V=0$,
the inequality in $(ii)$ of Theorem \ref{theorem2.1} is exactly the
Lebeau-Robbiano spectral inequality (see \cite{LR}, \cite{JL} or \cite{LZ}).
Here, we get, in the case that $V\neq 0$, the same inequality, and find how
the constant (on the right hand side of the inequality) depends on
\left\Vert V\right\Vert _{\infty }$.
\item It deserves to mention that all constant terms on the right hand side
of inequalities in Theorem \ref{theorem2.1} have explicit expressions in
terms of the norm of $V$ and time, but not $\Omega $ and $\omega $.
\item It was realized that when $V=0$, the inequality in $(i)$ of Theorem
\ref{theorem2.1} can imply the inequality $(ii)$ of Theorem \ref{theorem2.1}
(see Remark 1 in \cite{AEWZ}).
\item The key ingredient why the inequality in $(i)$ and the one in $(ii)$
of Theorem \ref{theorem2.1} are equivalent is that the evolution $e^{tA}$ is
time-invariant. We would like to mention that the inequality in $(i)$ of
Theorem \ref{theorem2.1} was proved in \cite{PWZ} (see also \cite{PW} and
\cite{PW1}) for the case when $V=V(x,t)$ (i.e., for the evolutions with
time-varying coefficients). However, for the time-varying evolutions, we do
not know whether $(iii)$, $(iv)$ and $(v)$ in Theorem \ref{theorem2.1} are
still valid and what should be the right alternative of the spectral
inequality in $(ii)$ of Theorem \ref{theorem2.1}. These should be
interesting open problems.
\end{enumerate}
\end{rem}
\bigskip
\bigskip
\begin{rem}
\label{remark2.3} When $V=0$, the function $\normalfont{\text{exp}}\left(
\sqrt{\normalfont{\text{ln}}^{+}\frac{1}{\varepsilon }}\right) $ is optimal
in the inequality in $(v)$ of Theorem \ref{theorem2.1} in the following
sense: For any function $\varepsilon \mapsto f\left( \varepsilon \right) $,
\varepsilon >0$, with $\underset{\varepsilon \rightarrow 0^{+}}{\overline
\normalfont{\text{lim}}}}\frac{f\left( \varepsilon \right) }{\sqrt
\normalfont{\text{ln}}^{+}\frac{1}{\varepsilon }}}=0$, the following
inequality is not true
\begin{equation*}
\left\Vert e^{t\Delta }\Phi \right\Vert \leq f\left( \varepsilon \right)
e^{f\left( \varepsilon \right) }\left\Vert 1_{\omega }^{\ast }e^{t\Delta
}\Phi \right\Vert _{\omega }+\varepsilon \left\Vert \Phi \right\Vert \quad
\text{for all }\Phi \in L^{2}(\Omega )\text{ and }\varepsilon >0\text{ .}
\end{equation*}
\end{rem}
\bigskip
We now explain why the optimality in Remark~\ref{remark2.3} is true.
According to Proposition 5.5 in \cite{LL}, there exist $C_{0}^{\prime }>0$
and $n_{0}>0$ so that for each $n\geq n_{0}$, there is $\left\{
a_{n,j}\right\} _{j=1}^{\infty }\subset \ell ^{2}\left\backslash \left\{
0\right\} \right. $ satisfying tha
\begin{equation*}
\left\Vert \sum_{\lambda _{j}\leq n}a_{n,j}\xi _{j}\right\Vert \geq
C_{0}^{\prime }e^{C_{0}^{\prime }\sqrt{n}}\left\Vert 1_{\omega }^{\ast
}\sum_{\lambda _{j}\leq n}a_{n,j}\xi _{j}\right\Vert _{\omega }\text{ .}
\end{equation*
Define, for each $n\in \mathbb{N}$
\begin{equation*}
\Phi _{n}:=\sum_{\lambda _{j}\leq n}e^{\lambda _{j}t}a_{n,j}\xi _{j}\text{
and }\delta _{n}:=\frac{1}{2}e^{-nt}\text{ .}
\end{equation*
Therefore, we find that for each $n\geq n_{0}$
\begin{equation*}
\begin{array}{ll}
\left\Vert e^{t\Delta }\Phi _{n}\right\Vert & =\left\Vert \displaystyl
\sum_{\lambda _{j}\leq n}a_{n,j}\xi _{j}\right\Vert =\left( 1-\delta
_{n}e^{nt}\right) \left\Vert \displaystyle\sum_{\lambda _{j}\leq
n}a_{n,j}\xi _{j}\right\Vert +\delta _{n}e^{nt}\left\Vert \displaystyl
\sum_{\lambda _{j}\leq n}a_{n,j}\xi _{j}\right\Vert \\
& \geq \displaystyle\frac{1}{2}\left\Vert \displaystyle\sum_{\lambda
_{j}\leq n}a_{n,j}\xi _{j}\right\Vert +\delta _{n}\left\Vert \displaystyl
\sum_{\lambda _{j}\leq n}e^{\lambda _{j}t}a_{n,j}\xi _{j}\right\Vert \\
& \geq \displaystyle\frac{1}{2}C_{0}^{\prime }e^{C_{0}^{\prime }\sqrt{n
}\left\Vert 1_{\omega }^{\ast }\displaystyle\sum_{\lambda _{j}\leq
n}a_{n,j}\xi _{j}\right\Vert _{\omega }+\delta _{n}\left\Vert \Phi
_{n}\right\Vert \text{ .
\end{array
\end{equation*
Meanwhile, it follows from the definition of $\delta _{n}$ that for each
n\in \mathbb{N}$
\begin{equation*}
\sqrt{\text{ln}\frac{1}{\delta _{n}}}=\sqrt{\text{ln}2+nt}\leq 1+\sqrt{n
\sqrt{t}\text{ .}
\end{equation*
Gathering all the previous estimates, we see tha
\begin{equation*}
\left\Vert e^{t\Delta }\Phi _{n}\right\Vert \geq \frac{1}{2}C_{0}^{\prime
}e^{-\frac{C_{0}^{\prime }}{\sqrt{t}}}e^{\frac{C_{0}^{\prime }}{\sqrt{t}
\sqrt{\text{ln}\frac{1}{\delta _{n}}}}\left\Vert 1_{\omega }^{\ast
}\sum_{\lambda _{j}\leq n}a_{n,j}\xi _{j}\right\Vert _{\omega }+\delta
_{n}\left\Vert \Phi _{n}\right\Vert \text{ ,}
\end{equation*
which leads to the following property: There exists $C_{0}>0$, $\left\{ \Phi
_{n}\right\} \subset L^{2}\left( \Omega \right) \left\backslash \left\{
0\right\} \right. $ and $\left\{ \delta _{n}\right\} \subset \left(
0,1\right) $, with $\underset{n\rightarrow \infty }{\text{lim}}\delta _{n}=0
, so tha
\begin{equation*}
\left\Vert e^{t\Delta }\Phi _{n}\right\Vert \geq C_{0}e^{C_{0}\sqrt{\text{ln
\frac{1}{\delta _{n}}}}\left\Vert 1_{\omega }^{\ast }e^{t\Delta }\Phi
_{n}\right\Vert _{\omega }+\delta _{n}\left\Vert \Phi _{n}\right\Vert \quad
\text{for all }n\text{ .}
\end{equation*}
\bigskip
The rest of this section is devoted to the proof of Theorem \ref{theorem2.1}.
\bigskip
Proof. We organize the proof of Theorem \ref{theorem2.1} by several steps.
\noindent\textit{Step 1: On the proposition }$(i)$.
The conclusion $(i)$ has been proved in \cite{PWZ} (see also \cite{PW} and
\cite{PW1}).
\noindent\textit{Step 2: To prove that }$(i)\Rightarrow(ii)$.
Let $C_{1}>0$ and $\beta \in (0,1)$ be given by $(i)$. Arbitrarily fix
\lambda \geq 0$ and $\{a_{j}\}\subset \mathbb{R}$. By applying the
inequality in $(i)$, with $\Phi =\sum_{\lambda _{j}<\lambda }a_{j}e^{\lambda
_{j}t}\xi _{j}$, we get that
\begin{equation*}
\sum_{\lambda _{j}<\lambda }\left\vert a_{j}\right\vert ^{2}\leq
e^{2C_{1}\left( 1+\frac{1}{t}+t\left\Vert V\right\Vert _{\infty }+\left\Vert
V\right\Vert _{\infty }^{2/3}\right) }\left( \sum_{\lambda _{j}<\lambda
}\left\vert a_{j}e^{\lambda _{j}t}\right\vert ^{2}\right) ^{\beta }\left(
\int_{\omega }\left\vert \sum_{\lambda _{j}<\lambda }a_{j}\xi
_{j}\right\vert ^{2}dx\right) ^{1-\beta }\text{ ,}
\end{equation*
which implies that
\begin{equation}
\sum_{\lambda _{j}<\lambda }|a_{j}|^{2}\leq e^{\frac{2}{1-\beta }C_{1}\left(
1+\frac{1}{t}+t\Vert V\Vert _{\infty }+\Vert V\Vert _{\infty }^{2/3}\right)
}e^{\frac{2\beta }{1-\beta }\lambda t}\int_{\omega }\left\vert \sum_{\lambda
_{j}<\lambda }a_{j}\xi _{j}\right\vert ^{2}dx\quad \text{\ for each }t>
\text{ .} \tag{2.1} \label{2.1}
\end{equation
Meanwhile, since $\Vert V\Vert _{\infty }^{1/2}\leq 1+\Vert V\Vert _{\infty
}^{2/3}$ and $\beta \in (0,1)$, we see tha
\begin{equation*}
\begin{array}{ll}
& \quad \underset{t>0}{\text{inf}}\left[ \displaystyle C_{1}\left( 1+\frac{
}{t}+t\Vert V\Vert _{\infty }+\Vert V\Vert _{\infty }^{2/3}\right) +\beta
\lambda t\right] \\
& =\underset{t>0}{\text{inf}}\left[ \displaystyle C_{1}(1+\Vert V\Vert
_{\infty }^{2/3})+\frac{C_{1}}{t}+\left( C_{1}\Vert V\Vert _{\infty }+\beta
\lambda \right) t\right] \\
& =C_{1}\left( 1+\Vert V\Vert _{\infty }^{2/3}\right) +2\sqrt{C_{1}\left(
C_{1}\Vert V\Vert _{\infty }+\beta \lambda \right) } \\
& \leq C_{1}\left( 1+\Vert V\Vert _{\infty }^{2/3}+2\Vert V\Vert _{\infty
}^{1/2}\right) +2\sqrt{C_{1}}\sqrt{\beta \lambda }\leq \text{max}\left\{
3C_{1},2\sqrt{C_{1}}\right\} \left( 1+\Vert V\Vert _{\infty }^{2/3}+\sqrt
\lambda }\right) \text{ .
\end{array
\end{equation*
This, along with (\ref{2.1}), leads to the conclusion $(ii)$, with $C_{2}=
max$\left\{ \frac{6C_{1}}{1-\beta },\frac{4\sqrt{C_{1}}}{1-\beta }\right\} $.
\noindent\textit{Step 3: To show that }$(ii)\Rightarrow(iii)$.
Arbitrarily fix $\lambda \geq 0$, $t>0$ and $\Phi =\sum_{j\geq 1}a_{j}\xi
_{j}$ with $\{a_{j}\}\subset \ell ^{2}$. Write
\begin{equation*}
e^{tA}\Phi =\sum_{\lambda _{j}<\lambda }a_{j}e^{-\lambda _{j}t}\xi
_{j}+\sum_{\lambda _{j}\geq \lambda }a_{j}e^{-\lambda _{j}t}\xi _{j}\text{ .}
\end{equation*
Then by $(ii)$, we find tha
\begin{equation*}
\begin{array}{ll}
\left\Vert e^{tA}\Phi \right\Vert & \leq \left\Vert \displaystyl
\sum_{\lambda _{j}<\lambda }a_{j}e^{-\lambda _{j}t}\xi _{j}\right\Vert
+\left\Vert \displaystyle\sum_{\lambda _{j}\geq \lambda }a_{j}e^{-\lambda
_{j}t}\xi _{j}\right\Vert \\
& \leq \left( \displaystyle\sum_{\lambda _{j}<\lambda }\left\vert
a_{j}e^{-\lambda _{j}t}\right\vert ^{2}\right) ^{1/2}+e^{-\lambda
t}\left\Vert \Phi \right\Vert \\
& \leq \left( e^{C_{2}\left( 1+\Vert V\Vert _{\infty }^{2/3}+\sqrt{\lambda
\right) }\displaystyle\int_{\omega }\left\vert \displaystyle\sum_{\lambda
_{j}<\lambda }a_{j}e^{-\lambda _{j}t}\xi _{j}\right\vert ^{2}dx\right)
^{1/2}+e^{-\lambda t}\left\Vert \Phi \right\Vert \text{ .
\end{array
\end{equation*
This, along with the triangle inequality for the norm $\left\Vert \cdot
\right\Vert _{\omega }$, yields tha
\begin{equation*}
\begin{array}{ll}
\left\Vert e^{tA}\Phi \right\Vert & \leq \left( e^{C_{2}\left( 1+\Vert
V\Vert _{\infty }^{2/3}+\sqrt{\lambda }\right) }\displaystyle\int_{\omega
}\left\vert \displaystyle\sum_{j\geq 1}a_{j}e^{-\lambda _{j}t}\xi
_{j}\right\vert ^{2}dx\right) ^{1/2} \\
& \quad +\left( e^{C_{2}\left( 1+\Vert V\Vert _{\infty }^{2/3}+\sqrt{\lambda
}\right) }\displaystyle\int_{\omega }\left\vert \displaystyle\sum_{\lambda
_{j}\geq \lambda }a_{j}e^{-\lambda _{j}t}\xi _{j}\right\vert ^{2}dx\right)
^{1/2}+e^{-\lambda t}\left\Vert \Phi \right\Vert \text{ .
\end{array
\end{equation*
Hence, it follows tha
\begin{equation*}
\begin{array}{ll}
\left\Vert e^{tA}\Phi \right\Vert & \leq e^{\frac{C_{2}}{2}\left( 1+\Vert
V\Vert _{\infty }^{2/3}+\sqrt{\lambda }\right) }\left\Vert 1_{\omega }^{\ast
}e^{tA}\Phi \right\Vert _{\omega }+e^{\frac{C_{2}}{2}\left( 1+\Vert V\Vert
_{\infty }^{2/3}+\sqrt{\lambda }\right) }e^{-\lambda t}\left\Vert \Phi
\right\Vert +e^{-\lambda t}\left\Vert \Phi \right\Vert \\
& \leq 2e^{\frac{C_{2}}{2}\left( 1+\Vert V\Vert _{\infty }^{2/3}\right) }e^
\frac{C_{2}}{2}\sqrt{\lambda }}\left( \left\Vert 1_{\omega }^{\ast
}e^{tA}\Phi \right\Vert _{\omega }+e^{-\lambda t}\left\Vert \Phi \right\Vert
\right) \text{ .
\end{array
\end{equation*
Combining the above estimate with the following inequality:
\begin{equation*}
\frac{C_{2}}{2}\sqrt{\lambda }\leq \frac{\rho }{2}\lambda t+\frac{1}{2t\rho
\left( \frac{C_{2}}{2}\right) ^{2}\quad \text{for any }\rho >0\text{ ,}
\end{equation*
we have that for all $\rho \in (0,2)$ and $\lambda \geq 0$,
\begin{equation*}
\left\Vert e^{tA}\Phi \right\Vert \leq 2e^{\frac{C_{2}}{2}\left( 1+\Vert
V\Vert _{\infty }^{2/3}\right) }e^{\frac{1}{2t\rho }\left( \frac{C_{2}}{2
\right) ^{2}}\left( e^{\frac{\rho }{2}\lambda t}\left\Vert 1_{\omega }^{\ast
}e^{tA}\Phi \right\Vert _{\omega }+e^{-\frac{2-\rho }{2}\lambda t}\left\Vert
\Phi \right\Vert \right) \text{ .}
\end{equation*
Since $\lambda $ was arbitrarily taken from $[0,\infty )$, we choose
\begin{equation*}
\lambda =\frac{1}{t}\text{ln}\left( \frac{e^{t\Vert V\Vert _{\infty
}}\left\Vert \Phi \right\Vert }{\left\Vert 1_{\omega }^{\ast }e^{tA}\Phi
\right\Vert _{\omega }}\right)
\end{equation*
to get
\begin{equation*}
\left\Vert e^{tA}\Phi \right\Vert \leq 2e^{\frac{C_{2}}{2}\left( 1+\Vert
V\Vert _{\infty }^{2/3}\right) }e^{\frac{1}{2t\rho }\left( \frac{C_{2}}{2
\right) ^{2}}\left( 2e^{t\Vert V\Vert _{\infty }}\left\Vert 1_{\omega
}^{\ast }e^{tA}\Phi \right\Vert _{\omega }^{1-\frac{\rho }{2}}\left\Vert
\Phi \right\Vert ^{\frac{\rho }{2}}\right)
\end{equation*
which is the inequality in $(iii)$ with $\theta =\rho /2$.
\noindent \textit{Step 4: To show that }$(iii)\Rightarrow (iv)$.
We write the inequality in $(iii)$ in the following way:
\begin{equation*}
\left\Vert e^{tA}\Phi \right\Vert \leq \left\Vert \Phi \right\Vert ^{\theta
}\left( e^{\frac{C_{3}}{1-\theta }\left( 1+\frac{1}{\theta t}+t\left\Vert
V\right\Vert _{\infty }+\left\Vert V\right\Vert _{\infty }^{2/3}\right)
}\left\Vert 1_{\omega }^{\ast }e^{tA}\Phi \right\Vert _{\omega }\right)
^{1-\theta }\text{ .}
\end{equation*
Notice that for any real numbers $E,B,D>0$ and $\theta \in (0,1)$
\begin{equation*}
\begin{array}{ll}
& E\leq B^{\theta }D^{1-\theta }\text{ } \\
\Leftrightarrow & E\leq \varepsilon B+\left( 1-\theta \right) \theta ^{\frac
\theta }{1-\theta }}\displaystyle\frac{1}{\varepsilon ^{\frac{\theta }
1-\theta }}}D\quad \forall \varepsilon >0\text{ .
\end{array
\end{equation*
Then by taking $\beta =\frac{\theta }{1-\theta }$ in the last inequality, we
are led to the inequality in $(iv)$ with the same constant $C_{3}$ as that
in $(iii)$. (The above equivalence can be easily verified by using the Young
inequality and by choosing $\varepsilon =\theta \left( \frac{D}{B}\right)
^{1-\theta }$.)
\noindent \textit{Step 5: To show that }$(iv)\Rightarrow (v)$.
We write the inequality in $(iv)$ in the following way
\begin{equation*}
\left\Vert e^{tA}\Phi \right\Vert \leq \text{exp}\left( \Lambda +\Upsilon
+\beta \left( \text{ln}\frac{1}{\varepsilon }+\Lambda \right) +\frac{1}
\beta }\Upsilon \right) \left\Vert 1_{\omega }^{\ast }e^{tA}\Phi \right\Vert
_{\omega }+\varepsilon \left\Vert \Phi \right\Vert \text{ ,}
\end{equation*
with $\Lambda =C_{3}\left( 1+\frac{1}{t}+t\left\Vert V\right\Vert _{\infty
}+\left\Vert V\right\Vert _{\infty }^{2/3}\right) $ and $\Upsilon =\frac
C_{3}}{t}$. Next, we optimize the above inequality with respect to $\beta >0$
by choosing $\beta =\sqrt{\frac{\Upsilon }{\text{ln}^{+}\frac{1}{\varepsilon
}+\Lambda }}$ to ge
\begin{equation*}
\begin{array}{ll}
\left\Vert e^{tA}\Phi \right\Vert & \leq \text{exp}\left( \Lambda +\Upsilon
+2\sqrt{\Upsilon \left( \text{ln}^{+}\frac{1}{\varepsilon }+\Lambda \right)
\right) \left\Vert 1_{\omega }^{\ast }e^{tA}\Phi \right\Vert _{\omega
}+\varepsilon \left\Vert \Phi \right\Vert \\
& \leq \text{exp}\left( \Lambda +\Upsilon +2\sqrt{\Upsilon \Lambda }+2\sqrt
\Upsilon \text{ln}^{+}\frac{1}{\varepsilon }}\right) \left\Vert 1_{\omega
}^{\ast }e^{tA}\Phi \right\Vert _{\omega }+\varepsilon \left\Vert \Phi
\right\Vert \\
& \leq \text{exp}\left( 4\Lambda +2\sqrt{\Upsilon \text{ln}^{+}\frac{1}
\varepsilon }}\right) \left\Vert 1_{\omega }^{\ast }e^{tA}\Phi \right\Vert
_{\omega }+\varepsilon \left\Vert \Phi \right\Vert \text{ .
\end{array
\end{equation*
This implies the inequality in $(v)$, with $c=4C_{3}$.
\noindent \textit{Step 6: to show that }$(v)\Rightarrow (i)$.
Since
\begin{equation*}
\sqrt{\frac{c}{t}\text{ln}^{+}\frac{1}{\varepsilon }}\leq \frac{c}{\alpha t
+\alpha \text{ln}\left( e+\frac{1}{\varepsilon }\right) \quad \forall \alpha
>0\text{ ,}
\end{equation*
the inequality in $(v)$ become
\begin{equation*}
\left\Vert e^{tA}\Phi \right\Vert \leq e^{c\left( 1+\frac{1}{t}+t\left\Vert
V\right\Vert _{\infty }+\left\Vert V\right\Vert _{\infty }^{2/3}\right) }e^
\frac{c}{\alpha t}}\left( e+\frac{1}{\varepsilon }\right) ^{\alpha
}\left\Vert 1_{\omega }^{\ast }e^{tA}\Phi \right\Vert _{\omega }+\varepsilon
\left\Vert \Phi \right\Vert \text{ .}
\end{equation*
Next, we choose
\begin{equation*}
\varepsilon =\frac{1}{2}\frac{\left\Vert e^{tA}\Phi \right\Vert }{\left\Vert
\Phi \right\Vert }
\end{equation*
and we use the fact that $\left\Vert e^{tA}\Phi \right\Vert \leq e^{t\Vert
V\Vert _{\infty }}\left\Vert \Phi \right\Vert $ to deduce the inequality in
(i)$.
This ends the proof.
\bigskip
\bigskip
\section{Impulse control}
\bigskip
In this section, we first state our key result on impulse approximate
controllability.
\bigskip
\begin{thm}
\label{theorem3.1} Let $0\leq T_{1}<T_{2}<T_{3}$. Let $\varepsilon >0$ and
z\in L^{2}\left( \Omega \right) $. Then the following conclusions are true:
\begin{description}
\item[$\left( i\right) $] There exists $f\in L^{2}\left( \omega \right) $\
such that the unique solution $y$ to the equation:
\begin{equation*}
\left\{
\begin{array}{ll}
y^{\prime }\left( t\right) -Ay\left( t\right) =0\text{ ,} & t\in \left(
T_{1},T_{3}\right) \backslash \left\{ T_{2}\right\} \ \text{,} \\
y\left( T_{1}\right) =z\text{ ,} & \\
y\left( T_{2}\right) =y\left( T_{2-}\right) +1_{\omega }f\text{ ,} &
\end{array
\right.
\end{equation*
satisfies
\begin{equation*}
\left\Vert y\left( T_{3}\right) \right\Vert \leq \varepsilon \left\Vert
z\right\Vert \text{ .}
\end{equation*
Moreover, it holds that
\begin{equation*}
\left\Vert f\right\Vert _{\omega }\leq \mathcal{C}_{\varepsilon }\left(
T_{3}-T_{2},T_{2}-T_{1}\right) \left\Vert z\right\Vert
\end{equation*
where $\mathcal{C}_{\varepsilon }:\mathbb{R}^{+}\times \mathbb{R
^{+}\rightarrow \mathbb{R}^{+}$ is given by
\begin{equation*}
\mathcal{C}_{\varepsilon }\left( t,s\right) =e^{4s\left\Vert V\right\Vert
_{\infty }}e^{c\left( 1+\frac{1}{t}+t\left\Vert V\right\Vert _{\infty
}+\left\Vert V\right\Vert _{\infty }^{2/3}\right) }\normalfont{\text{exp}
\left( \sqrt{\frac{c}{t}\normalfont{\text{ln}}^{+}\frac{1}{\varepsilon }
\right) \text{ ,}
\end{equation*
and $c>0$ is the same constant than in $(v)$ of Theorem 2.1 .
\item[$\left( ii\right) $] There is a unique $f$ solving the following
problem:
\begin{equation*}
\normalfont{\text{inf}}\left\{ \left\Vert h\right\Vert _{\omega };h\in
L^{2}\left( \omega \right) \text{ such that }\left\Vert y\left( T_{3}\right)
\right\Vert \leq \varepsilon \left\Vert z\right\Vert \text{ with }f\text{
replaced by }h\text{ in }\left( i\right) \right\} \text{ .}
\end{equation*}
\end{description}
\end{thm}
\bigskip
\bigskip
The proof of Theorem \ref{theorem3.1} will be given in Subsections 3.1-3.2.
We now apply Theorem \ref{theorem3.1} to the eigenfunctions $\left\{ \xi
_{j}\right\} _{j=1}^{\infty }$. More precisely, by choosing $z=\xi _{j}$ in
Theorem \ref{theorem3.1}, we have the following corollary:
\begin{cor}
\label{corollary3.2} For any $0\leq T_{1}<T_{2}<T_{3}$ and any $\varepsilon
>0$, $j\in \mathbb{N}$, there is a pair $\left( y_{j},f_{j}\right) $ such
that
\begin{equation*}
\left\{
\begin{array}{ll}
y_{j}^{\prime }\left( t\right) -Ay_{j}\left( t\right) =0\text{ ,} & t\in
\left( T_{1},T_{3}\right) \backslash \left\{ T_{2}\right\} \ \text{,} \\
y_{j}\left( T_{1}\right) =\xi _{j}\text{ ,} & \\
y_{j}\left( T_{2}\right) =y_{j}\left( T_{2-}\right) +1_{\omega }f_{j}\text{
} &
\end{array
\right.
\end{equation*
and
\begin{equation*}
\left\{
\begin{array}{ll}
\left\Vert y_{j}\left( T_{3}\right) \right\Vert \leq \varepsilon \text{ ,} &
\\
\left\Vert f_{j}\right\Vert _{\omega }\leq \mathcal{C}_{\varepsilon }\left(
T_{3}-T_{2},T_{2}-T_{1}\right) \text{ ,} &
\end{array
\right.
\end{equation*
where $\mathcal{C}_{\varepsilon }$ is given in Theorem \ref{theorem3.1}.
Further, the control function $f_{j}$\ can be taken as the unique solution
of the problem
\begin{equation*}
\normalfont{\text{inf}}\left\{
\begin{array}{c}
\left\Vert h\right\Vert _{\omega };h\in L^{2}\left( \omega \right) \text{
and\text{ }the\text{ }property\text{ }\left( i\right) \text{ }of\text{
Theorem\text{ }\ref{theorem3.1} \\
holds\text{ }with\text{ }\left( z,f\right) \text{ }replaced\text{ }by\text{
\left( \xi _{j},h\right
\end{array
\right\} \text{ .}
\end{equation*}
\end{cor}
\bigskip
\bigskip
Further we will apply Theorem \ref{theorem3.1} to a finite combination of
eigenfunctions. More precisely, we have the following consequence:
\bigskip
\begin{thm}
\label{theorem3.3} Let $\omega _{2}$ be a non-empty open subset of $\Omega $
and $K\in \mathbb{N}$. Let $0\leq T_{1}<T_{2}<T_{3}$ and $\varepsilon >0$.
Then for any $b=\left( b_{j}\right) _{j=1,\cdot \cdot ,K}$, there is a pair
\left( \widetilde{y},\widetilde{f}\right) $ such that
\begin{equation*}
\left\{
\begin{array}{ll}
\widetilde{y}^{\prime }\left( t\right) -A\widetilde{y}\left( t\right) =
\text{ ,} & t\in \left( T_{1},T_{3}\right) \backslash \left\{ T_{2}\right\}
\ \text{,} \\
\widetilde{y}\left( T_{1}\right) =\displaystyle\sum_{j=1,\cdot \cdot
,K}b_{j}\xi _{j}\text{ ,} & \\
\widetilde{y}\left( T_{2}\right) =\widetilde{y}\left( T_{2-}\right)
+1_{\omega _{2}}\widetilde{f}\text{ ,} &
\end{array
\right.
\end{equation*
and
\begin{equation*}
\left\{
\begin{array}{ll}
\left\Vert \widetilde{y}\left( T_{3}\right) \right\Vert \leq \varepsilon
\sqrt{K}\left\Vert b\right\Vert _{\ell ^{2}}\text{ ,} & \\
\widetilde{f}=\displaystyle\sum_{j=1,\cdot \cdot ,K}b_{j}f_{j}\text{ ,} &
\end{array
\right.
\end{equation*
where $f_{j}$ is given by Corollary \ref{corollary3.2} with $\omega =\omega
_{2}$ and satisfie
\begin{equation*}
\left\Vert f_{j}\right\Vert _{\omega _{2}}\leq \mathcal{C}_{\varepsilon
}\left( T_{3}-T_{2},T_{2}-T_{1}\right) \text{ .}
\end{equation*}
\end{thm}
\bigskip
In the study of our stabilization, we will use Corollary \ref{corollary3.2}
and Theorem \ref{theorem3.3}. The rest of this section is devoted to the
proof of Theorem \ref{theorem3.1}, and the studies on some minimal norm
control problem.
\bigskip
\subsection{Existence of impulse control functions and its cost}
\bigskip
The aim of this subsection is to prove the conclusion $(i)$ of Theorem \re
{theorem3.1}. It deserves mentioning what follows: The existence of controls
with $(i)$ of Theorem \ref{theorem3.1} is indeed the existence of impulse
controls (with a cost) driving the solution of the equation in $\left(
i\right) $ of Theorem \ref{theorem3.1} from the initial state $z$ to the
closed ball in $L^{2}(\Omega )$, centered at the origin and of radius
\varepsilon \left\Vert z\right\Vert $, at the ending time $T_{3}$.
\bigskip
To prove the conclusion $(i)$ of Theorem \ref{theorem3.1}, we let
\varepsilon >0$ and $z\in L^{2}(\Omega )$. Denote $\hbar =\varepsilon ^{2}$
and $k=\left( \mathcal{C}_{\varepsilon }\left(
T_{3}-T_{2},T_{2}-T_{1}\right) \right) ^{2}$. Consider the strictly convex
C^{1}$ functional $\digamma $ defined on $L^{2}\left( \Omega \right) $ given
b
\begin{equation*}
\digamma \left( \Phi \right) :=\frac{k}{2}\left\Vert 1_{\omega }^{\ast
}e^{\left( T_{3}-T_{2}\right) A}\Phi \right\Vert _{\omega }^{2}+\frac{\hbar
}{2}\left\Vert \Phi \right\Vert ^{2}+\left\langle z,e^{\left(
T_{3}-T_{1}\right) A}\Phi \right\rangle \text{ .}
\end{equation*
Notice that $\digamma $ is coercive and therefore $\digamma $ has a unique
minimizer $w\in L^{2}\left( \Omega \right) $, i.e. $\digamma (w)=\underset
\Phi \in L^{2}\left( \Omega \right) }{\text{min}}\digamma (\Phi )$. Since
\digamma ^{\prime }(w)\Phi =0$ for any $\Phi \in L^{2}\left( \Omega \right)
, we hav
\begin{equation}
k\left\langle 1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}w,1_{\omega
}^{\ast }e^{\left( T_{3}-T_{2}\right) A}\Phi \right\rangle _{\omega }+\hbar
\left\langle w,\Phi \right\rangle -\left\langle z,e^{\left(
T_{3}-T_{1}\right) A}\Phi \right\rangle =0\quad \forall \Phi \in L^{2}\left(
\Omega \right) \text{ .} \tag{3.1.1} \label{3.1.1}
\end{equation
But by multiplying by $e^{\left( T_{3}-t\right) A}\Phi $ the system solved
by $y$, one get
\begin{equation*}
\left\langle y\left( T_{3}\right) ,\Phi \right\rangle =\left\langle
z,e^{\left( T_{3}-T_{1}\right) A}\Phi \right\rangle +\left\langle 1_{\omega
}f,e^{\left( T_{3}-T_{2}\right) A}\Phi \right\rangle \quad \forall \Phi \in
L^{2}\left( \Omega \right) \text{ .}
\end{equation*
By choosing $f=-k1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}w$, the
above two equalities yield that
\begin{equation*}
y\left( T_{3}\right) =\hbar w\text{ .}
\end{equation*
Further, one can deduce that
\begin{equation*}
\frac{1}{k}\left\Vert f\right\Vert _{\omega }^{2}+\frac{1}{\hbar }\left\Vert
y\left( T_{3}\right) \right\Vert ^{2}=k\left\Vert 1_{\omega }^{\ast
}e^{\left( T_{3}-T_{2}\right) A}w\right\Vert _{\omega }^{2}+\hbar \left\Vert
w\right\Vert ^{2}\text{ .}
\end{equation*
Next, notice that by taking $\Phi =w$ in (\ref{3.1.1}) and by Cauchy-Schwarz
inequality, we hav
\begin{equation*}
k\left\Vert 1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}w\right\Vert
_{\omega }^{2}+\hbar \left\Vert w\right\Vert ^{2}=\left\langle z,e^{\left(
T_{3}-T_{1}\right) A}w\right\rangle \leq \frac{1}{2}\left\Vert z\right\Vert
^{2}+\frac{1}{2}\left\Vert e^{\left( T_{3}-T_{1}\right) A}w\right\Vert ^{2
\text{ .}
\end{equation*
Now, we claim that
\begin{equation*}
\left\Vert e^{\left( T_{3}-T_{1}\right) A}w\right\Vert ^{2}\leq k\left\Vert
1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}w\right\Vert _{\omega
}^{2}+\hbar \left\Vert w\right\Vert ^{2}\text{ .}
\end{equation*
When it is proved, we can gather the previous three estimates to yiel
\begin{equation*}
\frac{1}{k}\left\Vert f\right\Vert _{\omega }^{2}+\frac{1}{\hbar }\left\Vert
y\left( T_{3}\right) \right\Vert ^{2}\leq \left\Vert z\right\Vert ^{2}\text{
.}
\end{equation*
As a consequence of our choice of $\left( \hbar ,k\right) $, we conclude
that
\begin{equation*}
\left\Vert y\left( T_{3}\right) \right\Vert \leq \sqrt{\hbar }\left\Vert
z\right\Vert =\varepsilon \left\Vert z\right\Vert
\end{equation*
and
\begin{equation*}
\left\Vert f\right\Vert _{\omega }\leq \mathcal{C}_{\varepsilon }\left(
T_{3}-T_{2},T_{2}-T_{1}\right) \left\Vert z\right\Vert \text{.}
\end{equation*
From these, we see that the above $f$ satisfies two properties in $(i)$ of
Theorem \ref{theorem3.1}.
It remains to prove the above claim. To this end, we write the inequality in
Theorem \ref{theorem2.1} $(iii)$ as follows.
\begin{equation*}
\left\Vert e^{tA}\Phi \right\Vert ^{2}\leq \left( \left\Vert \Phi
\right\Vert ^{2}\right) ^{\theta }\left( e^{\frac{2C_{3}}{1-\theta }\left( 1
\frac{1}{\theta t}+t\left\Vert V\right\Vert _{\infty }+\left\Vert
V\right\Vert _{\infty }^{2/3}\right) }\left\Vert 1_{\omega }^{\ast
}e^{tA}\Phi \right\Vert _{\omega }^{2}\right) ^{1-\theta }\text{ .}
\end{equation*
Since $\left\Vert e^{LA}\Phi \right\Vert \leq e^{\left( L-t\right)
\left\Vert V\right\Vert _{\infty }}\left\Vert e^{tA}\Phi \right\Vert $ for
L\geq t$, it holds that
\begin{equation*}
\left\Vert e^{LA}\Phi \right\Vert ^{2}\leq \left( \left\Vert \Phi
\right\Vert ^{2}\right) ^{\theta }\left( e^{\frac{2}{1-\theta }\left(
L-t\right) \left\Vert V\right\Vert _{\infty }}e^{\frac{2C_{3}}{1-\theta
\left( 1+\frac{1}{\theta t}+t\left\Vert V\right\Vert _{\infty }+\left\Vert
V\right\Vert _{\infty }^{2/3}\right) }\left\Vert 1_{\omega }^{\ast
}e^{tA}\Phi \right\Vert _{\omega }^{2}\right) ^{1-\theta }\text{ .}
\end{equation*
Following the same technique as that used in the proof of Theorem \re
{theorem2.1} (step 5), using the Young inequality, we have that for any
\varepsilon >0$ and any $\Phi \in L^{2}\left( \Omega \right) $,
\begin{equation*}
\left\Vert e^{LA}\Phi \right\Vert ^{2}\leq \text{exp}\left( 2\Lambda
_{L}+2\Upsilon +\beta \left( \text{ln}\frac{1}{\varepsilon }+2\Lambda
_{L}\right) +\frac{1}{\beta }2\Upsilon \right) \left\Vert 1_{\omega }^{\ast
}e^{tA}\Phi \right\Vert _{\omega }^{2}+\varepsilon \left\Vert \Phi
\right\Vert ^{2}\text{ ,}
\end{equation*
that is equivalent t
\begin{equation*}
\left\Vert e^{LA}\Phi \right\Vert ^{2}\leq \text{exp}\left( 2\left( \Lambda
_{L}+\Upsilon +\beta \left( \text{ln}\frac{1}{\varepsilon }+\Lambda
_{L}\right) +\frac{1}{\beta }\Upsilon \right) \right) \left\Vert 1_{\omega
}^{\ast }e^{tA}\Phi \right\Vert _{\omega }^{2}+\varepsilon ^{2}\left\Vert
\Phi \right\Vert ^{2}\text{ ,}
\end{equation*
with $\Lambda _{L}=\left( L-t\right) \left\Vert V\right\Vert _{\infty
}+C_{3}\left( 1+\frac{1}{t}+t\left\Vert V\right\Vert _{\infty }+\left\Vert
V\right\Vert _{\infty }^{2/3}\right) $ and $\Upsilon =\frac{C_{3}}{t}$.
Next, we choose $\beta =\sqrt{\frac{\Upsilon }{\text{ln}^{+}\frac{1}
\varepsilon }+\Lambda _{L}}}$ to get that
\begin{equation*}
\left\Vert e^{LA}\Phi \right\Vert ^{2}\leq \text{exp}\left( 8\Lambda _{L}+
\sqrt{\Upsilon \text{ln}^{+}\frac{1}{\varepsilon }}\right) \left\Vert
1_{\omega }^{\ast }e^{tA}\Phi \right\Vert _{\omega }^{2}+\varepsilon
^{2}\left\Vert \Phi \right\Vert ^{2}\text{ ,}
\end{equation*
for any $\varepsilon >0$, $L\geq t>0$ and $\Phi \in L^{2}\left( \Omega
\right) $. Setting $c=4C_{3}$ (here $c>0$ is the same constant than in $(v)$
of Theorem 2.1) and applying the above, with the choices $L=T_{3}-T_{1}$,
t=T_{3}-T_{2}$, and $\Phi =w$, give the desired claim.
This completes the proof of the conclusion $\left( i\right) $ in Theorem \re
{theorem3.1}.
\bigskip
\bigskip
\bigskip
\subsection{Uniqueness of minimal norm impulse control and its construction}
\bigskip
The aim of this subsection is to study a minimal norm problem, which is
indeed given in $\left( ii\right) $ of Theorem \ref{theorem3.1}. We will
present some properties on this problem (see Theorem~\ref{theorem3.4}), and
give the proof of $\left( ii\right) $ of Theorem \ref{theorem3.1} (see
\left( a\right) $ of Remark~\ref{remark3.5}).
Arbitrarily fix $z\in L^{2}\left( \Omega \right) \left\backslash
\{0\}\right. $ and $\varepsilon >0$. Recall the following impulse controlled
equation over $\left[ T_{1},T_{3}\right] $
\begin{equation}
\left\{
\begin{array}{ll}
y^{\prime }\left( t\right) -Ay\left( t\right) =0\text{ ,} & t\in \left(
T_{1},T_{3}\right) \backslash \left\{ T_{2}\right\} \ \text{,} \\
y\left( T_{1}\right) =z\text{ ,} & \\
y\left( T_{2}\right) =y\left( T_{2-}\right) +1_{\omega }f\text{ .} &
\end{array
\right. \tag{3.2.1} \label{3.2.1}
\end{equation
In this subsection, we discuss the following minimal norm impulse control
problem $(\mathcal{P})$
\begin{equation}
\mathcal{N}_{z}:=\text{inf}\left\{ \left\Vert f\right\Vert _{\omega };f\in
L^{2}\left( \omega \right) \text{ and }\left\Vert y\left( T_{3}\right)
\right\Vert \leq \varepsilon \left\Vert z\right\Vert \right\} \text{ .}
\tag{3.2.2} \label{3.2.2}
\end{equation
This problem is to ask for a control which has the minimal norm among all
controls (in $L^{2}\left( \omega \right) $) driving solutions of equation
\ref{3.2.1}) from the initial state $z$ to the closed ball in $L^{2}(\Omega
) $, centered at the origin and of radius $\varepsilon \left\Vert
z\right\Vert $, at the ending time $T_{3}$. In this problem, $\mathcal{N
_{z} $ is called the minimal norm, while $f^{\ast }\in L^{2}\left( \omega
\right) $ is called a minimal norm control, if the solution $y$ of (\re
{3.2.1}) with $f=f^{\ast }$ satisfies
\begin{equation*}
\left\Vert y\left( T_{3}\right) \right\Vert \leq \varepsilon \left\Vert
z\right\Vert \text{ and }\left\Vert f^{\ast }\right\Vert _{\omega }=\mathcal
N}_{z}\text{ .}
\end{equation*}
\bigskip
The main result of this subsection is as:
\bigskip
\begin{thm}
\label{theorem3.4} The following conclusions are true:
\begin{description}
\item[$\left( i\right) $] The problem $(\mathcal{P})$ has a unique minimal
norm control.
\item[$\left( ii\right) $] The minimal norm control $f^{\ast }$\ to $
\mathcal{P})$ satisfies that
\begin{equation*}
f^{\ast }=0\text{ if and only if the solution }y^{0}\text{ of (\ref{3.2.1})
with }f=0\text{ satisfies }\left\Vert y^{0}\left( T_{3}\right) \right\Vert
\leq \varepsilon \left\Vert z\right\Vert \text{ .}
\end{equation*}
\item[$\left( iii\right) $] The minimal norm control $f^{\ast }$\ to $
\mathcal{P})$ is given by
\begin{equation*}
f^{\ast }=1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) }w
\end{equation*
where $w$ is the unique minimizer to $J:L^{2}\left( \Omega \right)
\rightarrow \mathbb{R}$ defined by
\begin{equation*}
J\left( \Phi \right) :=\frac{1}{2}\left\Vert 1_{\omega }^{\ast }\,e^{\left(
T_{3}-T_{2}\right) A}\Phi \right\Vert _{\omega }^{2}+\left\langle
z,e^{\left( T_{3}-T_{1}\right) A}\Phi \right\rangle +\varepsilon \left\Vert
z\right\Vert \left\Vert \Phi \right\Vert \text{ .}
\end{equation*}
\end{description}
\end{thm}
\bigskip
\begin{rem}
\label{remark3.5} $\left( a\right) $ The conclusion $(i)$ of Theorem \re
{theorem3.4} clearly gives the uniqueness in $(ii)$ of Theorem \re
{theorem3.1}. This, along with $(i)$ of Theorem \ref{theorem3.1}, shows the
conclusion $(ii)$ in Theorem \ref{theorem3.1}. $\left( b\right) $ The
construction of the control in $(iii)$ of Theorem \ref{theorem3.4} is
inspired by a standard duality strategy used in \cite{FPZ} for the
distributed controlled heat equations with a control in $L^{2}\left( \omega
\times \left( T_{1},T_{3}\right) \right) $.
\end{rem}
\bigskip
Proof of Theorem \ref{theorem3.4} .-
\noindent \textit{Proof of }$(i)$\textit{: }Writ
\begin{equation*}
\mathcal{F}_{ad}:=\left\{ f\in L^{2}\left( \omega \right) ;\left\Vert
y\left( T_{3}\right) \right\Vert \leq \varepsilon \left\Vert z\right\Vert
\right\} \text{ .}
\end{equation*
By the previous subsection, we see that $\mathcal{F}_{ad}\neq \emptyset $.
Meanwhile, one can easily check that $\mathcal{F}_{ad}$ is weakly closed in
L^{2}\left( \omega \right) $. From these, it follows that $(\mathcal{P})$
has a minimal norm control.
Suppose that $f_{1}$ and $f_{2}$ are two minimal norm controls to $(\mathcal
P})$. Then we have that
\begin{equation*}
0\leq \Vert f_{1}\Vert _{\omega }=\Vert f_{2}\Vert _{\omega }=\mathcal{N
_{z}<\infty \text{ ,}
\end{equation*
where $\mathcal{N}_{z}$ is given by (\ref{3.2.2}). Meanwhile, one can easily
check that $(f_{1}+f_{2})/2$ is also a minimal norm control to $(\mathcal{P
) $. This, along with the Parallelogram Law, yields that
\begin{equation*}
\left( \mathcal{N}_{z}\right) ^{2}=\Vert (f_{1}+f_{2})/2\Vert _{\omega }^{2}
\frac{1}{2}\left( \Vert f_{1}\Vert _{\omega }^{2}+\Vert f_{2}\Vert _{\omega
}^{2}\right) -\Vert (f_{1}-f_{2})/2\Vert _{\omega }^{2}\text{ .}
\end{equation*
From the two above identities on $\mathcal{N}_{z}$, we find that
f_{1}=f_{2} $. Thus, the minimal norm control to $(\mathcal{P})$ is unique.
This ends the proof.
\noindent \textit{Proof of }$(ii)$\textit{: }The second conclusion in
Theorem \ref{theorem3.4} follows from the definition of Problem $(\mathcal{P
)$ (see (\ref{3.2.2})) at once.
\noindent \textit{Proof of }$(iii)$\textit{:} To prove the last conclusion
in Theorem \ref{theorem3.4}, we need the next Lemma \ref{lemma3.6} whose
proof will be given at the end of the proof of Theorem \ref{theorem3.4}.
\bigskip
\begin{lem}
\label{lemma3.6} The functional $J$ in Theorem \ref{theorem3.4} has the
following properties:
\begin{description}
\item[$\left( a\right) $] It satisfies that
\begin{equation}
\underset{q\rightarrow \infty }{\normalfont{\text{lim}}}\underset{\left\Vert
\Phi \right\Vert =q}{\normalfont{\text{inf}}}\frac{J\left( \Phi \right) }
\left\Vert \Phi \right\Vert }\geq \varepsilon \left\Vert z\right\Vert \text{
.} \tag{3.2.3} \label{3.2.3}
\end{equation}
\item[$\left( b\right) $] It has a unique minimizer over $L^{2}(\Omega )$ .
\item[$\left( c\right) $] Write $w$ for its minimizer. The
\begin{equation*}
w=0\text{ if and only if the solution }y^{0}\text{ of (\ref{3.2.1}) with }f=
\text{ satisfies }\left\Vert y^{0}\left( T_{3}\right) \right\Vert \leq
\varepsilon \left\Vert z\right\Vert \text{ .}
\end{equation*}
\end{description}
\end{lem}
\bigskip
We now show the third conclusion of Theorem \ref{theorem3.4}. Notice that
when $\left\Vert y^{0}\left( T_{3}\right) \right\Vert \leq \varepsilon
\left\Vert z\right\Vert $ where $y^{0}$ is the solution of (\ref{3.2.1})
with $f=0$, it follows respectively from the second conclusion of Theorem
\ref{theorem3.4} and the conclusion $(c)$ of Lemma \ref{lemma3.6} that
f^{\ast }=0$ and $w=0$. Hence, the third conclusion of Theorem 3.4 is true
in this particular case.
We now consider the case where
\begin{equation}
\left\Vert y^{0}\left( T_{3}\right) \right\Vert >\varepsilon \left\Vert
z\right\Vert \tag{3.2.4} \label{3.2.4}
\end{equation
where $y^{0}$ is the solution of (\ref{3.2.1}) with $f=0$. Let $w\in
L^{2}(\Omega )$ be the minimizer of the functional $J$ (see $(b)$ of Lemma
\ref{lemma3.6}). Write
\begin{equation}
\widehat{f}=1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}w\text{ .}
\tag{3.2.5} \label{3.2.5}
\end{equation
We first claim that
\begin{equation}
\widehat{f}\in \mathcal{F}_{ad}:=\left\{ f\in L^{2}\left( \omega \right)
;\left\Vert y\left( T_{3}\right) \right\Vert \leq \varepsilon \left\Vert
z\right\Vert \right\} \text{ .} \tag{3.2.6} \label{3.2.6}
\end{equation
In fact, by (\ref{3.2.4}) and $(c)$ of Lemma \ref{lemma3.6}, we find that
w\neq 0$. Then the Euler-Lagrange equation associated to $w$ reads
\begin{equation}
e^{\left( T_{3}-T_{2}\right) A}\chi _{\omega }e^{\left( T_{3}-T_{2}\right)
A}w+e^{\left( T_{3}-T_{1}\right) A}z+\varepsilon \left\Vert z\right\Vert
\frac{w}{\left\Vert w\right\Vert }=0\text{ .} \tag{3.2.7} \label{3.2.7}
\end{equation
Meanwhile, since $1_{\omega }1_{\omega }^{\ast }=\chi _{\omega }$, it
follows from (\ref{3.2.5}) that the solution $y$ of (\ref{3.2.1}) with $f
\widehat{f}$ satisfies
\begin{equation*}
y\left( T_{3}\right) =e^{\left( T_{3}-T_{1}\right) A}z+e^{\left(
T_{3}-T_{2}\right) A}1_{\omega }\widehat{f}=e^{\left( T_{3}-T_{2}\right)
A}\chi _{\omega }e^{\left( T_{3}-T_{2}\right) A}w+e^{\left(
T_{3}-T_{1}\right) A}z\text{ .}
\end{equation*
This, together with (\ref{3.2.7}), indicates that
\begin{equation*}
y\left( T_{3}\right) =-\varepsilon \left\Vert z\right\Vert \frac{w}
\left\Vert w\right\Vert }\text{ ,}
\end{equation*
from which, (\ref{3.2.6}) follows at once.
We next claim that
\begin{equation}
\left\Vert \widehat{f}\right\Vert _{\omega }\leq \left\Vert f\right\Vert
_{\omega }\quad \text{for all }f\in \mathcal{F}_{ad}\text{ .} \tag{3.2.8}
\label{3.2.8}
\end{equation
To this end, we arbitrarily fix an $f\in \mathcal{F}_{ad}$. Then we have
that the solution $y$ of (\ref{3.2.1}) satisfies
\begin{equation}
\left\Vert y\left( T_{3}\right) \right\Vert \leq \varepsilon \left\Vert
z\right\Vert \text{ .} \tag{3.2.9} \label{3.2.9}
\end{equation
Since $\widehat{f}:=1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}w$,
\begin{equation}
\begin{array}{ll}
\left\Vert \widehat{f}\right\Vert _{\omega }^{2} & =\left\Vert 1_{\omega
}^{\ast }e^{\left( T_{3}-T_{2}\right) A}w\right\Vert _{\omega }^{2} \\
& =\left\Vert 1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}w\right\Vert
_{\omega }^{2}+\varepsilon \left\Vert z\right\Vert \left\Vert w\right\Vert
-\varepsilon \left\Vert z\right\Vert \left\Vert w\right\Vert \\
& =-\varepsilon \left\Vert z\right\Vert \left\Vert w\right\Vert
+\left\langle e^{\left( T_{3}-T_{2}\right) A}\chi _{\omega }e^{\left(
T_{3}-T_{2}\right) A}w+\varepsilon \left\Vert z\right\Vert \frac{w}
\left\Vert w\right\Vert },w\right\rangle \\
& =-\varepsilon \left\Vert z\right\Vert \left\Vert w\right\Vert
-\left\langle e^{\left( T_{3}-T_{1}\right) A}z,w\right\rangle \\
& \leq \left\langle y\left( T_{3}\right) ,w\right\rangle -\left\langle
e^{\left( T_{3}-T_{1}\right) A}z,w\right\rangle =\left\langle e^{\left(
T_{3}-T_{2}\right) A}1_{\omega }f,w\right\rangle \\
& =\left\langle 1_{\omega }f,\chi _{\omega }e^{\left( T_{3}-T_{2}\right)
A}w\right\rangle =\left\langle 1_{\omega }f,\,1_{\omega }\widehat{f
\right\rangle \leq \frac{1}{2}\left\Vert f\right\Vert _{\omega }^{2}+\frac{
}{2}\left\Vert \widehat{f}\right\Vert _{\omega }^{2}\text{ .
\end{array}
\tag{3.2.10} \label{3.2.10}
\end{equation}
Notice that we used $1_{\omega }1_{\omega }^{\ast }=\chi _{\omega }$, $\chi
_{\omega }1_{\omega }=1_{\omega }$ and Cauchy-Schwarz inequality in the last
line in (\ref{3.2.10}); the equality (\ref{3.2.7}) is applied in the fourth
equality of (\ref{3.2.10}); the inequality (\ref{3.2.9}) and Cauchy-Schwarz,
as well as the formula $y\left( T_{3}\right) =e^{\left( T_{3}-T_{1}\right)
A}z+e^{\left( T_{3}-T_{2}\right) A}1_{\omega }f$, are used in the fifth line
of (\ref{3.2.10}). Now, (\ref{3.2.10}) clearly leads to (\ref{3.2.8}).
From (\ref{3.2.6}) and (\ref{3.2.8}), we find that $\widehat{f}$ is a
minimal norm control to $(\mathcal{P})$. Since the minimal norm control of $
\mathcal{P})$ is unique, we have that $\widehat{f}=f^{\ast }$. So the third
conclusion of Theorem 3.4 is true.
\bigskip
Finally, we are ready to prove Lemma \ref{lemma3.6}.
\bigskip
Proof of Lemma \ref{lemma3.6} .- We will prove conclusions $(a),\left(
b\right) ,(c)$ one by one.
\noindent \textit{Proof of }$(a)$\textit{:} By contradiction, suppose that
\ref{3.2.3}) was not true. Then there would be an $\sigma \in (0,\varepsilon
)$ and a sequence $\{\Phi _{n}\}_{n=1}^{\infty }$ in $L^{2}(\Omega )$ so
that
\begin{equation}
\underset{n\rightarrow \infty }{\text{lim}}\left\Vert \Phi _{n}\right\Vert
=\infty \text{ } \tag{3.2.11} \label{3.2.11}
\end{equation
an
\begin{equation}
\frac{J\left( \Phi _{n}\right) }{\left\Vert \Phi _{n}\right\Vert }\leq
\left( \varepsilon -\sigma \right) \left\Vert z\right\Vert \text{\quad for
all }n\in \mathbb{N}\text{ .} \tag{3.2.12} \label{3.2.12}
\end{equation
From (\ref{3.2.11}), we can assume, without loss of generality, that $\Phi
_{n}\neq 0$ for all $n$. Thus we can set
\begin{equation}
\varphi _{n}=\frac{\Phi _{n}}{\Vert \Phi _{n}\Vert }\quad \text{for all
n\in \mathbb{N}\text{ .} \tag{3.2.13} \label{3.2.13}
\end{equation
From (\ref{3.2.13}), we see that $\left\{ e^{\left( T_{3}-T_{1}\right)
A}\varphi _{n}\right\} _{n=1}^{\infty }$ is bounded in $L^{2}(\Omega )$.
Then, from the definition of $J$ in Theorem \ref{theorem3.4} , (\ref{3.2.13
), (\ref{3.2.11}) and (\ref{3.2.12}), we find that
\begin{equation}
\begin{array}{ll}
& \quad \underset{n\rightarrow \infty }{\overline{\text{lim}}}\displaystyl
\frac{1}{2}\left\Vert 1_{\omega }^{\ast }\,e^{\left( T_{3}-T_{2}\right)
A}\varphi _{n}\right\Vert _{\omega }^{2} \\
& =\underset{n\rightarrow \infty }{\overline{\text{lim}}}\displaystyle\frac{
}{\Vert \Phi _{n}\Vert }\left[ \displaystyle\frac{J\left( \Phi _{n}\right) }
\left\Vert \Phi _{n}\right\Vert }-\left\langle z,e^{\left(
T_{3}-T_{1}\right) A}\varphi _{n}\right\rangle -\varepsilon \left\Vert
z\right\Vert \right] \\
& \leq \underset{n\rightarrow \infty }{\overline{\text{lim}}}\displaystyl
\frac{-\sigma \left\Vert z\right\Vert }{\Vert \Phi _{n}\Vert }+\underset
n\rightarrow \infty }{\overline{\text{lim}}}\displaystyle\frac{-\left\langle
z,e^{\left( T_{3}-T_{1}\right) A}\varphi _{n}\right\rangle }{\Vert \Phi
_{n}\Vert }=0\text{ .
\end{array}
\tag{3.2.14} \label{3.2.14}
\end{equation
Meanwhile, by (\ref{3.2.13}), there is a subsequence of $\left\{ \varphi
_{n}\right\} $, denoted in the same manner, so that
\begin{equation*}
\varphi _{n}\rightarrow \varphi \quad \text{weakly in }L^{2}\left( \Omega
\right) \text{ ,}
\end{equation*
for some $\varphi \in L^{2}\left( \Omega \right) $. Since the semigroup
\left\{ e^{tA}\right\} _{t\geq 0}$ is compact, the above convergence leads t
\begin{equation}
e^{\left( T_{3}-T_{2}\right) A}\varphi _{n}\rightarrow e^{\left(
T_{3}-T_{2}\right) A}\varphi \quad \text{strongly in }L^{2}\left( \Omega
\right) \text{ } \tag{3.2.15} \label{3.2.15}
\end{equation
and
\begin{equation}
1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}\varphi _{n}\rightarrow
1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}\varphi \quad \text
strongly in }L^{2}\left( \omega \right) \text{ .} \tag{3.2.16}
\label{3.2.16}
\end{equation
From (\ref{3.2.14}) and the convergence in (\ref{3.2.16}), we find that
\begin{equation*}
1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}\varphi =0\text{ ,}
\end{equation*
which, along with the unique continuation property of heat equations (see
(1)$ of Remark \ref{remark2.2}) yields that $\varphi =0$. Then from the
definition of $J$ in Theorem \ref{theorem3.4} and the convergence in (\re
{3.2.15}), we see that
\begin{equation*}
\begin{array}{ll}
\underset{n\rightarrow \infty }{\underline{\text{lim}}}\displaystyle\frac
J\left( \Phi _{n}\right) }{\left\Vert \Phi _{n}\right\Vert } & \geq \underse
{n\rightarrow \infty }{\underline{\text{lim}}}\left[ \left\langle
z,e^{\left( T_{3}-T_{1}\right) A}\varphi _{n}\right\rangle +\varepsilon
\left\Vert z\right\Vert \right] \\
& =\left\langle z,e^{\left( T_{3}-T_{1}\right) A}\varphi \right\rangle
+\varepsilon \left\Vert z\right\Vert =\varepsilon \left\Vert z\right\Vert
\text{ .
\end{array
\end{equation*
This, along with (\ref{3.2.12}), leads to a contradiction. Therefore, (\re
{3.2.3}) is true.
\noindent \textit{Proof of }$(b)$\textit{:} From (\ref{3.2.3}), we see that
the functional $J$ is coercive on $L^{2}(\Omega )$. Further $J$ is
continuous and convex on $L^{2}(\Omega )$. Thus, it has a minimizer on
L^{2}(\Omega )$.
Next, we show the uniqueness of the minimizer. It suffices to prove that the
functional $J$ is strictly convex. For this purpose, we arbitrarily fix
\Phi _{1},\Phi _{2}\in L^{2}\left( \Omega \right) \left\backslash
\{0\}\right. $, with $\Phi _{1}\neq \Phi _{2}$. There are only three
possibilities: $(a)$ $\Phi _{1}\neq \mu \Phi _{2}$ for any $\mu \in \mathbb{
}$; $(b)$ $\Phi _{1}=-\mu _{0}\Phi _{2}$ for some $\mu _{0}>0$; $(c)$ $\Phi
_{1}=\mu _{0}\Phi _{2}$ for some $\mu _{0}>0$. In the cases $(a)$ and $(b)$,
one can easily check that
\begin{equation}
\Vert \lambda \Phi _{1}+(1-\lambda )\Phi _{2}\Vert <\lambda \Vert \Phi
_{1}\Vert +(1-\lambda )\Vert \Phi _{2}\Vert \quad \text{for all }\lambda \in
(0,1)\text{ .} \tag{3.2.17} \label{3.2.17}
\end{equation
In the case $(c)$, we let
\begin{equation*}
H(\lambda )=J\left( \lambda \Phi _{2}\right) \text{ ,}\quad \lambda >0\text{
.}
\end{equation*
Since $\Phi _{2}\neq 0$ in $L^{2}(\Omega )$, it follows by the unique
continuation property of heat equations (see $(1)$ of Remark \ref{remark2.2
) that $\left\Vert 1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}\Phi
_{2}\right\Vert _{\omega }\neq 0$. Thus, $H$ is a quadratic function with a
positive leading coefficient. Hence, $H$ is strictly convex. This, along
with (\ref{3.2.17}), yields the strict convexity of $J$.
\noindent \textit{Proof of }$(c)$\textit{:} Let $y^{0}$ be the solution of
\ref{3.2.1}) with $f=0$. We first show that
\begin{equation}
\left\Vert y^{0}\left( T_{3}\right) \right\Vert \leq \varepsilon \left\Vert
z\right\Vert \Rightarrow w=0\text{ .} \tag{3.2.18} \label{3.2.18}
\end{equation
In fact, by multiplying by $e^{\left( T_{3}-t\right) A}\Phi $ the system
solved by $y^{0}$, we see that
\begin{equation*}
\left\langle z,e^{\left( T_{3}-T_{1}\right) A}\Phi \right\rangle
=\left\langle y^{0}\left( T_{3}\right) ,\Phi \right\rangle \quad \text{for
all}\;z,\Phi \in L^{2}(\Omega )\text{ .}
\end{equation*
This, along with the definition of $J$ in Theorem \ref{theorem3.4} and the
inequality on the left hand side of (\ref{3.2.18}), yields that for all
\;\Phi \in L^{2}(\Omega )$
\begin{equation*}
J\left( \Phi \right) \geq \left\langle y^{0}\left( T_{3}\right) ,\Phi
\right\rangle +\varepsilon \left\Vert z\right\Vert \left\Vert \Phi
\right\Vert \geq 0=J\left( 0\right) \text{ .}
\end{equation*
This implies the equality on the right hand side of (\ref{3.2.18}). Hence,
\ref{3.2.18}) is true.
We next show that
\begin{equation}
w=0\Rightarrow \left\Vert y^{0}\left( T_{3}\right) \right\Vert \leq
\varepsilon \left\Vert z\right\Vert \text{ .} \tag{3.2.19} \label{3.2.19}
\end{equation
By contradiction, suppose that (\ref{3.2.19}) were not true. Then we would
have that
\begin{equation}
\left\Vert y^{0}\left( T_{3}\right) \right\Vert >\varepsilon \left\Vert
z\right\Vert \text{ and }w=0\text{ .} \tag{3.2.20} \label{3.2.20}
\end{equation
Set $\psi :=-y^{0}\left( T_{3}\right) $, which clearly belongs to
L^{2}\left( \Omega \right) \left\backslash \{0\}\right. $. Then we have that
\begin{equation*}
\left\langle z,e^{\left( T_{3}-T_{1}\right) A}\psi \right\rangle
=\left\langle y^{0}\left( T_{3}\right) ,\psi \right\rangle =-\left\Vert
y^{0}\left( T_{3}\right) \right\Vert \left\Vert \psi \right\Vert \text{ .}
\end{equation*
This, along with the first inequality in (\ref{3.2.20}), yields that
\begin{equation*}
\left\langle z,e^{\left( T_{3}-T_{1}\right) A}\psi \right\rangle
+\varepsilon \left\Vert z\right\Vert \left\Vert \psi \right\Vert <0\text{ .}
\end{equation*
Thus, there is an $\sigma >0$ so that
\begin{equation*}
J\left( \sigma \psi \right) =\sigma ^{2}\frac{1}{2}\left\Vert 1_{\omega
}^{\ast }e^{\left( T_{3}-T_{2}\right) A}\psi \right\Vert _{\omega
}^{2}+\sigma \left( \left\langle z,e^{\left( T_{3}-T_{1}\right) A}\psi
\right\rangle +\varepsilon \left\Vert z\right\Vert \left\Vert \psi
\right\Vert \right) <0\text{ .}
\end{equation*
This, along with the second equation in (\ref{3.2.20}), indicates that
\begin{equation*}
0=J\left( 0\right) =\underset{\Phi \in L^{2}(\Omega )}{\text{min}}J\left(
\Phi \right) <0\text{ ,}
\end{equation*
which leads to a contradiction. So we have proved (\ref{3.2.19}). Finally,
the conclusion $\left( c\right) $ of Lemma \ref{lemma3.6} follows from (\re
{3.2.18}) and (\ref{3.2.19}) at once.
This ends the proof of Lemma \ref{lemma3.6} and completes the proof of
Theorem \ref{theorem3.4}.
\bigskip
\bigskip
\subsection{Best connection between Theorem 3.4 and Theorem 2.1}
\bigskip
What we study in this subsection will not have influence on the study of our
stabilization. However, it is independently interesting. Consider the
following problem $(\mathcal{NP})$ (with arbitrarily fixed $\varepsilon >0$.
Recall that $y$ is the solution of (\ref{3.2.1}) associated with the initial
datum $z$ and control $f$):
\begin{equation}
\mathcal{N}:=\underset{\left\Vert z\right\Vert \leq 1}{\text{sup}}\mathcal{N
_{z}=\underset{\left\Vert z\right\Vert \leq 1}{\text{sup}}\text{inf}\left\{
\left\Vert f\right\Vert _{\omega };f\in L^{2}\left( \omega \right) \text{
and }\left\Vert y\left( T_{3}\right) \right\Vert \leq \varepsilon \left\Vert
z\right\Vert \right\} \text{ .} \tag{3.3.1} \label{3.3.1}
\end{equation
The quantity $\mathcal{N}$ is called the value of the problem $(\mathcal{NP})
$. Next, let $C>0$ and introduce the following property $(\mathcal{Q}_{C})$:
For any $z\in L^{2}\left( \Omega \right) $, there is a control $f\in
L^{2}\left( \omega \right) $ so that
\begin{equation}
\text{max}\left\{ \frac{1}{C}\left\Vert f\right\Vert _{\omega },\frac{1}
\varepsilon }\left\Vert y\left( T_{3}\right) \right\Vert \right\} \leq
\left\Vert z\right\Vert \text{ .} \tag{3.3.2} \label{3.3.2}
\end{equation
We would like to mention that the property $(\mathcal{Q}_{C})$ may not hold
for some $C>0$ and $\varepsilon >0$. However, we have seen in Theorem \re
{theorem3.1} that given $\varepsilon >0$, there is $C=\mathcal{C
_{\varepsilon }\left( T_{3}-T_{2},T_{2}-T_{1}\right) >0$ so that the
property $(\mathcal{Q}_{C})$ is true.
\bigskip
The main result of this subsection is as follows: The value $\mathcal{N}$ is
the optimal coefficient $C$ so tha
\begin{equation}
\left\Vert e^{\left( T_{3}-T_{1}\right) A}\Phi \right\Vert \leq C\left\Vert
1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}\Phi \right\Vert _{\omega
}+\varepsilon \left\Vert \Phi \right\Vert \quad \text{for any }\Phi \in
L^{2}\left( \Omega \right) \text{ .} \tag{3.3.3} \label{3.3.3}
\end{equation
Precisely, we have the following result:
\bigskip
\begin{thm}
\label{theorem3.7} Let $\varepsilon >0$. It holds that
\begin{equation*}
\normalfont{\text{inf}}\left\{ C>0;C\text{ \textit{satisfies (\ref{3.3.3})}
\right\} =\mathcal{N}\text{ .}
\end{equation*}
\end{thm}
\bigskip
Further the connections among the problem $(\mathcal{NP})$, the property $
\mathcal{Q}_{C})$ and the observation inequalities in Theorem \re
{theorem2.1} are presented in the next Theorem~\ref{theorem3.8}, which will
be used in the proof of the above Theorem~\ref{theorem3.7}.
\bigskip
\begin{thm}
\label{theorem3.8} Let $\varepsilon >0$ and $C>0$. The following statements
are equivalent:
\begin{description}
\item[$\left( i\right) $] Let $\mathcal{N}$ be given by (\ref{3.3.1}). Then
\mathcal{N}\leq C$.
\item[$\left( ii\right) $] The property $(\mathcal{Q}_{C})$ defined by (\re
{3.3.2}) is true.
\item[$\left( iii\right) $] For any $\Phi \in L^{2}\left( \Omega \right) $,
the following estimate holds:
\begin{equation*}
\left\Vert e^{\left( T_{3}-T_{1}\right) A}\Phi \right\Vert \leq C\left\Vert
1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}\Phi \right\Vert _{\omega
}+\varepsilon \left\Vert \Phi \right\Vert \text{ .}
\end{equation*}
\end{description}
\end{thm}
\bigskip
Proof of Theorem \ref{theorem3.8} .- We organize the proof by three steps as
follows:
\noindent\textit{Step 1. To show that }$(i)\Leftrightarrow(ii)$.
We first prove that $(i)\Rightarrow (ii)$. Assume that $(i)$ is true. When
z=0$ in $L^{2}(\Omega )$, we find that (\ref{3.3.2}) holds for $f=0$. Thus,
it suffices to show $(ii)$ with an arbitrarily fixed $z\in L^{2}\left(
\Omega \right) \left\backslash \{0\}\right. $. For this purpose, we write
\widehat{z}=z/\left\Vert z\right\Vert $. Let $\widehat{f}$ be the solution
to $(\mathcal{P})$ associated to $\widehat{z}$. Then the solution $\widehat{
}$ of (\ref{3.2.1}) associated with initial data $\widehat{z}$ and control
\widehat{f}$ satisfies that
\begin{equation*}
\left\Vert \widehat{y}\left( T_{3}\right) \right\Vert \leq \varepsilon
\left\Vert \widehat{z}\right\Vert =\varepsilon \text{ .}
\end{equation*
Setting $f=\left\Vert z\right\Vert \widehat{f}$, the solution $y$ of (\re
{3.2.1}) have the following property:
\begin{equation*}
\left\Vert y\left( T_{3}\right) \right\Vert =\left\Vert z\right\Vert
\left\Vert \widehat{y}\left( T_{3}\right) \right\Vert \leq \varepsilon
\left\Vert z\right\Vert \text{ .}
\end{equation*
Thus, to show that the above $f$ satisfies (\ref{3.3.2}), we only need to
prove that $\left\Vert f\right\Vert _{\omega }\leq C\left\Vert z\right\Vert
. This will be done in what follows: Since $\widehat{f}$ is the solution to
(\mathcal{P})$ associated to $\widehat{z}$, we have that $\left\Vert
\widehat{f}\right\Vert _{\omega }=\mathcal{N}_{\widehat{z}}$. This, along
with (\ref{3.3.1}) and $(i)$ of Theorem \ref{theorem3.8}, yields that
\begin{equation*}
\left\Vert f\right\Vert _{\omega }=\left\Vert z\right\Vert \left\Vert
\widehat{f}\right\Vert _{\omega }=\left\Vert z\right\Vert \mathcal{N}_
\widehat{z}}\leq \left\Vert z\right\Vert \mathcal{N}\leq C\left\Vert
z\right\Vert \text{ .}
\end{equation*
Hence, $(ii)$ is true.
We next show that $(ii)\Rightarrow (i)$. Assume that $(ii)$ is true. By
contradiction, suppose that $(i)$ were false. Then there would be $z\neq 0$
with $\left\Vert z\right\Vert \leq 1$ so that $\mathcal{N}_{z}>C$. Let
\widehat{z}=z\left/ \left\Vert z\right\Vert \right. $. Then we have that
\begin{equation*}
\mathcal{N}_{\widehat{z}}=\frac{1}{\Vert z\Vert }\mathcal{N}_{z}\geq
\mathcal{N}_{z}>C\text{ .}
\end{equation*
Therefore, we see that there is no $f\in L^{2}(\Omega )$ so that the
solution $y$ of (\ref{3.2.1}), associated with the initial datum $\widehat{z}
$ and the control $f$, has the property:
\begin{equation*}
\left\Vert y\left( T_{3}\right) \right\Vert \leq \varepsilon \left\Vert
\widehat{z}\right\Vert \text{ and }\left\Vert f\right\Vert _{\omega }\leq
C=C\left\Vert \widehat{z}\right\Vert \text{ .}
\end{equation*
This contradicts $(ii)$. Hence, $(i)$ stands.
\noindent\textit{Step 2. To show that }$(ii)\Rightarrow(iii)$.
Suppose that $(ii)$ holds. Then, given $z\in L^{2}(\Omega )$, there is $f\in
L^{2}(\Omega )$ so that (\ref{3.3.2}) holds. Meanwhile, by multiplying by
e^{\left( T_{3}-t\right) A}\Phi $ the system solved by $y$, one gets
\begin{equation*}
\left\langle y\left( T_{3}\right) ,\Phi \right\rangle -\left\langle
z,e^{\left( T_{3}-T_{1}\right) A}\Phi \right\rangle =\left\langle 1_{\omega
}f,e^{\left( T_{3}-T_{2}\right) A}\Phi \right\rangle \quad \forall \Phi \in
L^{2}\left( \Omega \right) \text{ .}
\end{equation*
This, along with the inequality (\ref{3.3.2}), yields that for each $\Phi
\in L^{2}(\Omega )$
\begin{equation*}
\begin{array}{ll}
& \quad \left\Vert e^{\left( T_{3}-T_{1}\right) A}\Phi \right\Vert =\underse
{\left\Vert z\right\Vert \leq 1}{\text{sup}}\left\langle e^{\left(
T_{3}-T_{1}\right) A}\Phi ,z\right\rangle \\
& =\underset{\left\Vert z\right\Vert \leq 1}{\text{sup}}\left[ \left\langle
y\left( T_{3}\right) ,\Phi \right\rangle -\left\langle 1_{\omega
}f,e^{\left( T_{3}-T_{2}\right) A}\Phi \right\rangle \right] \\
& \leq \underset{\left\Vert z\right\Vert \leq 1}{\text{sup}}\left[
\left\Vert y\left( T_{3}\right) \right\Vert \left\Vert \Phi \right\Vert
+\left\Vert f\right\Vert _{\omega }\left\Vert 1_{\omega }^{\ast }e^{\left(
T_{3}-T_{2}\right) A}\Phi \right\Vert _{\omega }\right] \\
& \leq \underset{\left\Vert z\right\Vert \leq 1}{\text{sup}}\left[ \left(
\varepsilon \left\Vert \Phi \right\Vert +C\left\Vert 1_{\omega }^{\ast
}e^{\left( T_{3}-T_{2}\right) A}\Phi \right\Vert _{\omega }\right)
\left\Vert z\right\Vert \right] \\
& =C\left\Vert 1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}\Phi
\right\Vert _{\omega }+\varepsilon \left\Vert \Phi \right\Vert \text{ ,
\end{array
\end{equation*
which leads to the desired observation estimate. Hence, $(iii)$ is true.
\noindent\textit{Step 3. To show that }$(iii)\Rightarrow(ii)$.
Suppose that $(iii)$ is true. Arbitrarily fix $z\in L^{2}\left( \Omega
\right) $. Denote by $y^{0}$ the solution of (\ref{3.2.1}) with $f=0$. In
the case that $\left\Vert y^{0}\left( T_{3}\right) \right\Vert \leq
\varepsilon \left\Vert z\right\Vert $, (\ref{3.3.2}) holds for $f=0$. Thus,
we only need to consider the case that
\begin{equation}
\left\Vert y^{0}\left( T_{3}\right) \right\Vert >\varepsilon \left\Vert
z\right\Vert \text{ .} \tag{3.3.4} \label{3.3.4}
\end{equation
In this case, we let $f:=1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}w
, where $w$ is the unique minimizer of the functional $J$, which is given in
Theorem \ref{theorem3.4}. Then according to $(iii)$ of Theorem \re
{theorem3.4}, $f$ is the minimal norm control to $(\mathcal{P})$. By Lemma
\ref{lemma3.6} and (\ref{3.3.4}), we see that $w\neq 0$. Then using the
Euler-Lagrange equation (\ref{3.2.7}) and noticing that $\chi _{\omega
}=1_{\omega }1_{\omega }^{\ast }$, we find tha
\begin{equation*}
\begin{array}{ll}
& \quad \left\langle z,e^{\left( T_{3}-T_{1}\right) A}w\right\rangle
=\left\langle e^{\left( T_{3}-T_{1}\right) A}z,w\right\rangle \\
& =-\left\langle e^{\left( T_{3}-T_{2}\right) A}\chi _{\omega }e^{\left(
T_{3}-T_{2}\right) A}w+\varepsilon \left\Vert z\right\Vert \frac{w}
\left\Vert w\right\Vert },w\right\rangle \\
& =-\left\Vert 1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right)
A}w\right\Vert _{\omega }^{2}-\varepsilon \left\Vert z\right\Vert \left\Vert
w\right\Vert \text{ .
\end{array
\end{equation*
Since $f=1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right) A}w$, the above
equality, along with the definition of $J$ in Theorem \ref{theorem3.4},
shows that
\begin{equation}
J\left( w\right) =\frac{1}{2}\left\Vert 1_{\omega }^{\ast }e^{\left(
T_{3}-T_{2}\right) A}w\right\Vert _{\omega }^{2}+\left\langle z,e^{\left(
T_{3}-T_{1}\right) A}w\right\rangle +\varepsilon \left\Vert z\right\Vert
\left\Vert w\right\Vert =-\frac{1}{2}\left\Vert f\right\Vert _{\omega }^{2
\text{ .} \tag{3.3.5} \label{3.3.5}
\end{equation
Meanwhile, it follows from the above and the observation estimate in $(iii)$
in Theorem \ref{theorem3.8} tha
\begin{equation}
\begin{array}{ll}
J\left( w\right) & \geq \frac{1}{2}\left\Vert 1_{\omega }^{\ast }e^{\left(
T_{3}-T_{2}\right) A}w\right\Vert _{\omega }^{2}+\varepsilon \left\Vert
z\right\Vert \left\Vert w\right\Vert -\left\Vert \,e^{\left(
T_{3}-T_{1}\right) A}w\right\Vert \left\Vert z\right\Vert \\
& \geq \frac{1}{2}\left\Vert 1_{\omega }^{\ast }e^{\left( T_{3}-T_{2}\right)
A}w\right\Vert _{\omega }^{2}+\varepsilon \left\Vert z\right\Vert \left\Vert
w\right\Vert -\left( C\left\Vert 1_{\omega }^{\ast }e^{\left(
T_{3}-T_{2}\right) A}w\right\Vert _{\omega }+\varepsilon \left\Vert
w\right\Vert \right) \left\Vert z\right\Vert \\
& \geq \frac{1}{2}\left\Vert f\right\Vert _{\omega }^{2}-C\left\Vert
f\right\Vert _{\omega }\left\Vert z\right\Vert \text{ .
\end{array}
\tag{3.3.6} \label{3.3.6}
\end{equation
From (\ref{3.3.5}) and (\ref{3.3.6}), it follows that
\begin{equation}
\left\Vert f\right\Vert _{\omega }\leq C\left\Vert z\right\Vert \text{ .}
\tag{3.3.7} \label{3.3.7}
\end{equation
On the other hand, since $f$ is the minimal norm control to $(\mathcal{P})$,
it holds that $\left\Vert y\left( T_{3}\right) \right\Vert \leq \varepsilon
\left\Vert z\right\Vert $. From this and (\ref{3.3.7}), we find that (\re
{3.3.2}) is true. Hence, $(ii)$ stands.
In summary, we complete the proof of Theorem \ref{theorem3.8}.
\bigskip
The rest of this subsection is devoted to the proof of Theorem \re
{theorem3.7}.
\bigskip
Proof of Theorem \ref{theorem3.7} .- Define
\begin{equation*}
C^{\ast }:=\text{inf}\left\{ C>0;C\text{ satisfies (\ref{3.3.2})}\right\}
\text{ .}
\end{equation*
\noindent \textit{Proof of} $\mathcal{N}\leq C^{\ast }$: It directly follows
from Theorem \ref{theorem3.8}.
\noindent \textit{Proof of} $C^{\ast }\leq \mathcal{N}$: It suffices to
prove that the property $(\mathcal{Q}_{\mathcal{N}})$ holds. Indeed, by
making use of the proof of \textquotedblleft $(ii)\Rightarrow \left(
iii\right) $\textquotedblright\ of Theorem \ref{theorem3.8}, we find that
\begin{equation*}
C^{\ast }\leq \mathcal{N}\text{ .}
\end{equation*
Therefore, the remainder is that for each $z\in L^{2}\left( \Omega \right)
, there is a control $f\in L^{2}\left( \omega \right) $ satisfying tha
\begin{equation}
\text{max}\left\{ \frac{1}{\mathcal{N}}\left\Vert f\right\Vert _{\omega }
\frac{1}{\varepsilon }\left\Vert y\left( T_{3}\right) \right\Vert \right\}
\leq \left\Vert z\right\Vert \text{ .} \tag{3.3.8} \label{3.3.8}
\end{equation
When $z=0$, we can easily get (\ref{3.3.8}) by taking $f=0$. So it suffices
to prove (\ref{3.3.8}) for an arbitrarily fixed $z\in L^{2}\left( \Omega
\right) \left\backslash \left\{ 0\right\} \right. $. To this end, we let
\widehat{z}=z/\left\Vert z\right\Vert $. Denote $y$ the solution of (\re
{3.2.1}) associated with the initial datum $\widehat{z}$. It follows from
\ref{3.3.1}) that
\begin{equation*}
\text{inf}\left\{ \left\Vert f\right\Vert _{\omega };f\in L^{2}\left( \omega
\right) \text{ and }\left\Vert y\left( T_{3}\right) \right\Vert \leq
\varepsilon \left\Vert \widehat{z}\right\Vert \right\} \leq \mathcal{N}\text{
.}
\end{equation*
Because the infimum on the left hand side of the above inequality can be
reached, there is $\widehat{f}\in L^{2}\left( \omega \right) $ so tha
\begin{equation*}
\left\Vert \widehat{y}\left( T_{3}\right) \right\Vert \leq \delta \left\Vert
\widehat{z}\right\Vert \quad \text{with }\left\Vert \widehat{f}\right\Vert
_{\omega }\leq \mathcal{N}\text{ .}
\end{equation*
where $\widehat{y}$ the solution of (\ref{3.2.1}) associated with the
initial datum $\widehat{z}$ and control $\widehat{f}$. From these, we see
that (\ref{3.3.8}) holds for $f=\left\Vert z\right\Vert \widehat{f}$. This
ends the proof.
\bigskip
\bigskip
\bigskip
\section{Inverse source problem}
\bigskip
This section concerns an inverse source problem: Suppose that we have a
solution $\varphi $ of $\varphi ^{\prime }-A\varphi =0$ with a priori bound
on the initial data in $L^{2}(\Omega )$. The question is how to recover
approximatively the initial data from the knowledge of the solution $\varphi
$ in the future. This can be done as follows thanks to the impulse control.
\bigskip
\begin{thm}
\label{theorem4.1} Let $\omega _{1}$ be a non-empty open subset of $\Omega $
and $K\in \mathbb{N}$. Let $0\leq T_{1}<T_{2}<T_{3}$ and let $\varphi $ be a
solution of
\begin{equation*}
\left\{
\begin{array}{ll}
\varphi ^{\prime }\left( t\right) -A\varphi \left( t\right) =0\text{ ,} &
t\in \left( T_{1},T_{3}\right) \ \text{,} \\
\varphi \left( T_{1}\right) \in L^{2}\left( \Omega \right) \text{ .} &
\end{array
\right.
\end{equation*
Then for any $\varepsilon >0$, there exists $\left\{ g_{j}\right\}
_{j=1,\cdot \cdot ,K}\in L^{2}\left( \omega _{1}\right) $ such that for any
j=1,\cdot \cdot ,K$,
\begin{equation*}
\left\vert \langle \varphi \left( T_{1}\right) ,\xi _{j}\rangle +e^{\left(
T_{3}-T_{1}\right) \lambda _{j}}\langle g_{j},1_{\omega _{1}}^{\ast }\varphi
\left( T_{1}+T_{3}-T_{2}\right) \rangle _{\omega _{1}}\right\vert \leq
e^{\left( T_{3}-T_{1}\right) \lambda _{j}}\varepsilon \left\Vert \varphi
\left( T_{1}\right) \right\Vert
\end{equation*
and
\begin{equation*}
\left\Vert g_{j}\right\Vert _{\omega _{1}}\leq \mathcal{C}_{\varepsilon
}\left( T_{3}-T_{2},T_{2}-T_{1}\right)
\end{equation*
where $\mathcal{C}_{\varepsilon }$ is given in Theorem \ref{theorem3.1}.
Further $g_{j}$ is the control function given in Corollary \ref{corollary3.2}
with $\omega =\omega _{1}$.
\end{thm}
\bigskip
Proof .-
\noindent \textit{Step 1:} We apply Corollary \ref{corollary3.2} with
\omega =\omega _{1}$ and get the existence of $\left( y_{j},g_{j}\right) $
such that
\begin{equation*}
\left\{
\begin{array}{ll}
y_{j}^{\prime }\left( t\right) -Ay_{j}\left( t\right) =0\text{ ,} & t\in
\left( T_{1},T_{3}\right) \backslash \left\{ T_{2}\right\} \ \text{,} \\
y_{j}\left( T_{1}\right) =\xi _{j}\text{ ,} & \\
y_{j}\left( T_{2}\right) =y_{j}\left( T_{2-}\right) +1_{\omega _{1}}g_{j
\text{ ,} &
\end{array
\right.
\end{equation*
and $\left\Vert y_{j}\left( T_{3}\right) \right\Vert \leq \varepsilon $
where $g_{j}$ has the desired bound.
\noindent \textit{Step 2:} Write $\varphi \left( T_{1}\right) =\displaystyl
\sum_{i=1,\cdot \cdot ,+\infty }a_{i}\xi _{i}$ with $a_{i}=\langle \varphi
\left( T_{1}\right) ,\xi _{i}\rangle $. Then we have that
\begin{equation*}
\varphi \left( T_{3}\right) =\displaystyle\sum_{i=1,\cdot \cdot ,+\infty
}a_{i}e^{-\left( T_{3}-T_{1}\right) \lambda _{j}}\xi _{i}\text{ .}
\end{equation*
Hence, $\langle y_{j}(T_{1}),\varphi \left( T_{3}\right) \rangle =\langle
\xi _{j},\varphi \left( T_{3}\right) \rangle =a_{j}e^{-\left(
T_{3}-T_{1}\right) \lambda _{j}}$.
\noindent \textit{Step 3:} Multiply the equation solved by $y_{j}$ by the
solution $\varphi \left( T_{1}+T_{3}-t\right) $, with $t\in \left[
T_{1},T_{3}\right] $, to get
\begin{equation*}
\langle y_{j}(T_{3}),\varphi \left( T_{1}\right) \rangle =\langle
y_{j}(T_{1}),\varphi \left( T_{3}\right) \rangle +\langle g_{j},1_{\omega
_{1}}^{\ast }\varphi \left( T_{1}+T_{3}-T_{2}\right) \rangle _{\omega _{1}
\text{ .}
\end{equation*
Therefore by step 2, it holds that
\begin{equation*}
\left\vert a_{j}+e^{\left( T_{3}-T_{1}\right) \lambda _{j}}\langle
g_{j},1_{\omega _{1}}^{\ast }\varphi \left( T_{1}+T_{3}-T_{2}\right) \rangle
_{\omega _{1}}\right\vert =e^{\left( T_{3}-T_{1}\right) \lambda
_{j}}\left\vert \langle y_{j}(T_{3}),\varphi \left( T_{1}\right) \rangle
\right\vert \text{ .}
\end{equation*
This, along with the Cauchy-Schwarz inequality and step 1, leads to the
desired result.
\bigskip
\bigskip
\section{Main result}
\bigskip
This section presents the main result of this paper, as well as its proof.
We first recall that $\omega _{1}$ and $\omega _{2}$ are two arbitrarily
fixed open and non-empty subsets of $\Omega $. We next recall that
\{\lambda _{j}\}_{j=1}^{\infty }$ is the family of all eigenvalues of $-A$
so that (\ref{1.2}) holds and that $\{\xi _{j}\}_{j=1}^{\infty }$ is the
family of the corresponding normalized eigenfunctions. For each $\gamma >0$,
we define a natural number $K$ in the following manner:
\begin{equation}
K:=\text{card}\left\{ j\in \mathbb{N},\text{ }\lambda _{j}<\gamma +\frac
\text{ln}2}{T}\right\} \text{ .} \tag{5.1} \label{5.1}
\end{equation
Next, we define
\begin{equation}
\varepsilon :=\frac{1}{6\left( 1+K\right) }e^{-\gamma T}e^{-\left\Vert
V\right\Vert _{\infty }T}e^{-\lambda _{K}T/2}\text{ .} \tag{5.2}
\label{5.2}
\end{equation
Denote by $\left\{ f_{j}\right\} _{j=1,\cdot \cdot ,K}\in L^{2}\left( \omega
_{2}\right) $ the minimal norm control functions obtained by applying
Corollary \ref{corollary3.2} with $T_{1}=\frac{T}{4}$, $T_{2}=T$, $T_{3}
\frac{5T}{4}$ and $\omega =\omega _{2}$. Denote by $\left\{ g_{j}\right\}
_{j=1,\cdot \cdot ,K}\in L^{2}\left( \omega _{1}\right) $ the minimal norm
control functions obtained by applying Corollary \ref{corollary3.2} with
T_{1}=\frac{T}{4}$, $T_{2}=\frac{T}{2}$, $T_{3}=\frac{3T}{4}$ and $\omega
=\omega _{1}$. Now for each $\gamma >0$, we define a linear bounded operator
$\mathcal{F}$ from $L^{2}\left( \omega _{1}\right) $ into $L^{2}\left(
\omega _{2}\right) $ in the following manner:
\begin{equation}
\mathcal{F}(p):=-\sum_{j=1,\cdot \cdot ,K}e^{\lambda _{j}T/2}\langle
g_{j},p\rangle _{\omega _{1}}f_{j}\left( x\right) \quad \text{for each }p\in
L^{2}(\omega _{1})\text{ .} \tag{5.3} \label{5.3}
\end{equation
The closed-loop equation under consideration reads:
\begin{equation}
\left\{
\begin{array}{ll}
y^{\prime }\left( t\right) -Ay\left( t\right) =0\text{ ,} & \text{in }\left(
0,+\infty \right) \backslash \mathbb{N}T\ \text{,} \\
y\left( 0\right) \in L^{2}\left( \Omega \right) \text{ ,} & \\
y\left( \left( n+1\right) T\right) =y\left( \left( n+1\right) T_{-}\right)
+1_{\omega _{2}}\mathcal{F}\left( 1_{\omega _{1}}^{\ast }y\left( \left( n
\frac{1}{2}\right) T\right) \right) \text{ ,} & \text{for }n\in \overline
\mathbb{N}}\text{ .
\end{array
\right. \tag{5.4} \label{5.4}
\end{equation}
\bigskip
The main result of this paper is the following theorem:
\bigskip
\begin{thm}
\label{theorem5.1} For each $\gamma >0$, let $\mathcal{F}$ be given by (\re
{5.3}). Then the following conclusions are true:
\begin{description}
\item[$\left( i\right) $] Each solution $y$ to the equation (\ref{5.4})
satisfies that
\begin{equation*}
\left\Vert y\left( t\right) \right\Vert \leq e^{T\left( \gamma +\left\Vert
V\right\Vert _{\infty }\right) }\left( 1+\left\Vert \mathcal{F}\right\Vert _
\mathcal{L}(L^{2}(\omega _{1}),L^{2}(\omega _{2}))}\right) e^{-\gamma
t}\left\Vert y(0)\right\Vert \quad \text{for all}\; \;t\geq 0\text{ .}
\end{equation*}
\item[$\left( ii\right) $] The operator $\mathcal{F}$ satisfies the
estimate:
\begin{equation*}
\left\Vert \mathcal{F}\right\Vert _{\mathcal{L}(L^{2}(\omega
_{1}),L^{2}(\omega _{2}))}\leq Ce^{C\gamma }\text{ ,}
\end{equation*
where $C$ is a positive constant independent of $\gamma $, depending on
\Omega $, $\omega _{1}$ ,$\omega _{2}$, $d$, $T$ and $\Vert V\Vert _{\infty
} $. Moreover, the manner how it depends on $T$, $d$, $\Vert V\Vert _{\infty
}$ is explicitly given.
\end{description}
\end{thm}
\bigskip
Proof .-
\noindent \textit{Step 1:}
Set $L_{n}=nT+\frac{T}{4}$. In order to have the conclusion $(i)$ in the
theorem, it suffices to prove that the solution $y$ of (\ref{5.4}) satisfies
\begin{equation*}
\left\Vert y\left( L_{n+1}\right) \right\Vert \leq e^{-\gamma T}\left\Vert
y\left( L_{n}\right) \right\Vert
\end{equation*
for any $n\geq 0$. Indeed, thanks to the above inequality, we find that when
$t\in \left[ L_{n},\left( n+1\right) T\right] $,
\begin{equation*}
\begin{array}{ll}
\left\Vert y\left( t\right) \right\Vert & \leq e^{\left( t-L_{n}\right)
\left\Vert V\right\Vert _{\infty }}\left\Vert y\left( L_{n}\right)
\right\Vert \leq e^{\left( t-L_{n}\right) \left\Vert V\right\Vert _{\infty
}}e^{-n\gamma T}\left\Vert y(0)\right\Vert \\
& \leq e^{T\left\Vert V\right\Vert _{\infty }}e^{-n\gamma T}\left\Vert
y(0)\right\Vert \leq e^{T\left( \gamma +\left\Vert V\right\Vert _{\infty
}\right) }e^{-\gamma t}\left\Vert y(0)\right\Vert \text{ ,
\end{array
\end{equation*
and when $t\in \left[ \left( n+1\right) T,L_{n+1}\right] $,
\begin{equation*}
\begin{array}{ll}
\left\Vert y\left( t\right) \right\Vert & \leq e^{\left( t-\left( n+1\right)
T\right) \left\Vert V\right\Vert _{\infty }}\left\Vert y\left( \left(
n+1\right) T\right) \right\Vert \\
& \leq e^{\left( t-\left( n+1\right) T\right) \left\Vert V\right\Vert
_{\infty }}\left\Vert y\left( \left( n+1\right) T\right) _{-}\right\Vert
+e^{\left( t-\left( n+1\right) T\right) \left\Vert V\right\Vert _{\infty
}}\left\Vert \mathcal{F}\left( 1_{\omega _{1}}^{\ast }y\left( \left( n+\frac
1}{2}\right) T\right) \right) \right\Vert _{\omega _{2}} \\
& \leq e^{\left( t-\left( n+1\right) T+3T/4\right) \left\Vert V\right\Vert
_{\infty }}\left\Vert y\left( L_{n}\right) \right\Vert +\left\Vert \mathcal{
}\right\Vert _{\mathcal{L}(L^{2}(\omega _{1}),L^{2}(\omega ))}e^{\left(
t-\left( n+1\right) T+T/4\right) \left\Vert V\right\Vert _{\infty
}}\left\Vert y\left( L_{n}\right) \right\Vert \\
& \leq e^{T\left( \gamma +\left\Vert V\right\Vert _{\infty }\right) }\left(
1+\left\Vert \mathcal{F}\right\Vert _{\mathcal{L}(L^{2}(\omega
_{1}),L^{2}(\omega _{2}))}\right) e^{-\gamma t}\left\Vert y(0)\right\Vert
\text{ .
\end{array
\end{equation*
From these and the time translation invariance of the equation (\ref{5.4}),
we see that the conclusion $(i)$ in Theorem \ref{theorem5.1} is true for any
$t\geq \frac{T}{4}$. But the case $t\leq \frac{T}{4}$ is trivial.
\noindent \textit{Step 2:}
Denote $y\left( L_{n}\right) =\displaystyle\sum_{j=1,\cdot \cdot ,+\infty
}a_{j}\xi _{j}$ and $a=\left( a_{j}\right) _{j=1,\cdot \cdot ,+\infty }$.
Then one deduces that $\langle y\left( L_{n}\right) ,\xi _{j}\rangle =a_{j}$
and $\left\Vert a\right\Vert _{\ell ^{2}}=\left\Vert y\left( L_{n}\right)
\right\Vert $.
For the rest of the proof, recall that $K$ and $\varepsilon $ are given by
\ref{5.1}) and (\ref{5.2}) respectively.
\noindent \textit{Step 3:}
Notice that the solution $y$ of (\ref{5.4}) evolves freely without a control
function between in $\left[ L_{n},nT+\frac{3T}{4}\right] $. Thus, we can
apply Theorem \ref{theorem4.1} with the choice $T_{1}=L_{n}$, $T_{2}=nT
\frac{T}{2}$, $T_{3}=nT+\frac{3T}{4}$ and $\varphi =y$ to get $\left\{
g_{j}\right\} _{j=1,\cdot \cdot ,K}\in L^{2}\left( \omega _{1}\right) $ such
that for any $j=1,\cdot \cdot ,K$,
\begin{equation*}
\begin{array}{ll}
\left\vert a_{j}+e^{\lambda _{j}T/2}\langle g_{j},1_{\omega _{1}}^{\ast
}y\left( \left( n+\frac{1}{2}\right) T\right) \rangle _{\omega
_{1}}\right\vert & \leq e^{\lambda _{j}T/2}\varepsilon \left\Vert y\left(
L_{n}\right) \right\Vert \\
& \leq e^{\lambda _{K}T/2}\varepsilon \left\Vert y\left( L_{n}\right)
\right\Ver
\end{array
\end{equation*
and
\begin{equation*}
\left\Vert g_{j}\right\Vert _{\omega _{1}}\leq \mathcal{C}_{\varepsilon
}\left( T/4,T/4\right)
\end{equation*
where $\mathcal{C}_{\varepsilon }$ is given in Theorem \ref{theorem3.1}.
Further such $g_{j}$ is given in Corollary \ref{corollary3.2} with $\omega
=\omega _{1}$. By the time translation invariance of the equation (\ref{5.4
), $\left\{ g_{j}\right\} _{j=1,\cdot \cdot ,K}\in L^{2}\left( \omega
_{1}\right) $ is the control function obtained by applying Corollary \re
{corollary3.2} with $T_{1}=\frac{T}{4}$, $T_{2}=\frac{T}{2}$, $T_{3}=\frac{3
}{4}$ and $\omega =\omega _{1}$.
\noindent \textit{Step 4:}
Denote $b_{j}:=-e^{\lambda _{j}T/2}\langle g_{j},1_{\omega _{1}}^{\ast
}y\left( \left( n+\frac{1}{2}\right) T\right) \rangle _{\omega _{1}}$ for
j=1,\cdot \cdot ,K$, and $b=\left( b_{j}\right) _{j=1,\cdot \cdot ,K}$. Then
by step 3 and step 2, we have
\begin{equation*}
\begin{array}{ll}
\left\Vert b\right\Vert _{\ell ^{2}} & \leq \left\Vert \left(
a_{j}+e^{\lambda _{j}T/2}\langle g_{j},1_{\omega _{1}}^{\ast }y\left( \left(
n+\frac{1}{2}\right) T\right) \rangle _{\omega _{1}}\right) _{j=1,\cdot
\cdot ,K}\right\Vert _{\ell ^{2}}+\left\Vert a\right\Vert _{\ell ^{2}} \\
& \leq \left( \sqrt{K}e^{\lambda _{K}T/2}\varepsilon +1\right) \left\Vert
y\left( L_{n}\right) \right\Vert \text{ .
\end{array
\end{equation*}
\noindent \textit{Step 5:}
We apply Theorem \ref{theorem3.3} with $T_{1}=L_{n}$, $T_{2}=\left(
n+1\right) T$, $T_{3}=L_{n+1}$ and the above choice of $b$. Then there is a
solution $\left( \widetilde{y},\widetilde{f}\right) $ such that
\begin{equation*}
\left\{
\begin{array}{ll}
\widetilde{y}^{\prime }\left( t\right) -A\widetilde{y}\left( t\right) =
\text{ ,} & t\in \left( L_{n},L_{n+1}\right) \backslash \left\{ \left(
n+1\right) T\right\} \ \text{,} \\
\widetilde{y}\left( L_{n}\right) =\displaystyle\sum_{j=1,\cdot \cdot
,K}b_{j}\xi _{j}\text{ ,} & \\
\widetilde{y}\left( \left( n+1\right) T\right) =\widetilde{y}\left( \left(
n+1\right) T_{-}\right) +1_{\omega _{2}}\widetilde{f}\text{ ,} &
\end{array
\right.
\end{equation*
and
\begin{equation*}
\left\{
\begin{array}{ll}
\left\Vert \widetilde{y}\left( L_{n+1}\right) \right\Vert \leq \varepsilon
\sqrt{K}\left\Vert b\right\Vert _{\ell ^{2}}\text{ ,} & \\
\widetilde{f}=\displaystyle\sum_{j=1,\cdot \cdot ,K}b_{j}f_{j}:=
\displaystyle\sum_{j=1,\cdot \cdot ,K}e^{\lambda _{j}T/2}\langle
g_{j},1_{\omega _{1}}^{\ast }y((n+\tfrac{1}{2})T)\rangle _{\omega _{1}}f_{j
\text{ ,} &
\end{array
\right.
\end{equation*
where $f_{j}$ is given by Corollary \ref{corollary3.2} with $\omega =\omega
_{2}$ and satisfie
\begin{equation*}
\left\Vert f_{j}\right\Vert _{\omega _{2}}\leq \mathcal{C}_{\varepsilon
}\left( T/4,3T/4\right)
\end{equation*
with $\mathcal{C}_{\varepsilon }$ given in Theorem \ref{theorem3.1}. By the
time translation invariance of the equation (\ref{5.4}), $\left\{
f_{j}\right\} _{j=1,\cdot \cdot ,K}\in L^{2}\left( \omega _{2}\right) $ is
the control function obtained by applying Corollary \ref{corollary3.2} with
T_{1}=\frac{T}{4}$, $T_{2}=T$, $T_{3}=\frac{5T}{4}$ and $\omega =\omega _{2}
.
\noindent \textit{Step 6:}
One can check that $\widetilde{f}=\mathcal{F}\left( 1_{\omega _{1}}^{\ast
}y\left( \left( n+\frac{1}{2}\right) T\right) \right) $ and $y=\widetilde{y}
\widehat{y}+\overline{y}$ where $\widetilde{y}$ is given in step 5,
\widehat{y}$ solves
\begin{equation*}
\left\{
\begin{array}{ll}
\widehat{y}^{\prime }\left( t\right) -A\widehat{y}\left( t\right) =0\text{ ,}
& t\in \left( L_{n},L_{n+1}\right) \backslash \left\{ \left( n+1\right)
T\right\} \ \text{,} \\
\widehat{y}\left( L_{n}\right) =\displaystyle\sum_{j=1,\cdot \cdot ,K}\left(
a_{j}-b_{j}\right) \xi _{j}\text{ ,} & \\
\widehat{y}\left( \left( n+1\right) T\right) =\widehat{y}\left( \left(
n+1\right) T_{-}\right) \text{ ,} &
\end{array
\right.
\end{equation*
and $\overline{y}$ satisfies
\begin{equation*}
\left\{
\begin{array}{ll}
\overline{y}^{\prime }\left( t\right) -A\overline{y}\left( t\right) =0\text{
,} & t\in \left( L_{n},L_{n+1}\right) \backslash \left\{ \left( n+1\right)
T\right\} \ \text{,} \\
\overline{y}\left( L_{n}\right) =\displaystyle\sum_{j>K}a_{j}\xi _{j}\text{
} & \\
\overline{y}\left( \left( n+1\right) T\right) =\overline{y}\left( \left(
n+1\right) T_{-}\right) \text{ .} &
\end{array
\right.
\end{equation*}
\noindent \textit{Step 7:} We estimate $\left\Vert y\left( L_{n+1}\right)
\right\Vert =\left\Vert \widetilde{y}\left( L_{n+1}\right) +\widehat{y
\left( L_{n+1}\right) +\overline{y}\left( L_{n+1}\right) \right\Vert $ as
follows.
First, by step 5 and step 4,
\begin{equation*}
\left\Vert \widetilde{y}\left( L_{n+1}\right) \right\Vert \leq \varepsilon
\sqrt{K}\left\Vert b\right\Vert _{\ell ^{2}}\leq \varepsilon \sqrt{K}\left(
\sqrt{K}e^{\lambda _{K}T/2}\varepsilon +1\right) \left\Vert y\left(
L_{n}\right) \right\Vert \text{ .}
\end{equation*
Second, by step
\begin{equation*}
\left\Vert \widehat{y}\left( L_{n+1}\right) \right\Vert \leq e^{\left\Vert
V\right\Vert _{\infty }T}\left\Vert \widehat{y}\left( L_{n}\right)
\right\Vert \leq e^{\left\Vert V\right\Vert _{\infty }T}\sqrt{K}e^{\lambda
_{K}T/2}\varepsilon \left\Vert y\left( L_{n}\right) \right\Vert \text{ .}
\end{equation*
Third,
\begin{equation*}
\left\Vert \overline{y}\left( L_{n+1}\right) \right\Vert =\left(
\sum_{j>K}\left\vert a_{j}e^{-\lambda _{j}\left( L_{n+1}-L_{n}\right)
}\right\vert ^{2}\right) ^{1/2}\leq e^{-\lambda _{K+1}T}\left\Vert y\left(
L_{n}\right) \right\Vert \text{ .}
\end{equation*
Gathering all the previous estimates, one concludes that
\begin{equation*}
\left\Vert y\left( L_{n+1}\right) \right\Vert \leq \left( e^{-\lambda
_{K+1}T}+3e^{\left\Vert V\right\Vert _{\infty }T}e^{\lambda _{K}T/2}\left(
1+K\right) \varepsilon \right) \left\Vert y\left( L_{n}\right) \right\Vert
\end{equation*
with $\varepsilon \in \left( 0,1\right) $. Finally, the choice of $K$ (see
\ref{5.1})) gives $e^{-\lambda _{K+1}T}\leq \frac{1}{2}e^{-\gamma T}$, and
the choice of $\varepsilon \in \left( 0,1\right) $ (see (\ref{5.2})) gives
3e^{\left\Vert V\right\Vert _{\infty }T}e^{\lambda _{K}T/2}\left( 1+K\right)
\varepsilon =\frac{1}{2}e^{-\gamma T}$, which implies the desired estimate
for $\left\Vert y\left( L_{n+1}\right) \right\Vert $, that is $\left\Vert
y\left( L_{n+1}\right) \right\Vert \leq e^{-\gamma T}\left\Vert y\left(
L_{n}\right) \right\Vert $.
\noindent \textit{Step 8:} We treat the boundedness of $\mathcal{F}$ as
follows
\begin{equation*}
\begin{array}{ll}
\left\Vert \mathcal{F}\left( w\right) \right\Vert _{\omega _{2}}^{2} &
\displaystyle\int_{\omega _{2}}\left\vert \displaystyle\sum_{j=1,\cdot \cdot
,K}e^{\lambda _{j}T/2}\langle g_{j},w\rangle _{\omega _{1}}f_{j}\left(
x\right) \right\vert ^{2}dx \\
& \leq \displaystyle\sum_{j=1,\cdot \cdot ,K}\left\vert e^{\lambda
_{j}T/2}\langle g_{j},w\rangle _{\omega _{1}}\right\vert ^{2}\displaystyl
\sum_{j=1,\cdot \cdot ,K}\displaystyle\int_{\omega _{2}}\left\vert
f_{j}\left( x\right) \right\vert ^{2}dx \\
& \leq \left\Vert w\right\Vert _{\omega _{1}}^{2}e^{\lambda _{K}T
\displaystyle\sum_{j=1,\cdot \cdot ,K}\left\Vert g_{j}\right\Vert _{\omega
_{1}}^{2}\displaystyle\sum_{j=1,\cdot \cdot ,K}\left\Vert f_{j}\right\Vert
_{\omega _{2}}^{2} \\
& \leq \left\Vert w\right\Vert _{\omega _{1}}^{2}e^{\lambda _{K}T}K^{2}\left[
\mathcal{C}_{\varepsilon }\left( T/4,T/4\right) \mathcal{C}_{\varepsilon
}\left( T/4,3T/4\right) \right] ^{2}\text{
\end{array
\end{equation*
which implies
\begin{equation*}
\left\Vert \mathcal{F}\right\Vert _{\mathcal{L}(L^{2}(\omega
_{1}),L^{2}(\omega _{2}))}\leq e^{\lambda _{K}T/2}K\mathcal{C}_{\varepsilon
}\left( T/4,T/4\right) \mathcal{C}_{\varepsilon }\left( T/4,3T/4\right)
\text{ .}
\end{equation*}
\noindent \textit{Step 9:} We estimate $1/\varepsilon $ and $\mathcal{C
_{\varepsilon }$:
By the Weyl's asymptotic law for the Dirichlet eigenvalues $\alpha _{j}$,
there is a constant $\overline{C}>0$ (depending only on $\Omega $ and $d$)
such that for any $\mu >0$
\begin{equation*}
\text{card}\left\{ j\in \mathbb{N},\text{ }\alpha _{j}<\mu \right\} \leq
\overline{C}\left( 1+\mu ^{d/2}\right) \text{ .}
\end{equation*
By the min-max formula, one has
\begin{equation*}
-\left\Vert V\right\Vert _{\infty }+\alpha _{j}\leq \lambda _{j}\leq \alpha
_{j}+\left\Vert V\right\Vert _{\infty }\text{ . }
\end{equation*
One deduces that there is a constant $\overline{C}>0$ (depending only on
\Omega $ and $d$) such that for any $\gamma >0$
\begin{equation*}
K:=\text{card}\left\{ j\in \mathbb{N},\text{ }\lambda _{j}<\gamma +\frac
\text{ln}2}{T}\right\} \leq \overline{C}\left( 1+\left\Vert V\right\Vert
_{\infty }^{d/2}+\left( \frac{\text{ln}2}{T}\right) ^{d/2}+\gamma
^{d/2}\right) \text{ .}
\end{equation*
Further, for some constant $\overline{C}>0$ (depending only on $\Omega $ and
$d$), we have
\begin{equation*}
\frac{1}{\varepsilon }:=6e^{\gamma T}e^{\left\Vert V\right\Vert _{\infty
}T}e^{\lambda _{K}T/2}\left( 1+K\right) \leq \overline{C}\left( \gamma
^{d/2}+\left\Vert V\right\Vert _{\infty }^{d/2}+\frac{1}{T^{d/2}}\right)
e^{\left\Vert V\right\Vert _{\infty }T}e^{2\gamma T}\text{ .}
\end{equation*
We finish the proof by gathering the previous estimates with the definition
of $\mathcal{C}_{\varepsilon }$, that is $\mathcal{C}_{\varepsilon }\left(
t,s\right) :=e^{4s\left\Vert V\right\Vert _{\infty }}e^{c\left( 1+\frac{1}{t
+t\left\Vert V\right\Vert _{\infty }+\left\Vert V\right\Vert _{\infty
}^{2/3}\right) }$exp$\left( \sqrt{\frac{c}{t}\text{ln}^{+}\frac{1}
\varepsilon }}\right) $.
Hence, we complete the proof of Theorem~\ref{theorem5.1}.
\bigskip
\bigskip
\bigskip
\bigskip
|
1,108,101,563,273 | arxiv | \section{Introduction}
\label{Sec:Introduction}
Collisions between heavy ions accelerated at ultra-relativistic
energies provide the necessary conditions to form a deconfined
state of matter, the Quark Gluon Plasma~\cite{Karsch:2003jg}.
In this phase, the fundamental constituents of quantum
chromodynamics (QCD), the quarks and gluons, are not anymore
confined inside their usual hadronic bags. The transition to
a QGP from normal hadronic matter is expected to take place
at a temperature of about 155~MeV, and an energy density of
about 0.5 GeV/fm$^3$, according to lattice QCD calculations
\cite{Bazavov:2009zn,Bazavov:2011nk,Borsanyi:2010cj}. These
conditions can be reached in collisions between Pb ions at the
Large Hadron Collider
(LHC)~\cite{Chatrchyan:2012mb,Adam:2015lda,Adam:2016thv}.
Heavy ion collisions also provide the possibility to study novel
QCD phenomena that are otherwise not accessible experimentally.
One characteristic example is related to local parity (P) as well as charge conjugation and parity (CP) symmetry violation in strong
interactions. The possibility to observe parity violation in the
strong interaction using relativistic heavy-ion collisions has
been discussed in~\cite{Lee:1973iz,Lee:1974ma,Morley:1983wr} and
was further
reviewed in~\cite{Kharzeev:1998kz,Kharzeev:1999cz,Kharzeev:2015kna,Kharzeev:2007tn,Kharzeev:2007jp,Kharzeev:2015znc,Li:2020dwr,Kharzeev:2020jxw}.
In QCD, this symmetry violation originates from the interaction
between the chiral fermions of the theory and topologically
non-trivial gluonic fields that induce net-chirality. In the
presence of a strong magnetic field, such as the one created in
peripheral heavy ion collisions with a magnitude of around
$10^{15}$~Tesla~\cite{Skokov:2009qp,Bzdak:2011yy,Deng:2012pc},
these interactions lead to an asymmetry between left and right-handed
quarks. The generated net-chirality, in turns, leads to an excess
of positively and negatively charged particles moving in opposite
directions relative to the system's symmetry plane. This introduces
an electromagnetic current and the creation of an electric dipole
moment of QCD matter. The experimental search for these effects
has intensified recently, following the realisation that the
subsequent creation of charged hadrons results in an experimentally
accessible magnitude of charge separation along the direction of
this magnetic field, and perpendicular to the symmetry plane.
This phenomenon is called the Chiral Magnetic Effect (CME)~\cite{Fukushima:2008xe}
and its existence was recently reported in semi-metals like zirconium
pentatelluride ($ZrTe_5$)~\cite{Li:2014bha}.
Early enough it was realised that a way to probe these effects
is to rely on measuring two-particle azimuthal correlations
relative to the reaction plane ($\Psi_{\mathrm{RP}}$)~\cite{Voloshin:2004vk},
the plane defined by the impact parameter and the beam axis.
Since then, intensive experimental efforts have been made to
identify unambiguously signals of the CME. The first measurements
using this approach were reported by the STAR Collaboration
in Au--Au collisions at $\sqrt{s_{\mathrm{NN}}} = 0.2$~TeV~\cite{Abelev:2009ac,Abelev:2009ad}
and were consistent with initial expectations for a charge
separation relative to the reaction plane due to the CME. Soon
after, the first results from the LHC in Pb--Pb collisions at
$\sqrt{s_{\mathrm{NN}}} = 2.76$~TeV~were reported and showed a
quantitatively similar effect~\cite{Abelev:2012pa}. This agreement
between the results is up until this moment hard to comprehend
considering the differences in the centre-of-mass energy and
consequently in the multiplicity density~\cite{Aamodt:2010cz}.
In addition, the magnetic field and the way it evolves is, in
principle, different between the two
energies~\cite{Skokov:2009qp,Bzdak:2011yy,Deng:2012pc}. Overall,
this agreement hinted at the dominant role of background effects
in both measurements. These background effects were, in parallel,
identified as coming from local charge conservation coupled
to the anisotropic expansion of the system in non-central
collisions~\cite{Schlichting:2010qia,Pratt:2010zn}. The field
turned its focus to finding a way to constrain and quantify the
background and the CME contribution to such measurements.
In Ref.~\cite{Acharya:2017fau}, the ALICE Collaboration presented
the first ever upper limit of 26--33$\%$ at $95\%$ confidence level
for the CME contribution, using an Event Shape Engineering (ESE)
technique~\cite{Schukraft:2012ah}. In parallel, new measurements
of the STAR Collaboration in Au--Au collisions at a centre-of-mass
energy $\sqrt{s_{\mathrm{NN}}} = 200$~GeV~\cite{Adamczyk:2013kcb,Adamczyk:2013hsi,Adamczyk:2014mzf}
as well as results obtained from the analysis of data collected
from the beam energy scan at $\sqrt{s_{\mathrm{NN}}} =$7.7, 11.5,
19.6, 27, 39 and 62.4~GeV~\cite{Adamczyk:2014mzf} were still qualitatively
consistent with expectations from parity violating effects
in heavy ion collisions. To study background effects the
CMS~\cite{Khachatryan:2016got} and the STAR~\cite{STAR:2019xzd}
collaborations studied charge dependent correlations in both p--Pb
collisions at $\sqrt{s_{\mathrm{NN}}} = 5.02$~TeV and in p--Au and
d--Au collisions at $\sqrt{s_{\mathrm{NN}}} = 0.2$~TeV, respectively.
Both results illustrate that these correlations are similar to
those measured in heavy-ion collisions. The authors concluded that these findings could have important
implications for the interpretation of the heavy-ion data since it
is expected that the results in these ``small'' systems are
dominated by background effects. However these latter studies are lacking
a quantitative estimate of the reaction plane independent
background~\cite{Abelev:2009ac,Abelev:2009ad} and therefore should
not be used to extract a definite conclusion. Finally, the ALICE
Collaboration recently reported their updated upper limits of 15--18$\%$
at 95$\%$ confidence level for the centrality interval 0--40$\%$ by
studying charge dependent correlations relative to the third order
symmetry plane ~\cite{Acharya:2020rlz}. Overall, the extraction of
the CME signal has been exceptionally challenging.
In this article we follow a different approach by performing a
systematic study of the correlators used in CME searches for
Pb--Pb and Xe--Xe collisions at $\sqrt{s_{\mathrm{NN}}} = 5.02$~TeV
(for Pb ions)~\cite{Abelev:2012pa,Acharya:2020rlz} and at
$\sqrt{s_{\mathrm{NN}}} = 5.44$~TeV (for Xe ions)~\cite{Aziz:2020nia}
with the Anomalous-Viscous Fluid Dynamics (AVFD) framework~\cite{Shi:2017cpu,Jiang:2016wve,Shi:2019wzi}.
This is a state-of-the-art model that describes the initial state
of the collision using a Glauber prescription, and accounts for the
development of the early stage electromagnetic fields as well as for
the propagation of anomalous fermion currents. The expanding medium
is treated via a 2+1 dimensional viscous hydrodynamics (VISH2+1)
code which is coupled to a hadron cascade model
(UrQMD)~\cite{Bass:1998ca}. The goal of this study is to extract
the relevant values that govern the CME signal and the background
in the AVFD model that will allow for a quantitative description
of the centrality dependence of the charged dependent correlations
measured in various colliding systems and energies at the LHC.
The article is organised as follows: Section~\ref{Sec:Observables}
presents the main observables, followed by a discussion on how the
model is calibrated in Section~\ref{Sec:Model}. The main results are
presented in Section~\ref{Sec:Results}. The article concludes with
a summary.
\section{Model calibration and parametrisation}
\label{Sec:Model}
The goal of this study is to extract the values that control
the CME signal and the background in the AVFD model that will
allow for a quantitative simultaneous description of the centrality
dependence of the charged dependent correlations, i.e.
$\Delta \delta_1$ and $\Delta \gamma_{1,1}$ measured in Pb--Pb
collisions at
$\sqrt{s_{\mathrm{NN}}} = 5.02$~TeV~\cite{Abelev:2012pa,Acharya:2020rlz}
and in Xe--Xe collisions at $\sqrt{s_{\mathrm{NN}}} = 5.44$~TeV~\cite{Aziz:2020nia}.
Here, $\Delta \delta_1$ and $\Delta \gamma_{1,1}$ denote the difference
of $\delta_1$ and $\gamma_{1,1}$ between opposite- and same-sign pairs.
Within the AVFD framework, the CME signal is controlled by the axial
current density $n_5/\mathrm{s}$ which dictates the imbalance between
right- and left-handed fermions induced in the initial stage of each
event. The parameter that governs the background is represented by the
percentage of local charge conservation (LCC) within an event. This
number can be considered as the amount of positive and negative charged
partners emitted from the same fluid element relative to the total
multiplicity of the event.
The first step in the whole procedure was to calibrate the model
without the inclusion of any CME or LCC effects, in what will be
referred to in the rest of the text as ``baseline''. This involved
tuning the input parameters to describe the centrality dependence
of bulk measurements, such as the charged particle multiplicity density
$dN/d\eta$~\cite{Abbas:2013bpa,Adam:2015ptt,Acharya:2018hhy} and
$v_2$~\cite{Aamodt:2010pa,Adam:2016izf,Acharya:2018ihu} in Pb--Pb
and Xe--Xe collisions at various LHC energies. Overall the model
was able to describe the experimental measurements within 15\%.
Finally, we also checked that the slopes of the transverse momentum
($p_T$) spectra of pions, kaons and protons, in the baseline sample
of AVFD have a similar centrality dependence as the one reported by
ALICE in Refs.~\cite{Abelev:2013vea,Acharya:2019yoi,Acharya:2021ljw}.
\begin{figure}[!h]
\begin{center}
\includegraphics[width = 0.5\textwidth]{figures/BFieldVsCentrality.eps}
\end{center}
\caption{The dependence of the average value of the magnetic
field perpendicular to the reaction plane ($\textbf{B}_y$) on centrality for Pb--Pb and Xe--Xe collisions at
$\sqrt{s_{\mathrm{NN}}} = 5.02$~and
$\sqrt{s_{\mathrm{NN}}} = 5.44$~TeV, respectively.}
\label{fig:BFieldVsCentrality}
\end{figure}
One of the key ingredients in the development of the CME in the final
state, is the early stage electromagnetic field. The
AVFD model performs an event-by-event simulation of the electromagnetic
field value projected along the symmetry plane, accounting for the
decorrelation between the field direction and the true reaction plane
due to fluctuations~\cite{Shi:2017cpu}. The initial strength of this
field mainly depends on the atomic number of the nuclei that collide
and the center-of-mass energy of the collision. Figure~\ref{fig:BFieldVsCentrality}
presents the centrality dependence of the magnitude of $\textbf{B}$,
as simulated by AVFD at $t_0 = 0$, for Pb--Pb and Xe--Xe collisions at
$\sqrt{s_{\mathrm{NN}}} = 5.02$~and $\sqrt{s_{\mathrm{NN}}} = 5.44$~TeV,
respectively. The values for both systems reach and for some centralities
even exceed $10^{16}$~T. In addition, the magnitude of $\textbf{B}$ for
a given centrality interval in collisions between Pb-ions is larger than
the corresponding value in Xe--Xe collisions by a factor which reflects
the ratio of the atomic numbers of the two nuclei.
\begin{figure}[!h]
\begin{center}
\includegraphics[width = 0.5\textwidth]{figures/BFieldVsTime.eps}
\end{center}
\caption{The time evolution of the average value of the magnetic field
perpendicular to the reaction plane ($\textbf{B}$) for the 40\%-50\%
centrality interval in Pb--Pb and Xe--Xe collisions at
$\sqrt{s_{\mathrm{NN}}} = 5.02$~and $\sqrt{s_{\mathrm{NN}}} = 5.44$~TeV,
respectively.}
\label{fig:BFieldVsTime}
\end{figure}
The magnitude of the field evolves as a function of time in the model
according to
\begin{equation}
B(\tau,x)=\frac{B_0}{1+\tau^2/\tau_B^2},
\end{equation}
where $\tau_B$ is the magnetic field lifetime which is set, in this work,
conservatively to 0.2~fm/$c$, for both collision systems. Figure~\ref{fig:BFieldVsTime}
presents the time evolution of the magnitude of $\textbf{B}$ for an
indicative centrality interval i.e. 40-50\% for both Pb--Pb (solid
line) and Xe--Xe collisions (dashed line).
The next step in the calibration of the model required extracting
the dependence of the correlators $\Delta \gamma_{1,1}$ based on
Eq.~\ref{Eq:3ParticleCorrelator} and $\Delta \delta_1$ (see
Eq.~\ref{Eq:2ParticleCorrelator}) on both the axial current density
$n_5/\mathrm{s}$ and the percentage of LCC. For this, new AVFD
samples were produced for all centralities of both systems and
energies, for which the amount of CME induced signal was
incremented i.e., using $n_5/\mathrm{s}$ = 0.05, 0.07 and 0.1,
while at the same time keeping the percentage of LCC fixed at zero.
In addition, to gauge the dependence of both correlators on the
background, similar number of events as before were produced where,
this time, $n_5/\mathrm{s}$ was fixed at zero but the percentage
of LCC was incremented every time. In particular, the values selected for the Pb-system were 33 and 50\%\footnote{Other values of LCC percentage were also checked, but due to technical reasons related to computing resources, not for all centrality intervals}.
\begin{figure}[!h]
\begin{center}
\includegraphics[width = 0.5\textwidth]{figures/centralityDependenceAVFD-PbPb.eps}
\end{center}
\caption{The centrality dependence of $\Delta \delta_1$ (upper
panel) and $\Delta \gamma_{1,1}$ (lower panel), the charge dependent
difference of $\delta_1$ and $\gamma_{1,1}$ between opposite-
and same-sign pairs, in Pb--Pb collisions at $\sqrt{s_{\mathrm{NN}}} = 5.02$~TeV.
The results from the analysis of the baseline sample are
represented by the green markers. The various bands show the
AVFD expectations for $\Delta \delta_1$ and $\Delta \gamma_{1,1}$
for various values of $n_5/\mathrm{s}$ (red bands) and percentage
of LCC (blue bands).}
\label{fig:ModelCalibPbPb}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width = 0.5\textwidth]{figures/centralityDependenceAVFD-XeXe.eps}
\end{center}
\caption{The centrality dependence of $\Delta \delta_1$ (upper
panel) and $\Delta \gamma_{1,1}$ (lower panel), the charge dependent
difference of $\delta_1$ and $\gamma_{1,1}$ between opposite-
and same-sign pairs, in Xe--Xe collisions at $\sqrt{s_{\mathrm{NN}}} = 5.44$~TeV.
The results from the analysis of the baseline sample
are represented by the green markers. The various bands show the
AVFD expectations for $\Delta \delta_1$ and $\Delta \gamma_{1,1}$
for various values of $n_5/\mathrm{s}$ (red bands) and percentage
of LCC (blue bands).}
\label{fig:ModelCalibXeXe}
\end{figure}
Figure~\ref{fig:ModelCalibPbPb} presents the centrality dependence
of $\Delta \delta_1$ and $\Delta \gamma_{1,1}$,
in the upper and lower panels, respectively. The plots show results
obtained from the analysis of events of Pb--Pb collisions at
$\sqrt{s_{\mathrm{NN}}} = 5.02$~TeV.
The green markers are extracted from the analysis of the baseline
sample and, in both cases, exhibit non-zero values for the majority
of the centrality intervals. These non-zero values are due to the
existence of hadronic resonances in the model whose decay products
are affected by both radial and elliptic flow. In addition, the same
plots present how the magnitude of these correlators develop for
various values of the axial current density $n_5/\mathrm{s}$ which
are represented by the red bands. It can be seen that with increasing
values of $n_5/\mathrm{s}$ the two correlators exhibit opposite
trends: while $\Delta \delta_1$ decreases, the values of
$\Delta \gamma_{1,1}$ increase. This opposite behaviour originates
from the different sign the CME contributes to $\delta_1$ (Eq.~\ref{Eq:2ParticleCorrelator}) and
$\gamma_{1,1}$ (Eq.~\ref{Eq:3ParticleCorrelator}) and, consequently, to $\Delta \delta_1$ and
$\Delta \gamma_{1,1}$. Finally, when fixing
the value of $n_5/\mathrm{s}$ to zero and progressively increasing
the percentage
of LCC in the sample (black curves in fig.~\ref{fig:ModelCalibPbPb}),
the values of both $\Delta \delta_1$ and $\Delta \gamma_{1,1}$
increase. However, the latter correlator exhibits a smaller sensitivity
than $\Delta \delta_1$ to the background owning to the fact that
it is constructed as the difference in the magnitude of background
effects in- and out-of-plane (see Eq.~\ref{Eq:3ParticleCorrelator}).
Similarly, fig.~\ref{fig:ModelCalibXeXe} presents the centrality
dependence of $\Delta \delta_1$ and $\Delta \gamma_{1,1}$, this
time in Xe--Xe collisions at $\sqrt{s_{\mathrm{NN}}} = 5.44$~TeV.
Also here the results for the baseline AVFD sample are represented
with the green markers, while the red and black bands correspond
to samples with progressively increasing values of $n_5/\mathrm{s}$
and percentage of LCC, respectively. The same qualitative observations
are also found in this system: the baseline sample exhibits non-zero
values for both $\Delta \delta_1$ and $\Delta \gamma_{1,1}$, these
two correlators have opposite trends with increasing $n_5/\mathrm{s}$
and $\Delta \delta_1$ exhibits bigger sensitivity on the LCC percentage
than $\Delta \gamma_{1,1}$.
\begin{figure}[!h]
\begin{center}
\includegraphics[width = 0.5\textwidth]{figures/deltaDeltaAVFD.eps}
\end{center}
\caption{The centrality dependence of $\Delta \delta_1$ grouped in two scenarios: zero $n_5/\mathrm{s}$ but various choices of LCC (upper panel) and non-zero $n_5/\mathrm{s}$ but LCC fixed to zero (lower panel). The various bands show the AVFD expectations for $\Delta \delta_1$
for Pb--Pb collisions and Xe--Xe collisions, with blue and green bands, respectively. The results of the baseline sample are represented by the filled and open markers.}
\label{fig:CompareDeltaD11Pb5andXe}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width = 0.5\textwidth]{figures/deltaGammaAVFD.eps}
\end{center}
\caption{The centrality dependence of $\Delta \gamma_{1,1}$ grouped in two scenarios: zero $n_5/\mathrm{s}$ but various choices of LCC (upper panel) and non-zero $n_5/\mathrm{s}$ but LCC fixed to zero (lower panel). The various bands show the AVFD expectations for $\Delta \delta_1$
for Pb--Pb collisions and Xe--Xe collisions, with blue and green bands, respectively. The results of the baseline sample are represented by the filled and open markers.}
\label{fig:CompareDeltaG112Pb5andXe}
\end{figure}
To directly compare the values of $\Delta\delta_1$ and $\Delta\gamma_{1,1}$ between these two collision systems, the results are organised based on the input parameters used: zero $n_5/\mathrm{s}$ but various choices of LCC and non-zero $n_5/\mathrm{s}$ but LCC fixed to zero.
Figures~\ref{fig:CompareDeltaD11Pb5andXe} and~\ref{fig:CompareDeltaG112Pb5andXe}, summarize the centrality dependence of the results for $\Delta\delta_1$ and $\Delta\gamma_{1,1}$.
In the first case, the baseline and LCC being 15\% and 50\% for Pb--Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV and for Xe--Xe collisions at $\sqrt{s_{NN}}=5.44$ TeV are chosen. The upper panel of fig.~\ref{fig:CompareDeltaD11Pb5andXe} illustrates that for a fixed LCC percentage, the values of $\Delta\delta_1$ are higher for the Xe--Xe than for the Pb--Pb samples. For a fixed centrality, while the effect of radial flow between these two systems is similar~\cite{Acharya:2021ljw}, the charged particle multiplicity in Pb--Pb is 60--70\% higher than the corresponding value in Xe--Xe collisions~\cite{Adam:2015ptt,Acharya:2018hhy}. This could lead to a faster dilution of the correlations induced by the LCC mechanism in the larger system, reflected in this difference of $\Delta\delta_1$. At the same time, the upper panel of fig.~\ref{fig:CompareDeltaG112Pb5andXe} shows that the values of $\Delta\gamma_{1,1}$ for the two systems do not exhibit any significant difference. This is in line with the expectation that the sensitivity of $\Delta\gamma_{1,1}$ to the background is significantly reduced with respect to $\Delta \delta_1$.
In the second case of non-zero axial current density, the samples containing $n_5/\mathrm{s}=0.05, 0.07$ and $0.10$ are chosen. The lower panel of fig.~\ref{fig:CompareDeltaD11Pb5andXe} shows that $\Delta\delta_1$ is similar between the two systems since it is primarily affected by background contributions. This correlator needs higher values of $n_5/\mathrm{s}$ (e.g. $n_5/\mathrm{s} = 0.1$ in the plot) to start observing some differences. Finally, the lower panel of fig.~\ref{fig:CompareDeltaG112Pb5andXe} illustrates that the magnitude of $\Delta\gamma_{1,1}$ is higher in the Xe--Xe than in the Pb--Pb samples. Although the value of the magnetic field is higher for the larger Pb-system, as shown in fig.~\ref{fig:BFieldVsCentrality}, the significantly larger multiplicity that this system has, leads to a larger dilution effect reflected in the ordering of the corresponding curves in the plot.
\begin{figure}[!h]
\begin{center}
\includegraphics[width = 0.5\textwidth]{figures/centrality40To50AVFDLCC.eps}
\end{center}
\caption{The dependence of $\Delta \delta_1$ and $\Delta \gamma_{1,1}$
in the upper and lower panels, respectively, on the percentage of local
charge conservation in the analysed samples of Pb--Pb collisions at
$\sqrt{s_{\mathrm{NN}}} = 5.02$~TeV for the 40\%-50\% centrality interval.}
\label{fig:ResultsVsLCC}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width = 0.5\textwidth]{figures/centrality40To50AVFDCME.eps}
\end{center}
\caption{The dependence of $\Delta \delta_1$ and $\Delta \gamma_{1,1}$
in the upper and lower panels, respectively, on the axial current density
$n_5/\mathrm{s}$ in the analysed samples of Pb--Pb collisions at
$\sqrt{s_{\mathrm{NN}}} = 5.02$~TeV for the 40\%-50\% centrality interval.}
\label{fig:ResultsVsCME}
\end{figure}
The previous results for each colliding system and energy can
be grouped in a different way that allows to parametrise the
dependence of each of the correlators on the LCC percentage
and on $n_5/\mathrm{s}$. Figures~\ref{fig:ResultsVsLCC}
and~\ref{fig:ResultsVsCME} present how $\Delta \delta_1$ and
$\Delta \gamma_{1,1}$ develop as a function of the LCC percentage
and $n_5/\mathrm{s}$, respectively. Results for the 40--50\%
centrality interval of Pb--Pb collisions at
$\sqrt{s_{\mathrm{NN}}} = 5.02$~TeV are indicatively chosen to
illustrate the procedure. An identical protocol was used for all
centrality intervals of both colliding systems. One can see that
both $\Delta \delta_1$ and $\Delta \gamma_{1,1}$ exhibit a linear
dependence on the percentage of LCC, with the latter being less
sensitive and thus having a smaller slope. Finally, these two
correlators exhibit a quadratic dependence on $n_5/\mathrm{s}$
with opposite trend, originating from the dependence of $\delta_1$
and $\gamma_{1,1}$ on $\langle a_{1,\alpha} a_{1,\beta}\rangle$ and
$-\langle a_{1,\alpha} a_{1,\beta}\rangle$ in Eq.~\ref{Eq:2ParticleCorrelator}
and Eq.~\ref{Eq:3ParticleCorrelator}, respectively. This $a_1$
coefficient, in turns, has been shown in Ref.~\cite{Shi:2017cpu,Jiang:2016wve}
to be proportional to the value of $n_5/\mathrm{s}$.
Following this procedure for all centrality intervals of Pb--Pb and
Xe--Xe collisions, one is able to parametrise the dependence of
$\Delta \delta_1$ and $\Delta \gamma_{1,1}$ according to:
\begin{equation}
\Delta \delta_1 = c_2 \cdot (n_5/\mathrm{s})^2 + c_1 \cdot (n_5/\mathrm{s}) + b_1 \cdot (\mathrm{LCC}) + b_0,
\label{Eq:DeltaParametrisation}
\end{equation}
\begin{equation}
\Delta \gamma_{1,1} = e_2 \cdot (n_5/\mathrm{s})^2 + e_1 \cdot (n_5/\mathrm{s}) + d_1 \cdot (\mathrm{LCC}) + d_0,
\label{Eq:GammaParametrisation}
\end{equation}
\noindent where $e_2$, $e_1$, $d_1$, $d_0$, $c_2$, $c_1$, $b_1$ and $b_0$
are real numbers constrained from the simultaneous fit of the corresponding
dependencies of $\Delta \delta_1$ and $\Delta \gamma_{1,1}$ for each centrality
interval of every collision system and energy. The parametrisation of
Eq.~\ref{Eq:DeltaParametrisation} and Eq.~\ref{Eq:GammaParametrisation}
assumes that the two components that control the CME signal and the background
are not correlated. This is a reasonable assumption considering that the
two underlying physical mechanism are independent and take place at different
times in the evolution of a heavy ion collision.
\section{Experimental observables}
\label{Sec:Observables}
A way to probe the parity violating effects is by introducing P-odd
coefficients $a_{n,\alpha}$ in the Fourier series frequently used in
studies of azimuthal anisotropy~\cite{Voloshin:1994mz}. This leads to
the expression
\begin{equation}
\frac{dN}{d\varphi} \approx 1 + 2\sum_{n} \Big[v_n \cos[n(\varphi - \Psi_n)] + a_n \sin[n(\varphi - \Psi_n)]\Big]
\label{Eq:Fourier}
\end{equation}
\noindent where $N$ is the number of particles, $\varphi$ is the
azimuthal angle of the particle and $v_n$ are the corresponding
flow coefficients ($v_1$: directed flow, $v_2$: elliptic flow,
$v_3$: triangular flow etc.). The n-th order symmetry plane of
the system, $\Psi_n$, is introduced to take into account that
the overlap region of the colliding nuclei exhibits an irregular
shape~\cite{Manly:2005zy,Bhalerao:2006tp,Alver:2008zza,Alver:2010gr,Alver:2010dn}. This originates from the initial density profile of
nucleons participating in the collision, which is not isotropic
and differs from one event to the other. In case of a smooth
distribution of matter produced in the overlap zone, the angle
$\Psi_{n}$ coincides with that of the reaction plane,
$\Psi_{\rm RP}$. In Eq.~\ref{Eq:Fourier}, $a_1$ is the leading
order P-odd term that reflects the magnitude, while higher
harmonics (i.e. $a_2$ and above) represent the specific shape
of the CME signal.
In Ref.~\cite{Voloshin:2004vk}, Voloshin proposed that the
leading order P-odd coefficient can be probed through the
study of charge-dependent two-particle correlations relative
to the reaction plane $\Psi_{\rm RP}$. In particular, the
expression discussed is of the form
$\langle \cos(\varphi_{\alpha} + \varphi_{\beta} - 2\Psi_{\rm RP}) \rangle$, where $\alpha$ and $\beta$ denote particles with
the same or opposite charge. This expression can probe
correlations between the leading P-odd terms for different
charge combinations $\langle a_{1,\alpha} a_{1,\beta} \rangle$.
This can be seen if one expands the correlator using
Eq.~\ref{Eq:Fourier} according to
\[\langle \cos(\varphi_{\alpha} + \varphi_{\beta} - 2\Psi_{\rm RP}) \rangle = \]
\[ \langle \cos\big[(\varphi_{\alpha} - \Psi_{\rm RP}) +
(\varphi_{\beta} - \Psi_{\rm RP})\big] \rangle =
\langle \cos(\Delta \varphi_{\alpha} + \Delta \varphi_{\beta}) \rangle = \]
\[\langle \cos \Delta \varphi_{\alpha} \cos \Delta \varphi_{\beta} \rangle -
\langle \sin \Delta \varphi_{\alpha} \sin \Delta \varphi_{\beta} \rangle =
\]
\begin{equation}
\langle v_{1,\alpha}v_{1,\beta} \rangle + \mathrm{B_{in}} -
\langle a_{1,\alpha} a_{1,\beta}\rangle - \mathrm{B_{out}},
\label{Eq:3ParticleCorrelator}
\end{equation}
where $\mathrm{B_{in}}$ and $\mathrm{B_{out}}$ represent the
parity-conserving correlations projected onto the in- and
out-of-plane directions~\cite{Voloshin:2004vk}. The terms
$\langle \cos \Delta \varphi_{\alpha} \cos \Delta \varphi_{\beta}\rangle$
and $\langle \sin \Delta \varphi_{\alpha}\sin \Delta\varphi_{\beta}\rangle$
in Eq.~\ref{Eq:3ParticleCorrelator} quantify the correlations
with respect to the in- and out-of-plane directions, respectively.
The term $\langle v_{1,\alpha}v_{1,\beta} \rangle$, i.e. the product
of the first order Fourier harmonics or directed flow, is expected
to have negligible charge dependence in the mid-rapidity
region~\cite{Gursoy:2018yai}. In addition, for a symmetric collision
system the average directed flow at mid-rapidity is zero.
A generalised form of Eq.~\ref{Eq:3ParticleCorrelator},
describing also higher harmonics, is given by the mixed-harmonics
correlations and reads
\begin{eqnarray}
\label{eq:moments}
\gamma_{\mathrm{m,n}} = \langle \cos(\mathrm{m}\varphi_{\alpha} +
\mathrm{n}\varphi_{\beta} - \mathrm{(m + n)}\Psi_{\mathrm{|m+n|}})\rangle ,
\label{Eq:Generalised3ParticleCorrelator}
\end{eqnarray}
\noindent where $\mathrm{m}$ and $\mathrm{n}$ are integers.
Setting $\mathrm{m} = 1$ and $\mathrm{n} = 1$ (i.e. $\gamma_{1,1}$)
leads to Eq.~\ref{Eq:3ParticleCorrelator}.
In order to independently evaluate the contributions from
correlations in- and out-of-plane, one can also measure a
two-particle correlator of the form
\[ \langle \cos(\varphi_{\alpha} - \varphi_{\beta}) \rangle = \]
\[ \langle \cos\big[(\varphi_{\alpha} - \Psi_{\rm RP}) -
(\varphi_{\beta} - \Psi_{\rm RP})\big] \rangle =
\langle \cos(\Delta \varphi_{\alpha} - \Delta \varphi_{\beta}) \rangle= \]
\[\langle \cos \Delta \varphi_{\alpha} \cos \Delta \varphi_{\beta} \rangle +
\langle \sin \Delta \varphi_{\alpha} \sin \Delta \varphi_{\beta} \rangle =
\]
\begin{equation}
\langle v_{1,\alpha}v_{1,\beta} \rangle + \mathrm{B_{in}} +
\langle a_{1,\alpha} a_{1,\beta}\rangle + \mathrm{B_{out}},
\label{Eq:2ParticleCorrelator}
\end{equation}
\noindent This provides access to the two-particle correlations
without any dependence on the symmetry plane angle which can be
generalised according to
\begin{eqnarray}
\delta_\mathrm{m} = \langle \cos[\mathrm{m}(\varphi_{\alpha} -
\varphi_{\beta})] \rangle .
\label{Eq:Generalised2ParticleCorrelator}
\end{eqnarray}
\noindent This correlator, owing to its construction, is affected
if not dominated by background contributions. Charge-dependent
results for $\delta_1$, together with the relevant measurements of
$\gamma_{1,1}$ were first reported in Ref.~\cite{Abelev:2012pa} and
made it possible to separately quantify the magnitude of correlations
in- and out-of-plane.
\section{Results}
\label{Sec:Results}
Having the dependence of both $\Delta \delta_1$ and
$\Delta \gamma_{1,1}$ on $n_5/\mathrm{s}$ and LCC parametrised
from Eq.~\ref{Eq:DeltaParametrisation} and
Eq.~\ref{Eq:GammaParametrisation}, one can deduce the values of
these two parameters that govern the CME signal and the background
for each centrality, colliding system and energy that allows, at
the same time, for a quantitative description of the measured
centrality dependence of $\Delta \delta_1$ and $\Delta \gamma_{1,1}$
at LHC energies.
\begin{figure}[!h]
\begin{center}
\includegraphics[width = 0.5\textwidth]{figures/cmeResultsVsModelsPbPb5TeV.eps}
\end{center}
\caption{The centrality dependence of $\Delta \delta_1$ and
$\Delta \gamma_{1,1}$ in the upper and lower panels, respectively.
The data points represent the experimental measurements in Pb--Pb collisions at $\sqrt{s_{\mathrm{NN}}} = 5.02$~TeV~\cite{Acharya:2020rlz}. The green band shows the results
obtained from the tuned AVFD sample (see text for details).}
\label{fig:DataVsModel}
\end{figure}
Figure~\ref{fig:DataVsModel} presents the results of such procedure
for Pb--Pb collisions at $\sqrt{s_{\mathrm{NN}}} = 5.02$~TeV. The
data points, extracted from Ref.~\cite{Acharya:2020rlz} for both
correlators are described fairly well by the tuned model. A similarly
satisfactory description is also achieved for the results
of Xe--Xe collisions at
$\sqrt{s_{\mathrm{NN}}} = 5.44$~TeV~\cite{Aziz:2020nia}.
\begin{figure}[!h]
\begin{center}
\includegraphics[width = 0.5\textwidth]{figures/modelResult.eps}
\end{center}
\caption{The centrality dependence of the LCC percentage (upper
panel) and the axial current density $n_5/\mathrm{s}$ that allows to describe simultaneously the experimental measurements of $\Delta \delta_1$
and $\Delta \gamma_{1,1}$~\cite{Abelev:2012pa,Acharya:2020rlz,Aziz:2020nia}
in all collision systems and energy studied in this article.}
\label{fig:ModelResults}
\end{figure}
Figure~\ref{fig:ModelResults} presents the final result of the whole
procedure. The plots show the centrality dependence of the pairs of
LCC percentage (upper panel) and $n_5/\mathrm{s}$ (lower panel) that
are needed to describe with AVFD the experimental measurements of
$\Delta \delta_1$ and $\Delta \gamma_{1,1}$. The different markers
represent results for different collision systems and energies. It
can be seen that all systems can be described by large values of LCC
that range from 40$\%$ for peripheral up to around 60$\%$ for more
central Pb--Pb collisions. There is no significant difference observed
in these values among the two sets of results.
Furthermore, the lower panel of fig.~\ref{fig:ModelResults}
illustrates that there is no significant centrality dependence
of $n_5/\mathrm{s}$. However, there is a dependence on the colliding
system. More particularly, the experimental results from the
analysis of Xe--Xe collisions lead to values of $n_5/\mathrm{s}$
which are compatible with zero within the uncertainties for all
centrality intervals. A fit with a constant function results
into values of $0.011 \pm 0.005$. At the same time, the results
for Pb--Pb collisions can be described by non-zero values of axial
current densities, again for the entire centrality region studied.
The corresponding fit leads to a value of $0.034 \pm 0.003$ i.e.,
significantly above the background-only scenario.
\section{Summary}
\label{Sec:Summary}
In this article we presented a systematic study of charge
dependent azimuthal correlations which are commonly used
experimentally to probe the Chiral Magnetic Effect using
the Anomalous-Viscous Fluid Dynamics
framework~\cite{Shi:2017cpu,Jiang:2016wve}. After tuning
the model to reproduce, within 15\%, basic experimental
measurements such
as the centrality dependence of the charged particle multiplicity
density and the elliptic flow we were able to parametrise
the dependence of both $\Delta \delta_1$ and $\Delta \gamma_{1,1}$
on the LCC percentage, the main contribution to the background,
and the axial current density $n_5/\mathrm{s}$ which dictates
the amount of CME signal. This procedure was followed for Pb--Pb
collisions at $\sqrt{s_{\mathrm{NN}}} = 5.02$~TeV, as well as
for Xe--Xe collisions at $\sqrt{s_{\mathrm{NN}}} = 5.44$~TeV.
This parametrisation allowed for the estimation of the values
of both the LCC percentage and $n_5/\mathrm{s}$ needed to describe
quantitatively and at the same time the centrality dependence
of $\Delta \delta_1$ and $\Delta \gamma_{1,1}$ measured
experimentally~\cite{Abelev:2012pa,Acharya:2020rlz,Aziz:2020nia}.
The measurements in Xe--Xe are consistent with a background only
scenario, with values of $n_5/\mathrm{s}$ compatible with zero.
On the other hand, the results of Pb--Pb collisions require
$n_5/\mathrm{s}$ with significantly non-zero values.
|
1,108,101,563,274 | arxiv | \section{MODELLING OF MULTI-ROTOR UAV}
The rigid body dynamics of the multi-rotor are given by
\begin{equation}
\label{eq:dyn_full}
\left\{
\begin{array}{lr}
m\ddot{\mathbf{X}}=\mathbf{F}+m\mathbf{g}=R(\mathbf{q})\mathbf{T}_t+m\mathbf{g}\\
J\dot{\mathbf{\Omega}}=\mathbf{T}_r-\mathbf{\Omega}\times J\mathbf{\Omega} \;,
\end{array}
\right.
\end{equation}
where $m$ is the mass of the multi-rotor, $\mathbf{X}=[x\ y\ z]^T\in \mathbb{R}^{3\times 1}$ is the position in the earth fixed frame, $\mathbf{F}=[F_x\ F_y\ F_z]^T\in\mathbb{R}^{3\times 1}$ is the three-dimensional translational force vector generated by the multi-rotor, $R(\mathbf{q})$ is the rotation matrix from the body frame to earth fixed frame, $\mathbf{q}=[\phi\ \theta\ \psi]^T\in\mathbb{R}^{3\times 1}$ is an attitude of the multi-rotor in the earth fixed frame, $\mathbf{T}_t=[0\ 0\ -T_t]^T\in\mathbb{R}^{3\times 1}$ is the thrust force vector in the body frame, $T_t\in\mathbb{R}$ is the magnitude of the total thrust, and $\mathbf{g}=[0\ 0\ g]^T\in\mathbb{R}^{3\times1}$ is a gravity vector. The parameter $J\in\mathbb{R}^{3\times3}$ is the moment of inertia (MOI) of the multi-rotor, $\mathbf{\Omega}=[p\ q\ r]^T\in\mathbb{R}^{3\times1}$ is an angular velocity vector defined in the body frame, and $\mathbf{T}_r=[\tau_r\ \tau_p\ \tau_y]^T\in\mathbb{R}^{3\times 1}$ is an attitude control torque vector. For attitude dynamics, simplified dynamics of
\begin{equation}
\label{eq:dyn_rot_simplify}
J\ddot{\mathbf{q}}=\mathbf{T}_r
\end{equation}
is commonly used, taking into account the small operation range of roll and pitch angle of multi-rotor and negligible Coriolis term \cite{mUAVDynamics,DOB_star_2,proof_of_small_gyroscopic}.
\section{TRANSLATIONAL FORCE/ACCELERATION CONTROL}
In order to control the translational force/acceleration of the multi-rotor, we need to convert the target acceleration $\ddot{\mathbf{X}}_d$ into the target attitude $\mathbf{q}_d$ and the target thrust $T_{t,d}$. Throughout this paper, notation $(*)_d$ denotes the desired value of the variable $*$. Also, we assume that the yaw $\psi$ of $\mathbf{q}$ always remains zero through a well-behaved independent controller to simplify the discussion. Now, we define $\mathbf{r}=[\theta\ \phi\ T_t]^T\in\mathbb{R}^{3\times1}$ as a set of states that needs to be controlled for generating the desired translational acceleration of the multi-rotor.
Once we choose $\mathbf{r} =[\theta\ \phi\ T_{t}]^T$ as a set of state variables to control the translational force/acceleration of multi-rotor, our next task should be finding a way to convert the desired acceleration $\ddot{\mathbf{X}}_d$ to $\mathbf{r}_d$. To figure out how to convert the signal, let us first investigate the relationship between $\mathbf{r}$ and $\ddot{\mathbf{X}}$.
\subsection{Relationship between $\mathbf{r}$ and $\tilde{\ddot{\mathbf{X}}}$}
In Equation (\ref{eq:dyn_full}), we have discussed the dynamics of the translational motion of multi-rotor. Going into detail, the corresponding translational dynamics are expressed as
\begin{equation}
\label{eq:dyn_trn_detail_rot}
m\ddot{\mathbf{X}}=-R(\psi)
\begin{bmatrix}
\cos{\phi}\sin{\theta}\\
-\sin{\phi}\\
\cos{\phi}\cos{\theta}
\end{bmatrix}
T_t+m\mathbf{g},
\end{equation}
where $R(\psi)\in\mathbb{R}^{3\times 3}$ is the yaw rotation matrix. Now, let us define a vector of state variables $\tilde{\ddot{\mathbf{X}}}$ named the pseudo-acceleration vector as
\begin{equation}
\label{eq:x_tilde_definition}
\tilde{\ddot{\mathbf{X}}}=
\begin{bmatrix}
\tilde{\ddot{x}}\\
\tilde{\ddot{y}}\\
\tilde{\ddot{z}}
\end{bmatrix}
=R^{-1}(\psi)
\left(\ddot{\mathbf{X}}-\mathbf{g}\right)=R^{-1}(\psi)\big({{1}\over{m}}\mathbf{F}\big).
\end{equation}
Applying Equation (\ref{eq:x_tilde_definition}) to (\ref{eq:dyn_trn_detail_rot}), we obtain the following relationship between $\mathbf{r}$ and $\tilde{\ddot{\mathbf{X}}}$:
\begin{equation}
\label{eq:accel-cmd}
m\tilde{\ddot{\mathbf{X}}}=-h(\phi,\theta)T_t=-
\begin{bmatrix}
\cos{\phi}\sin{\theta}\\
-\sin{\phi}\\
\cos{\phi}\cos{\theta}
\end{bmatrix}T_t.
\end{equation}
\subsection{Calculation of $\mathbf{r}_d$ from $\tilde{\ddot{\mathbf{X}}}_d$ considering dynamics}
From Equation (\ref{eq:accel-cmd}), we begin a discussion on how to calculate $\mathbf{r}_d$ based on $\tilde{\ddot{\mathbf{X}}}_d$. First, Equation (\ref{eq:accel-cmd}) yields the following expression on $\mathbf{r}$:
\begin{equation}
\label{eq:cmd-accel}
\mathbf{r}=
\begin{bmatrix}
\theta\\
\phi\\
T_t
\end{bmatrix}
=
\begin{bmatrix}
\arctan{\left({\tilde{\ddot{x}}}\over{\tilde{\ddot{z}}}\right)}\\
\arctan{\left(-{{\tilde{\ddot{y}}\cos{\theta}}\over{\tilde{\ddot{z}}}}\right)}\\
-{{m\tilde{\ddot{z}}}\over{\cos{\phi}\cos{\theta}}}
\end{bmatrix}.
\end{equation}
Equation (\ref{eq:cmd-accel}) represents the required states $\mathbf{r}$ to generate such translational acceleration. From this, one might try to find the input to the controller to create the desired acceleration by replacing $\tilde{\ddot{\mathbf{X}}}$ and $\mathbf{r}$ with $\tilde{\ddot{\mathbf{X}}}_d$ and $\mathbf{r}_d$, respectively, as follows.
\begin{eqnarray}
\label{eq:rdes_candidate_theta}
\theta_d&\hspace{-.1cm}=\hspace{-.1cm}&\arctan{\left({\tilde{\ddot{x}}_d}\over{\tilde{\ddot{z}}_d}\right)}
\\
\label{eq:rdes_candidate_phi}
\phi_d&\hspace{-.1cm}=\hspace{-.1cm}&
\arctan{\left(-{{\tilde{\ddot{y}}_d\cos{\theta_d}}\over{\tilde{\ddot{z}}_d}}\right)}=\arctan{\left({{\tilde{\ddot{y}}_d}\over{\sqrt{\tilde{\ddot{x}}_d^2+\tilde{\ddot{z}}_d^2}}}\right)}
\\
\label{eq:rdes_candidate_T}
T_{t,d}&\hspace{-.1cm}=\hspace{-.1cm}&-{{m\tilde{\ddot{z}}_d}\over{\cos{\phi_d}\cos{\theta_d}}}=-m\sqrt{\tilde{\ddot{x}}_d^2+\tilde{\ddot{y}}_d^2+\tilde{\ddot{z}}_d^2}
\end{eqnarray}
However, this method can severely degrade control performance when multi-rotor is larger than a certain size as we discuss below.
\begin{figure}[t]
\begin{center}
\includegraphics[width=9cm]{dynamics}
\end{center}
\vspace{-0.5cm}
\caption{A block diagram of the relationship between $\mathbf{r}_d$ and $\mathbf{r}$, where $\mathbf{r}=[\theta, \phi, T_t]^T$, and $\mathbf{u}=[\tau_r, \tau_p, \tau_y, T_t]^T$.}
\label{gp:platform_structure}
\end{figure}
Fig. \ref{gp:platform_structure} shows the internal structure between $\mathbf{r}_d$ and $\mathbf{r}$. In this figure, we can see that $\phi_d$ and $\theta_d$ are realized to $\phi$ and $\theta$ through attitude controller, rotor dynamics, and attitude dynamics. In contrast, $T_{t,d}$ only passes through the rotor dynamics to become $T_t$. Here, we treat $\mathbf{u}_d=\mathbf{u}$, where $\mathbf{u}=[\mathbf{T}_r\ T_t]^T\in\mathbb{R}^{4\times1}$, since rotor dynamics are mostly negligible. Assuming that the attitude controller is properly designed, we can model the relationship between $\mathbf{r}_d$ and $\mathbf{r}$ as the following equation:
\begin{equation}
\label{eq:iorelationship}
\mathbf{r}(t)=
\begin{bmatrix}
\theta(t)\\
\phi(t)\\
T_t(t)
\end{bmatrix}
\approx
\begin{bmatrix}
\theta_d\left(t-\gamma_\theta\right)\\
\phi_d\left(t-\gamma_\phi\right)\\
T_{t,d}(t)
\end{bmatrix}.
\end{equation}
Here, $\gamma_*\in[0,\infty)$ are time-varying non-negative delay factors. Applying Equation (\ref{eq:iorelationship}) into (\ref{eq:accel-cmd}), we have
\begin{equation}
\label{eq:dynamics_false}
m\tilde{\ddot{\mathbf{X}}}=h\Big(\phi_d\big(t-\gamma_\phi\big),\theta_d\big(t-\gamma_\theta\big)\Big)T_{t,d}(t).\\
\end{equation}
In Equation (\ref{eq:dynamics_false}), the desired attitude and total thrust are realized asynchronously due to $\gamma_\phi$ and $\gamma_\theta$. Applying Equation (\ref{eq:rdes_candidate_T}) to Equation (\ref{eq:dynamics_false}), the result is follows.
\begin{equation}
\label{eq:dynamics_false_detail}
\begin{bmatrix}
\tilde{\ddot{x}}(t)\\
\tilde{\ddot{y}}(t)\\
\tilde{\ddot{z}}(t)
\end{bmatrix}
=
\begin{bmatrix}
\left({{\cos{\phi_d(t-\gamma_\phi)}}\over{\cos{\phi_d(t)}\cos{\theta_d(t)}}}\tilde{\ddot{z}}_d(t)\right)\sin{\theta_d(t-\gamma_\theta)}\\
-\left({{1}\over{\cos{\phi_d(t)\cos{\theta_d(t)}}}}\tilde{\ddot{z}}_d(t)\right)\sin{\phi_d(t-\gamma_\phi)}\\
\left({{\cos{\phi_d(t-\gamma_\phi)\cos{\theta_d(t-\gamma_\theta)}}}\over{\cos{\phi_d(t)}\cos{\theta_d(t)}}}\right)\tilde{\ddot{z}}_d(t)
\end{bmatrix}
\end{equation}
In the $\tilde{\ddot{z}}(t)$ equation of Equation (\ref{eq:dynamics_false_detail}), the parenthesized part can continuously change if $\gamma_\phi$ and $\gamma_\theta$ are too large to be ignored.
This indicates that $z$-directional control performance can be significantly reduced if the delay between the desired and actual attitude signals becomes large, for example in situations when the MOI of the multi-rotor increases, such as large multi-rotor or multi-rotor with large cargo.
{When the Z-directional control performance degrades, a high-level controller (e.g., position controller) or the operator may need to constantly modify the $\tilde{\ddot{z}}_d$ value to correct the poor Z-directional control performance.
As a result, this degrades the X and Y direction control performance because the values in parentheses of the $\tilde{\ddot{x}}(t)$ and $\tilde{\ddot{y}}(t)$ equations in \eqref{eq:dynamics_false_detail} also constantly change.}
The decline in control performance due to this control scheme will be shown in Fig. \ref{gp:pos_acc_comparison}.
To address this issue, we next consider two candidate solutions.
\subsubsection{Solution candidate 1}
The first candidate is to time-synchronize the attitude and total thrust output by adding an artificial time delay to $T_{t,d}$ in Equation (\ref{eq:rdes_candidate_T}) as
\begin{equation}
\label{eq:rdes_candidate_T2}
T_{t,d}=-{{m\tilde{\ddot{z}}_d(t-\gamma_v)}\over{\cos{\phi_d(t-\gamma_\phi)}\cos\theta_d(t-\gamma_\theta)}}.
\end{equation}
Here, $\gamma_v$ is a delay element deliberately applied to $\tilde{\ddot{z}}_d$.
Applying Equation (\ref{eq:rdes_candidate_T2}) to Equation (\ref{eq:dynamics_false}), the equation of motion is changed from Equation (\ref{eq:dynamics_false_detail}) to
\begin{equation}
\label{eq:dynamics_false_detail2}
\begin{bmatrix}
\tilde{\ddot{x}}(t)\\
\tilde{\ddot{y}}(t)\\
\tilde{\ddot{z}}(t)
\end{bmatrix}
=
\begin{bmatrix}
\tan{\theta_d(t-\gamma_\theta)}\\
-{{\tan{\phi_d(t-\gamma_\phi)}}\over{\cos{\theta_d(t-\gamma_\theta)}}}\\
1
\end{bmatrix}
\tilde{\ddot{z}}_d(t-\gamma_v).
\end{equation}
Through Equations (\ref{eq:rdes_candidate_theta}), (\ref{eq:rdes_candidate_phi}) and (\ref{eq:iorelationship}), $\phi(t)$ and $\theta(t)$ can be described as
\begin{equation}
\label{eq:attitude-attitude_d relationship}
\left\{
\begin{array}{lr}
\theta(t)=\arctan{\left({{\tilde{\ddot{x}}_d(t-\gamma_\theta)}\over{\tilde{\ddot{z}}_d(t-\gamma_\theta)}}\right)}\\
\phi(t)=\arctan{\left(-{{\tilde{\ddot{y}}_d(t-\gamma_\phi)\cos{\theta_d(t-\gamma_\phi)}}\over{\tilde{\ddot{z}}_d(t-\gamma_\phi)}}\right)}.
\end{array}
\right.
\end{equation}
Let us assume that $\gamma_\phi$ and $\gamma_\theta$ have the same value of $\gamma_h$ since most multi-rotors have nearly the same roll and pitch behavior due to the symmetrical mechanical structure.
Then, Equation (\ref{eq:dynamics_false_detail2}) with Equations (\ref{eq:iorelationship}) and (\ref{eq:attitude-attitude_d relationship}) becomes as
\begin{equation}
\label{eq:tdynamics_final_candidate}
\begin{bmatrix}
\tilde{\ddot{x}}(t)\\
\tilde{\ddot{y}}(t)\\
\tilde{\ddot{z}}(t)
\end{bmatrix}
=
\begin{bmatrix}
\tilde{\ddot{x}}_d(t-\gamma_h)\big({{{\tilde{\ddot{z}}_d(t-\gamma_v)}\over{\ddot{z}}_d(t-\gamma_h)}}\big)\\
\tilde{\ddot{y}}_d(t-\gamma_h)\big({{{\tilde{\ddot{z}}_d(t-\gamma_v)}\over{\ddot{z}}_d(t-\gamma_h)}}\big)\\
\tilde{\ddot{z}}_d(t-\gamma_v)
\end{bmatrix}.
\end{equation}
Now, we can solve the problem in Equation (\ref{eq:dynamics_false_detail}) by setting $\gamma_v$ equal to $\gamma_h$.
However, this method is not easily applicable in a real-world situation because it is difficult to determine the value of $\gamma_h$ that changes continuously during the flight.
Therefore, the control method through Equation (\ref{eq:rdes_candidate_T2}) cannot be a practical method.
\subsubsection{Solution candidate 2}
Alternatively, we can find a reasonable solution that is applicable in the real world by selectively delaying $\phi_d(t)$ and $\theta_d(t)$ in Equation (\ref{eq:rdes_candidate_T2}) by $\gamma_\phi$ and $\gamma_\theta$, but keeping $\gamma_v$ at zero.
As we can see from Equation (\ref{eq:iorelationship}), the values of $\phi_d(t)$ and $\theta_d(t)$ delayed by $\gamma_\phi$ and $\gamma_\theta$ seconds are $\phi(t)$ and $\theta(t)$.
Applying this idea to Equation (\ref{eq:rdes_candidate_T2}), we can obtain $T_{t,d}$ as
\begin{equation}
\label{eq:Tfd_determination}
T_{t,d}=-{{m\tilde{\ddot{z}}_d(t)}\over{\cos{\phi(t)}\cos{\theta(t)}}},
\end{equation}
where the values $\phi(t)$ and $\theta(t)$ can be measured from the built-in inertial measurement unit (IMU) sensor.
Then, by setting $\gamma_v$ to zero, we can determine the input/output relationship of the translational accelerations dynamics of the multi-rotor as
\begin{equation}
\label{eq:tdynamics_final}
\tilde{\ddot{\mathbf{X}}}=
\begin{bmatrix}
\tilde{\ddot{x}}_d(t-\gamma_h)\left({{\tilde{\ddot{z}}_d(t)}\over{\tilde{\ddot{z}}_d(t-\gamma_h)}}\right)\\
\tilde{\ddot{y}}_d(t-\gamma_h)\left({{\tilde{\ddot{z}}_d(t)}\over{\tilde{\ddot{z}}_d(t-\gamma_h)}}\right)\\
\tilde{\ddot{z}}_d(t)
\end{bmatrix}\approx
\begin{bmatrix}
\tilde{\ddot{x}}_d(t-\gamma_h)\\
\tilde{\ddot{y}}_d(t-\gamma_h)\\
\tilde{\ddot{z}}_d(t)
\end{bmatrix}\;,
\end{equation}
where we assume ${{\tilde{\ddot{z}}_d(t)}\over{\tilde{\ddot{z}}_d(t-\gamma_h)}}\approx1$.
This assumption is valid in most cases, except in situations where the change in target vertical acceleration is abnormally {large and} rapid.
Through the control techniques of solution candidate 2 \big(Equations (\ref{eq:rdes_candidate_theta}), (\ref{eq:rdes_candidate_phi}) and (\ref{eq:Tfd_determination})\big), we obtained a three-dimensional translational acceleration control method applicable to actual multi-rotor control.
In order to compare the performance of multi-rotor control using Equations (\ref{eq:rdes_candidate_T}) and (\ref{eq:Tfd_determination}), a brief simulation is conducted as shown in Fig. \ref{gp:pos_acc_comparison}.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=18cm]{Sim_Pos_Acc}
\end{center}
\vspace{-0.5cm}
\caption{{[Simulation] A comparison of cases where acceleration command is converted into a target attitude and a thrust signal using Equations (\ref{eq:rdes_candidate_theta}), (\ref{eq:rdes_candidate_phi}) and (\ref{eq:rdes_candidate_T}) (Case 1), and using Equations (\ref{eq:rdes_candidate_theta}), (\ref{eq:rdes_candidate_phi}) and (\ref{eq:Tfd_determination}) (Case 2) for multi-rotors with different MOI. Acceleration motions are simulated for two scenarios : in the first scenario, an arbitrary target acceleration command is applied (top), and the target acceleration is generated via a position controller that tracks the predefined desired trajectory in the second scenario (bottom).}}
\label{gp:pos_acc_comparison}
\end{figure*}
{The simulation shows the comparison of the target acceleration tracking performance of Case 1 with Equations (\ref{eq:rdes_candidate_theta}), (\ref{eq:rdes_candidate_phi}), (\ref{eq:rdes_candidate_T}) and Case 2 with Equations (\ref{eq:rdes_candidate_theta}), (\ref{eq:rdes_candidate_phi}), (\ref{eq:Tfd_determination}).
The upper set of figures show the acceleration tracking performance of Cases 1 and 2 with arbitrary acceleration command.
Here, we can see that there are no differences in performance between Cases 1 and 2 when MOI of the multi-rotor has small value of 0.1.
On the other hand, when the MOI of the multi-rotor increases, both Cases 1 and 2 show delayed responses in the X and Y direction acceleration tracking as expected.
However, we can observe that the Z-directional performance of the Case 2 remains the same regardless of the magnitude of the MOI, unlike Case 1 where the performance degradation is observed.
The effect of the decline in Z-directional control performance on the system is evident when controlling the position of the multi-rotor.
The bottom set of figures is the situation where the high-level position controller generates the desired acceleration command to track the predefined trajectory.
In Case 1, we can observe a decrease in acceleration tracking performance in both the X and Y directions as well as the Z direction as the MOI increases.
On the other hand, in Case 2, the Z-directional control performance remains constant regardless of the MOI of the platform, stabilizing the X and Y-directional control performance faster than Case 1.}
This phenomenon can be understood in other ways by considering the role of the denominator term of the $T_t$ equation in Equation (\ref{eq:cmd-accel}), which is to compensate for the reduction of the vertical thrust component in the sense of inertial coordinates when the multi-rotor is tilted.
When $T_{t,d}$ is calculated based on the desired attitude as Equation (\ref{eq:rdes_candidate_T}), the situation is similar to compensating for the future event after $\gamma_h$ seconds.
Instead, it is intuitive to use the current attitude as in Equation (\ref{eq:Tfd_determination}) to correct the vertical thrust reduction.
From the flight results using Equation (\ref{eq:Tfd_determination}) in Fig. \ref{gp:pos_acc_comparison}, we can confirm that the control performance in all directions is satisfactory.
\section{DISTURBANCE OBSERVER}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=18cm]{full_diagram}
\end{center}
\vspace{-0.5cm}
\caption{Overall system diagram with DOB structure. $C(t)$: Outer-loop controller, $\mathbf{F}_d$: Desired translational force vector, $\tilde{\mathbf{F}}_d$: Sum of $\mathbf{F}_d$ and disturbance cancellation signal $-\hat{\mathbf{d}}_{EID}$, $\mathbf{B}\scriptscriptstyle{[\tilde{\ddot{\mathbf{X}}}_d\rightarrow\mathbf{r}_d]}$: $\tilde{\ddot{\mathbf{X}}}_d$ to $\mathbf{r}_d$ translator \big(eq. (\ref{eq:rdes_candidate_theta}), (\ref{eq:rdes_candidate_phi}), (\ref{eq:Tfd_determination})\big), $\it{\Lambda(t)}$: Plant dynamics \big(Fig. \ref{gp:platform_structure}, Eq. (\ref{eq:dyn_full})\big), $\mathbf{B}\scriptscriptstyle{[\mathbf{r}\rightarrow\mathbf{F}]}$: $\mathbf{r}$ to $\mathbf{F}$ translator \big(Eq. (\ref{eq:x_tilde_definition}), (\ref{eq:accel-cmd})\big), $\mathbf{F}$: Force vector generated by the multi-rotor, $\tilde{\mathbf{F}}$: Sum of $\mathbf{F}$ and actual disturbance $\mathbf{d}_{actual}$, ${P_n(t)}$: Nominal model of ${P(t)}$, ${Q_{1,2}(t)}$: $Q$-filters for DOB. }
\label{gp:full_diagram}
\end{figure*}
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{FtoK}
\end{center}
\vspace{-0.5cm}
\caption{Configuration of ${P_n^{-1}(t)}$. The ${P_n^{-1}(t)}$ block is composed of the opposite order of ${P(t)}$, where ${\it{\Lambda}}_n(t)$ is the nominal model of ${\it{\Lambda}}(t)$.}
\label{gp:FtoK}
\end{figure}
External disturbances applied to multi-rotor act not only in the form of translational disturbances but also in the form of rotational torques. However, given that a number of solutions for overcoming the rotational torque disturbances \cite{Attitude_DOB_1}$\sim$\cite{suseong_DOB} have already been proposed, this section concerns only translational disturbances applied to the system for straightforward discussion and analysis.
\subsection{An overview of the disturbance-merged overall system}
Fig. \ref{gp:full_diagram} shows the overall configuration of the system. First, the position controller ${C(t)}$ generates the target acceleration input $\ddot{\mathbf{X}}_d$. This signal is then transformed into the target force input $\mathbf{F}_d$ through the following force-acceleration relationship:
\begin{equation}
\label{eq:X-F relationship}
\mathbf{F}=m(\ddot{\mathbf{X}}-\mathbf{g}).
\end{equation}
Then, $\mathbf{F}_d$ signal passes through ${{1}\over{m}}R^{-1}(\psi)$ block to transform the signal into the $\tilde{\ddot{\mathbf{X}}}_d$ \big(refer Equation (\ref{eq:x_tilde_definition})\big). The signal $\tilde{\ddot{\mathbf{X}}}_d$ then passes through the $\mathbf{B}\scriptscriptstyle{[\tilde{\ddot{\mathbf{X}}}_d\rightarrow\mathbf{r}_d]}$ block, which converts the target acceleration $\tilde{\ddot{\mathbf{X}}}_d$ to $\mathbf{r}_d$, the input to the multi-rotor controller, based on Equations (\ref{eq:rdes_candidate_theta}), (\ref{eq:rdes_candidate_phi}) and (\ref{eq:Tfd_determination}).
Once $\mathbf{r}_d$ passes through the dynamics described in Fig. \ref{gp:platform_structure} and outputs $\mathbf{r}$, it passes through $\mathbf{B}\scriptscriptstyle{[\mathbf{r}\rightarrow\mathbf{F}]}$ block to produce $\mathbf{F}$ \big(refer Equation (\ref{eq:x_tilde_definition}) and (\ref{eq:accel-cmd})\big).
Right after $\mathbf{F}$ is generated, the external disturbance force $\mathbf{d}_{actual}$ immediately compromises the thrust and results in $\tilde{\mathbf{F}}$ and $\ddot{\mathbf{X}}$.
\subsection{Disturbance observer}
In Fig. \ref{gp:full_diagram}, the translational force disturbance $\mathbf{d}_{actual}$ is combined with $\mathbf{F}$ to become $\tilde{\mathbf{F}}$. However, canceling $\mathbf{d}_{actual}$ is only possible by adding an appropriate disturbance cancellation term to the $\mathbf{F}_d$ signal. Therefore, it is preferable to assume that there is an equivalent input disturbance $\mathbf{d}_{EID}$ that has the same effect on the system as $\mathbf{d}_{actual}$ \cite{EID}. Then $\mathbf{d}_{actual}$ is replaced by $\mathbf{d}_{EID}$, making $\mathbf{F}=\tilde{\mathbf{F}}$. As we can see in Fig. \ref{gp:full_diagram}, the $\mathbf{d}_{EID}$ signal is merged into $\tilde{\mathbf{F}}_d$, which is the translational acceleration control input with disturbance cancellation signal. Now, let us construct the DOB based on the above settings.
\subsubsection{$\mathbf{d}_{EID}$ estimation algorithm}
For the estimation of $\mathbf{d}_{EID}$, we first estimate $\kappa$ the sum of $\tilde{\mathbf{F}}_d$ and $\mathbf{d}_{EID}$ by
\begin{equation}
\label{eq:k_hat}
\hat{\kappa}(s)=\hat{\tilde{\mathbf{F}}}_d(s)+\hat{\mathbf{d}}_{EID}(s)={P_n^{-1}}(s)\tilde{\mathbf{F}}(s).
\end{equation}
We can easily achieve the $\tilde{\mathbf{F}}$ signal from Equation (\ref{eq:X-F relationship}) where $\ddot{\mathbf{X}}$ is measured by the IMU sensor. The transfer function ${P_n(s)}$ is the nominal model of ${P(s)}$, and $\hat{(*)}$ is the representation of the estimation of $(*)$ signal throughout this paper. Once we estimate $\hat{\kappa}$, we then obtain $\hat{\mathbf{d}}_{EID}$ by
\begin{equation}
\label{eq:dhat_eid}
\hat{\mathbf{d}}_{EID}={Q_1}(s)\hat{\kappa}(s)-{Q_2}(s)\tilde{\mathbf{F}}_d(s).
\end{equation}
The signal $\hat{\kappa}(s)$ passes through the ${Q_1}$ block, which is basically a low pass filter, to overcome both the causality violation issue due to the improperness of ${P_n^{-1}}(s)$ and the potential instability issue caused by the non-minimum phase characteristic of ${P_n}(s)$. The filter ${Q_2}(s)$ is used to match the phase with ${Q_1}(s)\hat{\kappa}(s)$ signal. In the end, we generate a disturbance-compensating control input $\tilde{\mathbf{F}}_d$ by
\begin{equation}
\label{eq:Ftilde_d}
\tilde{\mathbf{F}}_d=\mathbf{F}_d-\hat{\mathbf{d}}_{EID}.
\end{equation}
This makes $\kappa$ become
\begin{equation}
\label{eq:kappa}
\kappa=\tilde{\mathbf{F}}_d+\mathbf{d}_{EID}=\mathbf{F}_d-\hat{\mathbf{d}}_{EID}+\mathbf{d}_{EID}\approx\mathbf{F}_d.
\end{equation}
The most important factor in the $\mathbf{d}_{EID}$ estimation process is the proper design of ${P_n}$ and ${Q}$. Of these, ${Q}$ is deeply related to the stability of the system and will be discussed in more detail in the next section. In the remainder of this section, we first discuss the design of the nominal model ${P_n}$ and then explain the structure of the $Q$-filter.
\subsubsection{Nominal model ${P_n}$}
The internal structure of ${P_n^{-1}}(t)$ is described as in Fig. \ref{gp:FtoK}, all of which are simple conversion blocks except for the ${{\it{\Lambda}}}^{-1}_n{\it(t)}$ block. The block ${\it{\Lambda}}(t)$ is the relationship between $\mathbf{r}_d$ and $\mathbf{r}$ depicted in Fig. \ref{gp:platform_structure}. The $\it{\Lambda_n(s)}$ is constructed from two parts: attitude and thrust dynamics. We denote these as $\it{\Lambda_{n,a}}(s)$ and $\it{\Lambda_{n,t}}(s)$ respectively.
As we see from Fig. \ref{gp:platform_structure}, $\it{\Lambda}_{n,a}(s)$ is constructed with attitude controller, rotor dynamics and attitudinal dynamics. Since rotor dynamics can be ignored, we only need to find the transfer function of the attitudinal dynamics and attitude controller. For attitude dynamics, let us refer to Equation (\ref{eq:dyn_rot_simplify}) and express it as
\begin{equation}
\label{eq:atti.-dyn_freq.}
{{\mathbf{q}_i(s)}\over{\mathbf{T}_{r,i}(s)}}={{1}\over{J_is^2}},
\end{equation}
where $i=1,2,3$ represent $\phi$, $\theta$, $\psi$ axis, respectively. For attitude control, PD control in the following form is used.
\begin{equation}
\label{eq:PID}
{{\mathbf{T}_{r,i}(s)}\over{\mathbf{q}_{i,d}(s)-\mathbf{q}_i(s)}}={P_i+D_is}
\end{equation}
The parameters $P_i$, $D_i$ represent control gains in each attitude component. Then, the overall transfer function ${\it{\Lambda}}_{n,a,i}$ between desired and current attitude becomes
\begin{equation}
\label{eq:lambda_na}
{\it{\Lambda}}_{n,a,i}(s)={{\mathbf{q}_i(s)}\over{\mathbf{q}_{i,d}(s)}}={{D_is+P_i}\over{J_is^2+D_is+P_i}}.
\end{equation}
In the case of ${\it{\Lambda}}_{n,t}$, the only dynamics involved is rotor dynamics, which we decided to neglect. Thus, it can be expressed as
\begin{equation}
\label{eq:lambda_nt}
{\it{\Lambda}}_{n,t}(s)=1.
\end{equation}
Now, we can construct the transfer matrix for ${\it{\Lambda}}_n=diag({\it{\Lambda}}_{n,1},{\it{\Lambda}}_{n,2},{\it{\Lambda}}_{n,3})$ using Equations (\ref{eq:lambda_na}) and (\ref{eq:lambda_nt}) as
\begin{equation}
\label{eq:lambda_n}
{\it{\Lambda}}_n(s)=
\begin{bmatrix}
{\it{\Lambda}}_{n,a,2}(s) & 0 & 0\\
0 & {\it{\Lambda}}_{n,a,1}(s) & 0\\
0 & 0 & {\it{\Lambda}}_{n,t}(s)
\end{bmatrix}.
\end{equation}
Equation (\ref{eq:lambda_n}) is a detailed representation of the relationship between $\mathbf{r}_d$ and $\mathbf{r}$, which was introduced in Equation (\ref{eq:iorelationship}).
On the other hand, $P_n$, which defines the nominal relationship between $\kappa$ and $\mathbf{F}$ \big(or $\tilde{\ddot{\mathbf{X}}}_d$ and $\tilde{\ddot{\mathbf{X}}}$\big), was introduced in Equation (\ref{eq:tdynamics_final}) \big(refer Equation (\ref{eq:x_tilde_definition}) for the relationship between $\tilde{\ddot{\mathbf{X}}}$ and $\mathbf{F}$\big). Here, we can see that both Equations (\ref{eq:iorelationship}) and (\ref{eq:tdynamics_final}) have the same input/output characteristics with time delay of $\gamma_h$ for the first and second channels and no time delay for the third channel.
Therefore, we can conclude that ${\it{\Lambda}}_n(s)$ in Equation (\ref{eq:lambda_n}) is also the transfer function between $\kappa$ and $\mathbf{F}$ as well as between $\mathbf{r}_d$ and $\mathbf{r}$, which is
\begin{equation}
\label{eq:PneqLn}
{P}_n(s)={\it{\Lambda}}_n(s).
\end{equation}
\subsubsection{$Q$-filter design}
In $Q$-filter design, we choose to make ${Q_1}(s){\it{\Lambda}}_n^{-1}(s)$, which is now identical to ${Q_1}(s){P}_n^{-1}(s)$, a proper function with relative degree of 1. Since ${P}_n(s)$ is composed of three channels in $X$, $Y$ and $Z$ directions, we need to design three separate $Q$-filters. As shown in Equation (\ref{eq:lambda_na}), ${\it{\Lambda}}_{n,1}(s)$\big($={\it{\Lambda}}_{n,a,2}(s)$\big) and ${\it{\Lambda}}_{n,2}(s)$\big($={\it{\Lambda}}_{n,a,1}(s)$\big) among the three transfer functions of ${\it{\Lambda}}_n(s)$ are systems with a relative degree of 1. The thrust transfer function ${\it{\Lambda}}_{n,3}(s)$\big($={\it{\Lambda}}_{n,t}(s)$\big) has a relative degree of 0, as can be seen from Equation (\ref{eq:lambda_nt}). Therefore, the $Q$-filters for making ${Q_1}(s){\it{\Lambda}}_n^{-1}(s)$ with a relative degree of 1 are designed as
\begin{equation}
\label{eq:Q_1}
{Q_1}(s)=diag\big({Q_{1,h}}(s),{Q_{1,h}}(s),{Q_{1,v}}(s)\big),
\end{equation}
\begin{equation}
\label{eq:Q-filter1}
{Q_{1,h}}(s)={{1}\over{(\tau_1 s)^2+\zeta(\tau_1 s)+1}},
\end{equation}
\begin{equation}
\label{eq:Q_filter2}
{Q_{1,v}}(s)={{1}\over{(\tau_2 s)+1}},
\end{equation}
where ${Q_{1,h}}$ and ${Q_{1,v}}$ are $Q$-filters corresponding to the horizontal ($\it{\Lambda}_{n,a}$) and vertical ($\it{\Lambda}_t$) models respectively. The symbol $\tau$ is the time constant and $\zeta$ is the damping ratio of the filter. The filter ${Q_1}$ is designed to have a gain of 1 when $s=0$ \cite{DOB_Q}. The filter ${Q_2}$ is set to ${Q_2}={Q_1}$, to easily achieve the purpose of phase matching.
\section{STABILITY ANALYSIS}
The design of $Q$-filter in the DOB structure should be based on rigorous stability analysis to ensure the overall stability. In particular, we note that there is always a difference between the nominal model $P_n(s)$ and the actual model $P(s)$, due to various uncertainties and applied assumptions.
Although the small-gain theorem (SGT) \cite{DOB_sjlazza} can still be a tool for stability analysis, the SGT analysis based on the largest singular value among uncertainties is likely to yield overly conservative results especially if multiple uncertain elements are involved. Instead, we use structured singular value analysis, or $\mu$-analysis \cite{SGTnmu, sariyildiz2013analysis, kim2018robust}, to reflect the combined effects of uncertainties.
\subsection{Modeling of ${P}(s)$ considering uncertainties}
The multi-rotor's actual transfer function ${P}(s)$ between $\kappa$ and $\mathbf{F}$ in Fig. \ref{gp:full_diagram} is
\begin{equation}
\label{eq:P}
{P}(s)=diag\big({P}_{1}(s),{P}_{2}(s),{P}_3(s)\big).
\end{equation}
Here, $P_1$, $P_2$ and $P_3$ represent the input/output translational force relationship in the $X$, $Y$, and $Z$ directions, respectively.
This research considers a small but nonzero DC-gain error, parametric error and phase shift error between ${P}_n(s)$ and ${P}(s)$.
Then each ${P}_{j}(s)$ can be expressed as the following equation:
\begin{equation}
\label{eq:P_model}
\begin{split}
{P}_{j}(s)&=K_{j}{P}_{n,j}(s)e^{-\delta_{j} s}\\
&=K_{j}{\it{\Lambda}}_{n,j}(s)e^{-\delta_{j} s}={P}_{p,j}(s)\Gamma_{j}(s),
\end{split}
\end{equation}
where $j=1,2,3$ represent $X$, $Y$, $Z$ axis. The symbols $K_{j},\delta_j\in\mathbb{R}$ represent the uncertain variable gain and time delay parameters, respectively. The nominal transfer function ${P}_{n,j}$ can be replaced by $\it{\Lambda}_{n,j}$ based on Equation (\ref{eq:PneqLn}). The portion containing only the parametric uncertainty is denoted by ${P}_{p,j}(s)=K_{j}{\it{\Lambda}}_{n,j}(s)$, and the time delay uncertainty is denoted by $\Gamma_{j}(s)=e^{-\delta_{j} s}$.
In Equation (\ref{eq:P_model}), each ${P}_{j}(s)$ contains three uncertain variables, which are $K_{j}$, $J_j$ and $\delta_{j}$. In the case of $K_j$, we define $K_j$ as
\begin{equation}
\label{eq:K_j}
K_j=1+K_{\Delta,j},
\end{equation}
where $K_{\Delta,j}\in\mathbb{R}$ is the error value of $K_j$. In the case of $J_j$, determining the actual value of $J_j$ is difficult compared to other physical quantities. We also define $J_j$ in the same manner as $K_j$ for the convenience of analysis as
\begin{equation}
\label{eq:J_j}
J_j=\bar{J}_j(1+J_{\Delta,j}),
\end{equation}
where $\bar{J}_j, J_{\Delta,j}\in\mathbb{R}$ are the nominal and error values of $J_j$. Because the term $\Gamma_{j}(s)$ containing $\delta_j$ is of an irrational form that is not suitable for analysis, we use an analytic approximation of the uncertain time-delay $\Gamma_{j}(s)$ to a rational function with unmodeled dynamic uncertainty \cite{kim2018robust}. First, we change the representation of the $P_j(s)$ model to a multiplicative uncertainty form that combines parametric uncertainties and unmodeled time-delay uncertainty as follows:
\begin{equation}
\label{eq:unmodeled_plant}
\begin{split}
{P}_{j}(s)={P}_{p,j}(s)\big(1+\Delta_{\delta,j}(s)W_{\delta,j}(s)\big),\\
||\Delta_{\delta,j}(s)||_{\infty} \leq 1.
\end{split}
\end{equation}
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{W_s_}
\end{center}
\vspace{-0.5cm}
\caption{Bode magnitude plots of $\Gamma_j(s)-1$ expressed by varying $\delta_j$ from $-0.12$ to $0.12$ (blue dashed line), maximum uncertainty $W_{\delta,j}(s)$ (red solid line).}
\label{gp:W(s)}
\end{figure}
A complex unstructured uncertainty $\Delta_{\delta,j}\in\mathbb{C}$ corresponds to unknown time delay $\delta_j$, and $W_{\delta,j}(s)$ is the maximum uncertainty that can be caused by $\Gamma_{j}(s)$. Here, we can obtain $W_{\delta,j}(s)$ using Equation (\ref{eq:P_model}) as
\begin{equation}
\label{eq:W(s)}
W_{\delta,j}(s)=\underset{\delta_j}\max{\Big|{{{P}_{j}(s)-{P}_{p,j}(s)}\over{{P}_{p,j}(s)}}\Big|}=\underset{\delta_j}\max|\Gamma_{j}(s)-1|.
\end{equation}
The maximum value of $|\Gamma_{j}(j\omega)-1|$ for each $\omega$ can be found using Euler's formula as
\begin{equation}
\label{eq:W_euler}
\underset{\delta_j}\max|\Gamma_{j}(j\omega)-1|
\underset{\delta_j}\max\sqrt{\big(\cos{(\omega\delta_{j})-1\big)^2+\big(\sin{(\omega\delta_{j})\big)^2}}}
\end{equation}
where
\begin{equation}
\label{eq:cossin}
\Gamma_{j}(j\omega)=e^{\delta_{j}(j\omega)}=\cos{(\omega\delta_{j})}+j\sin{(\omega\delta_{j})}.
\end{equation}
As a result of analyzing a large amount of actual experimental data, we confirmed that the time delay between ${P}_n(s)$\big($={\it{\Lambda}}_n(s)$\big) and ${P}(s)$ does not exceed 0.1 second in all three channels. We put 20 percent margin so that $|\delta_j|\leq 0.12$. Fig. \ref{gp:W(s)} is multiple Bode magnitude plots of $|\Gamma(s)-1|$ generated by varying $\delta$ from $-0.12$ to $+0.12$. From Fig. \ref{gp:W(s)}, we can extract
\begin{equation}
\label{eq:W_max(S)}
W_{\delta,j}(s)={{2.015s^3+52.88s^2+431.6s+0.415}\over{s^3+36.7s^2+606.8s+3521}}
\end{equation}
for all $j$, which is the upper boundary of $|\Gamma_{j}(s)-1|$ sets represented by the red solid line.
The uncertainties of $K_{j}$ and $J_j$ can also be modeled in the same manner as in Equation (\ref{eq:unmodeled_plant}) as
\begin{equation}
\label{eq:Pp_rewritten}
\left\{
\begin{array}{lr}
P_{p,j}(s)=K_j {\it{\Lambda}}_{n,j}(s)={\it{\Lambda}}_{n,j}(1+{\Delta}_{K,j}{W}_{K,j})\\
{\it{\Lambda}}_{n,j}(s)={\it{\Lambda}}_{n,n,j}(s)(1+{\Delta}_{J,j}{W}_{J,j}),
\end{array}
\right.
\end{equation}
where $||{\Delta}_{K,j}||_\infty,||{\Delta}_{J,j}||_\infty\leq1$. The transfer function ${\it{\Lambda}}_{n,n,j}$ is basically the same as ${\it{\Lambda}}_{n,j}$, except that $J$ in Equation (\ref{eq:lambda_na}) is replaced to the nominal MOI value $\bar{J}$. The transfer functions $W_{K,j}$ and $W_{J,j}$ are
\begin{eqnarray}
\label{eq:Wk,Wj}
W_{K,j}&\hspace{-.1cm}=\hspace{-.1cm}&\max|K_{\Delta,j}|\\
W_{J,j}&\hspace{-.1cm} =\hspace{-.1cm}& \left\{
\begin{array}{ll} \underset{J_{\Delta,j}}\max \Big|{{-\bar{J}_jJ_{\Delta,j}s^3}\over{\bar{J}_j(1+J_{\Delta,j})s^3+D_js^2+P_js+I_j}}\Big| &
(j=1,2)\\
0 & (j=3).
\end{array} \nonumber
\right.
\end{eqnarray}
\subsection{$\tau$-determination through $\mu$-analysis}
\subsubsection{$\mu$-robust stability analysis}
In \cite{muDoyle}, the structured singular value $\mu$ is defined as
\begin{equation}
\label{eq:mu}
\mu_{\Delta}(M_{11})={{1}\over{\underset{\Delta\in\mathbf{\Delta}}\min\big(\bar{\sigma}(\Delta):\det(I-M_{11}\Delta)=0\big)}}
\end{equation}
where $\Delta$ is a complex structured block-diagonal unmodeled uncertainty block which gathers all model uncertainties \cite{mu-analysis}. Following the common notation, the symbol $\mathbf{\Delta}$ represents a set of all stable transfer matrices with the same structure (full, block-diagonal, or scalar blocks) and nature (real or complex) as $\Delta$. The $\bar{\sigma}\big(\Delta\big)$ is the maximum singular value of uncertainty block $\Delta$. The matrices $M$ and $\Delta$ are defined by collapsing the simplified overall system to upper LFT uncertainty description as
\begin{equation}
\label{eq:LFT uncertainty description}
\begin{bmatrix}
z\\
y
\end{bmatrix}
=
\begin{bmatrix}
M_{11} & M_{12}\\
M_{21} & M_{22}
\end{bmatrix}
\begin{bmatrix}
w\\
r
\end{bmatrix}
,\ w=\Delta z,
\end{equation}
where $M$ is the known part of the system, $r$ is a reference input and $y$ is an output of the overall system. In the theory of the $\mu$-analysis, it is well-known that the system is robustly stable if $\mu$ satisfies the following conditions
\begin{equation}
\label{eq:mu_stable_condition}
\mu_\Delta(M_{11})<1,\ \forall\omega
\end{equation}
\cite{SGTnmu}\cite{muDoyle}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8.5cm]{LFT_model}
\end{center}
\vspace{-0.5cm}
\caption{Compressed block digram of the DOB-included transfer function from $\mathbf{F}_{d,j}$ to $\tilde{\mathbf{F}}_j$, whose original form was shown in Fig. \ref{gp:full_diagram} (top), further collapsed form expressed as a nominal closed-loop system $M_j$ and a complex unstructured uncertainty block $\Delta_j$ as in Equation (\ref{eq:Delta_ourcase}) (bottom).}
\label{gp:lft_model}
\end{figure}
The $\mu$-analysis is performed separately for each channel of $X$, $Y$, $Z$ thanks to the structure of the platform described by Equation (28), but since $X$ and $Y$ channels are composed of the same structure, they share the identical analysis result. As we can see from Fig. \ref{gp:lft_model}, the system is collapsed in the form of Equation (\ref{eq:LFT uncertainty description}) by using MATLAB's Robust Control Toolbox$^{\textrm{TM}}$, where $r_j=[\mathbf{F}_{d,j}\ \mathbf{d}_{EID,j}]^T\in\mathbb{R}^{2\times1}$ and $y_j=\tilde{\mathbf{F}}_j\in\mathbb{R}$ in our case. As a reminder, subscript $j$ refers to each channel of $X$, $Y$, and $Z$. Also, structured uncertainty ${\Delta}_j(s)\in\mathbb{C}^{3\times3}$ is constructed as
\begin{equation}
\label{eq:Delta_ourcase}
{\Delta}_j(s)=diag\big({\Delta}_{J,j}(s), {\Delta}_{K,j}(s), {\Delta}_{\delta,j}(s)\big),
\end{equation}
which includes unmodeled MOI uncertainty, time and gain uncertainty in our system.
\subsubsection{Results of analysis}
\begin{table}[b]
\renewcommand{\arraystretch}{1.3}
\caption{PHYSICAL QUANTITIES AND CONTROLLER GAINS.}
\label{table_1}
\centering
\begin{tabular}{ |l|l|l|l| }
\hline
Name &Value & Name & Value\\ \hline\hline
$P_{\phi,\theta}$ & 3 & Mass & 3.24 \si{Kg}\\ \hline
$D_{\phi,\theta}$ & 1 & $\bar{J}_{1,2}$ & 0.82 \si{Kg\cdot m^2}\\ \hline
$\ddot{\mathbf{X}}$ Limit & \si{\pm 3\ m/s^2} & $\bar{J}_3$ & 1.49 \si{Kg\cdot m^2}\\ \hline
$\max|\delta_j|$ & $0.12$ & $\max|J_{\Delta,j}|$ & $0.3$\\ \hline
$\max|K_{\Delta,j}|$ & $0.1$ & $\zeta$ & 0.707\\ \hline
\end{tabular}
\end{table}
Table \ref{table_1} shows the multi-rotor's physical quantities and controller gains used both in the simulation and the experiment.
The gains $P_{\phi, \theta}$ and $D_{\phi, \theta}$ are predefined values set during the primary gain-tuning process to obtain the ability to control the attitude of the platform.
The translational acceleration limit is set to prevent flight failure due to excessive acceleration control inputs and is set at $\pm 3\ \si{m/s^2}$ to have a roll and pitch limit of approximately $\pm0.3\ \si{rad}$ in level flight condition.
As previously mentioned, the unmodeled time delay $\delta_j$ is set to 0.1, and the gain error is assumed to be a maximum error of 10 percent.
For MOIs that are difficult to estimate, we assumed a wider 30 percent uncertainty.
The damping ratio $\zeta$ of the second order filter is set to 0.707, which is the critical damping ratio, to balance the overshoot and late response.
Fig. \ref{gp:mu_result} shows the results of $\mu$-analysis.
From the analysis, we can see that the system is stable when $\tau_1>0.12$ and $\tau_2>0.09$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8.5cm]{mu_result}
\end{center}
\vspace{-0.5cm}
\caption{$\mu$-analysis results for $X$, $Y$ channel (left), and $Z$ channel (right).}
\label{gp:mu_result}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=8.5cm]{small_gain_theorem}
\end{center}
\vspace{-0.5cm}
\caption{SGT-based analysis results for $X$, $Y$ channel (left), and $Z$ channel (right).}
\label{gp:small_gain}
\end{figure}
Fig. \ref{gp:small_gain} shows the results of the SGT-based stability analysis, performed in the same manner as \cite{DOB_sjlazza}.
The analysis is based on the following model:
\begin{equation}
\label{eq:plant_SGT}
\begin{split}
{P}_j(s)={\it{\Lambda_{n,n,j}}}(s)(1+{\Delta}_{l,j}{\it{{W}}}_{l,j}),\ ||{\Delta}_{l,j}||_\infty\leq1,
\end{split}
\end{equation}
where all uncertainties due to $\delta_j$, $K_j$ and $J_j$ are lumped using the functions $W_{l,h} (=W_{l,1}, W_{l,2})$ and $W_{l,v} (=W_{l,3})$, whose magnitude increases over frequency as shown in blue curves of Fig. \ref{gp:small_gain}. The stability condition of the SGT-based analysis in this case is
\begin{equation}
\label{eq:SGT_condition}
\bar{\sigma}\big({Q}_j(j\omega)\big)\bar{\sigma}\big({\it{W}}_{l,j}(j\omega)\big)<1
\end{equation}
\cite{DOB_sjlazza,SGTnmu,Small_Gain_Theorem}. In the SGT-based analysis, the bode plots of the $Q$-filter with $\tau_1=0.12$ and $\tau_2=0.09$ indicate that system with those $\tau$ values could be unstable. However, through the $\mu$-analysis, those $\tau$ values are still in the stable region. From this, we can confirm that the $\mu$-analysis provides more rigorous $\tau$ boundary values than SGT-based analysis.
\section{SIMULATION AND EXPERIMENTAL RESULT}
This section reports simulation and experimental results to validate the performance of our three-dimensional force controller and the disturbance cancellation performance of the DOB technique.
The comparison of the acceleration tracking performance of the force control methods according to the MOI variation is already shown in the simulation of Fig. \ref{gp:pos_acc_comparison}.
Therefore, in this section, we provide
\begin{enumerate}
\item experimental result to demonstrate the performance of the proposed force control technique for the actual plant, and
\item simulation and experimental results to demonstrate the capability of the DOB in overcoming the translational force disturbance.
\end{enumerate}
Based on the results from the previous section, the cutoff frequencies of the $Q$-filter are set to $\tau_1=0.15$ and $\tau_2=0.12$ in both simulation and actual experiment with additional margins to ensure additional stability.
\subsection{Validation of acceleration tracking performance}
In the experiment, arbitrary desired acceleration commands for $X$ and $Y$ directions are given by the operator-controlled radio controller. Fig. \ref{gp:Xd2X} shows the multi-rotor accurately following the target acceleration. From this result, we can confirm that our three-dimensional translational acceleration control technique functions effectively even in the actual flight.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8.5cm]{Xd2X}
\end{center}
\vspace{-0.5cm}
\caption{[Experiment] Desired 3-D acceleration generated by the operator through the R/C controller (blue), and the actual acceleration (red dash) generated by multi-rotor.}
\label{gp:Xd2X}
\end{figure}
\subsection{Validation of DOB performance}
\subsubsection{Simulation result}
\begin{figure}[t]
\begin{center}
\includegraphics[width=8.5cm]{_Sim_Position}
\end{center}
\vspace{-0.5cm}
\caption{[Simulation] Comparison of trajectory tracking performance before (left) and after (right) applying the DOB algorithm.}
\label{gp:sim_position}
\end{figure}
In the simulation, the multi-rotor follows a circular trajectory with radius of 3 \si{m} and height of 5 \si{m}. Meanwhile, the multi-rotor is exposed to periodic disturbances with accelerations up to 5.5 \si{m/s^2} in each axis. Fig. \ref{gp:sim_position} compares the multi-rotor's position tracking performance before and after applying DOB. On the left graphs of Fig. \ref{gp:sim_position}, the target trajectory tracking results are not smooth due to the unexpected disturbances, whereas the trajectory deviation is drastically reduced in the right graphs where the DOB algorithm is applied.
\subsubsection{Experimental Result}
\begin{figure}[t]
\begin{center}
\includegraphics[width=8.72cm]{Posforce}
\end{center}
\vspace{-0.5cm}
\caption{Comparison of the target position tracking performance before (left) and after (right) the DOB algorithm is applied.}
\label{gp:exp_position_force}
\end{figure}
In the experiment, the multi-rotor is commanded to hover at a specific point in three-dimensional space but connected to the translational force measurement sensor via the tether to measure the applied disturbance force. As we can see in Fig. \ref{gp:experiment}, the operator aligns the force sensor in the $X$-axis and pulls and releases the force sensor periodically to apply a disturbance to the multi-rotor.
Fig. \ref{gp:exp_position_force} is a comparison of hovering performance before (left) and after (right) applying the DOB algorithm. The graphs in the left column are the case when the DOB is not applied, which has a larger $X$ directional position shift than other axes. Unlike the DOB-off case, the DOB-on case shows a significant reduction in position error.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8.8cm]{Experiment}
\end{center}
\vspace{-0.5cm}
\caption{Experiment for DOB performance validation with disturbance using a tether. A force sensor is attached to the tether only to check the disturbance estimation performance. }
\label{gp:experiment}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=8.8cm]{wind_DOB}
\end{center}
\vspace{-0.4cm}
\caption{Comparison of the target position tracking performance in wind blast environment using an industrial fan.}
\label{gp:wind_DOB}
\end{figure}
Two graphs at the forth row shows the acceleration tracking results.
When DOB is not applied, an acceleration signal is generated by the position error, but we can see that the target acceleration cannot be followed due to the disturbance.
Meanwhile, we can see that the acceleration of the platform (yellow solid line) well tracks the target acceleration (blue dash-single dotted line).
This is because the well-behaved DOB algorithm generated control input including the disturbance compensation signal (orange dash-single dotted line) and applied to the platform.
The effect of the DOB can be confirmed by significantly reduced position error.
Four graphs at the bottom of the figure show the difference between the signal $\vec{\mathcal{F}}_d$ and $\tilde{\vec{\mathcal{F}}}_d$ (fifth row), and the comparison between $d_{actual,x}$ measured by force sensor and $\hat{d}_{EID,x}$ estimated by DOB algorithm (sixth row). When DOB is not applied, $\hat{d}_{EID}$ estimation process is working internally but the signal is not merged into $\tilde{\vec{\mathcal{F}}}_d$ signal, making $\vec{\mathcal{F}}_d$ and $\tilde{\vec{\mathcal{F}}}_d$ have the same value. On the other hand, we can see the difference between the $\vec{\mathcal{F}}_d$ and the $\tilde{\vec{\mathcal{F}}}_d$ signal when DOB is applied, because the $\hat{d}_{EID}$ signal is merged into the $\tilde{\vec{\mathcal{F}}}_d$ signal. Two graphs in the last row show the comparison between the measured disturbance and the estimated disturbance, and we can confirm that the estimates are fairly accurate in both cases.
An extra flight experiment is conducted under wind disturbance to validate the DOB performance in a more realistic environment. As we can see in Fig. \ref{gp:wind_DOB}, the target location of the multi-rotor is set on the centerline of a strong wind generator that generates wind speed of 7 \si{m/s}. The performance of DOB is visualized by comparing the position difference between DOB-on and DOB-off situations. the multi-rotor has a position error of about 1 \si{m} in the DOB-off case and about 0.3 \si{m} in the DOB-on case. Through the experiment, we can confirm that the proposed DOB algorithm works effectively even against a wind disturbance.
\section{CONCLUSION}
In this paper, we introduced 1) a new method of converting the target acceleration command to the desired attitude and total thrust, and 2) a DOB method for overcoming the disturbance that obstructs the translational motion, to more accurately control the translational acceleration of the multi-rotor UAV.
In the control input conversion process, we reflect the different dynamic characteristics of attitude and thrust, so that more precise control is possible than the existing methods.
Then, by using the DOB-based robust control algorithm based on the nominal translational force system, the magnitude of the disturbance force applied to the fuselage is estimated and compensated.
For the robust stability guarantee, the $Q$-filter of the DOB is designed based on the $\mu$-stability analysis.
The validity of the proposed method is confirmed through simulation and actual experiments.
The proposed technique is useful in various applications such as aerial parcel delivery service or drone-based industrial operations where precise acceleration control is required. For example,
in a multi-rotor-based parcel delivery service, the proposed DOB algorithm can maintain the nominal flight performance by considering the additional force due to the weight of the cargo attached to the multi-rotor as a disturbance to be estimated. Also, the proposed algorithm is suitable for situations that require precise trajectory tracking performance even in windy conditions such as maritime operations or human-rescue missions.
For industrial applications involving collaborative flight of multiple multi-rotors, the proposed algorithm can be used to estimate and stabilize internal forces caused in between physically-coupled multi-rotors.
\appendices
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{Bibliography/IEEEtran}
|
1,108,101,563,275 | arxiv | \section{Introduction}
\label{introduction}
Despite its numerous phenomenological successes,
the framework of the Standard Model coupled to general relativity
is not believed to provide a fundamental description of nature;
it is rather viewed as the low-energy limit
of some encompassing quantum theory
in which the gravitational, strong, and electroweak interactions are unified.
The energy scale of such an encompassing theory
is expected to be of the order of
the Planck mass $M_{Pl}\simeq10^{19}\,$GeV.
This expectation presents an enormous experimental challenge
because the emerging effects
are likely to be suppressed
by one or more powers of $M_{Pl}$
at presently attainable energies.
However,
minute Lorentz and CPT breakdown
has recently been identified as a promising signal in this context:
such effects may arise in various approaches to fundamental physics,
and they are amenable to ultrahigh-precision tests \cite{cpt04,reviews}.
The low-energy effects of Lorentz and CPT breakdown
are described by an effective-field-theory
framework called the Standard-Model Extension (SME) \cite{ck97,sme,gravext}.
Besides the usual Standard-Model and Einstein--Hilbert actions,
this framework contains all leading-order contributions
to the action
that violate particle Lorentz and CPT symmetry
while maintaining coordinate independence.
Of particular phenomenological interest
is the minimal SME.
In addition to conventional physics,
it only contains those Lorentz- and CPT-violating terms
that are expected to be dominant
and satisfy a few other physically desirable requirements.
The Minkowski-spacetime limit of the minimal SME
has been the focus of various investigations,
including ones
with photons \cite{photonexpt,radiative,photonth2,km02},
electrons \cite{trap,spinpol,eexpt3},
protons and neutrons \cite{cc,spaceexpt,bnsyn},
mesons \cite{hadronexpt},
muons \cite{muexpt},
neutrinos \cite{neutrinos,nuexpt},
and the Higgs \cite{higgs}.
Bounds in the gravity sector have also been obtained recently \cite{bk06,shore}.
The Lorentz- and CPT-violating coefficients in the minimal SME
are non-dynamical background vectors and tensors,
which are coupled to Standard-Model fields.
They are assumed to be generated by underlying physics.
Various specific mechanisms
for such effects
have been proposed in the literature.
For instance,
mechanisms compatible with spontaneous Lorentz and CPT breakdown
have been studied in models based on
string theory
\cite{ksp,gyb},
noncommutative geometry
\cite{ncqed},
spacetime-varying fields
\cite{spacetimevarying,ghostcond},
quantum gravity
\cite{qg},
nontrivial spacetime topology
\cite{klink},
random-dynamics models
\cite{fn02},
multiverses
\cite{bj},
and brane-world scenarios
\cite{brane}.
There are a number of known mappings between SME fields
that relate different SME coefficients for Lorentz violation \cite{ck97,sme,km02,redef}.
In some cases,
the effects of Lorentz breakdown
can be moved from one SME sector to another.
In other cases,
the coefficient can be removed from the SME altogether
implying it is unobservable.
Moreover,
for certain dimensionless Lorentz-violating parameters for fermions
a spinor redefinition is needed to obtain a hermitian Hamiltonian.
It therefore follows
that such field redefinitions play an important role
for the analysis of experimental Lorentz and CPT tests
and for the interpretation of Lorentz violation in the SME.
The form of possible field redefinitions
is essentially only constrained by
the requirement of invertibility.
On one hand,
this leaves a substantial amount of freedom in
the identification of useful field redefinitions.
On the other hand,
the large number of possibilities
hampers a systematic and comprehensive study of such redefinitions.
Previously analyzed mappings between SME fields
fall into two categories \cite{ck97,sme,km02,redef}:
redefinitions of the field variables
and coordinate rescalings.
In both cases,
the mapping is local
and applies on- as well as off-shell.
The present work is intended to launch an investigation
of a set of field redefinitions
characterized by non-locality.
We will focus on
a mapping within the SME's free $b^{\mu}$ model
that scales
(and hence eliminates in a certain limit)
the $b^{\mu}$ coefficient from this model.
With the exception of special cases,
the structure of the non-local mapping is such
that only on-shell applicability is guaranteed.
Although the $b^{\mu}$ coefficient
cannot be removed from a realistic interacting model,
this field redefinition is nevertheless interesting
for the following reasons.
In many situations,
the extraction of the physical content of a field theory,
such as the SME,
requires an initial investigation of its quadratic sectors.
For instance,
perturbation theory in quantum field theory
typically amounts to an expansion about known free-field physics.
Since our non-local field redefinition
can be employed to establish a one-to-one correspondence
between the usual free Dirac field
and the non-interacting $b^{\mu}$ model,
it can be used to gain insight into the free-field physics
of this model.
In particular,
this idea allows a complete
characterization of the solutions to the free $b^{\mu}$ model.
The present work illustrates this with a specific example:
the previously unknown $b^{\mu}$ eigenspinors
are generated from the conventional ones
via our non-local field redefinition.
We finally remark
that a generalization of this idea to the full SME,
if possible,
would completely characterize its free sectors.
The paper is organized as follows.
A brief review of the $b^{\mu}$ model
is given in Sec.\ \ref{model}.
Section \ref{plvector} discusses the Pauli--Lubanski vector,
which is an essential ingredient in our non-local field redefinition.
In Secs.\ \ref{shiftoperator} and \ref{Rinverse},
the field redefinition is constructed
and some of its properties including invertibility are established.
Section \ref{eigespinors}
employs these results
for the determination
of explicit expressions for the $b^{\mu}$-model
momentum-space eigenspinors.
A summary and outlook
is contained in Sec.\ \ref{sum}.
Throughout we employ the notation and conventions
of Ref.\ \cite{kl01}.
\section{Review: fermions with a ${\bm \gamma_5 /\hspace{-1.6mm}b}$ term}
\label{model}
This section
reviews various results derived in Ref.\ \cite{ck97}
on the relativistic quantum mechanics
of spin-$\frac{1}{2}$ fermions
with a $\gamma_5 /\hspace{-1.7mm}b$ term.
The starting point is the Lagrangian
\beq{blagrangian}
{\cal L}_b=
\overline{\psi}\left(\frac{1}{2} i/\hspace{-2.75mm}
\stackrel{\leftrightarrow}{\hspace{-1mm}\partial}\hspace{-1.75mm}{}
-m-\gamma_5 /\hspace{-1.7mm}b\right)\psi\;,
\end{equation}
where $m$ is the fermion's mass.
The last term of this Lagrangian
contains a nondynamical spacetime-constant vector $b^\mu$.
This vector coefficient has mass dimensions,
and it explicitly breaks Lorentz and CPT symmetry.
The ordinary Dirac Lagrangian ${\cal L}_0$
is recovered for zero $b^\mu$.
Lagrangian \rf{blagrangian} has recently
received substantial attention
in Lorentz-violation research.
It has been studied
in the context of radiative corrections in Lorentz-breaking electrodynamics \cite{radiative},
finite-temperature field theory \cite{finiteT},
as well as particle-decay processes \cite{decay},
and it might be generated within the ghost-condensate model \cite{ghostcond}.
We also mention
that experiments with spin-polarized matter
have constrained the order of magnitude of $b^\mu$
to be less than $10^{-30}\,$GeV for electrons \cite{spinpol}.
Additional bounds of $10^{-27}\,$GeV for both electrons and protons,
as well as $10^{-31}\,$GeV for neutrons
have been obtained by clock-comparison tests \cite{cc}.
The clean limit of $10^{-25}\,$GeV
on a component of $b^\mu$ for the electron
has been extracted from Penning-trap experiments \cite{trap}.
Under certain assumptions,
the order of magnitude of $b^\mu$
is less than $10^{-20}\,$GeV
for neutrinos \cite{neutrinos}.
Throughout this work,
we shall thus take $b^2\ll m^2$.
The modified Dirac equation
determined by Lagrangian \rf{blagrangian}
is
\beq{bequation}
\left(i/\hspace{-1.87mm}
\partial\hspace{-.2mm}{}
-m-\gamma_5 /\hspace{-1.7mm}b\right)\psi_b(t,\vec{r})=0\;,
\end{equation}
which can be rearranged
to give the Schr\"odinger equation
$i\partial_0\psi_b(t,\vec{r})=H_b\,\psi_b(t,\vec{r})$
with the hermitian Hamiltonian
\beq{bhamiltonian}
H_b=\gamma^0(-i\;\vec{\gamma}\!\cdot\!\vec{\nabla}+m+\gamma_5 /\hspace{-1.7mm}b)\;.
\end{equation}
We denote the
space
of solutions $\psi_b(t,\vec{r})$
to Eq.\ \rf{bequation}
by ${\cal S}_b$.
In the $b^\mu=0$ limit,
the ordinary Dirac case with solutions $\psi_0(t,\vec{r})$
and solution space ${\cal S}_0$ emerges.
For later convenience,
we abbreviate the modified Dirac operator
appearing above by
\beq{DiracOpDef}
D_b\equiv i/\hspace{-1.87mm}\partial\hspace{-.2mm}{}
-m-\gamma_5 /\hspace{-1.7mm}b\;.
\end{equation}
Reversing the sign of the mass term in $D_b$
and applying the resulting operator
from the left
to Eq.\ \rf{bequation}
yields the following modified Klein--Gordon equation:
\beq{bkgequation}
\left(\Box+m^2+b^2+2\gamma_5 \sigma^{\mu\nu}b_\nu\partial_{\nu}\right)\psi_b=0\;.
\end{equation}
With a second operator-squaring procedure,
one can derive the following equation
diagonal in spinor space:
\beq{4equation}
\left[(\Box+m^2+b^2)^2-4b^2\Box+4(b\!\cdot\!\partial)^2\right]\psi_b=0\;.
\end{equation}
A plane-wave ansatz $w(\lambda)\exp(-ix\!\cdot\!\lambda)$ for $\psi_b(x)$
in the above equation yields
\beq{bdr}
(\lambda^2-m^2-b^2)^2+4b^2\lambda^2-4(\lambda\!\cdot\! b)^2=0
\end{equation}
for the fermion's dispersion relation.
For any fixed wave vector $\vec{\lambda}$,
this dispersion relation constitutes a quartic equation
in the plane-wave frequency.
We denote its
four roots by
$(\lambda^{\pm}_a)^{\mu}=\big((\lambda^{\pm}_a)^{0}(\vec{\lambda}),\vec{\lambda}\big)$.
These roots are associated with fermion (superscript $+$) and antifermion (superscript $-$) solutions,
each with two possible spin states ($a=1,2$).
In Sec.\ \ref{plvector},
we will see
that $a$ labels spins parallel and antiparallel to $b^\mu$.
Explicit expressions for $(\lambda^{\pm}_a)^{0}(\vec{\lambda})$
are given in Appendix \ref{roots}.
Jointly with Eq.\ \rf{bequation},
the dispersion relation \rf{bdr}
determines the corresponding eigenspinors $w^{\pm}_a(\vec{\lambda})$.
Most analyses in this work
do not require
a full distinction between all the roots.
For notational convenience,
we then only display the necessary labels.
\section{The Pauli--Lubanski vector}
\label{plvector}
Up to the momentum-operator factor $\sqrt{P^{\mu}P_{\mu}}$,
the Pauli--Lubanski vector $W^{\mu}$
is the relativistic generalization
of a particle's spin.
For Dirac fermions,
this vector is given in position space by \cite{IZ}
\beq{plv}
W^\mu = \frac{1}{2}\gamma_5\sigma^{\mu\nu}\partial_\nu\;.
\end{equation}
Although Lorentz breakdown
precludes the conservation of total angular momentum
in the $b^\mu$ model \rf{blagrangian},
$W^{\mu}W_{\mu}=\frac{1}{2}(\frac{1}{2}+1)\Box$
is evidently conserved:
its gamma-matrix structure is trivial,
and so it commutes with $H_b$.
We remark in passing
that this argument does not employ the modified Dirac Eq.\ \rf{bequation}.
Hence,
$[H_b,W^2]=0$ is not only valid on solutions $\psi_b(t,\vec{r})$
but actually on any sufficiently well-behaved spinor $\psi(t,\vec{r})$.
This essentially means
that the Lorentz-violating $b^\mu$ interaction
leaves unchanged the spin-$\frac{1}{2}$ character
of the particle $\psi_b$,
as expected.
Note
that the above argument only uses
the spacetime independence of the Hamiltonian.
The result therefore generalizes to fermions
with translation-invariant Lorentz violation
in the full SME,
which incorporates the minimal SME.
In the usual Dirac case,
an analogous argument
establishes the conservation of the individual spin components $\sim\!W^{\mu}$.
In the present case,
such an argument fails
due to the absence of Lorentz symmetry.
However,
Lorentz transformations in the plane orthogonal to $b^\mu$
still determine a symmetry of the Lagrangian \rf{blagrangian},
leading us to investigate $W\!\!\cdot\!b$
as a candidate for a conserved spin component.
We begin by analyzing the eigenvalues of $W\!\!\cdot\!b$.
Let $\chi \exp(-ix\!\cdot\!\lambda)$ be an arbitrary momentum eigenspinor
that does not necessarily obey an equation of motion,
i.e.,
both $\chi$ and $\lambda^{\mu}$ are unconstrained.
Any eigenvalue $\Omega$
then satisfies $\det(-\frac{1}{2} i \gamma_5\sigma^{\mu\nu}b_{\mu}\lambda_{\nu}-\Omega)=0$.
We can obtain an explicit expression
for the square of this determinant
as follows.
Note that $\det(M)=\det(CMC^{-1})$
for any $M$ and any invertible $C$,
so that $\det(M)^2=\det(MCMC^{-1})$.
In our case, $M=-\frac{1}{2} i \gamma_5\sigma^{\mu\nu}b_{\mu}\lambda_{\nu}-\Omega$.
If now $C$ is chosen to be the usual charge-conjugation matrix
(e.g., $C=i\gamma^2\gamma^0$ in the Dirac representation),
a diagonal expression for $MCMC^{-1}$ emerges.
This yields a positive and a negative eigenvalue,
both twofold degenerate:
\beq{genev}
\Omega_{a}=\frac{1}{2}(-1)^{a}\sqrt{(\lambda\!\cdot\! b)^2-b^2\lambda^2}\;,
\end{equation}
where $a=1,2$.
It follows
that $W\!\!\cdot\!b$ and $P^{\mu}$ have simultaneous eigenstates $\chi_a(\lambda)\exp(-ix\!\cdot\!\lambda)$.
In order to employ the general result \rf{genev}
within the $b^\mu$ model \rf{blagrangian} and the ordinary Dirac case,
we must still verify
that $W\!\!\cdot\!b$ commutes with $H_b$ and $H_0$,
respectively.
As a first step,
note that $[D_b,W\!\!\cdot\!b]=0$.
We remark in passing
that this shows $W\!\!\cdot\!b\;{\cal S}_b\subset {\cal S}_b$,
a necessary condition for our claim.
Employing $D_b=\gamma^0(i\partial_0-H_b)$
in this commutator gives after some algebra
\beq{commutator}
[H_b,W\!\!\cdot\!b]=\gamma^0[\gamma^0,W\!\!\cdot\!b]D_b\;.
\end{equation}
It is apparent
that this commutator
is in general nonzero,
when it acts on arbitrary spinors $\psi(t,\vec{r})$.
However,
we only need $[H_b,W\!\!\cdot\!b]\,\psi_b(t,\vec{r})=0$
for all $\psi_b(t,\vec{r})$ satisfying Eq.\ \rf{bequation},
which is ensured by the presence of the Dirac operator $D_b$
on the right-hand side of Eq.\ \rf{commutator}.
Thus,
on the solution space ${\cal S}_b$ we indeed have
\beq{rcommutator}
[H_b,W\!\!\cdot\!b]_{{\cal S}_b}=0\;.
\end{equation}
A similar result holds for $H_0$.
We can therefore conclude
that $W\!\!\cdot\!b$ is conserved.
In particular,
simultaneous energy--momentum eigenspinors of $W \!\cdot b$ and $H_b$
as well as simultaneous energy--momentum eigenspinors of $W \!\cdot b$ and $H_0$
exist.
The eigenvalue formula \rf{genev}
is thus applicable in the $b^\mu$ model \rf{blagrangian}
and in the ordinary Dirac case.
We may now employ
the appropriate dispersion relations
to reduce the general eigenvalue expression \rf{genev}
in each of the two specific cases:
for the $b^\mu$ model,
the dispersion relation \rf{bdr} yields
\beq{bev}
\Omega_a=\pm\frac{1}{4}(-1)^a(\lambda_a^2-m^2-b^2)\;,
\end{equation}
and for the usual Dirac field,
$\lambda^2_a=m^2$ gives
\beq{diracev}
\Omega_a=\frac{1}{2}(-1)^a\sqrt{(\lambda_a\!\cdot\! b)^2-m^2b^2}\;.
\end{equation}
Note
that in the $b^{\mu}$ case
the correspondence of the signs is left open.
This issue can be resolved as follows.
The last term in the modified Klein--Gordon equation \rf{bkgequation}
is equal to $4\,W\!\!\cdot\!b$.
With this observation,
the momentum-space version of Eq.\ \rf{bkgequation} becomes
\beq{kgmeq}
(\lambda^2-m^2-b^2-4W\!\!\cdot\!b)w(\lambda)=0\;.
\end{equation}
This fixes the sign ambiguity in Eq.\ \rf{bev},
and one can now write
\beq{bevfixed}
\Omega_a=\frac{1}{4}(\lambda_a^2-m^2-b^2)
\end{equation}
for the eigenvalues of $W\!\!\cdot\!b$
in the $b^\mu$ model.
We remark
that this argument also provides
an independent proof of the fact
that the momentum eigenspinors of $H_b$
are at the same time
eigenstates of $W\!\!\cdot\!b$.
The gamma-matrix structure of $W\!\!\cdot\!b$
is determined by $\sigma^{\mu\nu}$.
Only the $\sigma^{jk}$,
where Latin indices range from 1 to 3,
are hermitian,
so the question arises
as to whether $W\!\cdot b$ is observable,
at least in principle.
It can be answered
with the help of the eigenvalues \rf{diracev} and \rf{bevfixed}.
For the $b^{\mu}$ model,
the eigenvalues \rf{bevfixed} of $W\!\cdot b$ are real
because the hermiticity of $H_b$ implies $\lambda^2\in\mathbb{R}$.
This is compatible
with the observability of $W\!\!\cdot\!b$ in models with nonzero $b^{\mu}$.
For the ordinary Dirac field,
only the second term under the square root in Eq.\ \rf{diracev}
together with a timelike $b^\mu$
can potentially lead to complex $\Omega_a$.
However,
in a coordinate system in which $b^{\mu}=(B,\vec{0})$
one can verify
that $\Omega_a^2\ge0$.
It follows
that also in the conventional Dirac model
the eigenvalues of $W\!\!\cdot\!b$ are consistent
with the observability of this operator.
Another question concerns the inverse of $W\!\!\cdot\!b$.
There are situations in which $\Omega_a=0$ and no inverse exists,
for example,
if $\lambda^{\mu}$ is parallel to $b^\mu$.
This issue is analyzed in more detail
in Appendix \ref{invert}.
But if we exclude such special cases,
we can determine $(W\!\!\cdot\!b)^{-1}$.
We begin
by observing
that $(\sigma_{\mu\nu}A^{\mu}B^{\nu})^2=A^2B^2-(A\!\cdot\! B)^2$
for any two 4-vectors $A^{\mu}$ and $B^{\nu}$.
Thus,
in momentum space we obtain
\bea{msinvwb}
(W\!\!\cdot\!b)^{-1}&=&\frac{-2i\,\gamma_5\sigma^{\mu\nu}b_{\mu}\lambda_{\nu}}{(\lambda\!\cdot\! b)^2-\lambda^2b^2}\nonumber\\
&=&\frac{4\,W\!\!\cdot\!b}{(\lambda\!\cdot\! b)^2-\lambda^2b^2}\;.
\end{eqnarray}
In position space,
we formally write
\beq{psinvwb}
(W\!\!\cdot\!b)^{-1}=\frac{4\,W\!\!\cdot\!b}{b^2\Box-(b\!\cdot\!\partial)^2}\;,
\end{equation}
where the action of the inverse derivative-type operator
on any (well-behaved) position-space function in $f$
is defined explicitly by
\beq{invdo1}
\frac{1}{b^2\Box-(b\!\cdot\!\partial)^2}\;f(x)\equiv
\int\! d^4y\;G(x-y)\;f(y)\;.
\end{equation}
Here, the Green function $G$ is given by
\beq{green}
G(x)=\int_{\cal C}\frac{d^4\lambda}{(2\pi)^4}\frac{e^{-i\lambda\cdot x}}{(\lambda\!\cdot\! b)^2-\lambda^2b^2}\;,
\end{equation}
as usual.
If $(W\!\!\cdot\!b)^{-1}$ acts on function spaces on which it can become singular
(i.e., $W\!\!\cdot\!b=0$),
the freedom in the choice of the contour ${\cal C}$
may be used to select certain boundary conditions.
In the present work,
we will only need to consider
the action of $(W\!\!\cdot\!b)^{-1}$ on ${\cal S}_0$ or ${\cal S}_b$ for $b^2\leq 0$.
The discussion in Appendix \ref{invert} demonstrates
that $(W\!\!\cdot\!b)^{-1}$ is nonsingular
in these situations.
Then,
the contour simply runs along the real-$\lambda^0$ axis,
and any ambiguities in the selection of ${\cal C}$ are absent.
Finally,
consider $\int\!d^4y\;G(x-y)\;\partial f(y)/\partial y^{\mu}$
and integrate by parts.
If $f$ falls off sufficiently fast
and is sufficiently smooth,
the boundary term can be dropped,
we may trade $\partial/\partial y^{\mu}$ for $-\partial/\partial x^{\mu}$,
and then pull this derivative outside the integral.
This shows explicitly
that $W\!\!\cdot\!b$ commutes with $[b^2\Box-(b\!\cdot\!\partial)^2]^{-1}$,
as expected for two spacetime-independent expressions
containing only the momentum operator.
It follows
that there are no ordering ambiguities in Eq.\ \rf{psinvwb}.
\section{The ${\bm b^{\bm \mu}}$-shift operator ${\cal{\bm R}}_{\bm \xi}$}
\label{shiftoperator}
Our goal is
to find an explicit representation of an operator ${\cal R}_\xi(x)$
that maps solutions of the $b^\mu$ model with coefficient $b^\mu$
to solutions of another $b^\mu$ model with shifted coefficient $b^{\mu}+\xi b^{\mu}$,
where $\xi\in\mathbb{R}$.
In other words,
we seek an operator
\beq{RProp}
{\cal R}_\xi: {\cal S}_b\rightarrow{\cal S}_{b+\xi b}\;,
\end{equation}
where the size (but not the direction) of $b^\mu$ is changed.
Only when ${\cal R}_\xi$ acts on solutions of the usual Dirac equation $b^{\mu}=0$,
the above stipulation is meaningless
and must be amended.
In this special case,
it is necessary
to first select a $b^\mu$ vector
that is nonzero but can otherwise be arbitrary.
We then require ${\cal R}_\xi$ to generate solutions of a model with coefficient $\xi b_{\mu}$,
so that ${\cal R}_\xi: {\cal S}_0\rightarrow{\cal S}_{\xi b}$.
For notational simplicity,
we have suppressed the vector character of various subscripts.
{\bf Definition.}
An operator with the properties of ${\cal R}_\xi$ indeed exists:
\beq{rdef}
{\cal R}_\xi(x)=\exp\left(-\xi x_{\mu}\vec{B}^{\mu\nu}\vec{\partial}_{\nu}\right)\;,
\end{equation}
where the projector-type quantity $\vec{B}^{\mu\nu}$
is given by
\beq{Bdef}
\vec{B}^{\mu\nu}=\frac{b^2\eta^{\mu\nu}-b^{\mu}b^{\nu}}{2\,W\!\!\cdot\!b}\;.
\end{equation}
Here,
the coefficient $b^{\mu}\neq 0$
is that of the space ${\cal S}_{b}$
the operator ${\cal R}_\xi$ acts on.
The only exception,
noted in the previous paragraph,
is the special case ${\cal R}_\xi\,{\cal S}_0$,
in which $b^\mu$ can be chosen freely.
At this stage,
the mathematical meaning of Eqs.\ \rf{rdef} and \rf{Bdef}
is somewhat vague.
In the remainder of this section,
we make the above definition more precise
and establish key properties of ${\cal R}_\xi$.
The exponential in Eq.\ \rf{rdef}
is to be understood as a short-hand notation
for the power-series expansion
\beq{Rseries}
\exp\left(-\xi x_{\mu}\vec{B}^{\mu\nu}\vec{\partial}_{\nu}\right)\equiv
\sum_{n=0}^\infty\frac{1}{n!}\left(-\xi x_{\mu}\vec{B}^{\mu\nu}\vec{\partial}_{\nu}\right)^n\;.
\end{equation}
The derivative $\vec{\partial}_{\nu}$
in this expression
is to be taken
with respect to $x_{\mu}$,
the position-space variable.
Since the commutator $[\vec{\partial}_{\nu},x_{\mu}]=\eta_{\mu\nu}$ is nonzero
and products of $x$ and $\partial$ appear above,
their order must be specified.
A similar issue arises for $\vec{B}^{\mu\nu}$ and $x_{\mu}$
because $\vec{B}^{\mu\nu}$ contains
the position-space expression for $W\!\!\cdot\!b$.
In Eq.\ \rf{Rseries},
the operator ordering is defined such
that none of the derivatives and integrations
are acting on the $x_{\mu}$s.
The $n$th term in the series looks therefore as follows:
\beq{nRseries}
\frac{\xi^n}{n!}\left(x_{\mu}\vec{B}^{\mu\nu}\vec{\partial}_{\nu}\right)^n\equiv
\frac{\xi^n}{n!}\prod_{j=1}^n x_{\mu_j}\prod_{k=1}^n \vec{B}^{\mu_j\nu_k}\vec{\partial}_{\nu_k}\;.
\end{equation}
Note
that the operator order
in the second product on the right-hand side of Eq.\ \rf{nRseries}
is irrelevant:
$\vec{B}^{\mu\nu}$ and derivatives commute
since $W\!\!\cdot\!b$ and $\partial$ have simultaneous eigenstates.
As a reminder for the ordering \rf{nRseries},
$\vec{\partial}_{\nu}$ and $\vec{B}^{\mu\nu}$ carry arrows
indicating the direction of action.
The definition of $\vec{B}^{\mu\nu}$
contains $(W\!\!\cdot\!b)^{-1}$,
which may be singular in certain cases.
Since $\vec{B}^{\mu\nu}$ appears always contracted with a 4-gradient,
we may address this issue
by considering $\vec{B}^{\mu\nu}\vec{\partial}_{\nu}$ instead.
Also,
our interest lies in mappings between models
with parallel $b^\mu$ coefficients of different length,
including the case $b^{\mu}=0$.
It therefore suffices
to specify the action of the above combined operator
on a complete set spanning
${\cal S}_b$ and ${\cal S}_0$.
We select the set of plane-wave eigenspinors
$\psi^{\lambda}_a(x)=w_a(\vec{\lambda})\exp(-ix\!\cdot\!\lambda_a)$ of $W\!\!\cdot\!b$,
where $a=1,2$ labels the eigenvalues of $W\!\!\cdot\!b$,
as before.
This gives
\beq{Baction}
\vec{B}^{\mu\nu}\partial_{\nu}\psi^{\lambda}_a(x)
=-iB^{\mu\nu}_a(\vec{\lambda})\;(\lambda_a)_{\nu}\;\psi^{\lambda}_a(x)\;,
\end{equation}
where we have introduced the momentum-space version of $\vec{B}^{\mu\nu}$:
\beq{Bdef}
B^{\mu\nu}_a(\vec{\lambda})\equiv
(-1)^{a}\frac{b^2\eta^{\mu\nu}-b^{\mu}b^{\nu}}{\sqrt{(\lambda_a\!\cdot\! b)^2-b^2\lambda_a^2}}\;.
\end{equation}
To arrive at this result,
we have used Eq.\ \rf{genev}.
Note
that $B^{\mu\nu}_a(\vec{\lambda})$ becomes singular,
when the square root vanishes.
In Appendix \ref{invert},
we show
that this requires
$\lambda^{\mu}$ to be parallel to a timelike $b^\mu$:
$(\lambda^\pm_a)^{\mu}(\vec{\lambda}_0)=\zeta b^{\mu}$,
where $\zeta$ is a dimensionless constant,
$b^2>0$,
and $\vec{\lambda}_0$ is the location of the singularity.
The most natural and straightforward definition of $B^{\mu\nu}_a(\vec{\lambda}_0)$
would employ the limit $\vec{\lambda}\to\vec{\lambda}_0$ at the $W\!\!\cdot\!b$ singularity:
\beq{singularitydef}
B^{\mu\nu}_a(\vec{\lambda}_0) \;\lambda_{\nu}
\equiv\lim_{\vec{\lambda}\to\vec{\lambda}_0}
B^{\mu\nu}_a(\vec{\lambda})\;\lambda_{\nu}\;.
\end{equation}
The remaining task is to determine the limit explicitly.
This is simplified
by working a coordinate system
in which $b^{\mu}=(B,\vec{0})$.
We can decompose any 4-momentum as
$p^{\mu}=\lambda^{\mu}(\vec{\lambda}_0)+\varsigma b^{\mu}+\varepsilon u^{\mu}$,
where $\varsigma$ and $\varepsilon$ are parameters
and $u^{\mu}=(0,\vec{u})$ obeys $u^2=-1$.
Employing this expression
in definition \rf{singularitydef}
where we have to take $\varsigma,\varepsilon\to0$ yields
\beq{singularity}
B^{\mu\nu}_a(\vec{\lambda}_0) \;\lambda_{\nu}
=(-1)^a|B|\;u^{\mu}\;.
\end{equation}
Although this result is finite,
the presence of $u^{\mu}$ indicates
that the limit depends upon the path
by which $(\lambda^\pm_a)^{\mu}(\vec{\lambda}_0)$ is approached.
This non-uniqueness means
that an inverse of $W\!\!\cdot\!b$
is ambiguous
for those states characterized
at the beginning of this paragraph.
Some results in the subsequent sections
require $W\!\!\cdot\!b$
to be invertible,
so that they are only valid
when these states are excluded.
Note,
however,
that this issue only arises for timelike $b^\mu$
and only for a subset of measure zero
in the respective ${\cal S}_b$.
{\bf Useful properties.}
We next establish two basic properties of ${\cal R}_{\xi}(x)$.
The first of these properties concerns the action
of ${\cal R}_{\xi}(x)$ on plane-wave spinors $\psi^{\lambda}_a(x)$
introduced earlier.
Starting from the power-series definition \rf{Rseries},
it is apparent
that the gradients $\vec{\partial}_{\mu}$
(including those in the denominator of $\vec{B}^{\mu\nu}$)
can be replaced by $-i\lambda_{\mu}$
when acting on $\psi^{\lambda}_a(x)$.
The resulting expression
contains no longer derivatives,
operator ordering becomes irrelevant,
and the series can be summed:
\beq{Raction}
{\cal R}_{\xi}(x)\;\psi^{\lambda}_a(x)
=\exp\big(i\xi x_{\mu}B_a^{\mu\nu}(\lambda_a)_{\nu}\big)\;\psi^{\lambda}_a(x)\;.
\end{equation}
The second property
concerns the derivative of ${\cal R}_{\xi}(x)$.
Beginning again with the series \rf{Rseries},
one can verify that
\beq{Rderivative}
\left[\partial^{\mu}{\cal R}_{\xi}(x)\right]=
-{\cal R}_{\xi}(x)\xi\vec{B}^{\mu\nu}\vec{\partial}_{\nu}\;.
\end{equation}
This essentially means
that the symbolic ``exp'' in Eq.\ \rf{rdef}
behaves as a true exponential
with regards to differentiation.
Note,
however,
that the operator ordering matters.
One can also show
that
\beq{prule}
\partial^{\mu}({\cal R}_{\xi}\psi)
=(\partial^{\mu}{\cal R}_{\xi})\psi
+{\cal R}_{\xi}(\partial^{\mu}\psi)\;,
\end{equation}
i.e.,
the usual product rule applies,
as expected.
{\bf Proof of Relation \rf{RProp}.}
We are now in the position
to establish the initial claim
that ${\cal R}_\xi$ changes
the magnitude of $b^\mu$.
It has to be verified
that
\beq{proposition}
D_{b+\xi b}\,{\cal R}_\xi\,\psi_b=0\quad
\textrm{if} \quad D_{b}\,\psi_b=0\;,
\end{equation}
where $D_{b}$ and $D_{b+\xi b}$ are Dirac operators
defined by Eq.\ \rf{DiracOpDef}.
We first use
that $W\!\cdot b$
(which determines the gamma-matrix structure of ${\cal R}_\xi$)
and $D_{b+\xi b}$
commute.
Displaying the spinor indices $c$, $d$, and $f$ for clarity,
one obtains
$(D_{b+\xi b})_{cd}({\cal R}_\xi)_{df}=
(D_{b+\xi b})_{df}({\cal R}_\xi)_{cd}$.
Since $D_{b+\xi b}$ contains the derivative
$i/\hspace{-1.87mm}\partial\hspace{-.2mm}{}$,
the product rule \rf{prule} generates
an additional term
when $(D_{b+\xi b})_{df}$ is moved past $({\cal R}_\xi)_{cd}$.
With the result \rf{Rderivative} at hand,
we then obtain
\beq{step2}
(D_{b+\xi b})_{df}({\cal R}_\xi)_{cd}=
({\cal R}_\xi)_{cd}(D_{b+\xi b}
-i\xi\gamma_{\mu}\vec{B}^{\mu\nu}\vec{\partial}_{\nu})_{df}\;.
\end{equation}
The position of the spinor indices
is now such
that we may convert back to matrix notation.
Moreover,
it can be verified
that $i\gamma_{\mu}\vec{B}^{\mu\nu}\vec{\partial}_{\nu}=-\xi\gamma_5 /\hspace{-1.7mm}b$
on any sufficiently well-behaved spinor $\psi(t,\vec{r})$.
This finally yields
\beq{step3}
D_{b+\xi b}\,{\cal R}_\xi\,\psi_b={\cal R}_\xi(D_{b+\xi b}+\xi\gamma_5 /\hspace{-1.7mm}b)\,\psi_b=D_{b}\,\psi_b=0\;,
\end{equation}
where we have employed the modified Dirac equation $D_{b}\,\psi_b=0$ in the last step.
This demonstrates that
\beq{result}
{\cal R}_\xi\,\psi_b=\psi_{b+\xi b}\;,
\end{equation}
i.e.,
the operator ${\cal R}_\xi$
maps any solution of a model with Lorentz-violating coefficient $b^\mu$
to some solution of a model with coefficient $b^{\mu}+\xi b^{\mu}$.
\section{Inverse of the ${\bm b^{\bm \mu}}$-shift operator}
\label{Rinverse}
Thus far,
we have found
that ${\cal R}_\xi\,{\cal S}_b\subset{\cal S}_{b+\xi b}$.
The goal of this section is
to sharpen this statement.
We will establish
that ${\cal R}_\xi$ determines,
in fact,
a one-to-one correspondence
between the elements of ${\cal S}_b$
and those of ${\cal S}_{b+\xi b}$.
Then,
the inverse of ${\cal R}_\xi$ exists,
and a number of useful insights and applications can be established.
For example,
certain properties and relations derived within a model with a specific $b^\mu$ coefficient
can be mapped to analogous results for other models with more general $b^\mu$.
The basic idea behind establishing the bijectivity of ${\cal R}_\xi$
is the following.
Both the range ${\cal S}_b$ and the domain ${\cal S}_{b+\xi b}$
are spanned by the plane-wave eigenspinors of the respective Dirac equations.
If we can show
that for each eigenspinor in ${\cal S}_b$
there is exactly one eigenspinor in ${\cal S}_{b+\xi b}$,
${\cal R}_\xi$ is one-to-one.
If we can further demonstrate
that each ${\cal S}_{b+\xi b}$ eigenspinor
can be obtained via this mapping,
${\cal R}_\xi$ is onto,
and the claim follows.
We will establish this result in three steps.
In the first step,
we show that eigenspinors are,
in fact,
mapped to eigenspinors.
The second step verifies
that ${\cal R}_\xi$ does not mix the four branches of eigenspinors.
This roughly means
that particles (antiparticles) are mapped to particles (antiparticles)
such that their spin state is left unaffected.
As the final third step,
we demonstrate
that for each of the resulting four maps between branches
the plane-wave momentum is mapped one-to-one and onto.
{\bf Eigenspinors are mapped to eigenspinors.}
Let
$\psi^{\lambda}_a(x)=w_a^{b}(\vec{\lambda})\exp(-ix\cdot\lambda_a)$.
Here,
the spinor superscript $b$
refers to the $b^\mu$ case.
With Eq.\ \rf{Raction},
one can establish
that ${\cal R}_\xi$
inserts an additional plane-wave exponential into the expression for $\psi^{\lambda}_a$:
\beq{Rbasis}
{\cal R}_\xi\;\psi^{\lambda}_a
=w_a^{b}(\vec{\lambda})\exp\big(i\xi x_{\mu} B^{\mu\nu}_a (\lambda_a)_{\nu}\big)\exp(-ix\!\cdot\!\lambda_a)\;.
\end{equation}
Combining the exponentials shows
that
\beq{ldef}
(\Lambda_{a'})^{\mu}\equiv(\lambda_a)^{\mu}-\xi B^{\mu\nu}_a(\lambda_a)_{\nu}
\end{equation}
must be interpreted as the new plane-wave momentum.
Since $(\lambda_a)_{\mu}$ satisfies the dispersion relation \rf{bdr},
$(\Lambda_{a'})^{\mu}$ is constrained as well:
one can verify
that it also satisfies Eq.\ \rf{bdr},
but with $b^\mu$ replaced by $b^{\mu}+\xi b^{\mu}$,
as expected.
We next use the fact \rf{result}
that ${\cal R}_\xi\psi^{\lambda}_a\in{\cal S}_{b+\xi b}$,
which gives
\beq{rel1}
[i/\hspace{-1.87mm}\partial\hspace{-.2mm}{}
-m-(1+\xi)\gamma_5 /\hspace{-1.7mm}b]\;
w_a^{b}(\vec{\lambda})\exp(-ix\!\cdot\!\Lambda_{a'})=0\;.
\end{equation}
It therefore follows
that in addition to its defining relation
$[/\hspace{-2.12mm}\lambda_a\hspace{-.2mm}{}
-m-\gamma_5 /\hspace{-1.7mm}b]\;
w_a^{b}(\vec{\lambda})=0$,
the momentum-space spinor $w_a^{b}(\vec{\lambda})$
also obeys
$[/\hspace{-2.1mm}\Lambda_{a'}\hspace{-.2mm}{}
-m-(1+\xi)\gamma_5 /\hspace{-1.7mm}b]\;
w_a^{b}(\vec{\lambda})=0$.
But this is the definition of $w^{b+\xi b}_{a'}(\vec{\Lambda})$.
We therefore have
\beq{Rspinor}
w_{a'}^{b+\xi b}(\Lambda)={\cal R}_{\xi}\;w_a^{b}(\lambda)\;,
\end{equation}
where $\Lambda_{a'}$ and $\lambda_{a}$
are related by Eq.\ \rf{ldef}.
The above results
lead to the conclusion
that ${\cal R}_{\xi}$
maps plane-wave eigenspinors of $W\!\!\cdot\!b$ for a model with coefficient $b^\mu$
into those for a model with coefficient $b^{\mu}+\xi b^{\mu}$.
The proof
that the map ${\cal R}_\xi:{\cal S}_b\rightarrow{\cal S}_{b+\xi b}$ is a bijection
is now reduced to the following.
We have to show
that each plane-wave eigenspinor in ${\cal S}_b$
corresponds to exactly one plane-wave eigenspinor in ${\cal S}_{b+\xi b}$
and that this correspondence is onto.
{\bf Branches are mapped to branches.}
Any solution space ${\cal S}_b$
contains four distinct branches of eigenspinors
labeled
by the sign of the plane-wave frequency $(\lambda^\pm_a)^0(\vec{\lambda})$
at large wave vectors
\footnote{For $b^{\mu}\neq 0$,
all dispersion-relation branches
have regions with spacelike $(\lambda^\pm_a)^{\mu}$,
and one can always select frames
in which part of a positive-frequency branch
dips below the frequency zero
and vice versa \cite{kl01}.
However,
the sign of the {\em asymptotic} plane-wave frequencies
$(\lambda^\pm_a)^0(|\vec{\lambda}|\to\infty)$
remains unchanged
under any finite coordinate boost.}
and the sign of the $W\!\!\cdot\!b$ eigenvalue $\Omega_a$.
In other words,
there are the usual particle and antiparticle solutions,
each with two possible spin projections along $b^{\mu}$,
as discussed in Sec.\ \ref{model}.
In what follows,
we will show
that ${\cal R}_\xi$ maps branches to branches
without mixing them.
More precisely,
the image of an ${\cal S}_b$ branch
lies on one and only one ${\cal S}_{b+\xi b}$ branch;
the images of any two distinct ${\cal S}_b$ branches
belong to distinct ${\cal S}_{b+\xi b}$ branches.
To this end,
we need to investigate
the behavior of the sign of $(\lambda^\pm_a)^0(\vec{\lambda})$
and the sign of $\Omega_a$
under the mapping ${\cal R}_\xi$.
We first consider the sign of the plane-wave frequency.
Its behavior under ${\cal R}_\xi$
is determined by Eq.\ \rf{ldef}.
For timelike $b^{\mu}$,
we immediately find difficulties.
The projector-type quantity $B^{\mu\nu}_a$ appearing in Eq.\ \rf{ldef}
contains $(W\!\!\cdot\!b)^{-1}$,
which may not exist for certain $\vec{\lambda}$,
as discussed in Sec.\ \ref{shiftoperator}.
For lightlike $b^\mu$,
on the other hand,
such issues are absent.
Equation \rf{ldef} reduces to
\beq{lighlike_map}
(\Lambda^{\pm}_{a'})^{\mu}=
(\lambda^{\pm}_a)^{\mu}-(-1)^a{\rm sgn}(b\cdot\!\lambda^{\pm}_a)\,\xi\, b^{\mu}\; .
\end{equation}
Up to a sign,
${\cal R}_\xi$ just adds the constant vector $b^{\mu}$ to $(\lambda^{\pm}_a)^{\mu}$.
For large $\vec{\lambda}$,
we thus have ${\rm sgn}(\Lambda^{\pm}_{a'})^{0}={\rm sgn}(\lambda^{\pm}_a)^{0}$,
which justifies the $\pm$ label on $(\Lambda^{\pm}_{a'})^{\mu}$.
For spacelike $b^\mu$,
we may select coordinates such that $b^{\mu}=(0,\vec{B})$.
Then, Eq.\ \rf{ldef} becomes
\beq{spacelike_map}
(\Lambda^{\pm}_{a'})^{\mu}=
(\lambda^{\pm}_a)^{\mu}+(-1)^a\xi\fr{\vec{B}^2(\lambda^{\pm}_a)^{\mu}-(\vec{\lambda}\!\cdot\!\vec{B})\,b^{\mu}}
{\sqrt{(\vec{\lambda}\!\cdot\!\vec{B})^2+\vec{B}^2{\lambda^\pm_a}^2}}\; .
\end{equation}
The second term on the right-hand side of Eq.\ \rf{spacelike_map}
has the structure $(\lambda^\pm_a)_{\nu}\,\epsilon^{\mu\nu}$,
where we have defined the tensor
\beq{etensor}
\epsilon_{\mu\nu}\equiv(-1)^a\xi\frac{\vec{B}^2\eta^{\mu\nu}+b^{\mu}b^{\nu}}
{\sqrt{(\vec{\lambda}\cdot\vec{B})^2+\vec{B}^2{\lambda^\pm_a}^2}}\;.
\end{equation}
The $\pm$ label on $(\Lambda^{\pm}_{a'})^{\mu}$ is justified,
if the components of $\epsilon^{\mu\nu}$ are small compared to 1.
The dispersion relation \rf{spacelike_roots} yields
\beq{drresult}
{\lambda^\pm_a}^2=m^2+B^2+2(-1)^a\sqrt{m^2B^2+(\vec{\lambda}\!\cdot\!\vec{B})^2}\;.
\end{equation}
Then,
the minimum of the square root in Eq.\ \rf{etensor} is given by $|\vec{B}|\big(m+(-1)^a|\vec{B}|\big)$,
where we have used our assumption $|b^2|\ll m^2$.
It follows
that the components of $\epsilon_{\mu\nu}$
are ${\cal O}\big(|\vec{B}|/m\big)\ll 1$,
which establishes the desired result.
We have seen above
that the ${\cal R}_\xi$ map leaves unchanged the $\pm$ label of the plane-wave frequencies,
at least for lightlike and spacelike $b^{\mu}$.
We now need to study the behavior of the $a$ label
under this map.
The eigenvalue equation for $\Omega_a$
reads
\beq{eqfev1}
\left[\frac{i}{2} \gamma_5\sigma_{\mu\nu}b^{\mu}(\lambda^{\pm}_{a})^{\nu}+\Omega_a\right]w^{\pm}_a(\vec{\lambda})=0\;.
\end{equation}
Since ${\cal R}_\xi$ just inserts an additional plane-wave exponential
into the position-space eigenspinors,
the eigenvalue equation changes under ${\cal R}_\xi$
according to $(\lambda^{\pm}_{a})^{\mu}\to(\Lambda^{\pm}_{a'})^{\mu}$.
In the case
when ${\cal R}_\xi$ connects two models with nontrivial $b^{\mu}$,
we also need to take $b^{\mu}\to(1+\xi)\,b^{\mu}$
in the expression for $W\!\!\cdot\!b$.
Then, the mapped eigenvalue $\Omega'_{a'}$ is given by
\beq{eqfev2}
\left[\frac{i}{2} \gamma_5\sigma_{\mu\nu}(1+\xi)b^{\mu}(\Lambda^{\pm}_{a'})^{\nu}
+\Omega'_{a'}\right]w^{\pm}_a(\vec{\lambda})=0\;.
\end{equation}
Comparison of the two eigenvalue equations \rf{eqfev1} and \rf{eqfev2} implies
\beq{a_map}
\Omega'_{a'}=(1+\xi)\left(\Omega_{a}-\frac{1}{2}\,\xi\, b^2\right)\;,
\end{equation}
where we have used Eq.\ \rf{ldef}.
The value of $(1+\xi)$ maybe positive or negative,
but it is fixed and does not change, e.g., along on a branch.
So if we can show
that $\Omega_a$ dominates the right-hand side of Eq.\ \rf{a_map},
it will determine the sign of $\Omega'_{a'}$.
In the lightlike $b^{\mu}$ case,
this feature is clear,
and for timelike $b^{\mu}$,
the aforementioned difficulties arise.
The discussion in the previous paragraph implies
$\Omega_a\ge\frac{1}{2}|\vec{B}|\big(m-|\vec{B}|\big)$
for $b^{\mu}=(0,\vec{B})$.
It follows
that the dominance of $\Omega_a$ is also
assured for spacelike $b^{\mu}$.
The above results establish
that ${\cal R}_\xi$ maps an $a$ branch
to a single other branch.
But one might wonder
why the other branch has a different label $a'\neq a$
when $(1+\xi)<0$.
The explanation for this fact simple.
With respect to a fixed $b^{\mu}$,
the spin alignment actually remains fixed under ${\cal R}_\xi$,
i.e., the spin stays either parallel or antiparallel to $b^{\mu}$.
However,
${\cal R}_\xi$ replaces $b^{\mu}\to(1+\xi)\,b^{\mu}$,
so that the projection axis
(and not the spin)
reverses direction
for $(1+\xi)<0$.
This leads to different signs
for $\Omega'_{a'}$ and $\Omega_{a}$.
As mentioned above,
we must slightly modify Eq.\ \rf{a_map}
when either the domain or the range of the ${\cal R}_\xi$ map
involves the space ${\cal S}_0$ of solutions to the conventional Dirac equation.
For example,
$\xi =-1$ maps a model with a nontrivial $b^\mu$
to the usual Dirac case,
but then the right-hand side of Eq.\ \rf{a_map} vanishes.
This issue arises
due the mapping $W\!\!\cdot\!b\to(1+\xi)\,W\!\!\cdot\!b$
in the above derivation of Eq.\ \rf{a_map}.
For mappings between ${\cal S}_0$ and ${\cal S}_b$,
we may instead chose to leave $W\!\!\cdot\!b$ unchanged.
We then obtain
\beq{a_map_special}
\Omega'_{a'}=\Omega_{a}\pm\frac{1}{2}\, b^2\;,
\end{equation}
where the upper and lower signs
refer to the cases with ${\cal S}_0$
as the range or domain,
respectively.
{\bf Momenta map is bijective for each branch.}
As claimed,
we have demonstrated above
that ${\cal R}_\xi$ maps branches to branches,
at least for lightlike and spacelike $b^{\mu}$.
This essentially
establishes the invertibility of ${\cal R}_\xi$
with regards to the spinor degrees of freedom.
It remains to study the $\vec{\lambda}$-momentum degrees of freedom,
a task
that can now be performed branch by branch.
The momentum map is given by Eq.\ \rf{ldef},
and we need to show
that it is onto and invertible.
The timelike $b^{\mu}$ case must again be excluded.
For a lightlike $b^{\mu}$,
Eq.\ \rf{lighlike_map} emerges and clearly shows
that the map is onto.
The map also implies
that $b\cdot\!\lambda^{\pm}_a=b\cdot\!\Lambda^{\pm}_{a'}$,
which ensures invertibility.
For spacelike $b^{\mu}$,
we need to study the second term on the right-hand side of Eq.\ \rf{spacelike_map},
which is given by $\epsilon^{\mu\nu}(\lambda^\pm_a)_{\nu}$.
We have established earlier
that $\epsilon^{\mu\nu}$ is a correction suppressed by at least $|\vec{B}|/m$.
Moreover,
$\epsilon^{\mu\nu}$ is smooth,
so that the map \rf{spacelike_map} must be onto.
When $(\lambda^{\pm}_{a})^{\mu}$ satisfies the usual dispersion relation ${\lambda^\pm_a}^2=m^2$,
the Jacobian for the map \rf{spacelike_map}
is given by
\beq{jacobian}
\left|\frac{\partial (\Lambda_a^\pm)^i}{\partial\lambda^j}\right|
=\left(1+(-1)^a\frac{\vec{B}^2}{\sqrt{(\vec{\lambda}\!\cdot\!\vec{B})^2+m^2\vec{B}^2}}\right)^2.
\end{equation}
Since this Jacobian is strictly nonzero,
the map \rf{spacelike_map} is invertible,
and thus bijective in this situation.
More general mappings $b^{\mu}\to(1+\xi)\,b^{\mu}$ for nonzero $b^{\mu}$
can always be decomposed as $b^{\mu}\to 0 \to (1+\xi)\,b^{\mu}$,
where each step is bijective by the above result.
It follows
that bijectivity is also guaranteed
for arbitrary spacelike $b^{\mu}$.
{\bf Explicit expression for inverse map.}
The above analysis has shown
that for both lightlike and spacelike $b^{\mu}$
the mapping generated by ${\cal R}_\xi$ is bijective.
This implies ${\cal R}_\xi$ has an inverse $R^{-1}_\xi$.
In cases not involving the usual Dirac model with ${\cal S}_0$,
it is natural to expect
that $R^{-1}_\xi=R_{\xi'}$,
where $\xi'$ is defined by $(1+\xi')(1+\xi)=1$:
\beq{inverse}
({\cal R}_\xi)^{-1}=R_{-\xi/(1+\xi)}\;.
\end{equation}
In situations with ${\cal S}_0$,
either as the domain or the range,
we anticipate
\bea{inverse_special}
(R_1)^{-1}&=&R_{-1}\;,\nonumber\\
(R_{-1})^{-1}&=&R_{1}\;,
\end{eqnarray}
where $R_1$
generates ${\cal S}_b$ from the conventional ${\cal S}_0$.
We may verify Eqs.\ \rf{inverse} and \rf{inverse_special}
by demonstrating
that $(R)^{-1}R\;\psi^{\lambda}_{a\pm}=\psi^{\lambda}_{a\pm}$
holds for any plane-wave eigenspinor
$\psi^{\lambda}_{a\pm}(x)=w_a^{\pm}(\vec{\lambda})\exp(-ix\!\cdot\!\lambda^{\pm}_a)$
in the appropriate ${\cal S}_b$ or ${\cal S}_0$.
In other words,
we have to show
that the frequency label $\pm$,
the spin label $a$,
and the wave 4-vector $\lambda^\pm_a$
remain unchanged under $(R)^{-1}R$.
We first note
that the frequency label $\pm$
is indeed unaffected by the map ${\cal R}_\xi$
for any $\xi$,
as established earlier in this section.
If either the domain or the range of ${\cal R}_\xi$
involves the conventional Dirac space ${\cal S}_0$,
the result \rf{a_map_special} shows
that the spin label $a$
is also left unaffected by ${\cal R}_\xi$.
In all other cases,
Eq.\ \rf{a_map} holds.
The twofold iteration of this equation
appropriate in the present situation
proves
that the label $a$ is invariant under $R_{-\xi/(1+\xi)}\,{\cal R}_\xi$,
as desired.
For the plane-wave vector,
we want to invert
\beq{ltransform_special}
(\Lambda_{a})^{\mu}=(\lambda_a)^{\mu}\pm
\frac{b^2(\lambda_a)^{\mu}-(\lambda_a\!\cdot\!\, b)\,b^{\mu}}{2\Omega_a}\;,
\end{equation}
in situations involving the usual Dirac case.
Here, the upper (lower) sign
refers to the case with ${\cal S}_0$
as the range (domain).
One can check that indeed
\beq{linvtransform_special}
(\lambda_{a})^{\mu}=(\Lambda_a)^{\mu}\mp
\frac{b^2(\Lambda_a)^{\mu}-(\Lambda_a\!\cdot\!\, b)\,b^{\mu}}{2\Omega'_a}\;,
\end{equation}
as anticipated.
For all other maps,
i.e., those not involving the conventional Dirac model,
we seek to invert
\beq{ltransform}
(\Lambda_{a})^{\mu}=(\lambda_a)^{\mu}-
\xi\,\frac{b^2(\lambda_a)^{\mu}-(\lambda_a\!\cdot\!\, b)\,b^{\mu}}{2\Omega_a}\;.
\end{equation}
One can again verify
that our expectation
\beq{linvtransform}
(\lambda_{a})^{\mu}=(\Lambda_a)^{\mu}+\frac{\xi}{1+\xi}\,
\frac{b^2(\Lambda_a)^{\mu}-(\Lambda_a\!\cdot\!\, b)\,b^{\mu}}{2\Omega'_{a'}}
\end{equation}
is correct,
which establishes Eqs.\ \rf{inverse} and \rf{inverse_special}.
{\bf Bottom line.}
For lightlike and spacelike $b^\mu$
we have now explicit expressions
for a $b^{\mu}$-shift operator $R_{\xi}$ and its inverse.
This operator connects solutions
of two $b^{\mu}$ models
with parameters $b^\mu$ and $(1+\xi)\,b^\mu$
in a one-to-one fashion.
In particular,
it is possible to map a $b^{\mu}$ model
to the conventional Dirac case,
and vice versa.
A composition of two $b^{\mu}$-shift operators
can thus be used
to map a $b^{\mu}$ model
to a $\tilde{b}^{\mu}$ model
via $b^{\mu}\to 0 \to \tilde{b}^\mu$
for {\em any} coefficients $b^2,\tilde{b}^2\leq 0$.
We remark
that even though a single ${\cal R}_\xi$ maps
a plane-wave eigenspinor in one model
to a single plane-wave eigenspinor in another model,
the above composition of ${\cal R}_\xi$ operators
will typically generate a linear superposition of eigenspinors.
This arises because a single ${\cal R}_\xi$ only scales
the spin-quantization axis $b^{\mu}$,
whereas a composition involving ${\cal S}_0$
can change the direction of $b^{\mu}$,
and thus lead to new spin-projection states.
The ${\cal R}_\xi$ map can be viewed as a field redefinition.
An example of another field redefinition discussed in the literature \cite{ck97}
is $\psi(x)\to e^{-ia\cdot x}\psi(x)$.
This field redefinition does not only remove $a^{\mu}$ from the free $a^{\mu}$ model,
but also from one-flavor QED.
Such a generalization is possible
because $\psi(x)\to e^{-ia\cdot x}\psi(x)$ has two key properties.
First,
the $a^{\mu}$ field redefinition is also defined off-shell.
Second,
it leaves the current $j^{\mu}=\overline{\psi}\gamma^{\mu}\psi$,
and thus the coupling to electrodynamics,
unchanged.
Our ${\cal R}_\xi$ field redefinition
does not seem to exhibit these properties in general.
First,
the action of ${\cal R}_\xi$ is only defined on ${\cal S}_b$,
where $b^{\mu}$ is lightlike, spacelike, or vanishing.
But interactions would require an off-shell extension of the ${\cal R}_\xi$ map.
However,
such an extension may face obstacles
similar to those encountered
in Secs.\ \ref{shiftoperator} and \ref{Rinverse} for timelike $b^{\mu}$.
Second,
the current $j^{\mu}=\overline{\psi}\gamma^{\mu}\psi$,
and thus the coupling to electrodynamics,
is altered by ${\cal R}_\xi$.
It therefore follows
that $b^{\mu}$ cannot be removed from QED.
It is,
in fact,
physical and can in principle be measured.
However,
${\cal R}_\xi$ does have other applications in the free $b^{\mu}$ model,
one of which is discussed in the next section.
\section{Explicit eigenspinors}
\label{eigespinors}
The $b^\mu$-model generalizations of various key features of the usual free Dirac case are known.
Instances of these are the dispersion relation,
the energy--momentum tensor,
certain expressions for spinor projectors,
and the Feynman propagator \cite{ck97,rl04}.
The determination of other generalizations
is often hampered by the complexity
that arises through the inclusion of Lorentz violation.
The momentum eigenspinors for the $b^\mu$ model
are one such example.
Besides approximations,
only the eigenspinors for $b^\mu=(B,\vec{0})$
have been obtained \cite{ck97}.
In this section,
we determine the momentum eigenspinors
for lightlike and spacelike $b^\mu$.
To this end,
we employ the ${\cal R}_\xi$ operator
to map the known eigenspinors in the conventional Dirac case
to those of the desired $b^\mu$ model.
{\bf Conventional eigenspinors.}
The conventional momentum-space eigenspinors
obey $(/\hspace{-2.1mm}\lambda^\pm_a - m)w^\pm_a (\vec{\lambda})=0$.
We may take $w^\pm_a(\vec{\lambda})=(/\hspace{-2.1mm}\lambda^\pm_a + m)W^\pm_a$,
where $W^\pm_a$ is an arbitrary spinor \cite{IZ}.
Since we have
$(/\hspace{-2.1mm}\lambda^\pm_a - m)(/\hspace{-2.1mm}\lambda^\pm_a + m)={\lambda^\pm_a}^2-m^2=0$,
this ansatz satisfies the above defining equation for $w^\pm_a(\vec{\lambda})$.
The choice
\beq{Wspinor}
W^+_a=\left(\begin{array}{c}
\phi_a\\0\\0
\end{array}\right)\;,
\quad W^-_a=\left(\begin{array}{c}
0\\0\\ \chi_a
\end{array}\right)
\end{equation}
ensures that $w^\pm_a(\vec{\lambda})$ is nonzero.
Here,
$\phi_a$ and $\chi_a$ are non-vanishing,
but otherwise arbitrary two-component spinors.
In the Dirac representation for the gamma matrices,
we obtain explicitly
\bea{conventional_spinor}
w^+_a(\vec{\lambda})&=&\frac{1}{\sqrt{2m(E+m)}}
\left(\begin{array}{c}
(E+m)\,\phi_a\\ (\vec{\lambda}\!\cdot\!\vec{\sigma})\,\phi_a
\end{array}\right)\;,\nonumber\\
w^-_a(\vec{\lambda})&=&\frac{1}{\sqrt{2m(E+m)}}
\left(\begin{array}{c}
-(\vec{\lambda}\!\cdot\!\vec{\sigma})\,\chi_a\\ (E+m)\,\chi_a
\end{array}\right)\;,
\end{eqnarray}
where we have set $E=(\lambda^+_a)^0=-(\lambda^-_a)^0=\sqrt{m^2+\vec{\lambda}^2}$.
The factor $1/\sqrt{2m(E+m)}$
has been included for normalization.
These spinors satisfy the orthogonality relations
\beq{orthonorm1}
{w^\pm_a}^\dagger(\vec{\lambda})\, w^\mp_{a'}(\vec{\lambda})=0\;,
\quad{w^\pm_a}^\dagger(\vec{\lambda})\, w^\pm_{a'}(\vec{\lambda})=\frac{E}{m}\,\delta_{aa'}\;,
\end{equation}
if $\phi_a$ and $\chi_a$ are chosen
such that ${\phi_{a}}^{\!\!\dagger}\phi_{a'}={\chi_a}^{\!\!\dagger}\chi_{a'}=\delta_{aa'}$.
The analogous relations
involving the Dirac conjugate spinors $\overline{w}^\pm_a = {w^\pm_a}^\dagger\gamma^0$
are
\beq{orthonorm2}
\overline{w}^\pm_a(\vec{\lambda})\, w^\mp_{a'}(-\vec{\lambda})=0\;,
\quad\overline{w}^\pm_a(\vec{\lambda})\, w^\pm_{a'}(-\vec{\lambda})=\pm\delta_{aa'}\;.
\end{equation}
We remark in passing
that the second of these equations
may also be written as
$\overline{w}^\pm_a(\vec{\lambda})\,w^\pm_{a'}(\vec{\lambda})=\pm\delta_{aa'}$.
Our sign choice in Eq.\ \rf{orthonorm2}
becomes the natural one
after the usual reinterpretation of the negative-energy solutions.
We intend to map the conventional spinors \rf{conventional_spinor}
to a Lorentz-violating model with coefficient $b^\mu=(b^0,\vec{b})$.
It is therefore convenient
to use the remaining freedom in $\phi_a$ and $\chi_a$
to require the $w^\pm_{a}(\vec{\lambda})$ spinors
to be eigenstates of
$W\!\!\cdot\!b$.
For the positive-frequency solutions,
the upper two components of the $W\!\!\cdot\!b$ eigenvalue equation give
\beq{pos_freq_Wb}
(\vec{n}_+\!\cdot\vec{\sigma})\,\phi_a=(-1)^a\,\phi_a\;.
\end{equation}
Here, we have used Eq.\ \rf{diracev}, and we have defined
\beq{lplus}
\vec{n}_+ \equiv \frac{m(E+m)\,\vec{b}-(\lambda\!\cdot\! b+b^0m)\,\vec{\lambda}}
{\sqrt{(\lambda\!\cdot\! b)^2-m^2b^2}\,(E+m)}\;.
\end{equation}
One can check
that $\vec{n}_+$ has unit length
and that the equation for the lower two components
can be derived from Eq.\ \rf{pos_freq_Wb},
as required by consistency.
It follows
that $\phi_a$ must be the eigenvector $\phi_a(\vec{n}_+)$
of $\vec{n}_+\!\cdot\vec{\sigma}$
with eigenvalue $(-1)^a$.
If the spherical-polar angles
that $\vec{n}_+$ subtends are specified as $(\theta,\varphi)$,
then we have explicitly:
\beq{explicitphi}
{}\hspace{-1.5mm}\phi_1(\vec{n}_+)=\left(\begin{array}{c}
e^{-i\varphi}\sin \frac{\theta}{2}\\ -\cos \frac{\theta}{2}
\end{array}\right),
\hspace{1.5mm}\phi_2(\vec{n}_+)=\left(\begin{array}{c}
\cos \frac{\theta}{2}\\e^{i\varphi}\sin \frac{\theta}{2}
\end{array}\right).
\end{equation}
An analogous reasoning for the negative-frequency spinors
yields $\chi_a=\chi_a(\vec{n}_-)$,
where
\beq{nminus}
\vec{n}_-\equiv -\frac{m(E+m)\,\vec{b}-(\lambda\!\cdot\! b-b^0m)\,\vec{\lambda}}
{\sqrt{(\lambda\!\cdot\! b)^2-m^2b^2}\,(E+m)}\;.
\end{equation}
{\bf Eigenspinors for ${\bm b^{\bm \mu}}$ model.}
We can now employ ${\cal R}_\xi$
to map these conventional momentum-space eigenspinors
to those for a $b^\mu$ model.
Equation \rf{Rspinor} shows
that the conventional spinors also satisfy the $b^\mu$ model,
but at a different momentum
determined by ${\cal R}_\xi$.
Hence,
the remaining task is to express the conventional-case $\vec{\lambda}$
in terms of the $b^\mu$-model momentum $\Lambda^{\pm}_{a}$.
The appropriate transformations are the lower-sign relations
in Eqs.\ \rf{ltransform_special} and \rf{linvtransform_special}.
To determine compact and explicit expressions for the ${\cal S}_b$ momentum-space spinors,
we write Eq.\ \rf{linvtransform_special} in the form
\beq{compact}
\lambda^{\pm}_{a}=\Lambda^{\pm}_{a}+\delta^{\pm}_{a}\;,
\end{equation}
where
\beq{deltadef}
(\delta^{\pm}_{a})^{\mu} \equiv (-1)^a\,
\frac{b^2(\Lambda^{\pm}_a)^{\mu}-(\Lambda^{\pm}_a\!\cdot\!\, b)\,b^{\mu}}{\sqrt{(\Lambda^{\pm}_a\!\cdot\!\, b)^2-b^2{\Lambda^{\pm\,2}_a} }}\;.
\end{equation}
The $b^\mu$-model momentum-space spinors $W^\pm_a(\vec{\Lambda})$
are then given by $W^\pm_a(\vec{\Lambda})=w^\pm_a(\vec{\Lambda}+\vec{\delta}^{\pm}_{a})$.
We explicitly obtain for these spinors:
\begin{widetext}
\beq{bmu_spinors}
W^+_a(\vec{\Lambda})=
\frac{1}{\sqrt{2m}}\!\left(\renewcommand{\arraystretch}{2.5}
\begin{array}{c}
\sqrt{m+(\Lambda^{+}_{a})^{0}+(\delta^{+}_{a})^{0}}\;\phi_a(\vec{N}^a_+)\\
\frac{\displaystyle (\vec{\Lambda}+\vec{\delta}^{+}_{a})\!\cdot\!\vec{\sigma}}
{\displaystyle \sqrt{m+(\Lambda^{+}_{a})^{0}+(\delta^{+}_{a})^{0}}}\;\phi_a(\vec{N}^a_+)
\end{array}\right),\quad
W^-_a(\vec{\Lambda})=\frac{1}{\sqrt{2m}}\!
\left(\renewcommand{\arraystretch}{2.5}
\begin{array}{c}
\frac{\displaystyle -(\vec{\Lambda}+\vec{\delta}^{-}_{a})\!\cdot\!\vec{\sigma}}
{\displaystyle \sqrt{m-(\Lambda^{-}_{a})^{0}-(\delta^{-}_{a})^{0}}}\;\chi_a(\vec{N}^a_-)\\
\sqrt{m-(\Lambda^{-}_{a})^{0}-(\delta^{-}_{a})^{0}}\;\chi_a(\vec{N}^a_-)
\end{array}\right).
\end{equation}
Here,
the vectors $\vec{N}^a_+$ and $\vec{N}^a_-$ are given by
\beq{Nvector}
\vec{N}^a_\pm= \pm\frac{m\big[m\pm (\Lambda^{\pm}_{a})^{0}\pm(\delta^{-}_{a})^{0}\big]\vec{b}
-(\Lambda^{\pm}_a\!\cdot\! b\pm b^0m)(\vec{\lambda}+\vec{\delta}^{\pm}_{a})}
{\sqrt{(\Lambda^{\pm}_a\!\cdot\! b)^2-m^2b^2}\,\big(m\pm (\Lambda^{\pm}_{a})^{0}\pm(\delta^{-}_{a})^{0}\big)}\;.
\end{equation}
\end{widetext}
For lightlike $b^{\mu}$,
we have $(\delta^{\pm}_a)^{\mu}=-(-1)^a\,{\rm sgn}(\Lambda^{\pm}_a\cdot b)\,b^{\mu}$,
so that in this case the $b^{\mu}$-model momentum-space spinors
take a relatively simply form.
{\bf Orthogonality relations in the ${\bm b^{\bm \mu}}$ model.}
Let us finally comment on
the orthogonality relations for $W^\pm_a(\vec{\Lambda})$.
Employing Eq.\ \rf{ltransform_special},
we may express $\vec{\Lambda}$ in terms of $\vec{\lambda}$ and write
\beq{Weq}
W^\pm_a(\vec{\lambda}-\vec{\kappa}^\pm_a)=w^\pm_a(\vec{\lambda})\;,
\end{equation}
where
\beq{kappadef}
(\kappa^{\pm}_{a})^{\mu}\equiv(-1)^a\,
\frac{b^2(\lambda^{\pm}_a)^{\mu}-(\lambda^{\pm}_a\!\cdot\!\, b)\,b^{\mu}}{\sqrt{(\lambda^{\pm}_a\!\cdot\!\, b)^2-m^2 b^2 }}\;.
\end{equation}
For notational consistency,
the we rename $\vec{\lambda}\to\vec{\Lambda}$,
which entails $(\lambda^{\pm}_a)^0\to\pm\sqrt{m^2+\vec{\Lambda}^2}\neq(\Lambda^{\pm}_a)^0$.
In what follows,
the dependence of $\kappa^{\pm}_{a}$
on $\vec{\Lambda}$ is understood.
The results of the mapping
of the conventional orthogonality relations \rf{orthonorm1} and \rf{orthonorm2}
to the $b^{\mu}$-model relations
are now given by:
\bea{orthonorm1mapped}
\!\!\!\!\!\!\!\!{W^\pm_a}^\dagger(\vec{\Lambda}-\vec{\kappa}^\pm_a)\,
W^\mp_{a'}(\vec{\Lambda}-\vec{\kappa}^\mp_{a'})&=&0\;,\nonumber\\
\!\!\!\!\!\!\!\!{W^\pm_a}^\dagger(\vec{\Lambda}-\vec{\kappa}^\pm_a)\,
W^\pm_{a'}(\vec{\Lambda}-\vec{\kappa}^\pm_{a'})&=&\frac{\sqrt{m^2+\vec{\Lambda}^2}}{m}\,\delta_{aa'}\;,
\end{eqnarray}
and
\bea{orthonorm2mapped}
\!\!\!\!\!\!\overline{W}^\pm_a(\vec{\Lambda}-\vec{\kappa}^\pm_{a,\vec{\Lambda}})\,
W^\mp_{a'}(-\vec{\Lambda}-\vec{\kappa}^\mp_{a',-\vec{\Lambda}})&=&0\;,\nonumber\\
\!\!\!\!\!\!\overline{W}^\pm_a(\vec{\Lambda}-\vec{\kappa}^\pm_{a,\vec{\Lambda}})\,
W^\pm_{a'}(-\vec{\Lambda}-\vec{\kappa}^\pm_{a',-\vec{\Lambda}})&=&\pm\delta_{aa'}\;.
\end{eqnarray}
For clarity,
we have made explicit the dependence of $\kappa^{\pm}_{a}$
on $\vec{\Lambda}$ in Eq.\ \rf{orthonorm2mapped}.
The four non-vanishing relations in the second line of Eq.\ \rf{orthonorm1mapped}
involve scalar products of two spinors with the {\em same} momentum argument.
It follows
that in these four equations we may shift the momentum arguments
to obtain simpler, more conventional expressions.
However,
the other orthogonality relations
involve spinors with {\em differing} momentum arguments.
The question arises
if these relations would also hold
at equal momentum arguments.
For Eq.\ \rf{orthonorm2mapped},
this is not the case \cite{ck97}.
On the other hand,
the vanishing relations in Eq.\ \rf{orthonorm1mapped}
do possess an equal-argument analogue:
they are eigenvectors of the momentum-space Hamiltonian,
and as such they are orthogonal for any fixed $\vec{\Lambda}$.
A possible degeneracy of the eigenenergies
does not invalidate this conclusion.
With our assumption $b^2\ll m^2$,
such a degeneracy is impossible
for eigenenergies with differing $\pm$ labels.
For energy degeneracies between states with differing $a$ label,
the corresponding spinors are orthogonal
by virtue of being eigenvectors of $W\!\!\cdot\!b$.
We thus have
\bea{orthonorm1mapped2}
\!\!\!\!\!\!\!\!{W^\pm_a}^\dagger(\vec{\Lambda})\,
W^\mp_{a'}(\vec{\Lambda})&=&0\;,\nonumber\\
\!\!\!\!\!\!\!\!{W^\pm_a}^\dagger(\vec{\Lambda})\,
W^\pm_{a'}(\vec{\Lambda})&=&\frac{\sqrt{m^2+(\vec{\Lambda}+\vec{\kappa}^\pm_a)^2}}{m}\,\delta_{aa'}\;.
\end{eqnarray}
We remark
that the above induced spinor normalization
differs from the choice in Refs.\ \cite{ck97,kl01,rl04}.
This is, however,
acceptable
because observables do not depend on
the choice of normalization \cite{ck01}.
We also note
that starting from Eq.\ \rf{orthonorm1mapped2} and mapping back to ${\cal S}_0$
produces orthogonality relations
for the conventional Dirac case.
They are nontrivial,
i.e., involve differing momentum arguments,
for the vanishing relations in Eq.\ \rf{orthonorm1mapped2}.
\section{Summary and outlook}
\label{sum}
The present work
has initiated the study of a novel type of field redefinitions
within the Lorentz- and CPT-violating SME.
As opposed to previously known SME field redefinitions,
the mappings considered here involved infinitely many derivatives
and integrations,
a feature associated with non-locality.
In the context of the Lorentz-violating $b^{\mu}$ model,
we have constructed a non-local operator ${\cal R}_\xi$,
given by Eq.\ \rf{rdef},
that induces an on-shell field redefinition,
such that the redefined field
satisfies an equation of motion with a scaled coefficient $(1+\xi)\,b^{\mu}$.
For lightlike and spacelike $b^{\mu}$,
this field redefinition is bijective.
The special choice $\xi=-1$
therefore permits the removal of Lorentz violation
from the free $b^{\mu}$ model for $b^2\leq 0$.
As for other field redefinitions considered in the literature,
${\cal R}_\xi$ cannot in general be applied in interacting models.
The significance of the ${\cal R}_\xi$ map
lies in the fact,
that it establishes a one-to-one correspondence
between the Lorentz-violating $b^{\mu}$ model
and the conventional Dirac case for $b^2\leq 0$.
This permits the determination and complete characterization
of the solutions to the $b^{\mu}$ model,
which is a prerequisite for many phenomenological studies
including perturbation theory.
As an example of this idea,
we have constructed the previously unknown explicit
momentum-space eigenspinors of the $b^{\mu}$ model
and their generalized orthogonality relations.
These results are contained in
Eq.\ \rf{bmu_spinors} and Eqs.\ \rf{orthonorm1mapped}--\rf{orthonorm1mapped2}.
The present work has opened several avenues for further research.
For instance,
it would be interesting to establish
whether ${\cal R}_\xi$
possesses an off-shell extension.
Such an extension would allow theoretical studies in an interacting model.
Another example for future investigations
is the determination of ${\cal R}_\xi$-type on-shell maps
for other Lorentz-violating SME coefficients.
This would yield
the full characterization of solutions
to the corresponding free SME sector.
Finally,
the spinor analysis in Sec.\ \ref{eigespinors}
could be extended to other conventional features
and their Lorentz-violating analogues in the $b^{\mu}$ model.
\acknowledgments
The author thanks Roman Jackiw for discussions.
This work is supported
by the U.S.\ Department of Energy
under cooperative research agreement No.\ DE-FG02-05ER41360
and by the European Commission
under Grant No.\ MOIF-CT-2005-008687.
|
1,108,101,563,276 | arxiv | \section{Introduction}
The success of machine learning and especially of deep neural networks is obvious and manifold. However, most deep learning algorithms model the underlying system as a \enquote{black box}---they lack explainability. According to the Turing award winner Judea Pearl \cite{pearlTheoreticalImpedimentsMachine2018}, the problem of state-of-the-art artificial intelligence is being \enquote{model-blind} which intrinsically limits its performance on cognitively more demanding tasks. Instead of only considering the observational distributions which current deep learning approaches replicate, he proposes to learn underlying causal mechanisms based on structural causal models (\ac{SCM}).
\paragraph{Causal Modelling and Causal Discovery}
There are recent influential works which take up this idea, for instance \cite{pawlowskiDeepStructuralCausal2020} in which a generative model is set up to replicate the generation of \ac{MNIST} digits by a simple causal model: the thickness a digit is drawn with, the intensity of the respective image and their influences on the final image are taken into account. Having a, in this case at least rudimentary, causal understanding of the generation process, questions like \enquote{How would the image have looked like if the thickness of drawing had been increased?} can be answered, known as counterfactuals. In this example, the causal relations are designed by a human which is of course undesired. It would be much more desirable if the underlying causal mechanisms could be inferred from pure data without any expert knowledge.
This problem is addressed by \emph{causal discovery}. In our work, we restrict ourselves to bivariate structural causal models which are also known as \enquote{cause-effect models}. It is the simplest case which can be seen as a base of multivariate causal graphs which model real-world data. Such a structural causal model \( \mathfrak{C} \) is defined by
\begin{align}
x &\sim p_{x}(x), \\
\label{eq:general-model}
y &= f_{y}(x, \varepsilon_{y}) \quad \text{with} \quad \varepsilon_{y} \sim p_{\varepsilon_{y}}(\varepsilon_{y}) \;\; \text{and} \;\; x \protect\mathpalette{\protect\independenT}{\perp} \varepsilon_{y},
\end{align}
where the common notation \( x \protect\mathpalette{\protect\independenT}{\perp} \varepsilon_{y} \) is used to denote the independence of the random variables \( x \) and \( \varepsilon_{y} \).
\paragraph{Identifiability}
Unfortunately, only having access to samples of the observational distribution \( p(x, y) \) is not sufficient to distinguish cause from effect because another structural causal model \( \tilde{\mathfrak{C}} \),
\begin{align}
y &\sim p_{y}(y), \\
x &= f_{x}(y, \varepsilon_{x}) \quad \text{with} \quad \varepsilon_{x} \sim p_{\varepsilon_{x}}(\varepsilon_{x}) \;\; \text{and} \;\; y \protect\mathpalette{\protect\independenT}{\perp} \varepsilon_{x},
\end{align}
exists which induces exactly the same observational distribution \cite[e.g\onedot} \def\Eg{E.g\onedot\footnote{Comprehensive introductions into causal reasoning are the textbooks \cite{petersElementsCausalInference2017} and \cite{pearlCausalInferenceStatistics2016}, Pearl's monograph \cite{pearlCausalityModelsReasoning2009a} can be considered for a detailed treatise of the subject. As an extensive review on the estimation of bivariate structural causal models we suggest to refer to \cite{mooijDistinguishingCauseEffect2016}.}][]{petersElementsCausalInference2017}. What the general structural causal model lacks is \emph{identifiability}: Only observing samples of \( x \) and \( y \) is not sufficient to distinguish \( x \to y \) from \( y \to x \) since both explanations are possible. But by restricting the functional form of \( f_{y} \) (and \( f_{x} \) for the non-causal \ac{SCM}) in an appropriate way, the correct causal mechanism can still be identified. The most common restriction is the (nonlinear) \emph{additive noise model} \cite{hoyerNonlinearCausalDiscovery2009} in which the effect is given by
\begin{equation}
\label{eq:additive-model}
y = f(x) + \varepsilon_{y}.
\end{equation}
Furthermore, we consider the \emph{post-nonlinear model} \cite{zhangIdentifiabilityPostNonlinearCausal2009} as an extension which introduces an additional (nonlinear) invertible function \( g \) as follows:
\begin{equation}
\label{eq:post-nonlinear-model}
y = g\bigl( \underbrace{f(x) + \varepsilon_{y}}_{\eqqcolon s} \bigr).
\end{equation}
Both classes of models are identifiable except for some special cases \cite{zhangIdentifiabilityPostNonlinearCausal2009}. One sufficient condition for a backward post-nonlinear model to not exist, for example, is \( f \) being non-invertible.
\paragraph{Model Selection Criteria}
There are several methods for identifying cause and effect given the previously stated models. Most of them build upon the \emph{Hilbert-Schmidt Independence Criterion} \cite{grettonKernelStatisticalTest2007}: As \( x \protect\mathpalette{\protect\independenT}{\perp} \varepsilon_{y} \) holds by definition, the estimated residuals
\begin{equation}
\label{eq:residuals}
\hat{r}_{y} = \hat{g}^{-1}(y) - \E \expcond[\big]{\hat{f}(x)}{x}
\end{equation}
need to be independent of \( x \) if \( p(x, y) \) follows a forward post-nonlinear model\footnote{Setting \( g(s) = s \) gives the residuals of the additive noise model. In the following, we always discuss the results for the more general post-nonlinear model.} \cite{hoyerNonlinearCausalDiscovery2009, zhangIdentifiabilityPostNonlinearCausal2009}. The estimates \( \hat{f} \) and \( \hat{g} \) are gained by appropriately regressing \( y \) on \( x \).
One straightforward alternative to the independence test is using the likelihood of \( \mathfrak{C} \) and \( \tilde{\mathfrak{C}} \) as a model selection criterion, i.e\onedot} \def\Ie{I.e\onedot following the maximum likelihood principle. Given a dataset \( \bigl\{ (x_{i}, y_{i}) \bigr\}_{i=1}^{N} \) of \( N \) mutually independent pairs \( (x_{i}, y_{i}) \), we can compare the log-likelihoods
\begin{equation}
\label{eq:L_xy}
L_{x \to y} = \sum_{i=1}^{N} \log \hat{p}^{\mathfrak{C}} \cond{y_{i}}{x_{i}; \vec{\theta}} + \sum_{i=1}^{N} \log \hat{p} (x_{i}; \vec{\phi})
\end{equation}
and
\begin{equation}
\label{eq:L_yx}
L_{y \to x} = \sum_{i=1}^{N} \log \hat{p}^{\tilde{\mathfrak{C}}} \cond{x_{i}}{y_{i}; \tilde{\vec{\theta}}} + \sum_{i=1}^{N} \log \hat{p} (y_{i}; \tilde{\vec{\phi}})
\end{equation}
with each other and decide for \( x \to y \) to be the causal mechanism if \( L_{x \to y} > L_{y \to x} \) and vice versa otherwise. \( \vec{\theta} \), \( \tilde{\vec{\theta}} \), \( \vec{\phi} \) and \( \tilde{\vec{\phi}} \) are those parameters maximising the respective likelihoods. Doing model selection based on the likelihoods is typically infeasible since most expressive models only enable sampling from the density \( \hat{p} \cond{y}{x} \) but not evaluating it. However, our estimation method provides a lower bound of \( \hat{p} \cond{y}{x} \) and thus enables (approximate) likelihood based causal discovery.
\paragraph{Previous Work}
A variety of methods to discover cause and effect from pure observational data have been proposed. Many of these are compared in the extensive review \cite{mooijDistinguishingCauseEffect2016} in which, additionally, new benchmark datasets are introduced for the task of causal discovery in the bivariate case.
Hoyer~et al\onedot, who originally proposed the additive noise model, use standard Gaussian process regression to estimate \( \hat{f} \) in a non-parametric way. This means, the additive noise \( \varepsilon_{y} \) is intrinsically assumed to be Gaussian \cite{hoyerNonlinearCausalDiscovery2009}. Mooij~et al\onedot slightly adapt the regression in their review \cite{mooijDistinguishingCauseEffect2016}. They use \ac{FITC} \cite{quinonero-candelaUnifyingViewSparse2005} as an approximation to exact Gaussian process regression in favour of a reduced computational complexity. The Gaussianity assumption is maintained.
Most influential for our work is \cite{zhangEstimationFunctionalCausal2016} in which post-nonlinear models are studied. The authors claim and illustrate that a wrong assumption on \( p_{\varepsilon_{y}} \) would lead to a biased estimate of \( f \) and \( g \). Therefore, they allow for a Gaussian mixture of the noise \( \varepsilon_{y} \). While \( \hat{f} \) is again the mean of a posterior Gaussian process, the inverse of \( g \) is a member of a parametric family of functions defined by
\begin{equation}
g^{-1}(y; \vec{a}, \vec{b}, \vec{c}) = y + \sum_{i=1}^{k} a_{i} \tanh \bigl( b_{i}(y + c_{i}) \bigr),
\end{equation}
where \( k \) is a hyperparameter and \( \vec{a} \), \( \vec{b} \), \( \vec{c} \) are all contained in \( \vec{\theta} \).
Because of the non-Gaussian noise \( \varepsilon_{y} \) the likelihood is not Gaussian anymore. Zhang~et al\onedot use Monte Carlo Expectation Maximisation (\ac{EM}) to learn the parameters of \( g^{-1} \), \( p_{\varepsilon_{y}} \) and of the process's kernel \cite{zhangEstimationFunctionalCausal2016}.
\paragraph{Our Contribution}
We present an alternative to this method, which uses variational inference and a noise distribution parametrised by normalising flows \cite{papamakariosNormalizingFlowsProbabilistic2019, durkanNeuralSplineFlows2019}. We therefore adapt the method \cite{hensmanScalableVariationalGaussian2015} which was originally proposed in the field of scalable Gaussian process classification. The main contributions of our work can be summarised under the following points:
\begin{itemize}
\item We propose a Gaussian regression method for data which follow post-nonlinear models. We use normalising flows to model an arbitrary noise distribution and combine these with a variational Gaussian process regression approach \cite{hensmanScalableVariationalGaussian2015}. Our method is considerably less time-consuming than the approach in \cite{zhangEstimationFunctionalCausal2016}. It reduces the computation time from more than ten hours to less than one minute.
\item Moreover, we show how the method can be applied to estimate bivariate structural causal models and demonstrate how causal discovery based on the marginal likelihood (more precisely a lower bound of the log marginal likelihood) or well-known independence tests can be performed.
\item We finally compare the usage of classical discovery methods based on the simple additive noise model to our approach on the Tübingen Database with Cause-Effect Pairs \cite{mooijDistinguishingCauseEffect2016} and present a heuristic method which combines both approaches.
\end{itemize}
\section{Estimation of Bivariate \ac{SCM}s}
In this section, we propose a novel regression method to estimate causal mechanisms which can be represented by bivariate post-nonlinear \ac{SCM}s
\begin{equation}
y = g \bigl( f(x) + \varepsilon_{y} \bigr).
\end{equation}
Our discussion starts with normalising flows which provide a framework to approximate complicated noise distributions \( \varepsilon_{y} \) while still enabling to evaluate the estimated density of \( \varepsilon_{y} \).
\subsection{Normalising Flows}
\label{sec:normalising-flows}
Most of the preliminary works on bivariate \ac{SCM}s as \cite{hoyerNonlinearCausalDiscovery2009} and \cite[Section~4.1]{mooijDistinguishingCauseEffect2016} assume the additive noise to be Gaussian distributed, motivated by the central limit theorem. However, Gaussian noise is a strong assumption not always fulfilled in practice\footnote{It is for example well known from a variety of engineering problems, as telecommunications, image processing, underwater acoustics, radar signal processing etc., that the densities of the noise variables are often heavy tailed \cite{nolanUnivariateStableDistributions2020}; not Gaussian but so called \emph{impulsive noise} is corrupting the respective signals.}. By transforming a Gaussian distributed random variable in an appropriate way, various non-Gaussian noise distributions can be realised.
One method to estimate the noise distribution from pure data is the concept of \emph{normalising flows} \cite{papamakariosNormalizingFlowsProbabilistic2019}. Let \( x \) be a continuous random variable with density \( p_{x}(x) \). \( x \) is a transformation \( T \) of a random variable \( u \) distributed according to the \emph{base distribution} \( p_{u}(u) \). Both \( T \) and \( p_{u} \) can be parametric, i.e. \( T(u) = T(u; \vec{\chi}) \) and \( p_{u}(u) = p_{u} (u; \vec{\psi}) \) with the parameter vectors \( \vec{\chi} \) and \( \vec{\psi} \). Typical parameters of the base distribution are the mean \( \mu \) and variance \( \sigma^{2} \) if we choose a normal base distribution as we will do in the following\footnote{We force the mean of the Gaussian base distribution to zero.}. In flow-based models, \( T \) is a diffeomorphism\footnote{i.e\onedot} \def\Ie{I.e\onedot \( T \) is bijective, \( T \) and its inverse \( T^{-1} \) are differentiable.}. It is possible to sample from the model by only sampling from the simple base distribution and applying \( T \). Furthermore, we can analytically evaluate our approximation \( q_{x}(x) \) of the density \( p_{x}(x) \) according to
\begin{equation}
\label{eq:change-of-variables}
q_{x}(x) = p_{u} \bigl( T^{-1}(x) \bigr) \abs*{\frac{\mathrm{d} T^{-1}(x)}{\mathrm{d} x}}
\end{equation}
if \( T^{-1} \) and its derivative are known.
This suffices to fit \( q_{x}(x) \) to given samples of \( x \) using maximum likelihood. There are several appropriate families of transformations \( T \). In particular \emph{linear rational spline flows} \cite{dolatabadiInvertibleGenerativeModeling2020} have proven useful during our investigations. Those define \( T \) piecewise: \( K \) bins of varying width and height span the region \( -B \leq u \leq B \) and \( -B \leq x \leq B \). The number of bins \( K \) and the boundary \( B \) are treated as hyperparameters. Within each bin, a monotonically increasing rational function
\begin{equation}
T(u) = \frac{au + b}{cu + d}
\end{equation}
is defined. The coefficients \( \{ a, b, c, d \} \) can be directly calculated from the parameter vector \( \vec{\phi} \) of the transform. \( \vec{\phi} \) contains the bin widths and heights, the derivatives at the knots (i.e\onedot} \def\Ie{I.e\onedot the intersections of neighbouring bins) and one additional parameter per bin which is necessary to obtain unique solutions for the coefficients of the rational function. See \cite[Algorithm~1]{dolatabadiInvertibleGenerativeModeling2020} for more details.
The estimation of the parameters can be performed iteratively by stochastic gradient descent or, for faster convergence, by Adam \cite{kingmaAdamMethodStochastic2014} or similar advanced stochastic optimisers. Though constructed for the use with high-dimensional data and in combination with other stacked transformations, one rational spline transformation as defined above is sufficiently expressive to (almost perfectly) replicate one-dimensional densities, even if these are multimodal as we will demonstrate in Section~\ref{sec:artificial-data}.
\subsection{Gaussian Processes}
\label{sec:gaussian-processes}
As an example of parametric modelling, a neural network can be used to approximate the nonlinear function \( f \). However, it is difficult to choose the \enquote{complexity} of such models appropriately: Under- or overestimating the required number of layers and neurons per layer results in under- or overfitting, respectively. Gaussian processes, on the contrary, are non-parametric and as all Bayesian methods, the additional prior information enables an intrinsic \enquote{smoothing}. Especially on small datasets, where the higher computational complexity can be afforded, Gaussian process regression is known to perform better than regression based on neural networks.
We therefore model the nonlinear function \( f \) by the mean function of a Gaussian process \cite{rasmussenGaussianProcessesMachine2006}. First, an important note regarding our notation: The function \( f \) forming the additive model \eqref{eq:additive-model} was considered deterministic. When modelling \( f \) by a Gaussian process, however, the values \( f_{i} \) which correspond to the given locations \( x_{i} \), i.e. \( f_{i} = f(x_{i}) \), are \emph{not} deterministic anymore. Moreover, we do not think of \( f(x_{i}) \) as a deterministic transform of some random \( x_{i} \). We rather place a Gaussian process prior on \( f \) which results in \( \rowvect{f_{i}, \dotsc, f_{N}}^{\mathsf{T}} \) being jointly Gaussian distributed. \( x_{i} \) additionally follows some distribution which is of less importance at this point as we only consider distributions conditioned on \( x_{i} \). The deterministic \( f \) addressed in \eqref{eq:additive-model} is estimated by the mean of the posterior. Though also denoting this function by \( f \) is slightly abusive, we are convinced that this does not impair the understanding.
For the prior \( f \sim \mathcal{GP}\bigl( m(x), k(x, x') \bigr) \), we choose \( m(x) \equiv 0 \) and the squared exponential kernel\footnote{We have also experimented with members of the Mat\'ern family because (the more popular) squared exponential kernels are known to lead to very smooth processes which rarely occur in physics
\cite{steinInterpolationSpatialData1999} (as cited in \cite{rasmussenGaussianProcessesMachine2006}). However, we could not observe a significant benefit from these kernels, so we left it at the squared exponential kernels.}
\begin{align}
\label{eq:kernel}
k(x, x') = \exp \left( -\frac{1}{2l^{2}} (x - x')^{2} \right).
\end{align}
If we restricted ourselves to Gaussian noise \( \varepsilon_{y} \sim \mathcal{N}(0, \sigma_{\varepsilon}^{2}) \), the log marginal likelihood
\begin{align}
\label{eq:log-marginal-likelihood}
\log p \cond{\vec{y}}{\vec{x}; \vec{\theta}}
\notag
&= -\frac{N}{2} \log (2\pi) - \frac{1}{2} \log\,\abs*{ \matr{K}_{\vec{x}, \vec{x}} + \sigma_{\varepsilon}^{2} \matr{I} } \\
& \qquad- \frac{1}{2} \vec{y}^{\mathsf{T}} \left[ \matr{K}_{\vec{x}, \vec{x}} + \sigma_{\varepsilon}^{2} \matr{I} \right]^{-1} \vec{y}
\end{align}
would be given by an analytical expression, where we subsumed \( l \) and \( \sigma^{2}_{\varepsilon} \) under the parameter vector \( \vec{\theta} = [l, \sigma_{\varepsilon}^{2}]^{\mathsf{T}} \). In the above expression, \( \vec{x} \) and \( \vec{y} \) contain all observed data, \( \matr{K}_{\vec{x}, \vec{x}'} \) denotes the Gram matrix with respect to the kernel \( k(x, x') \). Equation~\eqref{eq:log-marginal-likelihood} could be maximised to determine \( \vec{\theta} \) which gives the best fit of the process to the data.
\subsection{Variational Gaussian Process Regression}
\label{sec:vari-gauss-proc}
However, in general we intend to model non-Gaussian additive noise. Motivated by applications in classification instead of regression, Hensman~et al\onedot proposed a variational lower bound on the aforementioned log marginal likelihood in equation~\eqref{eq:log-marginal-likelihood} \cite{hensmanScalableVariationalGaussian2015}. We can directly apply their results when modelling \( \varepsilon_{y} \) by normalising flows. For a better understanding, we shortly review \cite{hensmanScalableVariationalGaussian2015}'s work. For sake of ease, we omit to explicitly state that the following densities are conditioned on the input locations \( \vec{x} \). Moreover, we first neglect the nonlinearity \( g \). In Section~\ref{sec:extens-post-nonl} we will discuss how to easily adapt the procedure to allow for estimating \( g \) hand in hand with \( f \) and \( \varepsilon_{y} \).
\begin{itemize}
\item Inference based on exact Gaussian process regression scales with \( \mathcal{O}(N^{3}) \). Sparse approximate methods \cite{quinonero-candelaUnifyingViewSparse2005} introduce \( M \ll N \) additional latent variables, called \emph{inducing variables}. They are stacked into \( \vec{u} \) and \enquote{compress} the information contained in \( \vec{f} \), the (unnoisy) function values of the Gaussian process at \( \vec{x} \). \( \vec{u} \) is in a similar way related to the inducing input points \( \vec{z} \) which are learned when maximising the lower bound on the marginal likelihood later on.
\item Further bounding the evidence lower bound (\ac{ELBO}), the authors derive
\begin{align}
\log p(\vec{y})
\label{eq:final-lower-bound}
\notag \geq \mathcal{L} &\coloneqq \sum_{i=1}^{N}\ \E_{q(f_{i})} \bigl[ \log p \cond{y_{i}}{f_{i}} \bigr] \\
&\qquad - D_{\mathrm{KL}} \divergence[\big]{q(\vec{u})}{p(\vec{u})}
\end{align}
where they introduced the abbreviation
\begin{displaymath}
q(f_{i}) \coloneqq \int p \cond{f_{i}}{\vec{u}} q(\vec{u}) \,\mathrm{d} \vec{u}.
\end{displaymath}
\( q(\vec{u}) \), the variational approximation of \( p \cond{\vec{u}}{\vec{y}} \), is chosen to be Gaussian,
\begin{equation}
q(\vec{u}) = \mathcal{N} (\vec{u}; \vec{m}, \matr{S}),
\end{equation}
with the mean \( \vec{m} \) and the covariance matrix \( \matr{S} \). \( q(f_{i}) \) is therefore Gaussian as well and the expectations with respect to it can be numerically determined using the \emph{Gauss-Hermite quadrature} \cite[Section~25.4.46]{abramowitzHandbookMathematicalFunctions1964}.
Furthermore, the second term of the bound, the \ac{KL}~divergence, can be analytically derived:
\begin{align}
D_{\mathrm{KL}} \divergence[\big]{q(\vec{u})}{p(\vec{u})}
\notag &= \frac{1}{2} \Bigl( \Tr \bigl( \matr{K}_{\vec{z}, \vec{z}} \matr{S} \bigr) - M \\
\notag &\qquad\quad + \log\, \abs{\matr{K}_{\vec{z}, \vec{z}}} - \log\, \abs{\matr{S}} \\
&\qquad\quad + \vec{m}^{\mathsf{T}} \matr{K}_{\vec{z}, \vec{z}}^{-1} \vec{m} \Bigr).
\end{align}
\end{itemize}
As we model \( \varepsilon_{y} \) by a spline flow, we have direct access to an analytical expression of
\begin{equation}
\log p \cond{y_{i}}{f_{i}} = \log p_{\varepsilon_{y}}\bigl( y_{i} - f_{i} \bigr).
\end{equation}
To fit the process, we iteratively maximise \eqref{eq:final-lower-bound} with respect to the following parameters:
\begin{itemize}
\item \( \vec{m} \) and \( \matr{S} \)\footnote{To enable an unconstrained optimisation, the Cholesky decomposition \( \matr{L} \) with \( \matr{S} = \matr{L} \matr{L}^{\mathsf{T}} \) is typically chosen as a parameter instead of \( \matr{S} \) itself.} which serve as the mean and covariance matrix of the variational approximation \( q(\vec{u}) \) of \( p \cond{\vec{u}}{\vec{y}} \);
\item \( \vec{z} \), the locations the inducing variables \( \vec{u} \) are associated with;
\item \( \vec{\theta} \) which comprises the parameters of the kernel and the likelihood, i.e. \( l \) in case of the kernel \eqref{eq:kernel}, the parameters \( \sigma^{2} \) and \( \vec{\chi} \) of our base distribution and of the linear rational spline transform, respectively.
\end{itemize}
To replicate data points by the learned model, we draw samples of the posterior predictive distribution
\begin{equation}
p \cond{\vec{f}_{*}}{\vec{y}}
\approx \int p \cond{\vec{f}_{*}}{\vec{u}} q(\vec{u}) \,\mathrm{d} \vec{u}.
\end{equation}
\( \vec{f}_{*} \) denotes the latent function values at (unseen) input locations \( \vec{x}_{*} \). As \( \vec{f}_{*} \) and \( \vec{u} \) are jointly Gaussian,
\begin{align}
p \cond{\vec{f}_{*}}{\vec{u}}
\notag &= \mathcal{N} \bigl( \vec{f}_{*}; \matr{K}_{\vec{x}_{*}, \vec{z}} \matr{K}_{\vec{z}, \vec{z}}^{-1} \vec{u}, \\
&\qquad\quad \matr{K}_{\vec{x}_{*}, \vec{x}_{*}} - \matr{K}_{\vec{x}_{*}, \vec{z}} \matr{K}_{\vec{z}, \vec{z}}^{-1} \matr{K}_{\vec{z}, \vec{x}_{*}} \bigr)
\end{align}
can be derived analytically.
\subsection{Extension to Post-Nonlinear Model}
\label{sec:extens-post-nonl}
Adapting the presented regression method such that it can cope with the post-nonlinear model is not demanding: The fact that \( g \) needs to be an invertible function enables us to parametrise it by another rational spline flow. The likelihood \( p \cond[\big]{y}{f(x)} \), which plays a major role during regression, is still tractable. To see this, we first rename the (now latent) variable \( f(x) + \varepsilon_{y} \) to \( s \). Under this new notation,
\begin{equation}
\log p_{s} \cond[\big]{s}{f(x)} = \log p_{\varepsilon_{y}} \bigl( s - f(x) \bigr)
\end{equation}
is already known. With \( g(s) \) being a parametric function \( g_{\vec{\varphi}}(s) \) with its inverse \( s = g_{\vec{\varphi}}^{-1}(y) \), the desired likelihood is given by
\begin{align}
\log p \cond[\big]{y}{f(x)}
\notag &= \log p_{\varepsilon_{y}} \Bigl( g_{\vec{\varphi}}^{-1}(y) - f(x) \Bigr) \\
&\qquad + \log\, \abs*{\frac{\mathrm{d} g_{\vec{\varphi}}^{-1}(y)}{\mathrm{d} y}}.
\end{align}
As we again choose linear rational splines to model \( g \), its parameters \( \vec{\varphi} \) contain, among others, the bin widths and heights and the derivatives at the knots. These modifications are all it takes to expand our regression method to the post-nonlinear model.
\section{Identification of Cause and Effect}
\label{sec:ident-cause-effect}
In the following, we discuss how the presented regression technique can be applied to the initial problem of identifying cause and effect in the bivariate case.
To this end, we regress \( y \) on \( x \) to estimate \( \mathfrak{C} \) and \( x \) on \( y \) to estimate \( \tilde{\mathfrak{C}} \) given a normalised\footnote{\( x \) and \( y \) have zero mean and unit variance. With normalised datasets, we benefit from hyperparameters of the spline flows that are easier to set, especially the bound \( B \).} dataset \( \bigl\{ (x_{i}, y_{i}) \bigr\}_{i=1}^{N} \). For this we maximise \eqref{eq:final-lower-bound} with respect to the model's parameters \( \vec{\theta} \) (and the auxiliary variables \( \vec{m} \), \( \matr{S} \), \( \vec{z} \) of the variational approach). This lower bound is the first contribution to the log-likelihood \( L_{x \to y} \) (or \( L_{y \to x} \) for the backward model). The other contribution is dependent on the marginal density of \( x \) (or \( y \) for the backward model) which we also estimate by rational spline flows. Given those, we can select \( \mathfrak{C} \) or \( \tilde{\mathfrak{C}} \) based on maximum likelihood.
After the optimisation, we have access to the posterior predictive density of the Gaussian process. Its mean is \( \E \expcond[\big]{\hat{f}(x_{i})}{x_{i}} \) which we need to test for the independence of the residuals \( \hat{r}_{y_{i}} \) \eqref{eq:residuals}
and \( x_{i} \) (in the case of \( \mathfrak{C} \); and accordingly \( \hat{r}_{x_{i}} \) and \( y_{i} \) for \( \tilde{\mathfrak{C}} \)). Checking for independence of \( \bigl\{ (x_{i}, r_{y_{i}}) \bigr\}_{i=1}^{N} \) is done by means of the \ac{HSIC}. It suffices to only consider the \( p \)-value \cite{hoyerNonlinearCausalDiscovery2009} (or, alternatively, the test statistic \cite{mooijDistinguishingCauseEffect2016}) of the test. A high \( p \)-value (a low test statistic) corresponds to a high probability of falsely rejecting the independence. Based on this criterion, we regard that model to be the causal one which leads to a higher \( p \)-value (a lower test statistic) in the hypothesis test.\footnote{Sometimes, the \( p \)-value for both \( \mathfrak{C} \) and \( \tilde{\mathfrak{C}} \) is high and comparable which can be interpreted as follows: The data fit to post-nonlinear models \( \mathfrak{C} \) \emph{and} \( \tilde{\mathfrak{C}} \). A possible reasoning might be that the true data generating process meets a non-identifiable post-nonlinear model as listed in \cite{zhangIdentifiabilityPostNonlinearCausal2009}.}
To sum up, our regression method enables causal discovery by means of two criteria: maximum likelihood and the independence of cause and residuals. Unfortunately, using the lower bound of the likelihood as the model selection criterion underperforms on practical examples as will be demonstrated in Section~\ref{sec:tubing-datab-with}.
\section{Experiments}
\label{sec:experiments}
The applicability of our estimation approach is now demonstrated on both artificial data and the real-world cause-effect pairs of the Tübingen database.
All of the following simulation results are generated using a squared exponential kernel as given in equation~\eqref{eq:kernel}. Furthermore, the hyperparameters of all involved rational splines are set to \( K = 21 \) and \( B = 10 \). We perform \num{1000} steps of stochastic gradient descent.
\subsection{Artificial Data}
\label{sec:artificial-data}
We start with the following artificial example: Let \( p(\varepsilon_{y}) \) be a multimodal density comprising two exponential densities:
\begin{equation}
\label{eq:ex-1}
p(\tilde{\varepsilon}_{y}) =
\begin{dcases}
0, & \tilde{\varepsilon}_{y} < -3, \\
0.7\mathrm{e}^{-(\tilde{\varepsilon}_{y}+3)}, & -3 \leq \tilde{\varepsilon}_{y} < 7, \\
0.7\mathrm{e}^{-(\tilde{\varepsilon}_{y}+3)} \\
\quad+ 0.3 \mathrm{e}^{-(\tilde{\varepsilon}_{y} - 7)}, & \tilde{\varepsilon}_{y} \geq 7.
\end{dcases}
\end{equation}
This density corresponds to the unnormalised noise variable \( \tilde{\varepsilon}_{y} \). We normalise our dataset such that both \( x \) and \( y \) have zero mean and unit variance. The density \( \tilde{\varepsilon}_{y} \) is shown in Figure~\ref{fig:regression-exponential-noise-3}.
The full process we want to analyse in this example reads
\begin{align}
\label{eq:ex-2}
\tilde{x}_{i} &\sim \mathcal{N}(2, 1), \\
\label{eq:ex-3}
\tilde{s}_{i} &= \tilde{x}_{i}^{2} + \tilde{\varepsilon}_{y, i} \quad \text{with} \quad \tilde{\varepsilon}_{y, i} \sim p(\tilde{\varepsilon}_{y}), \\
\label{eq:ex-4}
\tilde{y}_{i} &= \log (\tilde{s}_{i} + 4).
\end{align}
We draw \( 500 \)~samples from the model. The resulting normalised dataset is shown in Figure~\ref{fig:regression-exponential-noise} along with the regression results of our method. Apparently, it leads to a suitable explanation of the data.
For the simulation, we utilised \( M = 20 \) inducing variables. Using more inducing variables leads to better results; \( M \lesssim 5 \), on the contrary, is a too high \enquote{compression}.
\begin{figure}
\centering
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\def-11mm{-11mm}
\begin{tikzpicture}
\begin{groupplot}[
group style = {
rows = 4, columns = 2,
horizontal sep = 1.5cm,
vertical sep = 1.8cm,
},
width = 0.5\linewidth, height = 0.42\linewidth,
ylabel style = {at = {(-0.22, 0.5)}}
]
\nextgroupplot[
xlabel = \( x \), ylabel = \( y \),
xtick distance = 3,
xmin = -4.5, xmax = 4.5, ymin = -4.5, ymax = 3.5]
\addplot[only marks, mark size=0.75pt, myblue, opacity=0.2, draw opacity=0] table {plot_data/example_dataset.txt};
\nextgroupplot[
xtick distance = 3,
xlabel = \( x \), ylabel = \( y \),
xmin = -4.5, xmax = 4.5, ymin = -4.5, ymax = 3.5]
\addplot[only marks, mark size=0.75pt, myred, opacity=0.2, draw opacity=0] table {plot_data/example_dataset_replication.txt};
\nextgroupplot[
xlabel = \( \tilde{x} \), ylabel = \( \tilde{f}(\tilde{x}) \),
xmin = -0.9, xmax = 3.9, ymin = -1.5, ymax = 12.9]
\addplot[myblue, samples=200] {x^2};
\nextgroupplot[
xlabel = \( x \), ylabel = \( f(x) \),
xmin = -2.9, xmax = 2, ymin = -1.5, ymax = 1.9]
\addplot[draw = none, name path = upper] table {plot_data/example_pred_lower.txt};
\addplot[draw = none, name path = lower] table {plot_data/example_pred_upper.txt};
\addplot[opacity = 0.1] fill between[of = lower and upper];
\addplot[myred] table {plot_data/example_pred_mean.txt};
\nextgroupplot[
xlabel = \( \tilde\varepsilon_{y} \), ylabel = \( p(\tilde\varepsilon_{y}) \),
xmin = -7, xmax = 14, ymin = -0.2, ymax = 0.9
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\addplot[myblue] table {plot_data/example_noise_pdf.txt};
\nextgroupplot[
xlabel = \( \varepsilon_{y} \), ylabel = \( p(\varepsilon_{y}) \),
xmin = -2.3, xmax = 1.9, ymin = -0.7, ymax = 3.5
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\addplot[myred] table {plot_data/example_est_noise_pdf.txt};
\nextgroupplot[
xlabel = \( \tilde s \), ylabel = \( \tilde g(\tilde s) \),
xmin = -3.5, xmax = 2.5, ymin = -1.5, ymax = 2.9]
\addplot[myblue, samples=200] {ln(x + 4)};
\nextgroupplot[
xlabel = \( s \), ylabel = \( g(s) \),
xmin = -2.5, xmax = 3.1, ymin = -3.5, ymax = 2.9]
\addplot[myred, samples=200] table {plot_data/example_post.txt};
\end{groupplot}
\node[text width = 10mm] at ([yshift=-11mm] group c1r1.south) {\subcaption{\label{fig:regression-exponential-noise-1}}};
\node[text width = 10mm] at ([yshift=-11mm] group c1r2.south) {\subcaption{\label{fig:regression-exponential-noise-2}}};
\node[text width = 10mm] at ([yshift=-11mm] group c1r3.south) {\subcaption{\label{fig:regression-exponential-noise-3}}};
\node[text width = 10mm] at ([yshift=-11mm] group c1r4.south) {\subcaption{\label{fig:regression-exponential-noise-4}}};
\node[text width = 10mm] at ([yshift=-11mm] group c2r1.south) {\subcaption{\label{fig:regression-exponential-noise-5}}};
\node[text width = 10mm] at ([yshift=-11mm] group c2r2.south) {\subcaption{\label{fig:regression-exponential-noise-6}}};
\node[text width = 10mm] at ([yshift=-11mm] group c2r3.south) {\subcaption{\label{fig:regression-exponential-noise-7}}};
\node[text width = 10mm] at ([yshift=-11mm] group c2r4.south) {\subcaption{\label{fig:regression-exponential-noise-8}}};
\end{tikzpicture}
\caption[Regression on post-nonlinear model with multimodal noise]{Regression on post-nonlinear model with multimodal noise. The (normalised) data points are highlighted by small transparent dots in panel~\subref{fig:regression-exponential-noise-1}; \subref{fig:regression-exponential-noise-2}--\subref{fig:regression-exponential-noise-4} visualise the (unnormalised) data generating process defined by equations~\eqref{eq:ex-1}--\eqref{eq:ex-4}. The results when using our variational approach are displayed in \subref{fig:regression-exponential-noise-5}--\subref{fig:regression-exponential-noise-8}. These panels show the replication of the dataset (i.e\onedot} \def\Ie{I.e\onedot \num{500}~samples drawn from the learned model), posterior predictive distribution (visualised by its mean and confidence interval) the estimated noise distribution and the estimated post-nonlinear function, respectively.
Remark: The different scale and shift of the estimated functions is only partly caused by the normalisation of the data. Because any affine function chained with \( g \) can be fully merged into \( f \) and \( \varepsilon_{y} \) without changing \( p(x, y) \), the original scale cannot be recovered.}
\label{fig:regression-exponential-noise}
\end{figure}
\subsection{Tübingen Database with Cause-Effect Pairs}
\label{sec:tubing-datab-with}
Whether causal discovery works for real-world problems, is typically assessed by means of a (growing) dataset presented in the article \cite[Appendix~D]{mooijDistinguishingCauseEffect2016}.
We first pick two cause-effect pairs of the database which can be regarded as representative for opportunities of the more general post-nonlinear model compared to the additive noise model, on the one hand, and corresponding difficulties with respect to causal discovery, on the other hand.
\begin{enumerate}[i), nosep, listparindent = \parindent, itemsep = 0.25\baselineskip, topsep = 0.5\baselineskip]
\item \label{item:1} \emph{Causal discovery based on the post-nonlinear model succeeds---causal discovery based on the additive noise model fails:}
In general, there are a variety of functional forms which \( f_{y} \) as defined in equation~\eqref{eq:general-model} can take. Whether a simple additive noise model \eqref{eq:additive-model} suffices, is not known a priori. If so, any more complicated function \( f_{y} \) can be regarded as less plausible according to Occam's Razor \cite{hoyerNonlinearCausalDiscovery2009} (see also \cite[Principle~1]{mooijDistinguishingCauseEffect2016}).
One cause-effect pair in which neither \( x \to y \) (the true causal direction) nor \( y \to x \) follows an additive noise model, is pair~2 where the altitude (\( x \)) of a weather station of the Deutscher Wetterdienst (\ac{DWD}) and the corresponding mean annual precipitation (\( y \)) were collected. Figure~\ref{fig:alt_prec} shows that a second nonlinearity (which is monotonically increasing) is required to model the data properly. The data points drawn from the estimated post-nonlinear model are apparently closer to the actual data than data points drawn from the estimated additive noise model. We can confirm this observation quantitatively by means of the respective likelihoods. The lower bound on \( L_{x \to y, \text{\subref{fig:alt_prec-1}}} \) for the post-nonlinear model is approximately \num{-1.72} and thus significantly higher than the respective (unbounded) log-likelihood \( L_{x \to y, \text{\subref{fig:alt_prec-2}}} = -1.93 \).
\begin{figure}[t]
\centering
\def10mm{10mm}
\def-11mm{-11mm}
\begin{tikzpicture}
\begin{groupplot}[
group style = {
rows = 1, columns = 2,
horizontal sep = 1.5cm,
vertical sep = 2cm,
},
width = 0.5\linewidth, height = 0.42\linewidth,
]
\nextgroupplot[
xlabel = \( x \), ylabel = \( y \),
xmin = -2.5, xmax = 8.5, ymin = -2.5, ymax = 4.5]
\addplot[only marks, mark size=0.75pt, myblue, opacity=0.2, draw opacity=0] table {plot_data/alt_prec_dataset.txt};
\addplot[only marks, mark size=0.75pt, myred, opacity=0.2, draw opacity=0] table {plot_data/alt_prec_dataset_replication.txt};
\nextgroupplot[
xlabel = \( x \), ylabel = \( y \),
xmin = -2.5, xmax = 8.5, ymin = -2.5, ymax = 4.5]
\addplot[only marks, mark size=0.75pt, myblue, opacity=0.2, draw opacity=0] table {plot_data/alt_prec_dataset.txt};
\addplot[only marks, mark size=0.75pt, myred, opacity=0.2, draw opacity=0] table {plot_data/alt_prec_dataset_replication_gauss.txt};
\end{groupplot}
\node[text width = 10mm] at ([yshift=-11mm] group c1r1.south) {\subcaption{\label{fig:alt_prec-1}}};
\node[text width = 10mm] at ([yshift=-11mm] group c2r1.south) {\subcaption{\label{fig:alt_prec-2}}};
\end{tikzpicture}
\caption[Investigation of pair~2 (altitude~\( \to \)~precipitation)]{Investigation of pair~2 (altitude~\( \to \)~precipitation): The data can be better explained by allowing for a bijective post-nonlinear function. The replication of the original dataset (small blue dots) by generating samples from the estimated post-nonlinear model using our proposed method (small red dots in panel~\subref{fig:alt_prec-1}) suits better compared to samples taken from the estimated additive noise model (small red dots in panel~\subref{fig:alt_prec-2}).
}
\label{fig:alt_prec}
\end{figure}
\item \label{item:2} \emph{Causal discovery based on the additive noise model succeeds---causal discovery based on the post-nonlinear model fails:}
Unfortunately, using the more expressive post-nonlinear model instead of the additive noise model does not always enhance causal discovery based on \ac{HSIC}. For a significant subset of the cause-effect pairs, the situation is as follows: there is an additive noise model for \( x \to y \), but the data also fit to post-nonlinear models for both \( x \to y \) and \( y \to x \), with the latter producing a higher \( p \)-value during the independence test.
One example is pair~4 which is again related to meteorological data from the \ac{DWD}: \( x \) describes the altitude of the weather station, \( y \) the mean annual sunshine duration. Figure~\ref{fig:alt_sun} shows that causal discovery based on the post-nonlinear model can fail due to an existing anti-causal (\( y \to x \)) post-nonlinear model. We again validate this subjective impression by considering the respective log-likelihoods: The bound on \( L_{y \to x, \text{\subref{fig:alt_sun-1}}} \) is approximately \num{-2.30} for the post-nonlinear model. The exact log-likelihood \( L_{y \to x, \text{\subref{fig:alt_sun-2}}} = -2.74 \) under the additive noise model is lower.
\begin{figure}[b]
\centering
\def10mm{10mm}
\def-11mm{-11mm}
\begin{tikzpicture}
\begin{groupplot}[
group style = {
rows = 1, columns = 2,
horizontal sep = 1.5cm,
vertical sep = 2cm,
},
width = 0.5\linewidth, height = 0.42\linewidth,
]
\nextgroupplot[
xlabel = \( x \), ylabel = \( y \),
xmin = -3.5, xmax = 5.5, ymin = -4.5, ymax = 3.5]
\addplot[only marks, mark size=0.75pt, myblue, opacity=0.3, draw opacity=0] table[x index = 1, y index = 0] {plot_data/alt_sun_dataset.txt};
\addplot[only marks, mark size=0.75pt, myred, opacity=0.3, draw opacity=0] table[x index = 1, y index = 0] {plot_data/alt_sun_dataset_replication.txt};
\nextgroupplot[
xlabel = \( x \), ylabel = \( y \),
xmin = -3.5, xmax = 5.5, ymin = -4.5, ymax = 3.5]
\addplot[only marks, mark size=0.75pt, myblue, opacity=0.3, draw opacity=0] table[x index = 1, y index = 0] {plot_data/alt_sun_dataset.txt};
\addplot[only marks, mark size=0.75pt, myred, opacity=0.3, draw opacity=0] table[x index = 1, y index = 0] {plot_data/alt_sun_dataset_replication_gauss.txt};
\end{groupplot}
\node[text width = 10mm] at ([yshift=-11mm] group c1r1.south) {\subcaption{\label{fig:alt_sun-1}}};
\node[text width = 10mm] at ([yshift=-11mm] group c2r1.south) {\subcaption{\label{fig:alt_sun-2}}};
\end{tikzpicture}
\caption[Investigation of pair~4 (altitude~\( \to \)~sunshine duration)]{Investigation of pair~4 (altitude~\( \to \)~sunshine duration): The anti-causal direction follows a post-nonlinear model, but not an additive noise model. This can be seen if comparing the replication of the original dataset (small blue dots) by generating samples from the estimated post-nonlinear model (small red dots in panel~\subref{fig:alt_sun-1}) and the samples taken from the estimated additive noise model (small red dots in panel~\subref{fig:alt_sun-2}).
}
\label{fig:alt_sun}
\end{figure}
Nevertheless, taking Occam's Razor into consideration, we would prefer the simpler additive noise-model for \( x \to y \) over the post-nonlinear model for \( y \to x \). Exploiting this reasoning, a possible two-step procedure could be to first test if the data can be described by an additive noise model and only consider the post-nonlinear model if not. The difficulty lies in defining a plausible threshold on the \( p \)-value which is required for data to follow an additive noise model.
\item \emph{Overall results on the whole database:} \nopagebreak
Our method is constructed to cope with data which are one-dimensional in \( x \) and \( y \). \num{102} of \num{108} pairs of the database satisfy this condition. We evaluated the test statistic of \ac{HSIC} and the estimated log-likelihoods \eqref{eq:L_xy}, \eqref{eq:L_yx} for \num{25} independent runs, comparing our method with the classic approach based on the additive noise model. To exclude outliers inside the \num{25} samples (which occur due to the more challenging optimisation problem compared to the simpler additive noise model), we use the median of the \num{25} samples for the final model selection.
Choosing the \ac{HSIC} test statistic as the model selection criterion, our method achieves an accuracy of \SI{68.8}{\percent} compared to \SI{59.8}{\percent} for the additive noise model approach. The authors of \cite{mooijDistinguishingCauseEffect2016} suggest to weight each pair of the database as some pairs are correlated to each other. Based on this weighting, our accuracy decreases to \SI{59.6}{\percent} compared to \SI{62.4}{\percent} for the additive noise model approach. The median test statistics for each pair are reported in Table~\ref{tab:results-tuebingen}. Overall, there are \num{24}~pairs which are similar to the one presented in \ref{item:1}, i.e\onedot} \def\Ie{I.e\onedot where our method correctly identifies cause and effect in contrast to the additive noise approach. On the contrary, \num{15}~pairs behave conversely, as the pair introduced in \ref{item:2}.
Following the reasoning in \ref{item:2}, we combine our approach with the additive noise model approach: Whenever one of the \( p \)-values we obtain after estimating additive noise models for \( x \to y \) and \( y \to x \) exceeds some threshold \( t \), we conclude that the additive noise model adequately describes the data. In this case, we perform causal discovery by means of the additive noise model. Otherwise, the more expressive post-nonlinear model is considered. Choosing \( t \) appropriately is demanding and requires further investigation. We leave it at evaluating how the (weighted) accuracy behaves for different thresholds. The results are plotted in Figure~\ref{fig:combining-approaches}. Choosing \( t \) within \SI{1}{\percent} and \SI{5}{\percent}, an accuracy of up to \SI{74.5}{\percent} (weighted accuracy of \SI{67.7}{\percent}) can be achieved which is an improvement of more than five percent points.
Model selection based on \( L_{x\to y} \) and \( L_{y \to x} \), the lower bounds on the log-likelihoods, fails with respect to causal discovery. The achieved weighted accuracy of \SI{53.3}{\percent} is not significantly higher than if deciding by chance. We assume this to be caused by bounding the log-likelihood. We therefore suggest to only consider \ac{HSIC} as a suitable criterion for bivariate causal discovery when using our estimation method for post-nonlinear models.
\begin{figure}
\centering
\begin{tikzpicture}
\begin{axis}[
xmode = log,
xmin = 3e-11, xmax = 3e0, ymin = 0.58, ymax = 0.78,
xlabel = {threshold \( t \) on \( p \)-value}, ylabel = {accuracy},
width = 0.9\linewidth, height = 0.6\linewidth,
legend pos = north west
]
\addplot[myblue] table {plot_data/combining_anm_pnm_acc.txt};
\addlegendentry{accuracy}
\addplot[myred] table {plot_data/combining_anm_pnm_acc_weighted.txt};
\addlegendentry{weighted accuracy}
\end{axis}
\end{tikzpicture}
\caption[Combining the additive noise model and the post-nonlinear model]{Combining the additive noise model and the post-nonlinear model. The extreme case \( t = 1 \) corresponds to always deciding based on the post-nonlinear model. Conversely, \( t = 0 \) only considers the additive noise model. Using a threshold within \( [\SI{1}{\percent}, \SI{5}{\percent}] \), the accuracy can be improved by approximately five percent points.}
\label{fig:combining-approaches}
\end{figure}
\item \emph{Missing quantitative comparison to \cite{zhangEstimationFunctionalCausal2016}'s method:}
What is still lacking, is a comparison to the method our approach comes closest to: \cite{zhangEstimationFunctionalCausal2016}. The authors report an (unweighted) accuracy of \SI{76}{\percent} which seems to be noticeably higher than our result. However, at the time when their paper was published, the whole Tübingen database consisted of only \num{77} pairs. Unfortunately, we could not examine \cite{zhangEstimationFunctionalCausal2016}'s work on today's full database as the provided algorithm is computationally time-consuming.\footnote{Moreover, we could not find out on which \num{77}~pairs of the database their reported accuracy rests on. The oldest publicly available version of the Tübingen Database (released in June 2014) already contains \num{86}~pairs of which \num{81} are one-dimensional in \( x \) and \( y \).} The estimation of each post-nonlinear model takes roughly ten~hours. In comparison, our estimation method takes less than one minute, using the same hardware environment.
\end{enumerate}
\section{Conclusion}
\label{sec:conclusion}
In this paper, we presented a new regression technique which can be used to model post-nonlinear causal mechanisms. The method has proven that it is capable of modelling artificially generated data which follow post-nonlinear models. With the established procedures to distinguish cause and effect, we could identify cause and effect for identifiable structural causal models.
However, inference on practical data, where the underlying type of structural causal models is hidden, is still a demanding problem: the post-nonlinear model is sometimes too expressive. Following Occam's Razor, causal discovery benefits from considering the simpler additive noise model in such cases. By introducing an intuitive criterion for when to apply which of the two models, we could improve the accuracy on the Tübingen database by more than five percent points.
\clearpage
\begin{table*}[h]
\caption[Results of causal discovery on the Tübingen database]{Results of causal discovery on the Tübingen database. Comparison of the \ac{HSIC} test statistic for the causal (\( x \to y \)) and the anti-causal (\( y \to x \)) direction. (Note that we always relabelled the cause-effect pairs such that \( x \) is the cause of \( y \). In the original database, \( x \) and \( y \) are interchanged for some pairs.) Our method based on the post-nonlinear model (\ac{PNM}) correctly identifies \num{70}~pairs compared to \num{61}~pairs when using exact Gaussian process regression with the additive noise model (\ac{ANM}) \cite{hoyerNonlinearCausalDiscovery2009, mooijDistinguishingCauseEffect2016}. Those cause-effect pairs where causal discovery succeeded are shaded.}
\centering
\small
\begin{tabular}{S[table-format=3.0]*{4}{S[table-format=2.3]}}
\toprule
& \multicolumn{2}{c}{\ac{PNM}} & \multicolumn{2}{c}{\ac{ANM}} \\
\cmidrule(lr){2-3} \cmidrule(lr){4-5}
{pair} & {\( x \to y \)} & {\( y \to x \)} & {\( x \to y \)} & {\( y \to x \)} \\
\midrule
1 & {\cellcolor{myred!20}} 0.514 & {\cellcolor{myred!20}} 1.221 & {\cellcolor{myblue!20}} 0.662 & {\cellcolor{myblue!20}} 2.292 \\
2 & {\cellcolor{myred!20}} 0.360 & {\cellcolor{myred!20}} 1.331 & 2.730 & 2.074 \\
3 & 0.416 & 0.285 & 0.789 & 0.753 \\
4 & 0.927 & 0.674 & {\cellcolor{myblue!20}} 0.691 & {\cellcolor{myblue!20}} 1.036 \\
5 & {\cellcolor{myred!20}} 0.383 & {\cellcolor{myred!20}} 0.656 & {\cellcolor{myblue!20}} 0.498 & {\cellcolor{myblue!20}} 2.469 \\
6 & {\cellcolor{myred!20}} 0.256 & {\cellcolor{myred!20}} 0.494 & 6.219 & 2.670 \\
7 & {\cellcolor{myred!20}} 0.380 & {\cellcolor{myred!20}} 1.098 & {\cellcolor{myblue!20}} 0.906 & {\cellcolor{myblue!20}} 2.962 \\
8 & {\cellcolor{myred!20}} 0.229 & {\cellcolor{myred!20}} 0.583 & {\cellcolor{myblue!20}} 1.074 & {\cellcolor{myblue!20}} 2.602 \\
9 & {\cellcolor{myred!20}} 0.363 & {\cellcolor{myred!20}} 0.833 & 4.863 & 1.574 \\
10 & {\cellcolor{myred!20}} 0.543 & {\cellcolor{myred!20}} 0.557 & 5.747 & 1.717 \\
11 & {\cellcolor{myred!20}} 0.271 & {\cellcolor{myred!20}} 0.620 & 6.109 & 3.023 \\
12 & 0.446 & 0.359 & 8.528 & 3.245 \\
13 & {\cellcolor{myred!20}} 0.172 & {\cellcolor{myred!20}} 0.459 & 4.661 & 3.485 \\
14 & {\cellcolor{myred!20}} 0.199 & {\cellcolor{myred!20}} 0.392 & 2.496 & 2.284 \\
15 & {\cellcolor{myred!20}} 0.388 & {\cellcolor{myred!20}} 0.432 & 4.007 & 1.370 \\
16 & {\cellcolor{myred!20}} 0.251 & {\cellcolor{myred!20}} 0.263 & {\cellcolor{myblue!20}} 1.077 & {\cellcolor{myblue!20}} 2.014 \\
17 & 0.821 & 0.382 & 37.234 & 0.997 \\
18 & 0.517 & 0.278 & {\cellcolor{myblue!20}} 1.660 & {\cellcolor{myblue!20}} 4.842 \\
19 & {\cellcolor{myred!20}} 0.137 & {\cellcolor{myred!20}} 0.239 & {\cellcolor{myblue!20}} 0.243 & {\cellcolor{myblue!20}} 1.711 \\
20 & {\cellcolor{myred!20}} 2.090 & {\cellcolor{myred!20}} 2.442 & 3.577 & 2.279 \\
21 & {\cellcolor{myred!20}} 0.387 & {\cellcolor{myred!20}} 0.860 & 1.616 & 0.645 \\
22 & 0.663 & 0.602 & {\cellcolor{myblue!20}} 0.215 & {\cellcolor{myblue!20}} 0.594 \\
23 & {\cellcolor{myred!20}} 0.728 & {\cellcolor{myred!20}} 0.854 & {\cellcolor{myblue!20}} 0.652 & {\cellcolor{myblue!20}} 1.429 \\
24 & {\cellcolor{myred!20}} 0.545 & {\cellcolor{myred!20}} 0.617 & {\cellcolor{myblue!20}} 0.716 & {\cellcolor{myblue!20}} 0.772 \\
25 & {\cellcolor{myred!20}} 0.413 & {\cellcolor{myred!20}} 0.504 & {\cellcolor{myblue!20}} 0.838 & {\cellcolor{myblue!20}} 1.706 \\
26 & {\cellcolor{myred!20}} 0.525 & {\cellcolor{myred!20}} 1.632 & {\cellcolor{myblue!20}} 0.878 & {\cellcolor{myblue!20}} 2.731 \\
27 & {\cellcolor{myred!20}} 0.821 & {\cellcolor{myred!20}} 0.847 & {\cellcolor{myblue!20}} 3.101 & {\cellcolor{myblue!20}} 5.314 \\
28 & {\cellcolor{myred!20}} 0.557 & {\cellcolor{myred!20}} 1.182 & {\cellcolor{myblue!20}} 1.751 & {\cellcolor{myblue!20}} 4.421 \\
29 & {\cellcolor{myred!20}} 0.311 & {\cellcolor{myred!20}} 2.644 & {\cellcolor{myblue!20}} 0.779 & {\cellcolor{myblue!20}} 4.015 \\
30 & {\cellcolor{myred!20}} 0.639 & {\cellcolor{myred!20}} 1.041 & {\cellcolor{myblue!20}} 0.410 & {\cellcolor{myblue!20}} 1.100 \\
31 & {\cellcolor{myred!20}} 0.453 & {\cellcolor{myred!20}} 0.823 & {\cellcolor{myblue!20}} 0.644 & {\cellcolor{myblue!20}} 1.489 \\
32 & {\cellcolor{myred!20}} 0.232 & {\cellcolor{myred!20}} 3.039 & {\cellcolor{myblue!20}} 0.912 & {\cellcolor{myblue!20}} 28.474 \\
33 & {\cellcolor{myred!20}} 0.443 & {\cellcolor{myred!20}} 1.298 & {\cellcolor{myblue!20}} 0.264 & {\cellcolor{myblue!20}} 3.095 \\
34 & {\cellcolor{myred!20}} 0.303 & {\cellcolor{myred!20}} 0.425 & {\cellcolor{myblue!20}} 0.300 & {\cellcolor{myblue!20}} 1.656 \\
35 & {\cellcolor{myred!20}} 0.632 & {\cellcolor{myred!20}} 0.873 & 1.343 & 1.223 \\
36 & {\cellcolor{myred!20}} 0.367 & {\cellcolor{myred!20}} 0.664 & {\cellcolor{myblue!20}} 0.740 & {\cellcolor{myblue!20}} 2.250 \\
37 & {\cellcolor{myred!20}} 0.409 & {\cellcolor{myred!20}} 1.337 & {\cellcolor{myblue!20}} 2.374 & {\cellcolor{myblue!20}} 3.643 \\
38 & {\cellcolor{myred!20}} 0.423 & {\cellcolor{myred!20}} 0.838 & {\cellcolor{myblue!20}} 0.447 & {\cellcolor{myblue!20}} 1.168 \\
39 & {\cellcolor{myred!20}} 0.241 & {\cellcolor{myred!20}} 0.806 & {\cellcolor{myblue!20}} 0.323 & {\cellcolor{myblue!20}} 2.281 \\
40 & {\cellcolor{myred!20}} 0.208 & {\cellcolor{myred!20}} 0.690 & {\cellcolor{myblue!20}} 0.290 & {\cellcolor{myblue!20}} 5.401 \\
41 & {\cellcolor{myred!20}} 0.320 & {\cellcolor{myred!20}} 0.598 & {\cellcolor{myblue!20}} 1.118 & {\cellcolor{myblue!20}} 3.389 \\
42 & {\cellcolor{myred!20}} 0.206 & {\cellcolor{myred!20}} 4.696 & {\cellcolor{myblue!20}} 0.338 & {\cellcolor{myblue!20}} 33.150 \\
43 & {\cellcolor{myred!20}} 0.240 & {\cellcolor{myred!20}} 0.308 & 8.944 & 8.896 \\
44 & {\cellcolor{myred!20}} 1.188 & {\cellcolor{myred!20}} 1.417 & 1.723 & 1.235 \\
45 & {\cellcolor{myred!20}} 0.809 & {\cellcolor{myred!20}} 1.022 & {\cellcolor{myblue!20}} 2.069 & {\cellcolor{myblue!20}} 4.043 \\
46 & {\cellcolor{myred!20}} 0.493 & {\cellcolor{myred!20}} 0.895 & {\cellcolor{myblue!20}} 3.545 & {\cellcolor{myblue!20}} 5.744 \\
47 & {\cellcolor{myred!20}} 0.222 & {\cellcolor{myred!20}} 1.637 & {\cellcolor{myblue!20}} 0.620 & {\cellcolor{myblue!20}} 7.832 \\
48 & 0.843 & 0.781 & {\cellcolor{myblue!20}} 0.883 & {\cellcolor{myblue!20}} 2.108 \\
49 & 0.568 & 0.215 & {\cellcolor{myblue!20}} 0.694 & {\cellcolor{myblue!20}} 1.684 \\
50 & 1.253 & 0.521 & {\cellcolor{myblue!20}} 1.563 & {\cellcolor{myblue!20}} 1.776 \\
51 & 1.195 & 0.935 & 2.026 & 1.273 \\
\bottomrule
\end{tabular}
\qquad\qquad
\begin{tabular}{S[table-format=3.0]*{4}{S[table-format=2.3]}}
\toprule
& \multicolumn{2}{c}{\ac{PNM}} & \multicolumn{2}{c}{\ac{ANM}} \\
\cmidrule(lr){2-3} \cmidrule(lr){4-5}
{pair} & {\( x \to y \)} & {\( y \to x \)} & {\( x \to y \)} & {\( y \to x \)} \\
\midrule
56 & 0.490 & 0.398 & 3.484 & 1.138 \\
57 & {\cellcolor{myred!20}} 0.291 & {\cellcolor{myred!20}} 0.325 & 2.761 & 0.967 \\
58 & {\cellcolor{myred!20}} 0.343 & {\cellcolor{myred!20}} 0.434 & 2.548 & 0.923 \\
59 & 0.464 & 0.301 & 2.832 & 1.023 \\
60 & 0.525 & 0.353 & 2.151 & 1.028 \\
61 & {\cellcolor{myred!20}} 0.268 & {\cellcolor{myred!20}} 0.334 & 1.762 & 0.733 \\
62 & {\cellcolor{myred!20}} 0.363 & {\cellcolor{myred!20}} 0.479 & 1.669 & 1.002 \\
63 & 0.547 & 0.391 & 2.272 & 1.118 \\
64 & {\cellcolor{myred!20}} 0.274 & {\cellcolor{myred!20}} 0.334 & {\cellcolor{myblue!20}} 3.131 & {\cellcolor{myblue!20}} 3.225 \\
65 & 0.476 & 0.469 & {\cellcolor{myblue!20}} 1.032 & {\cellcolor{myblue!20}} 1.294 \\
66 & 0.330 & 0.258 & {\cellcolor{myblue!20}} 0.822 & {\cellcolor{myblue!20}} 1.121 \\
67 & {\cellcolor{myred!20}} 0.290 & {\cellcolor{myred!20}} 0.679 & {\cellcolor{myblue!20}} 1.372 & {\cellcolor{myblue!20}} 1.584 \\
68 & 1.089 & 0.759 & 14.627 & 1.546 \\
69 & 0.516 & 0.333 & 4.290 & 1.611 \\
70 & 15.139 & 0.889 & 54.621 & 4.725 \\
72 & {\cellcolor{myred!20}} 0.280 & {\cellcolor{myred!20}} 0.617 & {\cellcolor{myblue!20}} 0.761 & {\cellcolor{myblue!20}} 3.707 \\
73 & 0.296 & 0.258 & {\cellcolor{myblue!20}} 19.834 & {\cellcolor{myblue!20}} 23.979 \\
74 & {\cellcolor{myred!20}} 0.264 & {\cellcolor{myred!20}} 0.376 & {\cellcolor{myblue!20}} 2.387 & {\cellcolor{myblue!20}} 3.632 \\
75 & {\cellcolor{myred!20}} 0.299 & {\cellcolor{myred!20}} 0.313 & 4.330 & 4.195 \\
76 & {\cellcolor{myred!20}} 0.231 & {\cellcolor{myred!20}} 0.493 & {\cellcolor{myblue!20}} 0.381 & {\cellcolor{myblue!20}} 1.060 \\
77 & {\cellcolor{myred!20}} 0.208 & {\cellcolor{myred!20}} 0.284 & {\cellcolor{myblue!20}} 1.623 & {\cellcolor{myblue!20}} 5.040 \\
78 & 0.403 & 0.359 & {\cellcolor{myblue!20}} 2.153 & {\cellcolor{myblue!20}} 5.510 \\
79 & 0.528 & 0.375 & {\cellcolor{myblue!20}} 2.095 & {\cellcolor{myblue!20}} 3.365 \\
80 & {\cellcolor{myred!20}} 0.404 & {\cellcolor{myred!20}} 0.488 & {\cellcolor{myblue!20}} 1.988 & {\cellcolor{myblue!20}} 5.465 \\
81 & {\cellcolor{myred!20}} 0.389 & {\cellcolor{myred!20}} 0.615 & {\cellcolor{myblue!20}} 0.657 & {\cellcolor{myblue!20}} 0.957 \\
82 & {\cellcolor{myred!20}} 0.231 & {\cellcolor{myred!20}} 0.443 & 4.446 & 0.916 \\
83 & {\cellcolor{myred!20}} 0.278 & {\cellcolor{myred!20}} 0.584 & 5.152 & 0.842 \\
84 & 0.276 & 0.269 & {\cellcolor{myblue!20}} 0.433 & {\cellcolor{myblue!20}} 1.183 \\
85 & {\cellcolor{myred!20}} 0.263 & {\cellcolor{myred!20}} 0.745 & {\cellcolor{myblue!20}} 0.516 & {\cellcolor{myblue!20}} 0.896 \\
86 & {\cellcolor{myred!20}} 0.306 & {\cellcolor{myred!20}} 0.494 & {\cellcolor{myblue!20}} 0.944 & {\cellcolor{myblue!20}} 3.280 \\
87 & 1.688 & 0.859 & 29.249 & 7.272 \\
88 & {\cellcolor{myred!20}} 0.208 & {\cellcolor{myred!20}} 0.382 & 2.824 & 1.450 \\
89 & 0.184 & 0.167 & {\cellcolor{myblue!20}} 0.223 & {\cellcolor{myblue!20}} 0.244 \\
90 & 0.426 & 0.337 & {\cellcolor{myblue!20}} 0.437 & {\cellcolor{myblue!20}} 0.493 \\
91 & {\cellcolor{myred!20}} 0.222 & {\cellcolor{myred!20}} 0.340 & {\cellcolor{myblue!20}} 1.223 & {\cellcolor{myblue!20}} 1.804 \\
92 & 0.348 & 0.295 & 1.331 & 0.839 \\
93 & 0.907 & 0.365 & 1.826 & 1.692 \\
94 & {\cellcolor{myred!20}} 0.288 & {\cellcolor{myred!20}} 1.200 & {\cellcolor{myblue!20}} 1.234 & {\cellcolor{myblue!20}} 3.192 \\
95 & {\cellcolor{myred!20}} 0.385 & {\cellcolor{myred!20}} 1.122 & {\cellcolor{myblue!20}} 0.754 & {\cellcolor{myblue!20}} 3.541 \\
96 & {\cellcolor{myred!20}} 0.391 & {\cellcolor{myred!20}} 0.719 & {\cellcolor{myblue!20}} 1.644 & {\cellcolor{myblue!20}} 5.700 \\
97 & {\cellcolor{myred!20}} 0.150 & {\cellcolor{myred!20}} 0.297 & 0.283 & 0.247 \\
98 & {\cellcolor{myred!20}} 0.212 & {\cellcolor{myred!20}} 0.390 & 0.743 & 0.742 \\
99 & {\cellcolor{myred!20}} 0.307 & {\cellcolor{myred!20}} 2.306 & {\cellcolor{myblue!20}} 0.871 & {\cellcolor{myblue!20}} 1.035 \\
100 & {\cellcolor{myred!20}} 0.204 & {\cellcolor{myred!20}} 0.330 & {\cellcolor{myblue!20}} 0.689 & {\cellcolor{myblue!20}} 1.161 \\
101 & {\cellcolor{myred!20}} 0.347 & {\cellcolor{myred!20}} 0.573 & {\cellcolor{myblue!20}} 0.811 & {\cellcolor{myblue!20}} 6.127 \\
102 & {\cellcolor{myred!20}} 0.276 & {\cellcolor{myred!20}} 0.416 & {\cellcolor{myblue!20}} 0.450 & {\cellcolor{myblue!20}} 0.689 \\
103 & 0.263 & 0.171 & 0.257 & 0.241 \\
104 & 0.170 & 0.142 & 0.381 & 0.223 \\
106 & 0.312 & 0.264 & {\cellcolor{myblue!20}} 0.529 & {\cellcolor{myblue!20}} 0.551 \\
107 & 2.658 & 0.492 & 19.405 & 1.006 \\
108 & {\cellcolor{myred!20}} 0.390 & {\cellcolor{myred!20}} 0.468 & 3.153 & 0.820 \\
\bottomrule
\end{tabular}
\label{tab:results-tuebingen}
\end{table*}
\clearpage
{\small
\bibliographystyle{ieee_fullname}
|
1,108,101,563,277 | arxiv | \section{The entry and early involvement of China with the Union}
China has a rich tradition in astronomical observations and studies and boasts the
longest and most comprehensive ancient records of astronomical phenomena dating
back to the Shang dynasty. In its 5000-year continuous civilization, China has
nurtured world-class astronomers including Heng Zhang (78--139 AD), Shoujing Guo
(1231--1316 AD) and Guangqi Xu (1562--1633 AD).
The works of Euclid of Alexandria, Claudius Ptolemy, Nicolas Copernicus and Tycho
Brah\'e were translated, and introduced to and practised by Chinese astronomers in the
late Ming dynasty, following the arrival of Jesuit priests (Matteo Ricci,
Nicolaus Longobardi, Giacomo Rho, Johann Schreck, Johann Adam Schall von Bell).
In 1644, Johann Adam Schall von Bell (1592--1666 AD) was appointed Director of
the Imperial Observatory, the first time the position was held by a foreigner.
From the \textit{First Opium War} in 1840, China was forced to open up ports under the
weapons of the rising Western powers, and was reduced gradually to a semi-colonial,
semi-feudal country which ushered in a warlord era. In 1912, the last feudal
dynasty Qing fell and the Republic of China (ROC) was founded. The Lugou (Marco
Polo) Bridge Incident on July 7, 1937 marked the beginning of full-out invasion
of China by Imperial Japan and the Eight-year Chinese War of Resistance which followed, and
which ended in 1945 along with the World War II. This was followed by a bitter
period of civil war between the ruling Nationalist Party (Kuomingtang -- KMT)
of Jieshi Jiang (Kai-shek Chiang) and the Communist Party of China (CPC) led by
Zedong Mao (Tse-tung Mao). The war ended in 1949 with the founding of the
People's Republic of China (PRC) and the retreat of KMT from mainland China
to the Chinese island of Taiwan.
The World War II ended with the emergence of two political blocs, the Western
Bloc led by the United States of America (to which the provincial KMT
government in Taipei belonged) and the Eastern Bloc (with which the PRC was
associated) led by the Soviet Union, which fought between them the Cold War that
only ended decades later in 1992. In 1950, the Korean War broke out between the
Eastern Bloc North Korea (the Democratic People's Republic of Korea -- DPRK)
and the Western Bloc South Korea (the Republic of Korea -- ROK). The war
evolved rapidly into one involving PRC forces on the DPRK side and U.S. forces
fighting for the ROK. The War ended with the \textit{Korean Armistice Agreement} signed
in 1953.
Several astronomical observatories were built in China by foreign Jesuit
priests in the late nineteenth century, including Shanghai Xujiahui (1873) and
Sheshan (1901) Observatories by the French, Hongkong Observatory (1882) by the British
and Qingdao Observatory (1898) by Germans. In 1922, Qingdao Observatory was
reclaimed by China. Chinese meteorologist and astronomer returning from
the Universit\'e libre de Bruxelles,
Bingran Jiang (1883--1966), became
Director, commencing modern Chinese astronomical observation and research.
Meanwhile, more Chinese intellectuals studying abroad returned back to China. These included Lu
Gao (1877--1947), inspired by French astronomer Camille Flammarion (1842--1925),
Qingsong Yu (1897--1978) returning from the Lick Observatory, University of
California and Yuzhe Zhang (Yu-Cheh Chang; 1902--1986) from the Yerkes
Observatory, University of Chicago. In 1926, the Mathematics Department of Sun
Yat-sen University (SYSU) was renamed the Mathematics and Astronomy Department
and began to offer astronomical courses, marking the beginning of astronomical
higher education in China. In 1929, Yun Zhang (1896--1958) who studied astronomy
at the University of Lyon built the SYSU Observatory and became its first Director.
In 1928, the Institute of Astronomy, Academia Sinica was founded. On October
30, 1922, the Chinese Astronomical Society (CAS) was founded with 47 members at the
Beijing Ancient Observatory (constructed in 1442 during the Ming Dynasty on the
original site first built by Shoujing Guo in 1279 of the Yuan Dynasty). Lu Gao
became its first President. By 1947, the CAS had 688 individual and 6
institutional members.
Initiated by Yuanpei Cai (1868--1940), the first Director (1928--1940) of Academia
Sinica and President (1916--1927) of Peking University, Purple Mountain
Observatory (PMO), the best astronomical observatory at the time in the Far East, was inaugurated in
1934. Since its establishment, the CAS took liaison with the Union and
collaboration with the international community was one of its vital tasks. CAS
member Yun Zhang attended in 1925 the 2nd IAU General Assembly (GA) in Cambridge as
an observer, and members Qingsong Yu and Jinyi Zhao participated in 1928 in the 3rd GA in
Leiden. All these individuals paved the way for China joining the IAU. In 1935, at the 5th GA
in Paris, China was formally admitted to the Union as its 26th National Member, with
the CAS located at PMO in Nanjing as the adhering organization. Four initial
individual members, Qingsong Yu, Lu Gao, Bingran Jiang and a Japanese astronomer
Shinjo Shinzo (1873--1938) were also admitted. At the 6th GA in 1938 in Stockholm, China's
individual members increased to 11.
In spite of the War, Chinese astronomers did their best to engage with
the Union and paid the arrears in 1947. After the civil war and the
establishment of PRC in 1949, a delegation of four astronomers from mainland China,
invited by President Otto Struve (1897--1963), and including Yuzhe Zhang,
attended in 1955 the Dublin GA. The PRC resumed its legitimate National
Membership in the Union and cleared the arrears.
\section{The conflict and the withdrawal of China from the Union}
The ``application'' for Union membership from the ROC, submitted in late April
and early May 1958, just two months before the Moscow GA, was not initiated by
astronomers from the island (there were few if any at the time), but was
orchestrated by the US government in order to further isolate the PRC after the
Korean War, at the height of the Cold War.\footnote{The ``Chinese Astronomical
Society'' in Taipei was only established in July 1958, two months after the
application submission.} It came at a time when the US National Committee for
Astronomy (NCA) was about to submit their invitation to the EC to host the 1961
GA in Berkeley, after the 1958 GA in Moscow. The plot by the US Department of
State, using visas to the US as a threat, was deliberately designed to (one
stone, three birds): 1) Block astronomers from the ``communist'' mainland
China; 2) Promote the status of ``Free China (Taiwan)''; and 3) Create the
so-called ``\textit{Two Chinas}'' or ``\textit{One China, one Taiwan}''
problem.
The plot was opposed by Leo Goldberg (US NCA), Otto Struve (former IAU
President and still consultant to the EC), and Detlev Wulf Bronk (President,
National Academy of Sciences), worrying that this would tarnish the US
scientific reputation. In spite of the pressure from the US government, the EC
decided to postpone any decision until after the Moscow GA, in recognition of
``\textit{the serious implications}'' that ``\textit{acceptance of the Taiwan application during
the Moscow meetings might have had: immediate withdrawal of mainland China from
the IAU, and possibly also that of USSR, the host}'' \citep[][p.\,193]{Blaauw}.
Goldberg contacted his representative in the U.S. Congress, George Meader, a
conservative and fair-minded Republican, who presented the case to John Foster Dulles (the US Secretary of State), who
referred it to his science advisor, Wallace Brode. Brode promptly demanded that
Taiwan be invited to the IAU.
\begin{quotation}``\textit{The fact that Taiwan then had no astronomers and would have to qualify for
IAU membership in the approved way meant nothing to the militant anti-Communist
Brode. Brode wanted Goldberg to go to the 1958 Moscow meeting and submit the
1961 invitation but with the condition that Taiwan be admitted at once. Such a
demand could well wreck the IAU. From Brode's point of view, if the astronomers
would not go along with his orders, so much the worse for them.}'' [Biographical
Memoirs, U.S. National Academy of Sciences, 1997; \cite[cited from][p.\,154]{FY09}].\end{quotation}
Unfortunately, after the Moscow GA, the attitude of the EC made a U-turn.
In spite of opposition by the IAU Vice-Presidents from the USSR and Czechoslovakia,
who stated that admission should be judged solely on scientific grounds, that
the astronomical activity in this applicant's country was too low, and that the
admission of Taiwan might risk the withdrawal of mainland China, the EC pushed
through the admission by the ballot, with five votes for and two against.
Yuzhe Zhang, President of the CAS in Nanjing, in his letter to J.H. Oort,
serving President of the IAU, expressed surprise and indignation:
\begin{quotation}``\textit{\ldots Taiwan
is an inseparable part of Chinese territory, it is a province of China\ldots
Should the report be authentic, I, on behalf of the Astronomical Society of the
People's Republic China, hereby lodge our strong protest with you and insist
that the Executive Council of IAU rescind the illegal decision\ldots Otherwise,
the Astronomical Society\ldots will resolutely and definitely withdraw from the
IAU''} \citep[][p.\,193]{Blaauw}.\end{quotation}
IAU \textit{Information Bulletin} (IB) No.\,2 of November 1959 announced the adherence of
Taiwan as a member of the IAU. The withdrawal of mainland China was announced in
IB No.\,3 of May 1960. The rapid announcements underscored the acceptance as a
``\textit{fait accompli}''.
The EC further brushed aside letters of protest from the Polish and Bulgarian
Academies sent in March 1960, as well as concerns raised by Vice-President O.
Heckmann during the 1961 GA. Also during the Assembly, President Oort, in an
unusual move, before the vote, asked representatives to vote against a combined
motion submitted by the USSR and Czechoslovakia Academies of Sciences
requesting the decision of the EC to admit Taiwan be revoked.
The decision of the EC, succumbed to the political pressure from the US
government, led to the two-decade absence of astronomers of mainland China from
the international stage of astronomical collaboration and exchange. Squeezed
between the demands of the then two superpowers, the United States of America
and the Soviet Union, the IAU survived, but unfortunately was clearly damaged.
\section{The return of China}
In the following two decades, in spite of the growing dissatisfaction and
concern, restrained by the then prevailing political environment, both
international and domestic, little progress was made on the restoration of the
legitimate position of China in the IAU until the 1979 Montreal Assembly. Several events before then eased the way forwards from the prevailing deadlock:
\begin{enumerate}
\item[1.] In 1971, UN Resolution No.\,2758 restored the membership of the PRC and expelled
the ROC from the UN; \item[2.] In 1972, Richard Nixon visited China. The joint communiqu\'e
recognized Taiwan as a part of China, thereby ushering in a new era of Sino-American
relations; \item[3.] In 1976, the ten-year Cultural Revolution in mainland China
ended; and \item[4.] In 1978, China started the economic reforms and opening up.
\end{enumerate}
All these paved the political way for China rejoining the Union.
The key was to find ``\textit{a way acceptable to the Chinese of reinstating their
membership in the IAU without expelling Taiwan, an action that would violate
the statutes of the Union}'' \citep[see p.20]{Goldberg}.
Initiated by President Adriaan Blaauw, under the invitation of the EC, a
six-member delegation from mainland China, including Yuzhe Zhang, Shuhua Ye
(Vice President of the IAU 1988--1994) and Zhaohua Yi, and a single-member
delegation from Taiwan represented by C.S. Shen, President of the Taiwan NCA,
arrived in Montreal to discuss the matter prior to the GA on August 13--24 1979,
and to explore the possibility of ``dual membership'', that, on the one hand,
expresses the indivisibility of China (emphasized and agreed by both parties
across the Straight), and, on the other hand, reinstates the membership of PRC
without blocking China Taiwan's further adherence.
The negotiations resulted in a proposal communicated to the GA at its closing
session, that was presented in the form of an exchange of letters dated
Montreal, August 22, 1979, between the President of the CAS, Yuzhe Zhang, and
the outgoing President of the IAU, Adriaan Blaauw. The two letters were reproduced in
full in IAU \textit{Transactions} Vol. XVII \citep[pp 48--50]{Bappu}.
The only problem left was the name of the adhering organization in Taiwan.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.8\textwidth]{Fig01.pdf}
\caption{Parts of the letter from Yuzhe Zhang, President of the Chinese
Astronomical Society (1943--1982), to Adriaan Blaauw, President of the IAU \citep[pp. 525--527]{ChangBlaauw}}
\label{fig1}
\end{center}
\end{figure}
The problem was shortly solved in 1980. The arrangements were ratified at the
1982 Patras GA XVIII. The solution, accepted by all the parties involved, is
that China is to have two adhering organizations, with mainland Chinese
astronomers represented by the Chinese Astronomical Society located in Nanjing
(\textit{China Nanjing}; since 1935) and \textit{China Taiwan} astronomers represented by the
Academia Sinica located in Taipei (China Taipei; since 1959). The
denominations, ``\textit{China Nanjing}'' and ``\textit{China Taipei}'' represent the IAU
official resolution and should be used in all IAU events. At the time of this
writing (August, 2018), ``\textit{China Nanjing}'' and ``\textit{China Taipei}'' have
respectively 666 and 74 IAU individual
members.\footnote{\href{https://www.iau.org/administration/membership/national/}{www.iau.org/administration/membership/national}}
\begin{quotation}``\textit{The Union had succeeded in overcoming political schism and in restoring
harmony again among its membership for its prime purpose: the unhampered
pursuit of scientific research and intercourse}'' \citep[][p.\,204]{Blaauw}.
\end{quotation}
Shouguan Wang (President of the CAS 1985--1989), on behalf of the CAS, gave a
warm speech after the ratification of the membership of China by the GA \citep[p. 26]{Wang}:
\begin{quotation}\textit{Dear friends \& colleagues,}
\textit{The Chinese Astronomical Society celebrates its 60th anniversary this year. Its
reunion with this international community today is an event that is highly
appreciated by all its 900 members. I and my colleagues here are very glad to
have this opportunity of speaking on behalf of our Society and its members to
express our most cordial greetings and most sincere thanks to you all. Thank
you!} \end{quotation}
Zden$\check{\rm e}$k Kopal, writing on \textit{The IAU -- the first 60 years.
Reminiscences and Reflections}, published in \textit{Astrocosmos}, the newspaper of the
Patras GA, wrote, \begin{quotation}``\textit{\ldots the Union has really never been free of political
interference from many directions ever since. Perhaps the most conspicuous
example of such an interference in recent years was the technical expulsion of
the (People's) Republic of China, which was eased out of our midst in 1955
(1960) by the United States (during the enlightened era of John Foster Dulles),
in collaboration with certain astronomers from Western Europe. Only God knows
what good should have come to the science of astronomy and to the International
Astronomical Union from severing (albeit temporarily) its official ties with
the most populous nation of the Earth; but such acts did happen, and will
continue to happen as long as the present structure of the IAU remains
unchanged}'' \citep{Kopal}. \end{quotation}
The resolution of the China crisis, combined with the continued improvement of
relation across the Taiwan Straight, has benefited astronomers on both sides,
stimulated and facilitated collaborations amongst them as well as with the
international community at large. Astronomy in both mainland China and in the
Island has entered an era of rapid development. Four decades after the crisis,
China is poised to make major contributions to the world astronomy research and
education development.
In 2012, the IAU GA XXVIII was held in Beijing, being the first time in China in the
Union's nearly hundred-year history. In his opening ceremony speech, Vice-President
Jinping Xi remarked:
\begin{quotation}``\textit{The development of science and technology
requires extensive international cooperation. Science and technology have no
nationality! The vast expanse of space is the common home of all humankind; to
explore this vast universe is the common goal of all humankind; astronomy in
fast development is the shared fortune of all humankind}'' \citep{Xi}. \end{quotation}
\section{The entry and early involvement of China with the Union}
China has a rich tradition in astronomical observations and studies and boasts the
longest and most comprehensive ancient records of astronomical phenomena dating
back to the Shang dynasty. In its 5000-year continuous civilization, China has
nurtured world-class astronomers including Heng Zhang (78--139 AD), Shoujing Guo
(1231--1316 AD) and Guangqi Xu (1562--1633 AD).
The works of Euclid of Alexandria, Claudius Ptolemy, Nicolas Copernicus and Tycho
Brah\'e were translated, and introduced to and practised by Chinese astronomers in the
late Ming dynasty, following the arrival of Jesuit priests (Matteo Ricci,
Nicolaus Longobardi, Giacomo Rho, Johann Schreck, Johann Adam Schall von Bell).
In 1644, Johann Adam Schall von Bell (1592--1666 AD) was appointed Director of
the Imperial Observatory, the first time the position was held by a foreigner.
From the \textit{First Opium War} in 1840, China was forced to open up ports under the
weapons of the rising Western powers, and was reduced gradually to a semi-colonial,
semi-feudal country which ushered in a warlord era. In 1912, the last feudal
dynasty Qing fell and the Republic of China (ROC) was founded. The Lugou (Marco
Polo) Bridge Incident on July 7, 1937 marked the beginning of full-out invasion
of China by Imperial Japan and the Eight-year Chinese War of Resistance which followed, and
which ended in 1945 along with the World War II. This was followed by a bitter
period of civil war between the ruling Nationalist Party (Kuomingtang -- KMT)
of Jieshi Jiang (Kai-shek Chiang) and the Communist Party of China (CPC) led by
Zedong Mao (Tse-tung Mao). The war ended in 1949 with the founding of the
People's Republic of China (PRC) and the retreat of KMT from mainland China
to the Chinese island of Taiwan.
The World War II ended with the emergence of two political blocs, the Western
Bloc led by the United States of America (to which the provincial KMT
government in Taipei belonged) and the Eastern Bloc (with which the PRC was
associated) led by the Soviet Union, which fought between them the Cold War that
only ended decades later in 1992. In 1950, the Korean War broke out between the
Eastern Bloc North Korea (the Democratic People's Republic of Korea -- DPRK)
and the Western Bloc South Korea (the Republic of Korea -- ROK). The war
evolved rapidly into one involving PRC forces on the DPRK side and U.S. forces
fighting for the ROK. The War ended with the \textit{Korean Armistice Agreement} signed
in 1953.
Several astronomical observatories were built in China by foreign Jesuit
priests in the late nineteenth century, including Shanghai Xujiahui (1873) and
Sheshan (1901) Observatories by the French, Hongkong Observatory (1882) by the British
and Qingdao Observatory (1898) by Germans. In 1922, Qingdao Observatory was
reclaimed by China. Chinese meteorologist and astronomer returning from
the Universit\'e libre de Bruxelles,
Bingran Jiang (1883--1966), became
Director, commencing modern Chinese astronomical observation and research.
Meanwhile, more Chinese intellectuals studying abroad returned back to China. These included Lu
Gao (1877--1947), inspired by French astronomer Camille Flammarion (1842--1925),
Qingsong Yu (1897--1978) returning from the Lick Observatory, University of
California and Yuzhe Zhang (Yu-Cheh Chang; 1902--1986) from the Yerkes
Observatory, University of Chicago. In 1926, the Mathematics Department of Sun
Yat-sen University (SYSU) was renamed the Mathematics and Astronomy Department
and began to offer astronomical courses, marking the beginning of astronomical
higher education in China. In 1929, Yun Zhang (1896--1958) who studied astronomy
at the University of Lyon built the SYSU Observatory and became its first Director.
In 1928, the Institute of Astronomy, Academia Sinica was founded. On October
30, 1922, the Chinese Astronomical Society (CAS) was founded with 47 members at the
Beijing Ancient Observatory (constructed in 1442 during the Ming Dynasty on the
original site first built by Shoujing Guo in 1279 of the Yuan Dynasty). Lu Gao
became its first President. By 1947, the CAS had 688 individual and 6
institutional members.
Initiated by Yuanpei Cai (1868--1940), the first Director (1928--1940) of Academia
Sinica and President (1916--1927) of Peking University, Purple Mountain
Observatory (PMO), the best astronomical observatory at the time in the Far East, was inaugurated in
1934. Since its establishment, the CAS took liaison with the Union and
collaboration with the international community was one of its vital tasks. CAS
member Yun Zhang attended in 1925 the 2nd IAU General Assembly (GA) in Cambridge as
an observer, and members Qingsong Yu and Jinyi Zhao participated in 1928 in the 3rd GA in
Leiden. All these individuals paved the way for China joining the IAU. In 1935, at the 5th GA
in Paris, China was formally admitted to the Union as its 26th National Member, with
the CAS located at PMO in Nanjing as the adhering organization. Four initial
individual members, Qingsong Yu, Lu Gao, Bingran Jiang and a Japanese astronomer
Shinjo Shinzo (1873--1938) were also admitted. At the 6th GA in 1938 in Stockholm, China's
individual members increased to 11.
In spite of the War, Chinese astronomers did their best to engage with
the Union and paid the arrears in 1947. After the civil war and the
establishment of PRC in 1949, a delegation of four astronomers from mainland China,
invited by President Otto Struve (1897--1963), and including Yuzhe Zhang,
attended in 1955 the Dublin GA. The PRC resumed its legitimate National
Membership in the Union and cleared the arrears.
\section{The conflict and the withdrawal of China from the Union}
The ``application'' for Union membership from the ROC, submitted in late April
and early May 1958, just two months before the Moscow GA, was not initiated by
astronomers from the island (there were few if any at the time), but was
orchestrated by the US government in order to further isolate the PRC after the
Korean War, at the height of the Cold War.\footnote{The ``Chinese Astronomical
Society'' in Taipei was only established in July 1958, two months after the
application submission.} It came at a time when the US National Committee for
Astronomy (NCA) was about to submit their invitation to the EC to host the 1961
GA in Berkeley, after the 1958 GA in Moscow. The plot by the US Department of
State, using visas to the US as a threat, was deliberately designed to (one
stone, three birds): 1) Block astronomers from the ``communist'' mainland
China; 2) Promote the status of ``Free China (Taiwan)''; and 3) Create the
so-called ``\textit{Two Chinas}'' or ``\textit{One China, one Taiwan}''
problem.
The plot was opposed by Leo Goldberg (US NCA), Otto Struve (former IAU
President and still consultant to the EC), and Detlev Wulf Bronk (President,
National Academy of Sciences), worrying that this would tarnish the US
scientific reputation. In spite of the pressure from the US government, the EC
decided to postpone any decision until after the Moscow GA, in recognition of
``\textit{the serious implications}'' that ``\textit{acceptance of the Taiwan application during
the Moscow meetings might have had: immediate withdrawal of mainland China from
the IAU, and possibly also that of USSR, the host}'' \citep[][p.\,193]{Blaauw}.
Goldberg contacted his representative in the U.S. Congress, George Meader, a
conservative and fair-minded Republican, who presented the case to John Foster Dulles (the US Secretary of State), who
referred it to his science advisor, Wallace Brode. Brode promptly demanded that
Taiwan be invited to the IAU.
\begin{quotation}``\textit{The fact that Taiwan then had no astronomers and would have to qualify for
IAU membership in the approved way meant nothing to the militant anti-Communist
Brode. Brode wanted Goldberg to go to the 1958 Moscow meeting and submit the
1961 invitation but with the condition that Taiwan be admitted at once. Such a
demand could well wreck the IAU. From Brode's point of view, if the astronomers
would not go along with his orders, so much the worse for them.}'' [Biographical
Memoirs, U.S. National Academy of Sciences, 1997; \cite[cited from][p.\,154]{FY09}].\end{quotation}
Unfortunately, after the Moscow GA, the attitude of the EC made a U-turn.
In spite of opposition by the IAU Vice-Presidents from the USSR and Czechoslovakia,
who stated that admission should be judged solely on scientific grounds, that
the astronomical activity in this applicant's country was too low, and that the
admission of Taiwan might risk the withdrawal of mainland China, the EC pushed
through the admission by the ballot, with five votes for and two against.
Yuzhe Zhang, President of the CAS in Nanjing, in his letter to J.H. Oort,
serving President of the IAU, expressed surprise and indignation:
\begin{quotation}``\textit{\ldots Taiwan
is an inseparable part of Chinese territory, it is a province of China\ldots
Should the report be authentic, I, on behalf of the Astronomical Society of the
People's Republic China, hereby lodge our strong protest with you and insist
that the Executive Council of IAU rescind the illegal decision\ldots Otherwise,
the Astronomical Society\ldots will resolutely and definitely withdraw from the
IAU''} \citep[][p.\,193]{Blaauw}.\end{quotation}
IAU \textit{Information Bulletin} (IB) No.\,2 of November 1959 announced the adherence of
Taiwan as a member of the IAU. The withdrawal of mainland China was announced in
IB No.\,3 of May 1960. The rapid announcements underscored the acceptance as a
``\textit{fait accompli}''.
The EC further brushed aside letters of protest from the Polish and Bulgarian
Academies sent in March 1960, as well as concerns raised by Vice-President O.
Heckmann during the 1961 GA. Also during the Assembly, President Oort, in an
unusual move, before the vote, asked representatives to vote against a combined
motion submitted by the USSR and Czechoslovakia Academies of Sciences
requesting the decision of the EC to admit Taiwan be revoked.
The decision of the EC, succumbed to the political pressure from the US
government, led to the two-decade absence of astronomers of mainland China from
the international stage of astronomical collaboration and exchange. Squeezed
between the demands of the then two superpowers, the United States of America
and the Soviet Union, the IAU survived, but unfortunately was clearly damaged.
\section{The return of China}
In the following two decades, in spite of the growing dissatisfaction and
concern, restrained by the then prevailing political environment, both
international and domestic, little progress was made on the restoration of the
legitimate position of China in the IAU until the 1979 Montreal Assembly. Several events before then eased the way forwards from the prevailing deadlock:
\begin{enumerate}
\item[1.] In 1971, UN Resolution No.\,2758 restored the membership of the PRC and expelled
the ROC from the UN; \item[2.] In 1972, Richard Nixon visited China. The joint communiqu\'e
recognized Taiwan as a part of China, thereby ushering in a new era of Sino-American
relations; \item[3.] In 1976, the ten-year Cultural Revolution in mainland China
ended; and \item[4.] In 1978, China started the economic reforms and opening up.
\end{enumerate}
All these paved the political way for China rejoining the Union.
The key was to find ``\textit{a way acceptable to the Chinese of reinstating their
membership in the IAU without expelling Taiwan, an action that would violate
the statutes of the Union}'' \citep[see p.20]{Goldberg}.
Initiated by President Adriaan Blaauw, under the invitation of the EC, a
six-member delegation from mainland China, including Yuzhe Zhang, Shuhua Ye
(Vice President of the IAU 1988--1994) and Zhaohua Yi, and a single-member
delegation from Taiwan represented by C.S. Shen, President of the Taiwan NCA,
arrived in Montreal to discuss the matter prior to the GA on August 13--24 1979,
and to explore the possibility of ``dual membership'', that, on the one hand,
expresses the indivisibility of China (emphasized and agreed by both parties
across the Straight), and, on the other hand, reinstates the membership of PRC
without blocking China Taiwan's further adherence.
The negotiations resulted in a proposal communicated to the GA at its closing
session, that was presented in the form of an exchange of letters dated
Montreal, August 22, 1979, between the President of the CAS, Yuzhe Zhang, and
the outgoing President of the IAU, Adriaan Blaauw. The two letters were reproduced in
full in IAU \textit{Transactions} Vol. XVII \citep[pp 48--50]{Bappu}.
The only problem left was the name of the adhering organization in Taiwan.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.8\textwidth]{Fig01.pdf}
\caption{Parts of the letter from Yuzhe Zhang, President of the Chinese
Astronomical Society (1943--1982), to Adriaan Blaauw, President of the IAU \citep[pp. 525--527]{ChangBlaauw}}
\label{fig1}
\end{center}
\end{figure}
The problem was shortly solved in 1980. The arrangements were ratified at the
1982 Patras GA XVIII. The solution, accepted by all the parties involved, is
that China is to have two adhering organizations, with mainland Chinese
astronomers represented by the Chinese Astronomical Society located in Nanjing
(\textit{China Nanjing}; since 1935) and \textit{China Taiwan} astronomers represented by the
Academia Sinica located in Taipei (China Taipei; since 1959). The
denominations, ``\textit{China Nanjing}'' and ``\textit{China Taipei}'' represent the IAU
official resolution and should be used in all IAU events. At the time of this
writing (August, 2018), ``\textit{China Nanjing}'' and ``\textit{China Taipei}'' have
respectively 666 and 74 IAU individual
members.\footnote{\href{https://www.iau.org/administration/membership/national/}{www.iau.org/administration/membership/national}}
\begin{quotation}``\textit{The Union had succeeded in overcoming political schism and in restoring
harmony again among its membership for its prime purpose: the unhampered
pursuit of scientific research and intercourse}'' \citep[][p.\,204]{Blaauw}.
\end{quotation}
Shouguan Wang (President of the CAS 1985--1989), on behalf of the CAS, gave a
warm speech after the ratification of the membership of China by the GA \citep[p. 26]{Wang}:
\begin{quotation}\textit{Dear friends \& colleagues,}
\textit{The Chinese Astronomical Society celebrates its 60th anniversary this year. Its
reunion with this international community today is an event that is highly
appreciated by all its 900 members. I and my colleagues here are very glad to
have this opportunity of speaking on behalf of our Society and its members to
express our most cordial greetings and most sincere thanks to you all. Thank
you!} \end{quotation}
Zden$\check{\rm e}$k Kopal, writing on \textit{The IAU -- the first 60 years.
Reminiscences and Reflections}, published in \textit{Astrocosmos}, the newspaper of the
Patras GA, wrote, \begin{quotation}``\textit{\ldots the Union has really never been free of political
interference from many directions ever since. Perhaps the most conspicuous
example of such an interference in recent years was the technical expulsion of
the (People's) Republic of China, which was eased out of our midst in 1955
(1960) by the United States (during the enlightened era of John Foster Dulles),
in collaboration with certain astronomers from Western Europe. Only God knows
what good should have come to the science of astronomy and to the International
Astronomical Union from severing (albeit temporarily) its official ties with
the most populous nation of the Earth; but such acts did happen, and will
continue to happen as long as the present structure of the IAU remains
unchanged}'' \citep{Kopal}. \end{quotation}
The resolution of the China crisis, combined with the continued improvement of
relation across the Taiwan Straight, has benefited astronomers on both sides,
stimulated and facilitated collaborations amongst them as well as with the
international community at large. Astronomy in both mainland China and in the
Island has entered an era of rapid development. Four decades after the crisis,
China is poised to make major contributions to the world astronomy research and
education development.
In 2012, the IAU GA XXVIII was held in Beijing, being the first time in China in the
Union's nearly hundred-year history. In his opening ceremony speech, Vice-President
Jinping Xi remarked:
\begin{quotation}``\textit{The development of science and technology
requires extensive international cooperation. Science and technology have no
nationality! The vast expanse of space is the common home of all humankind; to
explore this vast universe is the common goal of all humankind; astronomy in
fast development is the shared fortune of all humankind}'' \citep{Xi}. \end{quotation}
|
1,108,101,563,278 | arxiv | \section{Introduction}\label{s:intro
Formalisms representing solutions of partial differential equations as the expected value of functionals of stochastic processes date back to Einstein and Feynman in physics, Kolmogorov and Kac in mathematics. The theory for linear parabolic equations is well developed \cite{friedman,ks}. The theory for nonlinear partial differential equations is not as well developed and it involves either branching processes \cite{legall}, or implicit, fixed point representations. Our representation is of this latter kind, `nonlinear in the sense of McKean
: the drift term in the stochastic differential equation is computed implicitly
as the expected value of an expression involving the flow it drives.
Connections between stochastic evolution and the deterministic Navier-Stokes equations have been established in seminal work of Chorin \cite{chorin}. In two dimensions, the nonlinear equation obeyed by the vorticity has the form a Fokker-Planck (forward Kolmogorov) equation. The Biot-Savart law relates the vorticity to the velocity in a linear fashion. These facts are used by Chorin to formulate the random vortex method to represent the vorticity of the Navier-Stokes equation using random walks and a particle limit. The random vortex method has been proved to converge by Goodman \cite{goodman} and Long \cite{long}, see also \cite{mb}. A stochastic representation of the Navier-Stokes equations for two dimensional flows using similar ideas but without
discretization is given in \cite{bf2d}.
A heuristic representation of the Navier-Stokes equations in three
dimensions using ideas of random walks and particles limits was done by Peskin \cite{peskin}. There are many examples of modeling approaches using stochastic
representations in physical situations, for instance \cite{chorin1, pope}.
The three dimensional situation is complicated by the fact that there are no obvious Fokker-Planck like equations describing the solutions. LeJan and Sznitman \cite{sznit} used a backward-in-time branching process in Fourier space to express the velocity field of a three dimensional viscous fluid as the average of a stochastic process. Their approach did not involve a limiting process, and this led to a new existence theorem. This was later generalized \cite{ossiander} to a physical space analogue.
More recently, Busnello, Flandoli and Romito developed a representation for the $3$-dimensional Navier-Stokes equations using noisy flow paths, similar to the ones considered here. They used Girsanov transformations to recover the velocity field of the fluid from magnetization variables, and generalized their method to work for any Fokker-Planck-type equation. Their formulation however does not admit a self contained local existence proof, which is possible using our formulation.
\smallskip
In this work we describe a stochastic representation of deterministic hydrodynamic equations that is based on the combination of the active vector formulation of the inviscid equation and the shift by a uniform Wiener process. The procedure consists thus of two steps. In the first step a Weber formula is used to express the velocity field of the inviscid equation in terms of the particle trajectories of the inviscid equation \textit{without} involving time derivatives. The second step is to replace classical Lagrangian trajectories by stochastic flows driven by the velocity field. Averaging the stochastic trajectories produces the solution. It is essential that time derivatives do not appear in the expressions to be averaged.
Perhaps the simplest illustration of this approach is the stochastic formulation of the viscous Burgers equation. The inviscid Burgers equation is
\begin{equation*}
\Dzero{u} = 0
\end{equation*}
with initial data
\begin{equation*}
u(x,0) = u_0(x).
\end{equation*}
The absence of a pressure term allows the velocity to be transported by the fluid flow. Thus if $X$ is the fluid flow map, $u \circ X$ is constant in time and $u = u_0 \circ (X_t^{-1})$ gives a Weber formula to recover the velocity from the instantaneous flow map. More explicitly, the system
\begin{align*}
\dot{X} &= u\\
A_t &= X_t^{-1}\\
u &= u_0(A)
\intertext{with initial data}
X(a,0) &= a
\end{align*}
is equivalent to the inviscid Burgers equation before the formation of shocks. By the notation $X_t ^{-1}$, we mean the spatial inverse of the map $X$ at time $t$.
It can be shown that if we add uniform noise to particle trajectories, then the average of the stochastic system satisfies the viscous Burgers equation. (We remark that we do not have to restrict our attention to gradient solutions $u = \nabla \phi$.)
\begin{theorem}\label{t:sl-burgers
Let $W$ be a $n$-dimensional Wiener process, $k \geqslant 1$ and $u_0 \in C^{k+1, \alpha}$. Let the pair $u, X$ be a solution of the stochastic system
\begin{align}
\label{e:burgers-xdef} dX &= u\,dt + \sqrt{2 \nu}\,dW\\
A &= X^{-1}\\
\label{e:burgers-udef} u &= \boldsymbol{\mathrm{E}} [u_0 \circ A]
\intertext{with initial data}
\label{e:burgers-xidata} X(a, 0) &= a.
\end{align}
For boundary conditions, we demand that either $u$ and $X - I$ are spatially periodic, or that $u$ and $X-I$ decay at infinity. Then $u$ satisfies the viscous Burgers equations
\begin{equation}
\label{e:burger-uevol} \D{u} = 0
\end{equation}
with initial data $u_0$. Here $\boldsymbol{\mathrm{E}}$ denotes the expected value with respect to the Wiener measure and $I$ is the identity function.
\end{theorem}
\begin{rem}\label{r:burgers-invert
The spatial invertibility of $X$ can be seen as follows: An elementary computation shows
$$ \det (\nabla X_t) = \exp{ \int_0^t \grad \cdot u_s\,ds}$$
showing $X_t$ is locally invertible for all time. Global invertibility now follows since our boundary conditions ensure $X_t$ is properly homotopic to the identity map and hence has degree $1$. We also remark that since the noise $W_t$ is spatially constant, the inverse map $A_t$ is as (spatially) regular as the flow map $X_t$. We refer the reader to \cite{sperturb, thesis, kunita} for the details.
\end{rem}
\begin{rem}\label{r:burgers-longtime
If a solution to the system \eqref{e:burgers-xdef}--\eqref{e:burgers-xidata}\ exists on the time interval $[0,T]$, then our proof will show that $u$ satisfies equation \eqref{e:burger-uevol} on this time interval. Though global existence for \eqref{e:burger-uevol} is known, fixed point methods will only yield a local existence result for \eqref{e:burgers-xdef}--\eqref{e:burgers-xidata}.
Conversely, given a (global) solution $u$ of \eqref{e:burger-uevol} which is either spatially periodic or decays at infinity, standard theory shows that \eqref{e:burgers-xdef} has a global solution. Now uniqueness of strong solutions for linear parabolic equations and Corollary \eqref{c:spde-v} shows that \eqref{e:burgers-udef} is satisfied for all time. We refer the reader to \cite{thesis} for the details.
\end{rem}
We provide two proofs of the corresponding result for the Navier-Stokes equations (Theorem \ref{t:sl-ns}), each of which can be adapted to yield the above theorem. We present one proof of Theorem \ref{t:sl-burgers} in Section \ref{s:ito-proof}, and leave the details of the second proof to the interested reader. In the next section we describe our stochastic formulation of the Navier-Stokes equations.
\iffals
Finally we remark that the above procedure can be used to provide stochastic representations for a more general class of nonlinear Fokker-Planck type equations.
\begin{theorem}\label{t:nl-fp
Let $(a_{ij})$ be a constant positive definite symmetric $n\times n$ matrix, $u$ a given vector field, and $N$ a nonlinear function. Let $\sigma = (\sigma_{ij})$ be a square root of the matrix $(a_{ij})$ (i.e. $a_{ij} = \sigma_{ik} \sigma_{jk}$), and $\vartheta$ be a solution of the ODE
$$ \dot{\vartheta} = N( \vartheta)$$
with initial data $\theta_0$. Let $X$ be a solution to the SDE
$$ dX = u\,dt + \sigma dW_t$$
with initial data $X(a,0) = a$. If $A = X^{-1}$ is the spatial inverse of $X$, then
$$ \theta = \boldsymbol{\mathrm{E}} \vartheta( A)$$
satisfies the nonlinear PDE
\begin{equation*}
\Dzero \theta - \tfrac{1}{2} a_{ij} \partial^2_{ij} \theta = N(\theta)
\end{equation*}
with initial data $\theta_0$.
\end{theorem
Our attention in this paper is primarily the Navier-Stokes equations, and thus we only present sketch the proof of Theorem \ref{t:nl-fp} in Section \ref{s:ito-proof}. A complete proof, and an extension of this result can be found in \cite{thesis}. We remark however that the arguments presented in this paper will not work if the the coefficients $a_{ij}$ are spatially non-constant.\medskip
The nonlocality of the pressure term in the Navier-Stokes equations will prevent us from using Theorem \ref{t:nl-fp} verbatim. However the procedure described earlier will still provide us with a stochastic representation of the Navier-Stokes equations. We describe this construction in the next section.
We provide two proofs showing our stochastic formulation indeed yields the Navier-Stokes equations. The first proof involves constructing a pointwise (in the probability space) solution to the SDE \eqref{e:X-def}. The second proof uses the generalized It\^o formula for stochastic transformations of stochastic processes \cite{kunita}, and a direct computation. These proofs can be found in sections \ref{s:ode-proof} and \ref{s:ito-proof} respectively.
\f
\section{The stochastic Lagrangian formulation of the incompressible Navier-Stokes equations}\label{s:sl-intro
In this section we show how the method described in the introduction can be used to produce a stochastic formulation of the Navier-Stokes equations. We begin by recalling a method of recovering the velocity field from the instantaneous flow map of an inviscid fluid. This is the Eulerian-Lagrangian formulation \cite{ele} which describes the Euler equations as an active vector system.
\begin{prop}\label{p:el-euler
Let $k \geqslant 0$ and $u_0 \in \holderspace{k+1}{\alpha}$ be divergence free. Then $u$ satisfies the incompressible Euler equations
\begin{gather}
\Dzero{u} + \nabla p = 0\\
\grad \cdot{u} = 0
\end{gather}
with initial data $u_0$ if and only if the pair of functions $u$, $X$ satisfies the system
\begin{align}
\label{e:ele-Xdef} \dot{X} &= u\\
\label{e:ele-Adef} A &= X^{-1}\\
\label{e:ele-udef} u &= \boldsymbol{\mathrm{P}}[ (\grad\transpose A) \, (u_0\circ A) ]
\intertext{with initial data}
\label{e:ele-Xidata} X(a,0) &= a.
\end{align}
Here $\boldsymbol{\mathrm{P}}$ is the Leray-Hodge projection \cite{cm,stein} on divergence free vector fields. We impose either periodic boundary conditions and demand that $u$ and $X - I$ are spatially periodic, or demand that $u$ and $X - I$ decay sufficiently rapidly at infinity ($I$ is the identity map).
\end{prop
The proof of one direction of this theorem follows immediately from Lemma \ref{l:w-evol}. A complete proof can be found in \cite{ele}.
Now to obtain a solution to the viscous system we use exactly the same Weber formula \eqref{e:ele-udef} for $u$, but consider noisy trajectories instead of deterministic ones and average the noise out.
\begin{theorem}\label{t:sl-ns
Let $\nu > 0$, $W$ be an $n$-dimensional Wiener process, $k\geqs1$ and $u_0 \in \holderspace{k+1}{\alpha}$ be a given deterministic divergence free vector field. Let the pair $u$, $X$ satisfy the stochastic system
\begin{align}
\label{e:X-def} dX &= u \,dt + \sqrt{2 \nu} \,dW\\
\label{e:A-def} A &= X^{-1}\\
\label{e:u-def} u &= \boldsymbol{\mathrm{E}} \boldsymbol{\mathrm{P}}\left[ (\grad\transpose A) \, (u_0 \circ A) \right]
\intertext{with initial data}
\label{e:X-idata} X(a,0) &= a.
\end{align}
We impose boundary conditions by requiring $u$ and $X-I$ are either spatially periodic, or decay sufficiently at infinity. Then $u$ satisfies the incompressible Navier-Stokes equations
\begin{gather*}
\D{u} + \nabla p = 0\\
\grad \cdot{u} = 0
\end{gather*}
with initial data $u_0$.
\end{theorem}
\begin{rem
Remarks \ref{r:burgers-invert} and \ref{r:burgers-longtime} are also applicable here. See also \cite{thesis}.
\end{rem}
\begin{rem}\label{r:sl-ns-force
In the presence of a deterministic external force $f$, we only need to replace $u_0$ in equation \eqref{e:u-def} with $\varphi$ defined by
$$\varphi_t = u_0 + \int_0^t (\grad\transpose X) f(X_s, s)\, ds.$$
We provide a proof of this in Section \ref{s:ito-proof}, however we remark that the proof of Theorem \ref{t:sl-ns} given in Section \ref{s:ode-proof} can also be adapted to yield this. Clearly, if the forcing is random and independent of the Wiener process $W$, then our procedure provides a representation of the stochastically forced Navier-Stokes equations.
\end{rem}
\iffals
This following remark is false! The construction below gives an extra $\grad\transpose (\varphi_\varepsilon * u) u$ term. The reason we miss it is because we do not have an inviscid Weber formula.
\begin{rem}\label{r:sl-alpha-model}
This construction can be modified to represent the Leray regularized Navier-Stokes equations. Let $\varphi$ be a smooth bump function. If we replace equation \eqref{e:X-def} with
$$ dX = ( \varphi_\varepsilon * u)(X) + \sqrt{2\nu} dW$$
then $u$ will satisfy the Leray regularized equation
$$ \D[ (\varphi_\varepsilon * u)]{u} + \nabla p = 0.$$
We leave the details of this modification to the interested reader.
\end{rem}
\fi
\begin{rem
The construction above can be modified to provide a stochastic representation of the LANS-alpha (or Camassa-Holm) equations. The inviscid Camassa-Holm \cite{chen, holm} equations are
\begin{gather*}
\Dzero{v} + (\grad\transpose u) v + \nabla p = 0\\
u = (1 - \alpha^2\triangle)^{-1} v\\
\grad \cdot v = 0
\end{gather*}
Lemma \ref{l:w-evol} gives a formula to recover $v$ from the inverse of the flow map. Thus we obtain a stochastic representation of the viscous Camassa-Holm equations by replacing \eqref{e:u-def} in \eqref{e:X-def}--\eqref{e:X-idata}\ with
\begin{gather}
\label{e:ch-vdef} v = \boldsymbol{\mathrm{E}}\boldsymbol{\mathrm{P}}\left[ (\grad\transpose A) \, u_0 \circ A \right]\\
\label{e:ch-udef} u = (1 - \alpha^2 \triangle)^{-1} v.
\end{gather}
The velocity $v$ will now satisfy the viscous equation
\begin{equation*}
\Dzero{v} + (\grad\transpose u) v - \nu \triangle v + \nabla p = 0.
\end{equation*}
We draw attention to the fact that the diffusive term is $\nu \triangle v$ and not $\nu \triangle u$. However, we do not derive the relation $u= (1-\alpha^2\triangle)^{-1} v$; and any other translation-invariant filter $u = Tv$ would work as well.
\end{rem
We provide two independent proofs of Theorem \ref{t:sl-ns}. We postpone these proofs to sections \ref{s:ode-proof} and \ref{s:ito-proof} respectively, and devote the remainder of this section to consequences of this theorem.
The first consequence we mention is that we have a self contained proof for the local existence of the stochastic system \eqref{e:X-def}--\eqref{e:X-idata}.
\begin{theorem}[Local existence]\label{t:snsexist
Let $k \geqslant 1$ and $u_0 \in \holderspace{k+1}{\alpha}$ be divergence free. There exists a time $T = T(k, \alpha, L, \hnorm{u_0}{k+1}{\alpha})$, but independent of viscosity, and a pair of functions $\lambda, u \in C([0,T], \holderspace{k+1}{\alpha})$ such that $u$ and $X = I + \lambda$ satisfy the system \eqref{e:X-def}--\eqref{e:X-idata}. Further $\exists U = U( k, \alpha, L, \hnorm{u_0}{k+1}{\alpha})$ such that for all $t \in [0,T]$ we have $\hnorm{u(t)}{k+1}{\alpha} \leqslant U$.
\end{theorem}
Here $L > 0$ is a given length scale, and \holderspace{k}{\alpha} is the space of $(k, \alpha)$ H\"older continuous functions which are periodic with period $L$. The theorem is still true if we consider the domain $\mathbb{R}^3$, and impose decay at infinity boundary conditions instead. We do not present the proof in this paper, but refer the reader to \cite{sperturb, thesis}.
We remark again that our estimates, and existence time are independent of viscosity. The theorem and proof also work when the viscosity $\nu = 0$, thus we have a proof that gives us local existence of both the Euler and Navier-Stokes equations.
The nature of our formulation causes most identities for the Euler equations (in the Eulerian-Lagrangian form) to be valid in the above stochastic formulation after averaging. We begin by presenting identities for the vorticity.
\begin{prop}\label{p:vort-rep
Let $\omega = \grad \times u$ be the vorticity, and $\omega_0 = \grad \times u_0$ be the initial vorticity. Then
\begin{equation}
\label{e:vort-trans} \omega = \boldsymbol{\mathrm{E}} \left[ \left( (\nabla X) \,\omega_0\right) \circ A \right].
\end{equation}
If the flow is two dimensional then the above formula reduces to
\begin{equation}
\label{e:2d-vort-trans} \omega = \boldsymbol{\mathrm{E}} \left[\omega_0 \circ A\right].
\end{equation}
\end{prop}
\begin{rem
In the presence of an external force $f$, we have to replace $\omega_0$ in equations \eqref{e:vort-trans} and \eqref{e:2d-vort-trans} with $\varpi$ defined by
$$ \varpi_t = \omega_0 + \int_0^t (\nabla X_s)^{-1} g(X_s, s) \, ds$$
where $g = \grad \times f$. For two dimensional flows this reduces to
$$ \varpi_t = \omega_0 + \int_0^t g(X_s, s) \, ds.$$
We draw attention to the fact that these are exactly the same as the expressions in the inviscid case.
\end{rem
We can prove this proposition in two ways: The first method is to directly differentiate \eqref{e:u-def}, and use the fact that $\boldsymbol{\mathrm{P}}$ vanishes on gradients. We leave the details of this to the interested reader. The other method (presented in Section \ref{s:ito-proof}) is to use Proposition \ref{p:spde-a} and the generalized It\^o formula to show that $\omega$ satisfies the vorticity equation
\begin{equation}
\label{e:vort-evol}\D{\omega} = (\omega \cdot \nabla) u.
\end{equation}
The two dimensional vorticity equation does not have the stretching term $(\omega \cdot \nabla) u$. With our formulation the two dimensional vorticity equation is immediately obtained by observing that the third component of the flow map satisfies $X_3(a,t) = a_3$.
Although it is evident from equations \eqref{e:u-def} and \eqref{e:vort-trans}, we explicitly point out that the source of growth in the velocity and vorticity fields arises from the gradient of the noisy flow map $X$. The Beale-Kato-Majda \cite{bkm} criterion guarantees if the vorticity $\omega$ stays bounded, then no blow up can occur in the Euler equations. In the case of the Navier-Stokes equations, well known criteria for regularity exist and they can be translated in criteria for the average of the stochastic flow map. As is well known, the vorticity of a two dimensional fluid stays bounded, which immediately follows from equation \eqref{e:2d-vort-trans}.
Finally we mention the conservation of circulation. For this we need to consider a stochastic velocity $\tilde{u}$ defined by
\begin{equation}
\label{e:urandom-def}\tilde{u} = \boldsymbol{\mathrm{P}} \left[(\grad\transpose A) \, (u_0 \circ A) \right].
\end{equation}
We remark that $\tilde{u}$ is defined for all realizations of the Wiener process, because the diffusion matrix in equation \eqref{e:X-def} is spatially constant. This can be seen from the techniques used by LeBris, Lions \cite{lebris} or those used in Section \ref{s:ode-proof}. Notice immediately that $u = \boldsymbol{\mathrm{E}} \tilde{u}$ and $\tilde{u}_0 = u_0$. The circulation of the stochastic velocity is conserved by the stochastic flow.
\begin{prop
If $\Gamma$ is a closed curve in space, then
$$ \oint_{X(\Gamma)} \tilde{u} \cdot dr = \oint_\Gamma u_0 \cdot dr.$$
\end{prop}
\begin{proof
By definition of $\boldsymbol{\mathrm{P}}$, there exists a function $q$ so that
\begin{alignat*}{2}
&& \tilde{u} &= (\grad\transpose A) (u_0\circ A) + \nabla q\\
&\implies\quad& \grad\transpose X\at{A} \tilde{u} &= u_0 \circ A + \grad\transpose X\at{A} \nabla q\\
&\implies& (\grad\transpose X) (\tilde{u} \circ X) &= u_0 + \nabla(q \circ X).
\end{alignat*}
Hence
\begin{align*}
\oint_{X(\Gamma)} \tilde{u} \cdot dr &= \int_0^1 (\tilde{u}\circ X \circ \Gamma) \cdot (\nabla X\at{\Gamma} \Gamma') \,dt\\
&= \int_0^1 (\grad\transpose X\at{\Gamma}) (\tilde{u} \circ X \circ \Gamma) \cdot \Gamma' \,dt\\
&= \oint_\Gamma (u_0 + \nabla( q \circ X) ) \cdot dr = \oint_\Gamma u_0 \cdot dr.
\end{align*}
We remark that the above proof is exactly the same as a proof showing circulation is conserved in inviscid flows.
\end{proof
The rest of this paper is devoted to proving the results stated in Sections \ref{s:intro} and \ref{s:sl-intro}. We provide two independent proofs of Theorem \ref{t:sl-ns}, however only adapt the second proof to yield Theorems \ref{t:sl-burgers}, \ref{t:sl-ns} and Proposition \ref{p:vort-rep}.
\section{Proof of the stochastic representation using pointwise solutions}\label{s:ode-proof
In this section we prove Theorem \ref{t:sl-ns} by constructing pointwise (in the probability space) solutions to the SDE \eqref{e:X-def}. This idea has been used by LeBris and Lions in \cite{lebris} using a generalization of the $W^{1,1}$ theory. In our context however, the velocity $u$ is spatially regular enough for us to explicitly construct the pointwise solution without appealing to the generalized $W^{1,1}$ theory.
We remark that the proof given here will also prove Theorem \ref{t:sl-burgers}, and leave the details to the interested reader. We begin with a few preliminaries.
\begin{defn
Given a (divergence free) velocity $u$, we define the operator $\mathcal{D}$ by
\begin{equation*}
\mathcal{D} v = \Dzero{v}
\end{equation*}
\end{defn}
\begin{lemma
The commutator $[\mathcal{D}, \nabla]$ is given by
\begin{equation*}
[\mathcal{D}, \nabla] f = \mathcal{D} (\nabla f) - \nabla( \mathcal{D} f) = -\grad\transpose u \nabla f
\end{equation*}
\end{lemma}
\begin{proof
By definition,
\begin{align*}
[\mathcal{D}, \nabla] f &= \mathcal{D} (\nabla f) - \nabla( \mathcal{D} f) \\
&= (u \cdot \nabla) \nabla f - \nabla\left[ (u \cdot \nabla) f \right]\\
&= (u \cdot \nabla) \nabla f - (u \cdot \nabla) \nabla f - \grad\transpose u \nabla f \qedhere
\end{align*}
\end{proof}
\begin{lemma}\label{l:w-evol
Given a divergence-free velocity $u$, let $X$ and $A$ be defined by
\begin{gather*}
\dot{X} = u(X)\\
X(a,0) = a\\
A = X^{-1}
\end{gather*}
We define $v$ by the evolution equation
\begin{equation*}
\mathcal{D} v = \Gamma.
\end{equation*}
with initial data $v_0$. If $w$ is defined by
\begin{equation*}
w = \boldsymbol{\mathrm{P}}\left[ \left(\grad\transpose A\right) \: v\right],
\end{equation*}
then the evolution of $w$ is given by the system
\begin{gather*}
\partial_t w + (u \cdot \nabla) w + (\grad\transpose u) w + \nabla p = (\grad\transpose A) \Gamma\\
\grad \cdot w = 0\\
w_0 = \boldsymbol{\mathrm{P}} v_0
\end{gather*}
\end{lemma}
\begin{proof
By definition of the Leray-Hodge projection, there exists a function $p$ such that
\begin{align*}
w &= \grad\transpose A \: v - \nabla p\\
&= v_i \nabla A_i - \nabla p\\
\implies\quad \mathcal{D} w &= (\mathcal{D} v_i) \nabla A_i + v_i \mathcal{D} \nabla A_i - \mathcal{D} \nabla p\\
&= \Gamma_i \nabla A_i - v_i (\grad\transpose u) \nabla A_i - \nabla \mathcal{D} p + (\grad\transpose u) \nabla p\\
&= (\grad\transpose A) \Gamma - (\grad\transpose u)( v_i \nabla A_i + \nabla p) - \nabla \mathcal{D} p\\
&= (\grad\transpose A) \Gamma - (\grad\transpose u)w - \nabla \mathcal{D} p. \qedhere
\end{align*}
\end{proof}
\begin{cor
If $u, X, A$ are as above, and we define $w$ by
\begin{equation*}
w = \boldsymbol{\mathrm{P}}\left[ (\grad\transpose A) \: u_0 \circ A \right].
\end{equation*}
Then $w$ evolves according to
\begin{gather*}
\mathcal{D} w + (\grad\transpose u) w + \nabla p = 0\\
\grad \cdot w = 0\\
w(x,0) = w_0(x)
\end{gather*}
\end{cor}
\begin{proof
The proof follows from Lemma \ref{l:w-evol} by setting $v=u_0 \circ A$ and $\Gamma = 0$.
\end{proof}
We now return to the proof of Theorem \ref{t:sl-ns}.
\begin{proof}[Proof of Theorem \ref{t:sl-ns}
For simplicity and without loss of generality we take $\nu = \frac{1}{2}$. Let $(\Omega, \mathcal{F}, P)$ be a probability space and $W:[0,\infty) \times \Omega \to \mathbb{R}^3$ a three dimensional Wiener process. Define $u^\omega$ and $Y^\omega$ by
\begin{equation*}
u^\omega(x,t) = u(x + W_t(\omega), t)
\end{equation*}
and
\begin{gather*}
\dot{Y}^\omega = u^\omega(Y^\omega)\\
Y^\omega(a,0) = a
\end{gather*}
Although $u^\omega$ is not Lipschitz in time, it is certainly uniformly (in time) Lipschitz in space. Thus the regular Picard iteration will produce solutions of this equations. Finally notice that the map $X$ defined by
\begin{equation*}
X(a, t, \omega) = Y^\omega(a, t, \omega) + W_t(\omega)
\end{equation*}
solves the SDE \eqref{e:X-def}.
Let $B^\omega$ be the spatial inverse of $Y^\omega$. Notice that
\begin{align*}
X(B^\omega(x - W_t, t), t) &= Y^\omega( B^\omega(x - W_t, t), t) + W_t\\
&= x
\end{align*}
and hence
\begin{equation}
A = \tau_{W_t} B^\omega
\end{equation}
where $\tau_x$ is the translation operator defined by
\begin{equation}
\tau_x f (y) = f(y - x)
\end{equation}
We define ${w^\omega}$ by
\begin{equation*}
{w^\omega} = \boldsymbol{\mathrm{P}}\left[ (\grad\transpose B^\omega) \; u_0 \circ B^\omega \right].
\end{equation*}
By Lemma \ref{l:w-evol}, the function ${w^\omega}$ evolves according to
\begin{gather}
\label{e:womega-evol} \partial_t {w^\omega} + (u^\omega \cdot \nabla) {w^\omega} + (\grad\transpose u^\omega) w + \nabla q^\omega = 0\\
\label{e:womega-dfree} \grad \cdot {w^\omega} = 0\\
\label{e:womega-idata} {w^\omega}(x,0) = u_0(x).
\end{gather}
Now using equation \eqref{e:u-def} we have
\begin{alignat}{2}
\nonumber && u &= \boldsymbol{\mathrm{E}} \boldsymbol{\mathrm{P}} \left[ (\grad\transpose A) \; u_0 \circ A \right]\\
\nonumber && &= \boldsymbol{\mathrm{E}} \boldsymbol{\mathrm{P}} \left[ (\grad\transpose \tau_{W_t} B^\omega) \; u_0 \circ \tau_{W_t} B^\omega \right]\\
\nonumber && &= \boldsymbol{\mathrm{E}} \boldsymbol{\mathrm{P}} \left[ \tau_{W_t} \left( (\grad\transpose B^\omega) \; u_0 \circ B^\omega \right)\right]\\
\nonumber && &= \boldsymbol{\mathrm{E}} \tau_{W_t} \boldsymbol{\mathrm{P}} \left[ (\grad\transpose B^\omega) \; u_0 \circ B^\omega \right]\\
\label{e:u=ew} && &= \boldsymbol{\mathrm{E}} \tau_{W_t} {w^\omega}
\end{alignat}
Our assumption $u_0 \in \holderspace{k+1}{\alpha}$ along with Theorem \ref{t:snsexist} guarantee that ${w^\omega}$ is spatially regular enough to apply the generalized It\^o formula \cite{kunita} to ${w^\omega}( x - W_t, t)$, and we have
\begin{multline*}
{w^\omega}( x - W_t, t) - u_0(x) = \int_0^t {w^\omega}( x - W_s, ds) - \int_0^t \nabla {w^\omega} \at{x-W_s, s} \, dW_s + \\
+ \tfrac{1}{2} \int_0^t \triangle {w^\omega}\at{x-W_s, s} \,ds + \qv{\int_0^t \partial_j {w^\omega}( x - W_s, ds)}{x_j - W^j_t}.
\end{multline*}
Notice that the process ${w^\omega}$ is $C^1$ in time (since the time derivative is given by equation \eqref{e:womega-evol}), and hence bounded variation. Thus the joint quadratic variation term vanishes. Taking expected values and using \eqref{e:u=ew} we conclude
\begin{equation}
\label{e:u1} u(x,t) - u_0(x) = \boldsymbol{\mathrm{E}} \int_0^t {w^\omega}( x - W_s, ds) + \tfrac{1}{2} \int_0^t \triangle u(x,s) \,ds
\end{equation}
Using equation \eqref{e:womega-evol} and the definition of the It\^o integral we have
\begin{align}
\nonumber \boldsymbol{\mathrm{E}} \int_0^t {w^\omega}( x - W_s, ds) &= \boldsymbol{\mathrm{E}} \int_0^t \partial_t {w^\omega}\at{x-W_s, s} ds\\
\nonumber & = - \boldsymbol{\mathrm{E}} \int_0^t \left[ ( u^\omega \cdot \nabla ) {w^\omega} + (\grad\transpose u^\omega) {w^\omega} + \nabla q^\omega \right]_{x-W_s, s} ds\\
\nonumber & = - \boldsymbol{\mathrm{E}} \int_0^t \Big[ \left( u(x,s) \cdot \nabla \right) {w^\omega}\at{x-W_s, s} + \left(\grad\transpose u(x,s)\right) {w^\omega}\at{x-W_s, s} +\\
\nonumber &\qquad\qquad + \nabla q^\omega\at{x-W_s, s}\Big]\, ds\\
\nonumber & = - \int_0^t \Big[ \left( u(x,s) \cdot \nabla \right) u\at{x, s} + \left(\grad\transpose u(x,t)\right) u\at{x, s} + \\
\nonumber & \qquad\qquad + \nabla \boldsymbol{\mathrm{E}} q^\omega(x - W_s, s) \Big] \, ds\\
\label{e:Eintwomega} & = - \int_0^t \left[ \left( u(x,s) \cdot \nabla \right) u\at{x, s} + \nabla q'\at{x,s} \right]\,ds
\end{align}
where $q'$ is defined by
\begin{equation*}
q' = \tfrac{1}{2} \nabla |u|^2 + \boldsymbol{\mathrm{E}} \tau_{W_t} q^\omega
\end{equation*}
Using equations \eqref{e:Eintwomega} in \eqref{e:u1} (along with the observation that the joint quadratic variation term is $0$) we obtain
\begin{equation*}
u(x,t) - u_0(x) = - \int_0^t \left[ (u \cdot \nabla) u + \tfrac{1}{2} \triangle u + \nabla q \right]_{x,s} \, ds
\end{equation*}
Equations \eqref{e:womega-dfree} and \eqref{e:u=ew} show that $u$ is divergence free, concluding the proof.
\end{proof}
\section{Proof of the stochastic representation using the It\^o formula}\label{s:ito-proof
In this section we provide a proof of Theorems \ref{t:sl-burgers}, \ref{t:sl-ns} and Proposition \ref{p:vort-rep} directly using the generalized It\^o formula. At the end of the section we provide a brief contrast with the diffusive Lagrangian formulation \cite{elns}.\smallskip
Before beginning our computations we remark that our assumptions $k\geqslant 1$, $u_0 \in \holderspace{k+1}{\alpha}$ and Theorem~\ref{t:snsexist} guarantee and that the processes $A$ and $X$ are spatially regular enough to apply the generalized It\^o formula~\cite{kunita}. Further, as shown in \cite{sperturb} the displacements $X-I$ and $A-I$ are spatially $\holderspace{k+1}{\alpha}$ (thus bounded in periodic domains) and hence the It\^o integrals (and expectations) that arise here are all well defined. This said, we assume subsequently that all processes are spatially regular enough for our computations to be valid.
We begin with a lemma leading up to computing the It\^o derivative of $A$.
\begin{lemma}\label{l:mpart-a
There exists a process $B$ of bounded variation such that almost surely
\begin{equation}
\label{e:mpart-a} A_t = B_t - \sqrt{2\nu}\int_0^t \nabla A_s \,dW_s
\end{equation}
\end{lemma}
\begin{proof
We apply the generalized It\^o formula to $A \circ X$ to obtain (almost surely)
\begin{align}
\nonumber 0 &= \int_{t'}^t A(X_s, ds) + \int_{t'}^t \nabla A\at{X_s,s} \,dX_s + \tfrac{1}{2}\int_{t'}^t \partial^2_{ij} A\at{X_s, s} d\qv{X_i}{X_j}_s + \\
\nonumber &\qquad\qquad + \qv{\int_{t'}^t \partial_i A(X_s, ds)}{X^i_t - X^i_{t'}}\\
\label{e:daox} &= \int_{t'}^t A(X_s, ds) + \int_{t'}^t \left[\nabla A\at{X_s,s} u + \nu\triangle A\at{X_s, s} \right]ds + \sqrt{2\nu}\int_{t'}^t \nabla A\at{X_s,s} \,dW_s + \\
\nonumber &\qquad\qquad + \qv{\int_{t'}^t \partial_i A(X_s, ds)}{X^i_t - X^i_{t'}}.
\end{align}
Notice that the second and fourth terms on the right are of bounded variation. Since the above equality holds for all $t', t$, and $X_s$ is a homeomorphism, the lemma follows.
\end{proof
We remark that the $\triangle A$ term in \eqref{e:daox} has a positive sign, which is the anti-diffusive sign forward in time. The quadratic variation term in \eqref{e:daox} brings in twice the negative Laplacian, thus correcting this sign, and we end up with a dissipative SPDE for $A$ as we should.
\iffalse
\begin{lemma}\label{l:mpart-coa
Let $C$ be an absolutely continuous process. Then there exists a process $B'$ of bounded variation such that
$$ C\circ A_t - C\circ A_{t'} = B'_t - \sqrt{2\nu} \int_{t'}^t \nabla (C\circ A_s) \,dW_s.$$
\end{lemma}
\begin{proof
Let $c$ be the derivative of $C$. We apply the generalized It\^o formula to $C\circ A$. Notice that the cross quadratic variation term vanishes because $C$ is of bounded variation. Thus we have
\begin{align*}
C\circ A_t - C\circ A_{t'} &= \int_{t'}^t c( A_s, s) \,ds + \int_{t'}^t \nabla C\at{A_s, s} \,dA_s + \tfrac{1}{2} \int_{t'}^t \partial^2_{ij} C\at{A_s, s} \,d\qv{A_i}{A_j}_s\\
&= \int_{t'}^t c( A_s, s) \,ds + \int_{t'}^t \nabla C\at{A_s, s} \,dB_s - \sqrt{2\nu} \int_{t'}^t \nabla C\at{A_s, s} \nabla A_s \,dW_s +\\
&\qquad\qquad + \tfrac{1}{2} \int_{t'}^t \partial^2_{ij} C\at{A_s, s} \,d\qv{A_i}{A_j}_s
\end{align*}
Since the first third and fourth terms have bounded variation, we are done.
\end{proof}
\f
\begin{prop}\label{p:spde-a
The process $A$ satisfies the stochastic partial differential equation
\begin{equation}
\label{e:spde-a} dA_t + (u \cdot \nabla) A \, dt - \nu \triangle A \,dt + \sqrt{2\nu} \nabla A \, dW_t = 0
\end{equation}
\end{prop}
\begin{proof
We apply the generalized It\^o formula to $A\circ X$ as in Lemma \ref{l:mpart-a}. Notice that differentiating equation \eqref{e:mpart-a}, we find the martingale part of $\partial_i A$. Since the joint quadratic variation term in equation \eqref{e:daox} depends only on the martingale part of $\partial_i A$, we can now compute it explicitly
\begin{align*}
\qv{\int_{t'}^t \partial_i A(X_s, ds)}{X^i_t - X^i_{t'}
&= -2\nu \qv{\int_{t'}^t \nabla \left[\partial_i A\right]_{X_s, s} dW_s }{W^i_t - W^i_{t'}}\\
&= -2\nu \int_{t'}^t \triangle A\at{X_s, s} ds \qquad\text{a.s.}
\end{align*}
Substituting this in equation \eqref{e:daox}, the proposition follows.
\end{proof}
\begin{cor}\label{c:spde-v
Let $\vartheta$ be spatially $C^2$, and differentiable in time. Then the process $v = \vartheta \circ A$ satisfies the stochastic PDE
\begin{equation}
\label{e:spde-v} dv_t + (u \cdot \nabla) v_t\, dt -\nu \triangle v_t \,dt + \sqrt{2\nu} \nabla v_t \,dW_t = \partial_t \vartheta\at{A_t} \,dt
\end{equation}
\end{cor}
\begin{rem}\label{r:traj-const}
Consider the case when $\vartheta$ is independent of time. Now when $\nu = 0$, we know that equation \eqref{e:spde-a} is equivalent to the fact that $A$ is constant along particle trajectories. Thus any function of $A$, in particular $v$, is also constant along trajectores and hence automatically satisfies equation \eqref{e:spde-v}. When $\nu \neq 0$, Proposition \ref{p:spde-a} and Corolary \ref{c:spde-v} make the same assertion for noisy trajectories.
\end{rem}
\begin{proof}
The corollary follows directly from Proposition \ref{p:spde-a} and the generalized It\^o formula:
\begin{align*}
dv_t &= \partial_t \vartheta\at{A_t}\,dt + \nabla \vartheta\at{A_t} \,dA_t + \tfrac{1}{2} \partial^2_{ij} \vartheta\at{A_t} d\qv{A_i}{A_j}_t\\
&= \left[ \partial_t \vartheta\at{A_t}-\nabla \vartheta\at{A_t} (\nabla A_t) u_t + \nu \nabla \vartheta\at{A_t} \triangle A_t + \nu \partial^2_{ij} \vartheta\at{A_t} \partial_k A^i_t \partial_k A^j_t \right] dt -\\
&\qquad\qquad - \sqrt{2\nu} \nabla \vartheta\at{A_t} \nabla A_t \,dW_t\\
&= \left[ \partial_t \vartheta\at{A_t}-(u_t \cdot \nabla) v_t + \nu \triangle v_t \right] dt- \sqrt{2\nu} \nabla v_t \,dW_t\qedhere
\end{align*}
\end{proof
The results from sections \ref{s:intro} and \ref{s:sl-intro} now follow as direct consequences of Proposition \ref{p:spde-a} and Corollary \ref{c:spde-v}.
\begin{proof}[Proof of Theorem \ref{t:sl-burgers}
The proof of this theorem follows by setting $\vartheta = u_0$, integrating \eqref{e:spde-v} and taking expected values.
\end{proof}
\iffals
\begin{proof}[Proof (sketch) of Theorem \ref{t:nl-fp}]
Analogous to Proposition \ref{p:spde-a}, we obtain
$$dA + (u \cdot \nabla) A \,dt - \tfrac{1}{2} a_{ij} \partial^2_{ij} A \,dt + (\nabla A) \sigma \,dW = 0 $$
Now applying the It\^o formula to $\vartheta \circ A$, the theorem follows.
\end{proof}
\fi
\begin{proof}[Proof of Theorem \ref{t:sl-ns}
We provide the proof in the presence of an external force $f$, as stated in Remark \ref{r:sl-ns-force}. Let $v = \varphi \circ A$, $w = (\grad\transpose A) v$ and we compute the It\^o derivative of $w$ using It\^o's formula:
\begin{align*}
dw_i &= (\partial_i A) \cdot dv + d( \partial_i A) \cdot v + d\qv{\partial_i A_j}{v_j}\\
&= \partial_i A \cdot \left[ (-u\cdot \nabla) v + \nu \triangle v + (\grad\transpose X)\at{A} f\right] dt - \sqrt{2 \nu} \partial_i A \cdot ( \nabla v \, dW) +\\
&\quad+ v \cdot \left[ -\left((\partial_i u) \cdot \nabla\right) A - ( u \cdot \nabla) \partial_i A + \nu \triangle \partial_i A\right] dt - \sqrt{2\nu} v \cdot ( \nabla \partial_i A \,dW)\\
&\quad+ 2 \nu \partial^2_{ki} A_j \partial_k v_j \,dt.
\end{align*}
Making use of the identities
\begin{gather*}
(u \cdot \nabla) w_i = \partial_i A \cdot [ (u \cdot \nabla) v] + [(u \cdot \nabla) \partial_i A] \cdot v\\
\triangle w_i = \partial_i A \cdot \triangle v + \triangle \partial_i A \cdot v + 2 \partial_{ki} A_j \partial_k v_j\\
\partial_i u_k w_k = v \cdot \left[ \left( \partial_i u \cdot \nabla\right) A \right]\\
\partial_i A \cdot \left[ (\grad\transpose X)\at{A} f\right] = f
\end{gather*}
and
$$\partial_k w_i = \partial_i A_j \partial_k v_j + v_j \partial_{ki} A_j$$
we conclude
\begin{equation}
\label{e:dw} dw = \left[ -(u \cdot \nabla) w + \nu \triangle w - (\grad\transpose u) w + f \right] dt - \sqrt{2\nu} \nabla w \,dW.
\end{equation}
Now from equation \eqref{e:u-def} we have (almost surely)
\begin{alignat*}{2}
&& u &= \boldsymbol{\mathrm{E}} w + \nabla q\\
&\implies& u - u_0 &= \boldsymbol{\mathrm{E}} \int_0^t \left[ -(u \cdot \nabla) w + \nu \triangle w - (\grad\transpose u) w + f\right] + \nabla q\\
&& &= \int_0^t \left[ -(u \cdot \nabla) (u - \nabla q) + \nu \triangle (u - \nabla q) - (\grad\transpose u) (u - \nabla q) + f\right] + \nabla q\\
&& &= \int_0^t \left[ -(u \cdot \nabla)u + \nu \triangle u + f \right] + \nabla p
\end{alignat*}
where
$$p = q - \int_0^t \left[ (u \cdot \nabla) q - \nu \triangle q + \tfrac{1}{2} |u|^2 \right].$$
Differentiating immediately yields the theorem.
\end{proof}
\begin{proof}[Proof of Proposition \ref{p:vort-rep}
Notice first that $\nabla X$ is differentiable in time. We set $\vartheta = (\nabla X) \omega_0$, $\tilde{\omega} = \vartheta \circ A$ and apply Corollary \ref{c:spde-v} to obtain
\begin{align*}
d\tilde{\omega} + (u \cdot \nabla) \tilde{\omega} \, dt - \nu \triangle \tilde{\omega}\, dt + \sqrt{2\nu} \nabla \tilde{\omega}\, dW &= \nabla \partial_t X\at{A_t} \omega_0(A) \,dt\\
&= (\nabla u) (\nabla X)\at A \omega_0(A) \, dt\\
&= (\nabla u) \tilde{\omega} \,dt
\end{align*}
Integrating and taking expected values shows that $\omega = \boldsymbol{\mathrm{E}} \tilde{\omega}$ satisfies the vorticity equation \eqref{e:vort-evol} with initial data $\omega_0$. The proposition follows now follows from the uniqueness of strong solutions.
\end{proof}
\subsection{A comparison with the diffusive Lagrangian formulation.
The computations above illustrate the connection between the stochastic Lagrangian formulation \eqref{e:X-def}--\eqref{e:X-idata}\ presented here, and the deterministic diffusive Lagrangian formulation. We briefly discuss this below. In \cite{elns} the Navier-Stokes equations were shown to be equivalent to the system
\begin{gather*}
\D{\bar{A}} = 0\\
u = \boldsymbol{\mathrm{P}}[ (\grad\transpose \bar{A}) v]\\
\D{\bar{v}_\beta} = 2 \nu C^i_{j,\beta} \partial_j \bar{v}_i\\
C^p_{j,i} = (\nabla \bar{A})^{-1}_{ki} \partial_k \partial_j \bar{A}_p
\end{gather*}
with initial data
\begin{gather*}
\bar{A}(x,0) = 0\\
\bar{v}(x,0) = u_0(x).
\end{gather*}
We see first that $\bar{A} = \boldsymbol{\mathrm{E}} A$. The commutator coefficients $C^\alpha_{ij}$ in the evolution of $\bar{v}$ compensate for the first order terms in $\triangle ( (\grad\transpose \bar{A}) \bar{v} )$. In the stochastic formulation these arises naturally from the generalized It\^o formula as the joint quadratic variation term $\qv{ \partial_i A_j}{ v_j}$. More explicitly, the equations
\begin{alignat*}{2}
&& 2\nu (\grad\transpose \bar{A}) C^j_{k, \cdot} \partial_k \bar{v}_j &= 2\nu \partial^2_{kj} \bar{A}_i \partial_k \bar{v}_j\\
&\text{and}\qquad& d\qv{\partial_i A_j}{ v_j} &= 2\nu \partial^2_{kj} A_i \partial_k v_j dt
\end{alignat*}
illustrate the connection between the two representations.
\iffals
This part needs to be fixed. The probability density needs to be scaled by $e^-\grad \cdot{u}$ to compensate for change in measure. After this, it's a fundamental solution of the desired equation.
\begin{rem*
The simplicity of the Weber formula allows a more illuminating and elementary proof of the above corollary. Let $p(x,t, a, 0) = P( X(a,t) = x )$ be the transition probabilities for the process $X$. We know $p$ is a fundamental solution for the Kolmogorov forward equation
$$\D{p} = 0.$$
Since $A$ is the spatial inverse of $X$, standard theory \cite{kunita} shows that the density of the process $A$ is given by
$$ P( A(x,t) = a) = p( x, t, a, 0).$$
Hence
$$\boldsymbol{\mathrm{E}}_{x,t} \, u_0\circ A = \int u_0(a) p(x, t, a, 0),$$
and we immediately obtain \eqref{e:berger-uevol}.
\end{rem*}
\fi
\section*{Acknowledgment
Stimulating discussions with L. Ryzhik and E. S. Titi are gratefully acknowledged. P.C.'s research was partially supported by NSF-DMS grant 0504213.
|
1,108,101,563,279 | arxiv | \section{Introduction}
Figure~\ref{pipeline} shows the image of a typical indoor scene. Overlaid on this image is the pose trajectory of a person, depicted here by renderings of her body skeleton over time instants, where Frames 1-3 are in the past, Frame 4 is the present, and Frames 5-12 are in the future.
In this paper, we study the following problem: \emph{Given the scene image and the person’s past pose and location history in 2D, predict her future poses and locations.}
Human movement is goal-directed and influenced by the spatial layout of the objects in the scene. For example, the person may be heading towards the window, and will find a path through the space avoiding collisions with various objects that might be in the way. Or perhaps a person approaches a chair with the intention to sit on it, and will adopt an appropriate path and pose sequence to achieve such a goal efficiently. We seek to understand such goal-directed, spatially contextualized human behavior, which we have formalized as a pose sequence and location prediction task.
With the advent of deep learning, there has been remarkable progress on the task of predicting human pose sequences ~\cite{fragkiadaki2015recurrent,martinez2017human,wei2019motion,zhang2019phd}. However, these frameworks do not pay attention to scene context. As a representative example, Zhang et al.~\cite{zhang2019phd} detect the human bounding boxes across multiple time instances and derive their predictive signal from the evolving appearance of the human figure, but do not make use of the background image. Given this limitation, the predictions tend to be short-term~(around 1 second), and local in space, e.g., walking in the same spot without global movement. If we want to make predictions that encompass bigger spatiotemporal neighborhoods, we need to make predictions conditioned on the scene context.
We make the following philosophical choices:
(1) To understand long term behavior, we must reason in terms of goals. In the setting of moving through space, the goals could be represented by the destination points in the image. We allow multi-modality by generating multiple hypotheses of human movement ``goals'', represented by 2D destinations in the image space.
(2) Instead of taking a 3D path planning approach as in the classical robotics literature~\cite{alexopoulos1992path,lavalle2006planning}, we approach the construction of likely human motions as a learning problem by constructing a convolutional network to implicitly learn the scene constraints from lots of human-scene interaction videos. We represent the scene using 2D images.
Specifically, we propose a learning framework that factorizes this task into three sequential stages as shown in Figure~\ref{fig2}. Our model sequentially predicts the motion goals, plans the 3D paths following each goal and finally generates the 3D poses.
In Section~\ref{sec:exp}, we demonstrate our model not only outperforms existing methods quantitatively but also generates more visually plausible 3D future motion.
To train such a learning system, we contribute a large-scale synthetic dataset focusing on human-scene interaction.
Existing real datasets on 3D human motion have either contrived environment~\cite{ionescu2013human3,wang2019geometric}, relatively noisy 3D annotations~\cite{savva2016pigraphs}, or limited motion range due to the depth sensor~\cite{hassan2019resolving,savva2016pigraphs}.
This motivates us to collect a diverse synthetic dataset with clean 3D annotations. We turn the Grand Theft Auto (GTA) gaming engine into an automatic data pipeline with control over different actors, scenes, cameras, lighting conditions, and motions.
We collect over one million HD resolution RGB-D frames with 3D annotations which we discuss in detail in Section~\ref{GTA}.
Pre-training on our dataset stabilizes training and improves prediction performance on real dataset~\cite{hassan2019resolving}.
In summary, our key contributions are the following:
(1) We formulate a new task of long-term 3D human motion prediction with scene context in terms of 3D poses and 3D locations.
(2) We develop a novel three-stage computational framework that utilizes scene context for goal-oriented motion prediction, which outperforms existing methods both quantitatively and qualitatively.
(3) We contribute a new synthetic dataset with diverse recordings of human-scene interaction and clean annotations.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\textwidth]{figures/sec1/pipeline_v5.pdf}
\vspace{-1.5em}
\caption{\small{\textbf{Overall pipeline}.
Given a single scene image and 2D pose histories, our method first samples (a) multiple possible future 2D destinations. We then predict the (b) 3D human path towards each destination. Finally, our model generates (c) 3D human pose sequences following paths, visualized with the ground-truth scene point cloud.}}
\figlabel{fig2}
\vspace{-5pt}
\end{figure}
\section{Related Work}
Predicting future human motion under real-world social context and scene constraints is a long-standing problem~\cite{alahi2016social,gupta2018social,helbing1995social,kitani2012activity,sadeghian2019sophie}. Due to its complexity, most of the current approaches can be classified into global trajectory prediction and local pose prediction. We connect these two components in a single framework for long-term scene-aware future human motion prediction.
\vspace{2pt}
\boldstart{Global trajectory prediction:} Early approaches in trajectory prediction model the effect of social-scene interactions using physical forces~\cite{helbing1995social}, continuum dynamics~\cite{treuille2006continuum}, Hidden Markov model~\cite{kitani2012activity}, or game theory~\cite{ma2017forecasting}. Many of these approaches achieve competitive
results even on modern pedestrian datasets~\cite{lerner2007crowds,pellegrini2009you}. With the resurgence of neural nets, data-driven prediction paradigm that captures multi-modal interaction between the scene and its agents becomes more dominant~\cite{alahi2016social,alahi2014socially,chai2019multipath,gupta2018social,makansi2019overcoming,sadeghian2019sophie,tai2018socially,yu2018one}.
Similar to our method, they model the influence of the scene implicitly.
However, unlike our formulation that considers images from diverse camera viewpoints, they make the key assumption of the bird-eye view image or known 3D information~\cite{alahi2016social,gupta2018social,kitani2012activity,sadeghian2019sophie}.
\vspace{2pt}
\boldstart{Local pose prediction:}
Similar to trajectory prediction, there has been plenty of interest in predicting future pose from image sequences both in the form of image generation~\cite{villegas2017learning,zhao2018learning}, 2D pose ~\cite{chao2017forecasting,walker2017pose}, and 3D pose~\cite{chiu2019action,ghosh2017learning,weng2019photo,zhang2019phd}.
These methods exploit the local image context around the human to guide the future pose generation but do not pay attention to the background image or the global scene context.
Approaches that focus on predicting 3D pose from 3D pose history also exist and are heavily used in motion tracking~\cite{elhayek2012spatio,vo2016spatiotemporal}. The goal is to learn 3D pose prior conditioning on the past motion using techniques such as Graphical Models~\cite{brand2000style}, linear dynamical systems~\cite{pavlovic2001learning}, trajectory basis~\cite{akhter2009nonrigid,akhter2012bilinear}, or Gaussian Process latent variable models~ \cite{tay2008modelling,urtasun2008topologically,wang2007gaussian,wang2007multifactor}, and more recently neural networks such as recurrent nets~\cite{fragkiadaki2015recurrent,jain2016structural,li2018auto,martinez2017human,pavllo2018quaternet}, temporal convolution nets~\cite{hernandez2019human,holden2015learning,Li_2018_CVPR}, or graph convolution net in frequency space~\cite{wei2019motion}. However, since these methods completely ignore the image context, the predicted human motion may not be consistent with the scene, i.e, waling through the wall. In contrast, we propose to utilize the scene context for future human motion prediction. This is similar in spirit to iMapper~\cite{monszpart2018imapper}. However, this approach relies on computationally expensive offline optimization to jointly reason about the scene and the human motion. Currently, there is no learning-based method that holistically models the scene context and human pose for more than a single time instance~\cite{chen2019holistic++,lee2018context,li2019putting,wang2017binge,wang2020predicting}.
\vspace{2pt}
\boldstart{3D Human Motion Dataset} Training high capacity neural models requires large-scale and diverse training data. Existing human motion capture datasets either contain no environment~\cite{CMUMocap}, contrive environment~\cite{ionescu2013human3,wang2019geometric}, or in the outdoor setting without 3D annotation~\cite{vonMarcard2018}. Human motion datasets with 3D scenes are often much smaller and have relatively noisy 3D human poses~\cite{hassan2019resolving,savva2016pigraphs} due to the limitations of the depth sensor. To circumvent such problems, researchers exploit the interface between the game engine and the graphics rendering system to collect large-scale synthetic datasets~\cite{fabbri2018learning,krahenbuhl2018free}. Our effort on synthetic training data generation is a consolidation of such work to the new task of future human motion prediction with scene context.
\section{Approach}
In this work, we focus on long-term 3D human motion prediction that is goal-directed and is under the influence of scene context. We approach this problem by constructing a learning framework that factorizes long-term human motions into modeling their potential goal, planing 3D path and pose sequence, as shown in Figure~\ref{network}.
Concretely, given a $N$-step 2D human pose history~${\bf X}_{1:N}$ and an 2D image\footnote[1]{We choose to represent the scene by RGB images rather than RGBD scans because they are more readily available in many practical applications.} of the scene~${\bf I}$ (the $N$th video frame in our case), we want to predict the next~$T$-step 3D human poses together with their locations, denoted by a sequence ${\bf Y}_{N+1:N+T}$.
We assume a known human skeleton consists of $J$ keypoints, such that ${\bf X} \in \mathbb{R}^{J\times2}, {\bf Y} \in \mathbb{R}^{J\times3}$.
We also assume a known camera model parameterized by its intrinsic matrix ${\bf K} \in \mathbb{R}^3$.
To denote a specific keypoint position, we use the superscript of its index in the skeleton, e.g., ${\bf X}^r$ refers to the 2D location of the human center (torso) indexed by $r \in [1, J]$.
We motivate and elaborate our modular design for each stage in the rest of the section.
Specifically, \textit{GoalNet} learns to predict multiple possible human motion goals, represented as 2D destinations in the image space, based on a 2D pose history and the scene image.
Next, \textit{PathNet} learns to plan a 3D path towards each goal -- the 3D location sequence of the human center (torso) -- in conjunction with the scene context.
Finally, \textit{PoseNet} predicts 3D human poses at each time step following the predicted 3D path.
In this way, the resulting 3D human motion has global movement and is more plausible considering the surrounding scene.
Thanks to this modular design, our model can have either deterministic or stochastic predictions. When deploying GoalNet, our model can sample multiple destinations, which results in stochastic prediction of future human motion. If not deploying GoalNet, our model generates single-mode prediction instead. We discuss them in more detail in the rest of the section and evaluate both predictions in our experiments.
\begin{figure*}[t]
\centering
\includegraphics[width=1.0\textwidth]{figures/sec3/network_v5.pdf} \\
\caption{\small \textbf{Network architecture}. Our pipeline contains three stages: \textit{GoalNet} predicts 2D motion destinations of the human based on the reference image and 2D pose heatmaps (Section~\ref{sec:GoalNet}); \textit{PathNet} plans the 3D global path of the human with the input of 2D heatmaps, 2D destination, and the image (Section~\ref{sec:pathnet}); \textit{PoseNet} predicts 3D global human motion, i.e., the 3D human pose sequences, following the predicted path (Section~\ref{sec:posenet}).
}
\figlabel{network}
\vspace{-5pt}
\end{figure*}
\subsection{\textit{GoalNet}: Predicting 2D Path Destination} \label{sec:GoalNet}
To understand long-term human motion, we must reason in terms of goals.
Instead of employing autoregressive models to generate poses step-by-step, we seek to first directly predict the destination of the motion in the image space.
We allow our model to express uncertainty of human motion by learning a distribution of possible motion destinations, instead of a single hypothesis.
This gives rise to our GoalNet denoted as~$\mathcal{F}$ for sampling plausible 2D path destination.
GoalNet learns a distribution of possible 2D destinations $\{\hat{{\bf X}}^{r}_{N+T}\}$ at the end of the time horizon conditioned on the 2D pose history ${\bf X}_{1:N}$ and the scene image ${\bf I}$. We parametrize each human keypoint ${\bf X}^j$ by a heatmap channel ${\bf H}^j$ which preserves spatial correlation with the image context.
We employ GoalNet as a conditional variational auto-encoder~\cite{kingma2013auto}. The model encodes the inputs into a latent ${\bf z}$-space, from which we sample a random ${\bf z}$ vector for decoding and predicting the target destination positions.
Formally, we have
\begin{equation}
{\bf z} \sim \mathcal{Q}({\bf z}|{\bf H}_{1:N}^{1:J},{\bf I})
\equiv
\mathcal{N}(\bm{\mu}, \bm{\sigma})
\text{, where }\bm{\mu}, \bm{\sigma} = \mathcal{F}_{\text{enc}} ({\bf H}_{1:N}^{1:J}, {\bf I}).
\end{equation}
In this way, we estimate a variational posterior $\mathcal{Q}$ by assuming a Gaussian information bottleneck using the decoder.
Next, given a sampled ${\bf z}$ latent vector, we learn to predict our target destination heatmap with our GoalNet decoder,
\begin{equation}
\hat{{\bf H}}^r_{N+T} = \mathcal{F}_{\text{dec}} ({\bf z}, {\bf I}),
\end{equation}
where we additionally condition the decoding process on the scene image. We use soft-argmax~\cite{sun2018integral} to extract the 2D human motion destination $\hat{{\bf X}}^r_{N+T}$ from this heatmap $\hat{{\bf H}}^r_{N+T}$.
We choose to use soft-argmax operation because it is differentiable and can produce sub-pixel locations.
By constructing GoalNet, we have
\begin{equation}
\hat{{\bf H}}^r_{N+T}
=
\mathcal{F}({\bf I}, {\bf H}_{1:N}^{1:J}).
\end{equation}
We train GoalNet by minimizing two objectives: (1) the destination prediction error and (2) the KL-divergence between the estimated variational posterior $\mathcal{Q}$ and a normal distribution $\mathcal{N}(\bm{0},\bm{1})$:
\begin{eqnarray}
\begin{aligned}
L_{\text{dest2D}}
&=
\|{\bf X}^{r}_{N+T}
-
\hat{{\bf X}}^{r}_{N+T}\|_1,
\\
L_{\text{KL}}
&=
\text{KL}
\big[
\mathcal{Q}
\big(
{\bf z}
|
{\bf H}_{1:N}^{1:J}
,
{\bf I}
\big)
||
\mathcal{N}(0, 1)
\big],
\end{aligned}
\end{eqnarray}
where we weigh equally between them.
During testing, our GoalNet is able to sample a set of latent variables $\{{\bf z}\}$ from $\mathcal{N}(\bm{0}, \bm{1})$ and map them to multiple plausible 2D destinations $\{\hat{{\bf H}}^r_{N+T}\}$.
\subsection{\textit{PathNet}: Planning 3D Path towards Destination} \label{sec:pathnet}
With predicted destinations in the image space, our method further predicts 3D paths (human center locations per timestep) towards each destination.
The destination determines where to move while the scene context determines how to move.
We design a network that exploits both the 2D destination and the image for future 3D path planning.
A key design choice we make here is that, instead of directly regressing 3D global coordinate values of human paths, we represent the 3D path as a combination of 2D path heatmaps and the depth values of the human center over time.
This 3D path representation facilitates training as validated in our experiments (Section~\ref{quantitative}).
As shown in \figref{network}, our PathNet $\Phi$ takes the scene image ${\bf I}$, the 2D pose history ${\bf H}_{1:N}^{1:J}$, and the 2D destination assignment $\hat{\bf H}_{N + T}^r$ as inputs, and predicts global 3D path represented as $(\hat{\bf H}_{N+1:N+T}^r, \hat{\bf d}_{1:N+T}^r)$, where $\hat{d}_t^r\in \mathbb{R}$ denotes the depth of human center at time $t$:
\begin{equation} \label{eq:pathnet}
\hat{\bf H}_{N+1:N+T}^r, \hat{\bf d}_{1:N+T}^r
=
\Phi({\bf I}, {\bf X}_{1:N}^{1:J}, {\bf X}_{N + T}^r).
\end{equation}
We use soft-argmax to extract the resulting 2D path $\hat{\bf X}_{N+1:N+T}^r$ from predicted heatmaps $\hat{\bf H}_{N+1:N+T}^r$.
Finally, we obtain the 3D path $\hat{{\bf Y}}_{1:N+T}^r$ by back-projecting the 2D path into the 3D camera coordinate frame using the human center depth $\hat{\bf d}_{1:N+T}^r$ and camera intrinsics ${\bf K}$.
We use Hourglass54~\cite{law2019cornernet,newell2016stacked} as the backbone of PathNet to encode both the input image and 2D pose heatmaps. The network has two branches where the first branch predicts 2D path heatmaps and the second branch predicts the depth of the human torso.
We describe the network architecture with more details in Appendix~\ref{sec:supp-network}.
We train our PathNet using two supervisions. We supervise our path predictions with ground-truth 2D heatmaps:
\begin{equation}
L_{\text{path2D}}
=
\|{\bf X}^{r}_{N+1:N+T}
-
\hat{\bf X}^{r}_{N+1:N+T} \|_1.
\end{equation}
We also supervise path predictions with 3D path coordinates, while encouraging smooth predictions by penalizing large positional changes between consecutive frames:
\begin{equation}
L_{\text{path3D}}
=
\| {\bf Y}^{r}_{1:N+T}
-
\hat{\bf Y}^{r}_{1:N+T} \|_1
+
\| \hat{\bf Y}^{r}_{1:N+T-1}
-
\hat{\bf Y}^{r}_{2:N+T} \|_1
.
\end{equation}
These losses are summed together with equal weight as the final training loss. During training, we use the ground-truth destination to train our PathNet, while during testing, we can use predictions from the GoalNet.
The GoalNet and PathNet we describe so far enable sampling multiple 3D paths during inference.
We thus refer to it as the stochastic mode of the model.
The modular design of GoalNet and PathNet is flexible.
By removing GoalNet and input ${\bf X}_{N + T}^r$ from Equation~\ref{eq:pathnet}, we can directly use PathNet to produce deterministic 3D path predictions.
We study these two modes, deterministic and stochastic mode, in our experiments.
\subsection{\textit{PoseNet}: Generating 3D Pose following Path}
\label{sec:posenet}
With the predicted 3D path~$\hat{{\bf Y}}^{r}_{1:N+T}$ and 2D pose history ${\bf X}_{1:N}$, we use the transformer network~\cite{vaswani2017attention} as our PoseNet~$\Psi$ to predict 3D poses following such path.
Instead of predicting the 3D poses from scratch, we first lift 2D pose history into 3D to obtain a noisy 3D human pose sequence~$\bar{{\bf Y}}_{1:N+T}$ as input, and further use~$\Psi$ to refine them to obtain the final prediction.
Our initial estimation consists of two steps.
We first obtain a noisy 3D poses~$\bar{\bf Y}_{1:N}$ by back-projecting 2D pose history~${\bf X}_{1:N}$ into 3D using the human torso depth~$\hat{\bf d}^{r}_{1:N}$ and camera intrinsics~${\bf K}$.
We next replicate the present 3D pose~$\bar{\bf Y}_{N}$ to each of the predicted future 3D path location for an initial estimation of future 3D poses~$\bar{\bf Y}_{N+1:N+T}$.
We then concatenate both estimations together to form~$\bar{\bf Y}_{1:N+T}$ as input to our PoseNet:
\begin{equation}
\hat{{\bf Y}}_{N+1:N+T}
=
\Psi(\bar{{\bf Y}}_{1:N+T}).
\end{equation}
The training objective for PoseNet is to minimize the distance between the 3D pose prediction and the ground-truth defined as:
\begin{equation}
L_{\text{pose3D}} =
\|{\bf Y}_{N+1:N+T} - \hat{{\bf Y}}_{N+1:N+T}\|_1.
\end{equation}
During training, ground-truth 3D path ${\bf Y}^{r}_{1:N+T}$ is used for estimating coarse 3D pose input.
During testing, we use the predicted 3D path $\hat{{\bf Y}}^{r}_{1:N+T}$ from PathNet.
\section{GTA Indoor Motion Dataset} \label{GTA}
We introduce the GTA Indoor Motion dataset (GTA-IM)\footnote[2]{Dataset available in \url{https://github.com/ZheC/GTA-IM-Dataset}} that emphasizes human-scene interactions.
Our motivation for this dataset is that
existing real datasets on human-scene interaction~\cite{hassan2019resolving,savva2016pigraphs} have relatively noisy 3D human pose annotations and limited long-range human motion limited by depth sensors.
On the other hand, existing synthetic human datasets~\cite{fabbri2018learning,krahenbuhl2018free} focus on the task of human pose estimation or parts segmentation and sample data in wide-open outdoor scenes with limited interactable objects.
More detailed comparison to these datasets is in Appendix~\ref{sec:supp-dataset}.
To overcome the above issues, we spend extensive efforts in collecting a synthetic dataset by developing an interface with the game engine for controlling characters, cameras, and action tasks in a fully automatic manner.
For each character, we randomize the goal destination inside the 3D scene, the specific task to do, the walking style, and the movement speed.
We control the lighting condition by changing different weather conditions and daytime.
We also diversify the camera location and viewing angle over a sphere around the actor such that it points towards the actor. We use in-game ray tracing API and synchronized human segmentation map to track actors.
The collected actions include climbing the stairs, lying down, sitting, opening the door, and etc. -- a set of basic activities within indoor scenes. For example, the character has 22 walking styles including 10 female and 12 male walking styles. All of these factors enable us to collect a diverse and realistic dataset with accurate annotations for our challenging task.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{figures/sec4/dataset-preview_v3.pdf} \\
\vspace{-5pt}
\caption{\small {\bf Sample RGBD images from GTA-IM dataset.} Our dataset contains realistic RGB images (visualized with the 2D pose), accurate depth maps, and clean 3D human pose annotations.
}
\figlabel{gta}
\vspace{-15pt}
\end{figure}
In total, we collect one million RGBD frames of $1920\times1080$ resolution with the ground-truth 3D human pose (98 joints), human segmentation, and camera pose.
Some examples are shown in~\figref{gta}. The dataset contains $50$ human characters acting inside $10$ different large indoor scenes. Each scene has several floors, including living rooms, bedrooms, kitchens, balconies, and etc., enabling diverse interaction activities.
\section{Evaluation}\label{sec:exp}
We perform extensive quantitative and qualitative evaluations of our future 3D human path and motion predictions.
The rest of this section is organized as follows:
We first describe the datasets we use in Section~\ref{datasets}.
We then elaborate on our quantitative evaluation metrics and strong baselines in Section~\ref{baseline}.
Further, we show our quantitative and qualitative improvement over previous methods in Section~\ref{quantitative}.
Finally, we evaluate our long-term predictions and show qualitative results of destination samples and final 3D pose results in Section~\ref{sec:stochastic}. We discussed some failure cases in Section~\ref{sec:failure}. Implementation details are described in Appendix~\ref{sec:details}.
\subsection{Datasets}
\label{datasets}
\vspace{2pt} \boldstart{GTA-IM:} We train and test our model on our collected dataset as described in Section~\ref{GTA}. We split $8$ scenes for training and $2$ scenes for evaluation. We choose $21$ out of $98$ human joints provided from the dataset. We convert both the 3D path and the 3D pose into the camera coordinate frame for both training and evaluation.
\vspace{2pt} \boldstart{PROX:} Proximal Relationships with Object eXclusion (PROX) is a new dataset captured using the Kinect-One sensor by Hassan et al.~\cite{hassan2019resolving}. It comprises of 12 different 3D scenes and RGB sequences of 20 subjects moving in and interacting with the scenes. We split the dataset with 52 training sequences and 8 sequences for testing. Also, we extract 18 joints from the SMPL-X model~\cite{SMPL-X:2019} from the provided human pose parameters.
\subsection{Evaluation Metric and Baselines}\label{baseline}
\vspace{2pt} \boldstart{Metrics:}
We use the Mean Per Joint Position Error (MPJPE)~\cite{ionescu2013human3} as a metric for quantitatively evaluating both the 3D path and 3D pose prediction. We report the performance at different time steps (seconds) in millimeters (mm).
\vspace{2pt} \boldstart{Baselines:} To the best of our knowledge, there exists no prior work that predicts 3D human pose with global movement using 2D pose sequence as input. Thus, we propose three strong baselines for comparison with our method. For the first baseline, we combine the recent 2D-to-3D human pose estimation method~\cite{pavllo20193d} and 3D human pose prediction method~\cite{wei2019motion}.
For the second baseline, we use Transformer~\cite{vaswani2017attention}, the state-of-the-art sequence-to-sequence model, to perform 3D prediction directly from 2D inputs treating the entire problem as a single-stage sequence to sequence task. For the third baseline, we compare with is constructed by first predicting the future 2D human pose using~\cite{vaswani2017attention} from inputs and then lifting the predicted pose into 3D using~\cite{pavllo20193d}.
Note that none of these baselines consider scene context or deal with uncertainty in their future predictions. We train all models on both datasets for two-second-long prediction based on 1-second-long history and report their best performance for comparison.
\subsection{Comparison with Baselines}
\label{quantitative}
In this section, we perform quantitative evaluations of our method in the two datasets. We also show some qualitative comparisons in~\figref{qualitative_comparison}. We evaluate the two modes of our model: the stochastic mode that can generate multiple future predictions by sampling different 2D destinations from the GoalNet; and the deterministic mode that can generate one identical prediction without deploying GoalNet
\begin{table}[h]
\centering
\resizebox{0.7\textwidth}{!}{\
\begin{tabular}{lccccccccc}
\toprule
& \multicolumn{4}{c}{3D path error (mm)} & \multicolumn{5}{c}{3D pose error (mm)} \\
\cmidrule(lr){2-5} \cmidrule(lr){6-10}
Time step (second) & 0.5 & 1 & 1.5 & 2 & 0.5 & 1 & 1.5 & 2 & \textbf{All} $\downarrow$\\
\midrule
TR~\cite{vaswani2017attention} & 277 & 352 & 457 & 603 & 291& 374& 489& 641 & 406\\
TR~\cite{vaswani2017attention} + VP~\cite{pavllo20193d} & 157 & 240 & 358 & 494 & 174 & 267 &388 & 526 & 211 \\
VP~\cite{pavllo20193d} + LTD~\cite{wei2019motion} & 124 & 194 & 276 & 367 & 121& 180& 249& 330 &193\\
\midrule
Ours (deterministic) &
\textbf{104} &
\textbf{163} &
\textbf{219} &
\textbf{297} &
\textbf{91} &
\textbf{158} &
\textbf{237} &
\textbf{328} &
\textbf{173}
\\
\midrule
Ours (samples=4) &114 &162 &227 &310 &94 &161 &236 &323 &173\\
Ours (samples=10) &110 &154 &213 &289 &90 &154 &224 &306 &165\\
\midrule
Ours w/ xyz. output &122 &179 &252 &336 &101 &177 &262 &359 &191\\
Ours w/o image &128 &177 &242 &320 &99 &179 &271 &367 &196\\
Ours w/ masked image &120 &168 &235 &314 &96 &170 &265 &358 &189 \\
Ours w/ RGBD input &100 &138 &193 &262 &93 &160 &235 &322 &172 \\
Ours w/ GT destination &104 &125 &146 &170 &85 &133 &178 &234 &137\\
\bottomrule
\end{tabular}
}
\vspace{5pt}
\caption{\small
{\bf Evaluation results in GTA-IM dataset}. We compare other baselines in terms of 3D path and pose error. The last column (All) is the mean average error of the entire prediction over all time steps. VP denotes Pavllo et al.~\cite{pavllo20193d}, TR denotes transformer~\cite{vaswani2017attention} and LTD denotes Wei et al.~\cite{wei2019motion}.
GT stands for ground-truth, xyz. stands for directly regressing 3D coordinates of the path. We report the error of our stochastic predictions with varying number of samples.
}
\label{tab:gta_all}
\vspace{-15pt}
\end{table}
\vspace{2pt} \boldstart{GTA-IM:}
The quantitative evaluation of 3D path and 3D pose prediction in GTA-IM dataset is shown in Table~\ref{tab:gta_all}.
Our deterministic model with image input can outperform the other methods by a margin, i.e., with an average error of $173$ mm vs. $193$ mm. When using sampling during inference, the method can generate multiple hypotheses of the future 3D pose sequence. We evaluate different numbers of samples and select the predictions among all samples that best matches ground truth to report the error. We find using four samples during inference can match the performance of our deterministic model ($173$ mm error), while using ten samples, we further cut down the error to $165$ mm. These results validate that our stochastic model can help deal with the uncertainty of future human motion and outperform the deterministic baselines with few samples.
As an ablation, we directly regress 3D coordinates (``Ours w/ xyz.'' in the Table~\ref{tab:gta_all}) and observe an overall error that is $18$ mm higher than the error of our deterministic model ($191$ mm vs. $173$ mm). This validates that representing the 3D path as the depth and 2D heatmap of the human center is better due to its strong correlation to the image appearance. We also ablates different types of input to our model. Without image input, the average error is $23$ mm higher. With only masked images input, i.e., replacing pixels outside human crop by ImageNet mean pixel values, the error is $16$ mm highe. This validates that using full image to encode scene context is more helpful than only observing cropped human image, especially for long-term prediction. Using both color and depth image as input (``Ours w/ RGBD input''), the average error is $172$ mm which is similar to the model with RGB input. This indicates that our model implicitly learns to reason about depth information from 2D input.
If we use ground-truth 2D destinations instead of predicted ones, and the overall error decreases down to $137$ mm. It implies that the uncertainty of the future destination is the major source of difficulty in this problem.
\begin{table}[h]
\vspace{-5pt}
\centering
\resizebox{0.75\textwidth}{!}{\
\begin{tabular}{lccccccccc}
\toprule
& \multicolumn{4}{c}{3D path error (mm)} & \multicolumn{5}{c}{3D pose error (mm)} \\
\cmidrule(lr){2-5} \cmidrule(lr){6-10}
Time step (second) & 0.5 & 1 & 1.5 & 2 & 0.5 & 1 & 1.5 & 2 & \textbf{All} $\downarrow$\\
\midrule
TR~\cite{vaswani2017attention} & 487 & 583 & 682& 783 & 512 & 603 & 698 & 801 & 615 \\
TR~\cite{vaswani2017attention} + VP~\cite{pavllo20193d} & 262 & 358 & 461 & 548 &297 & 398 & 502 & 590 & 326\\
VP~\cite{pavllo20193d} + LTD~\cite{wei2019motion} &
194 &
263 &
332 &
394 &
216 &
274 &
335 &
\textbf{394} &
282
\\
\midrule
Ours w/o GTA-IM pretrain &192 &258 &336 &419 &192 &273 &352 &426 &280\\
Ours (deterministic) &
\textbf{189} &
\textbf{245} &
\textbf{317} &
\textbf{389} &
\textbf{190} &
\textbf{264} &
\textbf{335} &
406 &
\textbf{270} \\
\midrule
Ours (samples=3) &192 &245 &311 &398 &187 &258 &328 &397 & 264 \\
Ours (samples=6) &185 &229 &285 &368 &184 &249 &312 &377 &254\\
Ours (samples=10) &181 &222 &273 &354&182 &244 &304 &367 &249 \\
\midrule
Ours w/ gt destination &193 &223 &234 &237 &195 &235 &276 &321 &237 \\
\bottomrule
\end{tabular}
}
\vspace{5pt}
\caption{\small
{\bf
Evaluation results in PROX dataset}. We compare other baselines in terms of 3D future path and pose prediction. VP denotes Pavllo et al.~\cite{pavllo20193d}, TR denotes transformer~\cite{vaswani2017attention} and LTD denotes Wei et al.~\cite{wei2019motion}. GT stands for ground-truth. We rank all methods using mean average error of the entire prediction (last column).
}
\label{tab:prox_all}
\vspace{-20pt}
\end{table}
\vspace{4pt} \boldstart{PROX:} The evaluation results in Table~\ref{tab:prox_all} demonstrate that our method outperforms the previous state of the art in terms of mean MPJPE of all time steps, $270$ mm vs. $282$ mm. Overall, we share the same conclusion as the comparisons in GTA-IM dataset. When using sampling during inference, three samples during inference can beat the performance of our deterministic model ($264$ mm vs. $270$ mm), while using ten samples, the error decreases to $249$ mm. Note that these improvements are more prominent than what we observe on GTA-IM benchmark. This is because the uncertainty of future motion in the real dataset is larger. Therefore, stochastic predictions have more advantage.
Moreover, we find that pre-training in GTA-IM dataset can achieve better performance ($270$ mm vs. $280$ mm). Our method exploits the image context and tends to overfit in PROX dataset because it is less diverse in terms of camera poses and background appearance (both are constant per video sequence). Pre-training in our synthetic dataset with diverse appearance and clean annotations can help prevent overfitting and boost the final performance.
\begin{figure}[h]
\vspace{-15pt}
\centering
\includegraphics[width=1.0\textwidth]{figures/sec5/qualitative_comparison_v3.pdf} \\
\vspace{-10pt}
\caption{\small \textbf{Qualitative comparison}. We visualize the input (a), the results of VP\cite{pavllo20193d} and LTD~\cite{wei2019motion} (b) and our results (c) in the ground-truth 3D mesh. The color of pose is changed over timesteps according to the color bar. The first example includes both top-down and side view. From the visualization, we can observe some collisions between the baseline results and the 3D scene, while our predicted motion are more plausible by taking the scene context into consideration.
}
\figlabel{qualitative_comparison}
\vspace{-15pt}
\end{figure}
\vspace{2pt} \boldstart{Qualitative comparison:} In~\figref{qualitative_comparison}, we show qualitative comparison with the baseline of VP~\cite{pavllo20193d} and LTD~\cite{wei2019motion}. This baseline is quantitatively competitive as shown in Table~\ref{tab:gta_all} and~\ref{tab:prox_all}. However, without considering scene context, their predicted results may not be feasible inside the 3D scene, e.g., the person cannot go through a table or sofa.
In contrast, our model implicitly learns the scene constraints from the image and can generate more plausible 3D human motion in practice.
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\linewidth]{figures/sec5/deterministic-vs-multimodal/curves_v2.pdf}
\vspace{-6pt}
\caption{\small
\textbf{
Comparison between our stochastic predictions and deterministic predictions.
}
We show error curves of predicted (a) 3D paths and (b) 3D poses with varying numbers of samples over varying timesteps on GTA-IM dataset. In all plots, we find that our stochastic model can achieve better results with a small number of samples, especially in the long-term prediction (within 2-3 second time span).
}
\label{fig:deter-vs-multi}
\end{figure*}
\begin{figure}[t]
\centering
\includegraphics[width=1.0\textwidth]{figures/sec5/destinations.pdf} \\
\vspace{-5pt}
\caption{\small \textbf{Destination sampling results}. In each image, the blue dots denote the path history, the green dots are ground-truth future destination, red dots are sample destinations from the GoalNet which we draw 30 times from the standard Gaussian. Our method can generate diverse plausible motion destination samples which leads to different activities, i.e., sitting still or standing up.
}
\figlabel{destinations}
\vspace{-10pt}
\end{figure}
\subsection{Evaluation and Visualization on Longer-term Predictions}
\label{sec:stochastic}
To demonstrate our method can predict future human motion for more than 2 seconds, we train another model to produce the 3-second-long future prediction.
In Figure~\ref{fig:deter-vs-multi}, we show the self-comparisons between our stochastic predictions and deterministic predictions.
Our stochastic models can beat their deterministic counterpart using 5 samples. With increasing number of samples, the testing error decreases accordingly. The error gap between deterministic results and stochastic results becomes larger at the later stage of the prediction, i.e., more than $100$ mm difference at $3$ second time step. This indicates the advantage of the stochastic model in long-term future motion prediction.
We show qualitative results of our stochastic predictions on movement destinations in~\figref{destinations}, and long-term future motion in~\figref{qualitative}. Our method can generate diverse human movement destination, and realistic 3D future human motion by considering the environment, e.g., turning left/right, walking straight, taking a U-turn, climbing stairs, standing up from sitting, and laying back on the sofa.
\begin{figure*}[!h]
\centering
\includegraphics[width=0.95\textwidth]{figures/supplement/failure.pdf} \\
\vspace{-5pt}
\caption{\small \textbf{Visualization of failure cases}. In each red circle area, we observe the intersection between the human feet and the 3D mesh, e.g., the bed.
}
\vspace{-15pt}
\figlabel{failure}
\end{figure*}
\subsection{Discussion of failure cases} \label{sec:failure}
Our model implicitly learns scene constraints in a data-driven manner from large amounts of training data, and can produce consistent 3D human paths without serious collision comparing to previous methods which do not take scene context into consideration as shown in \figref{qualitative_comparison}. However, without assuming we have access to the pre-reconstructed 3D mesh and using expensive offline optimization as~\cite{hassan2019resolving}, the resulting 3D poses may not strictly meet all physical constraints of the 3D scene geometry. Some failure cases are shown in \figref{failure}. In the red circled area, we observe small intersections between the human feet and the 3D scene mesh, e.g., the ground floor or the bed. This issue could be relieved by integrating multi-view/temporal images as input to the learning system to recover the 3D scene better. The resulting 3D scene could be further used to enforce explicit scene geometry constraints for refining the 3D poses. We leave this to the future work.
\section{Conclusion}
In this work, we study the challenging task of long-term 3D human motion prediction with only 2D input. This research problem is very relevant to many real-world applications where understanding and predicting feasible human motion considering the surrounding space is critical, e.g., a home service robot collaborating with the moving people, AR glass providing navigation guide to visually impaired people, and autonomous vehicle planning the action considering the safety of pedestrians. We present an initial attempt in attacking the problem by contributing a new dataset with diverse human-scene interactions and clean 3D annotations. We also propose the first method that can predict long-term stochastic 3D future human motion from 2D inputs, while taking the scene context into consideration. There are still many aspects in this problem that can be explored in the future, such as how to effectively evaluate the naturalness and feasibility of the stochastic human motion predictions, and how to incorporate information of dynamic objects and multiple moving people inside the scene.
\vspace{8pt}
\boldstart{Acknowledgements} We thank Carsten Stoll and Christoph Lassner for their insightful feedback. We are also very grateful to the feedback from the BAIR community members including but not limited to Angjoo Kanazawa, Xiaolong Wang, Ashish Kumar, Allan Jabri, Ilija Radosavovic, Sasha Sax, Lerrel Pinto, and Deepak Pathak.
\section*{APPENDIX}
\vspace{-0.6em}
In the supplement, we first compare relevant datasets in Section~\ref{sec:supp-dataset}. We also cover the implementation details in Section~\ref{sec:details}. Further, we evaluate GoalNet in Section~\ref{sec:supp-goalnet} and long-term predictions on PROX in Section~\ref{sec:supp-prox-long}. We also show qualitative comparison results in Section~\ref{sec:supp-qualitative}.
Finally, we describe the network architecture in details in Section~\ref{sec:supp-network}. We include more visual results on 3D path and 3D pose prediction in our {\bf video} which can be found at \url{https://people.eecs.berkeley.edu/~zhecao/hmp/}.
\section{Dataset Comparison} \label{sec:supp-dataset}
\vspace{-15pt}
\begin{table}[!h]
\scriptsize
\begin{center}
\setlength{\tabcolsep}{4pt}
\label{table:datasets}
\begin{tabular}{lllccccccc}
\hline\noalign{\smallskip}
Dataset & \#Clips & \#Frames & \#View per clip & \#Characters &\#Scenes & Pose jittering & Depth Range & 3D scene\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
H3.6M \cite{ionescu2013human3} & 80 & 3.6 M & 4 & 11 (6M 5F) & 1 & & 0 -- 6&\\
PiGraph \cite{savva2016pigraphs} & 63 & 0.1 M & 1 & 5 (4M 1F) & 30 & \checkmark & 0 -- 4 & \checkmark\\
PROX \cite{hassan2019resolving} & 60 & 0.1 M & 1 & 20 (16M 4F) & 12 & \checkmark &0 -- 4 & \checkmark\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Ours & 119 & 1 M & 14-67 & 50 (25M 25F) & 49 & &0 -- 200 & \checkmark\\
\hline
\end{tabular}
\vspace{6pt}
\caption{\small Overview of the publicly available datasets on 3D human motion.}
\label{supp:dataset}
\end{center}
\vspace{-30pt}
\end{table}
In Table~\ref{supp:dataset}, we list some representative datasets on 3D human motion including Human3.6M (H3.6M)~\cite{ionescu2013human3}, PiGraphs~\cite{savva2016pigraphs}, and Proximal Relationships with Object eXclusion (PROX)~\cite{hassan2019resolving}. H3.6M~\cite{ionescu2013human3} is a large-scale dataset with accurate 3D pose annotations using a multi-camera capturing system.
However, all recordings were captured in the lab environment (mostly empty space) and thus it lacks diverse human interaction with the indoor environment, e.g., sitting on a sofa or climbing stairs.
PiGraphs~\cite{savva2016pigraphs} and PROX~\cite{hassan2019resolving}, on the other hand, are dedicated datasets with extensive efforts for modeling the interaction between 3D human and 3D scene.
Due to extensive efforts required for manually collecting the RGBD sequence of human activities, both datasets have a relatively small number of frames, scenes, and characters.
They are also less diverse in terms of camera poses and background appearance (only one static camera viewpoint for each entire video clip).
As shown in our experiments (Table 1 in the main paper), models trained on these datasets tend to be overfitting to the training data.
Their 3D human poses are also relatively noisy, e.g., temporal jittering, due to the difficulty of obtaining accurate 3D poses in the real-world setting.
In contrast, we collect a large-scale and more diverse dataset with clean annotations by developing an automatic data collection pipeline based on the gaming engine. We diversify the camera location and viewpoint over a sphere around the actor such that it points towards the actor. We use in-game ray tracing API and synchronized human segmentation map to track actors. When the actor walks outside the field of view, the camera will be resampled immediately. We believe our synthetic dataset with clean annotations can be complementary to real data for stable training and rigorous evaluation.
\section{Implementation Details}
\label{sec:details}
In this section, we describe the implementation details for each module of the model. All modules are implemented in PyTorch~\cite{paszke2017automatic} and trained using the ADAM optimizer~\cite{kingma2014adam}. We set the input image size and the heatmap size to 256$\times$448; the resolution of output future heatmap prediction to 64$\times$112; all depth dimension values are caped by 10 and normalized by a factor 4 during training. We train all 3 modules separately and find it works better than joint training. The multi-modal nature of this problem makes it hard to train the model with intermediate prediction, e.g., it is not quite reasonable to supervise PathNet with ground-truth (GT) 3D path towards the GT destination, when taking the input of a very different destination predicted by GoalNet.
\vspace{2pt} \boldstart{GoalNet:}
We use a $10^{-4}$ learning rate without weight decay. For both datasets, we train for 2 epochs with a batch size of 128.
\vspace{2pt} \boldstart{PathNet:}
We train our PathNet with ground-truth destination input, while during inference, we use the prediction from GoalNet instead.
The learning rate is set to $2.5\times10^{-4}$ with a $10^{-4}$ weight decay. Our models are trained for $10$ epochs for GTA-IM and $6$ epochs for PROX where learning rates decay by a factor of 10 at $7$th and $4$th epochs, respectively. We use a batch size of 32.
\vspace{2pt} \boldstart{PoseNet:}
We train the PoseNet ground-truth 3D path, while during inference, we use the prediction from PoseNet instead. We train the model for $80$ epochs using a learning rate of $1\times10^{-3}$, an attention dropout rate of $0.2$, and batch size $1024$.
\section{GoalNet Evaluation} \label{sec:supp-goalnet}
\begin{table}[h]
\vspace{-23pt}
\centering
\resizebox{0.65\textwidth}{!}{\
\begin{tabular}{lcccc}
\toprule
& \multicolumn{2}{c}{GTA-IM dataset} & \multicolumn{2}{c}{PROX dataset} \\
\cmidrule(lr){2-3} \cmidrule(lr){4-5}
Dataset &
min &
avg{\tiny $\pm$std} &
min &
avg{\tiny $\pm$std} \\
\midrule
Ours (deterministic) &
23.7 &
- &
27.7 &
- \\
\midrule
Ours (samples=3) &
25.3 &
41.6{\tiny $\pm10.3$} &
30.3 &
38.3{\tiny $\pm7.1$}
\\
Ours (samples=5) &
23.6 &
40.3{\tiny $\pm12.7$} &
27.7 &
37.2{\tiny $\pm8.3$}
\\
Ours (samples=10) &
17.6 &
34.6{\tiny $\pm14.0$} &
24.9 &
35.3{\tiny $\pm9.1$}
\\
Ours (samples=30) &
\textbf{12.2} &
35.4{\tiny $\pm17.2$} &
\textbf{21.7} &
31.3{\tiny $\pm9.7$}
\\
\bottomrule
\end{tabular}
}
\vspace{10pt}
\caption{\small
{\bf
Evaluation of 2D goal predictions in GTA-IM and PROX dataset}.
We compare our results of directly predicting 2D destination using GoalNet with those obtained by our deterministic PathNet.
We compare them in terms of the least and average error among all samples.
}
\label{tab:2d_goal}
\vspace{-15pt}
\end{table}
In Table~\ref{tab:2d_goal}, we evaluate 2D future destination predictions of GoalNet.
We use the metric of Mean Per Joint Position Error (MPJPE)~\cite{ionescu2013human3} in the 2D image space.
We compare the stochastic results sampled from GoalNet with the deterministic results.
We vary the number of samples during the evaluation and present results on both datasets. Our findings are twofold.
(1) Directly predicting 2D destinations is beneficial.
Our GoalNet can achieve similar performance with the deterministic baseline using as few as 5 samples on both datasets.
(2) With more samples, our prediction performance increases monotonously.
When using 30 samples, our GoalNet can outperform the deterministic baseline by large margins, bringing around 40\% less error on GTA-IM dataset and 20\% less error on PROX dataset.
\section{Long-term Evaluation on PROX} \label{sec:supp-prox-long}
\begin{figure*}[t]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\linewidth]{figures/supplement/path3d_prox-v31.pdf}
\caption{predicted 3D paths}
\end{subfigure}%
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\linewidth]{figures/supplement/pose3d_prox-v31.pdf}
\caption{predicted 3D poses}
\end{subfigure}%
\vspace{-6pt}
\caption{\small
\textbf{
Comparison between our stochastic predictions and deterministic predictions of long-term prediction on PROX.
}
We show error curves of predicted (a) 3D paths and (b) 3D poses with varying numbers of samples over varying timesteps on GTA-IM dataset. In all plots, we find that our stochastic model can achieve better results with a small number of samples, especially in the long-term prediction (within 2-3 seconds time span).
}
\label{fig:supp-deter-vs-multi}
\end{figure*}
In Figure~\ref{fig:supp-deter-vs-multi}, we evaluate long-term prediction results on PROX dataset as we previously showed on GTA-IM dataset.
Specifically, we show results on predicted 3D paths and predicted 3D poses using our deterministic model and stochastic model with varying numbers of samples.
We note similar trends as previously seen on GTA-IM dataset.
More interestingly, we find that our stochastic models can beat their deterministic counterpart using only 3 samples on PROX, compared to 5 samples on GTA-IM.
\section{Qualitative Results} \label{sec:supp-qualitative}
We show additional qualitative comparison results in \figref{qualitative_comp2}.
More results on 3D path and 3D pose prediction are in our {\bf video} which can be found at \url{https://people.eecs.berkeley.edu/~zhecao/hmp/}.
\section{Network Architecture} \label{sec:supp-network}
We outline our network architectures in this section.
Specifically, we define our GoalNet in Table~\ref{tab:supp-goalnet}, PathNet in Table~\ref{tab:supp-pathnet}. For PoseNet, please refer to~\cite{vaswani2017attention}, we modified the architecture by removing the input embedding layer, output embedding layer, positional encoding layer and the softmax layer.
\begin{figure*}[!h]
\centering
\includegraphics[width=1.0\textwidth]{figures/supplement/supp_qualitative_comparison_v5.pdf} \\
\caption{\small \textbf{Qualitative comparison}. We visualize the input (a), the results of VP\cite{pavllo20193d} and LTD~\cite{wei2019motion} (b) and our results (c) in the ground-truth 3D mesh. The color of pose is changed over timesteps from purple to dark blue and finally light blue. The first example (the 1st and 2nd row) includes both top-down view and side view of the results. From the visualization, we can observe some collisions between the baseline results and the 3D scene, while our predicted motion are more plausible by taking the scene context into consideration.
}
\vspace{-5pt}
\figlabel{qualitative_comp2}
\end{figure*}
\begin{table}
\centering
\begin{tabular}{ccccc}
\toprule
Index & Input & Data & Operator & Output shape \\
\midrule
(1) & - & scene image & - & $3\times256\times448$ \\
(2) & - & stacked heatmaps & - & $(N \times J)\times64\times112$ \\
\midrule
(3) & (1) & & $7\times7$, stride 2 & $64\times128\times224$ \\
(4) & (3) & & MaxPool, stride 2 & $64\times64\times112$ \\
(5) & (4) & &
$\begin{bmatrix}
3 \times 3, \text{stride 1} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $64\times64\times112$ \\
(6) & (2) & &
$\begin{bmatrix}
3 \times 3, \text{stride 1} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $64\times64\times112$ \\
(7) & (5), (6) & &
$\begin{bmatrix}
3 \times 3, \text{stride 2} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $128\times32\times56$ \\
(8) & (7) & &
$\begin{bmatrix}
3 \times 3, \text{stride 2} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $256\times16\times28$ \\
(9) & (8) & &
$\begin{bmatrix}
3 \times 3, \text{stride 2} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $512\times8\times14$ \\
(10) & (9) & encoder feat. & GlobalAvgPool & $512\times1\times1$ \\
\midrule
(11) & (10) & $\mathbf{\mu}$ & Linear & $Z\times1\times1$ \\
(12) & (10) & $\mathbf{\sigma}$ & Linear & $Z\times1\times1$ \\
\midrule
(13) & (11), (12) & ${\bf z}$ &
Sample from $\mathcal{N}(\mathbf{\mu}, \mathbf{\sigma})$
& $Z\times8\times14$ \\
(14) & (13) & &
$3\times 3$, stride 1
& $512\times8\times14$ \\
(15) & (14) & &
$\begin{bmatrix}
3 \times 3, \text{stride 1} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $512\times8\times14$ \\
(16) & (15) & &
Upsample $2\times$
& $512\times16\times28$ \\
(17) & (16) & &
$\begin{bmatrix}
3 \times 3, \text{stride 1} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $256\times16\times28$ \\
(18) & (17) & &
Upsample $2\times$
& $256\times32\times56$ \\
(19) & (18) & &
$\begin{bmatrix}
3 \times 3, \text{stride 1} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $128\times32\times56$ \\
(20) & (19) & &
Upsample $2\times$
& $128\times64\times112$ \\
(21) & (20) & decoder feat. &
$\begin{bmatrix}
3 \times 3, \text{stride 1} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $64\times64\times112$ \\
\midrule
(22) & (21) & goal heatmap pred. & $1\times1$, stride 1 & $1\times64\times112$ \\
\bottomrule
\end{tabular}
\vspace{2mm}
\caption{\small
\textbf{Overall architecture for our GoalNet.}
Each convolutional block denoted in the bracket has an internal skip connection with appropriate strides.
Each convolutional operator is followed by a batch normalization and ReLU layer, except the one before heatmap prediction.
We denote $N$ as input time frames, $J$ as the number of human keypoints, $Z$ as the dimension of latent space.
We set $Z = 30$ in our experiments.
We use nearest upsampling operator.
}
\label{tab:supp-goalnet}
\vspace{-2mm}
\end{table}
\begin{table}
\centering
\begin{tabular}{ccccc}
\toprule
Index & Input & Data & Operator & Output shape \\
\midrule
(1) & - & scene image & - & $3\times256\times448$ \\
(2) & - & stacked heatmaps & - & $(N \times J)\times256\times448$ \\
(3) & - & goal heatmap & - & $1\times256\times448$ \\
(4) & - & initial depth. & - & $N\times1\times1$ \\
(5) & - & 2D pose sequence & - & $N \times J \times 2$ \\
\midrule
(6) & (1), (2), (3) & & $7\times7$, stride 2 & $128\times128\times224$ \\
(7) & (6) & &
$\begin{bmatrix}
3 \times 3, \text{stride 2} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $256\times64\times112$ \\
(8) & (7) & backbone feat$_1$. & HourglassStack & $256\times64\times112$ \\
(9) & (8) & backbone feat$_2$. & HourglassStack & $256\times64\times112$ \\
(10) & (9) & backbone feat$_3$. & HourglassStack & $256\times64\times112$ \\
\midrule
(11) & (8) or (9) or (10) & &
$\begin{bmatrix}
3 \times 3, \text{stride 1} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $256\times64\times112$ \\
(12) & (11) & heatmap pred. & $1\times1$, stride 1 & $T\times64\times112$ \\
\midrule
(13) & (8) or (9) or (10) & &
$\begin{bmatrix}
3 \times 3, \text{stride 2} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $384\times32\times56$ \\
(14) & (13) & &
$\begin{bmatrix}
3 \times 3, \text{stride 2} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $512\times16\times28$ \\
(15) & (14) & & GlobalAvgPool & $512\times1\times1$ \\
(16) & (4), (5) & & Linear & $256\times1\times1$ \\
(17) & (15), (16) & & Linear & $256\times1\times1$ \\
(18) & (17) & depth pred. & Linear & $(N+T)\times1\times1$ \\
\bottomrule
\end{tabular}
\vspace{2mm}
\caption{\small
\textbf{Overall architecture for our PathNet.}
Each convolutional block denoted in the bracket has an internal skip connection with appropriate strides.
Each convolutional operator is followed by a batch normalization and ReLU layer, except the one before heatmap prediction.
Each linear operator is followed by a layer normalization and ReLU layer, except the one before depth prediction.
We denote $N$ as input time frames, $T$ as output time frames, $J$ as the number of human keypoints.
We obtain initial depth as input by scaling the size of the human bounding box~\cite{moon2019camera}.
We define HourglassStack in Table~\ref{tab:supp-hgstack}.
After each stack output, we use two separate branches for predicting heatmaps and human center depth.
During training, we backpropagate gradient through all stacks, while during inference, we only use the predictions from final stack.
}
\label{tab:supp-pathnet}
\vspace{-2mm}
\end{table}
\begin{table}
\centering
\begin{tabular}{ccccc}
\toprule
Index & Input & Data & Operator & Output shape \\
\midrule
(1) & - & feat. & - & $256\times64\times112$ \\
\midrule
(2) & (1) & &
$\begin{bmatrix}
3 \times 3, \text{stride 1} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $256\times64\times112$ \\
(3) & (2) & &
$\begin{bmatrix}
3 \times 3, \text{stride 2} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $384\times32\times56$ \\
(4) & (3) & &
$\begin{bmatrix}
3 \times 3, \text{stride 1} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $384\times32\times56$ \\
(5) & (4) & &
$\begin{bmatrix}
3 \times 3, \text{stride 2} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $512\times16\times28$ \\
(6) & (5) & &
$\begin{bmatrix}
3 \times 3, \text{stride 2} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $512\times8\times14$ \\
(7) & (6) & &
$\begin{bmatrix}
3 \times 3, \text{stride 1} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $512\times8\times14$ \\
(8) & (7) & &
$\begin{bmatrix}
3 \times 3, \text{stride 1} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $512\times8\times14$ \\
(9) & (8) & &
Upsample $2\times$
& $512\times16\times28$ \\
(10) & (7), (9) & &
Sum
& $512\times16\times28$ \\
(11) & (10) & &
$\begin{bmatrix}
3 \times 3, \text{stride 1} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $384\times16\times28$ \\
(12) & (10) & &
Upsample $2\times$
& $384\times32\times56$ \\
(13) & (4), (12) & &
Sum
& $384\times32\times56$ \\
(14) & (13) & &
$\begin{bmatrix}
3 \times 3, \text{stride 1} \\
3 \times 3, \text{stride 1}
\end{bmatrix}$
& $256\times32\times56$ \\
(15) & (14) & &
Upsample $2\times$
& $256\times64\times112$ \\
\midrule
(16) & (2), (15) & refined feat. &
Sum
& $256\times64\times112$ \\
\bottomrule
\end{tabular}
\vspace{2mm}
\caption{\small
\textbf{Modular architecture for one HourglassStack.}
We follow the design and implementation of \cite{law2019cornernet}.
Each convolutional block denoted in the bracket has an internal skip connection with appropriate strides.
We use nearest upsampling operator.
}
\label{tab:supp-hgstack}
\vspace{-2mm}
\end{table} |
1,108,101,563,280 | arxiv | \section{Introduction}
Anderson localization (AL)~\cite{Anderson1958,RMP1,RMP2}, namely that eigenfunctions are exponentially localized in space because of the quantum interference in disordered systems, is a fundamental quantum phenomenon in nature. The transition between metal (extended) phase and insulator (localized) phase can occur for sufficiently strong disorder in three dimensional systems, and near the transition point, there exist mobility edges (MEs) which mark the critical energy separating the extended and localized states~\cite{Lagendijk2009,Evers2008}.
MEs lie at the heart of studying various fundamental localization phenomena such as the disorder induced metal-insulator transition. Since the effect of suppressing diffusion via quantum interference is particularly pronounced in one and two dimensions, in which the eigenstates are always localized for arbitrarily small disorder strengths~\cite{Anderson1979}, and thus, no MEs exist. Besides the random disorder, quasiperiodic potentials can also induce the extended-AL transition, and bring about different physics, e.g., the existence of Anderson transition and MEs even in one dimensional (1D) systems~\cite{AA,Xie1988,Biddle0,Biddle,Pu2013,Pin2014,Ganeshan2015,Santos2019,YaoH2019,YuWang2020,JBiddle,XiaoLi,YuchengME,Basu2021,TLiu2022,Ribeiro} and multifractal critical states~\cite{Hatsugi1990,Takada2004,Liu2015,YuchengC1,Cai2013,WangYC2016,WangJ2016,YuchengC2,BoYan2011}.
Critical phase is a third type of phases, and is fundamentally different from the localized and extended phases
in the spectral statistics~\cite{Geisel,Jitomirskaya}, wave functions' distributions~\cite{Halsey,Mirlin}, and dynamical behaviors~\cite{Hiramoto,Ketzmerick}.
Quasiperiodic systems have been realized in ultracold atomic gases by superimposing two 1D optical lattices with incommensurate wavelengths, and the extended-localized transition and MEs have been observed~\cite{Roati2008,BlochME,Gadway2018,BlochME2,Gadway2020}.
However, the critical phase has not been strictly realized in experiment until now.
In recent works, we have proposed to realize the critical phase in the optical Raman lattice~\cite{YuchengC2}, which possesses the spin-orbit coupling term and an incommensurate Zeeman potential. Further, we have predicted a coexisting phase consisting of three different energy-dependent regions, i.e., the extended, localized and critical regions~\cite{YuchengC3}, which shows the abundant transport features. Recently, T. Shimasaki et. al. reported the experimental observation of the critical states and anomalous localization in a kicked quasiperiodic Aubry-Andr\'{e} (AA) lattice~\cite{Shimasaki2022}. In the kicked AA model, there is not the critical phase but the phase with coexisting critical and localized (or extended) regions~\cite{YuZhang}. An important question is whether the critical phase consisting of solely critical eigenstates and the most nontrivial coexisting phase composed of three different regions can be realized in kicked systems.
Motivated by the recent experimental realizations of the optical Raman lattices~\cite{LiuXJ2013,LiuXJ2014,Lepori2016,WangBZ2018,Liu2016,Song2018,Song2019,JWPan,RamanReview2018} and kicked systems with AL in ultracold atomic gas~\cite{Shimasaki2022,Weld2022}, we propose a scheme to realize the critical phase and the coexisting phase based on the 1D optical Raman lattice with periodically kicked quasiperiodic Zeeman potential. This system displays extremely rich localization phenomena as the change of the driven period.
\begin{figure*}[t]
\hspace*{-0.6cm}
\includegraphics[width=0.95\textwidth]{fig1.pdf}
\caption{\label{01}
(a) The mean fractal dimension $\bar{\Gamma}$ with $L=F_{14}=610$ as a function of the kicked quasiperiodic potential strength $M_z$ and the period $T$. The dashed lines correspond to $M^c_z/T=2|J_0-J_{so}|$ and $M^c_z/T=2(J_0+J_{so})$, respectively. $\Gamma$ for the wave-function of each mode at (b) $M_z=0.09, 0.29$, (c) $0.11, 0.31$ with different sizes and fixed $T=0.1$. The index of energy mode $n_E$ runs from $1$ to $N=2L$. $\Gamma$ as a function of $n_E/N$ for the sizes $L=F_{13}=377$ and $L=F_{15}=987$ with (d) $M_z=0.8$, $T=0.8$, (e) $M_z=3$, $T=0.8$, and (f) $M_z=2.3$, $T=1.7$. Here we set $J_0=1$ and $J_{so}=0.5$.}
\end{figure*}
\section{Model and phase diagram}
We propose the periodically kicked quasiperiodic optical Raman lattice model described by
\begin{equation}\label{Hsum}
H=H_0+H_{\rm SOC}+H_{K},
\end{equation}
with
\begin{subequations}
\begin{eqnarray}
H_0= -J_0\sum_{\langle i\rangle}(c^{\dagger}_{i,\uparrow}c_{i+1,\uparrow}-c^{\dagger}_{i,\downarrow}c_{i+1,\downarrow})+H.c., \quad\\
H_{\rm SOC}=J_{so} \sum_i(c^{\dagger}_{i,\uparrow}c_{i+1,\downarrow}-c^{\dagger}_{i,\uparrow}c_{i-1,\downarrow})+H.c., \quad\\
H_{K}=\sum_n\delta(t-nT)\sum_{i}\mu_i(n_{i,\uparrow}-n_{i,\downarrow}), \quad \qquad
\end{eqnarray}
\end{subequations}
where $c_{i,\sigma}$, $c^{\dagger}_{i,\sigma}$ and $n_{i,\sigma}=c^{\dagger}_{i,\sigma}c_{i,\sigma}$ are the annihilation, creation and particle number operators at lattice site $i$, respectively, and $\sigma=\uparrow,\downarrow$ denotes the spin.
The term $H_0$ ($H_{\rm SOC}$) presents the nearest neighbor spin-conserved (spin-flip) hopping with strength $J_0$ ($J_{so}$), and for convenience, we set $J_0=1$ as the energy unit. $H_K$ denotes the kicking part with
\begin{equation}
\mu_i=M_z\cos(2\pi\alpha i+\phi),
\label{mu}
\end{equation}
where $\alpha$ and $\phi$ are the irrational number and phase shift, respectively. Without loss of generality, we set
$J_{so}=0.5$, $\phi=0$~\cite{explain1,explain2} unless otherwise stated, and $\alpha=(\sqrt{5}-1)/2$, which is approached by $\alpha=\lim_{m\rightarrow\infty}\frac{F_{m-1}}{F_m}$. Here $F_m$ is the Fibonacci number defined by $F_{m+1}=F_{m-1}+F_{m}$ with the starting values $F_0=F_1=1$~\cite{Kohmoto1983}. For a finite system with size $L=F_m$, we take $\alpha=\frac{F_{m-1}}{F_m}$ when using periodic boundary conditions. When the Zeeman potential is constantly turned on, i.e., $H_K=\sum_{i}\mu_i(n_{i,\uparrow}-n_{i,\downarrow})$, there are three distinct phases: extended, critical and localized phases~\cite{YuchengC2}. The phase boundary between the extended and critical phases satisfies $M^c_z=2|J_0-J_{so}|$ and the phase boundary between the critical and localized phases satisfies $M^c_z=2(J_0+J_{so})$.
The dynamical evolution of this kicked system is described by the Floquet unitary propagator over one period, i.e.,
\begin{equation}
U(T)=e^{-i(H_0+H_{\rm SOC})T}e^{-i\sum^L_{j=1}\mu_j(n_{j,\uparrow}-n_{j,\downarrow})}.
\label{Floquet}
\end{equation}
Here we have set $\hbar=1$. In the basis of $|j,\sigma\rangle$, $\langle i,\sigma|U|j,\sigma'\rangle$ is a $2L\times 2L$ matrix.
For a initial state $|\psi(0)\rangle$, the evolution state after $N_K$ kicked periods is given by $|\psi(N_KT)\rangle=[U(T)]^{N_K}|\psi(0)\rangle$. Thus, the distribution of the eigenstate $|\psi_{\beta}\rangle$ of the
propagator $U(T)$ with the Floquet energy $E_{\beta}$, i.e., $U(T)|\psi_{\beta}\rangle=e^{-iE_{\beta}T}|\psi_{\beta}\rangle$, can reflect the dynamical property of this kicked system. To describe the distribution, we introduce the fractal dimension, which for an arbitrary eigenstate $|\psi_{\beta}\rangle=\sum_{j=1}^L[u_{\beta,j}c^{\dagger}_{j,\uparrow}+v_{\beta,j}c^{\dagger}_{j,\downarrow}]|0\rangle$ is defined as
\begin{equation}
\Gamma=-\lim_{L\rightarrow\infty}\frac{\ln(IPR)}{\ln L},
\label{mu}
\end{equation}
where $IPR=\sum_{j=1}^L(u^4_{\beta,j}+v^4_{\beta,j})$ is the inverse participation ratio (IPR)~\cite{RMP1}. It is known that $\Gamma\rightarrow 0 (1)$ for the localized (extended) states, while $0<\Gamma<1$ for the critical state. To sketch out the phase diagram, we define the mean fractal dimension over all eigenstates: $\bar{\Gamma}=(2L)^{-1}\sum^{2L}_{\beta=1}\Gamma(\beta)$. Fig.~\ref{01} (a) shows $\bar{\Gamma}$ as a function of $M_z$ and $T$ with fixed $J_{so}=0.5$. In the high-frequency regime $T\ll 1$, it is shown that the phase boundaries between the critical and extended or localized phases of this system can be well described by the dashed lines, which correspond to
\begin{eqnarray}\label{boundary}
M_{z}^{c}/T= \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \nonumber\\
\begin{cases}
2|J_0-J_{so}|, \textrm{between extended and critical phases}, \\
2(J_0+J_{so}), \textrm{between critical and localized phases}.
\end{cases}
\end{eqnarray}
To see it clearly, in Figs.~\ref{01} (b) and (c), we fix $T=0.1$ and show $\Gamma$ of different eigenstates as a function of $n_E/N$ for different sizes, where $N=2L$ is the number of the total eigenstates. One can observe that $\Gamma$ tends to $1$ for all states at $M_z=0.09$ (satisfying $M_z/T< 2|J_0-J_{so}|$) with increasing the system size, meaning that they are extended, while $\Gamma$ tends to $0$ for all states at $M_z=0.31$ (satisfying $M_z/T> 2(J_0+J_{so})$) when increasing the size, implying that all states are extended. In contrast, when $M_z=0.11$ and $0.29$ (satisfying $2|J_0-J_{so}|< M_z/T< 2(J_0+J_{so})$), $\Gamma$ is clearly different from $0$ and $1$, and almost independent of the system size, showing that all states are critical. To understand this result, we derive the effective Hamiltonian $H_{eff}$ in the high-frequency regime, namely
\begin{equation}
U(T)=exp(-iH_{eff}T).
\label{Ut}
\end{equation}
By using the Baker-Campbell-Hausdorff formula~\cite{BCH} and combining Eq.~(\ref{Floquet}) and Eq.~(\ref{Ut}), one can obtain,
\begin{eqnarray}
H_{eff}=H_0+H_{\rm SOC}+\frac{M_z}{T}V-i\frac{M_z}{2}[H_0+H_{\rm SOC}, V] \nonumber\\
+\frac{TM_z}{12}[H_0+H_{\rm SOC}, [H_0+H_{\rm SOC}, V]]+\cdots,
\label{Lambda6}
\end{eqnarray}
where $V=\sum^{L}_{j=1}\cos(2\pi\alpha j)(n_{j,\uparrow}-n_{j,\downarrow})$. When $1/T\gg 1$ and $M_z\ll 1$, the effective Hamiltonian is simplified as $H_{eff}=H_0+H_{\rm SOC}+\frac{M_z}{T}V$, which is equivalent to that obtained by transforming the $H_K$ in Eq.~(\ref{Hsum}) into
$H_K=\sum_{i}\mu_i/T(n_{i,\uparrow}-n_{i,\downarrow})$ and leaving $H_0$ and $H_{\rm SOC}$ unchanged. Compared with the
non-kicked case~\cite{YuchengC2}, this effective model includes three distinct phases, and the phase boundaries satisfy Eq.~(\ref{boundary}).
\begin{figure}[t]
\hspace*{-0.2cm}
\includegraphics[width=0.49\textwidth]{fig2.pdf}
\caption{\label{02S}
$\bar{\Gamma}$ as a function of $1/m$ for different regions with (a) $M_z=0.8$, $T=0.8$ (corresponding to Fig.~\ref{01} (d)), and (b) $M_z=2.3$, $T=1.7$ (corresponding to Fig.~\ref{01} (f)). Here $m$ are the Fibonacci indices. }
\end{figure}
With increasing $T$, the high-order terms in Eq.~(\ref{Lambda6}) can't be neglected, and thus, the effective Hamiltonian includes the non-neighbor hopping term, which will induce the occurrence of MEs~\cite{Biddle0,Biddle,Santos2019}. Figs.~\ref{01} (d) and (e) show $\Gamma$ for different sizes and $M_z$ with fixed $T=0.8$. We see that $\Gamma$ tends to $1$ ($0$) for the states in center of energy spectra of the system with $M_z=0.8$ ($M_z=3$) when increasing the system size, suggesting that they are extended (localized). In contrast, in the tails of the energy spectra in both Figs.~\ref{01} (d) and (e), the fractal dimension $\Gamma$ is clearly different from $1$ and $0$ , and is almost
independent of system sizes, implying that all states are critical. Thus, there exist energy-dependent extended and critical regions when $M_z=0.8$, and energy-dependent localized and critical regions when $M_z=3$, meaning that there are the MEs separating the extended and localized states from the critical states, which are different from the conventional MEs separating the extended states from the localized ones. With the further increasing of $T$, there will be the quantum phase with three coexisting fundamentally different regions, i.e., the localized, extended, and critical regions, as shown in Fig.~\ref{01} (f).
Now we consider the finite size effect of the fractal dimension $\Gamma$. When changing the system size, the number and magnitudes of the eigenvalues will change accordingly. Thus, it is difficult to carry out the finite size scaling analysis for a fixed eigenstate. We take a coarse graining on the spectrum and investigate the average $\Gamma$ over the eigenstates in a single region, and accordingly we define
\begin{equation}
\bar{\Gamma}=\frac{1}{N_r}\sum_{same\ region}\Gamma,
\label{meanG}
\end{equation}
where $N_r$ is the number of eigenstates in the region and can be obtained by comparing the fractal dimension with different sizes, as shown in Figs.~\ref{01} (b-f). Since all eigenstates in the same region have the same properties, the average fractal dimension in an
arbitrary small sub-region of a region can also be similarly defined, and will display the same scaling behavior with the region. Figs.~\ref{02S} (a) and (b) show $\bar{\Gamma}$, which are obtained by computing the average $\Gamma$ of all states in the same region of the systems corresponding to Figs.~\ref{01} (d) and (f), respectively. $\bar{\Gamma}$ extrapolates to $1$ and $0.75$ in the extended and critical regions of Fig.~\ref{01} (d), which confirms that the corresponding states in these regions are extended and critical, respectively. $\bar{\Gamma}$ respectively extrapolates to $0$, $1$ and the value far from $0$ and $1$ in the three different regions of Fig.~\ref{01} (f), which confirms the corresponding system with three coexisting energy-dependent regions, i.e., the extended, localized, and critical regions.
\begin{figure*}[t]
\includegraphics[width=0.92\textwidth]{fig3.pdf}
\caption{\label{02}
(a) $N_e/N$, (b) $N_c/N$, (c) $N_l/N$, (d) $N_e/N\times N_l/N$, (e) $N_e/N\times N_c/N$, (f) $N_c/N\times N_l/N$ and (g) $\kappa$ as a function of $M_z$ and $T$. (h) Schematic figure of the density of states $\rho(E)$ as a function of energy $E$ for a system with coexisting extended, critical, and localized regions. We note that (h) is a sketch map, and the positions of the three different regions depend on specific systems. Here we fix $J_0=1$ and $J_{so}=0.5$. From figures (a-c), we see that $N_e/N=1$, $N_c/N=1$ and $N_l/N=1$ in three phases with the boundaries satisfying Eq.~(\ref{boundary}) when $T<0.4$. With increasing $T$ to $T\in(0.4, 0.9)$, from figure (f), $N_c/N\times N_l/N>0$ corresponds to the phase with coexisting critical and localized states, which suggests the existence of MEs separating critical states from localized ones. In this region, from figures (a,b,e), we see $N_e=1$ and $N_c=1$ when the parameters slightly away from the boundary described as Eq.~(\ref{boundary}), but $N_e\times N_c>0$ at the boundary. The change occurs only at the region that is very close to boundary, meaning that the states in this region are more susceptible. When $T\in(0.9, 1.5)$, there are the regions corresponding to $N_e/N\times N_c/N>0$ [figure (e)] or $N_c/N\times N_l/N>0$ [figure (f)], but no regions are $\kappa>0$, meaning that there exists the phase with coexisting extended (or localized) and critical regions, but no the phase with coexisting three different regions exists. When $T\in(1.5, 2.5)$, $\kappa>0$ means the existence of the phase with coexisting extended, critical and localized regions. We summarize these results in the figure (i).}
\end{figure*}
To clearly and completely characterize the phase diagram of this system, we introduce the extended-state fraction $N_e/N$, localized-state fraction $N_l/N$, critical-state fraction $N_c/N$, and their product~\cite{YuchengC3},
\begin{equation}\label{kappa}
\kappa=\frac{N_e}{N}\times\frac{N_l}{N}\times\frac{N_c}{N},
\end{equation}
where $N_e$, $N_l$ and $N_c$ are the numbers of the extended, localized and critical eigenstates, respectively. These diagnostic quantities can characterize all different phases. $N_e/N=1$, $N_c/N=1$ and $N_l/N=1$ correspond to the extended, critical and localized phases, respectively. In the large $L$ limit, $N_e/N\times N_l/N >0$ and $\kappa=0$ characterize the conventional ME separating localized states from extended ones. $N_e/N\times N_c/N >0$ ($N_c/N\times N_l/N >0$) and $\kappa=0$ describes the ME separating critical states from extended (localized) states. The phase with coexisting localized, extended, and critical regions corresponds to $\kappa>0$.
Figs.~\ref{02} (a), (b) and (c) show the $N_e/N$, $N_c/N$ and $N_l/N$, respectively.
We see that when $T\ll 1$, this system possesses three phases with solely extended, critical and localized eigenstates, which correspond to $N_e/N=1$, $N_c/N=1$ and $N_l/N=1$, respectively, and the phase boundaries satisfy Eq.~(\ref{boundary}). With increasing $T$, eigenstates with different properties overlap each other. Figs.~\ref{02} (d), (e), (f) and (g) display the behavior of $N_e/N\times N_l/N$, $N_e/N\times N_c/N$, $N_c/N\times N_l/N$ and $\kappa$, respectively. We see that when $T>0.4$, there is a broad phase region corresponding to $N_c/N\times N_l/N>0$ and $\kappa=0$ [Figs.~\ref{02}(f) and (g)], meaning that there is the phase with coexisting critical and localized regions, and thus there are MEs separating the critical states from localized ones. Further increasing $T$, there will appear the phase with coexisting extended and critical regions [Fig.~\ref{02}(e)] and the phase with three coexisting regions [Fig.~\ref{02}(g)]. Fig.~\ref{02}(h) is a sketch of the phase with coexisting three fundamentally different regions, and one can see two types of MEs separating the localized and extended
regions from critical regions, respectively. Although there exists a broad region corresponding to $N_e/N\times N_l/N>0$ [Fig.~\ref{02}(d)], $\kappa$ is non-zero in this region [Fig.~\ref{02}(g)], meaning that there is not a phase with coexisting extended and localized regions but no critical region here. The different behaviors as the change of the driven period are summarized in Fig.~\ref{02}(i).
Further, in the range of $T\in (0.4, 0.9)$, from Figs.~\ref{02}(a-c,d,e), we see that the phases on both sides of the phase boundary between the extended and critical phases remain extended and critical, respectively, but the phases on either side of the phase boundary between the critical and localized phases are more easily influenced by the period $T$ and they no longer remain solely critical or localized. This phenomenon can be understood from Eq.~(\ref{Lambda6}), since $M_z$ near the phase boundary between the critical and localized phases is larger, which induces that the high-order terms are larger and impact the original phases more easily in the process of increasing $T$.
\section{two interesting phenomena: fragility of phase boundaries and reentrant localization transition}
Besides the rich physical properties about the MEs and the critical phase or regions in the phase diagram, there are also two interesting phenomena.
From the above section, the phases on both sides of the extended-critical phases boundary are unaffected by the periodical kick when $T\in(0.4, 0.9)$, i.e., they remain the extended or critical behaviors. However, the states on the phase boundary are easily affected.
All eigenstates are originally critical on the phase boundary between the extended and critical phases when the quasiperiodic potential is non-kicked, i.e., $N_c/N=1$ and $N_e/N=0$ on the boundary. When $T\ll 1$, the effective Hamiltonian can be described by the Hamiltonian without the kicked case. Thus, the boundary is unaffected and has $N_c/N=1$. With increasing $T$, $N_e/N\times N_c/N$ becomes non-zero, as shown in Fig.~\ref{02}(e)~\cite{explainX}, which suggests that the boundary becomes from the situation with all eigenstates being critical to the situation with extended and critical states being coexisted. For the parameters slightly away from the boundary, the extended and critical phases remain unaffected. To illustrate this, we fix $T=0.8$ and show the fractal dimensions of all eigenstates with $M_z=0.7$ and $M_z=0.9$ in Figs.~\ref{03}(a) and (b), respectively. It can be seen that $N_e/N=1$ for $M_z=0.7$ [Fig.~\ref{03}(a)] and $N_c/N=1$ for $M_z=0.9$ [Fig.~\ref{03}(a)], which are slightly away from the boundary $M_{z}^{c}/T=2|J_0-J_{so}|=1$ (we have fixed $J_0=1$ and $J_{so}=0.5$) and show the similar properties with the non-kicked case. In comparison, on the boundary with $M_z=0.8$, $N_e/N\times N_c/N$ is larger than $0$, as shown in Fig.~\ref{01}(d), suggesting that the states are no longer solely critical. Thus, the phase boundary is more susceptible to the periodical kick, which shows the fragility of the phase boundary.
\begin{figure}[t]
\includegraphics[width=0.49\textwidth]{fig4.pdf}
\caption{\label{03}
Fractal dimension $\Gamma$ as a function of $n_E/N$ for the sizes $L=377$ and $L=987$ with (a) $M_z=0.7$, $T=0.8$, (b) $M_z=0.9$, $T=0.8$, (c) $M_z=2.6$, $T=1$, and (d) $M_z=2.9$, $T=1$. (e) (f) $\bar{\Gamma}$ as a function of $1/m$. (e) The average $\Gamma$ over all eigenstates of (a) and different regions of (c). (f) The average $\Gamma$ over all eigenstates of (b) and (b). Here we set $J_0=1$ and $J_{so}=0.5$.}
\end{figure}
Another interesting phenomenon is the reentrant localization transition, namely that with increasing the quasiperiodic potential strength, after the AL transition, some of the localized
states become extended for a range of intermediate potential strengths, and eventually, these states undergo the second localization transition at a higher quasiperiodic potential strength~\cite{Basu2021}. Figs.~\ref{03}(c) and (d) show the fractal dimension of this system with $M_z=2.6$ and $M_z=2.9$ for the fixed $T=1$. We see that when $M_z=2.6$, there exist the critical and localized regions, but when $M_z=2.9$, all eigenstates become critical, meaning that with increasing the quasiperiodic potential strength, some localized states become delocalized. Naturally, further increasing the potential strength, these states once again become localized. The phenomenon of the reentrant localization transition can only occur when $T>0.8$, namely in the low-frequency region. We note that there is not the reentrant localization transition when the quasiperiodic potential is non-kicked~\cite{YuchengC2}, the occurrence of this phenomenon originates from that the potential is added in the kicked way.
To further confirm the extended, critical or localized properties in different regions, we carry out the finite size analysis by calculating $\bar{\Gamma}$, as shown in Figs.~\ref{03}(e) and (f). The mean fractal dimension $\bar{\Gamma}$ averaged over all eigenstates in Fig.~\ref{03}(a) tends to 1 [blue spheres in Fig.~\ref{03}(e)], suggesting that all eigenstates are extended. Similarly, we can confirm that the system in Fig.~\ref{03}(c) includes the localized and critical regions [red squares and green triangles in Fig.~\ref{03}(e)], and all eigenstates in Figs.~\ref{03}(b) and (d) are critical [see Fig.~\ref{03}(f)].
\
\
\section{experimental realization and detection}
\subsection{Experimental realization}
We propose to realize the Hamiltonian (\ref{Hsum}) based on apodized Floquet engineering techniques~\cite{Shimasaki2022,Weld2022} and optical Raman lattices~\cite{LiuXJ2013,LiuXJ2014,Lepori2016,WangBZ2018,Liu2016,Song2018,Song2019,JWPan,RamanReview2018}. Fig.~\ref{04}(a) shows the schematic diagram, where ${\bf E}_1$ with $z$ polarization is a standing-wave beam and ${\bf E}_{3}$ with $x$ polarization
is a plane wave. They are applied to generate the spin-independent main lattice $V_1(x)=V_{\rm m}\cos^2({k}_1x)$ with the depth $V_{\rm m}$, which
induces the spin-conserved hopping ($H_0$), and a Raman coupling potential to generate the spin-flip hopping ($H_{\rm SOC}$).
The periodically kicked quasiperiodic potential potential ($H_K$) is realized by periodically applying another standing wave ${\bf E}_{2}$, which is used to generate a spin-dependent lattice $V_2(x)\sigma_x=V_{\rm s}\cos^2({k}_2x)\sigma_x$ with the depth $V_{\rm s}$.
In this setting, the lattice wave numbers $k_{1,2}$ are easily tunable in experiment and making them incommensurate to product the irrational number $\alpha=k_2/k_1$. In the tight-binding approximation, the realized Hamiltonian is given by
\begin{equation}\label{Hsumexp}
H=H_0+H_{\rm SOC}+F(t)\Delta\sum_{i}\cos(2\pi\alpha i)(n_{i,\uparrow}-n_{i,\downarrow}),
\end{equation}
with
\begin{equation}\label{HFt}
F(t)=\sum_ng_{\tau}(t-nT).
\end{equation}
being the waveform of the periodic pulse train. Here $g_{\tau}$ describes the shape of the pulse, $T$ is the pulse interval, and $\tau$ is the effective width of the single pulse: $\tau=\int^{\infty}_{-\infty}g_{\tau}(t)dt$, as shown in Fig.~\ref{04}(b), where we take the square pulses as an example. In the limit of small $\tau$~\cite{Shimasaki2022}, $\Delta F(t)=\sum_nM_z\delta(t-nT)$, where $M_z=\Delta\cdot\tau$, and then, the Hamiltonian (\ref{Hsumexp}) becomes the Hamiltonian (\ref{Hsum}).
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{fig5.pdf}
\caption{\label{04}
(a) Schematic of the experimental setup. ${\bf E}_1$ is a standing wave with $z$ polarization, which generates the spin-independent main lattice.
${\bf E}_2$ is the kicked standing wave giving the kicked quasiperiodic potential. ${\bf E}_{3}$ is a plane wave, which is used to form the Raman coupling potential. (b) Experimental sequence composed of the finite-width, unit-height pulse with the effective pulse width $\tau$ and the pulse interval $T$.}
\end{figure}
\begin{figure}[t]
\hspace*{-0.2cm}
\includegraphics[width=0.48\textwidth]{fig6.pdf}
\caption{\label{06}
Long-time survival probability ($t=5\times 10^7T$) with (a) $M_z=0.05$ (extended phase), $M_z=0.2$ (critical phase), $M_z=0.4$ (localized phase), and the fixed $T=0.1$, (b) $T=1$ and $M_z=2.6$ corresponding to the phase with coexisting critical and localized regions [see Fig.~\ref{03}(c)], and $T=1.7$ and $M_z=2.3$ corresponding to the phase with coexisting extended, critical and localized regions [see Fig.~\ref{01}(f)]. For (a) and (b), we take $20$ samples with a sample being specified by choosing an initial phase $\phi$. The dashed line is plotted by fixing $D_2=0.75$ and changing $c_0$ to fit the data points as Eq.~(\ref{Pr}). Log-log plot of $W$ versus the time $t$ for (c) $T=1$ and $M_z=2.6$, and (d) $T=1.7$ and $M_z=2.3$. We choose $L=987$ for all figures.}
\end{figure}
\subsection{Experimental detection}
Next we study the detections of the different phases based on the expansion dynamics. We consider a wave packet with spin up initially at the center of the lattice, i.e., $|\psi(0)\rangle=c^{\dagger}_{(L+1)/2,\uparrow}|0\rangle$ (let the size $L$ be odd), and the final state is set as $|\psi(t)\rangle=\sum_{j=1}^L[u_{j}(t)c^{\dagger}_{j,\uparrow}+v_{j}(t)c^{\dagger}_{j,\downarrow}]|0\rangle$. We firstly focus on the survival probability $P(r)$ defined as
\begin{equation}\label{SP}
P(r)=\sum_{|j-\frac{L+1}{2}|\leq r/2}|(u_{j}(t)|^2+|v_{j}(t)|^2),
\end{equation}
which describes the probability of finding the particle after a given time $t$ in the sites within the region $[-r/2, r/2]$~\cite{Santos2019}. After a long time evolution ($t\rightarrow\infty$), $P(r)$ is proportional to $(r/L)^{D_2}$ with $D_2$ being the generalized dimension of spectral measures~\cite{Santos2019,Geisel1992,Xu2020}. For the extended phase, the distribution of the final state will be uniform, and thus, $P(r)$ linearly increases as $r$ increases. For the localized phase, the particle will localize at the position near the initial point, and thus, $P(r)$ quickly reaches $1$ within a small $r$. For the critical phase, the distribution is delocalized and nonergodic, and thus, $P(r)$ reaches $1$ when $r\rightarrow L$ but the increasing rate is not linear. Fig.~\ref{06}(a) shows the typical distributions of $P(r)$ with long times in the extended phase (green line), critical phase (red line) and localized phase (blue line). For sufficiently large $r/L$, we have $P(r)\approx (r/L)^{D_2}$ with $D_2=0, 1$ and $0<D_2<1$ for the localized, extended and critical phases, respectively, and $0<D_2<1$ reflects the nonergodic character of the critical phase. For a system with MEs, the distribution of $P(r)$ will become complex.
Fig.~\ref{06}(b) shows the $P(r)$ of the phase with coexisting localized and critical regions (blue line) and the phase with coexisting three different regions (red line). $P(r)$ dramatically increases for a small $r$, suggesting the existence of localized regions, but reaches $1$ when $r\rightarrow L$, meaning that there also exist the delocalized regions. For the phase with coexisting localized and critical regions, the increasing rate of $P(r)$ is determined by the states in the critical region, and the average fractal dimension can be extracted by
\begin{equation}\label{Pr}
P(r)=(r/L)^{D_2}(1-c_0)+c_0,
\end{equation}
where $c_0$ is the constant that depends on the proportion of the localized states in all eigenstates. Fig.~\ref{03}(e) tells us $\bar{\Gamma}\approx0.75$, and thus we plug $D_2=0.75$ into Eq.~(\ref{Pr}) to well fit the $P(r)$. From Fig.~\ref{06}(b), Eq.~(\ref{Pr}) with $D_2=0.75$ can also be well fit to the coexisting phase with three different regions. Thus, it is difficult to further distinguish whether the delocalized regions are critical or the co-existing of critical and extend regions from $P(r)$ with $t\rightarrow\infty$.
To see the differences between the two cases in Fig.~\ref{06}(b) in dynamics, we should not consider the distributions after a long time evolution. Instead, we should consider the process of the expansion of the wave packet. To characterize the expansion of the above initial state, we consider the mean square displacement~\cite{YuchengC2,Hiramoto,Ketzmerick,Roati2008},
\begin{equation}\label{MSD}
W(t)=\sqrt{\sum_{j}[j-(L+1)/2]^2 (u_{j}(t)|^2+|v_{j}(t)|^2)},
\end{equation}
which measures the width of the wave packet after the evolution time $t$. $W(t)$ can be expressed as $W(t)\sim t^{\gamma}$ with $\gamma$ being the dynamical index. For AA model, $\gamma=0$, $\gamma=1$ and $\gamma\approx\frac{1}{2}$ in the localized phase,
extended phase and critical point, respectively, meaning that the corresponding expansion is localized, ballistic,
and normal diffusive, respectively. For the coexisting phase, $W(t)$ is not straightforward to $t^{\gamma}$, as shown in Fig.~\ref{06}(c) and (d). It is obvious that the coexisting phase including the extended region expands more quickly and reaches the boundary faster.
Further, from the viewpoint of the transport~\cite{Purkay,Saha2019}, the conductivity is independent of the system size in the extended region, while it decreases in the power-law and exponential fashion with the system size in the critical and localized region, respectively. Thus, by shifting the position of the Fermi energy across different regions, one can detect the corresponding transport properties, and further obtain the more precise information of the coexisting phases.
\section{summary}
We have investigated the critical and localized properties in the 1D periodically kicked quasiperiodic optical Raman lattice by comparing the fractal dimensions with different sizes. This system shows a rich phase diagram. In the high frequency regime ($T\ll 1$), the transition between the extended and critical phases occurs at $M_z/T=2|J_0-J_{so}|$, and the transition between the critical and localized phases occurs at $M_z/T=2(J_0+J_{so})$, which can be interpreted from the effective Hamiltonian of this system. With increasing $T$, there are the phase with coexisting critical and localized regions, and the phase with coexisting extended and critical regions. The two phases exhibit two types of MEs which separate the localized states from critical ones, and the extended states from critical ones, respectively. Further increasing $T$, there is the coexisting phase of extended, critical and localized regions. We have also found the fragility of the phase boundary, namely that the phase boundary is more susceptible to the dynamical kick, and the phenomenon of the reentrant localization transition. Finally, we have studied in detail the experimental realization, which can be immediately achieved in the current experiments, and the experimental detection based on the expansion dynamics of the wave packet.
Our results show that the periodically kicked incommensurate optical lattice is a new effective way to study and detect the novel critical phase, MEs, coexisting quantum phases and some other interesting phenomena.
\begin{acknowledgments}
We thank C. Yang and Y. Peng for reading our manuscript carefully. This work is supported by National Key R\&D Program of China under Grant No.2022YFA1405800, the National Natural Science Foundation of China (Grant No.12104205), the Key-Area Research and
Development Program of Guangdong Province (Grant No. 2018B030326001), Guangdong Provincial Key Laboratory (Grant No.2019B121203002).
\end{acknowledgments}
|
1,108,101,563,281 | arxiv | \section{Introduction}
\par \noindent In the past few years, a new approach in understanding and constructing quantum geometry had emerged from linking geometric quantities to information-theoretic one. This idea first emerged from the calculation of entropy of spherically symmetric isolated horizons~(SSIH) in Loop quantum gravity~\cite{immirzi1997real,PhysRevLett.80.904}. Later, the emergence of holographic principle to include the `entanglement entropy' of the vacuum to be linked to the are of the surface enclosing a geodesic-generated ball~\cite{reznik2003entanglement}. A deeper link link between quantum geometry and information theory was established~\cite{ER1}, as a solution to AMPS / Firewall paradox~\cite{almheiri2013black}. This solution suggested that for two black holes collapsed from an entangled matter, these black holes would be connected by an Einstein-Rosen Bridge~(ERB). This conjecture became widely known as the~$ER= EPR$ \textit{conjecture}. One can go even further with conjecture and claim that for any pair of entangled particles, there is a Plankian ERB, that matter will not fall into it. However, this can considered in an abstract geometric sense~\cite{ER2}.
\par \noindent The development of this conjecture has raised the possibility that space-time could be interwoven by entanglement, and wormholes could be holding the space-time together, This idea has been applied in category theory and topological quantum field theory to show that quantum fluctuations correspond to formation of wormholes and vice versa~\cite{baez2014wormholes}. Technically speaking, the~$ER=EPR$ raises the possibility that space-time could multiply connected. However, how one could start doing physics in multiply-connected space-time ? More importantly, how to formulate the physic of multiply connected space-time; and would it help to do so ?
\par \noindent The most basic way to start is by replacing the points of the space-time manifold~$ \mathcal{M}$ by a non-Abelian algebra generators,establishing non-commutative geometry~\cite{Nicolini:2005vd,Gingrich:2010ed},
\begin{equation}
\label{qgeo}
[q^\mu , q^\nu] = i \ell_p^2 g^{\mu \nu}.
\end{equation}
This is a direct result of combining gravitation theory with uncertainty relation of quantum mechanics. It is immediately recognised that~$ \ell_p$ is the Plank length, and the relation~\eqref{qgeo} implies that space-time at the Plank scale consists of what-so-called quantum foam~\cite{wheeler1}. First described by J. Wheeler and later studied in detail by S. Hawking~\cite{hawking1996virtual}, this quantum foam consists of a sea of virtual black holes, which Hawking described them as~\textit{bubbles}. These bubbles make the space-time posses a large second Betti number. Moreover, in his paper, Hawking argued that the space-time is simply connected and admits a topology~$ S^2 \times S^2$ or~$ K_3 \times K_3$ \dots etc.
\par \noindent Nevertheless, these bubbles can form the multiply-connected space-time we are looking for if we considered the~$ ER= EPR $ conjecture. In this paper, we shall discuss how to define a multiply-connected space-time combing both ideas discussed above. Next, show how this picture could help in the theory of quantum gravity, after that an application for this conjecture is discussed at the hand-waving level by recovering the Bekenstein-Hawking formula for Black hole entropy. Finally, we will point out the possible steps for this programme that ought to be taken in order to make the argument in this paper more formal.
\section{The space-time at the Plank scale}
\par \noindent Let~$\mathcal{M}$ be a 4-dimensional~$C^k$-manifold that admits a topology~$ \mathbb{R} \times \Sigma$, we deal with globally hyperbolic space-time in order to prepare for canonical formalism of gravity. If any field was put on~$ \mathcal{M}$, by assigning smooth sections~$ \varsigma(t,x) ; t,x \in \mathbb{R} \times \Sigma$ over~$\mathcal{M}$ to an associated bundle~$ \mathcal{F}$. This field is quantised by standard canonical quantisation rules, taking the field to be a scalar field~$ \phi (t,x) = \varsigma(t,x)$, the quantisation for the field is given by the commutation relation for~$\phi$ and its conjugate momentum~$ \pi (t,x)$ which is itself a section ~$ \varpi(t,x)= \frac{d\varsigma(t,x)}{dt} $,
\begin{align}
[\phi(t_0, x), \phi(t_0,x')] = 0, \nonumber \\
[\pi(t_0, x), \pi(t_0,x')]= 0 ,\nonumber \\
[\phi(t_0, x), \pi(t_0,x')]= i \delta(x-x').
\end{align}
These rules imply a Fock space \footnote{ Since we are dealing with quantum field theory on curved manifold, the Fock space and vacuum spates are by no means unique} and thereby we can define a Hamiltonian~$H$, whose expectation value over any vacuum state diverges as it is known from standard quantum field theory. The only non- perturbative method to remove such divergence is to impose some UV-cutoff on the modes of the filed. This cutoff condition can be made because of the gravity theory present on~$\mathcal{M}$. For general relativity~(GR) one can find this cutoff term to be for modes with wavelengths less than or equal to the plank length $ \ell_p =1$ in Plankian units
In the semi-classical GR, the geometry itself fluctuates like the quantum fields that lie on it to show this take a small 3-space like surface $\sigma$, let it be a geodesic-generated ball with radius $l$. We now integrate Einstein field equations over this ball~(in some chart defined over that small ball),
\begin{equation}
\frac{1}{4\pi} \int_\sigma d\sigma^\nu G_{\mu\nu} = 2 \int_\sigma d\sigma^\nu \langle\ \hat{T}_{\mu\nu}\rangle .
\end{equation}
The LHS is equal to the radius of curvature domain of small region of space time~$ R_\mu $. For the RHS, we consider the stress energy tensor of the scalar field~$ \phi$ we get,
\begin{equation}
\label{radofcurve}
R_\mu =\int d^3k \left( \omega_k +k_i\right) \langle 0| \hat{a}^\dagger_k \hat{a}_k+ \hat{a}_k\hat{a}^\dagger_k |0\rangle.
\end{equation}
The operators~$ \hat{a}^\dagger_k ,\hat{a}_k$ are the creation and inhalation operators of the defined Fock space. Here, the Latin indices run from $1$ to $3$ and the Greek ones run from $0$ to $3$, and $ \omega_k$ is the normal mode given by $\sqrt{k_j k^j}$. The expression~\eqref{radofcurve} can be further treated yielding,
\begin{equation}
\label{div}
R_\mu =\int d^3k \sum_{k=0}^{\infty} \omega_k
\end{equation}
Thus the radius of curvature domain is divergent, implying that the geometry is not supposed to exist with field fluctuating like that. Hence, locality and flat geometry cannot coexist. To get rid of that divergence we use the uncertainty relation~$ \Delta R_\mu x_\mu \sim 1$ . This is derived from Hawking's assumption about the quantum structure of space-time at the micro scale~\cite{hawking1996virtual}, as a sea of virtual blackholes. By this assumption, we cannot define the field nor the geometry at one point, but rather a region with minimal length. Thus geometry fluctuation is a key element in the stability of classical geometry. Concluding geometry must me quantised, i.e. the space-time is made from discrete bubbles of geometry. This result is obtained without assumptions outside GR and quantum theory.Virtual black holes were also studied in third quantised formalism of canonical quantum gravity~\cite{faizal2012some}, in dilatorn gravity theories~\ref{Grumiller:2002dm,Grumiller:2001rg,Grumiller:2000ah}, and in other alternative theories as in generalised uncertainty principle and their phenomenological implications in proton decay~\cite{alsaleh2017virtual} and higher spin theories from Teukolsky equations \cite{Prestidge:1998bk}. Virtual black holes are also important in understanding the black hole information paradox~\cite{Calmet:2014uaa}, for example, it has been shown that the S-matrix of scattering processes by virtual black hole is incoherent~\cite{Hawking:1997ia}, leading to a possible indication that information paradox could be resolved by studying the phenomenology of scattering by virtual black holes~\cite{Grumiller:2004yq}.
\par \noindent The predictions of quantum mechanics, these nearby virtual black holes are entangled. In order to show this, consider black holes in a box , each black hole will emit Hawking radiation to be absorbed by the others, even when the system reaches thermal equilibrium, this process will continue to happen. If we assign a density matrix for this system, it will evolve to a mixed states density matrix over time, even if we started with a pure state (each state describes a Black hole.). Viz., black holes will get entangled by absorbing each other's Hawking radiation~\cite{ER1}.
\par \noindent Similar argument can be made to the virtual black holes, that make up the quantum foam. Another way to show that the quantum foam is indeed entangled, and hence is filled with Plankian Wormholes, is the work done by~\cite{baez2014wormholes}, using category theory, they concluded that the pair production of quantum fluctuation correspond to formation of a wormhole. Although their argument was for topological field theories, it can be proven that this argument could be extended to general relativity if we considered it as a form of BF theory. Identifying the form $B =\ast [e\wedge e]$, where $e$ are the triebein fields corresponding to the 3 metric $\gamma$ of $\Sigma$, then the Palatini action reads,
\begin{equation}
S_P = \int_{\Sigma} \text{Tr} (\ast [e\wedge e] \wedge F ).
\end{equation}
However, it remains to be proven explicitly that these results will hold for full relativistic gravity theory.
\par \noindent Assigning to each unit volume on the 3 spacelike surface~$ \Sigma$ a quantum state~$ | \psi\rangle$, this unit area can carry only 1 bit of information, either it contains a virtual black hole~$| 1\rangle$ or not~$|0 \rangle$ Hence, we can write the quantum state for such unit area,
\begin{equation}
| \psi\rangle = \frac{1}{\sqrt{2}} \left( | 1\rangle + |0\rangle\right).
\end{equation}
We observe that given any 2-surface~$ \Delta$, enclosing a region of~$ \Sigma$ , we can establish near the boundary states of the volumes in the `in' and `out' regions. These states describe entangled states,
\begin{equation}
| \Psi\rangle = \frac{1}{\sqrt{2}} \left( | 1_{in}\rangle \otimes |1_{out}\rangle+ | 0_{in}\rangle \otimes |0_{out}\rangle\right).
\end{equation}
The state~$ | \Psi\rangle$ belongs to the Hilbert space~$ \mathcal{H} = \mathcal{H}_{in} \otimes \mathcal{H}_{out}$ of the space-time volumes.
This construction of space-time from entanglement does not require holography as in~\cite{van2010building}, but basically shares the same aim. In order to do so, it is needed to define the topology of the space-time having this construction.
\section{ The topology of the entangled quantum foam}
\par \noindent Since these volume elements~(making up) the quantum foam are entangled, using the $ER=EPR$ conjecture, each bubble of geometry is connected to another by a wormhole/ ERB. Making the space-time multiply connected. However, in 3 manifolds, we cannot clearly define the first Betti number corresponding the the first fundamental group. The first fundamental group, appears (well defined) for 2 surfaces. Taking the surface~$\Delta$ in the previous section, and analysing its homotopy we find that, and due to the $ER=EPR$. We can define loops between the `in' and `out' regions that cannot be shrunk to a point. These loops are made by passing through the wormholes made by the entangled bubbles. The surface~$\Delta$ is them homeomorphic to an~$N$-punctured sphere, $N+1$ being proportional to the area of the surface in Plank-area units.
\par \noindent The first fundamental group for the surface~$\Delta$ is generated by $N$ generators, each go around each puncture made by a wormhole. Note that we should only consider bubbles entangled in the region~$ Ar (\Delta) \pm 1$, the entanglement for bubbles further from Plank length is negligible. Hence the rank of this free group is proportional to~$ Ar( \Delta)$. The calculation of the proportionality constant is rather difficult, because we need to take into an account that the classical area is obtained only by filling the surface with infinite circles, while in quantum geometry, the 'quantum' area is a bit less, due to packing factor~(see Tammes Problem, Ref~\cite{tammes1930origin}). The exact number of generator is the number of wormholes forming between bubbles filling the area~$ Ar( \Delta-1)$ and~$ Ar( \Delta+1)$, assuming all the black holes in these two layers are entangled, the number of the wormholes forming is the area of the surface~$\Delta$ that comes between the layers multiplied by a filling factor, i.e. how many bubbles one can fit in the volume covered by~$ Ar( \Delta+1)$ minus the volume covered by~$ Ar( \Delta-1)$.
\section{Calculation of path integrals }
\par \noindent The topology of space-time could allow a non perturbative calculation of path integrals, automatically removing divergences that appear in the standard path integrals. Since we have found$ \mathcal{M}$ to be multiply connected, we can look at its universal covering space, and preform path integration on the covering space. Instead of integrating over all paths of the same homotopy class, one can instead integrate over homotopy classes in the universal cover. Nevertheless, this is not helpful in unbounded spaces, as one still gets infinite classes to integrate over. Rather, one may take two points in~$\mathcal{M}$, $ p,q$. Surround $p$ with a 2-spacelike surface~$ \sigma_p$ and~$ q$ with another~$ \sigma_q$. Homotopy classes of paths between~$p$ and~$q$ will be finite due to the topology of the surfaces $\sigma_p$ and $ \sigma_q$. One can preform a non perturbative integration over the homotopy classes.
\par \noindent Nevertheless, there is a rather peculiar feature of this technique, that is the dependence of the value of the path integral of the choice of surfaces. It remains an unsolved problem that needs special attention. This `freedom' in choosing the surfaces, could correspond to gauge freedom, that one can mod out. Alternatively, there is no freedom and it is fixed by the theory of gravity, like the case with the horizon of a black hole, this surface is fixed by gravity, and one needs to use - only- for the calculation of action on that surface.
Hence the general technique for calculating path integrals in a multiply-connected space-time could be summarised in the following points,
\begin{itemize}
\item Pick two points in the space-time manifold $\mathcal{M}$.
%
\item Surround these points by 2 spacelike surfaces.
%
\item Identify the homotopy classes and loops on these surfaces, then left them up to the associated bundle in which the field of concern 'lives'. Such than one can preform parallel transport along the loops.
%
\item Define the fundamental groups for the surfaces, and find a proper representation for these groups (loop representation).
%
\item Using the above representation, preform the standard technique for path integration in multiply connected space-time,
%
\begin{equation}
\int \mathcal{D}[f] e^{ -i S[f, f,_\mu]}\longrightarrow \sum_{\alpha \in \pi_1 (\sigma_p \cup \sigma_q)} \chi(\alpha) K^\alpha.
\end{equation}
%
Where $ \alpha$ is the homotopy class, $ \chi$ is a representation of the fundamental group and $ K^{\alpha}$ is the transition amplitude via the homotopy class $ \alpha$.
\end{itemize}
\section{Conclusion}
\par \noindent The action path integrals over a multiply connected spaces could be shown to be non-peturbative and yield exact results as one might sum over equivalent classes of paths that have the same winding number. In fact, in this paper, we have seen that merging the ideas of minimal length and creation of pair black holes~\cite{hawking1996virtual}, along with the ER=EPR conjecture \cite{ER1,ER2} could yield a multiply-connected space-time. Hence path integrals over space-time with gravity could be non-peturbative.
\par \noindent It could be shown in greater detail that the creation and inhalation of black-holes as a result of third-quantised theory of geometry~\cite{faizal2012some}. Hence one could also apply the non-peturbative techniques to geometry itself. There is a lot to add to this proposal in order to use it for exact calculation of path integrals , and/or quantisation of geometry. Based on the microscopic structure of space-time, and to generalise the calculations in topological field theory~\cite{baez2014wormholes} to the 4D relativistic space-time, that is non-topological.
\par \noindent As the topology is scale-independent,the non trivial topology of scape-time caused by the virtual wormholes forming the quantum foam, could have a low-energy effects. It would be interesting to investigate such effects in future work, such as the Aharonov-Bohm effect, corrections to scattering amplitudes in QED and QCD and shifts in energy levels of nuclear and atomic systems.
\section*{Acknowledgements}
{ \fontfamily{times}\selectfont
\noindent
This research project was supported by a grant from the `` Research Center of the Female Scientiffic and Medical Colleges'' , Deanship of Scientiffic Research, King Saud University.
|
1,108,101,563,282 | arxiv | \section{Introduction}
This paper introduces a novel class of sequence pairs called
``quasi-orthogonal Z-complementary pairs (QOZCPs)" and their applications in designing Doppler resilient waveforms in polarimetric radar systems. In what follows, we first review the state-of-the-art
on pairs of sequences and Doppler resilient waveforms in polarimetric radar systems, then introduce our contributions
in this work.
\subsection{Sequence pairs}
Research on designing sequence pairs with good correlation properties started in early 1950's when M. J. Golay proposed Golay complementary pairs (GCPs), in his work on multislit spectrometry \cite{golay1951static}. GCPs are sequence pairs, each having zero aperiodic auto-correlation sums at every out-of-phase time-shifts \cite{golay1951static}. Since then a lot of works have been done on analysing the properties and systematic constructions of the GCPs \cite{golay1961complementary,Borwein2003,Davis1999,Feng1999}. However, binary GCPs are available only for lengths of the form $2^\alpha 10^\beta 26^\gamma$ (where $\alpha$, $\beta$ and $\gamma$ are non-negative integers) \cite{Borwein2003}. In search of binary sequence pairs of other lengths, depicting similar properties to that of the GCPs, Fan \textit{et al.} \cite{fan2007z} proposed binary Z-complementary pairs (ZCPs). ZCPs are sequence pairs having zero auto-correlation sums for each time-shifts within a certain region around the in-phase position, called the zero-correlation-zone (ZCZ) \cite{fan2007z}. ZCPs are available for many more lengths as compared to GCPs \cite{fan2007z}. Systematic constructions of ZCPs based on insertion method and generalised Boolean functions for even and odd-lengths have also been extensively studied \cite{Liu2014,Adhikary2020a,Adhikary2020,Chen2017,Shen2019,Gu2019}. In 2013, Gong \textit{et al.} considered the periodic auto-correlation of a single Golay sequence and proposed a systematic construction such that each of the sequence have a zero auto-correlation zone \cite{Gong2013}. This property plays a very important role in the synchronization and detection of signals. Recently in 2018, Chen \textit{et al.} \cite{Chen2018} further studied the zero periodic cross-correlation of the GCPs and proposed a new class of sequence sets, namely Golay-ZCZ sets. However, the aperiodic cross-correlation of the GCPs was never been considered till date.
Beside the theoretical approaches for the systematic constructions of the sequence pairs, numerous numerical approaches are also proposed till date, beginning with the work of Groot \textit{et al.} \cite{doi:10.1080/02331939208843771} in 1992. Groot \textit{et al.} related the problem of optimizing the merit factor of a sequence with thermodynamics and introduced some evolutionary strategies using optimization tools to get binary sequences having low auto-correlation. Since the merit factor of a sequence is highly multimodal (i.e., it
may have multiple local maxima) stochastic optimization algorithms had been used for its maximization \cite{doi:10.1080/02331939208843771}. However, for large values of $N$, the computational complexity of these algorithms become very high. To overcome this, Stoica \textit{et al.} made a remarkable progress and introduced several cyclic algorithms (CAs), namely CA-pruned (CAP) \cite{stoica2009new}, CA-new (CAN) \cite{stoica2009new}, and Weighted CAN (We-CAN) \cite{stoica2009new} to design sequences with low aperiodic autocorrelation. To generate the sequences with low periodic correlations, the authors also proposed periodic CAN (PeCAN) \cite{Stoica2009}. To design waveforms with arbitrarily spectral shapes, PeCAN has been modified as the SHAPE algorithm \cite{Rowe2014}. Inspired by the ideas of \cite{stoica2009new}, Soltanalian \textit{et al.} \cite{soltanalian2013fast} proposed a CAN algorithm for complementary sets and termed it as CANARY. In 2012, Soltanalian \textit{et al.} proposed a computational framework based on an iterative twisted approximation (ITROX) \cite{Soltanalian2012} and
a set of associated algorithms to generate sequences with good periodic/aperiodic correlation properties. In another work, to minimize the correlation magnitudes in desired intervals, in 2013, Li \textit{et al.} \cite{li2014waveform} proposed an approach based on iterative spectral approximation algorithm (ISAA) and derivative-based non-linear optimization algorithms. Gradient based algorithms were proposed in \cite{zhang2016cognitive,arlery2016efficient,arlery2016eo,tan2016phase} to design sequences with good correlation properties, in the frequency domain.
Meanwhile, in 2015, Song \textit{et al.} \cite{song2015optimization} proposed an algorithm to directly minimize the periodic/aperiodic correlation magnitudes at out-of-phase time-shifts, using the general majorization-minimization (MM) method. On the other hand Liang \textit{et al.} \cite{liang2016unimodular} proposed unimodular sequence design using alternating direction method of multipliers (ADMM). ADMM is a superior optimization technique as it decomposes a
constrained convex optimization problem into multiple smaller
sub-problems whose solutions are coordinated to find the global
optimum. This form of decomposition-coordination procedure
allows parallel and/or distributed processing, and thus is well
suited to the handling big data. Second, in spite of employing iterations in the parameter updating process, it provides superior
convergence properties.
It should be noted that, designing of unimodular sequences using various optimization techniques is a decade old problem. However, the problem of designing complementary sequences considering the aperiodic cross-correlation among the sequences have not been considered before.
\subsection{Doppler resilient waveforms in polarimetric radar systems}
Fully polarimetric radar systems are equipped with vertically/
horizontally $(V/H)$ dual-dipole elements at every antenna
to make sure the simultaneous occurrence of transmitting and
receiving on two orthogonal polarizations \cite{calderbank2006instantaneous,pezeshki2008doppler,cui2017broadband}. The
essential ability of polarimetric radar systems is to capture the
scattering matrix which contains polarization properties of the
target. The scattering matrix can be given as follows,
\begin{equation}\label{eq:sm}
\mathbf{H}=\begin{bmatrix}
h_{VV} & h_{VH}\\
h_{HV} & h_{HH}
\end{bmatrix},
\end{equation}
where $h_{VH}$ denotes the target scattering coefficient that indicates the polarization change from $H$ (horizontally polarized incident field) into $V$ (vertical polarization channel).
The elements of the scattering matrix are estimated by analysing the auto-ambiguity functions (AAF) and cross-ambiguity functions (CAF) which are the matched filter outputs of the received signal with the transmitted
waveforms. Owing to its ideal ambiguity plot at desired delays waveforms with
impulse-like autocorrelation functions play an important role in radar applications. Phase coding is a commonly used to generate waveforms with impulse-like auto-correlations. Due to its ideal auto-correlation sum properties Howard \textit{et al.} \cite{howard2007simple} and Calderbank \textit{et al.} \cite{calderbank2006instantaneous} combined Golay complementary waveforms with Alamouti signal processing to enable pulse compression for multichannel and fully polarimetric radar systems.
One of the main drawbacks of waveforms phase coded with complementary sequences is that its effective ambiguity function is highly sensitive to Doppler shifts. Since then, several works have been done \cite{4339447,4784163,1405387} to design waveforms using sequences with good correlation properties which can exhibit some tolerance to Doppler shift. Working in this direction Pezeshki \textit{et al.} \cite{pezeshki2008doppler} made a remarkable progress in 2008, by designing Doppler resilient waveforms using Golay complementary sets. In \cite{pezeshki2008doppler} the transmission is determined by Alamouti coding and Prouhet-Thue-Morse (PTM) sequences. Extending further Tang \textit{et al.} \cite{tang2014construction} proposed Doppler resilient complete complementary code in multiple-input multiple-output (MIMO) radar by using generalised PTM sequences. Since, complementary sequences are not available for all lengths, in search of other sequences to design Doppler resilient waveforms, Wang \textit{et al.} \cite{Wang2017} proposed Z-complementary waveforms using equal sums of (like) powers (ESP) Sequences.
However, one of the major drawbacks of \cite{howard2007simple,calderbank2006instantaneous,pezeshki2008doppler} is that, the authors did not consider to design the dual-orthogonal waveforms for polarimetric radar systems. Although fully polarimetric radar systems simultaneously transmit and receive waveforms on two orthogonal polarizations, however, only this property does not help to extract the co- and cross-polarized scatter matrix elements \cite{224131,TITINSCHNAIDER2003633}. Therefore in polarimetric radar with simultaneous measurement of scattering matrix elements
the waveform need to have an extra orthogonality in addition to polarization orthogonality. Waveforms having two such orthogonality are called dual orthogonal waveforms.
\subsection{Contributions}
One objective of this paper is to propose a novel class of sequence pairs, called ``quasi-orthogonal Z-complementary pairs (QOZCPs)", each depicting Z-complementary property for their aperiodic auto-correlation sums and also having a low correlation zone when their aperiodic cross-correlation is considered. Construction of QOZCPs based on Successively Distributed Algorithms under Majorization Minimization (SDAMM) are presented.
Another objective of this paper is to apply the proposed QOZCPs in fully polarimetric radar systems and analyse the corresponding ambiguity functions. To be more precise, the contributions of this paper can be listed as follows:
\begin{enumerate}
\item New pairs of sequences called QOZCPs are proposed.
\item An efficient successively distributed algorithm under the MM framework is proposed. SDAMM transform a difficult optimization problem of two variables into two parallel sub-problems of a single variable. Each sub-problem has a closed-form so that the complexity is reduced significantly. Using SDAMM, we construct QOZCPs of any lengths.
\item Extending the works of Pezeshki \textit{et al.} we propose a Doppler resilient dual orthogonal waveform based on QOZCPs which can be used to efficiently estimate the co- and cross- polarised scatter matrix elements.
\item We compare the ambiguity plots of the proposed QOZCPs with the ambiguity plots of existing DR-GCPs and show that QOZCPs performs better while comparing the CAF plot.
\end{enumerate}
\subsection{Organization}
The rest of the paper is organised as follows. In section II, along with the preliminaries the definition of the QOZCP is proposed. The objective function for the construction of the proposed QOZCPs is derived. In Section III, SADMM algorithm is proposed to solve the optimization problem. In Section IV, we have explained the application of the proposed QOZCPs in radar waveform design. In section V, the numerical experiments are given, where we have compared the ambiguity plots of the proposed QOZCPs with the ambiguity plots of DR-GCPs. Finally, we have given some concluding remarks in section VI.
\section{QOZCP and Problem Formation}
In this section, we will propose the definition and the corresponding properties of QOZCPs. Throughout this paper, the entries of QOZCPs are complex $q$-th roots of unity (unimodular). Then we formally define QOZCPs as follows.
\subsection{Notations}
\begin{itemize}
\item $\mathbf x^T$ and $\mathbf x^H$ denote the transpose and the Hermitian transpose of vector $\mathbf x$, respectively.
\item $|x_l|$ and $x_l^*$ denote the modulus of $x_l$ and conjugate of $x_l$ respectively, where $x_l$ is the entries of $\mathbf x$.
\item $ \circ $ denotes the Hadamard product.
\item $C_{xy} (k) $ denotes the aperiodic cross-correlation function of $\mathbf x=[x_0,x_1,\cdots, x_{L-1}]$ and $\mathbf y=[y_0,y_1,\cdots, y_{L-1}]$, i.e.,
\begin{equation}
C_{xy} (k) =\left\{
\begin{aligned}
\sum_{l=0}^{L-k-1}x_l y_{l+k}^*,\quad k\geq 0\\
\sum_{l=0}^{L+k-1}x_{l-k}y_l^*,\quad k<0.\\
\end{aligned}
\right.
\end{equation}
\item $C_x (k) $ denotes the aperiodic auto-correlation function of $\mathbf x=[x_0,x_1,\cdots, x_{L-1}]$, i.e., $C_x(k) = C_{xx}(k)$.
\item $X(z)$ denotes z-transform of $\mathbf x$, i.e., $X(z) = x_0+x_1z^{-1}+\cdots+x_{L-1}z^{-(L-1)}$.
\end{itemize}
\begin{definition}
A pair of length-$L$ sequences $(\mathbf x,\mathbf y)$ is called a $(L,Z)$-QOZCP, if
\begin{eqnarray*}
\begin{split}
&\textit{C}1: |C_x (k)+C_y(k)| \leq \epsilon, \mathrm{for\,\, any} \,\, 0<|k|<Z,\\
&\textit{C}2: |C_{xy} (k)|\leq \epsilon, \mathrm{for\,\, any} \,\, |k|<Z,
\end{split}
\end{eqnarray*}
where $\epsilon$ is a very small positive real number which is very close to zero.
\end{definition}
Remark: since $C_{yx}(-k) = C_{xy}^*(k)$ and $ |C_{xy} (k)|\leq \epsilon$, then $|C_{yx}(k)|\leq \epsilon$ for any $|k|<Z$.
According to Definition 1, it can be observed that each QOZCP has low zero correlation zones when the ACFs' sum and the CCF are considered. It is noted that the nonzero values in the low zero autocorrelation zone are very close to zero. We illustrate the correlation properties of $(L,Z)$-QOZCP with $\epsilon=0$ in Fig. \ref{fig: C}. Following this definition of sequence, the task of constructing QOZCPs is transformed into equivalent optimization problems. The optimization problem will be formulated after introducing the objective function and constraints.
\begin{figure}[htbp]
\centering
\subfigure[$|C_x(\tau)+C_y(\tau)|$]{
\begin{minipage}{7cm}
\centering
\includegraphics[width=7cm]{pic/fig1a.pdf}\\
\end{minipage}
}
\subfigure[$|C_{xy}(\tau)|$]{
\begin{minipage}{7cm}
\centering
\includegraphics[width=7cm,height = 3.4cm]{pic/fig1b.pdf}
\end{minipage}
}
\caption{ Illustrative plots for the correlation properties of (L,Z)-QOZCP}
\label{fig: C}
\end{figure}
\subsection{Objective Function}
We have already introduced the desired sequence pair. In order to find it, unified metrics named the Weighted Complementary Integrated Sidelobe Level (WCISL) and the Weighted Cross-Correlation Integrated Level (WCCIL) are proposed in the following definitions.
\emph{Definition 2:} The Weighted Complementary Integrated Sidelobe Level (WCISL) of a sequence pair $(\mathbf x, \mathbf y)$ is defined as
\begin{equation}
\mathrm{WCISL} = \sum_{k=1}^{L-1}w_k|C_x (k)+C_y (k)|^2.
\end{equation}
\emph{Definition 3:} The weighted cross-correlation integrated level(WCCIL) of $\mathbf x$, $\mathbf y$ is given by
\begin{equation}
\mathrm{WCCIL} =\sum_{k=0}^{L-1 }\tilde{w}_k | {C_{xy}}(k)|^2,
\end{equation}
where $C_{x}(k)$ and $C_{xy}(k)$ are the auto-correlation function of $\mathbf x$ and the cross-correlation function of $\mathbf x,\mathbf y$, respectively. Besides, $w_{-k}=w_k$, $w_0 = 0$, $\tilde{w}_{-k}=w_k$, $\tilde{w}_0 \ne 0$.
The expressions of WCISL and WCCIL can also be transformed into the following expressions,
\begin{equation}
\begin{split}
&\mathrm{WCISL} =\sum_{k=1}^{L-1} w_k( |\mathbf x^H \mathbf{U}_k \mathbf x+\mathbf y^H \mathbf{U}_k \mathbf y|^2),\\
&\mathrm{WCCIL} =\sum_{k=0}^{L-1} w_k |\mathbf x^H \mathbf{U}_k \mathbf y|^2,
\end{split}
\end{equation}
where $\mathbf{U}_{k}$ \cite{song2016sequence} is an $N\times N$ Toeplitz matrix with 1 in $k$-th diagonal and 0 in the other positions.
To satisfy the conditions, i.e., C1 and C2 at the same time, WCISL and WCCIL should be considered together.
Then the objective function is written as
\begin{equation}
\begin{split}
\alpha \sum_{k=1}^{L-1} w_k( |\mathbf x^H \mathbf{U}_k \mathbf x&+\mathbf y^H \mathbf{U}_k \mathbf y|^2) +(1-\alpha)\sum_{k=0}^{L-1} w_k |\mathbf x^H \mathbf{U}_k \mathbf y|^2,\label{obj: 1}
\end{split}
\end{equation}
which will be minimized within a constraint set.
\subsection{Constraints of Interest}
Usually, the sequences to be designed have limited energy \cite{zhao2016unified}. In addition, the large PAPR results in a difficult dilemma between power efficiency and signal distortion \cite{wang2018optimized}. Therefore, constraints of energy and PAPR should be considered as follows:
1) Energy Constraint: The energy of $\mathbf x$ and $\mathbf y$ should be constrained to a given power $p_e$, i.e.,
\begin{equation}
||\mathbf x||^2=p_e, ||\mathbf y||^2=p_e, \label{con: pe}
\end{equation}
where $p_e$ is not larger than $L$(L is the length of $\mathbf x$ or $\mathbf y$).
2) PAPR Constraint: The PAPR is the ratio of the largest signal magnitude to its average power \cite{tropp2005designing,zhao2016unified}:
\begin{equation}
\mathrm{PAPR}(\mathbf x) = \frac{\max_{l}|x_l |^2}{||x||^2/L} \mathrm{and}\,\, \mathrm{PAPR}(\mathbf y) = \frac{\max_{l}|y_l |^2}{||\mathbf y||^2/L},
\end{equation}
where $1\leq \mathrm{PAPR}(\mathbf x)\leq L$ and $1\leq \mathrm{PAPR}(\mathbf y)\leq L$. We require a threshold $p_r(<L)$ which is determined by power amplifiers of the system, and let $\mathrm{PAPR}(x)<p_r$ and $\mathrm{PAPR}(x)<p_r$, so that the sequences can be with high power efficiency and small signal distortion. Since we have already set $||\mathbf x||^2=p_e$ and $||\mathbf y||^2=p_e$, the PAPR constraints are equivalent to: for $l = 1,2,\cdots, L$,
\begin{equation}
|x_l|\leq p_c,\, |y_l|\leq p_c,\label{con: pc}
\end{equation}
where $p_c = \sqrt{p_r p_e/L}$.
\subsection{Problem Formulation}
The problem formation is composed of the minimization of the objective shown in (\ref{obj: 1}) subject to the constraints (\ref{con: pe}) (\ref{con: pc}), and it reads
\begin{equation}
\mathcal{P}_0\left\{
\begin{aligned}
\begin{split}
\min \limits_{\mathbf x,\mathbf y}\quad &\alpha \sum_{k=1}^{L-1} w_k( |\mathbf x^H \mathbf{U}_k \mathbf x+\mathbf y^H \mathbf{U}_k \mathbf y|^2) +(1-\alpha)\sum_{k=0}^{L-1} w_k |\mathbf x^H \mathbf{U}_k \mathbf y|^2,\label{obj: 1}\\
s.t.\quad & ||\mathbf x||^2=p_e, ||\mathbf y||^2=p_e;\\
& |x_l|\leq p_c, |y_l|\leq p_c, \, \mathrm{for}\,\, l=0,1,2,\cdots, L-1.
\end{split}
\end{aligned}
\right.
\end{equation}
Besides, if the $\mathrm{PAPR}(x)$ and $\mathrm{PAPR}(y)$ are equal to 1, the optimization problem $\mathcal{P}_0$ can be changed into $\mathcal{P}_1$:
\begin{equation}
\mathcal{P}_1\left\{
\begin{aligned}
\begin{split}
\min \limits_{\mathbf x,\mathbf y}\quad &\alpha \sum_{k=1}^{L-1} w_k( |\mathbf x^H \mathbf{U}_k \mathbf x+\mathbf y^H \mathbf{U}_k \mathbf y|^2) +(1-\alpha)\sum_{k=0}^{L-1} w_k |\mathbf x^H \mathbf{U}_k \mathbf y|^2,\label{obj: 1}\\
s.t.\quad & | x_l |=1, | y_l |=1, \mathrm{for}\,\, l=0,1,2,\cdots, L-1.\\
\end{split}
\end{aligned}
\right.
\end{equation}
The optimization problems $\mathcal{P}_0$ and $\mathcal{P}_1$ are difficult to solve, since:\\
1) The objective function is non-convex and quartic for $\mathcal{P}_0$ and $\mathcal{P}_1$.\\
2) The variables $\mathbf x$ and $\mathbf y$ are difficult to be separated.\\
3) The constraint set is not a convex set.
In the following, we will pay more attention to dealing with $\mathcal{P}_0$, and later solve $\mathcal{P}_0$ without extra effort.
\section{QOZCP Design under Majorization Minimization Framework}
In this section, we will propose a Successively Distributed Algorithms under the Majorization Minimization (SDAMM) framework to solve $\mathcal{P}_0$ and $\mathcal{P}_1$.
\subsection{SDAMM for $\mathcal{P}_0$}
In order to analyze the proposed optimization problem conveniently, we combine the two optimization variables $\mathbf x$, $\mathbf y$ into one variable, i.e., $\mathbf{z} = [\mathbf{x}^T , \mathbf{y}^T ]^T$.
Then the optimization problem can be changed into
\begin{equation}
\begin{split}
\begin{aligned}
\min \limits_{\mathbf z,\mathbf x, \mathbf y}&\quad \alpha \sum_{k=-L+1}^{L-1} w_k |{\mathbf z^H \mathbf A_k \mathbf z}|^2+(1-\alpha)\!\! \sum_{k=-L+1}^{L-1} \tilde{w}_k | {\mathbf z^H \mathbf B_k \mathbf z}|^2\\
s.t. &\,\, \mathbf z = [\mathbf x^T, \mathbf y^T]^T\\
& ||\mathbf x||^2=p_e, ||\mathbf y||^2=p_e\\
& |x_l|\leq p_c, |y_l|\leq p_c, \, \mathrm{for}\,\, l=0,1,2,\cdots, L-1,
\end{aligned}\label{opt:p00}
\end{split}
\end{equation}
where
$\mathbf {A_k} = \begin{bmatrix} \mathbf U_k& \mathbf O\\ \mathbf O& \mathbf U_k \end{bmatrix},\quad \mathbf{B_k} = \begin{bmatrix} \mathbf O& \mathbf U_k\\ \mathbf O& \mathbf O \end{bmatrix}.$
\begin{proposition}
The optimization problem (\ref{opt:p00}) can be transformed as
\begin{equation}
\mathcal{P}_{0,1}\left\{
\begin{aligned}
\min \limits_{\mathbf Z,\mathbf z,\mathbf x, \mathbf y}&\quad \text{vec}(\mathbf Z)^H \mathbf J \text{vec}(\mathbf Z)\\
s.t.& \quad \mathbf Z= \mathbf {zz}^*\\
&\quad ||\mathbf x||^2=p_e, ||\mathbf y||^2=p_e;\\
& \quad |x_l|\leq p_c, |y_l|\leq p_c, \, \mathrm{for}\,\, l=0,1,2,\cdots, L-1.
\end{aligned}\label{opt:p01}
\right.
\end{equation}
where $\mathbf J= \alpha \mathbf J_A+(1-\alpha)\mathbf J_B$,
\begin{eqnarray} \mathbf{J}_{A} =\sum_{k=-L+1}^{L-1} w_{k} \operatorname{vec}\left(\mathbf{A}_{k}\right) \operatorname{vec}\left(\mathbf{A}_{k}\right)^{H}, \mathbf{J}_{B} =\sum_{k=-L+1}^{L-1} w_{k} \operatorname{vec}\left(\mathbf{B}_{k}\right) \operatorname{vec}\left(\mathbf{B}_{k}\right)^{H}. \end{eqnarray}
\end{proposition}
\begin{IEEEproof}
Please see Appendix A
\end{IEEEproof}
Now, we can use the framework of MM to dispose of the problem.
\begin{proposition}
Optimization problem $\mathcal{P}_{0,1}$ can be majorized by the following problem at $\mathbf z^{(t)}$
\begin{equation}
\mathcal{P}_{0,2}\left\{
\begin{aligned}
\min \limits_{\mathbf Z,\mathbf z,\mathbf x, \mathbf y}&\quad \text{Re}\left\{\text{vec}(\mathbf Z)^H (\mathbf J-\lambda_{\mathbf J}\mathbf I) \text{vec}(\mathbf Z^{(t)})\right\}\\
s.t.& \, \mathbf Z= \mathbf {zz}^*\\
&\,\, \mathbf z = [\mathbf x^T, \mathbf y^T]^T\\
& ||\mathbf x||^2=p_e, ||\mathbf y||^2=p_e\\
& |x_l|\leq p_c, |y_l|\leq p_c, \, \mathrm{for}\,\, l=0,1,2,\cdots, L-1,
\end{aligned}\label{opt:p02}
\right.
\end{equation}
where $\lambda_{\mathbf J}$ is the largest eigenvalue of $\mathbf J$.
\end{proposition}
\begin{IEEEproof}
Please see Appendix B
\end{IEEEproof}
\begin{theorem}
$\lambda_{\mathbf J} $, the largest eigenvalue of $\mathbf J$, is equal to
\begin{equation}
\begin{split}
&\lambda_{\mathbf J} = \max_k \left\{\max\{\lambda_{A}(k), \lambda_{B}(k) \right\}|k=-L+1,\cdots,L-1\},
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}
&\lambda_{A}(k) = w_k \alpha (2N-2|k|); \lambda_{B}(k) = w_k (1-\alpha) (N-|k|).
\end{split}
\end{equation}
\end{theorem}
\begin{IEEEproof}
Please see Appendix C.
\end{IEEEproof}
\begin{proposition}
Optimization problem $\mathcal{P}_{0,2}$ can be transformed into the following problem
\begin{equation}
\mathcal{P}_{0,3}\left\{
\begin{aligned}
\min \limits_{\mathbf z,\mathbf x, \mathbf y}&\quad \text{Re}\left\{\mathbf z^H \left( \mathbf Q -\lambda_{max}(\mathbf J) \mathbf z^{(t)}(\mathbf z^{(t)})^H\right)\mathbf z\right\}\\
s.t. &\,\, \mathbf z = [\mathbf x^T, \mathbf y^T]^T\\
& ||\mathbf x||^2=p_e, ||\mathbf y||^2=p_e\\
& |x_l|\leq p_c, |y_l|\leq p_c, \, \mathrm{for}\,\, l=0,1,2,\cdots, L-1,
\end{aligned}\label{opt:p03}
\right.
\end{equation}
where
\begin{eqnarray}
\begin{split}
&\mathbf {Q}
= \begin{bmatrix} \alpha\!\!\displaystyle \sum_{k=-L+1}^{L-1} \!\! w_k r_{-k}^{(l)}\mathbf U_k&\quad (1\!-\!\alpha)\!\displaystyle\sum_{k=-L+1}^{L-1} w_k c_{-k}^{(l)}\mathbf U_k\\
\mathbf O&\!\! \alpha\displaystyle \sum_{k=-L+1}^{L-1} w_k r_{-k}^{(l)}\mathbf U_k \end{bmatrix},
\end{split}\label{eq:Q}
\end{eqnarray}
in which $r_{-k}=C_x (-k)+C_y (-k)$ and $c_{-k}=C_{xy}(-k)$,
\end{proposition}
\begin{IEEEproof}
Please see Appendix D
\end{IEEEproof}
There are many operations in $\mathbf Q$ shown in (\ref{eq:Q}). In order to decrease the complexity of computing $\mathbf Q$, FFT (IFFT) is used.
\begin{theorem}
$r_{k}^{(l)}$ and $c_{k}$ in $\mathbf Q$ can be computed by FFT(IFFT) operations as follows, respectively,
\begin{eqnarray}
\begin{split}
&\mathbf r = [r_0^{(l)},r_1^{(l)},\cdots,r_{L-1}^{(l)},0,r_{1-L}^{(l)},\cdots,r_{-1}^{(l)},]\\
&= \mathbf F^H\!|\mathbf F[(\mathbf x^{(t)})^T\!,\mathbf 0_{1\times L}]^T|^2\!+\!\mathbf F^H\!|\mathbf F[(\mathbf y^{(t)})^T\!,\mathbf 0_{1\times L}]^T|^2,
\end{split}
\end{eqnarray}
and
\begin{eqnarray}
\begin{split}
&\mathbf c = [c_0^{(l)},c_1^{(l)},\cdots, c_{L-1}^{(l)},0,c_{1-L}^{(l)},\cdots,c_{-1}^{(l)}]^T\\
&= \mathbf F^H \left((\mathbf F[(\mathbf x^{(t)})^T,\mathbf 0_{1\times L}]^T)^*\circ (\mathbf F[(\mathbf y^{(t)})^T,\mathbf 0_{1\times L}]^T)\right),
\end{split}
\end{eqnarray}
where $\mathbf F$ is a $2L\times2L$ discrete Fourier matrix whose element is $F_{il} = e^{-j2\pi\omega_i l/L}$ and $|\cdot|^2$ denotes the element-wise absolute-squared operation.
\end{theorem}
\begin{IEEEproof}
Please see Appendix E.
\end{IEEEproof}
According to Lemma 4 in \cite{song2016sequence} and some simple operations, we have the following equations,
\begin{equation}
\begin{split}
&\mathbf{Q} = \left[\begin{matrix}
\frac{\alpha}{2L}\mathbf F_{:,1:L}^H diag(\mathbf \mu_r) \mathbf F_{:,1:L}& \frac{1-\alpha}{2L}\mathbf F_{:,1:L}^H Diag(\mathbf \mu_c)\mathbf F_{:,1:L}\\
\mathbf O & \frac{\alpha}{2L}\mathbf F_{:,1:L}^H diag(\mathbf \mu_r) \mathbf F_{:,1:L}
\end{matrix}\right],\label{Q}
\end{split}
\end{equation}
where $\mathbf {\mu_r = F t_r}$, $\mathbf \mu_c = \mathbf F \mathbf t_c$, $\mathbf f_x = \mathbf F[(\mathbf x^{(t)})^T,\mathbf 0_{1\times L}]^T $, $\mathbf f_y = \mathbf F[(\mathbf y^{(t)})^T,\mathbf 0_{1\times L}]^T $,
\begin{equation}
\begin{split}
&\mathbf {t_r} = [0,w_1 r_1^{(l)},\cdots, w_{L-1}r_{L-1}^{(l)},0,w_{L-1}r_{1-L}^{(l)},\cdots,w_1 r_{-1}^{(l)}],\\
&\mathbf {t_c} = [w_0 c_0^{(l)}\!\!,w_1 c_1^{(l)}\!\!,\cdots\!\!, w_{L-1}c_{L-1}^{(l)}, 0, w_{L-1}c_{1-L}^{(l)},\!\cdots\!\!,w_1 c_{-1}^{(l)}].
\end{split}
\end{equation}
Since $\mathbf{Q} -\lambda_{max}(\mathbf J) \mathbf z^{(t)}$ in (\ref{opt:p03}) is not Hermitian, the optimization problem (\ref{opt:p03}) is not a traditional Unimodular Quadratic Programming (UQP) defined in \cite{soltanalian2014designing}. Then we have to address the problem and transform (\ref{opt:p03}) into a UQP in next theorem.
\begin{proposition}
The optimization problem $\mathcal{P}_{0,3}$ can be equivalently transformed into the following optimization problem:
\begin{equation}
\mathcal{P}_{0,4}\left\{
\begin{aligned}
\min \limits_{\mathbf z,\mathbf x,\mathbf y}&\quad \mathbf z^H \left( \mathbf Q +\mathbf Q^H-2\lambda_{\mathbf J} \mathbf z^{(t)}(\mathbf z^{(t)})^H\right)\mathbf z\\
s.t. &\,\, \mathbf z = [\mathbf x^T, \mathbf y^T]^T\\
& ||\mathbf x||^2=p_e, ||\mathbf y||^2=p_e\\
& |x_l|\leq p_c, |y_l|\leq p_c, \, \mathrm{for}\,\, l=0,1,2,\cdots, L-1.
\end{aligned}\label{opt:p04}
\right.
\end{equation}
\end{proposition}
\begin{IEEEproof}
We can transform (\ref{opt:p03}) into a UQP by adding a conjugate term $\mathbf{Q}^H-\lambda_{\mathbf{J}}\mathbf{z}^{(t)}(\mathbf{z}^{(t)})^H$ to the objective of (\ref{opt:p03}), then the objective function of UQP is written as
\begin{equation}
\begin{aligned}
\text{Re}\left\{\mathbf z^H \left( \mathbf Q +\mathbf Q^H-2\lambda_{\mathbf J} \mathbf z^{(t)}(\mathbf z^{(t)})^H\right)\mathbf z \right\}\\
\end{aligned}
\end{equation}
whose result of optimal variable is not changed compared with (\ref{opt:p03}).
Also, the operation $\text{Re}(\cdot)$ can be removed, because $\left( \mathbf Q +\mathbf Q^H-2\lambda_{\mathbf J} \mathbf z^{(t)}(\mathbf z^{(t)})^H\right)$ has been Hermitian and
\begin{eqnarray}
\mathbf z^H\left( \mathbf Q +\mathbf Q^H-2\lambda_{\mathbf J} \mathbf z^{(t)}(\mathbf z^{(t)})^H\right)\mathbf z
\end{eqnarray}
is a real number. Then the optimization problem can be changed into $\mathcal{P}_{0,4}$.
\end{IEEEproof}
\begin{proposition}
The optimization problem $\mathcal{P}_{0,4}$ can be majorized by the majorization problem at $\mathbf z^{(t)}$:
\begin{equation}
\mathcal{P}_{0,5}\left\{
\begin{aligned}
\min \limits_{\mathbf z}&\quad \text{Re}\left\{\mathbf z^H \left( \mathbf Q +\mathbf Q^H-2\lambda_{\mathbf J} \mathbf z^{(t)}(\mathbf z^{(t)})^H-\lambda_u\right)\mathbf z^{(t)} \right\}\\
s.t. &\,\, \mathbf z = [\mathbf x^T, \mathbf y^T]^T\\
& ||\mathbf x||^2=p_e, ||\mathbf y||^2=p_e\\
& |x_l|\leq p_c, |y_l|\leq p_c, \, \mathrm{for}\,\, l=0,1,2,\cdots, L-1,
\end{aligned}
\right.
\end{equation}
where
$\lambda_u = 4L (\max \limits_{1\le i,j\le L}|\mathbf Q_{i,j}|)$, $\lambda_{\mathbf J}$ is the largest eigenvalue of $\mathbf J= \alpha \mathbf J_A+(1-\alpha)\mathbf J_B$.
\end{proposition}
\begin{IEEEproof}
Please see Appendix F.
\end{IEEEproof}
The objective of $\mathcal{P}_{0}$ can be majorized by the objective of $\mathcal{P}_{0,5}$ at $\mathbf z^{(t)}$, i.e., $-\text{Re}\{\mathbf z^H \mathbf P(\mathbf z^{(t)})\}$,
where
\begin{eqnarray}
\begin{split}
\mathbf P({\mathbf z^{(t)}}) &= -(\mathbf Q+\mathbf Q^H-2 \lambda_{\mathbf J} \mathbf z^{(t)}(\mathbf z^{(t)})^H-\lambda_u \mathbf I)\mathbf z^{(t)}\\
&= (\lambda_{\mathbf J}\cdot 4L+\lambda_u)\mathbf z^{(t)}-(\mathbf Q+\mathbf Q^H) \mathbf z^{(t)},
\end{split}
\end{eqnarray}
and
\begin{eqnarray}
\begin{split}
&(\mathbf Q+\mathbf Q^H) \mathbf z^{(t)} = \left[\begin{matrix} \frac{\alpha}{2L}\mathbf F_{:,1:L}^H ((\mathbf \mu_r+\bar{\mathbf \mu}_r) \circ \mathbf f_x)+\frac{1-\alpha}{2L}\mathbf F_{:,1:L}^H (\mathbf \mu_c\circ \mathbf f_y)\\
\frac{\alpha}{2L}\mathbf F_{:,1:L}^H ((\mathbf \mu_r+\bar{\mathbf \mu}_r) \circ \mathbf f_y)+\frac{1-\alpha}{2L}\mathbf F_{:,1:L}^H ( \bar{\mathbf \mu}_c\circ \mathbf f_x)\end{matrix}\right].
\end{split}\label{eq:qqz}
\end{eqnarray}
Besides,
\begin{eqnarray*}
\text{Re}\{\mathbf z^H \mathbf P(\mathbf z^{(t)}) \} = \text{Re}\{\mathbf x^H \mathbf P_x(\mathbf z^{(t)})\} +\text{Re}\{\mathbf y^H \mathbf P_y(\mathbf z^{(t)})\} ,
\end{eqnarray*}
where $\mathbf P_x(\mathbf z^{(t)}) = \mathbf P(\mathbf z^{(t)})(1:L)$ and $\mathbf P_y(\mathbf z^{(t)})=\mathbf p(\mathbf z^{(t)})(L+1:2L)$
In other words, $\mathcal{P}_0$ can be majorized by the following optimization at $\mathbf z^{(t)}$, i.e.,
\begin{eqnarray}
\mathcal{P}_{0,6}\left\{
\begin{aligned}
\begin{split}
\min \limits_{\mathbf x,\mathbf y}\quad &-\text{Re}\{\mathbf x^H \mathbf P_x(\mathbf z^{(t)})\}-\text{Re}\{\mathbf y^H \mathbf P_y(\mathbf z^{(t)})\}\\
s.t.\quad & ||\mathbf x||^2=p_e, ||\mathbf y||^2=p_e;\\
&|x_l|\leq p_c,|y_l|\leq p_c, \, \mathrm{for}\,\, l=0,1,2,\cdots, L-1.
\end{split}
\end{aligned}
\right.
\end{eqnarray}
The variables $\mathbf x$ and $\mathbf y$ are obviously separate in $\mathcal{P}_{0,6}$. Therefore, $\mathcal{P}_{0,6}$ can be divided into two optimization problems $\mathcal{P}_{0,x}$ and $\mathcal{P}_{0,y}$
\begin{eqnarray}
\mathcal{P}_{0, x}\left\{\begin{array}{cl}\max\limits_{\mathbf x} & \operatorname{Re}\left\{\mathbf{x}^{H} \mathbf{P}_{x}\left(\mathbf{z}^{(t)}\right)\right\} \\ \text { s.t. } & \|\mathbf{x}\|^{2}=p_{e}, \\ & \left|x_{l}\right| \leq p_{c},\end{array},
\quad \mathcal{P}_{0, y}\left\{\begin{array}{cc}\max \limits_{\mathbf{y}} & \operatorname{Re}\left\{\mathbf{y}^{H} \mathbf{P}_{y}\left(\mathbf{z}^{(t)}\right)\right\} \\ \text {s.t.} & \|\mathbf{y}\|^{2}=p_{e}, \\ & \left|y_{k}\right|^{2} \leq p_{c}.\end{array}\right.\right.
\end{eqnarray}
This kind of problem like $\mathcal{P}_{0,x}$ or $\mathcal{P}_{0,y}$ has been solved in \cite{yang2018cognitive} with a closed form denoted as $\mathbf x = \mathrm{Proj}_0 (\mathbf P_x(\mathbf z^{(t)})$ which is shown in Appendix G.
Then the proposed algorithm based on MM framework for $\mathcal{P}_0$ is summarized in Algorithm 1.
\subsection{SDAMM for $\mathcal{P}_1$}
Since $\mathcal{P}_1$ is a special case of $\mathcal{P}_0$ and the objective function of $\mathcal{P}_1$ is the same with that of $\mathcal{P}_0$, we can directly use the objective functions of $\mathcal{P}_{0,x}$ and $\mathcal{P}_{0,y}$ as the majorized functions of $\mathcal{P}_1$ at $\mathbf x^{(t)}$ and $\mathbf y^{(t)}$, respectively. Therefore, the two optimization problems are
\begin{eqnarray}
\mathcal{P}_{1,x}\left\{
\begin{aligned}
\max \limits_{\mathbf x}\quad &\text{Re}\{\mathbf x^H \mathbf P_x(\mathbf z^{(t)})\}\\
s.t.\quad &|x_l| = 1,
\end{aligned}
\right.
\quad
\mathcal{P}_{1,y}\left\{
\begin{aligned}
\begin{split}
\max \limits_{\mathbf y}\quad &\text{Re}\{\mathbf y^H\mathbf P_y(\mathbf z^{(t)})\}\\
s.t.\quad & |y_l|= 1.
\end{split}
\end{aligned}
\right.
\end{eqnarray}
According to \cite{soltanalian2014designing}, the minimizer $\mathbf x$ of $\mathcal{P}_{1,x}$ can be the equivalent minimizer of the following optimization problem (\ref{opt:p8})
\begin{equation}
\begin{aligned}
\min \limits_{\mathbf x}&\quad ||\mathbf x- \mathbf P_x({\mathbf z^{(t)}})||\\
s.t. &\quad |x_i| =1,\quad i=1,2,\ldots,2L, \\
\end{aligned}\label{opt:p8}
\end{equation}
similarly for $\mathbf y$.
It is obvious that the problem (\ref{opt:p8}) has a closed form
\begin{eqnarray}
\mathbf x = e^{arg(\mathbf P_x(\mathbf z^{(t)}))}
\end{eqnarray}
where $arg(\cdot)$ represents the argument. We denote $\mathbf x = e^{arg(\mathbf P_x(\mathbf z^{(t)}))}$ as $\mathbf x = \mathrm{Proj}_1 (\mathbf P_x(\mathbf z^{(t)}))$. Also, $\mathbf y=\mathrm{Proj}_1(\mathbf P_y(\mathbf z^{(t)}))=e^{arg(\mathbf P_y(\mathbf z^{(t)}))}$.
Then the proposed algorithm based on MM framework is summarized in Algorithm 1.
\begin{algorithm}[htb]
\caption{SDAMM: Successively Distributed Algorithms under Majorization Minimization for $\mathcal{P}_i$, $i=0,1$:}
\label{alg:Framwork}
\begin{algorithmic}[1]
\REQUIRE ~~\\
sequence length $L$;\\
weights $\{w_k \ge 0 \}_{k=1}^{L-1}$, $\{\tilde{w}_k\ge 0\}_{k=0}^{L-1}$;\\
scalar $\alpha=\frac{1}{2}$.
\STATE Set $l=0$, initialize $\mathbf z^{(0)}=[(\mathbf x^{(0)})^T, (\mathbf y^{(0)})^T]^T$.
\label{ code:fram:extract }
\STATE $\lambda_A (k) = w_k\alpha (2L-2|k|)$, $ \lambda_B (k) = w_k (1-\alpha)(L-|k|)$,\\
$\lambda_{\mathbf J} = \max_k\left\{ \max(\lambda_A (k), \lambda_B (k) )| k=1,\cdots, L \right\}$.
\label{code:fram:trainbase}
\STATE {\bf repeat}
\STATE $\mathbf x_1 = \mathrm{Proj}_i(\mathbf P_x (\mathbf z^{(t)}))$, $\mathbf y_1 = \mathrm{Proj}_i(\mathbf P_y (\mathbf z^{(t)}))$, \\
$\mathbf z_1 = [\mathbf x_1^T, \mathbf y_1^T]^T$
\STATE $\mathbf x_2 = \mathrm{Proj}_i(\mathbf P_x (\mathbf z^{(t)}))$, $\mathbf y_2 = \mathrm{Proj}_i(\mathbf P_y (\mathbf z^{(t)}))$,\\
$\mathbf z_2 = [\mathbf x_2^T, \mathbf y_2^T]^T$
\STATE $\mathbf v_1 = \mathbf z_1-\mathbf z^{(t)}$, $\mathbf v_2 = \mathbf z_2-\mathbf z_1- \mathbf v_1$
\STATE Compute the step length $\alpha_{sl}=-\frac{||\mathbf v_1||}{||\mathbf v_2||}$;
\STATE $\mathbf z^{(l+1)} =\mathrm{Proj}_i{(\mathbf P(\mathbf z^{(t)}-2\alpha_{sl}\mathbf v_1 +\alpha_{sl}^2 \mathbf v_2 ))}$
\STATE {\bf while} $\mathrm{obj}(\mathbf z)>\mathrm{obj}(\mathbf z^{(t)})$\\
{\bf do}\\
\quad$\alpha_{sl} = (\alpha_{sl}-1)/2$,\\
\quad $\mathbf z^{(l+1)} =\mathrm{Proj}_i{(\mathbf P(\mathbf z^{(t)}-2\alpha_{sl}\mathbf v_1 +\alpha_{sl}^2 \mathbf v_2 ))}$\\
end while
\STATE $l\leftarrow l+1$
\STATE {\bf{until}} convergence.
\end{algorithmic}
\end{algorithm}
\section{Applications of QOZCP to Radar Waveform Design}
\subsection{QOZCPs for SISO Radar Waveform}
Suppose from a transmit antenna is transmitted a sequence vector $\mathbf s^T=[\mathbf s_0, \mathbf s_1,\cdots,\mathbf s_{N-2},\mathbf s_{N-1}]$, where $\{\mathbf s_n\}_{n=0}^{N-1}$ are length-$L$ sequences over $N$ pulse repetition intervals (PRIs). It is noted that the sequences $\{\mathbf s_n\}_{n=0}^{N-1}$ are formed from QOZCP $(\mathbf x, \mathbf y)$ and their variants $\pm\mathbf x, \pm\mathbf y, \pm \tilde{\mathbf x}, \pm\tilde{\mathbf y}$, where $\tilde{\mathbf x}=(\tilde{x}[0], \tilde{x}[1], \cdots, \tilde{x}[L-1])$, and $\tilde{\cdot}$ denotes reversed complex conjugate, i.e., $\tilde{x}[l]=x^*[-l]$ for $l=0,1,\cdots,L-1$. Let $S_n (z) = \mathcal{Z}\{\mathbf s_n\}$ be the $z$-transform of $\mathbf s_n$ so that
\begin{eqnarray}
S_n(z) = s_n [0]+s_n [1]z^{-1}+\cdots+s_n [L-1]z^{-(L-1)}
\end{eqnarray}
for $n=0,1,\cdots,N-1$. Then $\mathbf S(z)$, the transmit sequence vector in $z$-domain, is written as
\begin{eqnarray}
\mathbf S^T(z)=[S_0(z),S_1(z),\cdots,S_{N-2}(z),S_{N-1}(z)].
\end{eqnarray}
As the assumption in \cite{pezeshki2008doppler}, the scatterer with a constant velocity has equal intra-Doppler in every PRI, whereas it has a relative Doppler shift $\theta_0$ between adjacent PRIs.
Then, in $n$-th PRI, $R_n(z)$, the returned sequence in $z$-domain, associate with a scatter at delay coordinate $d_0$, is denoted as
\begin{eqnarray}
R_n(z) = h_0 z^{-d_0}S_n(z) e^{jn\theta_0}+W_n(z)
\end{eqnarray}
where $h_0$ is a scattering coefficient and $w_n (z)$ is a noise in $n$-th PRI.
The returned sequence vector in $z$-domain is written as
\begin{eqnarray}
\mathbf R^T(z) = h_0 z^{-d_0}\mathbf S^T (z) \mathbf{D}(\theta_0)+\mathbf W^T(z),
\end{eqnarray}
where $\mathbf R^T(z) = [R_0 (z), R_1(z),\cdots, R_{N-2}(z), R_{N-1}(z)]$ , $\mathbf W^T(z)=[W_0 (z), W_1(z),\cdots, W_{N-2}, W_{N-1}]$, and $\mathbf{D}(\theta)$ is a diagonal Doppler modulation matrix which can be denoted as
\begin{eqnarray}
\mathbf{D}(\theta) = \mathrm{diag}(1,e^{j\theta},\cdots,e^{j(N-1)\theta}).
\end{eqnarray}
The received sequence vector $\mathbf R^T(z)$ is processed by the receiver vector $\tilde{\mathbf S}(z)$, i.e.,
\begin{eqnarray}
\tilde{\mathbf S}(z) = [\tilde{S}_0(z),\tilde{S}_1(z),\cdots,\tilde{S}_{N-2}(z),\tilde{S}_{N-1}(z)]^T
\end{eqnarray}
where $\tilde{S}_n(z)=S^*(1/z^*)$ is the $z$-transform of $\tilde{\mathbf s}_n$.
Then the output is written as
\begin{eqnarray}
\mathbf R^T(z) \tilde{\mathbf S}(z) = h_0 z^{-d_0} G(z,\theta_0)+\mathbf W^T(z) \tilde{\mathbf S}(z),
\end{eqnarray}
where $G(z,\theta)$ is given by
\begin{eqnarray}
G(z,\theta) = \mathbf S(z) \mathbf{D}(\theta) \mathbf S(z) = \sum_{n=0}^{N-1}e^{jn\theta}|S_n(z)|^2,
\end{eqnarray}
and $|S_n (z)|^2 = S_n (z) \tilde{S}_n(z)$.
According to the definition in \cite{pezeshki2008doppler}, $G(z,\theta)$ is called a $z$-transform of the ambiguity function:
\begin{eqnarray}
g(k,\theta) = \sum_{n=0}^{N-1} e^{jn\theta}c_n (k),
\end{eqnarray}
where $c_n (k) = \sum_{l=0}^{L-1} s_n [l] s^*_n [l+k]$ is the auto-correlation function (ACF) of $\mathbf s_n$.
We hope that for any $\theta$ among a modest Doppler shift interval, $G(z,\theta)$ has very low range sidelobes in a proper range interval $[-Z_{max}, Z_{max}]$, $Z_{max}\leq L$ . In other words, the desired $G(z,\theta)$ should be
\begin{eqnarray}\label{eq: ghope}
G(z,\theta)= \alpha (\theta)+ \sum_{l=-Z_{max}+1\atop l\neq 0}^{Z_{max}-1} v_l z^l + \hat {G}(z),
\end{eqnarray}
where $|v_l|\leq \delta$ for $l=-Z_{max}+1,\cdots,-1,1,\cdots,Z_{max}-1$, and $\delta$ is a very small positive real number. Also, $\hat {G}(z)$ is given by
\begin{equation}
\hat {G}(z)=\sum_{l=-L+1}^{-Z_{max}+1}v_l z^l + \sum_{l=Z_{max}-1}^{L-1}v_l z^l,
\end{equation}
which is what we do not pay attention on, because the range is outside the range interval $[-Z_{max}, Z_{max}]$. Besides, if $L=Z_{max}$, $\hat {G}(z)$ will vanish.
Now, we consider what is the key ingredient that can eliminate the Doppler effect to achieve the formula (\ref{eq: ghope}). Indeed, Taylor expansion used in \cite{pezeshki2008doppler} is a very important tool that can transform the wish into finding a solution to make the first $M$ Taylor coefficients almost vanish at all desired nonzero delays.
The Taylor expansion of $G(z,\theta)$ around $\theta=0$ is given by
\begin{eqnarray}
G(z,\theta) = \sum_{m=0}^{\infty} C_m(z)(j\theta)^m, \label{eq:taylor}
\end{eqnarray}
where $C_m (z)$ is the $m$-th Taylor coefficient which is given by
\begin{eqnarray}
C_m(z) = \sum_{n=0}^{N-1}n^m |S_{n}(z)|^2 \label{eq:taylorc}
\end{eqnarray}
for $ m=0,1,2,\cdots$.
In order to make $C_0(z), C_1(z),\cdots, C_M(z)$ almost vanish at all desired non-zero delays, $S_n(z)$ should be chosen carefully.
Note that, here PTM sequence is used to determine the sequence formed using PTM sequence is still used to determine the sequence formed using ($\mathbf x$, $\mathbf y$). Therefore, in the $n$-th PRI, $S_n(z)$ is given by
\begin{eqnarray}
S_n (z)=(1-a_n)X(z)+a_n Y(z) \label{eq:PTMSn}
\end{eqnarray}
where $ \{a_n\}_{n=0}^{N-1}$ is the PTM sequence defined as the following recursions:
\begin{equation}
\begin{cases}
a_0=0;\\
a_{2k}=a_k;\\
a_{2k+1}=1-a_k;
\end{cases}
\end{equation}
for all $k>0$.
\begin{theorem}
If $|X|^2+|Y|^2$ satisfies the following
\begin{eqnarray}
|X|^2+|Y|^2=2L+\hat{\zeta}(z)+\check{\zeta}(z),
\end{eqnarray}
then $C_0(z), C_1(z),\cdots, C_M(z)$ can be almost vanished at all desired nonzero delays.
Here
\begin{eqnarray}
\zeta(z)= \sum_{l=-Z_{max}+1\atop l\neq 0}^{Z_{max}-1} \hat{v}_l z^l,
\end{eqnarray}
and
\begin{eqnarray}
\check{\zeta}(z)=\sum_{l=-L+1}^{Z_{max}-1}\check{v}_l z^l + \sum_{l=Z_{max}-1}^{L-1}\check{v}_l z^l,
\end{eqnarray}
where $\hat{v}_l\leq \hat{\delta}$, $\hat{\delta}<\delta$ and $\check{v}_l$ is any value.
\end{theorem}
\begin{IEEEproof}
By substituting (\ref{eq:PTMSn}) into $C_m (z)$ shown in (\ref{eq:taylorc}), it is easy to verify
\begin{eqnarray}
\begin{split}
&C_m(z) =\sum_{n=0}^{N-1} n^m|(1-a_n)X+a_n Y|^2\\
&\quad=\sum_{n=0}^{N-1}n^m\left((1-a_n)^2|X|^2+a_n^2 |Y|^2\right)\\
&\quad\quad+\sum_{n=0}^{N-1} n^m\left((1-a_n)a_n^*XY^*+(1-a_n)^*a_n X^*Y \right)\\
&\quad=\left( \sum_{n=0}^{N-1}(1-a_n)n^m\right)|X|^2+\left(\sum_{n=0}^{N-1}a_n n^m\right)|Y|^2\\
&\quad=\beta_m (|X|^2+|Y|^2), \,\mathrm{for} \,\, m=0,1,2,\cdots,M,
\end{split} \label{eq:cm2}
\end{eqnarray}
where $\beta_m=\sum_{n=0}^{N-1} (1-a_n)n^m=\sum_{n=0}^{N-1}a_n n^m$ according to the Prouhet theorem (Theorem 1 in \cite{pezeshki2008doppler}), and $N=2^{M+1}-1$. It is noted that the third equation is satisfied based on the fact that the second part of the second equation can be vanished as $a_n$ can only be 0 or 1. Now, substituting $2L+\hat{\zeta}(z)+\check{\zeta}(z)$ into $C_m (z)$, then $C_m(z)$ is given by
\begin{eqnarray}
\begin{split}
& C_m (z) = \beta_m (|X|^2+|Y|^2)= \beta_m (2L+\hat{\zeta}(z)+\check{\zeta}(z))= 2L\beta_m+\hat{\zeta}(z)\beta_m+\check{\zeta}(z)\beta_m.
\end{split}
\end{eqnarray}
The desired nonzero delays of $C_m (z)$ is $\hat{\zeta}(z)\beta_m$, which is written as
\begin{eqnarray}
\hat{\zeta}(z)\beta_m = \sum_{l=-Z_{max}+1\atop l\neq 0}^{Z_{max}-1} \beta_m \hat{v}_l z^l,
\end{eqnarray}
Since $\hat{v}_l$ is small enough, then $\beta_m \hat{v}_l$ is still very small. Therefore, we can say that
$C_0(z)$, $C_1(z)$,$\cdots$, $C_M(z)$ can be almost vanished at all desired nonzero delays.
\end{IEEEproof}
\begin{remark}
The definition of quasi-Z-complementary pair $(\mathbf{x},\mathbf{y})$ in $z$-field is
\begin{eqnarray}
|X|^2+|Y|^2=2L+\hat{\zeta}(z)+\check{\zeta}(z),
\end{eqnarray}
which is shown in Theorem 1.
\end{remark}
\begin{remark}
The amplitudes of
$C_{M+1}(z),C_{M+2}(z),\cdots, C_{\infty}(z)$ are not almost equal to zero at every desired nonzero delay, since
\begin{eqnarray}
\sum_{n=0}^{N-1} (1-a_n)n^m \neq \sum_{n=0}^{N-1}a_n n^m,
\end{eqnarray}
when $m>M$. However, $\theta$ is small enough to make $ \sum_{m=M+1}^{\infty} C_m(z) (j\theta)^m$ convergent.
\end{remark}
\subsection{Application of QOZCPs in fully polarimetric radar}
Fully polarimetric radar systems are equipped with vertically/horizontally (V/H) dual-dipole elements at every antenna to make sure the simultaneous occurrence of transmitting and receiving on two orthogonal polarizations\cite{calderbank2006instantaneous}\cite{pezeshki2008doppler}\cite{cui2017broadband}. Sequences $\mathbf s_V=\{\mathbf s_{V,n} \}_{n=0}^{N-1}$ from vertically polarization and $\mathbf s_H=\{\mathbf s_{H,n}\}_{n=0}^{N-1}$ from horizontally polarization are transmitted together over $N$ pulse repetition intervals (PRIs). It is noted that the two sequence $\mathbf s_V$ and $\mathbf s_H$ are formed from the designed sequence pair $\mathbf x,\mathbf y$ which will be obtained in this paper. In other words, $\mathbf s_{V,n} $ and $\mathbf s_{H, n}$ can be chosen from $\left\{\pm\mathbf x, \pm\mathbf y, \pm \tilde{\mathbf x}, \pm\tilde{\mathbf y}\right\}$, where $\widetilde{\cdot}$ denotes reversed complex conjugate. The $z$-transform of transmit matrix formed from $\mathbf s_V$ and $\mathbf s_H$ can be denoted as
\begin{equation}
\begin{split}
\mathbf{S}(z)
&=\left[\begin{matrix}
\mathbf S_V (z)\\
\mathbf S_H(z)
\end{matrix}\right]=\left[\begin{matrix}
S_{V,0} (z)&S_{V,1} (z)&\cdots&S_{V,N-1} (z)\\
S_{H,0} (z)&S_{H,1} (z)&\cdots&S_{H,N-1} (z)
\end{matrix}\right].
\end{split}
\end{equation}
The received matrix can be written as
\begin{eqnarray}
\mathbf{R}(z) = z^{-d_0}\mathbf{H S}(z)\mathbf{D}(\theta_0)+\mathbf{W}(z).
\end{eqnarray}
where $\mathbf{H}$ is the scattering matrix given by (\ref{eq:sm}), $\mathbf{W}(z)$ is a noise matrix, and $\mathbf{D}(\theta)$ is the Doppler modulation matrix which can be written as $\mathbf{D}(\theta) = \mathrm{diag}(1,e^{j\theta},\cdots,e^{j(N-1)\theta})$.
If the received matrix $\mathbf{R}(z)$ is processed by a filter matrix $\mathbf{\widetilde{S}}(z)$ written as
\begin{eqnarray}
\mathbf{\widetilde{S}}(z)=
\left[\begin{matrix}
\widetilde S_{V,0} (z)& \widetilde S_{V,1} (z)&\cdots&\widetilde S_{V,N-1} (z)\\
\widetilde S_{H,0} (z)&\widetilde S_{H,1} (z)&\cdots&\widetilde S_{H,N-1} (z)
\end{matrix}\right]^T,
\end{eqnarray}
then the receiver output is
\begin{eqnarray}
\mathbf{R}(z) \mathbf{\widetilde{S}}(z)= z^{-d_0}\mathbf{H}\mathbf{G}(z,\theta_0)+\mathbf{W}(z)\widetilde{\mathbf{S}}(z),
\end{eqnarray}
where $d_0$ is the delay coordinate of a point target.
The matrix $\mathbf{G}(z,\theta_0) = \mathbf{S}(z)\mathbf{D}(\theta_0)\widetilde{\mathbf{S}}(z)$ is defined as $z$-transform of a matrix-valued ambiguity function for $\mathbf{S}(z)$.
The scattering matrix $\mathbf{H}$ can be easily obtained on a pulse-by-pulse basis\cite{pezeshki2008doppler}, if $\mathbf{G}(z,\theta_0)$
has the following expression:
\begin{equation}
\begin{split}
&\mathbf{G}(z,\theta) = \mathbf{S}(z)\mathbf{D}(\theta)\widetilde{\mathbf{S}}(z)\\
&=\left[\begin{matrix}
G_{VV}(z,\theta) & G_{VH}(z,\theta)\\
G_{HV}(z,\theta) & G_{HH}(z,\theta)
\end{matrix}\right]\approx \alpha(\theta)\left[\begin{matrix}
NL&0\\
0&NL
\end{matrix}\right],\label{eq: purpose}
\end{split}
\end{equation}
where $\alpha(\theta)$ is a function of $\theta$ independent of delay. Because $G_{VV}$ is equivalent to $G_{HH}$, and $G_{VH}=\widetilde{G}_{HV}$, it is sufficient to analyze $G_{VV}$ and $G_{VH}$ which are given by
\begin{eqnarray}
\begin{split}
&G_{VV} (z,\theta)= \sum_{n=0}^{N-1} e^{jn\theta} |S_{V,n}(z)|^2,\\
&G_{VH} (z,\theta)= \sum_{n=0}^{N-1} e^{jn\theta} S_{V,n}(z) \widetilde{S}_{H,n}(z),
\end{split}
\end{eqnarray}
where $|S_{V,n}(z)|^2$ is a $z$-transform of auto-correlation function of $s_{V,n}$ and $S_{V,n}(z) \widetilde{S}_{H,n}(z)$ is a $z$-transform of cross-correlation function of $s_{V,n}$ and $s_{H,n}$. The auto-correlation function of $s_{V,n}$ and cross-correlation function of $s_{V,n}$ and $s_{H,n}$ are respectively given by
$C_{s_{V,n}} (k) = \sum_{l=0}^{L-1} \mathbf s_{V,n}(l) \mathbf s_{V,n}(l+k) ^*$ and $C_{s_{V,n},s_{H,n}} (k) = \sum_{l=0}^{L-1} \mathbf s_{V,n}(l) \mathbf s_{H,n}(l+k) ^*$, $l=0,1,\cdots,L-1$, where * denotes conjugate.
Again, we consider what is the key ingredient that can eliminate the Doppler effect to achieve the formula (\ref{eq: purpose}). Taylor expansion is again used to transform the wish into finding a solution to make the first $M$ Taylor coefficients vanish at all desired nonzero delays.
The Taylor expansions of $G_{VV}(z,\theta)$ and $G_{VH}(z,\theta)$ around $\theta=0$ are respectively given by
\begin{eqnarray}
G_{VV}(z,\theta) = \sum_{m=0}^{\infty} CV_m(z)(j\theta)^m,G_{VH}(z,\theta) = \sum_{m=0}^{\infty} B_m(z)(j\theta)^m,
\end{eqnarray}
where
\begin{eqnarray}
CV_m(z) = \sum_{n=0}^{N-1}n^m |S_{V,n}(z)|^2,\\ \quad B_m(z) = \sum_{n=0}^{N-1}n^m S_{V,n}(z)\widetilde{S}_{H,n}(z), \label{eq:cm}
\end{eqnarray}
for $ m=0,1,2,\cdots$.
Combining the PTM sequence with the Alamouti matrix, we define a PTM-A matrix as follows
\begin{eqnarray}
\begin{split}
&\left[\begin{matrix}
S_{V,2k} & S_{V,2k+1}\\
S_{H,2k} & S_{H,2k+1}
\end{matrix}\right]=\left[\begin{matrix}
(1\!-\!a_{2k})X \!+\! a_{2k} (\!-\widetilde{Y}) & a_{2k+1}(\!-\! \widetilde{Y})\! \!+\!\! (1\!-\! a_{2k+1}) (\!-\!X)\\
\!\!(1-a_{2k})Y \!+\! a_{2k} \widetilde{X} & a_{2k+1}\widetilde{X} \! +\!(1-a_{2k+1}) (-Y)
\end{matrix}\right],
\end{split}\label{eq:PTM-A}
\end{eqnarray}
where $ \{a_n\}_{n=0}^{N-1}$ is the PTM sequence defined above.
By substituting (\ref{eq:PTM-A}) into $CV_m (z)$ shown in (\ref{eq:cm}), and following the proof of theorem 1, it is easy to verify
\begin{eqnarray}
\begin{split}
&CV_m(z) = \beta_m (|X|^2+|Y|^2), \,\mathrm{for} \,\, m=0,1,2,\cdots,M,
\end{split} \label{eq:cm2}
\end{eqnarray}
where $\beta_m=\sum_{n=0}^{N-1} (1-a_n)n^m=\sum_{n=0}^{N-1}a_n n^m$ according to the Prouhet theorem (Theorem 1 in \cite{pezeshki2008doppler}), and $N=2^{M+1}-1$. Indeed, $CV_m (z)$ is definitely equivalent to $C_m(z)$.
Similarly, $B_m(z)$ can be written as
\begin{eqnarray}
\begin{split}
&B_m(z) \!\!= \!\!\!\!\sum_{k=0}^{N/2-1} \!\!(\!(1\!-\!2a_{2k})(2k)^m \!+\!(1\!-\!2a_{2k\!+\!1})(2k\!+\!1)^m \!)X\widetilde{Y}=\left[\sum_{n=0}^{N-1}(-1)^{a_{n}} n^{m}\right] X \widetilde{Y}.
\end{split}\label{eq:bm2}
\end{eqnarray}
Again, by Prouhet theorem, we can easily obtain
\begin{eqnarray}
\sum_{n=0}^{N-1}(-1)^{a_{n}} n^{m}=0, \, \mathrm{for}\, \,m=0,1,\cdots, M\label{eq:pt}.
\end{eqnarray}
The Taylor coefficients $CV_m$ and $B_m$ are clearly shown in (\ref{eq:cm2}) and (\ref{eq:bm2}), respectively.
From (\ref{eq:pt}), we know that $B_{0},B_{1},\cdots,B_{M}$ can vanish thoroughly for any $\mathbf x, \mathbf y$, which attributes to a PTM-A matrix. However, the $B_{M+1},B_{M+2},\cdots$ can not be equal to zero when $N=2^{M+1}-1$. Therefore, for eliminating all $\{B_m\}_{m=0}^{\infty}$, the $X\widetilde{Y}$ should be equal to zero. But one can hardly achieve the purpose. To compromise on it, the proposed QOZCP can help $X\widetilde{Y}$ meet the following low local cross-correlation demand:
\begin{eqnarray}
X\widetilde{Y} = \sum_{l=-L+1}^{L+1} u_l z^l, \label{eq:qo}
\end{eqnarray}
where $|u_l|\leq \delta$ for $l=-Z_{max}+1,\cdots,-1,0,1,\cdots,Z_{max}-1$, $Z_{max}\leq L$, and $\delta$ is very close to zero.
From (\ref{eq:cm2}), in order to make $CV_m=\beta_m$ with $m = 0,1,2,\cdots,M$, $|X|^2+|Y|^2$ should be equal to $2L$, which is exactly the property of Golay complementary pair. However, all GCPs cannot meet the low local cross-correlation demand, i.e., (\ref{eq:qo}). To avoid the problem, we want $X\widetilde{Y}$ to meet the following property of low local sum of autocorrelations:
\begin{eqnarray}
|X|^2+|Y|^2= 2L+ \sum_{l=-L+1\atop l\neq 0}^{L+1} v_l z^l\label{eq:qzc}
\end{eqnarray}
where $|v_l|\leq \delta$ for $l=\pm 1,\pm 2, \cdots, \pm (Z_{max}-1)$, $Z_{max}\leq L$, and $\delta$ is very close to zero.
\begin{theorem}
If $X, Y$ satisfy the following equations
\begin{eqnarray}
|X|^2+|Y|^2=2L+\hat{\zeta}(z)+\check{\zeta}(z),X\widetilde{Y} = \xi(z)+\hat{\xi}(z),
\end{eqnarray}
then $CV_0(z), CV_1(z),\cdots, CV_M(z)$ tends to zero at all desired nonzero delays.
Here
\begin{eqnarray}
\zeta(z)= \sum_{l=-Z_{max}+1\atop l\neq 0}^{Z_{max}-1} \hat{v}_l z^l,
\xi(z) = \sum_{l=-Z_{max}+1}^{Z_{max}-1} \hat{u}_l z^l,
\end{eqnarray}
and
\begin{eqnarray}
\check{\zeta}(z)=\sum_{l=-L+1}^{Z_{max}-1}\check{v}_l z^l + \sum_{l=Z_{max}-1}^{L-1}\check{v}_l z^l,
\check{\xi}(z)=\sum_{l=-L+1}^{Z_{max}-1}\check{u}_l z^l + \sum_{l=Z_{max}-1}^{L-1}\check{u}_l z^l,
\end{eqnarray}
where $\hat{v}_l,\hat{u}_l\leq \hat{\delta}$, and $\hat{\delta}<\delta$ and $\check{v}_l,\check{u}_l$ are any value.
\end{theorem}
\section{Numerical Experiments and Discussions}
In this section, we aim at comparing the proposed Doppler resilient (DR) QOZCPs with DR-GCP sequences on the performance of Doppler resilience. Besides, the complementary sidelobes and cross correlation of $(L,Z)$-QOZCPs ($\mathrm{PAPR}=5$) obtained by SDAMM are also compared with those of length-$L$ GCPs .
The two DR-QOZCP sequences and the two DR-GCP sequences are generated by the $(L,Z)$-QOZCP $(\mathbf x_Q, \mathbf y_Q)$, and length-$L$ GCP $(\mathbf x_G, \mathbf y_G)$, respectively. The PTM-A matrix is chosen as (\ref{eq:PTM-A}) with $N=8$. Then the transmit matrix for DR-QOZCP sequences and DR-GCP sequences is chosen as follows,
\begin{eqnarray}
\mathbf{s}
=\left[\begin{matrix}
\mathbf {s_V}\\
\mathbf {s_H}
\end{matrix}\right]
=\left[\begin{matrix}
\mathbf x&- \tilde{\mathbf y}&- \tilde{\mathbf y}&-\mathbf x&- \tilde{\mathbf {y}}&-\mathbf x&\mathbf x&- \tilde{\mathbf y}\\
\mathbf {y}&\tilde{\mathbf {x}}& \tilde{\mathbf {x}}&-\mathbf {y}& \tilde{\mathbf {x}}&-\mathbf {y}&\mathbf{y}& \tilde{\mathbf {x}}
\end{matrix}\right],\end{eqnarray}
where $\mathbf {s_V}$ and $\mathbf {s_H}$ are Doppler resilient sequences sent from V polarization direction and from H polarization direction, respectively.
Their auto-ambiguity function (AAF) $g_{VV} (k,\theta)$ of $\mathbf s_V$, and cross-ambiguity function (CAF) $g_{VH} (k,\theta)$ of $\mathbf s_V$, $\mathbf s_H$ are respectively given by
\begin{eqnarray}
\begin{split}
&g_{VV} (k,\theta)= \sum_{n=0}^{N-1} e^{jn\theta} C_{VV,n}(k)=\sum_{n_0={0,3,5,7}}e^{jn_0\theta}C_{\mathbf x} (k)+\sum_{n_1={1,2,4,6}}e^{jn_1\theta}C_{\mathbf y} (k),
\end{split}\label{eq:gvv}
\end{eqnarray}
and
\begin{eqnarray}
\begin{split}
&g_{VH} (k,\theta)= \sum_{n=0}^{N-1} e^{jn\theta} C_{VH,n}(k)=\sum_{n_0={0,3,5,7}}e^{jn_0\theta}C_{xy} (k)-\sum_{n_1={1,2,4,6}}e^{jn_1\theta}C_{xy} (k).
\end{split}\label{eq:gvh}
\end{eqnarray}
\begin{table*}[]
\centering
\caption{\normalsize\\ \normalsize Maximun Complementary Sidelobe and Maximun Cross-Correlation of $(\mathbf x, \mathbf y)$ with $\mathrm{PAPR}=5$}
\label{tab:my-table}
\begin{tabular}{|l|l|r|l|l|l|}
\hline
\multicolumn{1}{|c|}{\multirow{2}{*}{L}} & \multicolumn{1}{c|}{\multirow{2}{*}{Z}} & \multicolumn{2}{r|}{Maximun Complementary Sidelobe in $[-Z+1,Z-1]$} & \multicolumn{2}{l|}{Maximun Cross-Correlation in $[-Z+1,Z-1]$} \\ \cline{3-6}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{\quad \quad\quad GCP \quad \quad\quad} & \multicolumn{1}{c|}{\quad \quad QOZCP \quad \quad} & \multicolumn{1}{c|}{\quad \quad \quad GCP \quad\quad \quad} & \multicolumn{1}{c|}{\quad \quad QOZCP \quad \quad} \\ \hline
64 & 10 & \multicolumn{1}{c|}0 & \multicolumn{1}{c|}{ $6.10\times10^{-11}$} & \multicolumn{1}{c|}{14} & \multicolumn{1}{c|}{$1.90\times10^{-10}$} \\ \hline
64 & 15 & \multicolumn{1}{c|}0 & \multicolumn{1}{c|}{ $1.31\times10^{-10}$} & \multicolumn{1}{c|}{15} & \multicolumn{1}{c|}{$4.56\times10^{-10}$} \\ \hline
64 & 20 & \multicolumn{1}{c|}0 & \multicolumn{1}{c|}{ $3.99\times10^{-10}$} & \multicolumn{1}{c|}{15} & \multicolumn{1}{c|}{$1.09\times10^{-9}$} \\ \hline
64 & 25 & \multicolumn{1}{c|}0 &\multicolumn{1}{c|}{ $6.18\times 10^{-10}$} & \multicolumn{1}{c|}{15} & \multicolumn{1}{c|}{$1.71\times10^{-9}$} \\ \hline
64 & 30 & \multicolumn{1}{c|}0 &\multicolumn{1}{c|}{ $9.46\times 10^{-10}$} & \multicolumn{1}{c|}{15} & \multicolumn{1}{c|}{$2.09\times 10^{-9}$} \\ \hline
\end{tabular}
\end{table*}
\begin{table*}[]
\centering
\caption{\\ \normalsize Maximun Auto-Ambiguity Function Sidelobes and Maximun Cross-Ambiguity Functions of $(\mathbf x, \mathbf y)$}
\label{tab:my-table}
\begin{tabular}{|l|l|r|l|l|l|}
\hline
\multicolumn{1}{|c|}{\multirow{2}{*}{L}} & \multicolumn{1}{c|}{\multirow{2}{*}{Z}} & \multicolumn{2}{r|}{Maximun Auto-Ambiguity Function Sidelobes in $\Omega_1$} & \multicolumn{2}{l|}{Maximun Cross-Ambiguity Function in $\Omega_2$} \\ \cline{3-6}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{\quad \quad\quad DR-GCP \quad \quad\quad} & \multicolumn{1}{c|}{\quad \quad DR-QOZCP \quad \quad} & \multicolumn{1}{c|}{\quad \quad \quad DR-GCP \quad \quad \quad} & \multicolumn{1}{c|}{\quad \quad DR-QOZCP \quad \quad} \\ \hline
64 & 10 & \multicolumn{1}{c|}{$5.39\times 10^{-2}$} & \multicolumn{1}{c|}{ $5.52\times10^{-2}$} & \multicolumn{1}{c|}{76.98} & \multicolumn{1}{c|}{$1.04\times10^{-9}$} \\ \hline
64 & 15 & \multicolumn{1}{c|}{$6.93\times 10^{-2}$} & \multicolumn{1}{c|}{ $6.76\times10^{-2}$} & \multicolumn{1}{c|}{76.98} & \multicolumn{1}{c|}{$2.51\times10^{-9}$} \\ \hline
64 & 20 & \multicolumn{1}{c|}{$6.93\times 10^{-2}$} & \multicolumn{1}{c|}{ $7.66\times10^{-2}$} & \multicolumn{1}{c|}{82.48} & \multicolumn{1}{c|}{$5.98\times 10^{-9}$} \\ \hline
64 & 25 & \multicolumn{1}{c|}{$1.00\times 10^{-1}$} &\multicolumn{1}{c|}{ $8.53\times 10^{-2}$} & \multicolumn{1}{c|}{82.48} & \multicolumn{1}{c|}{$9.40\times 10^{-9}$} \\ \hline
64 & 30 & \multicolumn{1}{c|}{$1.00\times 10^{-1}$} &\multicolumn{1}{c|}{ $8.61\times 10^{-2}$} & \multicolumn{1}{c|}{$82.48$} & \multicolumn{1}{c|}{$1.15\times 10^{-8}$} \\ \hline
\end{tabular}
\end{table*}
\begin{figure} \centering
\subfigure[The sum of auto-correlation functions, i.e., $|C_x(k)+C_y(k)|$.] {
\label{fig:a}
\includegraphics[width=0.45\columnwidth]{pic/acf}
}
\subfigure[The cross-correlation function, i.e., $|C_{xy}|(k)$] {
\label{fig:b}
\includegraphics[width=0.45\columnwidth]{pic/ccf}
}
\caption{ Two figures about the QOZCP $(\mathbf x,\mathbf y)$ by the Algorithm 1 }
\label{fig:acf}
\end{figure}
\begin{figure} \centering
\subfigure[The AAF of a DR-GCP sequence.] {
\label{fig:a}
\includegraphics[width=0.475\columnwidth]{pic/aambgcp}
}
\subfigure[The AAF of the a DR-QOZCP sequence] {
\label{fig:b}
\includegraphics[width=0.475\columnwidth]{pic/aambqozcp}
}
\caption{ AAFs of the DR-GCP sequence and the DR-QOZCP sequence, respectively}
\label{fig:amb}
\end{figure}
Remark: For computing their AAF and CAF of DR-QOZCP sequence(s), $\mathbf x_Q$, $\mathbf y_Q$, and $\mathbf x_G$, $\mathbf y_G$ can be substituted into $\mathbf x$, $\mathbf y$ in (\ref{eq:gvv}) and (\ref{eq:gvh}).
Some conclusions can be drawn based on Table I, Table II, and Fig. \ref{fig:acf} to Fig. \ref{fig:camb}.
\begin{itemize}
\item At first, we will compare $(L,Z)$-QOZCP ($\mathrm{PAPR}=5$) with length-$L$ GCP on maximun complementary sidelobe and cross correlation in delay interval $[-Z+1,Z-1]$. As Table I shows, there is a significant difference between QOZCP and GCP. Although QOZCP and GCP both have good complementary sidelobe in delay interval $[-Z+1,Z-1]$, QOZCP has much lower cross correlation than GCP in $[-Z+1,Z-1]$.
\item As shown in Fig. \ref{fig:acf}, two figures about the obtained $(L,Z)$-QOZCP $(\mathbf x,\mathbf y)$ by SDAMM, where $L=64$,$Z=30$, $\mathrm{PAPR}=5$. Fig. \ref{fig:acf}(a) is the sum of correlation functions $C_x (k)+C_y (k)$ and Fig. \ref{fig:acf}(b) is the cross-correlation function $C_{xy}(k)$. The sidelobe in Fig. \ref{fig:acf}(a) is very low in delay interval $[-Z+1,Z-1]$. The cross-correlation level in interval $[-Z+1,Z-1]$ in Fig. \ref{fig:acf}(b) is also very low.
\item Besides, we will compare DR-QOZCP with DR-GCP on Maximun Auto-Ambiguity Function Sidelobes in $\Omega_1$ and Maximun Cross-Ambiguity Function in $\Omega_2$ , where $\Omega_1 = \{(k,\theta)| k\in [-Z+1,Z-1], \theta \in [0,0.1]\}$ and $\Omega_2 = \{(k,\theta)| k\in [-Z+1,Z-1], \theta \in [0,3]\}$. What is interesting in table II is that DR-QOZCPs derived by $(L,Z)$-QOZCP have much lower Maximun Cross-Ambiguity Function than DR-GCPs', while Maximun Auto-Ambiguity Function Sidelobes are low for both DR-GCP and DR-QOZCP.
\item Fig. \ref{fig:amb}(a) and Fig. \ref{fig:amb}(b) show an AAF of the DR-GCP sequence and that of the DR-QOZCP sequence in the Delay interval $[-Z+1, Z-1]$ and Doppler interval $[0, 0.1]$. The sidelobes of the designed sequence in delay interval $[-Z+1,Z-1]$ are almost as good as that of the DR-GCP sequence along any Doppler shift in $[0, 0.1]$.
\item Fig. \ref{fig:camb}(a) and Fig. \ref{fig:camb}(b) show an AAF of a DR-GCP sequence and that of the designed DR-QOZCP sequence, respectively, in the Delay interval $[-20,20]$ and Doppler interval $[0, 3]$. The level of Fig. \ref{fig:camb}(b) is significantly lower than that of Fig. \ref{fig:camb}(a). For the system, the cross polarization aliasing is sufficiently vanished along any Doppler shift in $[0,3]$ when the proposed sequence pair is used.
\end{itemize}
In short, the Doppler resilient performance of our DR-QOZCP sequences is significantly better than that of DR-GCP sequences.
\begin{figure}[htbp]
\centering
\subfigure[The CAF of two DR-GCP sequences]{
\begin{minipage}{7cm}
\centering
\includegraphics[width=8cm]{pic/cambgcp}\\
\end{minipage}
}
\subfigure[The CAF of two DR-QOZCP sequences]{
\begin{minipage}{8cm}
\centering
\includegraphics[width=8cm]{pic/cambqozcp}
\end{minipage}
}
\caption{ CAFs of DR-GCP sequences and DR-QOZCP sequences, respectively}
\label{fig:camb}
\end{figure}
\section{Concluding Remarks and Open Problems}
In this paper we have given two contributions. Firstly, we have proposed new sets of sequence pairs called ``Quasi-Orthogonal Z-Complementary Pairs" (QOZCPs). Secondly, we have applied the proposed QOZCPs to design Doppler resilient waveforms for polarimetric radar systems with are also dual orthogonal. We have also solved an optimization problem efficiently by proposing SADMM algorithm in MM framework, which eventually constructs the proposed QOZCPs of any length. Finally, we have compared the ambiguity plots of the proposed QOZCPs with the ambiguity plots of the existing DR-GCPs. The comparison shows that the proposed QOZCPs are more efficient as compared to DR-GCPs when CAF plots are considered.
Owing to the efficiency of the QOZCPs in application scenarios, the reader is invited to propose systematic constructions of QOZCPs and analyze the structural properties of these pairs.
\section*{Appendix A\\Proof of Proposition 1}
\begin{IEEEproof}
Calculate
\begin{eqnarray}
\begin{split}
\sum_{k=-L+1}^{L-1} w_k |\mathbf {z^H A_k z}|^2&=\sum_{k=-L+1}^{L-1} w_k \text{vec}(\mathbf Z)^H \text{vec}(\mathbf A_k) \text{vec}(\mathbf A_k)^H \text{vec}(\mathbf Z)\\
&= \text{vec}(\mathbf Z)^H \left[\sum_{k=-L+1}^{L-1} w_k \text{vec}(\mathbf A_k) \text{vec}(\mathbf A_k)^H \right]\text{vec}(\mathbf Z),
\end{split}
\end{eqnarray}
\begin{eqnarray}
\begin{split}
&\sum_{k=-L+1}^{L-1} \tilde{w}_k |\mathbf {z^H B_k z}|^2= \text{vec}(\mathbf Z)^H \left[\sum_{k=-L+1}^{L-1} \tilde{w}_k \text{vec}(\mathbf B_k) \text{vec}(\mathbf B_k)^H \right]\text{vec}(\mathbf Z).
\end{split}
\end{eqnarray}
We set $J_A = \sum_{k=-L+1}^{L-1} w_k \text{vec}(\mathbf A_k) \text{vec}(\mathbf A_k)^H$, and
$J_B = \sum_{k=-L+1}^{L-1} w_k \text{vec}(\mathbf B_k) \text{vec}(\mathbf B_k)^H$.
Then, It is easy to verify that the objective of (\ref{opt:p00}) can be transformed into that of (\ref{opt:p01})
\end{IEEEproof}
\section*{Appendix B\\ Proof of proposition 2}
\begin{IEEEproof}
The objective of (\ref{opt:p01}) can be majorized by
\begin{eqnarray}
\begin{split}
&M_{f_1} (\mathbf Z, \mathbf Z^{(t)})\\
&=\lambda_{\mathbf J} \text{vec}(\mathbf Z)^H \text{vec}(\mathbf Z)+2 \text{Re}(\text{vec}(\mathbf Z)^H(\mathbf J-\lambda_{\mathbf J}\mathbf I)\text{vec}(Z^{(t)}))+\text{vec}(\mathbf Z^{(t)})^H(\lambda_{\mathbf J}\mathbf I-\mathbf J)vec(Z^{(t)})
\end{split}\label{eq:mf1}
\end{eqnarray}
The first term is equal to a constant $\lambda_{\mathbf J} L^2$, and the last term is also a constant obviously. Therefore, we ignore the both terms and only keep the second term, and the objective of (\ref{opt:p01}) can be majorized by the second term. It is noted that the coefficient 2 cannot make any effect on the optimization, so we remove it.
\end{IEEEproof}
\section*{Appendix C\\ Proof of Theorem 1}
\begin{IEEEproof}
Calculate
\begin{eqnarray}
J\text{vec}(\mathbf A_k)& =& (\alpha J_A+(1-\alpha)J_B) \text{vec}(\mathbf A_k)=\alpha J_A \text{vec}(\mathbf A_k)\\
&=&\sum_{i=1-L}^{L-1} w_i \alpha \text{vec}(A_i) \text{vec}(A_i)^H\text{vec}(A_k)\\
&=&w_k \alpha \text{vec}(A_k) \text{vec}(A_k)^H\text{vec}(A_k)\\
&=&w_k \alpha (2N-2|k|) \text{vec}(A_k).
\end{eqnarray}
Therefore,
\begin{eqnarray}
\lambda_{A}(k) = w_k \alpha (2N-2|k|).
\end{eqnarray}
Calculate
\begin{eqnarray}
J\text{vec}(\mathbf B_k) &=& (J_A+J_B) \text{vec}(\mathbf B_k)=J_B \text{vec}(\mathbf B_k)\\
&=&\sum_{i=1-L}^{L-1} w_i \alpha \text{vec}(B_i) \text{vec}(B_i)^H\text{vec}(B_k)\\
&=&w_k \alpha \text{vec}(B_k) \text{vec}(B_k)^H\text{vec}(B_k)\\
&=&w_k (1-\alpha) (N-|k|) \text{vec}(A_k).
\end{eqnarray}
Therefore,
\begin{eqnarray}
\lambda_{B}(k) = w_k (1-\alpha) (N-|k|);
\end{eqnarray}
Because $\lambda_A (k)$ and $\lambda_B(k)$ are the eigenvalue of $\mathbf J$, then
\begin{eqnarray}
\lambda_{\mathbf J} = \max_k \left\{\max\{\lambda_{A}(k), \lambda_{B}(k) \right\}|k=-L+1,\cdots,L-1\}.
\end{eqnarray}
\end{IEEEproof}
\section*{Appendix D\\Proof of Proposition 3}
\begin{IEEEproof}
The main operations are in (\ref{eq:bigproof})
\begin{figure*}
\begin{eqnarray}
\begin{split}
&\text{Re}\left\{\text{vec}(\mathbf Z)^H (\mathbf J-\lambda_{\text{max}}(\mathbf J)\mathbf I)\text{vec}(\mathbf Z^{(t)})\right\} \\
&= \text{Re}\left\{ \text{vec}(\mathbf Z)^H \left( \alpha \sum_{k=-L+1}^{L-1} w_k \text{vec}(\mathbf A_k) \text{vec}(\mathbf A_k)^H \right.\right.\\
&\qquad\left.\left.+ (1-\alpha)\sum_{k=-L+1}^{L-1} \tilde{w}_k \text{vec}(\mathbf B_k) \text{vec}(\mathbf B_k)^H-\lambda_{\mathbf J}\mathbf I \right) \cdot\text{vec}(\mathbf Z^{(t)}) \right\} \\
&= \text{Re}\left\{ \sum_{k=-L+1}^{L-1} \alpha w_k (\text{Tr}(\mathbf A_k \mathbf Z) \text{Tr}(\mathbf A_{-k} \mathbf Z^{(t)}))
\right.\\
&\left.\qquad+ \sum_{k=-L+1}^{L-1} (1-\alpha)\tilde{w}_k (\text{Tr}(\mathbf B_k \mathbf Z) \text{Tr}(\mathbf B_{-k} \mathbf Z^{(t)}))-\lambda_{\mathbf J} \text{Tr} (\mathbf Z^{(t)} \mathbf Z)\right\}\\
&= \text{Re}\left\{ \alpha \text{Tr} ( \sum_{k=-L+1}^{L-1} w_k r_{-k}^{(l)} \mathbf A_k \mathbf Z)
+(1-\alpha) \text{Tr} ( \sum_{k=-L+1}^{L-1} \tilde{w}_k c_{-k}^{(l)} \mathbf B_k \mathbf Z))-\lambda_{\mathbf J} \text{Tr}(\mathbf Z^{(t)}\mathbf Z)\right\} \\
&= \text{Re}\left\{ \alpha Tr(\mathbf Q_r \mathbf Z)+(1-\alpha) Tr(\mathbf Q_c \mathbf Z)-\lambda_{\text{max}}(\mathbf J) \text{Tr}(\mathbf Z^{(t)}\mathbf Z) \right\} \\
&= \text{Re}\left\{ \mathbf z^H \left(\alpha \mathbf Q_r +(1-\alpha)\mathbf Q_c -\lambda_{max}(\mathbf J) \mathbf z^{(t)}(\mathbf z^{(t)})^H\right)\mathbf z \right\} \\
&=\text{Re}\left\{\mathbf z^H \left( \mathbf Q -\lambda_{max}(\mathbf J) \mathbf z^{(t)}(\mathbf z^{(t)})^H\right)\mathbf z
\right\}
\end{split}\label{eq:bigproof}
\end{eqnarray}
\hrulefill
\vspace*{0pt}
\end{figure*}
where
\begin{eqnarray}
\begin{split}
&\mathbf Q_r = \sum_{k=-L+1}^{L-1} w_k r_{-k}^{(l)} \mathbf A_k =\begin{bmatrix} \sum_{k=-L+1}^{L-1} w_k r_{-k}^{(l)}\mathbf U_k&\mathbf O\\ \mathbf O& \sum_{k=-L+1}^{L-1} w_k r_{-k}^{(l)}\mathbf U_k \end{bmatrix},
\end{split}
\end{eqnarray}
\begin{eqnarray}
\begin{split}
&\mathbf Q_c = \sum_{k=-L+1}^{L-1} \tilde{w}_k c_{-k}^{(l)} \mathbf B_k
&=\begin{bmatrix} \mathbf O& \sum_{k=-L+1}^{L-1} w_k c_{-k}^{(l)}\mathbf U_k\\ \mathbf O& \mathbf O \end{bmatrix}
\end{split}
\end{eqnarray}
\begin{eqnarray}
\mathbf Q = \alpha\mathbf Q_r + (1-\alpha) \mathbf Q_c.
\end{eqnarray}
\end{IEEEproof}
\section*{Appendix E\\Proof of theorem 2}
\begin{IEEEproof}
It is well known that FFT(IFFT) computed autocorrelation function is given by \cite{song2016sequence}
\begin{equation}
\begin{split}
&[C_x^{(t)}(0),C_x^{(t)}(1),\cdots,C_x^{(t)}(L-1),0,C_x^{(t)}(1-L),\cdots,C_x^{(t)}(-1)]^T= \mathbf F^H\!|\mathbf F[(\mathbf y^{(t)})^T\!,\mathbf 0_{1\times L}]^T|^2.
\end{split}
\end{equation}
Similarly,
\begin{equation}
\begin{split}
&[C_y^{(t)}(0),C_y^{(t)}(1),\cdots,C_y^{(t)}(L-1),0,C_y^{(t)}(1-L),\cdots,C_y^{(t)}(-1)]^T= \mathbf F^H\!|\mathbf F[(\mathbf y^{(t)})^T\!,\mathbf 0_{1\times L}]^T|^2
\end{split}
\end{equation}
Also, $r_{k}^{(t)}=C_x^{(t)}(k)+C_y^{(t)}(k)$ holds, then the following equation is obviously obtained.
\begin{eqnarray}
\begin{split}
&\mathbf r = [r_0^{(t)},r_1^{(t)},\cdots,r_{L-1}^{(t)},0,r_{1-L}^{(t)},\cdots,r_{-1}^{(t)},]\\
&= \mathbf F^H\!|\mathbf F[(\mathbf x^{(t)})^T\!,\mathbf 0_{1\times L}]^T|^2\!+\mathbf F^H\!|\mathbf F[(\mathbf y^{(t)})^T\!,\mathbf 0_{1\times L}]^T|^2.
\end{split}
\end{eqnarray}
The discrete Fourier transform of $x$ and $y$ can be written as
\begin{eqnarray}
\sum_{l=1}^L x(l)e^{-j2\pi \omega_i l/L} \,\,\mathrm{and}\,\,
\sum_{m=1}^L y(l)e^{-j2\pi \omega_i m/L},
\end{eqnarray}
respectively. It is easy to verify the following equation holds
\begin{eqnarray}
\begin{split}
&(\sum_{l=1}^L x(l)e^{-j2\pi \omega_i l/L} )^*(\sum_{m=1}^L y(l)e^{-j2\pi \omega_i m/L})=\sum_{l=1}^L \sum_{m=1}^L x^*(l) y(m) e^{-j2\pi \omega_i m/L}\\
&=\sum_{k=-L+1}^{L-1} \sum_{l=0}^{L-1} x^*(l)y(k+l)e^{-j2\pi \omega_i k/L}=\sum_{k=-L+1}^{L-1} c_k e^{-j2\pi \omega_i k/L}
\end{split}
\end{eqnarray}
where $k$ in second equation is to let $k=m-l$.
Then it is easy to know the cross correlation $c_k=C_{xy}(k)$ can be obtained by inverse Fourier transform.
\end{IEEEproof}
\section*{Appendix F\\Proof of Proposition 5}
\begin{IEEEproof}
The objective of (\ref{opt:p04}) can be majorized by
\begin{eqnarray}
\begin{split}
&M_{f_2} (\mathbf z, \mathbf z^{(t)})=\lambda_u \mathbf z^H \mathbf z+2 \text{Re}(\mathbf z^H(\hat{\mathbf Q}-\lambda_u\mathbf I)\mathbf z^{(t)})+(\mathbf z^{(t)})^H(\lambda_u\mathbf I-\mathbf J)\mathbf z^{(t)}
\end{split}
\end{eqnarray}
where $\hat{\mathbf Q} = \mathbf Q +\mathbf Q^H-2\lambda_{\mathbf J} \mathbf z^{(t)}(\mathbf z^{(t)})^H $. Similarly,
Since $\lambda_u$ should be larger than $\lambda_{max} (\hat{\mathbf Q})$, and $\lambda_{max}(\mathbf Q +\mathbf Q^H)$ is larger than $\lambda_{max} (\hat{\mathbf Q})$, then we can let $\lambda_u > \lambda_{max}(\mathbf Q +\mathbf Q^H)$ so that $\lambda_u>\lambda_{max} (\hat{\mathbf Q})$.
Based on the fact in \cite{song2015optimization}, the following inequality holds
\begin{eqnarray}
\lambda_{max}(\mathbf Q+\mathbf Q^H)\leq 2L\max|a_{ij}|, \quad i,j\in\{1,2,\cdots,2L\}
\end{eqnarray}
where $a_{ij}$ is the elements of the matrix $\mathbf Q+\mathbf Q^H$. Besides, the following inequality holds
$$2\max|Q_{ij}|\geq \max|Q_{ij}|+\max|Q_{ji}^*|\geq \max|a_{ij}|,$$ and we have $4L (\max \limits_{1\le i,j\le L}| Q_{i,j}|)>2L \max|a_{ij}|$, where $Q_{ij}$ is the elements of $\mathbf Q$.
Hence, $\lambda_u = 4L (\max \limits_{1\le i,j\le L}|\mathbf Q_{i,j}|)$
\end{IEEEproof}
\section*{Appendix G\\
for calculating $x = \mathrm{proj}_0(\cdot)$}
Let $$x_k^{(t+1)}=|x_k^{(t+1)}|e^{j \,\mathrm{arg}(\mathbf P_x(\mathbf z^{(t)}))}$$
\begin{itemize}
\item if $m p_c^2 \leq p_e$,
$$\left|x_{k}^{(t+1)}\right|=\left\{\begin{array}{ll}{p_c} & {k=1, \cdots, m} \\ {\sqrt{\frac{p_e-m p_c^{2}}{p_e-m}}} & {k=m+1, \cdots, L}\end{array}\right.$$
\item else if $m p_c^2 > p_e$
$$ \left|x_{k}^{(t+1)}\right|=\min \left\{\delta\left|v_{k}^{(t)}\right|, \gamma\right\}$$
where $\delta \in\left[0, \frac{\gamma}{\min \left\{\left|v_{k}^{(t)}\right|,\left|v_{k}^{(t)}\right| \neq 0\right\}}\right]$ can be obtained by solving the following equation by bisection method.
\begin{eqnarray}
\sum_{n=1}^{N} \min \left\{\delta^{2}\left|v_{k}^{(t)}\right|^{2}, \gamma^{2}\right\}=p_e.
\end{eqnarray}
\end{itemize}
\renewcommand{\refname}{References}
|
1,108,101,563,283 | arxiv | \section{Introduction}
In the nearby Universe, compact dwarf galaxies are a rare class of objects. The majority of low-mass early-type galaxies (preferentially called dwarf ellipticals, dEs) do not follow the near-linear Log(size)-magnitude distribution defined by giant ellipticals but have nearly constant sizes over large luminosity ranges \citep{Janz08,Kormendy09}. A continuous relation joining bright and faint galaxies exists, but this is more complex than a simple power law \citep{Graham03,Graham13,Ferrarese06,Janz08}. There exist a very compact, high-surface brightness and non-star-forming elliptical galaxy \citep[cE,][]{Chilingarian09,Faber73}, which seems fall on the extension of the relation defined by massive ellipticals (Es) alone \citep{Kormendy09,Misgeld11,Paudel14}.
M32 and VCC1297 are classical examples of a cE in the nearby Universe \citep{Faber73}. These compact objects were thought to be extremely rare. For example, in the Virgo cluster there are only a handful of possible cE candidates while more then 500 dEs are confirmed members by redshift measurements. An increasing number of cE populations located in different environments have been reported in recent works \citep{Chilingarian09,Chilingarian15,Huxor11,Huxor13,Mieske05,Paudel14,Price09}. Most of them, however, are found in dense environments, particularly around massive galaxies. Such proximity to a massive host favors a scenario in which they arise through the tidal stripping of larger galaxies, where the inner bulge component becomes a naked compact galaxy after losing its outer disk component during the interaction with its massive host \citep{Bekki01,Choi02}. Indeed, the discovery of cEs with tidal debris near a massive host confirms this picture \citep{Huxor11}.
\begin{figure}
\includegraphics[width=8cm]{sdss.pdf}
\caption{Optical outlook of cE\_AGN within a field of view 50"$\times$50" centered at RA = 133.62990 and Dec = 17.62513. The SDSS g-r-i color composite
and CFHT $i-$band images are shown in the top and bottom panel, respectively. }
\label{gpic}
\end{figure}
\begin{table*}
\caption{ Global properties of SDSS J085431.18+173730.5 }
\begin{tabular}{lccccccccccc}
\hline
Galaxy & M$_{k}$ & M$_{r}$ & R$_{e}$&n & $<$$\mu_{r}$$>$& $g-r$ & z & Age & log(Z/Z${\sun}$) & M$_{\bullet}$ & M$_{*}$\\
& mag & mag & pc & & mag/arcsec$^{2}$ & mag & & Gyr & dex & log(M$_{\sun}$)& log(M$_{\sun}$)\\
\hline
cE\_AGN & $-$20.03 & $-$18.08& 490 &1.4 & 19.36 & 0.83 & 0.0140 & 9.95$\pm$0.4 &$-$0.12$\pm$0.01 & 6.3 & 9.2 \\
CGCG 036-042 (P14) & $-$20.78 & $-$18.21 &559 & 1.7 &19.64 & 0.77 & 0.0068 & 7.15$\pm$1.2 & $-$0.18$\pm$0.07 & & 9.5\\
\hline
\end{tabular}
\\
\\
The global photometric parameters - total brightness and half light radius-- are derived using Petrosian photometry where the galactic extinction is corrected using \cite{Schlafly11}. We analyzed the SDSS-optical spectroscopy to derive stellar population parameters and BH mass, see the text.
\label{htb}
\end{table*}
Galaxies follow several scaling relations between their total mass and structural, kinematic and stellar population properties. It has been argued, with observational support, that supermassive black holes (SMBHs) are present at the center of galaxies, and a correlation between the masses of SMBHs and host galaxies bulge masses exists. However, there has been a debate on the universality of this relation at the low mass end. It is also not conclusive whether a majority of dwarf galaxies possess BHs at their centers, in addition to that most dwarf galaxies are bulge-less (or pseudo-bulge) galaxies. Instead, nuclear star clusters (NSCs) are common in low mass galaxies galaxies \citep{Cote06,Georgiev14,Graham03}. \cite{Ferrarese06} show that the scaling relation between BH mass and bulge mass can be extended to the low-mass regime when taking the NSC mass and total host galaxy mass instead, but see also \cite{Graham14}.
\cite{Paudel14}, hereafter P14, report a cE galaxy located in an isolated environment that possesses an extended low-surface brightness feature; a likely sign of tidal interaction. In this work, we report the discovery of another cE type galaxy, SDSS J085431.18+173730.5, located in a nearly-isolated environment that possesses a prominent broad-line emission in the optical spectrum; a likely sign of nuclear accretion by the central BH. In this work we assume a cosmology with H$_{0}$ = 71 km s$^{-1}$ Mpc$^{-1}$, $ \Omega_{M}$ = 0.3, and $\Omega_{\Lambda}$ = 0.7.
\section{The data analysis}
SDSS J085431.18+173730.5 (hereafter cE\_AGN) has been serendipitously discovered in SDSS-DR12 \citep{Alam15} in the course of searching for low-luminosity early-type galaxies in isolated or low density environments. It is located at z = 0.014 with no significantly massive neighbor. Based on NED query, the nearest companion is a star-forming dwarf galaxy (M$_{r}$ = -17.33 mag) which is located at a sky-projected distance of 279 kpc in south-east direction and the relative line-of-sight velocity between cE\_AGN and its companion is 194 km/s whereas ARK 186 is the nearest giant (M$_{r}$ = -19.82 mag) neighbor galaxy. The sky-projected physical distance between ARK 186 and cE\_AGN is 639 kpc and the relative line-of-sight velocity between them is 38 km/s. This part of the sky, within the redshift range 0.011 to 0.019, is sparsely populated by galaxies where we find only 13 galaxies within the radius of one degree (corresponding to a physical radius of 1.01 Mpc) centered on ARK 186. The NGC 2672 group, with z = 0.014487, at a sky-projected distance of $\sim$2Mpc from cE\_AGN is the nearest large concentration of galaxies.
\begin{figure}
\includegraphics[width=8.5cm]{model1.pdf}
\caption{The unsharp-masked and GALFIT residual images are shown in the left and right top panel, respectively. IRAF $ellipse$ output parameters -- azimuthally averaged surface brightness, position angle and ellipticity -- are shown in the bottom figure where the observed major axis light profile (black dots) is fitted with a double S\'ersic function. The solid black line represents the best fit combined S\'ersic function. }
\label{galfit}
\end{figure}
\begin{figure*}
\includegraphics[width=16cm]{Model_fits1.pdf}
\caption{ The SDSS optical spectrum of cE\_AGN (blue), together with its best-fitting SSP model spectrum (red). The fit residuals are shown in the lower panel. The fit is generally consistent within 5 per cent of the observed flux. In the inset, we show modeling of the broad emission lines with multi component gaussian profiles, see text for more detail.}
\label{sspfits}
\end{figure*}
\subsection{Imaging}
We primarily made use of the SDSS-III \citep{Ahn12} imaging data and catalog to explore the parameter space in searching for compact objects that we are interested in. We present the an optical images of cE\_AGN in Figure \ref{gpic}, a $g-r-i$ color composite taken from the SDSS skyserver at the top and a CFHT $i-$band image at the bottom. We find that the SDSS image quality is rather poor with an average seeing in $r-$band of $\sim$1{\arcsec}. The SDSS measured half-light radius of cE\_AGN is 1.98{\arcsec}. Therefore, to avoid any uncertainty in the derived structural parameters due to the resolution limit, we searched for higher quality images in other publicly available archival imaging databases. We found higher quality imaging data in the Canada France Hawaii Telescope (CFHT\footnote{http://www.cfht.hawaii.edu}) archive, with an average seeing of 0.6", which was observed in the $i-$band.
We acquired the pre-processed, level 2 calibration, fits files which include dark current subtraction, flat normalization and the magnitude zero-point supplemented in the header. The observations were made as a part of the CFHT Very Wide survey. With the seeing of 0.6{\arcsec} and a pixel scale of 0.18{\arcsec}/pix, CFHT data provides significantly higher image quality compared to the SDSS observations. However, note that the CFHT MegaCam images are known to have severe issues with sky background subtraction due to the mosaic pattern of multiple CCD chips. Thanks to its compactness, our object extends entirely in one chip, thus sky-background subtraction becomes straightforward.
As we can see in Figure \ref{gpic}, a background (z = 0.105) galaxy is located next to cE\_AGN. We removed the background contamination by subtracting its light from the fits image using the IRAF\footnote{http://iraf.noao.edu} $ellipse$ and $bmodel$ tasks. An unsharp masking technique has been applied to examine the presence of any substructural features such as spiral arm/bar or inner disk in cE\_AGN. Finally, we used {\sc galfit} \citep{Peng02} to carry out a 2D multicomponent structural analysis where we modeled the light distribution with several different possible combinations, i.e one with a simple S\'ersic function and the other with a double S\'ersic function with or without a central gaussian point source. For the double S\'ersic function, we fixed the outer component to be an exponential, i.e., $n = 1$. A PSF image, required by {\sc galfit}, was prepared by stacking the stars found in the same field of view. Foreground and background sources were masked by providing a mask image that we prepared manually.
With visual inspection of these residual images, we find that the use of a double S\'ersic function provides a better fit, see top right panel of Figure \ref{galfit}. The inner component has a S\'ersic index n = 1.4 and an effective radius of 1.58{\arcsec}. The outer component has an effective radius of 4.9{\arcsec}. The flux ratio between the inner and outer components, i.e., bulge to disk ratio (B/T), is 2.36.
Within the spatial resolution limit of the CFHT image, the presence of the central nucleus is not obvious, however, the addition of a central gaussian component in the {\sc galfit} modeling slightly improves the fit in the inner region. The gaussian component light contribution is less than 1/50 of the bulge. A careful inspection of the unsharp-masked image and the model-subtracted residual images seems indicate the presence of an inner bar/disk-like structure. The ellipticity profile along the major axis shows a mild change and the maximum ellipticity coincides with the maximum change in position angle.
The global photometric parameters (e.g., half-light radius and total luminosity) are derived using a non-parametric approach. We follow a similar procedure as described in \cite{Janz08} where the fluxes are summed up within a Petrosian aperture. The K-band luminosity is derived from the 2MASS
\footnote{http://irsa.ipac.caltech.edu} archival images. The bulge luminosity is derived using the inner and outer component flux ratio. Since the detailed decomposition has been done in CFHT $i-$band image, here we assume that the stellar population in cE\_AGN is homogenous. The results of photometry and structure modeling are listed in Table \ref{htb}.
\begin{figure*}
\includegraphics[width=18cm]{szm_u3.pdf}
\caption{Left: Relation between Log(Re) and M$_{r}$. The plot is reproduced from P14, see their Figure 4. The additional point, cE\_AGN, is shown in red. Here, we used the global photometric parameters, i.e. half light radius and total brightness. Right: Relations between BH mass and host galaxy bulge properties where \cite{Kormendy13} and \cite{Graham07} samples are used to show the correlation between BH mass and bulge K-band magnitude and BH mass and bulge S\'ersic index, respectively. }
\label{scl}
\end{figure*}
\subsection{Spectroscopy}
The SDSS spectroscopic observation of cE\_AGN was done as part of the BOSS\footnote{https://www.sdss3.org/surveys/boss.php} survey. The redshift measured by the SDSS pipeline is z = 0.0140. We exploit the optical spectrum of cE\_AGN for twofold. First, we aim to derive the properties of the underlying old stellar population using the information that is available in absorption features. Second, we measure the mass of the central BH by analyzing the stellar continuum subtracted emission lines. For this, we first fitted the observed galaxy spectrum in a wavelength range from 4000 to 7000 \AA{} with the Simple Stellar Population (SSP) models from \cite{Vazdekis10}. In doing so, prominent emission line regions were masked. For this purpose, we used a publicly available full spectrum fitting tool ULySS\footnote{http://ulyss.univ-lyon1.fr} \citep{Koleva09}. An additive continuum of third order was added to account for a non-thermal continuum while matching the galaxy and SSP spectra. The quality of fit is shown in the lower panel Figure \ref{sspfits}, where the observed spectrum matches within 5\% of the flux; excluding the masked emission-line regions. The best fit SSP parameters are Age = 9.95 Gyr and log(Z/Z$_{\sun}$) = $-$0.12 dex.
Finally, we reanalyzed the residual spectrum that mainly contains emission lines (Figure \ref{sspfits} lower panel). The emission lines were de-blended into a narrow and a broad component by fitting multi-Gaussian models. A single Gaussian was used to model the narrow component of H$\alpha$, assuming the narrow component has the same profile and redshift as the [NII] doublet as shown in the inset of Figure \ref{sspfits}. However, for the [O III] doublet ($\lambda$ 4959 and $\lambda$ 5007) and the narrow component of H$\beta$, we adopt double Gaussians to account for the blue wing \citep{Greene05}. For the broad components we use multiple Gaussians, which yields FWHMs of 1550, 2400 km/s for the broad H$\beta$ and H$\alpha$ components, respectively. Finally we measured H$\alpha$ and H$\beta$ narrow and broad emission fluxes from the best-fit narrow and broad component lines, respectively.
Having carefully measured the H$\alpha$ and H$\beta$ flux from stellar continuum subtracted spectrum, the central black hole mass is measured by using the calibration provided by Eq. 4 in \cite{Ho15} where we replace H$\beta$ FWHM with H$\alpha$ FWHM and L5100 is converted from the L(H$\alpha$), using the conversion factor in \cite{Greene05}. The different normalization factors $f$ are assumed for classical bulge and pseudo bulge galaxies. With a S\'ersic index $n = 1.4$, cE AGN may be neither a de Vaucouleurs bulge ($n = 4$) nor an exponential disk ($n = 1$). Therefore, we used the scaling factor that was derived from combining both types of galaxies. The derived BH mass is $2.1 \times 10^6 M_{\sun}$. The uncertainty for the BH mass measurement is approximately 0.4 dex which is mainly due to the uncertainty in the scaling factor.
\section{Scaling Relations}
The distribution of sizes and luminosities for early-type galaxies is shown in the left panel of Figure \ref{scl} left. This plot is reproduced from \cite{Paudel14} where the data points were compiled from \citet[][J08]{Janz08} for dEs, Es and S0s, and \citet[][C09]{Chilingarian09} for cEs. The solid line is a power-law fit to the size and luminosity within the brightness range $M_{r} < -19.2$ mag. The dashed line is derived from the observed correlation between the central surface brightness and (global) S\'ersic index with luminosity, and this, not surprisingly, provides a common scaling relation for Es and dEs \citep{Graham03}, albeit showing significant systematic offsets at intermediate luminosities \citep{Janz08}. The two solid black circles are the isolated cEs previously identified in \cite{Paudel14}.
The red symbol is newly added from this work. We can clearly see that cE\_AGN overlaps with the previously identified cEs (black dots and solid gray squares). cE\_AGN falls significantly below the curved size-luminosity relation of Es/dEs, and rather continues the sequence of giant ellipticals shown by a solid black line. In other words, with the high mean surface brightness, $<\mu_{r}> $ = 19.36 mag/arcsec$^{2}$ for its total brightness of M$_{r}$ = -18.08 mag, it also follows the so-called Kormendy relation (magnitude vs surface brightness) of giant ellipticals \citep{Kormendy77}.
In the right panel of Figure \ref{scl}, we show the two scaling relations between the BH masses and the host galaxies properties. In the left, a well-known correlation between BH mass and K-band bulge luminosity of the host galaxy is shown \citep{Kormendy13}. In addition to this, a relation between BH mass and S\'ersic index, as proposed by \cite{Graham07}, is shown in the right panel. In both plots, cE\_AGN seems to roughly follow the observed trends, and interestingly, it almost overlaps with M32 which is shown by a large filled-gray square.
\section{Discussion and Conclusion}
We present a new compact early-type galaxy, cE\_AGN with optical signature of an accreting BH. Among the cE or dE class galaxies, cE\_AGN is the first which possesses the broad-line emission of width larger than 2000 km/s. Below, we discuss its distinction and the possible origin.
\subsection{Black holes in low mass early-type galaxies}
AGNs are not common in low-mass early-type galaxies (dE or cE classes). According to a common wisdom these galaxies are devoid of gas and have less chance of acquiring gas from external sources as massive galaxies do, thus there is no fuel left to trigger the nuclear activity. Nevertheless, the number of detections of BHs in low-mass galaxies has been increased in recent literature with the advent of large scale surveys, high precision instruments and the sophisticated modeling of the host galaxy and BH kinematics \citep[e.g.,][]{Brok15,Greene04,Seth14}.
\cite{Barth04} identified a dwarf Seyfert 1 AGN candidate, POX 52, at redshift z = 0.021. They classified the host galaxy morphology to be a dwarf elliptical and reported a BH mass of 1.5 $\times$ 10$^{5}$ M$_{\sun}$. In the last few years, number of searches have been devised to list dwarf galaxies with central massive BHs \citep{Greene04,Greene07,Moran14,Reines13}. In particular, the study of \cite{Reines13} presents the largest number of nearby dwarf galaxies which exhibit optical spectroscopic signatures of accreting BHs. Visually classifying these BH-host dwarf galaxies located within z $<$ 0.02\footnote{We have chosen this redshift cut because beyond this the visual classification of dwarf galaxies from the SDSS color possibly is not reliable.}, we find that 13 out of 33 can be classified as dE or dS0. Only one, J122342.82+581446.4, has a broad H$\alpha$ emission similar to that of cE\_AGN. \cite{Reines13} estimate its BH mass $\sim$10$^{6}$ M$_{\sun}$.
\subsection{Origin of cEs}
The majority of cEs are found in the vicinity of giant galaxies, e.g., M32, and many of them possess morphological features which might have originated from tidal stripping activity \citep{Huxor11}. With the discovery of an isolated cE by \cite{Huxor13}, an alternative scenario, other than stripping, has gained much attention in the recent literature \citep{Dierickx14,Paudel14}. cE\_AGN is also located away from the nearest massive galaxy. Recently, \cite{Chilingarian15} proposed a fly-by scenario where those isolated compact ellipticals may be tidally-stripped systems that ran away from their hosts.
The discovery of a cE with AGN signature might help us to explore another aspects of compact early-type galaxies. Bulge-dominated early-type galaxies follow a scaling relation between bulge mass and BH mass. Measuring the BH mass allows us to locate the position of cEs in this relation. We find nothing uncommon, as cE\_AGN is placed near M32, following the overall relation between BH mass and bulge mass (see Figure \ref{scl} right). M32 is a Local Group cE located at the vicinity of M31 which has been an iconoclast example of cE type galaxy \citep{Faber73}. Based on X-ray and radio observations, presence of the nuclear activity has also been reported in M32 \citep{Ho03}. It has been long argued that M32 is a stripped bulge of a disk galaxy \citep{Choi02,Faber73,Graham02} -- but also see \cite{Dierickx14}. As a hypothesis proposed by \cite{Chilingarian15}, the possibility that cE\_AGN might also be a tidally stripped bulge that ran away from its hosts, perhaps from NGC 2672 group, can not be ruled out. NGC 2672 is a relatively massive galaxy group with a viral mass of $\sim$10$^{13}$M$\sun$ \citep{Ramella02}. It is expected that the member galaxies could be spread out as far as a couple of Mpc in projected distance \citep{Sales07}. However, given the large physical separation between cE\_AGN and NGC 2672, the stripping event might have happened either very early in time or at fast speed with a very eccentric orbit. The small relative velocity to NGC 2672 might be explained by an orbit in the plane of the sky and cE\_AGN being close to apogalacticon.
Other hand, It is remarkable to see that while following the extension of mass--size relation of massive early-type galaxies, cE\_AGN also seems to follow the observed empirical relation between the bulge S\'ersic index and the BH mass suggested by \cite{Graham07}. Interestingly, a detailed decomposition of the light profile by \cite{Graham02} reveals that M32's light profile resembles an almost perfect bulge - exponential disk system where the bulge component has a S\'ersic index of $n = 1.5$, which is in agreement with the relation between the S\'ersic
index and the BH mass. Considering the location in the scaling relations, we suggest cE\_AGN is a low luminosity extension of Es. It is, though, not sufficient to rule out the stripping scenario and to support the alternative scenario which assumes a similar evolutionary path as Es, formed possibly via mergers \citep[e.g.,][]{Naab07}. Furthermore, we expect that a detailed study of the underlying stellar population and central kinematics from high quality 2-dimensional spectroscopy may allow us to test the above conclusions which we plan to do for the coming observing semester.
\acknowledgements
We thank Alister Graham for fruitful discussions and comments on the draft version of this paper. We also thank the referee, Igor Chilingarian, for helpful comments that improved the paper. This study is based on the archival images and spectra from the Sloan Digital Sky Survey (the full acknowledgment can be found at http://www.sdss.org/collaboration/credits.html). We also made use of archival data from Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institute National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France and the University of Hawaii.
|
1,108,101,563,284 | arxiv | \section{Introduction}
Exclusive processes mediated by the weak force are
an area of investigation which will gather, we believe, a wide interest
in the forthcoming years due to the various experimental proposals to detect
neutrino oscillations at intermediate energy using neutrino factories and superbeams,
\cite{Schaefer1}.
Also, neutrino scattering on nucleons in the deeply virtual kinematics
is important in order to draw a better picture of the hadronic interactions mediated by the weak currents.
This is one of the simplest processes, among all, described by the Generalized Parton
distribution functions (GPDFs) which unify the parton model description of both inclusive ed exclusive
processes \cite{Ji,Radyushkin}. Exclusive processes in the region of small momentum transfer
(few GeVs) are well described by hadronic wave functions - also called "distribution amplitudes" -
using the ERBL formalism of Efremov-Radyushkin-Brodsky-Lepage (see \cite{CorianoLi} \cite{Sterman}
for an overview),
while inclusive processes at high energy are described by ordinary parton distributions.
GPDs are the appropriate tool to describe the so called ``Generalized Bjorken region''
which is the region where, beside Bjorken's x variable, additional scaling variables appear.
Nonforward parton distributions are the simplest constructs belonging to this class
and describe true light-cone correlators in the nucleon state, not built by unitarity as
ordinary parton distributions, since the optical theorem does not hold. An asymmetry scaling parameter,
in this case, $\xi$, defined below, characterizes the longitudinal momentum exchange
between the initial and final momenta of the nucleon. The study of the electromagnetic interaction
in this kinematical region, the Deeply Virtual Compton Scattering (DVCS) region, by now amounts to
several hundreds of papers written in the last few years. Our objective, in this contribution, is to
briefly outline the extension to the electroweak sector of these findings.
\section{DVNS kinematics}
A pictorial description of the process we are going to illustrate
is given in Fig~\ref{DVCS_1.eps} where a neutrino of momentum $l$
scatters off a nucleon of momentum $P_1$ by an interaction with a neutral current;
from the final state a photon and a nucleon emerge, of momenta $q_2$ and $P_2$ respectively,
while the momenta of the final lepton is $l'$
\begin{figure}[t]
{\par\centering \resizebox*{8cm}{!}{\includegraphics{DVCS_1.eps}} \par}
\caption{Leading hand-bag diagrams for the process}
\label{DVCS_1.eps}
\end{figure}
\begin{equation}
P_{1,2}= \bar{P} \pm \frac{\Delta}{2}\,\,\,\,\,\,\,\, q_{1,2}= \bar{q} \mp \frac{\Delta}{2},
\end{equation}
with $-\Delta=P_2-P_1$ being the momentum transfer. Clearly
\begin{equation}
\bar{P}\cdot \Delta=0,\,\,\,\,\, t=\Delta^2 \,\,\,\,\,\,\, \bar{P}^2=M^2 - \frac{t}{4}
\end{equation}
and $M$ is the nucleon mass. There are two scaling variables which are identified in the process,
since 3 scalar products can grow large in the generalized Bjorken limit:
$\bar{q}^2$, $\Delta\cdot q$, $\bar{P}\cdot \bar{q}$.
The momentum transfer $t=\Delta^2$ is a small parameter in the process.
Momentum asymmetries between the initial and the final state nucleon are measured
by two scaling parameters, $\xi$ and $\eta$, related to ratios of the former invariants
\begin{equation}
\xi=-{\bar{q}^2\over 2 \bar{P}\cdot {\bar q}} \,\,\,\,\,\,\,\,\,\,\,\,
\eta={\Delta\cdot \bar{q}\over 2 \bar{P}\cdot \bar{q}},
\end{equation}
where $\xi$ is a variable of Bjorken type, expressed in terms of average
momenta rather than nucleon and Z-boson momenta.
The standard Bjorken variable $x= - q_1^2/( 2 P_1\cdot q_1)$ is trivially
related to $\xi$ in the $t=0$ limit.
In the DIS limit $(P_1=P_2)$ $\eta=0$ and $x=\xi$, while in the DVCS limit
$\eta=\xi$ and $x=2\xi/(1 +\xi)$, as one can easily deduce from the relations
\begin{equation}
q_1^2=\left(1 +\frac{\eta}{\xi}\right) \bar{q}^2 +\frac{t}{4},\,\,\,
q_2^2=\left(1 -\frac{\eta}{\xi}\right) \bar{q}^2 +\frac{t}{4}.
\end{equation}
We introduce also the inelasticity parameter $y=P_1\cdot l/(P_1\cdot q_1)$
which measures the fraction of the total energy that is transferred to the final state photon.
\section{The Amplitudes}
The Compton amplitude is described by the following correlator
involving neutral and charged currents
\begin{small}
\begin{eqnarray}
T_{\mu \nu}=i\int d^{4}x e^{i qx} \langle P_2|T\left(J_{\nu}^{\gamma}(x/2)
J_{\mu}^{W^{\pm},Z_{0}}(-x/2)\right)|P_1\rangle.
\end{eqnarray}
\end{small}
The parameterizations of the non-forward light cone correlators in terms of
GPD's is of the form given by Ji at leading twist \cite{Ji}
\begin{small}
\begin{eqnarray}
&&\int\frac{d\lambda }{(2\pi)}e^{i\lambda z}\langle P'|\overline {\psi}
\left(-\frac{\lambda n}{2}\right)\gamma^\mu \psi\left(\frac{\lambda n}{2}\right)|P\rangle=
\nonumber\\\nonumber\\
&&H(z,\xi,\Delta^2)\overline{U}(P')\gamma^\mu U(P)\nonumber\\
&&\hspace{1cm}+ E(z,\xi,\Delta^2)\overline{U}(P')\frac{i\sigma^{\mu\nu}
\Delta_{\nu}}{2M} U(P) + .....\nonumber\\
&&\int\frac{d\lambda }{(2\pi)}e^{i\lambda z}\langle P'|\overline {\psi}
\left(-\frac{\lambda n}{2}\right)\gamma^\mu \gamma^5 \psi\left(\frac{\lambda n}{2}\right)|P\rangle=
\nonumber\\\nonumber\\
&&\tilde{H}(z,\xi,\Delta^2)\overline{U}(P')\gamma^\mu\gamma^5 U(P)\nonumber\\
&&\hspace{1cm}+ \tilde{E}(z,\xi,\Delta^2)\overline{U}(P')\frac{\gamma^5 \Delta^{\mu}}{2M}U(P) + .....
\end{eqnarray}
\end{small}
which have been expanded in terms of functions $H,E, \tilde{H},
\tilde{E}$ which are the GPDs and the ellipses are meant to denote
the higher-twist contributions. It is interesting to observe that the amplitude
is still described by the same light-cone correlators as in the electromagnetic
case (vector, axial vector) but now parity is not conserved.
In our calculations we use a simplified model for the GPD's
where the $\Delta^2$ dependence can be factorized as follows \cite{Schaefer2}
for a generic GPD
\begin{equation}
H^{i}(z,\xi,\Delta^2,Q^2)=F^{i}_{1}\,(\Delta^2)q^{i}(z,\xi,Q^2),
\end{equation}
where $F^{i}_{1}\,(\Delta^2)$ is a form factor, and the construction of the input distributions,
in correspondence of an input scale $Q_0$, is performed using
\begin{eqnarray}
&&\hspace{-0.5cm}q(z,\xi,Q^2)=\int_{-1}^{1}dx'\int_{-1+|x'|}^{1-|x'|}dy'\delta(x'+\xi y'-z)\nonumber\\
&&\hspace{0.5cm}\times f(y',x',Q^2).
\label{reduction}
\end{eqnarray}
As an example we show a plot of some GPDs in Fig~(\ref{Hud}), for the up and the down quarks as a function
of the variable $z$ (a generalization of Bjorken-x, with $-1 < z < 1$) and at a fixed value of the
variables $\Delta$ and $\xi$.
\begin{figure}[t]
{\par\centering \resizebox*{6cm}{!}{\rotatebox{-90}{\includegraphics{Hprova.ps}}} \par}
\caption{GPD's $H_u$ and $H_d$ generated by the diagonal parton distribution
at an initial $0.26$ GeV$^2$}
\label{Hud}
\end{figure}
\section{The cross sections}
One possible differential cross section to study, in analogy to the DVCS case, is the following
\begin{eqnarray}
&&\frac{d\sigma}{dx d Q^2 d|\Delta^2| d\phi_{r}}=
\frac{y}{Q^2}\frac{d\sigma}{dx dy d|\Delta^2| d\phi_{r}}=\nonumber\\
&&\frac{x y^2}{8\pi Q^4}
\left(1+ \frac{4M^2 x^2}{Q^2}\right)^{-\frac{1}{2}}|{\cal M}_{fi}|^2,
\end{eqnarray}
where $\phi_{r}$ is the angle between the lepton and the hadron
scattering planes. $|{\cal M}_{fi}|^2$, as a functions of the invariants of the
process, can be found in \cite{DVNS1}.
\begin{figure}
{\centering \resizebox*{7cm}{!}{\rotatebox{-90}{\includegraphics{set2.ps}}} \par}
\caption{DVCS cross section at $x=0.1$ and center of mass energy $M E=10$ GeV$^2$.}
\label{set2.ps}
\end{figure}
\section{Amplitudes in the charged sector}
We have also analized the expressions for the correlators of
the charged electroweak current and derived their expressions in terms of the GDPs \cite{DVNS2}.
These are formally given by the correlators
\begin{small}
\begin{eqnarray}
&&\hspace{-1cm}T^{W^{+}}_{\mu \nu}=i\int d^4 x \frac{e^{iqx}x^{\alpha}U_{u d}}{2\pi^2(x^2-i\varepsilon)^2}\langle P_2|
\left[i S_{\mu\alpha\nu\beta}\left(\tilde{O}_{u d}^{\beta}+O_{u d}^{5 \beta}\right)
\right.\nonumber\\
&&\hspace{2cm}\left.+\varepsilon_{\mu\alpha\nu\beta}\left(O_{u d}^{\beta}+\tilde{O}_{u d}^{5 \beta}\right)\right] |P_1\rangle,
\nonumber\\
&&\hspace{-1cm}T^{W^{-}}_{\mu \nu}=i\int d^4 x \frac{e^{iqx}x^{\alpha}U_{d u}}{2\pi^2(x^2-i\varepsilon)^2}\langle P_2|
\left[-i S_{\mu\alpha\nu\beta}\left(\tilde{O}_{d u}^{\beta}+O_{d u}^{5 \beta}\right)
\right.\nonumber\\
&&\hspace{2cm}\left.-\varepsilon_{\mu\alpha\nu\beta}\left(O_{d u}^{\beta}+\tilde{O}_{d u}^{5 \beta}\right)\right] |P_1\rangle\,,
\nonumber\\
\end{eqnarray}
\end{small}
where the partonic content is extracted from the $O_{d u}$ operators while $S_{\mu\alpha\nu\beta}$
denote suitable combinations of Dirac gamma matrices. We refer to \cite{DVNS2} for more details.
We have suppressed the $x$-dependence of the operators in the former equations.
The identification of the relevant partonic description in terms of GPDs has been obtained by us \cite{DVNS2}
using the leading-twist decomposition worked out in \cite{blumlein}. Related work, using a different
approach, can be found also in \cite{Psaker}.
\section{Conclusions}
We have presented an extension of the standard DVCS process to the case
of one neutral current exchange, describing the scattering of a neutrino
off a proton in the parton model.
We have described the leading twist behaviour of the cross section;
we have found that this is comparable to other typical
neutrino cross sections.
The process is the natural generalization of DIS with neutral currents and relies on
the notion of Generalized Parton Distributions, new constructs in the parton model
which have received considerable attention in recent years.
From the theoretical and experimental viewpoints
the study of these processes is of interest,
since very little is known of the neutrino interaction at intermediate energy
in these more complex kinematical domains.
|
1,108,101,563,285 | arxiv | \section{Introduction}
In \cite{MUY}, the authors constructed a complete bounded
minimal immersion of the unit disk $\mathbb{D}$ into $\mathbb{R}^3$ whose conjugate
is also bounded.
As applications of our results, we show in this paper that the technique
developed by F. J. L\'opez in \cite{Lopez} can be suitably modified
to give the following three things:
\begin{itemize}
\item The first examples of complete bounded complex submanifolds
with arbitrary genus immersed in $\mathbb{C}^2$;
\item A new and simple
method to construct complete, bounded minimal surfaces
with arbitrary genus in the Euclidean 3-space;
\item A method to construct weakly complete, bounded maximal surfaces
with arbitrary genus in the Lorentz-Minkowski 3-spacetime.
\end{itemize}
Actually, a complete and bounded conformal minimal immersion of $\mathbb{D}$ into
$\mathbb{R}^3$ whose conjugate is also bounded is the real part of a complete and
bounded null holomorphic immersion
\[
X_0:\mathbb{D}\longrightarrow \mathbb{C}^3.
\]
As it was shown in \cite{MUY}, if we consider the projection
\begin{equation}\label{eq:proj2}
\pi:\mathbb{C}^3\ni (z_1,z_2,z_3)\longmapsto (z_1,z_2)\in \mathbb{C}^2,
\end{equation}
the map $\pi\circ X_0:\mathbb{D}\to \mathbb{C}^2$ gives a complete bounded
complex submanifold immersed in $\mathbb{C}^2$.
By a perturbation of $X_0$, considering a similar deformation like
in \cite{Lopez}, we shall prove the following
\begin{introtheorem}\label{thm:main}
Let $M$ be a closed Riemann surface.
For an arbitrary positive integer $N$,
there exist points $Q_0$, $Q_1$, \dots, $Q_e$ $(e\ge N)$ on $M$,
neighborhoods $U_{l}$ of $Q_{l}$ which are homeomorphic to
the unit disc $\mathbb{D}$, and a holomorphic immersion
\[
X:M\setminus \bigcup_{l=0}^e \overline{U_{l}}
\longrightarrow \mathbb{C}^2
\]
which is complete and bounded.
\end{introtheorem}
If we do not specify the choice of $M$ with the fixed arbitrary genus,
we can choose $M$ so that the number of ends is at most $3$
(see \cite[Lemma 3]{Lopez}).
Among other things, L\'opez in \cite{Lopez} gave a method to
construct complete minimal surfaces between two
parallel planes of arbitrary genus. He constructed these examples
as a perturbation of the minimal disk given by Jorge and Xavier
in \cite{jorge-xavier}.
However, the global boundedness was not treated in \cite{Lopez}, because
L\'opez' technique is not of use when you apply it directly to
Nadirashvili's minimal disk \cite{nadi}.
So, the aim of this paper is to introduce some simple but
appropriate changes in L\'opez'
machinery in order to obtain complete bounded minimal surfaces with
nontrivial topology.
Also the deformation parameter given in \cite{Lopez} is not sufficient to kill the
periods of the holomorphic immersions in $\mathbb{C}^2$, and so we must
consider another deformation space and calculate the new periods.
On the other hand, the proof of the above theorem allow us to conclude
that
\begin{introcorollary}\label{cor:cor}
Let $M$ be a closed Riemann surface.
For an arbitrary positive integer $N$,
there exist points $Q_0$,\dots,$Q_e$ $(e\ge N)$ on $M$,
neighborhoods $U_l$ of $Q_l$ which are homeomorphic to
the unit disc $\mathbb{D}$, and a complete conformal
minimal immersion into the Euclidean $3$-space $\mathbb{R}^3$
{\rm (}resp.\
a weakly complete maximal surface in the sense of \cite{uy3},
which may have singular points,
whose first fundamental form is conformal
on the regular set,
in the Lorentz-Minkowski $3$-spacetime $\L^3${\rm)}
\[
x:M\setminus \bigcup_{l=0}^e \overline{U_l}\longrightarrow
\mathbb{R}^3\quad(\text{resp.\ $\L^3$})
\]
whose image is bounded.
\end{introcorollary}
Though general methods to produce bounded complete minimal
immersions into $\mathbb{R}^3$
with higher genus are already known
(cf.\ \cite{AFM}, \cite{FMM}, \cite{LMM-Handles} and \cite{LMM-Nonorientable} ),
this corollary gives a new and short way to construct such examples.
On the other hand, for a maximal surface in $\L^3$,
this corollary produces the first examples
of weakly complete bounded maximal surfaces with arbitrary genus.
The {\em weak completeness\/} of maximal surfaces, which may admit
certain kind of singularities was defined in \cite{uy3}.
In \cite{alarcon} a weakly complete disk satisfying
a certain kind of boundedness was given.
After that, a bounded example of genus zero with one end was
shown in \cite{MUY}.
Finally, we remark that the technique in this paper does not
produce bounded null holomorphic curves immersed in $\mathbb{C}^3$
with arbitrary genus because of the difficulty in
obtaining the required
deformation parameters.
If we do succeed to find an example, the above three examples could be all
realized as projections of it.
So it is still an open problem to show the existence of complete bounded
null holomorphic immersion into $\mathbb{C}^3$ with arbitrary genus.
\section{Preliminaries}\label{sec:prelim}
Let $M$ be a compact Riemann surface of genus $\kappa$,
and fix a point $Q_0\in M$.
Let
\begin{equation}\label{eq:gap}
1=d_1 < d_2 < \dots < d_{\kappa} \leq 2\kappa-1
\end{equation}
be the {\em Weierstrass gap series\/} at $Q_0$, that is,
there exists a meromorphic function on $M$ which is holomorphic
on $M\setminus\{Q_0\}$ such that $Q_0$ is the pole of order $d$
if and only if $d\not\in \{d_1,\dots,d_{\kappa}\}$.
Then there exists a meromorphic function $f$ on $M$ which is
holomorphic on $M\setminus\{Q_0\}$ and $Q_0$ is the pole of order
$m_0$ ($> d_\kappa$).
Let $\{Q_1,\dots,Q_e\}(\subset M)$ be the set of branch points of $f$.
Then the divisor $(df)$ of $df$ is written as
\begin{equation}\label{eq:df}
(df) = \frac{
\prod_{l=1}^e Q_l^{m_l}
}{Q_0^{m_0+1}}\qquad (m_0>d_{\kappa}),
\end{equation}
where $m_l$ ($l=1,\dots,e$) are positive integers, and the divisor
is expressed by the multiplication of these branch points.
\begin{remark}\label{rmk:N}
In the latter construction of surfaces in Theorem~\ref{thm:main}
and Corollary~\ref{cor:cor},
the ends of surfaces in $\mathbb{C}^2$ or $\mathbb{R}^3$ correspond to
the points $Q_0$, \dots, $Q_e$.
We set
\[
f_{j+1}:=(f_j-c_j)^2,\quad f_0=f \qquad (j=1,2,3,\dots)
\]
inductively, where the $c_j$'s are complex numbers.
Then, by replacing $f$ by
$f_1$, $f_2$, $f_3$,\dots, we can make an example
such that $e$ is greater than the given number $N$.
\end{remark}
We write
\begin{equation}\label{eq:M0}
M_0 := M \setminus \{Q_0,\dots,Q_e\}.
\end{equation}
Denoting by $H^1_{\op{hol}}(M)$ and $H^1_{\op{hol}}(M_0)$ the
(first) holomorphic de Rham
cohomology group of $M$ and $M_0$
(as the vector space of holomorphic
differentials on $M$ factored by the subspace
of exact holomorphic differentials), respectively, then
\[
\dim H^1_{\op{hol}}(M) = \kappa,\qquad
\dim H^1_{\op{hol}}(M_0) = n
\]
hold, where we set, for the sake of simplicity,
\begin{equation}\label{eq:n}
n:=2\kappa + e.
\end{equation}
Take a basis $\{\zeta_1,\dots,\zeta_{\kappa}\}$ of $H^1_{\op{hol}} (M)$.
\begin{lemma}[cf.\ III.5.13 in \cite{Kra}]
\label{lem:basis}
One can take a basis of $H^1_{\op{hol}}(M_0)$
\[
\{\zeta_1,\dots,\zeta_{\kappa};\xi_1,\dots,\xi_{\kappa};
\eta_1,\dots,\eta_e\},
\]
where
$\xi_j$ is a meromorphic $1$-form on $M$ which is holomorphic on
$M\setminus\{Q_0\}$,
and $Q_0$ is a pole of order $m_j+1$,
and
$\eta_l$ is a meromorphic $1$-form on $M$ which
is holomorphic on
$M\setminus\{Q_0,Q_l\}$,
and $Q_0$ and $Q_l$ are poles of order $1$.
\end{lemma}
For simplicity, we set
\begin{equation}\label{eq:zeta-gen}
\begin{aligned}
\zeta_{\kappa+j} &:= \xi_j\qquad &(j=1,\dots,\kappa), \\
\zeta_{2\kappa+l} &:= \eta_l \qquad &(l=1,\dots,e).
\end{aligned}
\end{equation}
Then $\{\zeta_1,\dots,\zeta_{n}\}$ ($n=2\kappa+e$)
is a basis of $H^1_{\op{hol}}(M_0)$.
\begin{lemma}[{\cite[Lemma 1]{Lopez}}]\label{lem:v}
There exists a meromorphic function $v$ on $M$ with the following
properties{\rm :}
\begin{enumerate}
\item $v$ is holomorphic on $M_0=M\setminus\{Q_0,\dots,Q_e\}$,
\item $Q_l$ $(l=1,\dots,e)$ is a pole of $v$ whose order
is greater than or equal to $m_l+2$, and
\item $Q_0$ is a pole of $v$ whose order is greater than $m_0$.
\end{enumerate}
\end{lemma}
\begin{proof}
For each $l=1,\dots,e$, let $v_l$ be a meromorphic function on $M$
which is holomorphic on $M\setminus\{Q_l\}$, and so that
$Q_l$ is a pole of order $\max\{m_l+2,2\kappa\}$.
On the other hand, take a meromorphic function $v_0$ on $M$
which is holomorphic on $M\setminus\{Q_0\}$ and
$Q_l$ is a pole of order $\max\{m_0+1,2\kappa\}$.
Then $v= v_0 + v_1 +\dots + v_e$
is a desired one.
\end{proof}
Using $f$ as in \eqref{eq:df} and $v$ as in Lemma~\ref{lem:v},
we define
\begin{equation}\label{eq:G-lambda}
\mathcal{G}_{\Lambda}:=
\lambda_0 v +
\frac1{df}
\sum_{j=1}^{n} \lambda_j \zeta_j
: M \longrightarrow \mathbb{C}\cup\{\infty\},
\end{equation}
where
\[
\Lambda:=(\lambda_0,\lambda_1,\dots,\lambda_{n})\in \mathbb{C}^{n+1}.
\]
\begin{lemma}[{\cite[Section 3]{Lopez}}]\label{lem:G-lambda}
The function $\mathcal{G}_{\Lambda}$ as in \eqref{eq:G-lambda} is a
meromorphic function on $M$ such that
\begin{enumerate}
\item\label{item:G:1}
$\mathcal{G}_{\Lambda}$ is holomorphic on
$M_0=M\setminus\{Q_0,\dots,Q_e\}$,
\item\label{item:G:2}
if $\Lambda\neq \vect{0}$, $\mathcal{G}_{\Lambda}$ is
nonconstant on $M$, and
\item\label{item:G:3}
if $\lambda_0\neq 0$, $\mathcal{G}_{\Lambda}$ has poles
at $Q_0,\dots,Q_e$.
\end{enumerate}
\end{lemma}
\begin{proof}
\ref{item:G:1} and \ref{item:G:2} are obvious.
\ref{item:G:3} follows from the fact that $v$ has
a pole of higher order than $\zeta_1/df$, \dots, $\zeta_{n}/df$
at $Q_j$ for each $j=0,\dots,e$.
\end{proof}
We write
\[
|\Lambda|=\sqrt{|\lambda_0|^2+|\lambda_1|^2+\dots+|\lambda_{n}|^2},
\]
and consider the unit sphere in the space of $\Lambda$:
\[
\S_1:=\{\Lambda\in \mathbb{C}^{n+1}\,;\, |\Lambda|=1\}.
\]
The following assertion is a modification of
\cite[Lemma 2]{Lopez}, which is much easier to prove.
For our purpose, this weaker assertion is sufficient.
\begin{proposition}\label{prop:inverse}
Let $\Lambda_0=(a_0,a_1,\dots,a_{n})$
be a point in $\S_1$ satisfying $a_0\ne 0$.
Then there exist $\varepsilon(>0)$ and a neighborhood $W$ of
$\Lambda_0$ in $\S_1$ such that if $0<|t|<\varepsilon$ and
$\Lambda\in W$,
the set $\mathcal{G}_{t\Lambda}^{-1}(\mathbb{D})$ is conformally equivalent
to a compact surface of genus $\kappa$ minus
$e+1$ pairwise disjoint discs with analytic regular boundaries.
In particular, there are no branch points of $\mathcal{G}_{t\Lambda}$ on
the boundary
$\partial\mathcal{G}^{-1}_{t\Lambda}(\mathbb{D})=\mathcal{G}^{-1}_{t\Lambda}(\partial \mathbb{D})$.
\end{proposition}
\begin{proof}
Since the poles of $\mathcal{G}_{\Lambda_0}$ are exactly
$Q_0$, \dots, $Q_e$ and
$\mathcal{G}_{t\Lambda}^{-1}(\mathbb{D})=\mathcal{G}_{\Lambda}^{-1}\bigl((1/t)\mathbb{D}\bigr)$,
for sufficiently small $t$, the inverse image $\mathcal{G}_{t\Lambda}^{-1}(\mathbb{D})$
of the unit disk $\mathbb{D}$ by $\mathcal{G}_{t\Lambda}$ is homeomorphic to
a compact surface of genus $\kappa$ minus
$e+1$ pairwise disjoint discs with piecewise analytic
boundaries.
Moreover, since the set of branch points of $\mathcal{G}_{\Lambda_0}$ does not
have any accumulation points, $\mathcal{G}_{t\Lambda_0}^{-1}(\partial \mathbb{D})$
has no branch points for sufficiently small $t$, and
$\mathcal{G}_{t\Lambda}^{-1}(\partial \mathbb{D})$ consists of real analytic regular
curves in $M$.
Furthermore, since $a_0\ne 0$, $\mathcal{G}_{\Lambda_0}/\mathcal{G}_{\Lambda}$
are holomorphic near $Q_0$, \dots, $Q_e$ for any $\Lambda$
which is sufficiently close to $\Lambda_0$.
Thus $\mathcal{G}_{t\Lambda}^{-1}(\mathbb{D})$
has the same properties as $\mathcal{G}_{t\Lambda_0}^{-1}(\mathbb{D})$.
\end{proof}
Under the situation in Proposition \ref{prop:inverse},
let $\{\gamma_1, \dots, \gamma_{2\kappa}\}$ be a family of loops on
$\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})$ which is a homology basis of $M$.
On the other hand,
take a loop $\gamma_{2\kappa+l}$ on $\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})$
for each $l=1,\dots,e$, surrounding a neighborhood of $Q_l$
(as in Proposition~\ref{prop:inverse}).
We define the {\em period matrix\/} as
\begin{equation}\label{eq:period}
P=(p_{kj}),\qquad
p_{kj}:=\int_{\gamma_k}\zeta_j,
\end{equation}
which is a nondegenerate $n\times n$ matrix.
\section{Proof of the main theorem}
In this section, we give a proof of Theorem~\ref{thm:main} in the
introduction.
\subsection*{The initial immersion}
Let $X_0\colon{}\mathbb{D}\to \mathbb{C}^3$ be a complete holomorphic null immersion
whose image is bounded in $\mathbb{C}^3$ (as in Theorem~A in \cite{MUY}),
where $\mathbb{D}\subset \mathbb{C}$ is the unit disk.
We write
\begin{equation}\label{eq:initial}
X_0(z) = \int_{0}^{z}
\bigl( \varphi_1(z),\varphi_2(z),\varphi_3(z) \bigr)\,dz,
\end{equation}
where $z$ is a canonical coordinate on $\mathbb{D}\subset \mathbb{C}$ and
$\varphi_j$ ($j=1,2,3$) are holomorphic functions on $\mathbb{D}$.
Since $X_0$ is null, it holds that
\begin{equation}\label{eq:null}
\bigl(\varphi_1\bigr)^2+
\bigl(\varphi_2\bigr)^2+
\bigl(\varphi_3\bigr)^2=0.
\end{equation}
Let $(g,\omega:=\omega_0\,dz)$ be the Weierstrass data of $X_0$,
that is,
\begin{equation}\label{eq:w-data}
\varphi_1 = (1-g^2)\omega_0,\quad
\varphi_2 = \mathrm{i}(1+g^2)\omega_0,\quad
\varphi_3 = 2 g \omega_0,
\end{equation}
where $\mathrm{i}=\sqrt{-1}$.
\newpage
\begin{lemma}\label{lem:phi}
Let $X_0\colon{}\mathbb{D}\to \mathbb{C}^3$ be a null holomorphic immersion
as in \eqref{eq:initial} whose image is not contained in any plane.
Then there exists a point $z_0\in \mathbb{D}$
and a complex orthogonal transformation $T:\mathbb{C}^3\to \mathbb{C}^3$
in $\O(3,\mathbb{C})=\{a\in \op{GL}(3,\mathbb{C})\,;\, \trans{a}=a^{-1}\}$
{\rm (}$\trans{a}$ is the transpose of $a${\rm)}
such that, up to replacing $X_0$ by $T\circ X_0$,
the following properties
hold{\rm :}
\begin{enumerate}
\item\label{item:phi:1}
$\varphi_1(z_0)=0$,
\item\label{item:phi:2}
$\varphi_3(z_0)\ne 0$ and $\varphi'_3(z_0)\ne 0$, where $'=d/dz$,
\item\label{item:phi:3}
$\varphi_2(z_0)=\mathrm{i} \varphi_3(z_0)$ and
$\varphi'_2(z_0)=-\mathrm{i} \varphi'_3(z_0)$.
\end{enumerate}
Moreover, if $X_0$ is complete and bounded, then so is $T\circ X_0$.
\end{lemma}
We shall now assume our initial $X_0$ satisfies the three properties
above, and set $z_0=0$ by a coordinate change of $\mathbb{D}$.
\begin{proof}[Proof of Lemma~\ref{lem:phi}]
Since the image of $X_0$ is not contained in any plane,
at least one of $\varphi_1$, $\varphi_2$ and $\varphi_3$,
say $\varphi_3$, is not constant.
Then we can take $z_0\in \mathbb{D}$ such that $\varphi_3(z_0)\ne 0$
and $\varphi'_3(z_0)\ne 0$.
Moreover, if $\varphi_1+\mathrm{i} \varphi_2$ or $\varphi_1-\mathrm{i} \varphi_2$
vanishes identically, this contradicts that
the image of $X_0$ is not contained in any plane.
So we may also assume that
\[
\varphi_1(z_0)\ne \pm \mathrm{i} \varphi_2(z_0).
\]
When $\varphi_1(z_0)\ne 0$,
we replace $X_0$ by $T\circ X_0$,
where $T$ is the linear map
associated with a complex orthogonal matrix
\[
\begin{pmatrix}
-c(1+c^2)^{-1/2} & \hphantom{-c}(1+c^2)^{-1/2} & 0 \\
- \hphantom{c}(1+c^2)^{-1/2} & -c(1+c^2)^{-1/2} & 0 \\
0 & 0 & 1
\end{pmatrix}
\qquad (c=\varphi_2(z_0)/\varphi_1(z_0)).
\]
Since $T$ is orthogonal,
$T\circ X_0$ is also a null holomorphic immersion.
So we get the property \ref{item:phi:1}.
Then \eqref{eq:null} implies that $\varphi_2(z_0)=\pm\mathrm{i}\varphi_3(z_0)$.
Replacing $\varphi_2$ by $-\varphi_2$ if necessary,
we may assume $\varphi_2(z_0) = \mathrm{i} \varphi_3(z_0)$.
Differentiating \eqref{eq:null}, we have
$\varphi_1(z_0)\varphi_1'(z_0)+ \varphi_2(z_0)\varphi_2'(z_0)+
\varphi_3(z_0)\varphi_3'(z_0)=0$.
Then
\[
-\varphi_3(z_0)\varphi'_3(z_0)=\varphi_1(z_0)
\varphi_1'(z_0) +
\varphi_2(z_0)\varphi_2'(z_0)
= \mathrm{i}\varphi_3(z_0)\varphi_2'(z_0)
\]
holds.
In particular, it holds that
$\varphi_2'(z_0) = -\mathrm{i} \varphi_3'(z_0)$.
Next we prove the boundedness and completeness of $T\circ X_0$
under the assumption that $X_0$ is complete and bounded.
Since $X_0$ is bounded and $T$ is continuous,
$X_0(\mathbb{D})$ and $T(X_0(\mathbb{D}))$ are both contained in the
closed ball $\overline{B_0}(R)$
in $\mathbb{C}^3$ of a certain radius $R>0$ centered at
the origin.
We denote by $h_0$ the canonical metric on $\mathbb{C}^3$, and
consider the pull-back metric $h_1=T^*h_0$ by $T$.
Now we apply the following Lemma 3.1 in \cite{MUY}
for $K:=\overline{B_0}(R)$ in $N:=\mathbb{C}^3$.
Then there exist positive numbers $a$ and $b$ ($0<a<b$)
such that $ah_0 < h_1 < bh_0$ on $\overline{B_0}(R)$.
Now, we consider the pull back of the metric $h_0$ and
$h_1$ by the immersion $X_0$.
Since
\[
(X_0)^*h_1=(X_0)^*(T^*h_0)
=(T\circ X_0)^*h_0=(X_1)^*h_0,
\]
we have that
\[
a (X_0)^*h_0 < (X_1)^*h_0 < b(X_0)^*h_0
\]
on $\mathbb{D}$.
Since $X_0$ is complete, $(X_0)^*h_0$ is a
complete Riemannian metric on $\mathbb{D}$.
Then the above relation
implies that $(X_1)^*h_0$ is also a complete Riemannian metric on
$\mathbb{D}$,
that is, $X_1$ is also a complete immersion.
\end{proof}
\subsection*{A family of holomorphic immersions}
Let
\[
\Lambda=(\lambda_0,\dots,\lambda_{n})\in\mathbb{C}^{n+1},\qquad
\Delta=(\delta_1,\dots,\delta_{n})\in\mathbb{C}^{n},
\]
and take a meromorphic function $\mathcal{G}_{\Lambda}$ as in
\eqref{eq:G-lambda}.
Define a meromorphic function $\mathcal{F}_{\Delta}$ as
\begin{equation}\label{eq:F-delta}
\mathcal{F}_{\Delta} =
\frac{1}{df}\sum_{j=1}^{n}\delta_j\zeta_j
\colon{}M\longrightarrow\mathbb{C}\cup\{\infty\}.
\end{equation}
and Weierstrass data $(\hat g,\hat\omega)$ on
$\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})$ as
\begin{equation}\label{eq:def-w-data}
\hat g=\hat g_{(\Lambda,\Delta)}:= h_{\Delta}\cdot (g\circ \mathcal{G}_{\Lambda}),\quad
\hat\omega =\hat\omega_{(\Lambda,\Delta)}:=
\frac{\omega_0\circ \mathcal{G}_{\Lambda}\,df}{h_{\Delta}} \quad
\bigl(h_{\Delta}:=\exp \mathcal{F}_{\Delta}\bigr),
\end{equation}
where $f$ is the meromorphic function as in \eqref{eq:df},
and define holomorphic $1$-forms on $\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})$ as
\begin{equation}\label{eq:mod-forms}
\Psi_1 = (1-\hat g^2)\hat\omega,\qquad
\Psi_2 = \mathrm{i}(1+\hat g^2)\hat\omega,\qquad
\Psi_3 = 2\hat g \hat\omega.
\end{equation}
The following lemma is a modified version of \cite[Theorem 3]{Lopez}.
(In fact, our data \eqref{eq:F-delta} and \eqref{eq:def-w-data} for the
surfaces are somewhat different from those in \cite{Lopez}.)
\begin{lemma}\label{lem:complete}
If $X_0$ as in \eqref{eq:initial} is a complete immersion,
the metric
\[
d\hat s^2:=(1+|\hat g|^2)^2 |\hat\omega|^2
\]
determined by
$(\hat g,\hat\omega)$ as in \eqref{eq:def-w-data} is a complete
Riemann metric on $\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})$ for a sufficiently small
$(\Lambda,\Delta)\neq (\vect{0},\vect{0})$.
\end{lemma}
\begin{proof}
As in the equations (15) and (17) in \cite{Lopez}, there exists a positive
constant
$a$ ($<1$) such that
\[
a<|h_{\Delta}|<\frac1a \qquad\text{and}\qquad
\left|\frac{df}{d\mathcal{G}_{\Lambda}}\right|>a\qquad
\text{on $\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})$}.
\]
Then, (setting $z=\mathcal{G}_{\Lambda}$)
\begin{align*}
(1+|\hat g|^2)|\hat\omega|
&=
(1+|gh_{\Delta}|^2)\left|\frac{\omega_0}{h_{\Delta}}\right|
\left|\frac{df}{d\mathcal{G}_{\Lambda}}\right|\,|dz|\\
&\geq (a^2+|ag|^2)|a\omega_0|\,(a\,|dz|)=a^4(1+|g|^2)|\omega|.
\end{align*}
Thus we have the conclusion.
\end{proof}
Thus, for each (sufficiently small)
$(\Lambda,\Delta)\in \mathbb{C}^{2n+1}\setminus\{\vect{0}\}$,
there exists a complete null immersion
\begin{equation}\label{eq:F-deform}
X_{(\Lambda,\Delta)}:=
\int_{z_0}^z
\bigl(\Psi_1,\Psi_2,\Psi_3\bigr)
\colon{}\widetilde{\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})}\longrightarrow
\mathbb{C}^3,\end{equation}
where $\widetilde{\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})}$ denotes the universal cover of
$\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})$.
In fact, the line integral $\int_{z_0}^z \Psi_j$ ($j=1,2,3$)
from a base point $z_0$ depends on the choice of the path,
but can be considered as a
single-valued function on $\widetilde{\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})}$.
Then we get the following assertion, which can be proven exactly
in the same way as Corollary B in \cite{MUY}:
\begin{proposition}\label{cor:complete-proj}
Let $\pi$ be the projection as in \eqref{eq:proj2}.
Then $\pi\circ X_{(\Lambda,\Delta)}$ is
a complete immersion of $\widetilde{\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})}$
into $\mathbb{C}^2$.
\end{proposition}
\subsection*{The period map}
Under the situations above, we define the {\em period map}
\begin{multline}\label{eq:period-map}
\op{Per}_1\colon{}\mathbb{C}^{2n+1}\ni(\Lambda,\Delta)\\
\longmapsto
\trans{\left(
\int_{\gamma_1}\Psi_1,\dots,
\int_{\gamma_{n}}\Psi_1,
\int_{\gamma_{1}}\Psi_2,\dots,
\int_{\gamma_{n}}\Psi_2
\right)}\in
\mathbb{C}^{2n},
\end{multline}
where $n=2\kappa+e$ (see \eqref{eq:n}),
``$\trans{(~)}$'' is the transposing operation for matrices,
$\gamma_j$'s are loops as in \eqref{eq:period}, and
$\Psi_1$ and $\Psi_2$ are as in \eqref{eq:mod-forms}.
The following assertion is an analogue of \cite[Theorem 2]{Lopez}:
\begin{proposition}\label{prop:jacobian}
Suppose that $X_0$ satisfies the three conditions as in
Lemma \ref{lem:phi}. Then
the $(2n)\times (2n)$ matrix
\begin{equation}\label{eq:jacobian}
J_1 := \left.
\left(
\frac{\partial \op{Per}_1}{\partial\lambda_1},\dots,
\frac{\partial \op{Per}_1}{\partial\lambda_{n}},
\frac{\partial \op{Per}_1}{\partial\delta_1},\dots,
\frac{\partial \op{Per}_1}{\partial\delta_{n}}
\right)
\right|_
{(\Lambda,\Delta)=(\vect{0},\vect{0})}
\end{equation}
is nondegenerate.
\end{proposition}
\begin{proof}
Note that
\begin{equation}\label{eq:FG-initial}
\left. \mathcal{G}_{\Lambda}\right|_{\Lambda=\vect{0}}
=\left. \mathcal{F}_{\Delta}\right|_{\Delta=\vect{0}}=0,\qquad
\left. h_{\Delta}\right|_{\Delta=\vect{0}}=1.
\end{equation}
By the definitions, we have
\[
\left.\frac{\partial \mathcal{G}_{\Lambda}}{\partial \lambda_j}\right|_
{\Lambda=\vect{0}}
= \frac{\zeta_j}{df},\qquad
\left.\frac{\partial h_{\Delta}}{\partial \delta_j}\right|_
{\Delta=\vect{0}}=
\left.\frac{\partial \exp \mathcal{F}_{\Delta}}{\partial \delta_j}\right|_
{\Delta=\vect{0}}= \frac{\zeta_j}{df}
\]
for $j=1,\dots,n$.
Then
\begin{align*}
\left.\frac{\partial \hat g}{\partial\lambda_j}
\right|_{(\Lambda,\Delta)=(\vect{0},\vect{0})} &
= \left(\left.\frac{dg}{dz}\right|_{z=0}\right)
\left(\left.\frac{\partial \mathcal{G}_{\Lambda}}{\partial\lambda_j}
\right|_{\Lambda=\vect{0}}\right)
= g'(0)\frac{\zeta_j}{df},\\
\left.\frac{\partial \hat \omega}{\partial\lambda_j}
\right|_{(\Lambda,\Delta)=(\vect{0},\vect{0})}
& = \omega_0'(0)\zeta_j,\\
\left.\frac{\partial \hat g}{\partial\delta_j}
\right|_{(\Lambda,\Delta)=(\vect{0},\vect{0})}
& = g(0)\frac{\zeta_j}{df},\qquad
\left.\frac{\partial \hat \omega}{\partial\delta_j}
\right|_{(\Lambda,\Delta)=(\vect{0},\vect{0})}
=-\omega_0(0)\zeta_j,
\end{align*}
where $'=d/dz$.
Hence we have
\begin{align*}
\left.\frac{\partial\Psi_1}{\partial\lambda_j}
\right|_{(\Lambda,\Delta)=(\vect{0},\vect{0})}
&=\left.
\left(-2\hat g\frac{\partial \hat g}{\partial \lambda_j}\hat\omega+
(1-\hat g^2)\frac{\partial\hat\omega}{\partial\lambda_j}\right)
\right|_{(\Lambda,\Delta)=(\vect{0},\vect{0})}\\
&= \bigl(-2 g(0)g'(0)\omega_0(0)+(1-g(0)^2)\omega_0'(0)\bigr)\zeta_j\\
&= \left.\bigl((1-g^2)\omega\bigr)'\right|_{z=0}\zeta_j
= \varphi_1'(0)\zeta_j,\\
\left.\frac{\partial\Psi_1}{\partial\delta_j}
\right|_{(\Lambda,\Delta)=(\vect{0},\vect{0})}
&= -(1+\{g(0)\}^2)\omega_0(0)=\mathrm{i}\varphi_2(0)\zeta_j,
\end{align*}
where $\varphi_j$'s are holomorphic functions on $\mathbb{D}$ as in \eqref{eq:w-data}.
Similarly, we have
\[
\frac{\partial\Psi_2}{\partial\lambda_j}
= \varphi'_2(0)\zeta_j,\quad
\frac{\partial\Psi_3}{\partial\lambda_j}
= \varphi'_3(0)\zeta_j,\quad
\frac{\partial\Psi_2}{\partial\delta_j}
= -\mathrm{i}\varphi_1(0)\zeta_j,\quad
\frac{\partial\Psi_3}{\partial\delta_j}
=0
\]
at $(\Lambda,\Delta)=(\vect{0},\vect{0})$.
Thus, we have that
\begin{equation}\label{eq:periods2}
\begin{alignedat}{2}
\frac{\partial}{\partial\lambda_j}\int_{\gamma_k}\Psi_1
&=\varphi_1'(0)\int_{\gamma_k}\zeta_j,\qquad
&\frac{\partial}{\partial\delta_j}\int_{\gamma_k}\Psi_1
&=\mathrm{i}\varphi_2(0)\int_{\gamma_k}\zeta_j,\\
\frac{\partial}{\partial\lambda_j}\int_{\gamma_k}\Psi_2
&=\varphi_2'(0)\int_{\gamma_k}\zeta_j,\qquad
&
\frac{\partial}{\partial\delta_j}\int_{\gamma_k}\Psi_2
&=-\mathrm{i}\varphi_1(0)\int_{\gamma_k}\zeta_j,\\
\frac{\partial}{\partial\lambda_j}\int_{\gamma_k}\Psi_3
&=\varphi_3'(0)\int_{\gamma_k}\zeta_j,\qquad
&
\frac{\partial}{\partial\delta_j}\int_{\gamma_k}\Psi_2
&=0
\end{alignedat}
\end{equation}
hold at $(\Lambda,\Delta)=(\vect{0},\vect{0})$,
for $j,k=1,\dots,n$.
Hence the matrix $J_1$ in \eqref{eq:jacobian} is written as
\[
J_1 = \begin{pmatrix}
\varphi_1'(0) P & \hphantom{-}\mathrm{i}\varphi_2(0) P \\
\varphi_2'(0) P & -\mathrm{i}\varphi_1(0) P
\end{pmatrix}=
\begin{pmatrix}
\varphi_1'(0) P & \hphantom{-}\mathrm{i}\varphi_2(0) P \\
\varphi_2'(0) P & O
\end{pmatrix}
\qquad (\varphi_2(0), \varphi'_2(0)\neq 0)
\]
because of Lemma~\ref{lem:phi},
where $P$ is the nondegenerate period matrix as in \eqref{eq:period}.
Hence $J_1$ is nondegenerate.
\end{proof}
\subsection*{The period-killing problem}
Since $\op{Per}_1(\vect{0})=\vect{0}$,
Proposition~\ref{prop:jacobian} yields that
there exists a holomorphic map
$c \mapsto \bigl(\lambda_1(c),\dots,\lambda_{n}(c),
\delta_1(c),\dots,\delta_{n}(c)
\bigr)$
such that
\[
\op{Per}_1\bigl(c,\lambda_1(c),\dots,\lambda_{n}(c),
\delta_1(c),\dots,\delta_{n}(c)
\bigr) = 0
\]
for sufficient small $c$.
We set
\[
\mathcal{G}_c = \mathcal{G}_{\Lambda(c)},\qquad
\text{where}\quad
\Lambda(c):=\bigl(c,\lambda_1(c),\dots,\lambda_{n}(c)\bigr).
\]
Since
$\Lambda(0)=\vect{0}$,
there exists an analytic function $\Lambda^*(c)$ such that
$\Lambda(c)=c\Lambda^*(c)$ near $c=0$.
Now we can apply Proposition \ref{prop:inverse} to
$\Lambda_0:=\Lambda^*(0)/|\Lambda^*(0)|\in\S_1$.
Then, for sufficiently small $c$, $\mathcal{G}_{c}^{-1}(\mathbb{D})$
is conformally equivalent to $M$ minus $e+1$
pairwise disjoint discs with analytic regular boundaries,
and the map
\begin{multline}\label{eq:result}
X_c:=\pi\circ X_{\bigl(\Lambda(c),\Delta(c)\bigr)},\\
\Lambda(c)=(c,\lambda_1(c),\dots,\lambda_{n}(c)),\quad
\Delta(c)=(\delta_1(c),\dots,\delta_{n}(c))
\end{multline}
is well-defined on $\mathcal{G}_{c}^{-1}(\mathbb{D})$.
Moreover, by Corollary~\ref{cor:complete-proj},
$X_c$ is a complete immersion for any sufficient small $c$.
\subsection*{Boundedness of $X_c$}
By Proposition~\ref{prop:inverse}, $d\mathcal{G}_c$ does not vanish
on $\partial\mathcal{G}_c^{-1}(\mathbb{D})$ for sufficiently small $c$.
Then if we choose
a real number $r\in (0,1)$ sufficiently close to $1$,
we have
\[
d\mathcal{G}_c\neq 0 \qquad \text{on}\quad \mathcal{G}_c^{-1}(\overline\mathbb{D}\setminus\mathbb{D}_r),
\]
where
$\mathbb{D}_r=\{z\in\mathbb{C}\,;\,|z|<r\}$.
Moreover, $\mathcal{G}_c^{-1}(\overline\mathbb{D}\setminus\mathbb{D}_r)$ is exactly
a union of $e+1$ closed annular domains surrounding
the points $Q_0,Q_1,\dots,Q_e$ in $M$.
To show the boundedness of $X_c$, it is sufficient to show that
the image of each annular domain by $X_c$ is bounded.
For the sake of simplicity, we show the boundedness
of $X_c$ at $Q_0$.
We denote by $\overline\Omega$ the closed
annular domain surrounding the point $Q_0$.
Then
\[
\mathcal{G}:=\mathcal{G}_c|_{\overline\Omega}:\overline\Omega
\longrightarrow \overline\mathbb{D}\setminus\mathbb{D}_r
\]
gives a holomorphic finite covering.
Since $\mathcal{G}_c^{-1}(\overline\mathbb{D}_r)\subset \mathcal{G}_c^{-1}(\mathbb{D})$ is compact and
$X_c\colon{}\mathcal{G}_c^{-1}(\mathbb{D})\to\mathbb{C}^2$ is holomorphic, there exists
a positive constant $K_0$ such that
\begin{equation}\label{eq:bound1}
|X_c| \leq K_0\qquad \text{on}\quad \mathcal{G}_c^{-1}(\overline\mathbb{D}_r).
\end{equation}
We denote by $\Omega$ the set of interior points of $\overline\Omega$,
and fix $q\in \Omega$ arbitrarily.
Let $\sigma(t)$ ($0\le t\le 1$)
be a line segment such that
$\sigma(0)=\mathcal{G}(q)$ and $\sigma(1)=r \mathcal{G}(q)/|\mathcal{G}(q)|\in \partial \mathbb{D}_r$,
that is,
\[
\sigma(t):=(1-t)\mathcal{G}(q)+t\frac{r\mathcal{G}(q)}{|\mathcal{G}(q)|}.
\]
Since $\mathcal{G}$ is a covering map, there exists a unique smooth curve
$\tilde \sigma:[0,1]\to \overline\Omega$
such that $\tilde \sigma(0)=q$ and $\mathcal{G}\circ \tilde\sigma=\sigma$.
Moreover, there exists a neighborhood $U$ of the line segment
$\sigma([0,1])$ and a holomorphic map
$\H:U\to M$ which gives the (local) inverse of $\mathcal{G}$.
By definition, we have
$\tilde \sigma=\H\circ \sigma$.
We set
\[
q_1:=\tilde \sigma(1)\in
\partial \Omega \setminus \mathcal{G}_c^{-1}(\partial \mathbb{D}),
\]
that is, $q_1$ lies on the connected component of $\partial\Omega$
which is further from $Q_0$, see Figure~\ref{fig:sigma}.
\begin{figure}
\begin{center}
\includegraphics{bounded.eps}
\end{center}
\caption{The segments $\tilde\sigma$ and $\sigma$}%
\label{fig:sigma}
\end{figure}
By \eqref{eq:bound1}, it is sufficient to show that
\begin{equation}\label{eq:small-part}
\bigl|
X_c(q_1)-X_c(q)
\bigr|
=\left|\int_{\tilde \sigma}(\Psi_1,\Psi_2)\right|
\end{equation}
is bounded from above by a constant which does not depend on $q$.
Here we have
\begin{multline}\label{eq:int-phi-one}
\left|\int_{\tilde \sigma}\Psi_1\right|=
\left|
\int_{\tilde \sigma}
\left(1-\bigl(h_{\Delta}\bigr)^2
\bigl(g\circ \mathcal{G}_c\bigr)^2\right)
\frac{\omega_0\circ \mathcal{G}_c}{%
h_{\Delta}}df
\right|\\
\begin{aligned}
&=
\left|
\int_0^1
\left(
\left(1-\bigl(g\circ \sigma(t)\bigr)^2\bigl(h(t)\bigr)^2\right)
\frac{\omega_0\circ \sigma(t)}{h(t)}
\left. \frac{d\bigl(f\circ \H(z)\bigr)}{dz}\right|_{z=\sigma(t)}
\frac{d\sigma}{dt}
\right)\,dt
\right|\\
&=|I_1+I_2|,
\end{aligned}
\end{multline}
where $h(t):=h_{\Delta}\circ \tilde \sigma(t)$ and
\begin{align*}
I_1 &= \frac{1}{2}\int_0^1
\left(
\left. \bigl(\varphi_1(z)-\mathrm{i}\varphi_2(z)\bigr)
\right|_{z=\sigma(t)}
\frac{d\sigma}{dt}
\right)
\left(
\left. \frac{1}{h_{\Delta}\circ \H(z)}
\frac{df\circ \H(z)}{dz}\right|_{z=\sigma(t)}
\right)\,dt,\\
I_2 &= \frac{-1}{2}\int_0^1
\left(
\left. \bigl(\varphi_1(z)+\mathrm{i}\varphi_2(z)\bigr)
\right|_{z=\sigma(t)}
\frac{d\sigma}{dt}
\right)
\left(
\left. h_{\Delta}\circ \H(z)
\frac{df\circ \H(z)}{dz}\right|_{z=\sigma(t)}
\right)\,dt.
\end{align*}
Here we used the relations $2\omega_0=\varphi_1-\mathrm{i}\varphi_2$ and
$2g^2\omega_0=-(\varphi_1+\mathrm{i}\varphi_2)$.
To estimate $I_1$,
we shall apply Lemma~\ref{lem:app} in the appendix for
\[
a(t) =
\left. \bigl(\varphi_1(z)-\mathrm{i}\varphi_2(z)\bigr)
\right|_{z=\sigma(t)}
\frac{d\sigma}{dt}
\quad\text{and}\quad
b(t) =
\left. \frac{1}{h_{\Delta}\circ \H(z)}
\frac{df\circ \H(z)}{dz}\right|_{z=\sigma(t)}.
\]
Let us check the hypotheses:
\[
(A(s):=)\int_0^s a(t)\,dt = \int_{\sigma([0,s])}
(\varphi_1(z)-\mathrm{i}\varphi_2(z))\,dz
= \left.\bigl(X_1(z)-\mathrm{i} X_2(z)\bigr)\right|_{z=\sigma(0)}^{z=\sigma(s)},
\]
where $X=X_0$ as in \eqref{eq:initial}.
Since $X$ is bounded, $|A(s)|$ is bounded for all $s$.
On the other hand,
\[
b(t) = \tilde b\bigl(\sigma(t)\bigr),\qquad\text{where}\quad
\tilde b(z) =
\frac{1}{h_{\Delta}\circ \H(z)}\frac{d(f\circ \H(z))}{dz}.
\]
Since $\H(z)$ can be considered as a single valued holomorphic
function on a certain finite covering of $\overline\mathbb{D}\setminus \mathbb{D}_r$,
both $\tilde b(z)$ and $\tilde b'(z)$ ($'=d/dz$) are
bounded by a constant.
Hence by the lemma, we have that $|I_1|$ is bounded.
Similarly, we can show that $|I_2|$ is bounded, and
we can conclude that
the integration of $\Psi_1$
(and similarly $\Psi_2$) along $\tilde\sigma$ is bounded.
The resulting immersion $X_c$ has an arbitrary number of
ends by setting $f=f_N$ as in Remark \ref{rmk:N}.
\section{Proof of the corollary}
In this section we shall prove Corollary~\ref{cor:cor} in the
introduction.
Define $\mathcal{G}_{\Lambda}$, $(\Psi_1,\Psi_2,\Psi_3)$ and
the null immersion
$X_{(\Lambda,\Delta)}$ as in the previous section,
and define real parameters $(s_j,t_j)$ as
\begin{equation}\label{eq:real-parameter}
\lambda_j = s_j + \mathrm{i} t_j,\qquad
\delta_j = s_{n+j}+ \mathrm{i} t_{n+j}.
\end{equation}
Note that for a holomorphic function $F(u)$ in $u=s+\mathrm{i} t$, one has:
\begin{equation}\label{eq:re-im}
\begin{alignedat}{2}
\frac{\partial\Re F}{\partial s} &=\Re \frac{dF}{du},\qquad
&\frac{\partial\Re F}{\partial t} &= -\Im \frac{dF}{du},\\
\frac{\partial\Im F}{\partial s} &=\Im \frac{dF}{du},\qquad
&\frac{\partial\Im F}{\partial t} &=\hphantom{-} \Re \frac{dF}{du}.
\end{alignedat}
\end{equation}
\subsection*{Minimal surfaces in $\mathbb{R}^3$}
First, we treat the case of minimal surfaces in $\mathbb{R}^3$.
Let
\[
x= x_{(\Lambda,\Delta)}:=\Re X_{(\Lambda,\Delta)}=
\Re\int(\Psi_1,\Psi_2,\Psi_3)\colon{}
\widetilde{\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})}\longrightarrow \mathbb{R}^3,
\]
where $\widetilde{\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})}$ is the universal cover of
$\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})$.
Then $x$ is a conformal minimal immersion, and the induced metric
is complete because of Lemma~\ref{lem:complete}.
If $x_{(\Lambda,\Delta)}$ were well-defined on $\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})$
for sufficiently small $(\Lambda,\Delta)$, boundedness of
the image of $x_{(\Lambda,\Delta)}$ can be proved in a similar way
as in the previous section.
Hence it is sufficient to solve the period-killing problem
to show the corollary.
We define the period map
\begin{multline}\label{eq:period-R}
\op{Per}_2\colon{}\mathbb{R}^{4n+2}
\ni(s_0,\dots,s_{2n};
t_0,\dots,t_{2n})
\\
\longmapsto
\trans{\left(
\left(\Re\!\!\int_{\gamma_k}\Psi_1\right)_{k=1,\dots,n},
\left(\Re\!\!\int_{\gamma_k}\Psi_2\right)_{k=1,\dots,n},
\left(\Re\!\!\int_{\gamma_k}\Psi_3\right)_{k=1,\dots,n}
\right)}\in
\mathbb{R}^{3n},
\end{multline}
where $(s_j,t_j)$ are real parameter as in \eqref{eq:real-parameter}.
Consider the $(3n)\times (4n)$ matrix
\begin{equation}\label{eq:Jacobi-R}
J_2:=\left.
\left(
\frac{\partial\op{Per}_2}{\partial s_1},\dots,
\frac{\partial\op{Per}_2}{\partial s_{2n}},
\frac{\partial\op{Per}_2}{\partial t_1},\dots,
\frac{\partial\op{Per}_2}{\partial t_{2n}}
\right)\right|_{(\Lambda,\Delta)=(\vect{0},\vect{0})}.
\end{equation}
To solve the period-killing problem,
it is sufficient to show that the rank of $J_2$ is $3n$.
By \eqref{eq:periods2} and \eqref{eq:re-im}, we have
\[
J_2 = \begin{pmatrix}
\Re \bigl(\varphi_1'(0) P\bigr) &
\hphantom{-}\Re \bigl(\mathrm{i}\varphi_2(0) P\bigr) &
-\Im \bigl(\varphi_1'(0) P\bigr) &
-\Im \bigl(\mathrm{i}\varphi_2(0) P \bigr)\\
\Re \bigl(\varphi_2'(0) P\bigr) &
-\Re \bigl(\mathrm{i}\varphi_1(0) P\bigr) &
-\Im \bigl(\varphi_2'(0) P\bigr) &
\hphantom{-}\Im \bigl(\mathrm{i}\varphi_1(0) P\bigr) \\
\Re \bigl(\varphi_3'(0) P \bigr)&
O &
-\Im \bigl(\varphi_3'(0) P\bigr) &
O
\end{pmatrix},
\]
where the $n\times n$ matrix $P$ is the period matrix as in \eqref{eq:period}.
Here, we remark that the real vectors are linearly independent over $\mathbb{R}$
if and only if they are linearly independent over $\mathbb{C}$.
Since we may assume that $X_0$ satisfies the three conditions as in
Lemma~\ref{lem:phi}, we have that
\begin{align*}
\op{rank} J_2 &=
\op{rank}\begin{pmatrix}
\varphi_1'(0) P &
\hphantom{-}\mathrm{i}\varphi_2(0) P &
\overline{\varphi_1'(0) P} &
\hphantom{-}\overline{\mathrm{i}\varphi_2(0)P}\\
\varphi_2'(0) P &
-\mathrm{i}\varphi_1(0) P &
\overline{\varphi_2'(0) P} &
-\overline{\mathrm{i}\varphi_1(0) P} \\
\varphi_3'(0) P & O &
\overline{\varphi_3'(0) P} & O
\end{pmatrix}\\
&=
\op{rank}\begin{pmatrix}
\hphantom{-\mathrm{i}}\varphi_1'(0) P &
\mathrm{i}\varphi_2(0) P &
\hphantom{\mathrm{i}}\overline{\varphi_1'(0) P} &
\overline{\mathrm{i}\varphi_2(0)P}\\
-\mathrm{i}\varphi_3'(0) P &
O &
\mathrm{i}\overline{\varphi_3'(0) P} &
O \\
\hphantom{-\mathrm{i}}\varphi_3'(0) P & O &
\hphantom{\mathrm{i}}\overline{\varphi_3'(0) P} & O
\end{pmatrix}\\
&=\op{rank}
\begin{pmatrix}
O & P & * & * \\
P & O & * & * \\
O & O & 2\overline P & *
\end{pmatrix}= 3n,
\end{align*}
since $P$ is nondegenerate. Then we can solve the period problem as we
did in the previous section.
\subsection*{Maximal surfaces in the Lorentz-Minkowski spacetime}
We denote by $\L^3$ the Lorentz-Minkowski $3$-spacetime, that is,
$\bigl(\mathbb{R}^3;(x_0,x_1,x_2)\bigr)$ endowed with the indefinite metric
$-(dx_0)^2+(dx_1)^2+(dx_2)^2$.
Under the same settings as above, we set
\[
y = y_{(\Lambda,\Delta)}
:= \Re\int (\mathrm{i} \Psi_1,\Psi_2,\Psi_3)\colon{}
\widetilde{\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})}\longrightarrow \L^3.
\]
Then $y$ gives a maximal surface (a mean curvature zero surface),
which could possibly have singular points.
In particular, since the holomorphic lift
\begin{equation}\label{eq:maxface-lift}
\int (\mathrm{i} \Psi_1,\Psi_2,\Psi_3)\colon{}
\widetilde{\mathcal{G}_{\Lambda}^{-1}(\mathbb{D})}\longrightarrow \mathbb{C}^3
\end{equation}
is an immersion, $y$ is a {\em maxface\/} in the sense of \cite{uy3}.
Moreover, the induced metric by \eqref{eq:maxface-lift} is complete
because of Lemma~\ref{lem:complete}.
Hence $y$ is a {\em weakly complete maxface\/} in the sense of
\cite{uy3}.
To show the $\L^3$ case of Corollary~\ref{cor:cor},
we consider the period map
\begin{multline}\label{eq:period-L}
\op{Per}_3\colon{}\mathbb{R}^{4n+2}
\ni(s_0,\dots,s_{2n};
t_0,\dots,t_{2n})
\\
\longmapsto
\trans{\left(
\left(\Im\!\!\int_{\gamma_k}\Psi_1\right)_{k=1,\dots,n},
\left(\Re\!\!\int_{\gamma_j}\Psi_2\right)_{k=1,\dots,n},
\left(\Re\!\!\int_{\gamma_j}\Psi_3\right)_{k=1,\dots,n}
\right)}\in
\mathbb{R}^{3n}.
\end{multline}
Then
\begin{equation}\label{eq:Jacobi-L}
J_3:=\left.
\left(
\frac{\partial\op{Per}_3}{\partial s_1},\dots,
\frac{\partial\op{Per}_3}{\partial s_{2n}},
\frac{\partial\op{Per}_3}{\partial t_1},\dots,
\frac{\partial\op{Per}_3}{\partial t_{2n}}
\right)\right|_{(\Lambda,\Delta)=(\vect{0},\vect{0})}
\end{equation}
has the expression
\begin{align*}
J_3 =\begin{pmatrix}
\Im \bigl(\varphi_1'(0) P\bigr) &
\hphantom{-}\Im \bigl(\mathrm{i}\varphi_2(0) P\bigr) &
\hphantom{-}\Re \bigl(\varphi_1'(0) P\bigr) &
\Re \bigl(\mathrm{i}\varphi_2(0) P \bigr)\\
\Re \bigl(\varphi_2'(0) P\bigr) &
-\Re \bigl(\mathrm{i}\varphi_1(0) P\bigr) &
-\Im \bigl(\varphi_2'(0) P\bigr) &
\Im \bigl(\mathrm{i}\varphi_1(0) P\bigr) \\
\Re \bigl(\varphi_3'(0) P \bigr)&
O &
-\Im \bigl(\varphi_3'(0) P\bigr) &
O
\end{pmatrix},
\end{align*}
and then it can be easily checked that
$J_3$ is of rank $3n$ like as in the case of $\mathbb{R}^3$.
Hence we conclude as in the previous cases.
|
1,108,101,563,286 | arxiv | \section*{R\'esum\'e}
\else \small
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\fancyhead[EC]{\sc{Fran\c{c}ois Dubois}}
\fancyhead[OC]{\sc{On hyperbolic systems with entropy velocity... }}
\begin{document}
\fancypagestyle{plain}{ \fancyfoot{} \renewcommand{\footrulewidth}{0pt}}
\fancypagestyle{plain}{ \fancyhead{} \renewcommand{\headrulewidth}{0pt}}
\title{\bf \LARGE ~ \vspace{.5 cm} ~\\
On hyperbolic systems with entropy velocity \\ covariant under the action of a group
\\ ~ }
\author { { \large Fran\c{c}ois Dubois~$^{ab}$} \\ ~\\
{\it \small $^a$ Conservatoire National des Arts et M\'etiers, Department of Mathematics, and } \\
{\it \small
Laboratoire de M\'ecanique des Structures et des Syst\`emes Coupl\'es, Paris, France. } \\
{\it \small $^b$ Department of Mathematics, University Paris-Sud,} \\
{\it \small B\^at. 425, F-91405 Orsay Cedex, France} \\
{\it \small [email protected]} }
\date{ 10 september 2013~\protect\footnote{~This contribution
in homage to Jean-Marie Souriau (1922 - 2012) is
issued from a lecture entitled ``Sur les syst\`emes hyperboliques
\`a vitesse entropique invariants sous l'action d'un groupe''
given on tuesday 28 august 2012 at the 56th
``Colloque International de Th\'eories Variationnelles'',
La Baume, Aix en Provence, France. }}
\maketitle
\begin{abstract}
For hyperbolic systems of conservation laws in one space dimension with a mathematical
entropy,
we define the notion of entropy velocity.
Then we give sufficient conditions for such a system to be covariant under the action of
a group of space-time transformations. These conditions naturally introduce a
representation of the group in the space of states.
We construct such hyperbolic system from the knowledge of data on
the manifold of null velocity. We apply these ideas for Galileo, Lorentz and circular
groups. We focus on particular non trivial examples for two
of two systems of conservation laws.
$ $ \\[4mm]
{\bf Keywords}: Galileo group, Lorentz group, circular group, mathematical entropy.
$ $ \\[4mm]
{\bf AMS classification}: 35L40, 58J45.
\end{abstract}
\bigskip \bigskip \newpage \noindent {\bf \large 1) \quad Introduction}
\fancyfoot[C]{\oldstylenums{\thepage}}
\monitem
The first link between covariance under the action of a group
and conservation laws is the classic result of
Emmy Noether \cite{No18}:
when the action is invariant under the action of a group,
an associated moment is conserved along the time evolution.
This result is well understood for systems that possess
some action.
This result is also fundamental for all the modern physics.
It has been intensively studied and generalized by Jean-Marie Souriau
in his fundamental book \cite{So70}
(see also the work of Kostant \cite{Ko70}).
The fact to consider Galilean relativity is
natural in this kind of work.
In order to explicit the mass as an invariant, the
Bargmann group \cite{Ba54} is necessary in this framework
and we refer to the work of de Saxc\'e and Vall\'ee
\cite{SV12} for the introduction of the
second principle of thermodynamics.
Observe also
that very early Souriau has considered
not only Galilean relativity but also
Einstein's relativity \cite{So64, So77, So78}.
An other contribution is due to Vall\'ee \cite {Va87} relative to continuum
media in the framework of special relativity.
\monitem
Our scientific background is first concerned with
the approximation of gas dynamics problems
with numerical methods \cite{DD05}. The notion of first order hyperbolic
system of conservation laws is fundamental for this study and the
contribution of Peter lax (see {\it e.g.} \cite {La06}) is
essential for all that follows.
The introduction of the second principle of thermodynamics in
this framework is due to Godunov \cite{Go61} and to
Friedrichs and Lax \cite{FL71} with the notion of
``mathematical entropy''. A fundamental result of existence
and uniqueness of an entropy
solution in the scalar case is due to Kru$\breve {\rm z} $kov \cite {Kr70}.
Galilean conservation for hyperbolic systems has been
proposed by Ruggeri \cite{Ru89}.
We have considered the same question with
enforcing the structure of the mathematical entropy in
\cite{Du01}. We consider an extension of this
approach in this contribution.
\monitem
The summary of our study is the following. We
introduce the entropy velocity at Section~2
for hyperbolic systems with a mathematical entropy.
Then we define in Section~3
the notion of covariance for an hyperbolic system with entropy velocity
under the action of a group
and propose sufficient conditions that make in evidence
an algebraic structure.
The manifold of null velocity defined at
Section~4 is a natural notion in ths context.
Then precise constraints for the entropy variables
are derived at Section~5. In Section~6, we study the particular case
of systems of order two. Some words of conclusion are proposed at Section~7.
\bigskip \bigskip \noindent {\bf \large 2) \quad Entropy velocity }
\monitem
Let $N$ be a positive integer and $\, \Omega \,$ a non void convex open part
of $ \, {\rm I}\! {\rm R}^N .$
The physical flux $ \, f(W) \in {\rm I}\! {\rm R}^N \,$ of a state
$\, W \in \Omega \,$ is a regular function
$ \, f : \, \Omega \longrightarrow {\rm I}\! {\rm R}^N . \,$
We study in the following the system of conservation laws in one space dimension
with unknown
$\, {\rm I}\! {\rm R} \times [0, \, +\infty[ \, \, \ni (x, \, t) \, \longmapsto \, W(x, \, t) \in \Omega \, $
associated with this nonlinear flux function:
\moneq \label{edp}
{{\partial W}\over{\partial t}} \,+\, {{\partial}\over{\partial x}} f(W) \,=\, 0 \,.
\vspace*{-6pt} \end{equation}
\monitem
As proposed by Godunov \cite{Go61}, Friedrichs and Lax \cite{FL71}
and Boillat \cite{Bo74},
a mathematical entropy for the system (\ref{edp}) is a regular strictly convex
function $ \, \eta : \, \Omega \longrightarrow {\rm I}\! {\rm R} \,$
such that for any regular solution $\, W(x, \, t) \,$ of the conservation law
(\ref{edp}), there exists a complementary conservation equation
\moneq \label{cons-entropy}
{{\partial \eta}\over{\partial t}} \,+\, {{\partial \zeta}\over{\partial x}}
\,=\, 0 \,.
\vspace*{-6pt} \end{equation}
The space derivative of the second term in (\ref{cons-entropy}) is
the entropy flux, a regular function $ \, \zeta : \Omega \longrightarrow {\rm I}\! {\rm R} \,$
that satisfies the identity
\moneq \label{flux-entropy}
{\rm d} \zeta \, \equiv \, {\rm d} \eta \, {\scriptstyle \bullet } \, {\rm d} f \,.
\vspace*{-6pt} \end{equation}
It is well known (see {\it e.g.} Godlewski - Raviart \cite{GR96} or \cite{DD05})
that a system of conservation law which admits a mathematical entropy is hyperbolic: the
jacobian $ \, {\rm d} f(W) \,$ is diagonalisable in $ \, {\rm I}\! {\rm R}^N .\,$
Observe also that the restriction to only one space dimension
is not restrictive in principle if we adopt a point of view founded
on mathematical entropy. The generalization to several space dimensions
does not add in priciple new fundamental difficulties.
As well known, the equality (\ref{cons-entropy}) has to be replaced by an
inequality (taken in an appropriate weak sense) when discontinuous solutions
of the conservation law (\ref{edp}) are considered.
The co-vector $ \, \varphi \,$ of entropy variables is the
jacobian of the entropy relatively to the variation of the state
$ \, W $:
\moneq \label{var-entropic}
{\rm d} \eta \, \equiv \, \varphi \,{\scriptstyle \bullet } \, {\rm d} W \,, \qquad \forall \, W \in \Omega \,.
\vspace*{-6pt} \end{equation}
The dual entropy $ \, \eta^* \,$ is the
Legendre \cite{Le87} - Fenchel \cite{Fe49} - Moreau \cite{Mo62}
transform of the convex function $ \, \eta $.
It is a function of the entropy variables:
\moneq \label{dual-entropy}
\eta^* (\varphi) \,\,\, \,\,\, \equiv \,\,\,\,\,\,
{\rm sup}_{_{_{_{_{ \!\!\!\!\!\!\!\!\!\!\!\!\!\! \normalsize W \in \Omega}}}}} \,
\big( \varphi \, {\scriptstyle \bullet }\, W - \eta(W) \big) \,
\vspace*{-6pt} \end{equation}
and we have
\moneq \label{dual-entropy-grad}
{\rm d} \eta^* (\varphi) \, = \, {\rm d} \varphi \, {\scriptstyle \bullet }\, W \, .
\vspace*{-6pt} \end{equation}
\monitem
In the following,
we define the ``entropy velocity'' $\, u \,$ as the quotient of the entropy flux
divided by the entropy. In other terms, we suppose that the system
(\ref{edp}) (\ref{cons-entropy}) admits an entropy velocity $ \, u \,$
if there exists a regular function
$ \, u : \, \Omega \longrightarrow {\rm I}\! {\rm R} \,$ such that
the entropy flux $ \, \zeta \, $ can be written under the form
$ \, \eta \, u $:
\moneq \label{def-vitesse}
\zeta(W) \, \equiv \, \eta(W) \,\, u(W) \,, \qquad \forall \, W \in \Omega \,.
\vspace*{-6pt} \end{equation}
This definition has been previously proposed in \cite{Du01}. For gas dynamics
this entropy velocity is the usual velocity because the specific entropy
is advected by the flow (see {\it e.g.} \cite{DD05} or \cite{GR96}).
Once the velocity is defined, it is natural, as suggested
by Ruggeri \cite{Ru89}, to decompose the physical flux $ \, f \, $ into a convective part
$ \, u \, W \,$ and a complementary contribution $ \, g $.
We define the ``thermodynamic flux'' $ \, g(W) \, $ by the relation
\moneq \label{def-flux-thermo}
f(W) \, \equiv \, u(W) \, W \,+\, g(W) \,, \qquad \forall \, W \in \Omega \,.
\vspace*{-6pt} \end{equation}
\newpage
\smallskip \monitem { \bf Proposition 1. Compatibility relation }
\noindent
With the above framework, the compatibility relation (\ref{flux-entropy})
is satisfied if and only if
\moneq \label{compati}
\varphi \, {\scriptstyle \bullet } \, {\rm d}g \,+ \, \eta^* \, {\rm d}u \, \equiv \, 0 \, .
\vspace*{-6pt} \end{equation}
\smallskip \noindent {\bf Proof of Proposition 1. }
\noindent
The proof has been explicited in \cite{Du01}; we detail it for completeness.
We have, due to the relations (\ref{flux-entropy}) to (\ref{def-flux-thermo}),
the following calculus:
\noindent $ \displaystyle
{\rm d} \zeta \, -\, {\rm d} \eta \, {\scriptstyle \bullet } \, {\rm d} f \, = \,
{\rm d} ( \eta \, u ) \, - \, {\rm d} \eta \,{\scriptstyle \bullet } \, {\rm d} \big( u \, W \,+\, g \big) \, = \,
\eta \, {\rm d} u \, + \, u \, {\rm d} \eta \, - \,
\varphi \, {\scriptstyle \bullet } \, \big[
( {\rm d} u ) \, W \, + \, u \, {\rm d} W \, + \, {\rm d} g \, \big]
$
\noindent $ \displaystyle \qquad = \,\,
\big( \eta \, - \, \varphi \, {\scriptstyle \bullet } \, W \big) \, {\rm d} u
\, + \, u \, \big( {\rm d} \eta \,-\, \varphi \, {\scriptstyle \bullet } \, {\rm d} W \big)
\, - \, \varphi \, {\scriptstyle \bullet } \, {\rm d} g \,\,\, = \, \,
- \eta^* \, {\rm d} u \, - \, \varphi \, {\scriptstyle \bullet } \, {\rm d} g $
\smallskip \noindent
and the conclusion is clear. $ \hfill \square $
\bigskip \bigskip \noindent {\bf \large 3) \quad Covariance under the action of a group }
\monitem
We introduce a group $ \, {\cal G} \, $ with one real parameter $ \, \theta \, $
composed by transformations $ \, G_\theta \, $ of space-time
defined by a two by two matrix:
\moneq \label{groupe-general}
G_\theta \, = \, \begin{pmatrix} \alpha_\theta & \delta_\theta \cr
\beta_\theta & \gamma_\theta \end{pmatrix} , \qquad G_\theta \in {\cal G}
\,, \quad \theta \in {\rm I}\! {\rm R} \, . \vspace*{-6pt} \end{equation}
At a fixed $ \, \theta $, the transformation
$ \, (x, \, t) \longmapsto (x', \, t') \,$ is defined by the relations:
\moneq \label{transfo-theta}
\begin{pmatrix} x' \cr t' \end{pmatrix} \, = \,
G_\theta \, \begin{pmatrix} x \cr t \end{pmatrix} \, ; \qquad
\left\{ \begin{array} [c]{l} \displaystyle
x' \, = \, \alpha_\theta \, x \, + \, \delta_\theta \, t
\\ \displaystyle
t' \,\, = \, \beta_\theta \, x \, + \, \gamma_\theta \, t \, .
\end{array} \right. \vspace*{-6pt} \end{equation}
In the following, we consider three particular groups,
defined by the conditions
\moneqstar
\alpha_\theta \,=\, \gamma_\theta \,=\, C_\theta \,, \quad
\beta_\theta \,=\, {{\varepsilon}\over{c}} \, S_\theta \,, \quad
\delta_\theta \,=\, c \, S_\theta \,
\vspace*{-6pt} \end{equation*}
{\it id est} by the following matrices
\moneq \label{groupe-particulier}
G_\theta \, = \, \begin{pmatrix} C_\theta & c\, S_\theta \cr
\displaystyle {{\varepsilon}\over{c}} \, S_\theta & C_\theta \end{pmatrix} \, . \vspace*{-6pt} \end{equation}
The Galileo group corresponds to
\moneq \label{gpe-galilee}
\varepsilon \, = \, 0 \,, \qquad C_\theta \, \equiv \, 1
\,, \qquad S_\theta \, \equiv \, \theta \,,
\vspace*{-6pt} \end{equation}
the Lorentz group to
\moneq \label{gpe-lorentz}
\varepsilon \, = \, 1 \,, \qquad C_\theta \, \equiv \, {\rm cosh} \, \theta
\,, \qquad S_\theta \, \equiv \, {\rm sinh} \, \theta \,,
\vspace*{-6pt} \end{equation}
and a third one here called the ``circular group'' to
\moneq \label{gpe-circular}
\varepsilon \, = \, -1 \,, \qquad C_\theta \, \equiv \, {\rm cos} \, \theta
\,, \qquad S_\theta \, \equiv \, {\rm sin} \, \theta \,.
\vspace*{-6pt} \end{equation}
We remark that we have the elementary relations,
\moneq \label{trigo}
C_\theta ^2 \, - \, \varepsilon \, S_\theta ^2 \, \equiv \, 1 \,, \quad
{{{\rm d} S_\theta}\over{{\rm d} \theta}} \, \equiv \, C_\theta \,, \quad
{{{\rm d} C_\theta}\over{{\rm d} \theta}} \, \equiv \, \varepsilon \, S_\theta \,, \quad
\varepsilon \in \{ -1 ,\, 0 , \, 1 \} \, .
\vspace*{-6pt} \end{equation}
\monitem
We say here that the system (\ref{edp}) associated with the conservation of entropy
\moneq \label{cons-entropy-vitesse}
{{\partial \eta}\over{\partial t}} \,+\, {{\partial}\over{\partial x}}
\big( \eta \, u \big) \,=\, 0 \,
\vspace*{-6pt} \end{equation}
is covariant under the action of the group $ \, \cal G \, $
if for each $ \, \theta \in {\rm I}\! {\rm R} \, $ and for each state $ \, W \in \Omega ,$
we can define a new state $ \, W' \,$
such that after the change of variables
(\ref{transfo-theta}), the system of equations (\ref{edp}) (\ref{cons-entropy-vitesse})
take the form
\moneq \label{edp-prime}
\left\{ \begin{array} [c]{l} \displaystyle
\, \, \, \, {{\partial W'}\over{\partial t'}} \, \quad + \quad
\, {{\partial}\over{\partial x'}} \, f(W') \quad \,=\, 0
\\ \displaystyle \vspace{-.5cm} ~ \\ \displaystyle
{{\partial}\over{\partial t'}} \, \eta(W') \,+\, {{\partial}\over{\partial x'}}
\, \big( \eta(W') \, u(W') \big) \,=\, 0 \,
\end{array} \right. \vspace*{-6pt} \end{equation}
with the {\it same} flux function $ \, f({\scriptstyle \bullet }) , \,$ the {\it same} mathematical entropy
$\, \eta({\scriptstyle \bullet }) \, $ and the {\it same} entropy velocity $\, u ({\scriptstyle \bullet }) . \, $
\monitem
We present now an elementary calculus. We have the chain rule
\moneqstar
\left\{ \begin{array} [c]{l} \displaystyle
{{\partial}\over{\partial x}} \,\,=\,\,
{{\partial x'}\over{\partial x}} \, {{\partial}\over{\partial x'}} \,+ \,
{{\partial t'}\over{\partial x}} \, {{\partial}\over{\partial t'}} \,\,=\,\,
\alpha_\theta \, {{\partial}\over{\partial x'}} \,+ \,
\beta_\theta \, {{\partial}\over{\partial t'}}
\\ \displaystyle \vspace{-.3cm} ~ \\ \displaystyle
{{\partial}\over{\partial t}} \, \,\,=\,\,
{{\partial x'}\over{\partial t}} \, {{\partial}\over{\partial x'}} \,+ \,
{{\partial t'}\over{\partial t}} \, {{\partial}\over{\partial t'}} \,\,=\,\,
\delta_\theta \, {{\partial}\over{\partial x'}} \,+ \,
\gamma_\theta \, {{\partial}\over{\partial t'}} \, .
\end{array} \right. \vspace*{-6pt} \end{equation*}
We inject these operators inside the relation (\ref{edp}):
\smallskip \noindent $ \displaystyle \qquad \qquad
\Big( \delta_\theta \, {{\partial}\over{\partial x'}} \,+ \,
\gamma_\theta \, {{\partial}\over{\partial t'}} \Big) \, W
\,+ \, \Big( \alpha_\theta \, {{\partial}\over{\partial x'}} \,+ \,
\beta_\theta \, {{\partial}\over{\partial t'}} \Big) \, f(W)
\,=\, 0 $
\smallskip \noindent and we put in evidence the new partial derivatives:
\smallskip \noindent $ \displaystyle \qquad \qquad
{{\partial}\over{\partial t'}} \Big( \gamma_\theta \, W \, + \,
\beta_\theta \, f(W) \Big) \, + \,
{{\partial}\over{\partial x'}} \Big( \delta_\theta \, W \, + \,
\alpha_\theta \, f(W) \Big) \,=\, 0 \, . $
\smallskip \noindent
We introduce a bijective linear operator $ \, Y_\theta \,$
depending on $ \, \theta \in {\rm I}\! {\rm R} , \,$
acting from $ \, {\rm I}\! {\rm R}^N \,$ into $ \, {\rm I}\! {\rm R}^N \,$ and independent on
$\, x' \,$ and $ \, t' $:
\moneq \label{Y-theta}
Y_\theta \in {\cal GL} ( {\rm I}\! {\rm R}^N ) \,, \qquad \theta \in {\rm I}\! {\rm R} \, .
\vspace*{-6pt} \end{equation}
After applying $ \, Y_\theta \,$ to the previous equation, we obtain
a new conservation law equivalent to (\ref{edp}):
\moneq \label{edp-prime-vec}
{{\partial}\over{\partial t'}} \Big( \gamma_\theta \,\, Y_\theta \,{\scriptstyle \bullet } \, W \, + \,
\beta_\theta \,\, Y_\theta \,{\scriptstyle \bullet } \, f(W) \Big) \, + \,
{{\partial}\over{\partial x'}} \Big( \delta_\theta \,\, Y_\theta \,{\scriptstyle \bullet } \, W \, + \,
\alpha_\theta \,\, Y_\theta \,{\scriptstyle \bullet } \, f(W) \Big) \,=\, 0 \, .
\vspace*{-6pt} \end{equation}
A {\it sufficient} condition to establish the first equation of
(\ref{edp-prime}) is to identify the arguments of the two partial
derivatives $ \, \partial_{t'} \,$ and $ \, \partial_{x'} \, $ in the previous equation:
\moneq \label{transfo-vecteurs}
\left\{ \begin{array} [c]{l} \displaystyle
W' \quad \,\,\, =\, \gamma_\theta \, \, Y_\theta \,{\scriptstyle \bullet } \, W \, + \,
\beta_\theta \,\, Y_\theta \,{\scriptstyle \bullet } \, f(W)
\\ \displaystyle
f(W') \,=\, \delta_\theta \,\, Y_\theta \,{\scriptstyle \bullet } \, W \, + \,
\alpha_\theta \,\, Y_\theta \,{\scriptstyle \bullet } \, f(W) \, .
\end{array} \right. \vspace*{-6pt} \end{equation}
The relation (\ref{transfo-vecteurs}) can be written in a more compact form:
\moneq \label{transfo-vecteurs-2}
\begin{pmatrix} f(W') \cr W' \end{pmatrix}
\,=\, G_\theta \,\, Y_\theta \,{\scriptstyle \bullet } \,
\begin{pmatrix} f(W) \cr W \end{pmatrix}
\, \equiv \, G_\theta \,\, \begin{pmatrix}
Y_\theta \,{\scriptstyle \bullet } \, f(W) \cr Y_\theta \,{\scriptstyle \bullet } \, W \end{pmatrix} \, .
\vspace*{-6pt} \end{equation}
Under the ``linearity hypothesis'' (\ref{Y-theta}), the relation
(\ref{transfo-vecteurs-2}) gives geometrical constraints
for the transformation
$ \, \Omega \ni W \longmapsto W' \in \Omega \,$
associated with the action of the group $ \, \cal G $.
\monitem
We make an analogous calculus for the second equation of
(\ref{edp-prime}) relative to entropy. After the transformation of partial derivatives,
the relation (\ref{cons-entropy-vitesse}) takes the form
\smallskip \noindent $ \displaystyle \qquad \qquad
\Big( \delta_\theta \, {{\partial}\over{\partial x'}} \,+ \,
\gamma_\theta \, {{\partial}\over{\partial t'}} \Big) \, \eta(W)
\,+ \, \Big( \alpha_\theta \, {{\partial}\over{\partial x'}} \,+ \,
\beta_\theta \, {{\partial}\over{\partial t'}} \Big) \,
\big( \eta(W) \, u(W) \big) \,=\, 0 $
\smallskip \noindent
and can be written as
\moneq \label{edp-prime-entrop}
{{\partial}\over{\partial t'}} \Big( \gamma_\theta \,\, \eta(W) \, + \,
\beta_\theta \, \, \eta(W) \, u(W) \Big) \, + \,
{{\partial}\over{\partial x'}} \Big( \delta_\theta \, \, \eta(W) \, + \,
\alpha_\theta \, \, \eta(W) \, u(W) \Big) \,=\, 0 \, .
\vspace*{-6pt} \end{equation}
This is a scalar conservation law. This equation is identical to the second equation
of (\ref{edp-prime})
if the following {\it sufficient} conditions are satisfied:
\moneq \label{transfo-entropie}
\left\{ \begin{array} [c]{l} \displaystyle
\quad \,\, \, \eta(W') \quad \,\,\, =\, \gamma_\theta \, \, \eta(W) \, \, + \,
\beta_\theta \,\, \eta(W) \, u(W)
\\ \displaystyle
\eta(W') \, u(W') \,=\, \delta_\theta \,\, \eta(W) \, + \,
\alpha_\theta \,\, \eta(W) \, u(W) \, .
\end{array} \right. \vspace*{-6pt} \end{equation}
With a notation that makes the action of the group $ \, \cal G \, $ explicit:
\moneq \label{transfo-entropie-2}
\begin{pmatrix} \eta(W') \, u(W') \cr \eta(W') \end{pmatrix}
\,=\, G_\theta \,\,
\begin{pmatrix} \eta(W) \, u(W) \cr \eta(W) \end{pmatrix} \, .
\vspace*{-6pt} \end{equation}
\smallskip \monitem { \bf Proposition 2. Linear representation }
\noindent
If the relation (\ref{transfo-vecteurs-2}) is satisfied for every $ \, \theta \in {\rm I}\! {\rm R}, \,$
then the application $ \, \, {\cal G } \ni G_\theta \longmapsto \, Y_\theta
\in {\cal GL } ({\rm I}\! {\rm R}^N) \, $ is a linear representation of the group $ \, {\cal G }$:
\moneq \label{rep-linear}
Y_{\theta + \theta'} \,=\, Y_{\theta} \,{\scriptstyle \bullet }\, Y_{\theta'} \, , \qquad
\theta , \, \theta' \in {\rm I}\! {\rm R} \, .
\vspace*{-6pt} \end{equation}
\smallskip \noindent {\bf Proof of Proposition 2. }
\noindent
We first consider the relation (\ref{transfo-vecteurs-2})
and an analogous relation obtained by replacing the variable $ \, \theta \,$
by the variable $ \, \theta' $:
\moneqstar
\begin{pmatrix} f(W'') \cr W'' \end{pmatrix}
\,=\, G_\theta'\,\, Y_\theta' \,{\scriptstyle \bullet } \,
\begin{pmatrix} f(W') \cr W' \end{pmatrix}
\, \equiv \, G_\theta' \,\, \begin{pmatrix}
Y_\theta' \,{\scriptstyle \bullet } \, f(W') \cr Y_\theta' \,{\scriptstyle \bullet } \, W' \end{pmatrix} \, .
\vspace*{-6pt} \end{equation*}
We compose this relation with (\ref{transfo-vecteurs-2}) and we obtain
\moneqstar
\begin{pmatrix} f(W'') \cr W'' \end{pmatrix}
\,=\, G_{\theta'}\,\,\, G_\theta\,\,
\begin{pmatrix} Y_{\theta'} \,{\scriptstyle \bullet }\, Y_{\theta} \, {\scriptstyle \bullet } \, f(W) \cr
Y_{\theta'} \,{\scriptstyle \bullet }\, Y_{\theta} \, {\scriptstyle \bullet } \, W \end{pmatrix} \, .
\vspace*{-6pt} \end{equation*}
When we apply the relation (\ref{transfo-vecteurs-2}) with the argument
$ \, \theta + \theta' $, we have directly
\moneqstar
\begin{pmatrix} f(W'') \cr W'' \end{pmatrix}
\,=\, G_{\theta + \theta'}\,\,\,
\begin{pmatrix} Y_{\theta + \theta'} \,{\scriptstyle \bullet }\, f(W) \cr
Y_{\theta + \theta'} \, {\scriptstyle \bullet } \, W \end{pmatrix} \, .
\vspace*{-6pt} \end{equation*}
We look precisely to the two previous relations. The scalars
$ \, \alpha_\theta $, $ \, \beta_\theta $, $ \, \gamma_\theta \, $ and
$ \, \delta_\theta \, $ commute with the linear operator
$ \, Y_\theta \,$ and we obtain the relation (\ref{rep-linear})
because the two previous equalities are satisfied for an arbitrary state $W$.
$ \hfill \square $
\bigskip \bigskip \noindent {\bf \large 4) \quad Manifold of null velocity }
\monitem
In the following we denominate by $ \, \Omega_0 \,$
the set of states with a velocity equal to zero.
Because the mapping $ \, \Omega \ni W \longmapsto u(W) \in {\rm I}\! {\rm R} \,$ is scalar, it is natural
to suppose that $ \, \Omega_0 \,$ is a manifold in $\, {\rm I}\! {\rm R}^N \,$ of codimension one.
We observe also that the flux function $ \, f({\scriptstyle \bullet }) \,$ is reduced to its
thermodynamic contribution $ \, g({\scriptstyle \bullet }) \,$ on the manifold of null velocity.
Our program is to construct the states $W$, the flux function $ f({\scriptstyle \bullet }) $
and the mathematical entropy $ \eta({\scriptstyle \bullet }) $
when we suppose that the flux function
$ \, \Omega_0 \ni W_0 \longmapsto g_0(W_0) \in {\rm I}\! {\rm R}^N \, $
and the mathematical entropy
$ \, \Omega_0 \ni W_0 \longmapsto \sigma(W_0) \in {\rm I}\! {\rm R} \, $
are given on the manifold of null
velocity.
The idea is to use the group invariance to link a given state $ \, W \, $
with an appropriate state $ \, W_0 \, $ on the manifold $ \, \Omega_0 \,$
as presented in Figure~1. The conditions (\ref{transfo-vecteurs-2}) and
(\ref{transfo-entropie-2}) are geometric constraints that will allow
essentially to solve the problem.
\bigskip
\smallskip \smallskip
\centerline { \includegraphics[width=.45 \textwidth] {citv03.eps} }
\noindent {\bf Figure 1}. \quad
Fiber above the manifold of null velocity $\, \Omega_0 .$ Thanks to the action of
the group, a state $ \, W_0 \, $ of null velocity $\, \Omega_0 \,$
generates a current state $W$.
\smallskip \smallskip
\smallskip \monitem { \bf Proposition 3. Velocity field }
\noindent
With the hypothesis (\ref{transfo-vecteurs}) and if the mathematical entropy
$ \, \sigma \,$ on the manifold of null velocity is not the null function,
the velocity field $ \, u(W) \,$
is necessarily given by the relation
\moneq \label{groupe-vitesse}
u(W) \,= \, {{\delta_\theta}\over{\gamma_\theta}} \, .
\vspace*{-6pt} \end{equation}
\smallskip \noindent {\bf Proof of Proposition 3. }
\noindent
We consider a given state $ \, W_0 \,$ on the manifold
$ \, \Omega_0 \, $ and a running state $ \, W \in \Omega $ obtained by the relations
(\ref{transfo-vecteurs}). With the actual notations, we have
\moneq \label{transfo-vecteurs-3}
W \, =\, \gamma_\theta \, \, Y_\theta \,{\scriptstyle \bullet } \, W_0 \, + \,
\beta_\theta \,\, Y_\theta \,{\scriptstyle \bullet } \, g_0(W_0) \,, \qquad
f(W) \,=\, \delta_\theta \,\, Y_\theta \,{\scriptstyle \bullet } \, W_0 \, + \,
\alpha_\theta \,\, Y_\theta \,{\scriptstyle \bullet } \, g_0(W_0) \, .
\vspace*{-6pt} \end{equation}
The analogous relations (\ref{transfo-entropie}) relative to the entropy
take a simple form
\moneq \label{transfo-entropie-3}
\eta(W) \, =\, \gamma_\theta \, \sigma(W_0) \,\,, \qquad
\eta(W) \, u(W) \,=\, \delta_\theta \,\, \sigma(W_0) \, .
\vspace*{-6pt} \end{equation}
because the velocity is also null on $ \, \Omega_0 $.
We consider a state $W_0$ such that $ \, \sigma(W_0) \not= 0 .$
If $ \, \gamma_\theta \,$ is null, then $\, \eta(W) = 0 \, $ is null and
$\, \delta_\theta = 0 \,$ because $ \, \sigma(W_0) \not= 0 .$
This contradicts the fact that the matrix
defining $\, G_\theta \,$ by the relation (\ref{groupe-general})
is invertible. Then $ \, \gamma_\theta \not = 0 \,$
and the relation
(\ref{groupe-vitesse}) is obtained by taking the ratio of the two
relations of (\ref{transfo-entropie-3}).
$ \hfill \square $
\smallskip \monitem
With the example proposed in (\ref{groupe-particulier}), we have
\moneq \label{vitesse-groupe}
u \,=\, c \, {{S_\theta}\over{C_\theta}} \, .
\vspace*{-6pt} \end{equation}
We observe that we have also established the following property:
\smallskip \newpage \monitem { \bf Proposition 4. Entropy field }
\noindent With the above hypotheses, if the state $ \, W \,$ is given
from $ \, W_0 \in \Omega_0 \,$
with the help of the relations (\ref{transfo-vecteurs-3}),
we have necessarily
\moneq \label{entropie}
\eta(W) \, =\, \gamma_\theta \, \sigma(W_0) \, .
\vspace*{-6pt} \end{equation}
\monitem { \bf Hypothesis 1.
The null-velocity manifold $ \Omega_0 \, $ is included in an hyperplane}
\noindent
We remark that if a convex restriction of the mathematical entropy
$ \, \Omega_0 \ni W_0 \longmapsto \sigma(W_0) \in {\rm I}\! {\rm R} \, $
is given on the manifold $ \, \Omega_0 $,
a natural hypothesis is to suppose $ \, \Omega_0 \, $
convex. We will do this hypothesis in the following. Moreover,
we suppose that $ \, \Omega_0 \, $ is flat; this is expressed
by our hypothesis~1.
\smallskip \monitem { \bf Proposition 5. Thermodynamic flux }
\noindent With the above framework (\ref{transfo-vecteurs-3}),
the thermodynamic flux $ \, g(W) \,$ is given from the
datum $ \, g_0(W_0) \,$ according to
\moneq \label{thermo-flux}
g (W) \, = \, \big( \alpha_\theta \, - \, u(W) \, \beta_\theta \, \big) \,
Y_\theta \,{\scriptstyle \bullet }\, g_0 (W_0) \, .
\vspace*{-6pt} \end{equation}
\smallskip \noindent {\bf Proof of Proposition 5. }
\noindent
The second relation of (\ref{transfo-vecteurs-3}) and the definition
of the thermodynamic flux (\ref{def-flux-thermo}) allows us to evaluate
the function $\, g({\scriptstyle \bullet }) .$ We have
\smallskip \noindent $ \displaystyle
g(W) \, = \, f(W) - u(W) \, W $
\smallskip \noindent $ \displaystyle \qquad \,\,\,\,
\, = \, \delta_\theta \,\, Y_\theta \,{\scriptstyle \bullet } \, W_0 \, + \,
\alpha_\theta \,\, Y_\theta \,{\scriptstyle \bullet } \, g_0(W_0) \, - \,
u(W) \, \big( \gamma_\theta \, \, Y_\theta \,{\scriptstyle \bullet } \, W_0 \, + \,
\beta_\theta \,\, Y_\theta \,{\scriptstyle \bullet } \, g_0(W_0) \big) \, $
\smallskip \noindent $ \displaystyle \qquad \,\,\,\,
\, = \, \big( \alpha_\theta \, - \, u(W) \, \beta_\theta \, \big) \,
Y_\theta \,{\scriptstyle \bullet } \, g_0(W_0) \, $ \qquad due to (\ref{groupe-vitesse})
\smallskip \noindent
and the relation (\ref{thermo-flux}) is established.
$ \hfill \square $
\monitem
For the groups proposed in (\ref{groupe-particulier}), we have
$ \,\, \, \alpha_\theta \, - \, u(W) \, \beta_\theta \,=\,
C_\theta - c \, {{S_\theta}\over{C_\theta}} \, {{\varepsilon}\over{c}} \, S_\theta
\,=\, {{1}\over{C_\theta}} $ \, \quad due to (\ref{trigo}).
Then the relation (\ref{thermo-flux}) takes the form
\moneq \label{thermo-flux-2}
g (W) \, =\, {{1}\over{C_\theta}} \, Y_\theta \,{\scriptstyle \bullet }\, g_0 (W_0) \, .
\vspace*{-6pt} \end{equation}
\bigskip \bigskip \noindent {\bf \large 5) \quad Entropy variables }
\monitem
Even if the state and the entropy
are only partially known with the help of relations (\ref{transfo-vecteurs-3})
and (\ref{entropie}), its jacobian $\, \varphi \,$ introduced in (\ref{var-entropic})
relative to the conserved variables $ \, W \, $ can be essentially determined.
We first consider the dynamical equations present in this study.
First, the relation
\moneq \label{evol-group}
G_{\theta + \theta'} \, =\, G_{\theta} \, {\scriptstyle \bullet } \, G_{\theta'}
\vspace*{-6pt} \end{equation}
and the relation (\ref{rep-linear}) for the matrices $ \, Y_\theta \,$
can be differentiated relatively to $ \, \theta $. We have
\moneq \label{derive-groupe}
{{{\rm d}G_\theta}\over{{\rm d}\theta}} \,=\, G'_0 \, G_\theta
\,=\, G_\theta \, G'_0 \,, \qquad
{{{\rm d}Y_\theta}\over{{\rm d}\theta}} \,=\, Y'_0 \, Y_\theta
\,=\, Y_\theta \, Y'_0 \, .
\vspace*{-6pt} \end{equation}
With the choice (\ref{groupe-particulier})
of one classical group, we have
\moneq \label{derivee-zero-groupe}
G'_0 \, = \, \begin{pmatrix} 0 & c \cr
\displaystyle {{\varepsilon}\over{c}} & 0 \end{pmatrix} \, \vspace*{-6pt} \end{equation}
and we can precise how to extend the differential relatations
(\ref{evol-group}) and (\ref{derive-groupe}).
\smallskip \monitem { \bf Proposition 6. Differential relations for the entropy variables }
\noindent With the above context, we have
\moneq \label{derive-theta-W0}
G'_0 \, \begin{pmatrix} u \, \eta^* + \varphi \, g \cr \eta^* \end{pmatrix}
\, {\rm d}\theta \,+\, \begin{pmatrix} \varphi \, Y'_0 \, f(W) \cr
\varphi \, Y'_0 \,\, W \end{pmatrix} \, {\rm d}\theta \,+\, G_\theta \,
\begin{pmatrix} \varphi \, Y_\theta \, {\rm d}g_0 \cr
\varphi \, Y_\theta \, {\rm d}W_0 - {\rm d}\sigma \end{pmatrix} \,=\, 0
\, \vspace*{-6pt} \end{equation}
\smallskip \noindent {\bf Proof of Proposition 6. }
\noindent
We write the relation (\ref{transfo-vecteurs-3}) under a matricial form
\moneq \label{transfo-vecteurs-4}
\begin{pmatrix} f(W) \cr W \end{pmatrix} \,=\, \, G_\theta \,
\begin{pmatrix} Y_\theta \, g_0(W_0) \cr Y_\theta \, W_0 \end{pmatrix} \,.
\vspace*{-6pt} \end{equation}
We differentiate the relation (\ref{transfo-vecteurs-4}), taking into account the
two contributions relative to
the group $\, {\cal G} \,$ on one hand and along the manifold
$ \, \Omega_0 \,$ on the other hand, as illustrated on the figure~1. Then
\smallskip \noindent $ \displaystyle
\begin{pmatrix} {\rm d}f \cr {\rm d} W \end{pmatrix} \,=\,
G_0' \,\, G_\theta \,
\begin{pmatrix} Y_\theta \, \, g_0 \cr Y_\theta \, \, W_0 \end{pmatrix} \, {\rm d}\theta
\,+\, G_\theta \, \begin{pmatrix} Y'_0 \,\, Y_\theta \,\, g_0 \cr
Y'_0 \,\, Y_\theta \,\, W_0 \end{pmatrix} \, {\rm d}\theta \,+\, G_\theta \,
\begin{pmatrix} Y_\theta \,\, {\rm d} g_0 \cr Y_\theta \,\, {\rm d} W_0 \end{pmatrix}
\,$
\smallskip \noindent
and because
\smallskip \noindent $ \displaystyle
G_\theta \, \begin{pmatrix} Y'_0 \, \, Y_\theta \,\, g_0 \cr
Y'_0 \, \,Y_\theta \,\, W_0 \end{pmatrix} \,=\,
\begin{pmatrix} \alpha_\theta \, Y'_0 \,\, Y_\theta \,\, g_0
\,+\, u\, \, \gamma_\theta \,\, Y'_0 \,\, Y_\theta \,\, W_0 \cr
\beta_\theta \ \,, Y'_0 \,\, Y_\theta \,\, g_0 \,+\,
\gamma_\theta \,\, Y'_0 \,\, Y_\theta \,\, W_0 \end{pmatrix} \,=\,
\begin{pmatrix} Y'_0 \, \big( \alpha_\theta \,\, Y_\theta \,\, g_0
\,+\, u\, \, \gamma_\theta \,\, Y_\theta \,\, W_0 \big) \cr
Y'_0 \, \big( \beta_\theta \, \, Y_\theta \,\, g_0 \,+\,
\gamma_\theta \,\, Y_\theta \,\, W_0 \big) \end{pmatrix}
$
\smallskip \noindent $ \displaystyle \qquad \qquad \qquad \quad \,=\,
\begin{pmatrix} Y'_0 \,\, f(W) \cr Y'_0 \,\, W \end{pmatrix} \, $ ,
\smallskip \noindent
we have
\moneq \label{derive-transfo-vecteurs-4}
\begin{pmatrix} {\rm d}f \cr {\rm d} W \end{pmatrix} \,=\,
G'_0 \, \begin{pmatrix} f \cr W \end{pmatrix} \, {\rm d}\theta \,+\,
\begin{pmatrix} Y'_0 \,\, f \cr Y'_0 \,\, W \end{pmatrix} \, {\rm d}\theta \,+\,
G_\theta \, \begin{pmatrix} Y_\theta \,\, {\rm d} g_0 \cr
Y_\theta \,\, {\rm d} W_0 \end{pmatrix} \, .
\vspace*{-6pt} \end{equation}
We multiply each line of the relation (\ref{derive-transfo-vecteurs-4})
by the line vector $ \, \varphi .$ We remark that
\smallskip \noindent $ \displaystyle
\varphi \, G'_0 \, \begin{pmatrix} {\rm d}f \cr {\rm d} W \end{pmatrix} \,=\,
\varphi \, \begin{pmatrix} \alpha'_0 & (u \, \gamma)'_0 \cr
\beta'_0 & \gamma'_0 \end{pmatrix} \,
\begin{pmatrix} {\rm d}f \cr {\rm d} W \end{pmatrix} \,=\,
\varphi \, \begin{pmatrix} \alpha'_0 \,\, f \,+\, (u \, \gamma)'_0 \,\, W \cr
\beta'_0 \, \, f + \gamma'_0 \,\, W \end{pmatrix} \, $
\smallskip \noindent $ \displaystyle \qquad \qquad \qquad \,=\,
\begin{pmatrix} \alpha'_0 \,\, \varphi \,\, f \,+\, (u \, \gamma)'_0 \,\, \varphi \,\, W \cr
\beta'_0 \,\, \varphi \, \, f + \gamma'_0 \,\, \varphi \,\, W \end{pmatrix} \,=\,
G'_0 \, \begin{pmatrix} \varphi \,\, f \cr \varphi \,\, W \end{pmatrix} \, $
\smallskip \noindent
and in a similar way,
\smallskip \noindent $ \displaystyle
\varphi \,\, G_\theta \,\, Y_\theta \,\, \begin{pmatrix} {\rm d}g_0 \cr
{\rm d} W_0 \end{pmatrix} \,=\, \varphi \,\, \begin{pmatrix}
\alpha_\theta \,\, Y_\theta \,\, {\rm d}g_0 \,+\, (u \, \gamma)_\theta \, \,
Y_\theta \,\, {\rm d}W_0 \cr \beta_\theta \,\, Y_\theta \,\, {\rm d}g_0 \,+\,
\gamma_\theta \,\, Y_\theta \,\, {\rm d}W_0 \end{pmatrix} $
\smallskip \noindent $ \displaystyle \qquad \qquad \qquad \quad \ \,=\,
\begin{pmatrix} \alpha_\theta \,\, \varphi \,\, Y_\theta \,\, {\rm d}g_0 \,+\,
(u \, \gamma)_\theta \,\, \varphi \,\, Y_\theta \,\, {\rm d}W_0 \cr
\beta_\theta \,\, \varphi \,\, Y_\theta \, \, {\rm d}g_0 \,+\,
\gamma_\theta \,\, \varphi \,\, Y_\theta \,\, {\rm d}W_0 \end{pmatrix} \,=\,
G_\theta \, \begin{pmatrix} \varphi\, \, Y_\theta \,\, {\rm d}g_0 \cr
\varphi \, \, Y_\theta \,\, {\rm d}W_0 \end{pmatrix} \, . $
\smallskip \noindent
Then
\moneq \label{entropic-derive-transfo-vecteurs-4}
\begin{pmatrix} \varphi \, {\rm d}f \cr \varphi \, {\rm d}W \end{pmatrix} \,=\,
G_0' \, \begin{pmatrix} \varphi \, \, f \cr \varphi \,\, W \end{pmatrix} \, {\rm d}\theta
\,+\, \begin{pmatrix} \varphi \,\, Y'_0 \,\, f \cr \varphi \, Y'_0 \,\, W
\end{pmatrix} \, {\rm d}\theta \,+\, G_\theta \, \begin{pmatrix} \varphi \,\, Y_\theta \,
\, {\rm d}g_0 \cr \varphi \,\, Y_\theta \, \, {\rm d}W_0 \end{pmatrix} \, .
\vspace*{-6pt} \end{equation}
\monitem
We develop the same calculus for the entropy. We start with a matricial form
of the relation (\ref{transfo-entropie-3}):
\moneq \label{transfo-entropie-4}
\begin{pmatrix} \eta \, u \cr \eta \end{pmatrix} \,=\,
G_\theta \, \begin{pmatrix} 0 \cr \sigma \end{pmatrix}
\vspace*{-6pt} \end{equation}
Then
\moneq \label{derive-transfo-entropie-4}
\begin{pmatrix} {\rm d} ( \eta \, u ) \cr {\rm d} \eta \end{pmatrix} \,=\,
G'_0 \, \begin{pmatrix} \eta \, u \cr \eta \end{pmatrix} \, {\rm d} \theta
\,+\, G_\theta \, \begin{pmatrix} 0 \cr {\rm d} \sigma \end{pmatrix} \,.
\vspace*{-6pt} \end{equation}
We make the difference between the relations
(\ref{derive-transfo-entropie-4}) and (\ref{entropic-derive-transfo-vecteurs-4}).
Due to (\ref{flux-entropy}) and (\ref{var-entropic})
the left hand side is equal to zero!
Then the right hand side can be written as
\smallskip \noindent \qquad \qquad $ \displaystyle
G_0' \, \begin{pmatrix} \varphi \, f - \eta \, u
\cr \varphi \, W - \eta \end{pmatrix} \, {\rm d}\theta
\,+ \, \begin{pmatrix} \varphi \,\, Y'_0 \,\, f \cr \varphi \,\, Y'_0 \,\, W
\end{pmatrix} \, {\rm d}\theta \,+\, G_\theta \, \begin{pmatrix} \varphi \,\, Y_\theta \,
\, {\rm d}g_0 \cr \varphi \,\, Y_\theta \, \, {\rm d}W_0 \, - \, {\rm d} \sigma
\end{pmatrix} \,. $
\smallskip \noindent
The relation (\ref{derive-theta-W0}) is now easily obtained by the explicitation of the dual
entropy introduced in (\ref{dual-entropy}) and the thermodynamic flux defined in
(\ref{def-flux-thermo}).
$ \hfill \square $
\smallskip \monitem { \bf Proposition 7. Constraints for the entropy variables }
\noindent
With the above framework, if we suppose moreover that the derivative
of the group $\, \cal G \,$ for $ \, \theta = 0 \,$ is given by the relation
(\ref{derivee-zero-groupe}),
we have the following relations for the entropy variables
\moneq \label{variables-entropiques-cn1}
c \, \eta^* \,+\, \varphi \,\, Y_0' \,\, f(W) \,=\, 0
\vspace*{-6pt} \end{equation} {\vspace*{-.7 cm}}
\moneq \label{variables-entropiques-cn2}
{{\varepsilon}\over{c}} \, \big( u \, \eta^* \,+\, \varphi \, g(W) \big)
\,+\, \varphi \,\, Y_0' \,\, W \,=\, 0
\vspace*{-6pt} \end{equation} {\vspace*{-.7 cm}}
\moneq \label{variables-entropiques-cn3}
\varphi \, Y_\theta \,\, {\rm d}g_0 \,=\, 0
\vspace*{-6pt} \end{equation} {\vspace*{-.7 cm}}
\moneq \label{variables-entropiques-cn4}
\varphi \,\, Y_\theta \,\, {\rm d}W_0 \, - \, {\rm d}\sigma \,=\, 0 \, .
\vspace*{-6pt} \end{equation}
\smallskip \noindent {\bf Proof of Proposition 7. }
\noindent
With the matrix $ \, G_0' \,$ chosen according to (\ref{derivee-zero-groupe}),
the relation (\ref{derive-theta-W0}) becomes
\smallskip \noindent \qquad \qquad $ \displaystyle
\begin{pmatrix} c \, \eta^* \cr
{{\varepsilon}\over{c}} \, \big( u \, \eta^* \,+\, \varphi \, g(W) \big)
\end{pmatrix} \, {\rm d}\theta \,+\, \begin{pmatrix} \varphi \, Y'_0 \, f(W) \cr
\varphi \, Y'_0 \,\, W \end{pmatrix} \, {\rm d}\theta \,+\, G_\theta \,
\begin{pmatrix} \varphi \, Y_\theta \, {\rm d}g_0 \cr
\varphi \, Y_\theta \, {\rm d}W_0 - {\rm d}\sigma \end{pmatrix} \,=\, 0 $
\smallskip \noindent
and the relations (\ref{variables-entropiques-cn1}) to (\ref{variables-entropiques-cn4})
are clear.
$ \hfill \square $
\monitem
The relation (\ref{variables-entropiques-cn4}) can be expressed with the help of the
gradient $ \, \varphi_0 \,$ of the entropy restricted on the manifold $ \, \Omega_0 $:
\moneq \label{variables-entropiques-vit-nulles}
{\rm d}\sigma \,\equiv \, \varphi_0 \, {\scriptstyle \bullet } \, {\rm d}W_0 \, .
\vspace*{-6pt} \end{equation}
Then the relation (\ref{variables-entropiques-cn4}) express that the
difference $ \, \varphi \, Y_\theta - \varphi_0 \,$ belongs to the
polar set $ \, \Omega_0^0 \,$ of $ \, \Omega_0 $: we can write
$ \, \varphi \, Y_\theta - \varphi_0 \,=\, \mu \, r_0 \,$
for some scalar $\, \mu \,$ and some non-null linear form $ \, r_0 $
identically equal to zero on $ \, \Omega_0$.
We make also a new hypothesis.
\monitem { \bf Hypothesis 2. The entropy $ \, \sigma \,$ and the thermodynamic flux
$ \, g_0 \, $ are weakly decoupled on the hyperplane $\Omega_0$}
\noindent
In other terms,
\moneq \label{bon-decouplage}
( \varphi_0 \,,\, {\rm d}g_0 \,{\scriptstyle \bullet }\, \rho_0 ) \,= \, 0 \,, \qquad \forall \rho_0 \in
\Omega_0 \, .
\vspace*{-6pt} \end{equation}
\smallskip \monitem { \bf Proposition 8. Explicitation of the entropy variables }
\noindent
We make the hypotheses 1 and 2 and suppose moreover
\moneq \label{pas-trivial}
\exists \, \rho_0 \in \Omega_0 \,, \quad {\rm d}g_0 \,{\scriptstyle \bullet }\, \rho_0 \not= 0 \, .
\vspace*{-6pt} \end{equation}
Then the relations (\ref{variables-entropiques-cn1}) to (\ref{variables-entropiques-cn4})
imply a simple form for the entropy variables:
\moneq \label{var-entrop-simple}
\varphi \, Y_\theta \,=\, \varphi_0 \, .
\vspace*{-6pt} \end{equation}
\smallskip \noindent {\bf Proof of Proposition 8. }
\noindent
Consider an arbitrary vector $ \, \rho_0 \in \Omega_0 $.
When we apply the relation
$ \, \varphi \, Y_\theta - \varphi_0 \,=\, \mu \, r_0 \,$
established previously to the vector $ \, {\rm d}g_0 \,{\scriptstyle \bullet }\, \rho_0 $,
we find zero due to the constraint (\ref{variables-entropiques-cn3}) and
the hypothesis~2 with (\ref{bon-decouplage}).
Then $ \, \mu \, < r_0 \,,\, {\rm d}g_0 \,{\scriptstyle \bullet }\, \rho_0 > \, = \, 0 .\,$
With the choice of $ \, \rho_0 \,$ suggested in (\ref{pas-trivial}),
the duality product $ \, < r_0 \,,\, {\rm d}g_0 \,{\scriptstyle \bullet }\, \rho_0 > \,$
is not null. So $ \, \mu = 0 \,$ and the relation (\ref{var-entrop-simple})
is established.
$ \hfill \square $
\monitem { \bf Hypothesis 3. The entropy $ \, \sigma \,$ and the thermodynamic flux
$ \, g_0 \, $ are strongly decoupled on the hyperplane $\Omega_0$}
\noindent
In other terms,
\moneq \label{fort-decouplage}
\varphi_0 (W_0) \, {\scriptstyle \bullet } \, g_0 (\widetilde{W_0}) \, \,= \,\, 0 \,,
\qquad \forall \, W_0, \, \widetilde{W_0} \in \Omega_0 \, .
\vspace*{-6pt} \end{equation}
We observe that this strong hypothesis (\ref{fort-decouplage})
implies (\ref{bon-decouplage}) by differentiation relatively to
the variable $ \, \widetilde{W_0} \,$ in $ \, \Omega_0 $.
In consequence, we have a new property for the entropy variables.
\smallskip \monitem { \bf Proposition 9. The entropy variables in the polar set of the thermo flux }
\noindent
We make the hypotheses 1 and 3 and suppose also the technical
condition (\ref {pas-trivial}) proposed previously.
Then we have a complete decoupling:
\moneq \label{varentrop-ortho-thermo-flux}
\varphi (W) \, {\scriptstyle \bullet } \, g({\widetilde{W}}) \, \equiv \, 0 \,
\qquad \forall \, W , \, \widetilde{W} \in \Omega \, .
\vspace*{-6pt} \end{equation}
\smallskip \noindent {\bf Proof of Proposition 9. }
\noindent
We just have to precise the expression (\ref{thermo-flux-2}) of the thermodynamic flux.
Then we have
\smallskip \noindent \qquad $ \displaystyle
\varphi (W) \, {\scriptstyle \bullet } \, g({\widetilde{W}}) \,= \,
\varphi (W) \, \,Y_\theta \,\, Y_{-\theta} \,\, g({\widetilde{W}}) \,= \,
\varphi_0 (W_0) \,\, {{1}\over{C_\theta}} \, g_0({\widetilde{W_0}}) \,= \, 0
$
\smallskip \noindent
due to (\ref{varentrop-ortho-thermo-flux}).
$ \hfill \square $
\smallskip \monitem { \bf Proposition 10. Constraints for the representation $Y$ }
\noindent
Under the same hypotheses that the ones done for Proposition~9, we have
\moneq \label{variables-entropiques-cn1bis}
c \, \eta^* \,+\, C_\theta \,\, \varphi_0 \,\, Y_0' \,\, g_0 \,=\, 0
\vspace*{-6pt} \end{equation} {\vspace*{-.7 cm}}
\moneq \label{variables-entropiques-cn2bis}
\varphi_0 \, \, Y_0' \,\, W_0 \,=\, 0 \, \quad {\rm on} \, \, \Omega_0 \, .
\vspace*{-6pt} \end{equation}
\smallskip \noindent {\bf Proof of Proposition 10. }
\noindent
We first have the following calculus:
\smallskip \noindent \qquad $ \displaystyle
\varphi \, \, Y_0' \, \, g
\,=\, \varphi \, \, Y_\theta \, \, Y_{-\theta} \,\, Y_0' \, \, g
\,=\, \varphi \, \, Y_\theta \, \, Y_0' \, \, Y_{-\theta} \,\, g
\,=\, \varphi_0 \, \, Y_0' \, \, {{1}\over{C_\theta}} \, g_0
\,=\, {{1}\over{C_\theta}} \, \varphi_0 \, \, Y_0' \, \, g_0 \, . $
\smallskip \noindent
Then we multiply the equation (\ref{variables-entropiques-cn2}) by
the opposite of the velocity $\, u \, $ and we add it to
(\ref{variables-entropiques-cn1}). With the definition (\ref{def-flux-thermo})
of the thermodynamic flux, we have, taking into account the
``trigonometrical relations'' (\ref{trigo}) and the expression (\ref{groupe-vitesse}) of
the velocity:
\smallskip \noindent $ \displaystyle
0 \,=\,
\eta^* \, \Big( c - {{\varepsilon \, u^2}\over{c}} \Big) \,+ \, \varphi \, \, Y_0' \, \, g
\,=\, \eta^* \, \Big( c - {{\varepsilon \, c \, S_\theta^2}\over{C_\theta^2}} \Big)
\,+ \, {{1}\over{C_\theta}} \, \varphi_0 \, \, Y_0' \, \, g_0
\,=\,
{{c\, \eta^*}\over{C_\theta^2}} \,+\, C_\theta \, \varphi_0 \, \, Y_0' \, \, g_0 $
\smallskip \noindent
and the relation (\ref{variables-entropiques-cn1bis}) is established.
We report now the expression (\ref{transfo-vecteurs-3})
of a current state $ \, W\, $
inside the equation (\ref{variables-entropiques-cn2}):
\smallskip \noindent $ \displaystyle
0 \,=\, {{\varepsilon}\over{c}} \, u \, \eta^* \,+\,
\varphi \, \, Y_0' \, \, Y_\theta \,{\scriptstyle \bullet } \,\Big(
C_\theta \, W_0 \, + \, {{\varepsilon}\over{c}} \, S_\theta \, g_0 \Big)
\,=\, C_\theta \,\varphi \, \, Y_\theta \, \, Y_0' \,\, W_0
\, + \, {{\varepsilon}\over{c}} \,\Big(
u \, \eta^* \, +\, S_\theta \,
\varphi \, \, Y_\theta \, \, Y_0' \,\, g_0 \Big) $
\smallskip \noindent $ \displaystyle \quad
= \, C_\theta \,\varphi \, \, Y_\theta \, \, Y_0' \,\, W_0
\, + \, {{\varepsilon}\over{c}} \,\Big(
u \, \eta^* \, - \, S_\theta \, {{c}\over{C_\theta}} \, \eta^* \Big)
\, = \, C_\theta \,\,\varphi \, \, Y_\theta \, \, Y_0' \,\, W_0
\,+\, \varepsilon \, \eta^* \, \Big( {{u}\over{c}} \,-\,
{{S_\theta}\over{C_\theta}} \Big) $
\smallskip \noindent $ \displaystyle \quad
= \, C_\theta \,\,\varphi \, \, Y_\theta \, \, Y_0' \,\, W_0 $
\smallskip \noindent
and the relation (\ref{variables-entropiques-cn2bis}) follows
from (\ref{var-entrop-simple}).
$ \hfill \square $
\smallskip \monitem { \bf Proposition 11. Dual entropy }
\noindent
Under the hypotheses proposed at Proposition~9, we denote by
$ \, \sigma^* \,$ the dual of the entropy $\, \sigma \,$
restricted on the manifold of null velocity:
\moneq \label{dual-entropie-omega-zero}
\sigma^* \,\equiv \, \varphi_0 \,{\scriptstyle \bullet } \, W_0 \,-\, \sigma (W_0) \,
\quad {\rm on} \, \, \Omega_0 \, , \qquad {\rm with } \quad \varphi_0 = \sigma'(W_0) \,.
\vspace*{-6pt} \end{equation}
Then we have
\moneq \label{dual-entropie}
\eta^* \,=\, C_\theta \, \, \sigma^* \,
\vspace*{-6pt} \end{equation}
and the relation (\ref{variables-entropiques-cn1bis})
takes the form
\moneq \label{variables-entropiques-cn1ter}
c \, \, \sigma^* \,+\, \, \varphi_0 \,\, Y_0' \,\, g_0 \,=\, 0
\qquad {\rm on} \, \, \Omega_0 \, .
\vspace*{-6pt} \end{equation}
\smallskip \noindent {\bf Proof of Proposition 11. }
\noindent
We have, thanks to (\ref{dual-entropy}), (\ref{entropie}), (\ref{transfo-vecteurs-4})
and (\ref{var-entrop-simple})~:
\smallskip \noindent \qquad $ \displaystyle
\eta^* \, = \, \varphi \, {\scriptstyle \bullet } \, W \, - \, \eta(W)
\,=\, \varphi_0 \, {\scriptstyle \bullet } \, Y_{- \theta} \, \, Y_{\theta} \, \big( C_\theta \, W_0
\,+\, {{\varepsilon}\over{c}} \, S_\theta \, g_0 \big)
\, - C_\theta \, \sigma
\, \, =\, C_\theta \, \big( \varphi_0 \, {\scriptstyle \bullet } \, W_0 \,-\, \sigma \big)
$
\smallskip \noindent due to Hypothesis 3 (relation (\ref{fort-decouplage})).
The end of the proof is clear.
$ \hfill \square $
\bigskip \bigskip \noindent {\bf \large 6) \quad Hyperbolic systems of order two }
\monitem
If we suppose $ \, N = 2 , \,$ the matrix $ \, Y_\theta \,$ is a two by two matrix. We
suppose here that this matrix has the same algebaic form that the form
(\ref{groupe-particulier}) suggested for $ \, G_\theta .\, $ We set for this
contribution
\moneq \label{matrice-Y-theta}
Y_\theta \, = \, \begin{pmatrix} {\widetilde C}_\theta & \displaystyle
{ {\widetilde \varepsilon}\over{a}} \, {\widetilde S}_\theta \cr
\displaystyle a \, {\widetilde S}_\theta & $\,\,$ {\widetilde C}_\theta
\end{pmatrix} \,, \qquad \theta \in {\rm I}\! {\rm R} \,, \quad a > 0 \, . \vspace*{-6pt} \end{equation}
In the relation (\ref{matrice-Y-theta}), the functions $ \, {\widetilde S}_\theta \,$
and $ \, {\widetilde C}_\theta \,$ are analogous to the ones proposed
in (\ref{gpe-galilee}), (\ref{gpe-lorentz}), (\ref{gpe-circular}) and
(\ref{trigo})
for $ \, S_\theta \,$ and $ \, C_\theta .\,$ We have in particular
\moneq \label{trigo-tilde}
{\widetilde C}_\theta^2 \, - \, {\widetilde \varepsilon} \,
{\widetilde S}_\theta^2 \,\, \equiv \,\, 1 \, .
\vspace*{-6pt} \end{equation}
The nilpotent representation of the group $\, {\cal G} \,$ corresponds to
$ \, {\widetilde \varepsilon} = 0 ,\,$ the hyperbolic representation
to $ \, {\widetilde \varepsilon} = 1 \,$ and the elliptic one
to $ \, {\widetilde \varepsilon} = -1 . \,$ The derivative $ \, Y_\theta ' \,$ of
the matrix $ \, Y_\theta \,$ proposed in (\ref{matrice-Y-theta}) at $ \,\theta
= 0 \,$ is given by
\moneq \label{matrice-Y-prime-zero}
Y_0' \, = \, \begin{pmatrix} 0 & \displaystyle { {\widetilde \varepsilon}\over{a}} \, \cr
\displaystyle a \, & 0 \end{pmatrix} \,.
\vspace*{-6pt} \end{equation}
\monitem
We particularize the system by a simple choice for the null-velocity manifold.
As in our previous work \cite{Du01}, we suppose
\moneq \label{variete-zero-Neq2}
W_0 \, = \, \begin{pmatrix} \rho_0 \cr 0 \end{pmatrix} \, \, \in \Omega_0 \,
\vspace*{-6pt} \end{equation}
and
\moneq \label{g-zero-Neq2}
g_0 (W_0) \, = \, \begin{pmatrix} 0 \cr p_0 \end{pmatrix} \,
\quad {\rm for} \,\, W_0 \in \Omega_0 \, .
\vspace*{-6pt} \end{equation}
With this choice, the hypotheses 2 and 3 are satisfied.
The variable $ \, p_0 \,$ in the right hand side of
the relation (\ref{g-zero-Neq2}) is the thermodynamic pressure.
\smallskip \monitem { \bf Proposition 12. Pressure as a thermodynamic variable }
\noindent With the above hypotheses, we have the relation
\moneq \label{pression}
\sigma^* \,+\, {{\widetilde \varepsilon}\over{a \, c}} \,\,\sigma' \,\, p_0 \,=\, 0 \, .
\vspace*{-6pt} \end{equation}
\smallskip \noindent {\bf Proof of Proposition 12. }
\noindent Just write the relation (\ref{variables-entropiques-cn1ter})
with the help of (\ref{matrice-Y-prime-zero}) and (\ref{variete-zero-Neq2})~:
\smallskip \noindent $ \displaystyle
c \, \, \sigma^* \,+\, \, \varphi_0 \,\, Y_0' \,\, g_0 \,=\,
c \, \, \sigma^* \,+\, \begin{pmatrix} \sigma' & 0 \end{pmatrix}
\,\, \begin{pmatrix} 0 & \displaystyle { {\widetilde \varepsilon}\over{a}} \, \cr
\displaystyle a \, & 0 \end{pmatrix} \,\,
\begin{pmatrix} 0 \cr p_0 \end{pmatrix} \,=\,
c \, \, \sigma^* \,+\, {{\widetilde \varepsilon}\over{a}} \, \sigma' \,\, p_0 $
\smallskip \noindent and the relation (\ref{pression}) is established.
$ \hfill \square $
\monitem
The nilpotent case ($ \, \widetilde \varepsilon = 0 \,$)
has no interest when $ \, N=2 \,$ because the dual of the entropy is
necessary null, as we have
remarked in \cite{Du01} in the particular case of the Galileo group ($\varepsilon = 0$).
We can explicit the first relations of (\ref{transfo-vecteurs-3}). With the notation
\moneq \label{variables-conservatives}
W \, \equiv \, \begin{pmatrix} \rho \cr J \end{pmatrix} \,
\vspace*{-6pt} \end{equation}
we have
\moneq \label{variables-conservatives-2}
\rho \, = \, C_\theta \, {\widetilde C}_\theta \, \rho_0
\,+\, {{\varepsilon \, \widetilde \varepsilon}\over{a \, c}} \,
S_\theta \, {\widetilde S}_\theta \, p_0 \,, \quad
{{J}\over{a}} \, = \, C_\theta \, {\widetilde S}_\theta \, \rho_0
\,+\, {{\varepsilon }\over{a \, c}} \, S_\theta \, {\widetilde C}_\theta \, p_0 \,.
\vspace*{-6pt} \end{equation}
\monitem
In the case of Galileo group, we have
$\varepsilon = 0$, $ C_\theta = 1$, $S_\theta = \theta$ and $ u = c \, \theta$.
The explicitation of
the parameter $ \, \theta \,$ and the state of null velocity $\, \rho_0 \,$
is simple by resolution of the system (\ref{variables-conservatives-2}).
We have
\moneqstar
\rho \, = \, {\widetilde C}_\theta \, \rho_0 \,, \quad
{{J}\over{a}} \, = \, {\widetilde S}_\theta \, \rho_0 \,, \quad
\varepsilon = 0 \,, \quad {\rm Galileo} \,\, {\rm group}.
\vspace*{-6pt} \end{equation*}
In the hyperbolic case, we have
\moneqstar
\tanh \theta \, = \, {{J}\over{a \, \rho}} \,, \quad
u \,=\, c \, \theta \,, \quad \varepsilon = 0 \,, \,\, \widetilde \varepsilon = 1 \,
\vspace*{-6pt} \end{equation*}
and in the elliptic one,
\moneqstar
{\rm tg } \, \theta \, = \, {{J}\over{a \, \rho}} \,, \quad
u \,=\, c \, \theta \,, \quad \varepsilon = 0 \,, \,\, \widetilde \varepsilon = -1 \,
\vspace*{-6pt} \end{equation*}
as we have explained in \cite{Du01}.
\monitem
In the following, we focus on hyperbolic ($ \widetilde \varepsilon = 1 $)
and elliptic ($ \widetilde \varepsilon = -1 $) representations of the Lorentz
($\varepsilon = 1$) and circular ($\varepsilon = -1$) groups. With other words,
\moneq \label{epsilon-tilde-carre}
\varepsilon^2 \,\,= \,\,
\widetilde \varepsilon^2 \,\,= \,\, 1 \, .
\vspace*{-6pt} \end{equation}
We can replace the pressure
$ \, p_0 \, $ with an appropriate thermodynamic quantity because
\smallskip \noindent \qquad \qquad \qquad \qquad \qquad \qquad $ \displaystyle
{{\sigma^*}\over{\sigma'}} \,=\, {{\rho_0 \, \sigma' - \sigma}\over{\sigma'}} \,=\,
\rho_0 \, - \, {{\sigma}\over{\sigma'}} \, . $
\smallskip \noindent
Then the relations (\ref{variables-conservatives-2}) take the form
\moneq \label{variables-conservatives-3}
\rho \, = \, \big( C_\theta \, {\widetilde C}_\theta
\,-\, \varepsilon \, S_\theta \, {\widetilde S}_\theta \big) \, \rho_0
\,+\, \varepsilon \, S_\theta \, {\widetilde S}_\theta \, {{\sigma}\over{\sigma'}} \,
\,, \quad
{{J}\over{a}} \, = \, \big( C_\theta \, {\widetilde S}_\theta
\,-\, \varepsilon \, \widetilde \varepsilon\, S_\theta \, {\widetilde C}_\theta
\big) \, \rho_0
\,+\, \varepsilon \, \widetilde \varepsilon \, \, S_\theta \, {\widetilde C}_\theta \,
{{\sigma}\over{\sigma'}} \, \,.
\vspace*{-6pt} \end{equation}
For the Lorentz ($\varepsilon = 1$)
or the circular ($\varepsilon = -1$) group, the resolution of the system
(\ref{variables-conservatives-3}) with unknowns $\, \theta \,$ and
$\, \rho_0 \, $ is absolutly non trivial.
In order to be able to differentiate the solution $\, (\theta ,\,
\rho_0 )\, $ relatively to $ \, (\rho \,, J) ,\,$ we have the following result.
\smallskip \monitem { \bf Proposition 13. Jacobian }
\noindent
We denote by $ \, \zeta_0 \, $ the partial derivative of the ratio $ \, \sigma / \sigma' \,$
relatively to the dentity $ \, \rho_0 \,$ at rest~:
\moneq \label{zeta_zero}
\zeta_0 \, \equiv \, {{\partial}\over{\partial \rho_0}}
\, \Big( {{\sigma}\over{\sigma'}} \Big) \, .
\vspace*{-6pt} \end{equation}
If the hypothesis (\ref{epsilon-tilde-carre}) is satisfied and
if the following jacobian determinant
\moneq \label{det-jaco}
\Delta \, \equiv \, {{\partial \, ( \rho , \, J)}
\over {\partial \, (\theta ,\, \rho_0 )}}
\,=\, a \, {{\sigma}\over{\sigma'}} \, \Big[ (\varepsilon \, \widetilde \varepsilon
-1) \,\, C_\theta^2 \,\, {{\sigma^*}\over{\sigma}} \, \,
+\,\, \widetilde \varepsilon \, S_\theta^2 \, \zeta_0
\,\, - \, \,( C_\theta^2 \,+ \, \widetilde \varepsilon \, S_\theta^2 ) \Big]
\vspace*{-6pt} \end{equation}
is not null, the solution $\, (\theta ,\, \rho_0 )\, $ of the system
(\ref{variables-conservatives-3}) is locally unique and we can differentiate
the parameters $ \, \theta \,$ and $ \, \rho_0 \,$ relatively
to the state $ \, W \, $ introduced in (\ref{variables-conservatives}).
\smallskip \noindent {\bf Proof of Proposition 13. }
\noindent
We have
\moneqstar \left\{ \begin{array} [c]{rcl} \displaystyle
{{\partial \rho}\over{\partial \theta}} \,&=&\, \displaystyle
( \widetilde \varepsilon \, - \, \varepsilon ) \,
C_\theta \, \widetilde S_\theta \, \, \rho_0
\,+\, \varepsilon \big( C_\theta \, \widetilde S_\theta \,
+ \, S_\theta \, \widetilde C_\theta \big) \, \, {{\sigma}\over{\sigma'}}
\\ \displaystyle \vspace{-.3cm} ~ \\ \displaystyle
{{\partial \rho}\over{\partial \rho_0 }} \,&=&\, \displaystyle
\big( C_\theta \, \widetilde C_\theta \,
- \, \varepsilon \, S_\theta \, \widetilde S_\theta \big) \,+\,
\varepsilon \, S_\theta \, \widetilde S_\theta \,\, \zeta_0
\\ \displaystyle \vspace{-.3cm} ~ \\ \displaystyle
{{1}\over{a}} \, {{\partial J}\over{\partial \theta}} \,&=&\, \displaystyle
( 1 \,-\, \varepsilon \, \widetilde \varepsilon ) \,
C_\theta \, \widetilde C_\theta \, \, \rho_0
\,+\, \big( \varepsilon \, \widetilde \varepsilon \, C_\theta \, \widetilde C_\theta \,
+ \, \varepsilon \, S_\theta \, \widetilde S_\theta \big) \, \, {{\sigma}\over{\sigma'}}
\\ \displaystyle \vspace{-.3cm} ~ \\ \displaystyle
{{1}\over{a}} \, {{\partial J}\over{\partial \rho_0}} \,&=&\, \displaystyle
\big( C_\theta \, \widetilde S_\theta \,
- \, \varepsilon \, \widetilde \varepsilon \, S_\theta \, \widetilde C_\theta \big) \,+\,
\varepsilon \, \widetilde \varepsilon \, S_\theta \, \widetilde C_\theta \, \, \zeta_0 \, .
\end{array} \right. \vspace*{-6pt} \end{equation*}
Then the determinant $ \, \Delta \, \equiv \, {{\partial \, ( \rho , \, J)}
\over {\partial \, (\theta ,\, \rho_0 ) }} \, \,$ can be evaluated as follows~:
\smallskip \noindent $ \displaystyle
\Delta \, = {{\partial \rho}\over{\partial \theta}} \, \,
{{\partial J}\over{\partial \rho_0}} \, \,-\, \,
{{\partial \rho}\over{\partial \rho_0 }} \, \,
{{\partial J}\over{\partial \theta}} \, $
\smallskip \noindent $ \displaystyle = \,
a \, \, \Big[ \, \Big(
( \widetilde \varepsilon \, - \, \varepsilon ) \,
C_\theta \, \widetilde S_\theta \, \, \rho_0
\,+\, \varepsilon \big( C_\theta \, \widetilde S_\theta \,
+ \, S_\theta \, \widetilde C_\theta \big) \, \, {{\sigma}\over{\sigma'}} \Big)
\, \Big( \big( C_\theta \, \widetilde S_\theta \,
- \, \varepsilon \, \widetilde \varepsilon \, S_\theta \, \widetilde C_\theta \big) \,+\,
\varepsilon \, \widetilde \varepsilon \, S_\theta \, \widetilde C_\theta \, \, \zeta_0
\Big) \, \Big] \, $
\smallskip \noindent $ \displaystyle \quad - \,
a \, \, \Big[ \, \Big(
\big( C_\theta \, \widetilde C_\theta \,
- \, \varepsilon \, S_\theta \, \widetilde S_\theta \big) \,+\,
\varepsilon \, S_\theta \, \widetilde S_\theta \,\, \zeta_0 \Big) \,
\Big( ( 1 \,-\, \varepsilon \, \widetilde \varepsilon ) \,
C_\theta \, \widetilde C_\theta \, \, \rho_0
\,+\, \big( \varepsilon \, \widetilde \varepsilon \, C_\theta \, \widetilde C_\theta \,
+ \, \varepsilon \, S_\theta \, \widetilde S_\theta \big) \, \, {{\sigma}\over{\sigma'}}
\Big) \, \Big] $
\newpage
\smallskip \noindent \qquad $ \displaystyle = \,
a \, \, \Big[ \, ( \varepsilon \, \widetilde \varepsilon - 1) \, C_\theta^2 \,
\big( \widetilde C_\theta^2 \,-\, \widetilde \varepsilon \, \widetilde S_\theta^2 \big) \, \rho_0
\,+\, \big( S_\theta^2 \,+\, \varepsilon \, C_\theta^2 \big)
\, \big( \widetilde S_\theta^2 \,-\, \widetilde \varepsilon \, \widetilde C_\theta^2 \big)
\, {{\sigma}\over{\sigma'}} $
\smallskip \noindent \qquad $ \displaystyle \qquad \qquad
+ \, \widetilde \varepsilon \, S_\theta^2 \,
\big( \widetilde C_\theta^2 \,-\, \widetilde \varepsilon \, \widetilde S_\theta^2 \big)
\, {{\sigma}\over{\sigma'}} \, \zeta_0 \, \Big] $
\smallskip \noindent
and due to the identity (\ref{trigo-tilde}) and the hypothesis (\ref{epsilon-tilde-carre}),
we have the intermediate result~:
\moneq \label{det-jaco-2}
\Delta \, \,=\,
a \, \, \Big[ ( \varepsilon \, \widetilde \varepsilon -1 ) \,
C_\theta ^2 \, \, \rho_0 \,\, - \,\, \widetilde \varepsilon \,
\big( S_\theta^2 \,+\, \varepsilon \, C_\theta^2 \big) \, {{\sigma}\over{\sigma'}}
\,\, + \,\, \widetilde \varepsilon \,
S_\theta ^2 \, \, {{\sigma}\over{\sigma'}} \, \zeta_0 \, \, \Big] \, .
\vspace*{-6pt} \end{equation}
Then
\smallskip \noindent $ \displaystyle
\Delta \, \,=\, \, a \, {{\sigma}\over{\sigma'}} \, \Big[ \, ( \varepsilon
\, \widetilde \varepsilon -1 ) \,
C_\theta ^2 \, \Big( {{\sigma^*}\over{\sigma}} \,+ \, 1 \Big)
\,\, - \,\, \widetilde \varepsilon \,
\big( S_\theta^2 \,+\, \varepsilon \, C_\theta^2 \big)
\,\, + \,\, \widetilde \varepsilon \,
S_\theta ^2 \, \, \zeta_0 \, \Big] $
\smallskip \noindent \quad $ \displaystyle \,\,
=\, a \, {{\sigma}\over{\sigma'}} \, \Big[ \, (\varepsilon \, \widetilde \varepsilon
-1) \,\, C_\theta^2 \,\, {{\sigma^*}\over{\sigma}}
\,\, - \, \,( C_\theta^2 \,+ \, \widetilde \varepsilon \, S_\theta^2 )
\, \, +\,\, \widetilde \varepsilon \, S_\theta^2 \, \zeta_0 \, \Big] $
\smallskip \noindent
and the expression (\ref{det-jaco}) is established.
$ \hfill \square $
\smallskip \monitem
After determining the parameters $ \, \theta \,$ and $ \, \rho_0 \,$
it is possible to evaluate the mathematical entropy $ \,\, \eta \,\,$ thanks to the relation
(\ref{entropie}). Then we can differentiate this mathematical entropy relatively
the density $ \, \rho \,$ and the momentum $ \, J . \,$ We have the following remarquable
coherence property.
\smallskip \monitem { \bf Proposition 14. Entropy variables }
\noindent
If the jacobian determinant (\ref{det-jaco}) is not null, then the entropy variables
$ \, \varphi \, \equiv \, ( \alpha , \, \beta ) \,$
can be determined according to the relations (\ref{var-entrop-simple}).
In other terms,
\moneq \label{var-entrop-simple-2}
\alpha \,\equiv \, {{\partial \eta}\over{\partial \rho}} \,\,=\,\,
\widetilde C_\theta \,\, \sigma'(\rho_0) \,, \qquad
\beta \,\equiv \, {{\partial \eta}\over{\partial J}} \,\,=\,\,
- \widetilde {{\varepsilon}\over{a}} \,\,
\widetilde S_\theta \,\, \sigma'(\rho_0) \,.
\vspace*{-6pt} \end{equation}
\smallskip \noindent {\bf Proof of Proposition 14. }
\noindent
We first invert the jacobian matrix~:
\smallskip \noindent \qquad \qquad \qquad $ \displaystyle
\begin{pmatrix} \displaystyle {{\partial \rho}\over{\partial \theta}} &
\displaystyle {{\partial \rho}\over{\partial \rho_0}} \cr \vspace{-.4cm} ~ & ~ \cr
\displaystyle {{\partial J}\over{\partial \theta}} &
\displaystyle {{\partial J }\over{\partial \rho_0}} \end{pmatrix}^{-1}
\,\, = \,\,
\begin{pmatrix} \displaystyle {{\partial \theta}\over{\partial \rho}} &
\displaystyle {{\partial \theta}\over{\partial J}} \cr \vspace{-.4cm} ~ & ~ \cr
\displaystyle {{\partial \rho_0}\over{\partial \rho}} &
\displaystyle {{\partial \rho_0}\over{\partial J}} \end{pmatrix}
\,\, = \,\, {{1}\over{\Delta}} \,\,
\begin{pmatrix} \displaystyle \,\, {{\partial J }\over{\partial \rho_0}} &
\displaystyle - {{\partial \rho}\over{\partial \rho_0}} \cr \vspace{-.4cm} ~ & ~ \cr
\displaystyle - {{\partial J}\over{\partial \theta}} &
\displaystyle \,\, {{\partial \rho }\over{\partial \theta}} \end{pmatrix} \, . $
\smallskip \noindent
Then
\moneqstar \left\{ \begin{array} [c]{rclcl} \displaystyle
{{\partial \theta}\over{\partial \rho}} \,&=&\,
\displaystyle {{1}\over{\Delta}} \, {{\partial J}\over{\partial \rho_0 }}
\,&=&\, \displaystyle {{a}\over{\Delta}} \, \Big[ ( C_\theta \, \widetilde S_\theta \,
- \, \varepsilon \, \widetilde \varepsilon \, S_\theta \, \widetilde C_\theta )
\,+\, \varepsilon \, \widetilde \varepsilon \, S_\theta \, \widetilde C_\theta
\, \zeta_0 \Big]
\\ \displaystyle \vspace{-.4cm} ~ \\ \displaystyle
{{\partial \rho_0}\over{\partial \rho }} \,&=&\, \displaystyle
\displaystyle - {{1}\over{\Delta}} \, {{\partial J}\over{\partial \theta }}
\,&=&\, \displaystyle - {{a}\over{\Delta}} \, \Big[ ( 1 - \varepsilon \, \widetilde \varepsilon )
\, C_\theta \, \widetilde C_\theta \, \rho_0 \,+\,
\big( \varepsilon \, \widetilde \varepsilon \, C_\theta \, \widetilde C_\theta \,
+ \, \varepsilon \, S_\theta \, \widetilde S_\theta \big) \, {{\sigma}\over{\sigma'}}
\Big]
\\ \displaystyle \vspace{-.4cm} ~ \\ \displaystyle
{{\partial \theta}\over{\partial J }} \,&=&\, \displaystyle
\displaystyle - {{1}\over{\Delta}} \, {{\partial \rho}\over{\partial \rho_0 }}
\,&=&\, \displaystyle - {{1}\over{\Delta}} \,
\Big[ ( C_\theta \, \widetilde C_\theta \, - \, \varepsilon \,
S_\theta \, \widetilde S_\theta ) \, + \, \varepsilon \,
S_\theta \, \widetilde S_\theta \, \zeta_0 \Big]
\\ \displaystyle \vspace{-.4cm} ~ \\ \displaystyle
{{\partial \rho_0 }\over{\partial J }} \,&=&\, \displaystyle
\displaystyle {{1}\over{\Delta}} \, {{\partial \rho}\over{\partial \theta }}
\,&=&\, \displaystyle {{1}\over{\Delta}} \, \Big[ ( \widetilde \varepsilon
\, - \, \varepsilon ) \, C_\theta \, \widetilde S_\theta \, \rho_0 \, + \,
\varepsilon \, ( C_\theta \, \widetilde S_\theta \, + \,
S_\theta \, \widetilde C_\theta ) \, {{\sigma}\over{\sigma'}} \Big] \, .
\end{array} \right. \vspace*{-6pt} \end{equation*}
We first observe that
\smallskip \noindent \qquad \qquad $ \displaystyle
{{\partial \eta}\over{\partial \theta}} \,=\,
{{\partial}\over{\partial \theta}} \big( C_\theta \, \sigma(\rho_0) \big) \,=\, \varepsilon
\, S_\theta \, \sigma \,, \qquad
{{\partial \eta}\over{\partial \rho_0}} \,=\,
{{\partial}\over{\partial \rho_0}} \big( C_\theta \, \sigma(\rho_0) \big) \,=\,
C_\theta \, \sigma' \, . $
\smallskip \noindent
Then with the chain rule, we have
\smallskip \noindent \qquad $ \displaystyle
\alpha \, = \, {{\partial \eta}\over{\partial \rho}} \,=\,
{{\partial \eta}\over{\partial \theta}} \, {{\partial \theta}\over{\partial \rho}} \, + \,
{{\partial \eta}\over{\partial \rho_0}} \, {{\partial \rho_0}\over{\partial \rho}} $
\smallskip \noindent \qquad $ \displaystyle \quad = \,
{{a}\over{\Delta}} \, \Big[
\varepsilon \, S_\theta \, \sigma \, \big( ( C_\theta \, \widetilde S_\theta \,
- \, \varepsilon \, \widetilde \varepsilon \, S_\theta \, \widetilde C_\theta )
\,+\, \varepsilon \, \widetilde \varepsilon \, S_\theta \, \widetilde C_\theta
\, \zeta_0 \big) $
\noindent \qquad $ \displaystyle \qquad \qquad
\,-\, C_\theta \, \sigma' \,
\big( ( 1 - \varepsilon \, \widetilde \varepsilon )
\, C_\theta \, \widetilde C_\theta \, \rho_0 \,+\,
( \varepsilon \, \widetilde \varepsilon \, C_\theta \, \widetilde C_\theta \,
+ \, \varepsilon \, S_\theta \, \widetilde S_\theta \, {{\sigma}\over{\sigma'}} \big)
\Big] $
\smallskip \noindent \qquad $ \displaystyle \quad = \,
{{a}\over{\Delta}} \, \Big[ ( \varepsilon \, \widetilde \varepsilon - 1 ) \,
C_\theta^2 \, \widetilde C_\theta \, \rho_0 \, \sigma' \, + \,
\widetilde \varepsilon \, S_\theta^2 \, \widetilde C_\theta \, \zeta_0 \, \sigma
\, - \, \varepsilon \, \widetilde C_\theta \, ( S_\theta^2 \, +
\varepsilon \, C_\theta^2 ) \, \sigma \Big] $
\smallskip \noindent \qquad $ \displaystyle \quad = \,
{{a}\over{\Delta}} \, \widetilde C_\theta \, \sigma' \,
\Big[ ( \varepsilon \, \widetilde \varepsilon - 1 ) \,
C_\theta^2 \, \rho_0 \, + \,
\widetilde \varepsilon \, S_\theta^2 \, \zeta_0 \, {{\sigma}\over{\sigma'}}
\, - \, \varepsilon \, ( S_\theta^2 \, + \varepsilon \, C_\theta^2 ) \,
{{\sigma}\over{\sigma'}} \Big] \,\, = \,\,
\widetilde C_\theta \, \sigma' \, $
\smallskip \noindent
due to (\ref{det-jaco-2}). In an analogous way, we have
\smallskip \noindent \qquad $ \displaystyle
\beta \, = \, {{\partial \eta}\over{\partial J }} \,=\,
{{\partial \eta}\over{\partial \theta}} \, {{\partial \theta}\over{\partial J }} \, + \,
{{\partial \eta}\over{\partial \rho_0}} \, {{\partial \rho_0}\over{\partial J }} $
\smallskip \noindent \qquad $ \displaystyle \quad = \,
{{1}\over{\Delta}} \, \Big[
- \varepsilon \, S_\theta \, \sigma \, \Big(
( C_\theta \, \widetilde C_\theta \, - \, \varepsilon \,
S_\theta \, \widetilde S_\theta ) \, + \, \varepsilon \,
S_\theta \, \widetilde S_\theta \, \zeta_0 \Big) $
\noindent \qquad $ \displaystyle \qquad \qquad
\,+\, C_\theta \, \sigma' \,
\Big( ( \widetilde \varepsilon
\, - \, \varepsilon ) \, C_\theta \, \widetilde S_\theta \, \rho_0 \, + \,
\varepsilon \, ( C_\theta \, \widetilde S_\theta \, + \,
S_\theta \, \widetilde C_\theta ) \, {{\sigma}\over{\sigma'}} \Big) \, \Big] $
\smallskip \noindent \qquad $ \displaystyle \quad = \,
{{\widetilde \varepsilon}\over{\Delta}} \, \Big[
( 1 - \varepsilon \, \widetilde \varepsilon ) \, C_\theta^2 \, \widetilde S_\theta \,
\rho_0 \, \sigma' \, \,+ \,\, \widetilde \varepsilon
\big( S_\theta^2 \,+\ \varepsilon \, C_\theta^2 \big) \, \widetilde S_\theta \, \sigma
\,\, - \, \widetilde \varepsilon \, S_\theta^2 \, \widetilde S_\theta \,
\zeta_0 \, \sigma \Big] $
\smallskip \noindent \qquad $ \displaystyle \quad = \,
{{\widetilde \varepsilon}\over{\Delta}} \, \widetilde S_\theta \, \sigma' \, \Big[
( 1 - \varepsilon \, \widetilde \varepsilon ) \, C_\theta^2 \,
\rho_0 \, \,+ \,\, \widetilde \varepsilon
\big( S_\theta^2 \,+\, \varepsilon \, C_\theta^2 \big) \, {{\sigma}\over{\sigma'}}
\,\, - \, \widetilde \varepsilon \, S_\theta^2 \, \zeta_0 \, {{\sigma}\over{\sigma'}} \Big]
\,\,= \,\,
- {{\widetilde \varepsilon}\over{a}} \,\, \widetilde S_\theta \,\, \sigma' \, $
\smallskip \noindent
due to (\ref{det-jaco-2}). The proof is completed.
$ \hfill \square $
\smallskip \monitem
We particularize our study and wish to construct nontrivial
hyperbolic systems with entropy velocity covariant under
the Lorentz group and / or the circular group.
It seems essential to maintain a constant sign for the determinant
$\, \Delta \,$ presented at the relation (\ref{det-jaco}).
Moreover, we have to enforce the strict convexity of the
mathematical entropy $ \, ( \rho , \, J ) \longmapsto
\eta( \rho , \, J ) .\,$
In this contribution, we only consider the particular case
\moneq \label{epsilon-epsilon-tilde}
\varepsilon \, = \, \widetilde \varepsilon \, .
\vspace*{-6pt} \end{equation}
We focus on ``circular elliptic'' and ``Lorentz hyperbolic''
two by two systems of conservation laws.
Observe first that due to the condition (\ref{epsilon-epsilon-tilde}),
the expression (\ref{variables-conservatives-3})
of the conserved variables simplify into
\moneq \label{variables-conservatives-4}
\rho \,\,=\,\, \rho_0 \,+\, \varepsilon \, S_\theta^2 \, {{\sigma}\over{\sigma'}}
\,, \quad
J \,\,=\,\, a \, S_\theta \, C_\theta \, {{\sigma}\over{\sigma'}} \, .
\vspace*{-6pt} \end{equation}
Then the mathematical entropy $ \, \eta \,$ constrained to the condition
(\ref{entropie}) is strictly convex if and only if the determinant
of its hessian
$\, {\rm det} \, ( {\rm d}^2 \eta ) \, $
is stricly positive and if the second derivative
$ \, {{\partial^2 \eta}\over{\partial \rho^2}} \,$
is also strictly positive.
\smallskip \monitem { \bf Proposition 15. Technical lemma for convexity conditions }
\noindent
Under the hypotheses (\ref{epsilon-tilde-carre}) and (\ref{epsilon-epsilon-tilde}),
the expression of the jacobian determinant $ \, \Delta \,$ is given according to
\moneq \label{det-jaco-3}
\Delta \, = \, a \, \delta \, {{\sigma}\over{\sigma'}} \,
\vspace*{-6pt} \end{equation}
with
\moneq \label{petit-delta}
\delta \,=\, \varepsilon \, S_\theta^2 \, \zeta_0
\,\, - \, \,( C_\theta^2 \,+ \, \varepsilon \, S_\theta^2 ) \, .
\vspace*{-6pt} \end{equation}
The two signed expressions are given according to
\moneq \label{det-hessian}
{\rm det} \, \big( {\rm d}^2 \eta \big) \, = \, {{\varepsilon \,
\sigma' \, \sigma''}\over{a \, \Delta}} \, \, = \, {{\varepsilon}\over{a^2}} \,
\big( \sigma' \big)^2 \, {{\sigma''}\over{\sigma}} \, {{1}\over{\delta}}
\vspace*{-6pt} \end{equation}
and
\moneq \label{d2eta-sur-d2rho}
{{\partial^2 \eta}\over{\partial \rho^2}} \,=\,
{{\partial \alpha}\over{\partial \rho}} \,= \, {{C_\theta}\over{\delta}}
\, \Big( \varepsilon \, S_\theta^2 \, \zeta_0 \, {{\sigma'^2}\over{\sigma}}
\,- \, \big( C_\theta^2 + \varepsilon \, S_\theta^2 \big) \, \sigma'' \Big) \, .
\vspace*{-6pt} \end{equation}
\smallskip \noindent {\bf Proof of Proposition 15. }
\noindent
The expression $\, \delta \,$ in (\ref{petit-delta}) is just
a specific name for the big braket in right hand side of
the expression (\ref{det-jaco}) and the relation (\ref{det-jaco-3})
is satisfied. Under the condition
(\ref{epsilon-epsilon-tilde}), the expression $\, \delta \,$
is clearly reduced to the expression (\ref{petit-delta}).
We establish now the expression (\ref{det-hessian}) of the
determinant of the hessian matrix.
From the chain rule, we have
\smallskip \noindent \qquad $ \displaystyle
{\rm det} \, \big( {\rm d}^2 \eta \big) \, = \,
{{\partial \, ( \alpha , \, \beta)} \over {\partial \, ( \rho ,\, J )}}
\,\,=\,\,
{{\partial ( \alpha , \, \beta)} \over {\partial( \theta ,\, \rho_0 )}}
\,\,
{{\partial ( \theta ,\, \rho_0 )} \over {\partial( \rho ,\, J )}}
\,\, = \,\, {{1}\over{\Delta}} \, \, {\rm det}
\begin{pmatrix} \displaystyle {{\partial \alpha}\over{\partial \theta}} &
\displaystyle {{\partial \alpha}\over{\partial \rho_0}} \cr \vspace{-.4cm} \cr
\displaystyle {{\partial \beta}\over{\partial \theta}} &
\displaystyle {{\partial \beta}\over{\partial \rho_0}} \end{pmatrix} $
\smallskip \noindent \qquad $ \displaystyle \qquad \qquad \,\, = \,\,
{{1}\over{\Delta}} \, \, {\rm det}
\begin{pmatrix} \displaystyle \varepsilon \, S_\theta \, \sigma' &
\displaystyle C_\theta \, \sigma'' \cr
\displaystyle - {{\varepsilon}\over{a}} \, C_\theta \, \sigma' &
\displaystyle - {{\varepsilon}\over{a}} \, S_\theta \, \sigma'' \end{pmatrix}
\,\, = \,\, - {{1}\over{\Delta}} \, {{\varepsilon}\over{a}} \, \sigma'
\, \sigma'' \, \, {\rm det}
\begin{pmatrix} \displaystyle \varepsilon \, S_\theta &
\displaystyle C_\theta \cr
\displaystyle \, C_\theta &
\displaystyle \, S_\theta \end{pmatrix} $
\smallskip \noindent
due to (\ref{var-entrop-simple-2}) and (\ref{epsilon-epsilon-tilde}). Then
$ \displaystyle \,\,
{\rm det} \, \big( {\rm d}^2 \eta \big) \, = \, {{1}\over{a \, \delta}}
\, {{\sigma'}\over{\sigma}} \, {{\varepsilon}\over{a}} \, \sigma'
\, \sigma'' \, $
is a consequence of the trigonometrical property (\ref{trigo})
and the expression (\ref{det-hessian}) follows.
The evaluation of
(\ref{d2eta-sur-d2rho}) is also elementary.
From the chain rule, we have
\smallskip \noindent \qquad $ \displaystyle
{{\partial \alpha}\over{\partial \rho}} \,\,= \,\,
{{\partial \alpha}\over{\partial \theta}} \,
{{\partial \theta}\over{\partial \rho}} \,+\,
{{\partial \alpha}\over{\partial \rho_0}} \,
{{\partial \rho_0}\over{\partial \rho}} \, \,\,= \,\,
{{a}\over{\Delta}} \, \Big[ \varepsilon \, S_\theta \, \sigma' \, S_\theta
\, C_\theta \, \zeta_0 \,-\, C_\theta \, \sigma'' \,
\big( C_\theta^2 \, + \, \varepsilon \, S_\theta^2 \, \big) {{\sigma}\over{\sigma'}}
\Big] $
\smallskip \noindent \qquad $ \displaystyle \quad \,\, \,\, \,\,= \,\,
{{1}\over{\delta}} \, {{\sigma'}\over{\sigma}} \, C_\theta
\, \Big[ \varepsilon \, S_\theta^2 \, \sigma' \, \zeta_0 \,-\, \sigma'' \,
\big( C_\theta^2 \, + \, \varepsilon \, S_\theta^2 \big) \, {{\sigma}\over{\sigma'}}
\Big] $ \quad
\smallskip \noindent
and the expression (\ref{d2eta-sur-d2rho}) is clear.
$ \hfill \square $
\smallskip \monitem
We detail now two explicit examples of hyperbolic system of order two.
The first one is invariant under the action of the circular group
and the second under the Lorentz group.
\smallskip \monitem { \bf Proposition 16. Exponential entropy for
a ``circular-elliptic'' system }
\noindent
We introduce two strictly positive constants $ \, \rho_* \,$
and $ \, \overline \sigma \,$ and an exponential entropy
on the manifold $\, \Omega_0 \,$ thanks to
\moneq \label{entropie-exponentielle}
\sigma (\rho_0) \,\, = \, \, \overline \sigma \, \exp \Big(
{{\rho_0}\over{\rho_*}} \Big) \, .
\vspace*{-6pt} \end{equation}
We set
\moneq \label{grand-psi}
\left\{ \begin{array} [c]{l} \displaystyle \displaystyle
\Psi \,=\, {{2\, J}\over{a \,\rho_*}} \,,\qquad
\rho_0 \,=\,
\rho + {{1}\over{2}} \big( 1 - \sqrt{1 - \Psi^2} \big)\, \rho_* \,, \qquad
\theta \,=\, {{1}\over{2}} \, {\rm Arcsin} \, \Psi \,,
\\ \vspace{-.4cm} \\ \displaystyle
u \, = \, c \, {{J}\over{\mid \! J \! \mid}} \,\,
\sqrt{ {{1 - \sqrt{1 - \Psi^2}}\over{1+\sqrt{1 - \Psi^2}}}}
\, = \, c \, {\rm sgn}(J) \, \tan \theta \,,\qquad
p_0 \,=\, a \, c \, \, \big( \rho_0 \,-\, \rho_* \big) \, .
\end{array} \right. \vspace*{-6pt} \end{equation}
The system of conservation laws
\moneq \label{circulaire-elliptique}
\left\{ \begin{array} [c]{rcl} \displaystyle
{{\partial \rho}\over{\partial t}} \,+\,
{{\partial}\over{\partial x}} \Big(
{{c}\over{a}} \, J
\Big) \,\, &=& \,\, 0
\\ \vspace{-.4cm} \\ \displaystyle
{{\partial J}\over{\partial t}} \,+\,
{{\partial}\over{\partial x}} \big( u \, J \,+ \, p_0
\big) \,\, &=& \,\, 0
\end{array} \right. \vspace*{-6pt} \end{equation}
is hyperbolic if we suppose
\moneq \label{hyp-circulaire-elliptique}
\mid \! \Psi \! \mid \,\, \leq \,\, 1 \, ,\qquad
\mid \! \theta \! \mid \,\, \leq \,\, {{\pi}\over{4}} \, .
\vspace*{-6pt} \end{equation}
It admits a mathematical entropy given by the expression
\moneq \label{entropie-circulaire-elliptique}
\eta ( \rho ,\, J) \,\, \equiv \, \,
\sqrt{ {{1}\over{2}} \, \big( 1 + \sqrt{1 - \Psi^2} \big) } \,\,\,\,
\sigma \Big(
\rho \,+\, {{\rho_*}\over{2}} \, \big( 1 - \sqrt{1-\Psi^2} \big) \Big)
\vspace*{-6pt} \end{equation}
and the dissipation of entropy of entropy solutions
is given by the inequality
\moneq \label{ineg-entropie-circulaire-elliptique}
{{\partial \eta }\over{\partial t}} \,+\,
{{\partial}\over{\partial x}} \big( u \, \eta \big) \,\, \leq \,\, 0 \, .
\vspace*{-6pt} \end{equation}
\smallskip \noindent {\bf Proof of Proposition 16. }
\noindent
With the choice (\ref{entropie-exponentielle})
of an exponential entropy, the parameter $\, \zeta_0 \,$ introduced at the relation
(\ref{zeta_zero}) is null. Because $ \, \overline \sigma \,$
is supposed strictly positive,
$\, \sigma \,$ is a convex function and the condition
$ \, {\rm det} \, \big( {\rm d}^2 \eta \big) \,> \, 0 \,$
is reduced to $ \, \delta < 0 \, $ if we look to a ``circular-elliptic''
hyperbolic system, {\it id est} if we suppose $ \, \varepsilon = -1 \,$
(see (\ref{det-hessian})).
Then with the notation $ \, \Psi \,$
introduced in (\ref{grand-psi}), the relations (\ref{variables-conservatives-4})
can be written
\moneqstar
\sin \, ( 2 \, \theta) \,\,=\,\, \Psi \, , \qquad
\rho \,\,=\,\, \rho_0 \,-\, \sin^2 \theta \, \rho_* \, .
\vspace*{-6pt} \end{equation*}
and the calculus of $ \, \theta \,$ proposed in (\ref{grand-psi})
is natural. We have the same remark for the expression
of $\, \rho_0 \,$ in (\ref{grand-psi})
as a consequence of the identity
$\, \sin^2 \, \theta \,=\, {{1}\over{2}} \, \big( 1 -
\sqrt{ 1 - \sin^2 (2 \, \theta) } \big) . \,$
The second condition in (\ref{hyp-circulaire-elliptique}) is a consequence
of the expression of $ \, \theta \,$ proposed in (\ref{grand-psi}).
This last condition is also necessary when we impose
$\, \delta < 0 \, $ because the relation (\ref{petit-delta})
can be now written $ \,\delta = -( \cos^2 \theta - \sin^2 \theta ) .\,$
The velocity in (\ref{grand-psi})
is simply the actual formulation of the relation (\ref{vitesse-groupe})
because
\moneqstar
\tan \, \theta \,=\,
{\rm sgn}(\theta) \, \sqrt {
{{1 - \sqrt{ 1 - \sin^2 (2 \, \theta) }
}\over { 1 + \sqrt{ 1 + \sin^2 (2 \, \theta) } }}} . \,
\vspace*{-6pt} \end{equation*}
\monitem
The static pressure $ \, p_0 \,$ is related to the entropy $ \, \sigma \,$
thanks to (\ref{pression}).
The expressions (\ref{def-flux-thermo}) and (\ref{thermo-flux-2})
of the physical and thermodynamix fluxes,
the algebraic forms (\ref{matrice-Y-theta}) of the matrix $ \, Y_\theta \,$
and the choice (\ref{g-zero-Neq2}) for $ \, g_0 \,$ show that
the first component $ \, f_1 \, $ of $ \, f(W) \,$ is given by
\smallskip \noindent \qquad $ \displaystyle
f_1 \,\,=\,\, \rho \, u \, - {{\sin \theta}\over{a\, \cos \, \theta }} \, p_0
\,\,=\,\, \rho \, u \, - {{1}\over{a \, c}} \, u \, a \, c \, ( \rho_0 - \rho_* )
\,\,=\,\, \rho_* \, u \, (1 - \sin^2 \theta) $
\smallskip \noindent \qquad $ \displaystyle \quad \,\,\, = \,
\rho_* \, c \, \sin \theta \, \cos \theta
\,\,=\,\, {{1}\over{2}} \, \rho_* \, c \, \sin (2 \, \theta)
\,\,=\,\, {{1}\over{2}} \, \rho_* \, c \, \Psi \,
\,\,=\,\, {{1}\over{2}} \, \rho_* \, c \, {{2\, J}\over{a \,\rho_*}}
\,\,=\,\, {{c}\over{a}} \, J \, . $
\smallskip \noindent
Then the relations (\ref{circulaire-elliptique}) are established.
The expression (\ref{entropie-circulaire-elliptique}) of the mathematical
entropy is then a consequence of (\ref{entropie})~:
$ \, \eta = \cos \theta \, \, \sigma(\rho_0) .\,$
A particular hyperbolic system with entropy velocity
invariant under the action of the circular group is constructed.
$ \hfill \square $
\smallskip \newpage \monitem { \bf Proposition 17. Homographic entropy for
a ``Lorentz-hyperbolic'' system }
\noindent
We consider two strictly positive constants $ \, \rho_* \,$
and $ \, \overline \sigma \,$ and an homographic entropy $\, \sigma \,$
on the manifold $\, \Omega_0 \,$ defined by the relation
\moneq \label{entropie-homographic}
\sigma (\rho_0) \,\, = \, \, - \, \overline \sigma \,
{{\rho_0}\over{\rho_0 + \rho_*}} \,, \qquad \rho_0 > 0 \, .
\vspace*{-6pt} \end{equation}
We set
\moneq \label{grand-phi-hyperbolic}
\Phi (\rho_0) \,\, \equiv \,\, {{2\, J}\over{a}} \, \,
{{\sigma'}\over{\sigma}}
\,\, = \,\, {{2\, J}\over{a}} \, {{\rho_*}\over{\rho_0 \, ( \rho_0 + \rho_*)}} \,
\vspace*{-6pt} \end{equation}
and we suppose
\moneq \label{hyp-lorentz-hyperbolic}
{{\mid \! J \! \mid}\over{a}} \,\, \leq \,\,
\rho \,\, \leq \,\, {{1}\over{2}} \, \rho_* \, .
\vspace*{-6pt} \end{equation}
Then the equation
\moneq \label{lorentz-equation}
F(\rho_0) \, \equiv \, \rho_0 + {{\sigma}\over{2 \, \sigma'}}
\Big( \sqrt{1 + \Phi (\rho_0)^2} - 1 \Big) \,\,=\,\, \rho
\vspace*{-6pt} \end{equation}
has a unique solution $ \, \rho_0 \,$ such that
\moneq \label{condition-rho-zero}
0 \,\leq \, \rho_0 \,\leq \, {{1}\over{2}} \, \rho_* \, .
\vspace*{-6pt} \end{equation}
We consider the expressions
\moneq \label{expressions-hyperbolic}
p_0 \,=\, a \, c \, {{\rho_0^2}\over{\rho_*}} \,, \quad
\theta \,=\, {{1}\over{2}} \, {\rm Argsh} \, \Phi(\rho_0) \,, \quad
u \,=\, a \, \tanh \theta \,=\, c \, {\rm sgn}(J) \,
\sqrt{ {{ \sqrt{1 + \Phi^2} - 1 }\over{ \sqrt{1 + \Phi^2} + 1 }} } \, .
\vspace*{-6pt} \end{equation}
Then the system of conservation laws
\moneq \label{lorentz-hyperbolic}
\left\{ \begin{array} [c]{rcl} \displaystyle
{{\partial \rho}\over{\partial t}} \,+\,
{{\partial}\over{\partial x}}
\Big[ \Big( \rho + {{\rho_0}\over{\rho_*}} \Big) \, u \Big]
\,\, &=& \,\, 0
\\ \vspace{-.4cm} \\ \displaystyle
{{\partial J}\over{\partial t}} \,+\,
{{\partial}\over{\partial x}} \big( u \, J \,+ \, p_0
\big) \,\, &=& \,\, 0
\end{array} \right. \vspace*{-6pt} \end{equation}
is hyperbolic and admits a mathematical entropy given by the expression
\moneq \label{entropie-lorentz-hyperbolic}
\eta ( \rho ,\, J) \,\, \equiv \, \,
\sqrt{ {{1}\over{2}} \, \big( 1 + \sqrt{1 + \Phi(\rho)^2} \big) }
\,\,\, \sigma ( \rho_0 ) \, .
\vspace*{-6pt} \end{equation}
The dissipation of entropy of entropy solutions
is still given by the inequality (\ref{ineg-entropie-circulaire-elliptique}).
\smallskip \noindent {\bf Proof of Proposition 17. }
\noindent
In this case of the Lorentz group, we have $\, \varepsilon = 1 , \,$
$ \, S_\theta \equiv \sinh \theta \,$ and
$ \, C_\theta \equiv \cosh \theta . \,$
We have chosen with (\ref{entropie-homographic})
a negative convex function $ \, \sigma . \,$
Then ({\it c.f.} (\ref{det-hessian})) the condition
$ \, \delta < 0 \,$ has to be satisfied to assume
$ \, {\rm det} \, \big( {\rm d}^2 \eta \big) \,> \, 0 .\,$
We have
\moneqstar
{{\sigma}\over{\sigma'}} \,=\, {{\rho_0}\over{\rho_*}} \,
( \rho_0 + \rho_* ) \,, \quad
\zeta_0 \,\equiv\, {{\partial}\over{\partial \rho_0}}
\Big( {{\sigma}\over{\sigma'}} \Big)
\,=\, 1 \,+\, 2 \, {{\rho_0}\over{\rho_*}} \, .
\vspace*{-6pt} \end{equation*}
The relations (\ref{variables-conservatives-4}) can now be written as
\moneqstar
\sinh \, ( 2 \, \theta) \,\,=\,\, \Phi (\rho_0) \, , \qquad
\rho \,\,=\,\, \rho_0 \,+\, \sinh^2 \theta \, \, {{\sigma}\over{\sigma'}} \, .
\vspace*{-6pt} \end{equation*}
The identity $ \, \sinh^2 \theta \,\equiv \, {{1}\over{2}} \big(
\sqrt{1 + \sinh^2 (2 \, \theta) } - 1 \big) \, $
shows that the density $ \, \rho_0 \,$ at null velocity
associated with an arbitrary state $ \, W \equiv (\rho , \, J)^{\rm \displaystyle t} \,$
has to solve the equation
\moneqstar
\rho \,-\, \rho_0
\,=\, {{1}\over{2}} \Big( \sqrt{1 + \Phi (\rho_0)^2} - 1 \Big)
\, \, {{\sigma}\over{\sigma'}}
\vspace*{-6pt} \end{equation*}
which is equivalent to (\ref{lorentz-equation}).
We remark that $ \, F(0) = \, \mid \! J \! \mid \! / a \,$
and the first condition of (\ref{condition-rho-zero}) is natural.
\monitem
We prove now that the equation (\ref{lorentz-equation})
has a unique solution if the conditions (\ref{hyp-lorentz-hyperbolic})
are satisfied. We have
\smallskip \noindent $ \displaystyle \qquad
{{\partial}\over{\partial \rho_0}} \Big[ {{\sigma}\over{2 \, \sigma'}}
\, \sqrt{ 1 + \Phi^2} \Big] \,\,= \,\,
{{1}\over{2}} {{\partial}\over{\partial \rho_0}}
\sqrt { \Big( {{\sigma}\over{\sigma'}} \Big)^2
\,+\, {{4 \, J^2}\over{a^2}} }
\,\,= \,\, {{2}\over{4 \, \sqrt { \Big( \displaystyle {{\sigma}\over{\sigma'}} \Big)^2
\,+\, {{4 \, J^2}\over{a^2}} \, } }} \, \, {{\sigma}\over{\sigma'}} \,
\, \zeta_0 \,\, \,\, > \,\, 0 $
\smallskip \noindent
if $ \, \rho_0 > 0 . \,$ We have also
$ \displaystyle \quad
{{\partial}\over{\partial \rho_0}} \Big( \rho_0 \,-\,
{{\sigma}\over{2 \, \sigma'}} \Big)
\,\,= \,\, 1 - {{\zeta_0}\over{2}}
\,\,= \,\, {1\over2} - {{\rho_0}\over{\rho_*}} \,\, > \,\, 0 \quad $
under the condition (\ref{condition-rho-zero}) for $\, \rho_0 .\, $
Then we have $ \, F'(\rho_0) > 0 \,$ and the equation (\ref{lorentz-equation})
has a unique solution when (\ref{condition-rho-zero}) is satisfied.
The maximum of the function $ \, F({\scriptstyle \bullet }) \,$ on the interval $ \, [
0 ,\, {{1}\over{2}} \, \rho_* ] \,$ can be estimated as follows. We have first
$ \quad {{\sigma}\over{\sigma'}} \big( {{\rho_*}\over{2}} \big) \,=\,
{{1}\over{2}} \, \big( {{1}\over{2}} + 1 \big) \, \rho_* \, = \,
{{3}\over{4}} \, \rho_* . \, $ Then
\smallskip \noindent $ \displaystyle \qquad
F \Big( {{\rho_*}\over{2}} \Big) \,\,= \,\,
{{\rho_*}\over{2}} \,-\, {{3}\over{8}} \, \rho_*
\,+\, {{\rho_*}\over{2}} \,
\sqrt{ \Big( {{3}\over{4}} \Big)^2 \,+\, \Big( {{2 \, J}\over{a \, \rho_*}} \Big)^2 \, }
\,\,\, \geq \, \,\, {{\rho_*}\over{2}} \,-\, {{3}\over{8}} \, \rho_*
\,+\, {{3}\over{8}} \, \rho_* \,\,= \,\, {{\rho_*}\over{2}}
$
\smallskip \noindent
and the second condition of (\ref{hyp-lorentz-hyperbolic}) is natural.
The pressure $ \, p_0 \,$ is given by the relation (\ref{pression})~:
\smallskip \noindent $ \displaystyle \qquad \qquad
p_0 \,=\, - \, a \, c \, {{\sigma^*}\over{\sigma'}} \,=\,
a \, c\, \Big( {{\sigma}\over{\sigma'}} - \rho_0 \Big) \,=\,
a \, c\, {{\rho_0^2}\over{\rho_*}} $
\smallskip \noindent
and the first relation of (\ref{expressions-hyperbolic}) is established.
The two other relations of (\ref{expressions-hyperbolic})
are an elementary consequence of the previous considerations.
\monitem
We now establish the algebraic form of the hyperbolic system (\ref{edp}).
From (\ref{def-flux-thermo}) and (\ref{thermo-flux-2}),
the particular expression (\ref{matrice-Y-theta}) of the matrix $ \, Y_\theta \,$
and the choice $ \, g_0 \equiv ( 0 , \, p_0 )^{\rm \displaystyle t} \,$
proposed in (\ref{g-zero-Neq2})
show that the first component $ \, f_1 \, $ of $ \, f(W) \,$ is given by
\smallskip \noindent $ \displaystyle \qquad \qquad
f_1 \,\,=\,\, \rho \, u \, + {{\sinh \theta}\over{a\, \cosh \, \theta }} \, p_0
\,\,=\,\, \rho \, u \, + {{1}\over{a \, c}} \, \,u \,\, a \, c \, \, {{\rho_0^2}\over{\rho_*}}
\,\,=\,\, \Big( \rho \, + \, {{\rho_0^2}\over{\rho_*}} \Big) \, u \, . $
\smallskip \noindent
Then the fist line of the relations (\ref{lorentz-hyperbolic}) is established.
The second line is straightforword to evidence.
The expression of the mathematical entropy
is a direct consequence of (\ref{entropie}) and the explitation
of $\, \cosh \theta \,$ as a function of $ \, \Phi(\rho_0) \, $
as proposed in (\ref{expressions-hyperbolic}).
This remark completes the construction of an hyperbolic system with entropy velocity
(\ref{lorentz-hyperbolic}) invariant under the action of the Lorentz group.
$ \hfill \square $
\bigskip \bigskip \noindent {\bf \large 7) \quad Conclusion}
\monitem
In this contribution, we have extended our previous work \cite{Du01}
relative to the construction of hyperbolic systems covariant under the Galileo group
and space reflection.
We have precised the notion of ``entropy velocity'' as an intrinsic velocity
associated with the conservation of entropy for regular solutions.
In particular, the usual velocity of gas dynamics system
is an entropy velocity.
An hyperbolic system of conservation laws and the
associated conservation of entropy are covariant under the action of
a group of space-time transformations if the algebraic form
of the equations is not changed after the action of the corresponding group.
We have proposed sufficient conditions
in order to satisfy a group covariance property.
These conditions introduce naturally a representation of the group
in the space of states that can be chosen in multiple ways.
We have also introduced a natural manifold of ``null velocity''.
When thermodynamic data are given on this manifold, the natural question
is the construction of the whole hyperbolic system from the covariance property.
We have also natural constraints for the entropy variables.
\monitem
After the study of the particular case
of Galileo covariance in \cite{Du01},
we have particularized in this contribution our study
for the two-dimensional case with the Lorentz and circular groups.
We have proposed nontrivial hyperbolic systems covariant under
each of these two groups.
For the circular example, we find
a condition (\ref{hyp-circulaire-elliptique}) that shows that the associated hyperbolic
system is not covariant under the action of the whole circular group.
The question is set \cite{Va13} to know if there exists a mathematical obstruction
to find hyperbolic systems covariant under the action of
the whole circular group.
A better knowledge of such systems of conservation laws
is a possible next step. Last but not least, the link with
relativistic gas dynamics (see {\it e.g.} Souriau \cite{So77, So78},
Chiu \cite{Ch73}
and Vall\'ee \cite{Va87}) is also a natural question.
The extension of this work to several space dimensions
is, in principle, straightforward.
\bigskip \bigskip \noindent {\bf \large Acknowledgments}
\noindent
The author thanks G\'ery de Saxc\'e and Claude Vall\'ee for their kind invitation to
participate to the 56th and 57th
CITV in august 2012 and august 2013,
and for stimulating discussions during these workshops.
\bigskip \bigskip \noindent {\bf \large References }
\vspace{-.2cm}
|
1,108,101,563,287 | arxiv | \section*{Abstract}
Social scientists and psychologists take interest in understanding how people express emotions and sentiments when dealing with catastrophic events such as natural disasters, political unrest, and terrorism. The COVID-19 pandemic is a catastrophic event that has raised a number of psychological issues such as depression given abrupt social changes and lack of employment. Advancements of deep learning-based language models have been promising for sentiment analysis with data from social networks such as Twitter. Given the situation with COVID-19 pandemic, different countries had different peaks where rise and fall of new cases affected lock-downs which directly affected the economy and employment. During the rise of COVID-19 cases with stricter lock-downs, people have been expressing their sentiments in social media. This can provide a deep understanding of human psychology during catastrophic events. In this paper, we present a framework that employs deep learning-based language models via long short-term memory (LSTM) recurrent neural networks for sentiment analysis during the rise of novel COVID-19 cases in India. The framework features LSTM language model with a global vector embedding and state-of-art BERT language model. We review the sentiments expressed for selective months in 2020 which covers the first major peak of novel cases in India. Our framework utilises multi-label sentiment classification where more than one sentiment can be expressed at once. Our results indicate that the majority of the tweets have been positive with high levels of optimism during the rise of the novel COVID-19 cases and the number of tweets significantly lowered towards the peak. We find that the optimistic, annoyed and joking tweets mostly dominate the monthly tweets with much lower portion of negative sentiments. The predictions generally indicate that although the majority have been optimistic, a significant group of population has been annoyed towards the way the pandemic was handled by the authorities.
\section{Introduction}
\label{S:1}
There has been unprecedented growth of information via social media which has been of interest to social scientists and psychologists in a better understanding of the human condition, psychology \cite{golbeck2011predicting,quercia2011our,bittermann2021mining,lin2015building} and mental health \cite{coppersmith2014quantifying}. Social media platforms such as Twitter has been used as a medium for data acquisition for research in psychology and behavioural science \cite{murphy2017hands}. It has also been used as a tool for predicting personality type \cite{golbeck2011predicting,quercia2011our}, and understanding trends and backgrounds of users online \cite{zhou2019comparative,wang2016twitter}. There has also been interest regarding how people express sentiments when dealing with catastrophic events such as natural disasters, extreme political viewpoints \cite{alizadeh2019psychology}, and terrorism \cite{garg2017sentiment}. For instance, in the case of a terror attack in Kenya \cite{garg2017sentiment}, Twitter became a crucial channel of communication between the government, emergency response team and the public.
Sentiment analysis \textcolor{black}{involves the use of} natural language processing (NLP) \cite{manning1999foundations} methods to systematically study affective states and emotion understanding of individuals or social groups \cite{liu2012survey,medhat2014sentiment,hussein2018survey}. \textcolor{black}{We note that deep learning, which is a machine learning method, has been prominently used for NLP tasks.}
Apart from research in psychology, sentiment analysis has a number of applications such as understanding customer behaviour \cite{ordenes2014analyzing}, clinical medicine \cite{greaves2013use}, building better prediction model for trading stocks \cite{mittal2012stock}, and elections such as the US Presidential campaign in 2012 \cite{wang2012system}. \textcolor{black}{Recently, there has been a trend of using deep learning-based language models \cite{zhang2018deep} with training data from
Twitter for sentiment analysis} \cite{kouloumpis2011twitter,giachanou2016like}. One of the earliest works began using NLP methods such as n-grams with hash tags for building training data and machine learning methods such as Adaboost for sentiment classification \cite{kouloumpis2011twitter}. Deep learning methods such as convolutional neural networks have been used for sentiment analysis on Twitter \cite{severyn2015twitter}.
The \textit{coronavirus disease 2019} (COVID-19) \cite{Gorbalenya2020species,monteil2020inhibition,world2020coronavirus,cucinotta2020declares} global pandemic has been a catastrophic event with major impact on the world's economy which created rise in unemployment, psychological issues, and depression. The abrupt social, economic and travel changes has motivated research from various fields \cite{siche2020impact,richards2020impact,tiwari2021delhi}, where computational modelling with machine learning has been prominent \cite{shinde2020forecasting}; however, it had a number of challenges due to the testing and reporting of cases \cite{rahimi2021review}. Deep learning models have played a significant role in forecasting COVID-19 infection treads for various parts of the world \cite{zeroual2020deep,ChandraCOVID2021,TiwariAir2021}.
During rise of COVID-19 cases, and stricter lock downs, people have been expressing different sentiments in social media such as Twitter. Social media has played a significant role during COVID-19 which has driven researchers for analysis with NLP and machine learning methods.
A study that used sentiment analysis via deep learning reported that World Health Organisation (WHO) tweets have been unsuccessful in providing public guidance \cite{chakraborty2020sentiment}. There has been a study on sentiment analysis to study the effect of nationwide lockdown due to COVID-19 outbreak in India where it was found that people took the fight against COVID19 positively and majority were in agreement with the government for the initial nation-wide lockdown \cite{barkur2020sentiment}. Social media posts and tweets brings another level of understanding when combined with sentiment analysis. An example of topic modelling examined tweets during the COVID-19 pandemic identified themes such as 'origin of the virus', and 'the economy' \cite{abd2020top}. Topic modelling has been used with Twitter based sentiment analysis during early stages of COVID-19 and sentiments such as fear was dominant \cite{xue2020public}. In region specific studies, tweets from the United States was used to determine the network of dominant topics and sentiments \cite{hung2020social}. Further work was done in case of China via \textit{bi-directional
encoder representations from transformers} (BERT) language model for trend, topic, and sentiment analysis \cite{wang2020covid}. Further region specific studies include community sentiment analysis in Australia \cite{zhou2020examination} and sentiment analysis in Nepal \cite{pokharel2020twitter}, where majority positive sentiments were found with elements of fear. In Spain, sentiment analysis reviewed how digital platforms created an impact during COVID-19 \cite{de2020sentiment}. A study of
cross-language sentiment analysis of European Twitter messages during the first few months of the COVID-19 pandemic found that the lockdown announcements correlate with a deterioration of moods, which recovers within a short time span \cite{kruspe2020cross}.
The sentiment analysis examples from Europe and India show how social media can play a powerful role in understanding the psychology \textcolor{black}{and the human condition} Recent advancements of deep learning models as a tool for building robust language models have provided further motivation in understanding the temporal nature of the sentiments during and after the first peak of COVID-19 in India. Different countries had different peaks where rise and fall of new cases implemented lock-downs which directly affected the economy and employment. India is special in this way (until February 2021), where a single nation-wide peak was seen and only certain states had multiple peaks (Delhi and Maharashtra) \cite{ChandraCOVID2021}.
In this paper, we use deep learning based language models via long short-term memory (LSTM) recurrent neural networks for sentiment analysis via tweets with a focus of rise of novel cases in India. \textcolor{black}{We use LSTM and bidirectional LSTM (BD-LSTM) model with global vector (GloVe) for word representation for building a language model. Moreover, we use the BERT model to compare the results from LSTM and BD-LSTM models and then use the best model for COVID-19 sentiment analysis for the case of India.} We use three datasets, which include India, along with the state of Maharashtra (includes Mumbai) and Delhi. We compare the monthly sentiments expressed covering the major peak of new cases in 2020. We present a framework that focuses on multi-label sentiment classification, where more than one sentiment can be expressed at once. We use Senwave COVID-19 sentiment dataset \cite{yang2020senwave} which features 10,000 tweets collected worldwide and hand-labelled by 50 experts for training LSTM models.
\textcolor{black}{We highlight that there is no study that uses language models for sentiment analysis during the rise of novel COVID-19 cases. Our framework compares the different types of sentiments expressed across the different months in relation to the rise of the number of cases, which impacted the economy and had different levels of lock downs. This had an effect on the psychology of the population given stress and fear. Hence, the study is a way to quantify and validate emotional and psychological conditions given uncertainty about the pandemic. The major contribution of the paper is in using state-of-art sentiment analysis methods for understanding the public behaviour in terms of psychological well being in relation to the rise of novel COVID-19 infections. This study presents a novel framework that makes use of information from social media for understanding public behavior during a major disruptive event of the century.}
The rest of the paper is organised as follows: Section 2 presents a background of related work, and Section 3 presents the proposed methodology with data analysis. Section 4 presents experiments and results. Section 5 provides a discussion and Section 6 concludes the paper with discussion of future work.
\section{Related Work}
Word embedding is the process of feature extraction from text for NLP tasks such as sentiment analysis \cite{li2018word,kutuzov2018diachronic,ruder2019survey}. Word embedding can be obtained using methods where words or phrases from the vocabulary are mapped to vectors of real numbers. The process generally involves a mathematical embedding from a large corpus with many dimensions per word to a vector space with a lower dimension that is useful for machine learning or deep learning models for text classification tasks \cite{li2018word}. Basic word embedding methods such as \textit{bag of words} \cite{zhang2010understanding} and \textit{ term frequency inverse document
frequency } \cite{ramos2003using} do not have context awareness and semantic information in embedding. This is also a problem for skip-grams \cite{goodman2001bit} that use n-grams (such as bigrams and tri-grams) to develop word embedding, and in addition allow adjacent sequences of words tokens to be “skipped” \cite{guthrie2006closer}.
Over the last decade, there has been phenomenal progress in the area of world embedding and language models. Mikolov et al. \cite{mikolov2013distributed} proposed \textit{word2vec} embedding which uses a feedforward neural network model to learn word associations from a text dataset which can detect synonymous words or suggest additional words given a partial sentence. It uses continuous bag-of-words (CBOW) or continuous skip-gram model architectures to produce a distributed representation of words. The method is used to create a large vector which represent each unique word in the corpus where semantic information and relation between the words are preserved. It has been shown that for two sentences that do not have much words in common, their semantic similarity can be captured using word2vec \cite{mikolov2013distributed}. The limitation of word2vec is that it does not well represent the context of a word. \textcolor{black}{Pennington et al. \cite{pennington2014glove} proposed global vectors (GloVe) for word embedding as an unsupervised learning algorithm} for obtaining vector representations for words by mapping words into a meaningful space where the distance between words is related to semantic similarity . GloVe uses matrix factorization to constructs a large matrix of co-occurrence information to obtain representation that showcase linear substructures of the word vector space. The embedding feature vectors with top list words that match with certain distance measures. GloVe \textcolor{black}{can be used to find relations} between words such as synonyms, company-product relations. \textcolor{black}{Due to the awareness in ethics in machine learning, there has been a major focus on ethical issues in NLP. } A recent study showed that GloVe can have gender biased information; hence, a gender neutral GloVe method has been proposed \cite{zhao2018learning}.
There are some studies that review the effectiveness of word embedding methods. Ghannay et al. \cite{ghannay2016evaluation} \textcolor{black}{provided an evaluation} of word embedding methods such as GloVe \cite{pennington2014glove}, skip-gram, and continuous space language models (CSLM) \cite{schwenk2007continuous}. The authors reported that skip-gram and GloVe outperformed CSLM in all the language tasks. Wang et al. \cite{wang2018comparison} evaluated word embedding methods such as GloVe for applications of biomedical text analysis where it was found that word embedding trained from clinical notes and literature better captured word semantics.
\section{Methodology}
\subsection{LSTM and BERT language models}
Recurrent neural networks (RNN) in general feature a context memory layer to incorporate previous state and current inputs for propagating information to the next state, and eventually the output layer for decision making. Canonical RNNs (simple RNNs) feature several different architectures, besides the Elman RNN \cite{elman_Zipser1988,Elman_1990} for modelling temporal sequences and dynamical systems \cite{Omlin_etal1996,Omlin_Giles1992,Chandra_Omlin2007}. One of the major challenges in training simple RNNs is due to the architectures properties of unfolding in time for long-term dependency problems. Backpropagation through time (BPTT), which is an extension of the backpropagation algorithm, has been prominent for training simple RNNs \cite{Werbos_1990}. Due to problem of learning long-term dependencies given vanishing and exploding gradients with simple RNNs \cite{Hochreiter_1998}, long short-term memory (LSTM) recurrent neural networks have been developed \cite{hochreiter1997long}. LSTM networks have better capabilities for learning long-term dependencies in temporal data using memory cells and gates.
In the last decade, with the deep learning revolution, several LSTM architectures have been developed. \textcolor{black}{ Bidirectional LSTM models \cite{graves2005} process information in two directions, rather than making use of only previous context state for determining the next states which are based on bidirectional RNNs \cite{schuster1997}}. In this way,
two independent LSTM models allow both backward and forward information about the sequence at every time step. This enables better access to \textcolor{black}{long range state information which have been useful for word embedding \cite{graves2005} and several other sequence processing problems \cite{graves2005,Fan2014TTSSW,graves2013hybrid}.}
\textcolor{black}{A transformer is an extended LSTM model that adopts the mechanism of attention which mimics cognitive attention to enhance important parts of the data while fading the rest \cite{vaswani2017attention}. Transformers also use an encoder-decoder architecture and have mostly been used for NLP tasks such as translation and text summarising \cite{vaswani2017attention,wolf2020transformers}. In comparison to conventional RNNs, transformers do not require data to be processed in a sequential order since the attention operation provides context for any position in the input sequence. BERT is a pre-trained transformer-based model for NLP tasks which has been used to better understand user behaviour in Google search engine \cite{devlin2018bert}. BERT can be used for a number of other NLP applications such as clinical data processing \cite{su2020application}. The original BERT \cite{devlin2018bert} has two models: 1.) BERT-base features 12 encoders with 12 bidirectional self-attention heads, and 2.) BERT-large features 24 encoders with 16 bidirectional self-attention heads. These are pre-trained from unlabeled data extracted from a corpus with 800 million words and English Wikipedia with 2,500 million words, respectively. Word2vec and GloVe are content-free models that generate a single word embedding representation for each word, whereas BERT takes into account the context for each occurrence of a given word which makes BERT one of the best language models. Hence, BERT is suitable for our sentiment analysis tasks using tweets during rise of COVID-19 cases in India. }
\subsection{Twitter-based Sentiment Analysis Framework for COVID-19}
Our framework for sentiment analysis with deep learning follows a series of steps with major components that involve: 1.) tweet extraction; 2.) tweet pre-processing; 3.) model development and training using LSTM, BD-LSTM and BERT; and 4.) prediction using selected COVID-19 data. Figure \ref{fig:framework} shows the framework diagram where we highlight the major components. We note that the framework features multi-label classification which has multiple outcomes at once, i.e the tweet can be optimistic and joking at the same time.
\begin{figure*}[htbp!]
\centering
\includegraphics[width=14cm]{figures/1.jpg}
\caption{\textcolor{black}{Framework: Twitter-based sentiment analysis for COVID-19 with model development via LSTM, BD-LSTM and BERT. After training, the best model is chosen for prediction of COVID-19 tweets from India.} Note that the framework features multi-label classification which has the ability to provide more than two outcomes at once; i.e. a tweet can be optimistic and joking at the same time. }
\label{fig:framework}
\end{figure*}
The first step involves processing on COVID-19 dataset from a selected region. \textcolor{black}{We choose COVID-19 tweets from while of India along with two states which have been amongst the highest states of COVID-19 cases in 2020 (Maharashtra and Delhi). } \textcolor{black}{We process Twitter data obtained for India and related state as follows. We note that special phrases, emotion symbols (emoji's) and abbreviations are present in tweets, since language in social media has been rapidly evolving. Hence, we have to transform them for building the language model. Table \ref{tab:process} shows other words and emotions that are transformed.} We note that since this is a case of India, where English is used in combination with some key indigenous Indian languages (such as Hindi). Hence, we transform certain words, emotions and character symbols expressed in other languages as well.
Note that we use hand-labelled sentiment dataset (Senwave COVID-19 dataset) \cite{yang2020senwave} which features 11 different sentiments labelled by a group of 50 experts for 10,000 tweets worldwide during COVID-19 pandemic in 2020. In this dataset, "official report" is classified as a sentiment although it is a topic.
\begin{center}
\begin{table}[htbp!]
\small
\begin{tabular}{l l }
\hline
Original Phrase/Emoji & Transformed Word \\
\hline
omg & oh my god \\
btw & by the way \\
socialdistancing & social distancing \\
\smiley{} & smiling face \\
\frownie{} & sad face \\
\Bed & bed \\
\Fire & fire \\
\Winkey & wink \\
\Laughey & laugh \\
\hline
\hline
\end{tabular}
\caption{Examples of word transformation that features changing symbols and emoji's to words that can have semantic representation. }
\label{tab:process}
\end{table}
\end{center}
The next step involves converting each word into its corresponding GLoVe embedding vector, where each word is converted into a vector (300 dimensions in our case). We select GLoVe embedding since it has shown good results for language models with sentiment analysis in the literature \cite{pennington2014glove}. The aforementioned vector from each word thus obtained is passed on to the respective LSTM models for training once the model architecture has been defined as shown in Figure \ref{fig:framework}. We first evaluate our trained models (LSTM, BD-LSTM, and BERT) and then use it for COVID-19 sentiment analysis using test data (India, Maharashtra, and Delhi).
As shown in Figure \ref{fig:framework}, the trained model is used to classify 11 sentiments such as ``anxious'', ``sad'', and ``joking''.
It is essential to have the right metric to evaluate the model for the given application. The outcome of a misclassification in a multi-label classification is no longer a correct or an incorrect instance as opposed to binary classification or multi-class classification \cite{dembczynski2012label}. In multi-label classification, a prediction that features a subset of the actual classes are considered better than a prediction that contains none of the actual classes; hence, this needs to be captured by the metric that captures the loss or gives a score. As given in the literature \cite{dembczynski2012label}, multi-label classification evaluation is typically based on 1.) binary cross-entropy (BCE) loss\cite{zhang2018generalized} which is a softmax activation combined with a cross-entropy loss; 2.) Hamming loss\cite{dembczynski2012label} which calculates loss generated in the bit string of class labels using \textit{exclusive or} (XOR) operation between the actual and predicted labels and then computes the average over the instances of the dataset. The Jaccard coefficient score\cite{hamers1989similarity} provides a measure of the overlap between actual and predicted labels with their attributes capturing similarity and diversity. Furthermore, label ranking average precision (LRAP) score \cite{furnkranz2008multilabel} finds the percentage of the higher-ranked labels that resemble true labels for each of the given samples. F1-score conveys the balance between the precision and the recall. F1-score has been prominent for understanding class imbalance problems \cite{jeni2013facing}. \textcolor{black}{In application to multi-label classification, two types of F1-scores are predominantly used. F1-macro is computed using the F1-score per class/label and then averaging them, whereas F1-micro computes the F1-score for the entire data considering the total true positives, false negatives and false positives} \cite{lewis1996training}. We use a combination of these scores to evaluate our model results for the first phase of the framework where the respective models are trained.
\subsection{Model Training}
We present an experimental study that compares multi-label classification using \textcolor{black}{LSTM, BD-LSTM, and BERT models} as shown in the framework (Figure \ref{fig:framework}). We use 90\% of the dataset for training and 10\% for testing. We use the respective models and train them using the Senwave COVID-19 dataset which is publicly available, but restricted and needs permission from the authors for usage \cite{yang2020senwave}. Therefore, we cannot provide the processed dataset, but rather provide the trained models via Github repository (\url{https://github.com/sydney-machine-learning/COVID19_sentinentanalysis}). The Senwave dataset features 10,000 Tweets collected from March to May 2020.
\textcolor{black}{In the LSTM model, we determine the model hyper-parameters based on how the model performs in trial experiments. The GloVe embedding uses a word vector of size 300 size to provide data representation which is a standard representation in the literature \cite{pennington2014glove}. We use a dropout regularisation probability of 0.65 for LSTM and BD-LSTM models which feature: 300 input units, two layers with 128 and 64 hidden units, and finally an output layer with 11 units for classifying the sentiments. In the case of BERT, we use the default hyper-parameters for the BERT-base (uncased) model and only tune the learning rate. The only major addition is a dropout layer at the end of the BERT architecture followed by a linear activation layer with 11 outputs corresponding to the 11 sentiments.}
Table \ref{tab:train} shows the model training results (mean) for 10 experiments with different initial weights and biases for the respective models with different performance metrics. \textcolor{black}{Generally, we find that BERT is better than LSTM and BD-LSTM models across the different performance metrics shown in Table \ref{tab:train}. Hence, we proceed with the BERT model for evaluating the test dataset with a visualization of the predictions.}
\begin{center}
\begin{table}[htbp!]
\small
\begin{tabular}{c c c c}
\hline
Metric & BD-LSTM & LSTM & \textcolor{black}{BERT} \\
\hline
\hline
BCE Loss & 0.281 & 0.255 & 0.372\\
Hamming Loss & 0.163 & 0.157 & 0.142\\
Jaccard Score & 0.417 & 0.418 & 0.510\\
LRAP Score & 0.503 & 0.511 & 0.766\\
F1 Macro & 0.434 & 0.430 & 0.530\\
F1 Micro & 0.495 & 0.493 & 0.587\\
\hline
\hline
\end{tabular}
\caption{Training performance for \textcolor{black}{BD-LSTM, LSTM and BERT model} using Senwave COVID-19 training dataset. Note that except for the BCE and Hamming loss, higher scores shows better performance.}
\label{tab:train}
\end{table}
\end{center}
\subsection{Application to COVID-19 pandemic in India}
India COVID-19 pandemic management had major challenges given large population with a large number of densely populated cities \cite{lancet2020india}. India reported first COVID-19 case on 30th January 2020 and entered a lock-down afterwards which was gradually changed.
India (22-nd March, 2021) had more than 11.6 million confirmed cases with more than 160 thousand deaths which made India the third highest confirmed cases in the world after the United States and Brazil. India was the 8th in world with more than 300 thousand active cases (prior to the second wave). India had a single major peak around middle of September in 2020 with close to 100 thousand cases daily, which gradually decreased to around 11 thousand cases daily end of January 2021, and since then it was gradually raising. In March 2021, the cases began rising faster and by 22nd March, India had 47 thousand daily new cases which was moving towards a second peak \cite{indiaworld}.
In the first six months, the states of Maharashtra (population of about 124 million), Delhi (population of 19 million), and Tamil Nadu (population of about 8 million) led COVID-19 infections \cite{indiapop}. \textcolor{black}{Note that the city of Mumbai is in the state of Maharashtra, which in terms of the population can be compared to some } of the highly populated countries. In the later half of the year, Delhi cases reduced but still remained in leading 8 states \cite{ChandraCOVID2021}. In 2021, Maharashtra continued as state of highest infections and in March, it was featuring more than half of new cases on a weekly basis and Delhi contained the situation with less than a thousand daily cases. Hence, our focus is to study whole of India with two states of interest that includes Maharashtra and Delhi.
We note that the proposed framework can be applied to any part of the world; however, we are choosing the case of India to show the effectiveness of the framework.
The final step is in applying the different datasets from COVID-19 in India which include, nation-wide COVID-19 sentiments, and two major states with COVID-19 cases. The trend in the cases shows that both states had a major peak followed by minor ones, whereas India as a whole had a single major peak which was around mid-September 2020 with close to 97,000 novel cases per day during the peak as shown in the next section (Figure 2).
\section{Results}
In this section, we provide results of the implementation of our proposed framework using Twitter dataset for COVID-19 in India.
\subsection{COVID-19 Data Visualisation}
Our test dataset \cite{lamsal2020design} features COVID-19 India which features tweets associated with COVID-19 from March to September 2020. It consists of more than 150,000 tweets from India. We generate two other datasets from this which feature the regions of Maharashtra (state) and Delhi (union territory) which contains around 18,000 tweets each, respectively.
We first provide visualisation of the dataset and report features of interest to language models such as bi-grams, and tri-grams and distribution of monthly tweets. Figure \ref{fig:cases} shows the distribution of number of tweets for selected months along with the number of cases for the respective datasets.
We notice that the number of tweets in the case of India follows the same trend as the number of novel COVID-19 cases until July, afterwards the number of tweets decline (Panel~a, Figure~\ref{fig:cases}). There is a similar pattern for case of Maharashtra (Panel~b). The case of Delhi (Panel~c) is slightly different as the first peak was reached in July with increasing tweets that declined afterwards and did not keep up with the second peak of cases in September. This indicates that as more cases were observed in the early months, there was much concern which eased before the major peak was reached and the number of tweets were drastically decreased. There could be elements of fear, depression and anxiety as the tweets decreased drastically after July with increasing cases, which we analyse in the following section.
Figure \ref{fig:ngrams_india} shows a visualisation of the number of bi-grams and tri-grams for the case of India. In case of bi-grams (Panel a), we find that the term ``corona - virus" is mostly used followed by ``covid - 19" which are essentially the same. Next, it is interesting that the terms ``folded - hand" are mostly used followed by ``positive - case" and ``social - distancing". The term ``folded - hand" shows Indian social and religious symbol that denotes keeping the faith, giving respect and also a sign of acknowledgement and appreciation. We note that the ``folded hand" is an emotion icon (emoji) used predominantly in social media, and hence been processed as a text during prepossessing (Table 1), which is actually is not a bi-gram from a semantic viewpoint. In order to provide better understanding of the context of the tweets, we give examples in Table \ref{tab:bigrams}. In the case of tri-grams (Panel b), we find more information in tweets such as "backhand - index - pointing" which is an emoji; hence, we provide some of the tweets that fall in this category in Table \ref{tab:trigrams}.
Figure \ref{fig:heatoccur_senwave} shows number of occurrence of a given sentiment in relation to the rest of the sentiments for 10,000 hand-labelled tweets in Senwave dataset \cite{yang2020senwave} used for training.
\begin{figure}[htbp!]
\centering
\subfigure[India]{
\includegraphics[scale =0.25]{figures/2a.png}
}
\subfigure[Maharashtra]{
\includegraphics[scale =0.25]{figures/2b.png}
}
\subfigure[Delhi]{
\includegraphics[scale =0.25]{figures/2c.png}
}
\caption{Novel COVID-19 tweets and cases in India, Maharashtra, and Delhi. \textcolor{black}{The red bars indicate the number of tweets while the black bars indicate the number of novel cases.} }
\label{fig:cases}
\end{figure}
\begin{figure}[htbp!]
\centering
\subfigure[Bi-grams]{
\includegraphics[scale =0.33]{figures/3a.png}
}
\subfigure[Tri-grams]{
\includegraphics[scale =0.33] {figures/3b.png}
}
\caption{Bi-grams and tri-grams for case of India. Note that a combination of emoji's such as ``backhand - index - pointing", ``smiling - face", and words have been used. }
\label{fig:ngrams_india}
\end{figure}
\begin{table*}[htbp!]
\small
\centering
\begin{tabular}{l l l}
\hline
Month & Tweet & Bi-gram (Emoji) \\
\hline
\hline
March & ``releasing today at 6:00 pm on my youtube channel! & ``folded hand" \\
& let's fight this together folded hands i need your support guys" & \\
\hline
July & ``that's really shameful and heinous folded hands" & ``folded hand" \\
\hline
August & ``please applaud this corona economy warrior. folded hands kudos." & ``folded hand" \\
\hline
March & ``india this backhand index pointing down" & ``index pointing" \\
\hline
May & ``corona : to everyone backhand index pointing down" & ``index pointing" \\
\hline
July & ``backhand index pointing right. ........lockdown time..... & ``index pointing" \\
& \#picture \#instagood \#instadaily \#insta" & \\
\hline
\hline
\end{tabular}
\caption{Selected examples of processed tweets that are captured in most prominent bi-grams.}
\label{tab:bigrams}
\end{table*}
\begin{table*}[htbp!]
\small
\centering
\begin{tabular}{l l l}
\hline
Month & Tweet & Tri-gram (Emoji) \\
\hline
\hline
May & ``google doc of resources for amphan and covid. & ``backhand - index - pointing" \\
& backhand index pointing down. & \\
& retweet for greater reach." & \\
\hline
August & "\#covid tips backhand index pointing down" & ``backhand - index - pointing" \\
\hline
September & "backhand index pointing right registers & ``backhand - index - pointing" \\
& highest single-day cases in the world - 95,529" & \\
\hline
\hline
\end{tabular}
\caption{Selected examples of processed tweets that are captured in most prominent tri-grams.}
\label{tab:trigrams}
\end{table*}
\begin{figure}[htbp!]
\includegraphics[width=10cm]{figures/4.png}
\caption{Heatmap showing number of occurrence of a given sentiment in relation to the rest of the sentiments for 10,000 tweets from Senwave dataset \cite{yang2020senwave} used for training.}
\label{fig:heatoccur_senwave}
\end{figure}
\subsection{Sentiment Prediction}
In this section, we present results of the predictions of the COVID-19 tweets in India, Maharashtra, and Delhi treated as separate datasets. We note that in our Senwave dataset analysed in previous section has been used for training. The label ``official report", which is not a sentiment was included in the Senwave dataset, and hence we use it for prediction. Figure \ref{fig:senti_distri} presents distribution of sentiments predicted by the LSTM and BERT models for the respective datasets for the entire time-span of interest (March to September 2020). In the case of India (Panel a), we notice that the ``optimistic", ``annoyed" and ``joking" sentiments are the most expressed which also follows in the case of Maharashtra and Delhi (Panel b and c). We notice that the BERT model seems to capture more sentiments expressed when compared to the LSTM model, particularly the ``optimistic", ``anxious", and ``annoyed" sentiments. We find that negative sentiments such as ``pessimistic", ``anxious" and ``sad" have been least expressed. We find ``optimistic" sentiment as the most prominent sentiment which is followed by ``annoyed" and ``joking". It is simpler to label "optimistic" and "thankful" as a positive sentiment, but it becomes increasing difficult when it comes to the sentiments ``joking" and ``surprise", when context information is not available. Hence, it is insightful to review the sentiments such as ``joking" and ``surprise" in relation to other sentiments expressed. We find a similar trend in the case of Delhi and Maharashtra (Panel b and c) which are subsets of data from India (Panel a), the only major difference is the number of tweets which is significantly lower due to their respective population. Since the BERT model provided the best results for the training data, we provide the results by the BERT model henceforward.
\begin{figure}[htbp!]
\centering
\subfigure[India]{
\includegraphics[scale =0.27]{figures/5a.png}
}
\subfigure[Maharashtra]{
\includegraphics[scale = 0.27] {figures/5b.png}
}
\subfigure[Delhi]{
\includegraphics[scale = 0.27] {figures/5c.png}
}
\caption{Distribution of sentiments predicted for the respective datasets by the \textcolor{black}{LSTM and BERT models.}}
\label{fig:senti_distri}
\end{figure}
\begin{figure}[htbp!]
\includegraphics[width=10cm]{figures/6.png}
\caption{Heatmap showing number of occurrence of a given sentiment in relation to the rest of the sentiments for the India dataset \textcolor{black}{using the BERT model.}}
\label{fig:heatoccur}
\end{figure}
\begin{figure}[htbp!]
\includegraphics[width=10cm]{figures/7.png}
\caption{Heatmap showing number of occurrence of a given sentiment in relation to the rest of the sentiments for the Maharashtra dataset \textcolor{black}{using the BERT model.}}
\label{fig:heatoccur_maharastra}
\end{figure}
\begin{figure}[htbp!]
\includegraphics[width=10cm]{figures/8.png}
\caption{Heatmap showing number of occurrence of a given sentiment in relation to the rest of the sentiments for the Delhi dataset \textcolor{black}{using the BERT model.}}
\label{fig:heatoccur_delhi}
\end{figure}
Next, we review the sentiments predicted using a heatmap to examine number of occurrence of a given sentiment in relation to the rest of the sentiments (Figures \ref{fig:heatoccur}, 7 and 8). These heatmaps essentially indicate how two sentiments have been expressed at once and provides more insights regarding the context of the negative and positive sentiments given in Figure \ref{fig:senti_distri}. Note that in Figure \ref{fig:senti_distri}, we found that the third most prominent sentiment has been ``joking"; however, we were not sure if it is positive or negative sentiment. As shown in Figure \ref{fig:heatoccur}, \textcolor{black}{we find that most tweets that are associated with ``joking" are either ``optimistic" (8680) or ``annoyed" (10323), and some are also ``thankful" (889). A much lower portion (below 500) are either ``empathetic" or ``official report" while ``joking". Next, we review the sentiment ``optimistic" and find that majority are either ``thankful" (14230) or ``joking" (8680). The ``optimistic" tweets with negative sentiments are relatively minor, such as ``pessimistic" (772), ``denial" (143), and ``sad" (349). Furthermore, a significant portion are also ``empathetic" (2533).} It is not common for one to make a statement that is optimistic and pessimistic at the same time; hence, this could be a wrong prediction by the model. However, it seems the model is making the right prediction when we look at the heatmap for the hand-labelled training datasets (Figure \ref{fig:heatoccur_senwave}), where such combinations of sentiments have been labelled by experts. We show examples of tweets of this case and compare with those that are optimistic and thankful in Table \ref{tab:optipessimis}.
Figure \ref{fig:heatoccur_maharastra} and Figure \ref{fig:heatoccur_delhi} show the number of occurrence of a given sentiment in relation to the rest of the sentiments for the case of Maharashtra and Delhi, which follows a similar pattern when compared to case of India (Figure \ref{fig:heatoccur})). We note that the Senwave dataset which shows tweets worldwide (Figure \ref{fig:heatoccur_senwave}) follow a similar pattern than the case in Indian datasets when it comes to sentiments such as ``joking" and being ``optimistic" or ``joking" and ``annoyed". Senwave dataset also features cases of being optimistic and pessimistic at the same time (226 cases). This could be due to sentiments expressed in two opposing sentences in a tweet. We can infer that since such patterns are part of the training data, it cannot be an error in predictions when looking at the Indian datasets.
\begin{table*}[htbp!]
\small
\centering
\begin{tabular}{l l l}
\hline
Month & Tweet & Sentiment combination\\
\hline
\hline
May & ``don’t care china only care indian covid19 news" & ``optimistic - pessimistic" \\
\hline
August & ``dear uddhav ji other than covid problem, & ``optimistic - pessimistic" \\
& what ever wrong is happening is not good & \\
& for your government" & \\
\hline
September & ``sir, plz look in have benefits to sr citizen." & ``optimistic - pessimistic" \\
\hline
April & ``thank you switzerland smiling face with smiling & ``optimistic - thankful" \\
& eyes especially zermatt for showing " & \\
& solidarity for india flag" & \\
\hline
May & ``big thanks to the cfpc....thoroughly enjoyed it!" & "optimistic - thankful" \\
\hline
June & ``doctors, activists urge pm to promote plant-based diet & ``optimistic - thankful" \\
& | india news - times of india" & \\
\hline
\hline
\end{tabular}
\caption{Selected example that show cases of tweets that are ``optimistic" and also ``pessimistic", along with cases that are ``optimistic" and also ``thankful".}
\label{tab:optipessimis}
\end{table*}
\begin{figure}[htbp!]
\centering
\subfigure[Senwave hand-labelled sentiments (Worldwide)]{
\includegraphics[scale =0.25]{figures/9a.png}
}
\subfigure[Predicted (India) \textcolor{black}{using BERT}]{
\includegraphics[scale =0.25]{figures/9b.png}
}
\subfigure[Predicted (India) \textcolor{black}{using LSTM}]{
\includegraphics[scale =0.25]{figures/9c.png}
}
\caption{Distribution of tweets with number of combination of sentiment types (labels) \textcolor{black}{for Senwave training dataset, and predictions by BERT and LSTM models.}}
\label{fig:numbsenti}
\end{figure}
Figure \ref{fig:numbsenti} provides a visualisation of the distribution of tweets with number of combination sentiments. We find that around 60\% of the tweets have a singular sentiment. Moreover, about 25\% of the tweets have two sentiment attached them and 14\% have no sentiment attached to them. Furthermore, a small number of tweets have 3 or more emotions attached to them which indicates that much often, people do not show multiple emotions at the same time.
Finally, we present results that visualises the trend of sentiments expressed over time in the three datasets. This is one of the key findings which can enable a understanding of the reaction in terms of emotions expressed by the population given rise and fall of COVID-19 novel cases as shown in Figure \ref{fig:cases}.
Figures \ref{fig:monindia}, \ref{fig:monmaha} and \ref{fig:mondelhi} presents visualisation of the monthly sentiments of India, Maharashtra and Delhi, respectively.
\begin{figure*}[htbp!]
\centering
\includegraphics[width=14cm]{figures/10.png}
\caption{Monthly COVID-19 sentiments in India \textcolor{black}{using the BERT model.}}
\label{fig:monindia}
\end{figure*}
\begin{figure*}[htbp!]
\centering
\includegraphics[width=14cm]{figures/11.png}
\caption{Monthly COVID-19 sentiments in Maharashtra \textcolor{black}{using the BERT model.}}
\label{fig:monmaha}
\end{figure*}
\begin{figure*}[htbp!]
\centering
\includegraphics[width=14cm]{figures/12.png}
\caption{Monthly COVID-19 sentiments in Delhi \textcolor{black}{using the BERT model.}}
\label{fig:mondelhi}
\end{figure*}
\section{Discussion}
Humans can express more than one sentiment at a time; however, there are variations in the number of sentiments that can be expressed by facial expressions \cite{jack2014dynamic}, when compared to spoken or written emotions such as tweets. As shown in Figure
\ref{fig:numbsenti}, majority of the tweets have one sentiment, both in hand-labelled (Panel a) and predicted datasets (Panel b). This is followed by two sentiments, while a minority have 3 sentiments. We note that there is a significant portion of tweets with no sentiments and there are no tweets of more than three sentiments at a time. The study of emotions with technology has gained much interested in last decade which gave much progress in understanding sentiments. Trampe et al. \cite{trampe2015emotions} presented a study of everyday emotional experiences through an experience sampling smartphone application that monitored real-time emotions of more than 11,000 participants and found the group experienced at least one emotion 90\% of the time with joy as the the most frequent emotion, followed by love and anxiety. The type of emotion would be highly dependent on the study region which featured Europeans with majority French participants. Cowen and Keltner reported \cite{cowen2017self} twenty seven distinct categories of emotions bridged by continuous gradients by data from emotionally evocative short videos with varying situational content. These studies contributed to better understanding of emotions given historical viewpoints about context and definitions of emotions and associations between them \cite{wintre1994developmental,kemper1987many,ekman1992argument,lucas2003measuring}. However, we did not find any study that reviews the number of emotions that can be expressed at a time in relation to catastrophic events that keep changing with time, such as the rise of COVID-19 cases.
We revisit the case of Indian dataset (Figure \ref{fig:monindia}), where the monthly tweets did not sharply increase with number of novel cases (Figure 2) with a nationwide peak of novel cases (Figure 2, Panel a). The number of tweets gradually increased with peak of tweets in July (Figure 2, Panel a). When India had a peak of novel cases, we found that the number of tweets significantly lowered. Hence, people have been alarmed by rising cases, but a month before the peak of novel cases was seen, the tweets were reduced. Moreover, we find that the ``optimistic", ``annoyed" and ``joking" tweets are mostly dominating the monthly tweets for India (Figure 10) and Maharashtra (Figure 11), with a mix of annoyed sentiments in case of Delhi (Figure 12). There is significantly lower number of negative sentiments for the respective datasets (Figure 10-12).
We note that a limitation of the framework is due to the Senwave training data which considered tweets worldwide during COVID-19 by a group of experts; however, there can be limitations on how experts perceive humour in different regions. Humour is expressed differently in different cultural groups, i.e. a tweet that may be humorous in USA may not be taken as humorous in India due to cultural and language dimensions. There are several studies about the effect of humour in changing cultural or regional settings \cite{friedman2014comedy,rossato2010audiences,kuipers2011politics}. A good example is that in orthodox Chinese literature, humour was not expressed due to religious taboo in ancient Buddhism which later eased with Zen Buddhism \cite{chey2011youmo,clasquin2001real}; however, the Hindu literature had a different or eased attitude towards humour as expressed in ancient Hindu texts \cite{elst2011humour}. Although historic textual composition cannot be related to how tweets are expressed in India, it is important to note the cultural and language differences in how humour has been expressed.
Another limitation of the study is regarding the uncertainty of the data. Our results are based on the tweets expressed; however, we note that only a small fraction of the population generally express their views on Twitter. Moreover, we need to be aware that due to restrictions on freedom of speech and political biases, not everyone can express themselves freely. Social media networks have been active in monitoring how people express themselves in order to limit the rise of anti-vaccine sentiments. Furthermore, our framework’s training data is based on tweets expressed worldwide, whereas the test data is based on tweets from India during the pandemic. There is a large language diversity in India and at times, people express themselves with a combination of languages with geographic and culture specific jargon. Although, we convert them to English, certain limitations in capturing the context will exist.
We note that there has not been much work in uncertainty quantification for the predictions and there are different level of uncertainties, particularly in model parameters and data. We note that the training data is hand-labelled, and at times two or three sentiments have been expressed at once. This could be something open to interpretation by experts as it is hard to formally detect more than one sentiment from a tweet of only thirty words. Hence, the expert labelled training dataset adds to uncertainty in model predictions. In future work, Bayesian deep learning can provide a way to address uncertainty in model predictions \cite{chandra2019langevin,
chandra2021revisiting,chandra2021bayesian}.
\section{Conclusions}
We presented a study with novel deep learning models for sentimental analysis during the rise of COVID-19 infections. We selected tweets from India as our case study and reviewed tweets from specific regions that included Maharastra and Delhi. We took advantage of COVID-19 dataset of 10,000 hand-labelled tweets for training the respective deep learning models. Our investigation revealed that majority of the tweets have been ``optimistic", ``annoyed" and ``joking" that expresses optimism, fear and uncertainty during the rise of the COVID-19 cases in India. The number of tweets significantly lowered towards the peak of new cases. Furthermore, the optimistic, annoyed and joking tweets mostly dominated the monthly tweets with much lower number of negative sentiments expressed. We found that most tweets that have been associated with ``joking" were either ``optimistic" or ``annoyed”, and minority of them were also ``thankful". In terms of the ``annoyed” sentiments in tweets, mostly were either ``surprised" or ``joking".
These predictions generally indicate that although the majority have been optimistic, a significant group of population has been annoyed towards the way the pandemic was handled by the authorities. The major contribution of the paper is the framework which provides sentiment analysis in a population given the rise of the COVID-19 cases. The framework can be used by officials for better COVID-19 management through policies and projects, such as support for depression and mental health issues.
Future work can use the framework for different regions, countries, ethnic and social groups to understand their behaviour given multiple peaks of novel cases. The framework can be extended to understand reactions towards vaccinations with the rise of anti-vaccine sentiments given fear, insecurity and unpredictability of COVID-19. Finally, the framework can incorporate topic modelling with sentiment analysis which can provide more details for emerging topics during the rise of COVID-19 cases in relation to various government protocols such as lock-downs and vaccination plans.
\section*{Data and Code}
We provide Python-based open source code and data for further research
(\url{https://github.com/sydney-machine-learning/COVID19_sentinentanalysis}).
|
1,108,101,563,288 | arxiv | \section{Introduction}
In this paper we apply the systematic theory of braided geometry introduced
during the last few years by the
author\cite{Ma:exa}\cite{Ma:lin}\cite{Ma:poi}\cite{Ma:fre}\cite{Ma:add} to the
problem of defining the totally antisymmetric tensor ${\epsilon}_{ijkl}$ and other
antisymmetrisers on quantum spaces of R-matrix type, for the first time in a
general way.
Braided-geometry differs from other approaches to q-deforming physics in that
the deformation is put directly into non-commutativity or `braid statistics' of
the tensor product of independent systems. Individual algebras also tend to be
non-commutative (as in non-commutative geometry) but this is a secon{{\rm d}} ary
phenomenon. The theory is modelled on ideas of super-geometry with a braiding
$\Psi$ (typically defined by a parameter $q$) in the role of $\pm 1$ for usual
bose or fermi statistics. It turns out that this point of view is rather
powerful and using it a great many problems encountered in other approaches
are immediately overcome.
The starting point of braided geometry is that quantum group covariance, unlike
usual group covariance, induces braid statistics\cite{Ma:exa}\cite{Ma:lin}. The
quantum group plays the role of ${\Bbb Z}_2$-grading in the theory of super-symmetry
even when the quantum group is very far from discrete (e.g. when it is the
$q$-deformed Lorentz group). The systematic development of braided geometry has
been a matter of going back to basics and re-inventing from scratch the most
fun{{\rm d}} amental concepts in physics on this basis, layer by layer. After
covariance, the next layer is that of coaddition on quantum spaces, introduced
in \cite{Ma:poi}. Once one can add vectors on braided or q-deformed vector
spaces, the next layer is differentiation, introduced in \cite{Ma:fre} as an
infinitesimal coaddition:
\eqn{diff}{ {\partial}^i f(\vecx)={\rm coeff\ of\ }a_i\ {\rm in}\ f(\veca+\vecx)}
where the addition is a braided addition (so $\veca$ and $\vecx$
braided-commute). Following this, there is also translation-invariant
integration\cite{KemMa:alg}. Braided matrices, traces etc were also introduced
in
\cite{Ma:exa}\cite{Ma:lin}. The approach also links up with the more usual
approach based on quantum forms and non-commutative geometry by pushing the
arguments of \cite{WesZum:cov} backwards (from partial derivatives ${\partial}^i$ to
exterior derivative ${{\rm d}}$). This is essentially known though some details are
included in the present paper for completeness. It provides a constructive
approach
to ${{\rm d}}$.
The antisymmetric tensor by contrast needs a conceptually new point of view in
order to be able to apply this existing braided geometry. Here we present a
novel and, we believe, powerful point of view about it. This point of view
is useful even when $q=1$ where it corresponds to the view that the exterior
algebra of forms can and should be viewed as a super-space with co-ordinates
$\theta_i$, say. Usually, one
realises super-spaces using exterior algebras, but our point of view is the
reverse of this. The braided geometry applies just as well to super-spaces and
their q-deformations as to bosonic-spaces and their deformations, so we can
apply it at once to the exterior algebra without effort. In particular it is
natural for us to define
\eqn{eps}{{\epsilon}^{i_1i_2\cdots i_n}={{\partial}\over{\partial} \theta_{i_1}}{{\partial}\over{\partial}
\theta_{i_2}}\cdots{{\partial}\over{\partial} \theta_{i_n}} \theta_1\theta_2\cdots
\theta_n}
on any reasonable n-dimensional braided space with top form
$\theta_1\theta_2\cdots\theta_n$. We also construct the Hodge $*$-operator
and interior products on forms in this setting.
Finally, important examples such as $q$-Euclidean and $q$-Minkowski spaces are
also known in this framework of braided geometry\cite{Mey:new}\cite{Ma:euc},
which
examples are compatible too with the earlier ideas of
\cite{CWSSW:lor}\cite{OSWZ:def}\cite{PodWor:def} based on spinors. Hence our
results apply at
once to these important braided spaces.
During the preparation of this paper there appeared \cite{Mey:wav} in which the
$q$-epsilon tensor in the case of
$q$-Minkowski space was found directly by computer and used to develop Hodge
theory and
scaler electrodynamics. Our general formulation in Section~3 is motivated in
part by
this. We would also like to mention \cite{Fio:det} where $q$-epsilon tensors
for
$SO_q(n)$-covariant Euclidean spaces were considered, again rather explicitly.
The tensor
for $GL_q$-covariant quantum planes is even more well known. By contrast with
such specific examples, we present here a uniform R-matrix approach.
\subsection*{Acknowledgements} This work was completed during a visit to the
Erwin Schr\"odinger Institute, Wien. I would like to thank the staff there for
their
help and support.
\subsection*{Preliminaries on braided vector spaces}
Here we recall the formulation in \cite{Ma:poi} of braided vector and covector
spaces, and strengthen their construction slightly for our purposes. The
position
co-ordinates $\vecx=\{x_i\}$ form a braided-covector space, while their
differentials ${\partial}^i$ form a braided vector space\cite{Ma:fre}. Throughout
this paper, we treat only spaces of this type, i.e. braided versions of ${\Bbb R}^n$.
The input {{\rm d}} ata for these constructions are a pair of R-matrices $R,R'\in
M_n\mathop{\otimes} M_n$ such that\cite{Ma:poi}
\[ R'_{12}R_{13}R_{23}=R_{23}R_{13}R'_{12},\quad
R'_{23}R_{13}R_{12}=R_{12}R_{13}R'_{23} \]
\[ R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12},\quad (PR+1)(PR-1)=0,\quad
R_{21}R'=R'_{21}R\]
where $P$ is the usual permutation matrix. These are enough to ensure that
there are braided vector and covector spaces
\[ {V\haj{\ }}(R',R)=\{x_i\}:\ \vecx_1\vecx_2=\vecx_2\vecx_1R',\quad
V(R',R)=\{v^i\}:\ \vecv_1\vecv_2=R'\vecv_2\vecv_1\]
with braided coaddition $\vecx''=\vecx+\vecx'$, $\vecv''=\vecv+\vecv'$ where
these obey the same relations provided $\vecx,\vecv$ and their identical primed
copies have braid statistics
\[ \vecx'_1\vecx_2=\vecx_2\vecx_1R,\quad \vecv'_1\vecv_2=R\vecv_2\vecv'_1.\]
There are also braid statistics between $\vecx$ and $\vecv$ etc.
Mathematically, they form braided-Hopf algebras in the braided category of
$A(R)$-comodules where $A(R)$ is the usual quantum group or bialgebra
associated to $R$. For regular R-matrices they also live in the braided
category of $\widetilde{A}$-comodules, where $A$ is a Hopf algebra quotient of
$A(R)$ and $\widetilde A$ is dilatonic extension\cite{Ma:poi}. The
transformation laws are $\vecx\to \vecx{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}\dila$ and
$\vecv\to\dila^{-1}{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}^{-1}\vecv$ where ${\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}$ is the quantum matrix
generator and $\dila$ the dilaton.
To this framework, we now add the additional conditions
\[ R_{12}R'_{13}R'_{23}=R'_{23}R'_{13}R_{12},\quad
R_{23}R'_{13}R'_{12}=R'_{12}R'_{13}R_{23} \]
\[ R'_{12}R'_{13}R'_{23}=R'_{23}R'_{13}R'_{12}\]
so that there is a certain symmetry between $R$ and $R'$. More precisely, we
have a symmetry
\[ R \leftrightarrow -R'\]
and can thereby define
\[ \Lambda(R',R)\equiv {V\haj{\ }}(-R,-R')=\{\theta_i\},\quad
\Lambda^*(R',R)\equiv V(-R,-R')=\{\phi^i\}\]
which we call respectively the braided covector space of antisymmetric tensors
or {\em forms} $\Lambda$ and braided vector space of {\em coforms} $\Lambda^*$.
As a braided-Hopf algebra, the
latter is the dual of the former. In our geometrical application, the
differentials $\theta_i={{\rm d}} x_i$ obey the algebra of forms, while the operators
${\partial}\over{\partial}\theta_i$ necessarily obey the algebra of coforms.
The forms and coforms are covariant under the transformation
$\theta\to\theta{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}\dila$ and $\phi\to\dila^{-1}{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}^{-1}\phi$ of a quantum
group obtained from $A(-R')=A(R')$. We assume for convenience that this is the
same as the quantum group obtained from $A(R)$. This is true in some
generality, for
example whenever $PR'=f(PR)$ for some function $f$. It is also true for our
$q$-Euclidean
and $q$-Minkowski examples. See \cite{Mey:new} for the latter.
\section{Symmetric and antisymmetric tensors by differentiation}
In \cite{Ma:fre} was introduced a general theory of partial differentiation
${\partial}^i$ on braided spaces of the type above. This recovered all known cases
and, moreover, works generally. If $\{x_i\}$ are the position co-ordinates,
then ${\partial}^i$ are given explicitly as the operators\cite{Ma:fre}
\eqn{delx}{ {{\partial}\over{\partial} x_i}(\vecx_1\cdots \vecx_m)={\bf
e}^i{}_1\vecx_2\cdots\vecx_m \left[m;R\right]_{1\cdots m}}
where ${\bf e}^i$ is a basis covector $({\bf e}^i){}_j=\delta^i{}_j$, i.e.
\[ {{\partial}\over{\partial} x_i}(x_{i_1}\cdots x_{i_m})=\delta^i{}_{j_1}x_{j_2}\cdots
x_{j_m}\left[m;R\right]^{j_1\cdots j_m}_{i_1\cdots i_m}.\]
Here
\eqn{braint}{\left[m;R\right]=1+(PR)_{12}+(PR)_{12}(PR)_{23}
+\cdots+(PR)_{12}\cdots (PR)_{m-1,m}}
are the {\em braided integers} which we introduced for this purpose. One of the
main theorems in \cite{Ma:fre} is that these differentiation operators on
$\{x_i\}$ obey the vector relations as for the $\{v^i\}$. One can say that the
partial-derivatives $R'$-commute. They also obey a braided-Leibniz rule with
braiding $R$\cite{Ma:fre}.
Moreover, since the result is quite general, it holds just as well for the
partial derivatives ${\partial}\over{\partial}\theta_i$,
\eqn{deltheta}{{{\partial}\over{\partial} \theta_i}(\theta_1\cdots \theta_m)={\bf
e}^i{}_1\theta_2\cdots\theta_m \left[m;-R'\right]_{1\cdots m}}
on the algebra of forms $\{\theta_i\}$. We deduce that these differential
operators obey the relations of the coforms $\Lambda^*$. This means that they
$-R$-commute and obey a braided Leibniz rule with braiding $-R'$.
These theorems about the partial-derivatives are quite powerful, and we use
them now. In particular we can differentiate any function $f$ and will know
that
\[ v^{i_1i_2\cdots i_m}={{\partial}\over{\partial} x_{i_1}}\cdots {{\partial}\over{\partial}
x_{i_m}}f\]
is an $R'$-symmetric tensor of rank $m$, in the sense
\eqn{R'-sym}{ R'^{i_p i_{p+1}}_{ab} v^{i_1\cdots i_{p-1}bai_{p+2}\cdots
i_m}=v^{i_1\cdots i_m},\quad \forall p=1,\cdots,m-1.}
If $f$ is a scaler function (quantum group covariant) then, because all our
constructions in \cite{Ma:fre} are covariant, we will know that this tensor is
likewise invariant. The same applies in the $\theta$ space, in which case the
tensors must be manifestly $-R$-symmetric i.e., $R$-antisymmetric in the sense
\eqn{R-asym}{ R^{i_p i_{p+1}}_{ab} \phi^{i_1\cdots i_{p-1}bai_{p+2}\cdots
i_m}=-\phi^{i_1\cdots i_m},\quad \forall p=1,\cdots,m-1.}
For a simple example of this idea, we suppose that the co-ordinate algebra
$\{x_i\}$ has a {\em radius function} $r^2$ which is quantum-group invariant (a
scaler under the transformation). Then
\[ \eta^{ij}={{\partial}\over{\partial} x_i}{{\partial}\over{\partial} x_j} r^2\]
is an $R'$-symmetric invariant tensor, which we call the {\em metric}
associated to the radius function. If $r^2$ is quadratic then $\eta$ is an
ordinary ${\Bbb C}$-number matrix. In nice cases it will be invertible. Moreover,
invariance implies at once the first half of the identities
\eqn{ggR}{ \eta_{ia}\eta_{jb}R^a{}_m{}^b{}_n=R^a{}_i{}^b{}_j
\eta_{am}\eta_{bn}, \quad \eta_{ia}\eta_{jb}R'^a{}_m{}^b{}_n=R'^a{}_i{}^b{}_j
\eta_{am}\eta_{bn}}
We adopt the second half too in order to keep the symmetry between $R$ and
$-R'$. They have the meaning that then\cite{Mey:new} the algebra of vectors and
covectors are isomorphic via
\[ x_i=\eta_{ia}v^a,\quad v^i=x_a\eta^{ai},\quad
\eta_{ja}\eta^{ia}=\delta^i{}_j=\eta_{aj}\eta^{ai}\]
so that the metric can be used to raise and lower indices for any operators
behaving like the vectors and covectors. It clearly does the same job for
raising and lowering indices of the forms and coforms by the symmetry.
We now use the same idea in the deformed super-space of forms. We say that the
braided space has {\em form dimension} $n$ if the algebra of forms has (up to
normalisation) a unique element of highest degree $n$, which we call the {\em
top form} $\omega$. In nice cases the form dimension will be the same as the
number $n$ of our co-ordinate generators and indeed, the top form
will be $\omega=\theta_1\cdots\theta_n$.
We then define
\[ {\epsilon}^{i_1 i_2\cdots i_n}={{\partial}\over{\partial} \theta_{i_1}}\cdots {{\partial}\over{\partial}
\theta_{i_n}} \omega={{\partial}\over{\partial} \theta_{i_1}}\cdots {{\partial}\over{\partial}
\theta_{i_n}}\theta_1\cdots\theta_n.\]
By the reasoning above, it will be $R$-antisymmetric.
If we want to have tensors with lower indices, we can obtain them also by
differentiation of monomials in the co-ordinates. Thus an $R'$-symmetric tensor
of rank $m$ with lower indices means
\eqn{symlower}{ x_{i_1\cdots i_{p-1}bai_{p+2}\cdots
i_m}R'{}^a{}_{i_p}{}^b{}_{i_{p+1}}=x_{i_1\cdots i_m},\quad \forall
p=1,\cdots,m-1}
and an $R$-antisymmetric tensor with lower indices means
\eqn{asymlower}{ \theta_{i_1\cdots i_{p-1}bai_{p+2}\cdots
i_m}R{}^a{}_{i_p}{}^b{}_{i_{p+1}}=-\theta_{i_1\cdots i_m},\quad \forall
p=1,\cdots,m-1.}
The first of these can be obtained by applying any m-th order differential
operator built from ${\partial}\over{\partial} x_i$ to monomials $x_{i_1}\cdots x_{i_m}$.
Likewise, we can follow the same idea in form-space and obtain an
$R$-antisymmetric tensor by applying
an $m$-th order operator built from ${\partial}\over{\partial}\theta_i$ to
$\theta_{i_1}\cdots\theta_{i_m}$. For example, we define
\[ {\epsilon}_{i_1\cdots i_n}={{\partial}\over{\partial} \theta_n}\cdots {{\partial}\over{\partial}
\theta_1} \theta_{i_1}\cdots\theta_{i_n}.\]
Its total R-antisymmetry is inherited this time from antisymmetry of the
$\theta_i$ co-ordinates in form-space.
\begin{propos} If the top form is $\omega=\theta_1\cdots\theta_n$ say, we have
an explicit formula:
\[ {\epsilon}^{i_1\cdots i_n}=([n;-R']!)^{i_n \cdots i_1}_{12\cdots n},\quad
{\epsilon}_{i_1\cdots i_n}=([n;-R']!)_{i_1 \cdots i_n}^{12\cdots n}\]
where
\align{ &&\kern -1.7em [n;-R']!=({\rm id}\mathop{\otimes} [2;-R'])({\rm id}\mathop{\otimes} [3;-R'])\cdots [n;-R']\\
&&=(1-PR'_{n-1\ n})(1-PR'_{n-2\ n-1}+PR'_{n-2\ n-1}PR'_{n-1\ n})\cdots\\
&&\cdots (1-PR'_{12}+PR'_{12}PR'_{23}-\cdots+(-1)^{n-1}PR'_{12}PR'_{23}\cdots
PR'_{n-1\ n})}
is built from braided-integers (\ref{braint}).
\end{propos}
\goodbreak\noindent{\bf Proof\quad} This follows directly from the above definitions by carefully iterating
the formula (\ref{deltheta})
for braided-differentiation on the $\theta$ co-ordinates. {\ $\lform$}\bigskip
For example, in four dimensions, the braided 4-factorial matrix is
\[ ([4;-R']!)^{i_1\cdots i_4}_{j_1\cdots j_4}=[2,-R']^{i_3
i_4}_{b_3b_4}[3,-R']^{i_2 b_3 b_4}_{a_2 a_3 a_4}[4; -R']^{i_1 a_2 a_3
a_4}_{j_1j_2j_2j_4}\]
and is totally $R$-antisymmetric in its lower indices and in its upper-indices.
If one wants totally antisymmetric tensors of lower rank, these are provided by
the lower braided-factorials $[m,-R']!$ in the same way. For example
\[
{{\partial}\over{\partial}\theta_{i_1}}{{\partial}\over{\partial}\theta_{i_2}}\theta_{j_1}\theta_{j_2}
=[2;-R']^{i_2i_1}_{j_1j_2},\quad [2;-R']=1-PR'\]
\[
{{\partial}\over{\partial}\theta_{i_1}}{{\partial}\over{\partial}\theta_{i_2}}
{{\partial}\over{\partial}\theta_{i_3}}
\theta_{j_1}\theta_{j_2}\theta_{j_3}=([3;-R']!)^{i_3i_2i_1}_{j_1j_2j_3},\quad
[3;-R']!=(1-PR'_{23})(1-PR'_{12}+PR'_{12}PR'_{23})\]
etc. One can also take different numbers of $\theta$ derivatives and
co-ordinates, giving tensors with different numbers of lower and upper indices,
but again totally $R$-antisymmetric among the lower and among the upper.
If one wants tensors with totally $R'$-symmetric inputs and outputs, these are
provided by braided-factorials $[n:R]!$ and other braided-integers, with $R$
in place of $-R'$. The symmetric and antisymmetric theory here are just the
same construction, with a different choice of R-matrix. In this context,
there is already proven a {\em braided-binomial theorem} in \cite{Ma:fre} for
`counting' such `braided permutations'.
\section{Application to Hodge $*$-operator}
One can obtain still more tensors with symmetric or antisymmetric inputs and
outputs by contraction along the lines
of \cite{Mey:wav}. For example, given our ${\epsilon}$ tensors it is natural to
consider the
contractions of $n-m$ indices,
\eqn{proj}{ P_{i_1\cdots i_m}{}^{j_1\cdots j_m}={\epsilon}_{i_1\cdots i_m
a_{m+1}\cdots a_n}{\epsilon}^{j_1\cdots j_ma_n\cdots
a_{m+1}}.}
These are typically proportional to projection operators, i.e.
\[ P_{i_1\cdots i_m}{}^{a_n\cdots a_{m+1}}P_{a_{m+1}\cdots a_n}{}^{j_m\cdots
j_1}=d_m P_{i_1\cdots i_m}{}^{j_m\cdots j_1}\]
for some constants $\{d_m\}$. This is verified in examples, where also these
constants can be determined. On the other hand, it appears to be a rather
general feature which can be expected for any suitably nice $R,R'$-matrices.
These $P$ project onto tensors with totally R-antisymmetric indices.
\begin{propos} There is a well-defined operator on forms given by
\[ P:\Lambda\to\Lambda,\quad P(\theta_{i_1}\cdots \theta_{i_m})=P_{i_1\cdots
i_m}{}^{a_m\cdots a_1}\theta_{a_1}\cdots\theta_{a_m}=d_m
\theta_{i_1}\cdots\theta_{i_m}\]
\end{propos}
\goodbreak\noindent{\bf Proof\quad} One can expect the diagonal form in view of the above remarks since the
products $\theta_{i_1}\cdots \theta_{i_m}$ are already $R$-antisymmetric. Here
we check that $P$ as an operator is indeed well-defined. Indeed, the relations
of $\Lambda$ are respected as
\[
P(\theta_{i_1}\cdots\theta_b\theta_a\cdots\theta_{i_m})R^a{}_{i_p}{}^b
{}_{i_{p+1}}={\epsilon}_{i_1\cdots ba\cdots i_m a_{m+1}\cdots a_n}
{\epsilon}^{b_1\cdots b_ma_n\cdots
a_{m+1}}\theta_{b_1}\cdots\theta_{b_m}=-P(\theta_{i_1}\cdots\theta_{i_m})\]
for all $p$ due to $R$-antisymmetry of ${\epsilon}$. We give this is in detail
because this and a similar consideration for the output of $P$ dictates the
ordering of the indices in the action of $P$. {\ $\lform$}\bigskip
As another immediate application of our epsilon tensor one can
write a general R-matrix formula for the quantum-determinant of the
symmetry quantum group of our theory, namely
\[ \det({\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v})=d_0^{-1} {\epsilon}_{i_1\cdots i_n}t^{i_1}{}_{j_1}\cdots
t^{i_n}{}_{j_n}{\epsilon}^{j_n\cdots j_1}=([n;-R']!)^{1\cdots n}_{i_1\cdots
i_n}t^{i_1}{}_{j_1}\cdots t^{i_n}{}_{j_n} ([n;-R']!)^{j_1\cdots j_n}_{1\cdots
n}.\]
There is no metric needed here, but if one exists then it is easy to see
from (\ref{ggR}) that it can used to turn any $R'$-symmetric or
$R$-antisymmetric tensor with upper-indices to one with lower indices.
In our setting with a
unique top form, one can also expect that
a totally antisymmetric tensor with $n$ indices is unique up to a scale. In
this
case one has
\eqn{epsud}{ {\epsilon}_{i_1i_2\cdots i_n}=\lambda
\eta_{i_1 a_1}\cdots \eta_{i_n a_n}{\epsilon}^{a_1\cdots a_n}=\lambda
{\epsilon}^{a_1\cdots a_n}\eta_{a_1 i_1}\cdots \eta_{a_n i_n}}
where $\lambda$ is a constant depending on the metric.
Finally, one can use the ${\epsilon}$ tensor in the usual way to define a Hodge
$*$-operator, along the lines in \cite{Mey:wav} for $q$-Minkowski space, where
${\epsilon}$ was found by hand. In the present formulation we have
\begin{propos} There is a well-defined operator on forms given by
\eqn{hodge}{*:\Lambda\to\Lambda,\quad (\theta_{i_1}\cdots
\theta_{i_m})^*={\epsilon}^{a_1\cdots a_m b_n\cdots b_{m+1}}\eta_{a_1
i_1}\cdots\eta_{a_m i_m}\theta_{b_{m+1}}\cdots \theta_{b_n}= H_{i_1\cdots
i_m}{}^{a_n\cdots a_{m+1}}\theta_{a_{m+1}}\cdots \theta_{a_n}.}
\end{propos}
\goodbreak\noindent{\bf Proof\quad} This time consistency with the relations of $\Lambda$ follows using
(\ref{ggR}) after which we can then use antisymmetry of ${\epsilon}$ as in the
preceding proposition. {\ $\lform$}\bigskip
The appropriate tensor $H$ here has square which is typically proportional to
the projectors in (\ref{proj}),
\[ H_{i_1\cdots i_m}{}^{a_n\cdots a_{m+1}}H_{a_{m+1}\cdots a_n}{}^{j_m\cdots
j_1}\propto P_{i_1\cdots i_m}{}^{j_m\cdots j_1}.\]
This too is verified in examples, were one also learns the constants of
proportionality. It holds in some generality and
means that $*^2\propto{\rm id}$ on forms of each degree. This is analogous to the
classical situation. Motivated by this one can also define the interior product
of forms by a form $\theta_i$ as the adjoint under $*$ of multiplication by
$\theta_i$ in the exterior algebra. It obeys a graded ${\Bbb Z}_2$-Leibniz rule, as
checked for $q$-Minkowski case in \cite{Mey:wav}.
It is possible also to make a much more radical formulation of the interior
product and Hodge $*$ operations, based on the idea of differentiation
on form space and not directly on ${\epsilon}$. Thus we define the braided-interior
product $i$ and braided-Hodge operator $\circ$ in the algebra of forms
$\{\theta_i\}$ by
\[ i_{f(\theta)} g(\theta)=f({\partial})g(\theta),\quad f(\theta)^\circ
=i_{f(\theta)}\theta_1\cdots\theta_n\]
where $f({\partial})$ consists of relacing $\theta_i$ by the operators
${\partial}_i=\eta_{ia}{{\partial}\over{\partial}\theta_a}$. For $\circ$ we use
$\theta_1\cdots\theta_n$ (say) as the top form. We have explicitly,
\alignn{newhodge}{ (\theta_{i_1}\cdots\theta_{i_m})^\circ\kern -1.7em&&=
\eta_{i_1a_1}\cdots \eta_{i_m a_m}{{\partial}\over{\partial}
\theta_{a_1}}\cdots{{\partial}\over{\partial}\theta_{a_m}}\theta_1\cdots\theta_n\nonumber\\
&&=
\eta_{i_1a_1}\cdots \eta_{i_m a_m}(({\rm id}\mathop{\otimes} [n-m+1;-R'])\cdots
[n;-R'])^{a_m\cdots a_1 b_{m+1}\cdots b_n}_{12\cdots n}\theta_{b_{m+1}}\cdots
\theta_{b_{n}}.}
For example, in four dimensions, the formulae are
\[
(\theta_{i_1}\theta_{i_2}\theta_{i_3}\theta_{i_4})^\circ
=\lambda^{-1}{\epsilon}_{i_1\cdots i_4}\]
\[
(\theta_{i_1}\theta_{i_2}\theta_{i_3})^\circ=\eta_{i_1a_1}
\eta_{i_2a_2}\eta_{i_3a_3}{\epsilon}^{a_3a_2a_1b}\theta_b\]
\[
(\theta_{i_1}\theta_{i_2})^\circ=\eta_{i_1a_1}\eta_{i_2a_2}
({\rm id}\mathop{\otimes}[3;-R'])[4;-R']^{a_2 a_1 b_1 b_2}_{1234}\theta_{b_1}\theta_{b_2}\]
\[
(\theta_{i})^\circ=\eta_{ia}[4;-R']^{ab_1b_2b_3}_{1234}\theta_{b_1}
\theta_{b_2}\theta_{b_3},\quad 1^\circ=\theta_1\theta_2\theta_3\theta_4.\]
Note that only the Hodge operation on $n$ and $n-1$ degree forms involve the
braided-factorial or ${\epsilon}$ tensor directly. The other degrees involving this
and normalisation integers $1\over (n-m)!$ etc. are replaced now by
differentiation.
This second approach to the Hodge operation is different from the first one,
though agrees when $q=1$ after suitable normalisations at each degree. In
general we do not have that $i$ is a graded derivation and we also do not have
that $\circ^2\propto{\rm id}$ on forms of a given degree. On the other hand, this
second approach is conceptually quite simple and can be thought of in fact as a
kind of `Fourier transform' in form-space $\{\theta_i\}$. This is suggested by
the interior product $i_f$ appearing as braided-differentiation in form-space.
Moreover, from this point of view one would expect $\circ^2$ to be something
like the braided-antipode $S$ on the braided-Hopf algebra $\{\theta_i\}$, which
is not simply $\pm1$ in the braided case. The technology for braided Fourier
transform is in \cite{KemMa:alg}.
\section{Differential forms}
Until now we have considered the algebra of forms $\theta_i$ in
isolation, as some q-deformed superspace. For completeness we now consider both
the co-ordinates
$\hbox{{$\cal M$}}} \def\CP{\hbox{{$\cal P$}}=\{x_i\}$ and the forms $\theta_i$ together with $\theta_i=dx_i$. Thus we
consider the {\em exterior algebra}
\eqn{Omega}{\Omega=\Lambda\und\mathop{\otimes} \hbox{{$\cal M$}}} \def\CP{\hbox{{$\cal P$}},\quad \vecx_1\theta_2=\theta_2\vecx_1R}
where the product is the braided tensor product with the cross-relations as
stated. The essence of the braided tensor product here is that it keeps all
constructions covariant, hence $\Omega$ remains a comodule algebra under our
background quantum group $\widetilde{A}$ (I would like to thank A. Sudbery for
this remark\cite{Sud:ort}).
Next we consider $\Omega$ as bi-graded with components
\[ \Omega^{p|q}={\rm span}\{\theta_{i_1}\cdots\theta_{i_p} x_{j_1}\cdots
x_{j_q}\},\quad \Omega^p=\oplus_{q=0}^{\infty}\Omega^{p|q},\quad
\Omega^{|q}=\oplus_{p=0}^{\infty}\Omega^{p|q}\]
where $\Omega^p$ are the usual $p$-forms in differential geometry. Actually,
one can proceed quite symmetrically with $\Omega^{|q}$ the `differential forms
in super-space' generated by $x_i=d\theta_i$.
\begin{propos} We define the exterior derivative ${{\rm d}}$ as
\[ {{\rm d}}: \Omega^p\to\Omega^{p+1},\quad d (\theta_{i_1}\cdots\theta_{i_p}
f(\vecx))=\theta_{i_1}\cdots\theta_{i_p} \theta_i {{\partial}\over{\partial} x_i}
f(\vecx).\]
It obeys a right-handed ${\Bbb Z}_2$-graded-Leibniz rule
\[ {{\rm d}}(fg)=(-1)^p({{\rm d}} f)g+f{{\rm d}} g,\quad\forall g\in\Omega,\ f\in\Omega^p.\]
\end{propos}
\begin{figure}
\begin{picture}(200,280)(-50,0)
\epsfbox{dleib.eps}
\end{picture}
\caption{(a) definition of exterior differential ${{\rm d}}$ in braided approach (b)
braided-Leibniz rule for action of partial derivatives and (c) proof of usual
graded-Leibniz rule for ${{\rm d}}$}
\end{figure}
\goodbreak\noindent{\bf Proof\quad} This is well-defined because the partial derivatives ${\partial}\over{\partial} x_i$
are
well-defined as operators on $\hbox{{$\cal M$}}} \def\CP{\hbox{{$\cal P$}}=V^*$\cite{Ma:fre}. I.e. our ${{\rm d}}$ is built up
from
well-defined operations. It is also covariant under the quantum group
$\widetilde{A}$
for the same reasons. Here the element
in
$\Lambda\mathop{\otimes} V$ (where $V$ is the
vector algebra) is invariant under the transformation law in \cite{Ma:lin} and
hence
behaves bosonically (i.e. with trivial braiding). We consider ${{\rm d}}$ as the
action of
the $V$ part of this element on $\hbox{{$\cal M$}}} \def\CP{\hbox{{$\cal P$}}$ by the action $\alpha$ defined by
differentiation\cite{Ma:fre},
followed by the product in $\Lambda$. This is shown diagrammatically in
Figure~1 (a). Part (b) recalls the
module-algebra property of the action $\alpha$\cite{Ma:fre} which comes out as
the braided-Leibniz
rule for the differentials ${{\partial}\over{\partial} x_i}=\alpha_{v^i}$ because the
coproduct $\Delta$ in the
degree one part $V_1$ of the vector algebra is just the linear one $v^i\mathop{\otimes}
1+1\mathop{\otimes} v^i$. Using these
facts about partial derivatives from \cite{Ma:fre} we can easily prove the
Leibniz rule for ${{\rm d}}$, which
we do in part (c). On the left is the braided tensor product\cite{Ma:bra} in
$\Lambda\und\mathop{\otimes} \hbox{{$\cal M$}}} \def\CP{\hbox{{$\cal P$}}$, followed by
${{\rm d}}$ in the box. We then use the braided-Leibniz rule from part (b) for the
first equality and functoriality
(rearrangement of braids) for the second, as well as associativity of the
products. In these diagrams we work in the braided category of $\tilde
A$-comodules
in which the braiding
is given by $R$. But the
commutation relations of $\Lambda$ are also given
by $-R$ so we have
\[
\cdot\circ\Psi(\theta_{i}\mathop{\otimes}\theta_{i_1}\cdots\theta_{i_p})
=(-1)^p\theta_{i_1}\cdots\theta_{i_p}\theta_i\]
which we use for the third equality. Note that we do not make use of the
coaddition on $\Lambda$, which would require
a different braiding (based on $-R'$) from the one we use here. Finally we use
functoriality and associativity again to
recognise the result. Conceptually, the element $\theta_i\mathop{\otimes}{{\partial}\over{\partial}
x_i}$ is bosonic (invariant) and
hence the resulting derivation property is the usual ${\Bbb Z}_2$-graded one (albeit
coming out from the right in our present conventions) and not a braided one.
{\ $\lform$}\bigskip
This is the construction of the exterior differential calculus on a quantum or
braided vector space
coming out of braided geometry. The resulting R-matrix formulae
\eqn{dx}{ {{\rm d}}\vecx_1{{\rm d}}\vecx_2=-{{\rm d}}\vecx_2{{\rm d}}\vecx_1R,\quad
\vecx_1{{\rm d}}\vecx_2={{\rm d}}\vecx_2\vecx_1 R}
here are essentially the same as in
\cite{WesZum:cov} but the difference is that we begin with our well-defined
partial
differential operators ${\partial}\over{\partial} x_i$ and define ${{\rm d}}$ from them in a
well-defined way, rather than arguing backwards by consistency requirements
within the axiomatic
framework of Woronowicz. In the braided approach the starting point is the
braided addition law\cite{Ma:poi}, which then defines partial
derivatives\cite{Ma:fre}, which in turn define ${{\rm d}} $ as above.
In nice cases, one will also have that ${{\rm d}}^2=0$. Using the above definition it
is clear that the essential requirement for this is the identity
\eqn{d2}{ \theta_1\theta_2{\partial}_2{\partial}_1=0}
which in turn is immediate at least when $PR'=f(PR)$ for some function $f$ such
that $f(-1)\ne 1$. For then
$\theta_1\theta_2{\partial}_2{\partial}_1=\theta_1\theta_2(PR'){\partial}_2{\partial}_1
=\theta_1\theta_2f(PR){\partial}_2{\partial}_1=f(-1)\theta_1\theta_2{\partial}_2{\partial}_1.$
This includes the Hecke case where $R'\propto R$ but also the more general
construction\cite{Ma:poi} where $R'$ is built from the minimal polynomial
of $R$. In cases where $PR'$ is not given explicitly in terms of $PR$, we can
check (\ref{d2}) directly using the same strategy. The same remarks apply
for the proof of covariance of the vector addition law in \cite{Ma:poi}.
Using these ingredients one can immediately write down a Laplacian
$L={{\rm d}}\delta+\delta {{\rm d}}$ where $\delta=*{{\rm d}}*$ and develop a general theory of
wave-equations and electromagnetism on general braided spaces associated to
$R$-matrices. In our setting we consider the gauge potential
as $A\in \Omega^1$ and let $F={{\rm d}} A$. The gauge theory is well-defined because
${{\rm d}}^2=0$.
It is compatible with the Maxwell equations $\delta F=j$ because
$*^2\propto{\rm id}$. In other
words, the properties needed have been covered in our theory above and hold in
some generality.
More specific applications will be studied elsewhere, but see \cite{Mey:wav}
where
these concepts are developed directly for $q$-Minkowski with some interesting
results.
\section{Examples}
A simple example is the quantum plane ${\Bbb C}_q^n$ associated to the usual
$GL_q(n)$ $R$-matrix. In this
case our general formula for the ${\epsilon}$ tensor recovers the usual one as in
\cite{FRT:lie}. There is no
invariant metric so we do not have a Hodge $*$-operator. The differential
calculus recovers the one of
\cite{PusWor:twi}\cite{WesZum:cov}. This is clear already from the comparison
of the
corresponding partial derivatives in \cite{Ma:fre}.
On the other hand, many other important algebras in q-deformed physics are in
fact braided spaces with a coaddition
law, so at once amenable to our machinery. Note that once the additive braid
statistics $R$ is known, we do not
need to do any more work for the differential calculus: we just write down
(\ref{dx}) with {\em the same $R$} as used in the
braid statistics for the addition law on the quantum space.
For example, we know from \cite{Ma:add} that the quantum matrices
$A(R)=\{t^i{}_j\}$ have an addition law, at least when $R$ is Hecke. We can put
it into the form of a braided space by
\[ R{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}_1{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}_2={\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}_2{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}_1R\quad\Leftrightarrow t_It_J=t_Bt_A{\bf
R'}^A{}_I{}^B{}_J,\quad {\bf
R'}^I{}_J{}^K{}_L=R^{-1}{}^{j_0}{}_{i_0}{}^{l_0}{}_{k_0}
R^{i_1}{}_{j_1}{}^{k_1}{}_{l_1}\]
\[ {\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}'_1{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}_2=R_{21}{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}_2{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}'_1R\quad\Leftrightarrow
t_It'_J=t'_Bt_A{\bf R}^A{}_I{}^B{}_J,\quad {\bf
R}^I{}_J{}^K{}_L=R{}^{l_0}{}_{k_0}{}^{j_0}{}_{i_0}
R^{i_1}{}_{j_1}{}^{k_1}{}_{l_1}\]
where $t_I=t^{i_0}{}_{i_1}$. We recover at once the vector algebra\cite{Ma:add}
and bicovariant differential calculus\cite{Sud:alg}
\eqn{dt}{ R{\partial}_2{\partial}_1={\partial}_1{\partial}_2 R,\quad {{\rm d}}{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}_1{{\rm d}}{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}_2=-R_{21}{{\rm d}}
{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}_2{{\rm d}}{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}_1R,\quad {\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}_1{{\rm d}}{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}_2=R_{21}{{\rm d}} {\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}_2{\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}_1R}
on $A(R)$ where ${\partial}^I={\partial}^{i_1}{}_{i_0}={{\partial}\over{\partial} t^{i_0}{}_{i_1}}$ is
written as a matrix. This includes the usual results for $M_q(n)$ and
multiparametric $M_{p,q}(n)$ etc. That $d^2=0$ for
this class is known so we omit the direct check of (\ref{d2}) which is needed
in our constructive approach of the last section. It is very similar to the
proof for $\bar A(R)$ below. The construction is covariant under
$\widetilde{A}$ built from
$A(R)^{\rm cop}\mathop{\otimes} A(R)$ as explained in\cite{Ma:add}, which corresponds to
bicovariance under $GL_q(n)$ etc. in the usual approach. The dilatonic
extension here is needed if one wants to go to quotient quantum groups rather
than working at the $GL_q$ level.
That $\bf R'$ obeys the QYBE etc. is easily checked and means that the theory
in Section~2 applies and gives us a $q$-epsilon tensor on $A(R)$ to go with
this differential calculus.
For a second class of examples, we have the algebras $\bar A(R)=\{x^i{}_j\}$
introduced in \cite{Ma:euc} as a variant of $A(R)$ and again with a braided
addition law when $R$ is Hecke. It forms a braided space with\cite{Ma:euc}
\[ R_{21}\vecx_1\vecx_2=\vecx_2\vecx_1R\quad\Leftrightarrow x_Ix_J=x_Bx_A{\bf
R'}^A{}_I{}^B{}_J,\quad {\bf
R'}^I{}_J{}^K{}_L=R^{-1}{}^{l_0}{}_{k_0}{}^{j_0}{}_{i_0}
R^{i_1}{}_{j_1}{}^{k_1}{}_{l_1}\]
\[ \vecx'_1\vecx_2=R\vecx_2\vecx'_1R\quad\Leftrightarrow x_Ix'_J=x'_Bx_A{\bf
R}^A{}_I{}^B{}_J,\quad {\bf R}^I{}_J{}^K{}_L=R{}^{j_0}{}_{i_0}{}^{l_0}{}_{k_0}
R^{i_1}{}_{j_1}{}^{k_1}{}_{l_1}\]
which gives at once the vector algebra and quantum differential calculus
\eqn{dxeuc}{ R{\partial}_2{\partial}_1={\partial}_1{\partial}_2 R_{21},\quad
{{\rm d}}\vecx_1{{\rm d}}\vecx_2=-R{{\rm d}}\vecx_2{{\rm d}}\vecx_1R,\quad
\vecx_1{{\rm d}}\vecx_2=R{{\rm d}}\vecx_2\vecx_1R}
on $\bar A(R)$. Since this is a new differential calculus, we formally verify
(\ref{d2}) as
\align{&&\kern -1.7em X={\rm Tr\, } {{\rm d}} \vecx_1 {{\rm d}} \vecx_2 {\partial}_2{\partial}_1={\rm Tr\, } {{\rm d}} \vecx_1
{{\rm d}} \vecx_2 R^{-1}R{\partial}_2{\partial}_1=-{\rm Tr\, } R{{\rm d}} \vecx_2 {{\rm d}} \vecx_1
{\partial}_1{\partial}_2R_{21}=-{\rm Tr\, } {{\rm d}} \vecx_2 {{\rm d}} \vecx_1 {\partial}_1{\partial}_2(1+\lambda PR)\\
&&\kern -1.7em 2X=-\lambda{\rm Tr\, } {{\rm d}} \vecx_2 {{\rm d}} \vecx_1 R_{21}^{-1}R_{21}{\partial}_1{\partial}_2
PR=-\lambda{\rm Tr\, } {{\rm d}} \vecx_2 {{\rm d}} \vecx_1 R_{21}^{-1}{\partial}_2{\partial}_1 P(1+\lambda
PR)\\
&&=\lambda{\rm Tr\, } R_{21}{{\rm d}} \vecx_1{{\rm d}}\vecx_2{\partial}_2{\partial}_1
P-\lambda^2{\rm Tr\, }{{\rm d}}\vecx_2{{\rm d}}\vecx_1{\partial}_1{\partial}_2=-2X-\lambda^2X}
where $\lambda=q-q^{-1}$ and ${\rm Tr\, }={\rm Tr\, }_1{\rm Tr\, }_2$ over the two sets of
matrix indices. Hence $X=0$ provided $4+\lambda^2\ne 0$, i.e. provided $q^2\ne
-1$. We just used the relations (\ref{dxeuc}) many times, cyclicity of the
trace and the Hecke assumption $R_{21}R=1+\lambda PR$.
It is also easy to see that $\bf R'$ obeys the QYBE etc. so that the theory
above applies and gives us a $q$-epsilon tensor on $\bar A(R)$ to go with this
differential calculus. Our constructions here are covariant under
$\widetilde{A}$ obtained from $A(R)\mathop{\otimes} A(R)$. The simplest case with $R$ the
standard $4\times 4$ $SU_q(2)$ R-matrix gives $q$-Euclidean space\cite{Ma:euc}
and is studied in detail in Section~5.1. It is covariant under
$\widetilde{SU_q(2)\mathop{\otimes} SU_q(2)}$, i.e. the $q$-Euclidean rotation group with
dilatonic extension.
For a third immediate class of examples, we know from \cite{Mey:new} that the
braided matrices $B(R)=\{u^i{}_j\}$ introduced in \cite{Ma:exa} also have a
braided addition law when $R$ is Hecke. It appears in the form of a braided
space with\cite{Ma:exa}\cite{Mey:new}
\[ R_{21}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1R{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2= {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2 R_{21} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1 R\ \Leftrightarrow\
u_Ju_L= u_Ku_I {\bf R'}^I{}_J{}^K{}_L,\quad {\bf
R'}^I{}_J{}^K{}_L=R^{-1}{}^{d}{}_{k_0}{}^{j_0}{}_{a}
R^{k_1}{}_{b}{}^{a}{}_{i_0}R^{i_1}{}_c{}^b{}_{l_1} {\widetilde
R}^c{}_{j_1}{}^{l_0}{}_d\]
\[ R^{-1}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1'R{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2= {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2 R_{21}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1'R\ \Leftrightarrow\
u_J'u_L= u_Ku_I' {\bf R}^I{}_J{}^K{}_L,\quad {\bf
R}^I{}_J{}^K{}_L=R^{j_0}{}_a{}^d{}_{k_0}
R^{k_1}{}_b{}^a{}_{i_0}
R^{i_1}{}_c{}^b{}_{l_1} {\widetilde R}^c{}_{j_1}{}^{l_0}{}_d\]
where $\widetilde{R}$ is given by transposition in the second two indices,
inversion and transposition again. The first multi-index R-matrix was
introduced by the author in \cite{Ma:exa} (as well as another one for
multiplicative braid statistics), while the second was introduced by Meyer in
\cite{Mey:new}. The algebra here is an important one and appears in other
contexts also as explained in \cite{Ma:skl}. For this algebra we have at once
the vector algebra and differential calculus
\eqn{du}{ R{\partial}_2\widetilde{R}{\partial}_1={\partial}_1\widetilde{R_{21}}{\partial}_2
R_{21},\quad R^{-1}{{\rm d}}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1R{{\rm d}}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2=-{{\rm d}}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2 R_{21}{{\rm d}}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1R,\quad
R^{-1}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1R{{\rm d}}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2= {{\rm d}}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2 R_{21}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1R}
as recently studied by several authors\cite{Vla:coa}\cite{Isa:int} and
references therein. We would like to stress that these relations themselves are
just (and necessarily) the same form as Meyer's additive braid statistics and
hence could not be considered as new. On the other hand, \cite{Vla:coa}
contains an interesting new result that $\Omega$ itself can have a braided
addition law in this and other cases. Meanwhile \cite{Isa:int} contains an
interesting observation about its braided-comodules. For completeness, we need
to add from our constructive point of view the formal proof of (\ref{d2}) and
hence of ${{\rm d}}^2=0$ as
\align{&&\kern -1.7em X={\rm Tr\, } {{\rm d}} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1 {{\rm d}}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2{\partial}_2{\partial}_1={\rm Tr\, }{{\rm d}}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1 R
{{\rm d}}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2{\partial}_2 \widetilde{R}{\partial}_1={\rm Tr\, }{{\rm d}}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1 R {{\rm d}}{\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2 R^{-1}R{\partial}_2
\widetilde{R}{\partial}_1\\
&&=-{\rm Tr\, } R{{\rm d}} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2 R_{21}{{\rm d}} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1 {\partial}_1
\widetilde{R_{21}}{\partial}_2R_{21}=-{\rm Tr\, }{{\rm d}} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2 R_{21}{{\rm d}} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1 {\partial}_1
\widetilde{R_{21}}{\partial}_2(1+\lambda PR)\\
&&\kern -1.7em 2X=-\lambda{\rm Tr\, } {{\rm d}} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2 R_{21}{{\rm d}} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1 R_{21}^{-1}R_{21}{\partial}_1
\widetilde{R_{21}}{\partial}_2 PR=-\lambda{\rm Tr\, } {{\rm d}} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2 R_{21}{{\rm d}} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1
R_{21}^{-1}{\partial}_2 \widetilde{R}{\partial}_1 P(1+\lambda PR)\\
&&=\lambda{\rm Tr\, } R_{21}{{\rm d}} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1 R{{\rm d}} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2 {\partial}_2 \widetilde{R}{\partial}_1
P-\lambda^2{\rm Tr\, } {{\rm d}} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_2 R_{21}{{\rm d}} {\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}_1 {\partial}_1
\widetilde{R_{21}}{\partial}_2=-2X-\lambda^2X}
which implies $X=0$ provided $q^2\ne -1$. In fact, there is a mathematically
precise equivalence between this proof and the one for $A(R)$ and its variant
above for $\bar A(R)$, provided respectively by the theory of
transmutation\cite{Ma:exa} or twisting\cite{Ma:euc}. This expresses products of
the ${\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}$ in terms of products of the ${\bf t}}\def\vecs{{\bf s}}\def\vecv{{\bf v}$ or $\vecx$ in a precise way and
in a corresponding way for the partial differentials ${\partial}$.
It is also known that $\bf R'$ here obeys the QYBE\cite{Ma:sol}, while the
mixed relations involving $\bf R,R'$ are also easily checked in the same way
and reduce to the QYBE for $R$. Hence we are in the symmetrical situation
needed for our theory of the ${\epsilon}$ tensor. Moreover, our construction is
manifestly covariant under a quantum group $\widetilde{A}$ obtained from
$A(R){\bowtie} A(R)$, where ${\bowtie}$ is the double cross product construction
from \cite{Ma:seq}. See also \cite[Sec. 4]{Ma:poi}. The standard $4\times 4$
R-matrix gives $q$-Minkowski space studied in detail in Section~5.2. Here the
covariance is under $\widetilde{SU_q(2){\bowtie} SU_q(2)}$ which is the
$q$-Lorentz group of \cite{PodWor:def}\cite{CWSSW:lor} with dilatonic
extension.
In both cases here the dilatonic extension is needed for the braiding to be
given by our categorical constructions with the correct
normalisation\cite{Ma:poi}. One could try to leave it out by adjusting the
normalisations in (\ref{dx}) etc. by hand but in this case one can expect an
inconsistency at some other point where both the braiding and the determinant
or other non-quadratic relations of the background quantum group are needed. An
alternative way out is to allow the $q$-Lorentz group to be treated with
anyonic or $q$-statistics\cite{Ma:csta}. This will be explained elsewhere. We
note also that the $\bar A(R)$ and braided matrices $B(R)$ are related by a
twisting construction for comodule algebras\cite{Ma:euc} which becomes a
`quantum Wick rotation' in the Euclidean/Minkowski case. We know also from
their original definition in \cite{Ma:exa} that the braided matrices $B(R)$ are
strictly related to $A(R)$ by transmutation. This is already reflected in the
similarity of the proofs above and it would be interesting to formalise it
further as a theory of twisting and transmutation for differential calculi and
${\epsilon}$ tensors.
We conclude with the simplest cases of the $\bar A(R)$ and $B(R)$ constructions
computed in detail using the symbol manipulation package REDUCE. This is needed
to determine the normalisations $d_m$ etc. concerning the projectors and Hodge
operators. Direct R-matrix methods like those above are not yet known for these
properties, but they are verified in both of these
examples as well as in similar ones based on other well-known R-matrices.
\subsection{q-Euclidean space}
For $q$-Euclidean space, we use the definition in \cite{Ma:euc} as twisting
$\bar M_q(2)$ of the usual $2\times 2$ quantum matrices. This is the simplest
example of the $\bar A(R)$ construction above. We have generators
$\vecx=\pmatrix{a&b\cr c&d}$ and relations
\ceqn{posneucl}{ba=qab,\quad ca=q^{-1}ac,\quad da=ad,\quad db=q^{-1}bd\quad
dc=qcd\\
bc=cb+(q-q^{-1})ad.}
This is actually isomorphic to the usual $M_q(2)$ by a permutation of the
generators, so one can
regard the following as results on this with its additive structure as
introduced in \cite{Ma:add}.
Vector algebra of derivatives is
\[ {{\partial}\over{\partial} d} {{\partial}\over{\partial} c}=q^{-1} {{\partial}\over{\partial} c} {{\partial}\over{\partial}
d}, \quad {{\partial}\over{\partial} d} {{\partial}\over{\partial} b} = {{\partial}\over{\partial} b}
{{\partial}\over{\partial} d} q,\quad
{{\partial}\over{\partial} d} {{\partial}\over{\partial} a} = {{\partial}\over{\partial} a} {{\partial}\over{\partial} d}\]
\[ {{\partial}\over{\partial} c} {{\partial}\over{\partial} b}= {{\partial}\over{\partial} b} {{\partial}\over{\partial} c} +
(q-q^{-1} ) {{\partial}\over{\partial} a} {{\partial}\over{\partial} d},\quad
{{\partial}\over{\partial} c} {{\partial}\over{\partial} a} = {{\partial}\over{\partial} a} {{\partial}\over{\partial} c}
q,\quad {{\partial}\over{\partial} b} {{\partial}\over{\partial} a} =q^{-1} {{\partial}\over{\partial} a}
{{\partial}\over{\partial} b}\]
The metric is \cite{Ma:euc}
\eqn{eucmetric}{\eta^{IJ}=\pmatrix{0&0&0&1\cr 0&0&-q&0\cr 0&-q^{-1}&0&0\cr
1&0&0&0}={{\partial}\over{\partial} x_I}{{\partial}\over{\partial} x_J}(ad-qcb).}
It has determinant $\det(\eta)=1$ and is already correctly normalised, so
$\lambda=1$ in (\ref{epsud}). Here
\[ x_Jx_I\eta^{IJ}=(1+q^{-2})(ad-qcb)\]
is a natural `radius function' according to the construction in \cite{Ma:euc}.
The algebra of forms is
\[ {{\rm d}} a {{\rm d}} a=0,\quad {{\rm d}} b{{\rm d}} b=0,\quad {{\rm d}} c{{\rm d}} c=0,\quad {{\rm d}} d{{\rm d}} d=0\]
\[ {{\rm d}} b {{\rm d}} a =-q^{-1} {{\rm d}} a {{\rm d}} b ,\quad {{\rm d}} c {{\rm d}} a =- {{\rm d}} a {{\rm d}} c q,\quad
{{\rm d}} d {{\rm d}} b =- {{\rm d}} b {{\rm d}} d q\]
\[ {{\rm d}} c {{\rm d}} b =- {{\rm d}} b {{\rm d}} c ,\quad {{\rm d}} d {{\rm d}} c =-q^{-1} {{\rm d}} c {{\rm d}} d ,\quad
{{\rm d}} d {{\rm d}} a =- (q-q^{-1} ) {{\rm d}} b {{\rm d}} c - {{\rm d}} a {{\rm d}} d \]
We have the $q$-epsilon tensor as:
\[{\epsilon}_{abcd}=-{\epsilon}_{acbd}={\epsilon}_{adbc}=-{\epsilon}_{adcb}
={\epsilon}_{bcad}=-{\epsilon}_{bcda}=1\]
\[-{\epsilon}_{cbad}={\epsilon}_{cbda}=-{\epsilon}_{dabc}={\epsilon}_{dacb}
=-{\epsilon}_{dbca}={\epsilon}_{dcba}=1\]
\[ {\epsilon}_{acdb}=-{\epsilon}_{cdba}=-{\epsilon}_{dcab}=q,\quad
-{\epsilon}_{abdc}=-{\epsilon}_{bacd}={\epsilon}_{bdca}={\epsilon}_{dbac}=q^{-1}\]
\[ -{\epsilon}_{cadb}={\epsilon}_{cdab}=q^2,\quad {\epsilon}_{bacd}=-{\epsilon}_{bdac}=q^{-2}\]
\[ -{\epsilon}_{adad}={\epsilon}_{dada}=(q-q^{-1})\]
The resulting raw (un-normalised) antisymmetriser projectors $P$ have
associated constants
\[ d_0=2q^4[2]^2[3],\quad d_1=-2q^2[3],\quad d_2=q^2[2]^2,\quad
d_3=-2q^2[3],\quad d_4=2q^4[2]^2[3]\]
where $[m]\equiv[m,q^{-2}]={1-q^{-2m}\over 1-q^{-2}}$. In each case, the
projections are on the space of totally $R$-antisymmetric tensors and have the
same ranks as classically.
The Hodge $*$-operator for this metric is:
\[ ({{\rm d}} a{{\rm d}} b{{\rm d}} c{{\rm d}} d)^*=1,\quad ({{\rm d}} a{{\rm d}} b{{\rm d}} c)^*={{\rm d}} a,\quad ({{\rm d}} a{{\rm d}} b{{\rm d}} d)^*={{\rm d}}
b\]
\[ ({{\rm d}} a{{\rm d}} c{{\rm d}} d)^*=-{{\rm d}} c,\quad ({{\rm d}} b {{\rm d}} c{{\rm d}} d)^*=-{{\rm d}} a\]
\[ ({{\rm d}} a{{\rm d}} b)^*=-q[2]{{\rm d}} a{{\rm d}} b,\quad ({{\rm d}} a{{\rm d}} c)^*=q[2]{{\rm d}} a{{\rm d}} c,\quad ({{\rm d}} a{{\rm d}}
d)^*=2{{\rm d}} b{{\rm d}} c-(q-q^{-1}){{\rm d}} a{{\rm d}} d\]
\[ ({{\rm d}} b{{\rm d}} c)^*=2{{\rm d}} a{{\rm d}} d+(q-q^{-1}){{\rm d}} b {{\rm d}} c,\quad ({{\rm d}} b{{\rm d}} d)^*=q[2]{{\rm d}} b{{\rm d}}
d,\quad ({{\rm d}} c{{\rm d}} d)^*=-q[2]{{\rm d}} c{{\rm d}} d\]
\[ ({{\rm d}} a)^*=2q^2[3]{{\rm d}} a{{\rm d}} b{{\rm d}} c,\quad ({{\rm d}} b)^*=2q^2[3]{{\rm d}} a{{\rm d}} b{{\rm d}} d,\quad ({{\rm d}}
c)^*=-2q^2[3]{{\rm d}} a{{\rm d}} c{{\rm d}} d,\quad ({{\rm d}} d)^*=-2q^2[3]{{\rm d}} b{{\rm d}} c{{\rm d}} d\]
\[ 1^*=2q^4[2]^2[3]\]
One can check that the square of this Hodge * operator is
\[ *^2=(-1)^mP=(-1)^md_m\]
on forms of degree $m$. The Hodge $*$-operator on 2-forms is a $6\times 6$
matrix with eigenvalues $\pm q[2]$ with multiplicity $3$. The self-dual and
antiself dual 2-forms with respect to it are characterised by
\[ F^*=q[2]F,\quad {\rm (self-dual\ form)}; \quad F^*=-2[q]F,\quad {\rm
(anti-self-dual\ form)}.\]
Of course, one may adjust the normalisation of $*$ to have the more usual
limiting form.
Note that our computations here have been for a matrix basis where the metric
$\eta$ has
the signature (2,2) in the $q=1$ limit. The ${\epsilon}$ tensor and value of $*^2$
are as one would expect for
this after bearing in mind the ordering of the indices in our conventions
(there is a reversal in (\ref{hodge})). There is a complex transformation with
real determinant which maps the matrix basis to the usual space-time basis
$t,x,y,z$ with Euclidean signature, so in this basis we still have $*^2$
positive. Again this is the right classical result for our index
conventions. The same remarks apply for the quantum case with $q$ real. The
noncommutative matrix generators transform to self-adjoint or `real' ones by a
complex linear transformation\cite{Ma:euc}. The ${\epsilon}$ computed in the new
basis is not just tensorially related to
the one in the matrix basis because the top form ${{\rm d}} t {{\rm d}} x {{\rm d}} y {{\rm d}} z$ is
different from ${{\rm d}} a {{\rm d}} b {{\rm d}} c {{\rm d}} d$ that we differentiated before. But these
top forms are proportional, up to a real constant and ${\epsilon}$ transforms
tensorially up to this.
By way of contrast, we include also the second Hodge $\circ$-operator:
\[ ({{\rm d}} a{{\rm d}} b{{\rm d}} c{{\rm d}} d)^\circ=1,\quad ({{\rm d}} a{{\rm d}} b{{\rm d}} c)^\circ=-{{\rm d}} a,\quad ({{\rm d}} a{{\rm d}}
b{{\rm d}} d)^\circ=-{{\rm d}} b\]
\[ ({{\rm d}} a{{\rm d}} c{{\rm d}} d)^\circ={{\rm d}} c,\quad ({{\rm d}} b {{\rm d}} c{{\rm d}} d)^\circ={{\rm d}} d\]
\[ ({{\rm d}} a{{\rm d}} b)^\circ=-q {{\rm d}} a{{\rm d}} b,\quad ({{\rm d}} a{{\rm d}} c)^\circ=q^{-1}{{\rm d}} a{{\rm d}} c,\quad ({{\rm d}}
a{{\rm d}} d)^\circ={{\rm d}} b{{\rm d}} c-(q-q^{-1}){{\rm d}} a{{\rm d}} d\]
\[ ({{\rm d}} b{{\rm d}} c)^\circ={{\rm d}} a{{\rm d}} d,\quad ({{\rm d}} b{{\rm d}} d)^\circ=q^{-1}{{\rm d}} b{{\rm d}} d,\quad ({{\rm d}}
c{{\rm d}} d)^\circ=-q{{\rm d}} c{{\rm d}} d\]
\[ ({{\rm d}} a)^\circ=-{{\rm d}} a{{\rm d}} b{{\rm d}} c,\quad ({{\rm d}} b)^\circ=-{{\rm d}} a{{\rm d}} b{{\rm d}} d,\quad ({{\rm d}}
c)^\circ={{\rm d}} a{{\rm d}} c{{\rm d}} d,\quad ({{\rm d}} d)^\circ={{\rm d}} b{{\rm d}} c{{\rm d}} d\]
On two-forms it has eigenvalues
$q^{-1},-q$ each with multiplicity 3. Hence with respect to $\circ$ we have
\[ F^\circ=q^{-1}F,\quad {\rm (self-dual\ form)}; \quad F^\circ =-qF,\quad
{\rm (anti-self-dual\ form)}\]
\subsection{q-Minkowski space}
We use for $q$-Minkowski space the $2\times 2$ braided-hermitian matrices
introduced in \cite{Ma:exa}. It is the simplest example of the $B(R)$
construction above.
The covector algebra of position co-ordinates
${\bf u}}\def\vecx{{\bf x}}\def\vecp{{\bf p}=\pmatrix{a&b\cr c&d}$ is:
\ceqn{posn}{ba=q^2ab,\quad ca=q^{-2}ac,\quad d a=ad,\qquad
bc=cb+(1-q^{-2})a(d-a)\\
d b=bd+(1-q^{-2})ab,\quad cd=d c+(1-q^{-2})ca}
This maps onto a braided tensor product of two copies of the quantum plane and
is thereby compatible with the approach of \cite{CWSSW:lor}\cite{OSWZ:def}
also. The additive
structure we need is from \cite{Mey:new}.
The vector algebra of differentiation operators is:
\[ {{\partial}\over{\partial} d}{{\partial}\over{\partial} c}=q^{-2}{{\partial}\over{\partial} c}{{\partial}\over{\partial}
d},\quad
{{\partial}\over{\partial} d}{{\partial}\over{\partial} b}= {{\partial}\over{\partial} b}{{\partial}\over{\partial} d}q^2\]
\[{{\partial}\over{\partial} d}{{\partial}\over{\partial} a}= {{\partial}\over{\partial} a}{{\partial}\over{\partial} d},\quad
{{\partial}\over{\partial} b}{{\partial}\over{\partial} a}={{\partial}\over{\partial} a}{{\partial}\over{\partial}
b}+{{\partial}\over{\partial} b}{{\partial}\over{\partial} d}(q^2-1)\]
\[{{\partial}\over{\partial} c}{{\partial}\over{\partial} a}={{\partial}\over{\partial} a}{{\partial}\over{\partial}
c}+{{\partial}\over{\partial} c}{{\partial}\over{\partial} d}(q^{-2}-1),\quad
{{\partial}\over{\partial} c}{{\partial}\over{\partial} b}={{\partial}\over{\partial} b}{{\partial}\over{\partial}
c}+{{\partial}\over{\partial} d}{{\partial}\over{\partial} d}(q^{-2}-1)+{{\partial}\over{\partial} a}{{\partial}\over{\partial}
d}(q^2-1)\]
The metric is \cite{Mey:new}:
\eqn{minkg}{\eta^{IJ}={{\partial}\over{\partial} u_I}{{\partial}\over{\partial}
u_J}(ad-q^2cb)=\pmatrix{q^{-2}-1&0&0&1\cr 0&0&-q^2&0\cr 0&-1&0&0\cr
1&0&0&0}.}
It has $\det(\eta)=q^{2}=\lambda$ as the required normalisation constant in
(\ref{epsud}). We get back the `radius function' from the metric as
\[ u_Ju_I\eta^{IJ}=(1+q^{-2})(ad-q^2cb).\]
The algebra of forms is
\[ {{\rm d}} c {{\rm d}} c=0,\quad {{\rm d}} a {{\rm d}} a=0,\quad {{\rm d}} b {{\rm d}} b=0,\quad {{\rm d}} b {{\rm d}} a=-{{\rm d}} a {{\rm d}} b\]
\[ {{\rm d}} c {{\rm d}} a=-{{\rm d}} a {{\rm d}} c,\quad {{\rm d}} c {{\rm d}} b=-{{\rm d}} b {{\rm d}} c,\quad{{\rm d}} d{{\rm d}} d={{\rm d}} b {{\rm d}}
c(1-q^{-2})\]
\[{{\rm d}} d {{\rm d}} c=-{{\rm d}} c{{\rm d}} d q^{-2}+{{\rm d}} a {{\rm d}} c(1-q^{-2}),\quad{{\rm d}} d {{\rm d}} b=- {{\rm d}} b{{\rm d}} d q^2
- {{\rm d}} a {{\rm d}} b(q^2 -1)\]
\[dd {{\rm d}} a=- {{\rm d}} b {{\rm d}} c (q^2 -1)- {{\rm d}} a{{\rm d}} d\]
The dimensions in each degree are the usual ones: 1:4:6:4:1 and we can take a
basis ${{\rm d}} x_{i_1}\cdots {{\rm d}} x_{i_m}$ with $i_1<i_2\cdots<i_m$ with top form ${{\rm d}}
a {{\rm d}} b {{\rm d}} c{{\rm d}} d$.
We have the $q$-epsilon tensor as:
\[{\epsilon}_{addd}=-{\epsilon}_{bdcd}=-{\epsilon}_{dadd}
={\epsilon}_{dbdc}={\epsilon}_{ddad}=-{\epsilon}_{ddda}=1-q^{-2}\]
\[ -{\epsilon}_{adad}=-{\epsilon}_{cdbd}={\epsilon}_{dada}={\epsilon}_{dcdb}=q^2-1\]
\[{\epsilon}_{abcd}=-{\epsilon}_{acbd}={\epsilon}_{adbc}=-{\epsilon}_{adcb}
=-{\epsilon}_{bacd}={\epsilon}_{bcad}=-{\epsilon}_{bcda}={\epsilon}_{cabd}=1\]
\[-{\epsilon}_{cbad}={\epsilon}_{cbda}=-{\epsilon}_{dabc}={\epsilon}_{dacb}
={\epsilon}_{dbac}=-{\epsilon}_{dbca}=-{\epsilon}_{dcab}={\epsilon}_{dcba}=1\]
\[ {\epsilon}_{acdb}=-{\epsilon}_{cadb}={\epsilon}_{cdab}=-{\epsilon}_{cdba}=q^2\]
\[ -{\epsilon}_{abdc}={\epsilon}_{badc}=-{\epsilon}_{bdac}={\epsilon}_{bdca}=q^{-2}.\]
The resulting raw (un-normalised) antisymmetriser projectors have associated
constants
\[ d_0=2q^4[2]^2[3],\quad d_1=-2q^2[3],\quad d_2=q^2[2]^2,\quad
d_3=-2q^2[3],\quad d_4=2q^4[2]^2[3]\]
as in the Euclidean case. The corresponding projections are on the space of
totally $R$-antisymmetric
tensors and have the same ranks as classically.
The Hodge $*$-operator for this metric is:
\[ ({{\rm d}} a{{\rm d}} b{{\rm d}} c{{\rm d}} d)^*=q^{-2},\quad ({{\rm d}} a{{\rm d}} b{{\rm d}} c)^*=q^{-2}{{\rm d}} a,\quad ({{\rm d}} a{{\rm d}}
b{{\rm d}} d)^*=q^{-2}{{\rm d}} b\]
\[ ({{\rm d}} a{{\rm d}} c{{\rm d}} d)^*=-{{\rm d}} c,\quad ({{\rm d}} b {{\rm d}} c{{\rm d}} d)^*=q^{-2}(1-q^{-2}){{\rm d}} a-q^{-2}{{\rm d}}
d\]
\[ ({{\rm d}} a{{\rm d}} b)^*=-[2]{{\rm d}} a{{\rm d}} b,\quad ({{\rm d}} a{{\rm d}} c)^*=[2]{{\rm d}} a{{\rm d}} c,\quad ({{\rm d}} a{{\rm d}}
d)^*=2{{\rm d}} b{{\rm d}} c-(1-q^{-2}){{\rm d}} a{{\rm d}} d\]
\[ ({{\rm d}} b{{\rm d}} c)^*=2q^{-2}{{\rm d}} a{{\rm d}} d+(1-q^{-2}){{\rm d}} b {{\rm d}} c,\quad ({{\rm d}} b{{\rm d}} d)^*=[2]({{\rm d}}
b{{\rm d}} d+2(1-q^{-2}){{\rm d}} a{{\rm d}} b),\quad ({{\rm d}} c{{\rm d}} d)^*=-[2]{{\rm d}} c{{\rm d}} d\]
\[ ({{\rm d}} a)^*=2q^2[3]{{\rm d}} a{{\rm d}} b{{\rm d}} c,\quad ({{\rm d}} b)^*=2q^2[3]{{\rm d}} a{{\rm d}} b{{\rm d}} d,\quad \]
\[({{\rm d}} c)^*=-2q^2[3]{{\rm d}} a{{\rm d}} c{{\rm d}} d,\quad ({{\rm d}} d)^*=-2q^2[3]({{\rm d}} b{{\rm d}} c{{\rm d}} d
-(1-q^{-2}){{\rm d}} a {{\rm d}} b {{\rm d}} c),\quad 1^*=2q^4[2]^2[3]\]
One can check that the square of this Hodge * operator is
\[ *^2=(-1)^mq^{-2}P=(-1)^mq^{-2}d_m\]
on forms of degree $m$. The Hodge $*$-operator on 2-forms has eigenvalues $\pm
[2]$ with multiplicity $3$. The self-dual and antiself dual 2-forms with
respect to it are characterised by
\[ F^*=[2]F,\quad {\rm (self-dual\ form)}; \quad F^*=-[2]F,\quad {\rm
(anti-self-dual\ form)}.\]
As before, one can adjust the normalisation of $*$ to have the more usual limit
when $q=1$.
Also, the same remarks apply as in the Euclidean case to the effect that there
is a natural $*$-structure and a complex transformation from our matrix basis
to self-adjoint or `real' space-time bases $t,x,y,z$\cite{Ma:mec}. This time
the top form, ${\epsilon}$ and $*$ change by an imaginary factor. This again brings
our results here in line with the classical situation for our indexing
conventions.
Finally, by way of contrast our alternative Hodge $\circ$-operator is
\[ ({{\rm d}} a{{\rm d}} b{{\rm d}} c{{\rm d}} d)^\circ=q^{-2},\quad ({{\rm d}} a {{\rm d}} b{{\rm d}} d)^\circ=- {{\rm d}} b q^{-2}\]
\[ ({{\rm d}} a {{\rm d}} c{{\rm d}} d)^\circ={{\rm d}} c,\quad ({{\rm d}} a{{\rm d}} d{{\rm d}} d)^\circ=- {{\rm d}} a
q^{-2}(1-q^{-2}),\quad
({{\rm d}} b {{\rm d}} c{{\rm d}} d)^\circ=ddq^{-2} - {{\rm d}} aq^{-2}(1-q^{-2})\]
\[ ({{\rm d}} a {{\rm d}} b)^\circ= - {{\rm d}} a {{\rm d}} b, \quad ({{\rm d}} a{{\rm d}} c)^\circ={{\rm d}} a {{\rm d}} c
q^{-2},\quad ({{\rm d}} a{{\rm d}} d)^\circ={{\rm d}} b {{\rm d}} c+ {{\rm d}} a{{\rm d}} d (q^{-2}-1), \quad ({{\rm d}} b {{\rm d}}
a)^\circ={{\rm d}} a {{\rm d}} b\]
\[ ({{\rm d}} b {{\rm d}} c)^\circ={{\rm d}} a{{\rm d}} d q^{-2},\quad ({{\rm d}} b{{\rm d}} d)^\circ={{\rm d}} b{{\rm d}} d q^{-2} +
{{\rm d}} a {{\rm d}} b(1-q^{-4}),\quad ({{\rm d}} c{{\rm d}} a)^\circ=- {{\rm d}} a {{\rm d}} c q^{-2}\]
\[ ({{\rm d}} a)^\circ= - {{\rm d}} a {{\rm d}} b {{\rm d}} c,\quad ({{\rm d}} b)^\circ= - {{\rm d}} a {{\rm d}} b{{\rm d}} d,\quad
(\circ {{\rm d}} c)={{\rm d}} a {{\rm d}} c{{\rm d}} d q^{-2},\quad ({{\rm d}} d)^\circ={{\rm d}} b {{\rm d}} c{{\rm d}} d - {{\rm d}} a {{\rm d}} b
{{\rm d}} c(1-q^{-2})\]
On two-forms it has eigenvalues
$q^{-1},-1$ each with multiplicity 3. Hence with respect to $\circ$ we have
\[ F^\circ =q^{-1}F,\quad {\rm (self-dual\ form)}; \quad F^\circ =-F,\quad
{\rm (anti-self-dual\ form)}\]
|
1,108,101,563,289 | arxiv | \section{Introduction}
\label{sec:intro}
Quantum computers and quantum simulators are nowadays
becoming a reality thanks to the advances in ion trapping and integrated
superconducting technology~\cite{2017arXiv171203773A,supercon,Linke3305,Monroe1164,Devoret1169}.
A possible device which is quickly being developed are
quantum annealers. Annealing, as opposed to quenching, is a method to
produce the ground state of a target Hamiltonian by slowly deforming/adjusting
a well-known ground state of a different Hamiltonian. Annealing is
in fact a concept originating from classical metallurgy, extended in the 1980s
to classical optimization problems, and known as simulated annealing~\cite{Kirkpatrick671,cerny1985}.
In the current quantum versions, quantum annealing is very much analogous to
quantum adiabatic computing, but is typically targeted towards the classical
optimization problems. The idea is to add a simple, noninteracting, but noncommuting
term to the original classical Hamiltonian. This simple additional term should
dominate the system at the initial time, so that the ground state will be easy to
find, since it will correspond to a non-interacting system. The non-commuting
nature of the additional term ensures that the initial and target ground states
are not symmetry protected. Then, the additional term is adiabatically removed
and the ground state is expected to go slowly from the initial one to the
one of the Hamiltonian of interest~\cite{Brooke779,PhysRevE.58.5355,Farhi472}, see also
the recent review~\cite{RevModPhys.90.015002}.
This scheme is nowadays plausible with a
large number of possible platforms, including trapped ions, cavity QED, circuit
QED, superconducting junctions~\cite{nature10012} and atoms in nanostructures.
The first commercially
accessible quantum annealers are in the market~\cite{Rnnow420,Boixo2014,youtube}.
Since the original proposals~\cite{PhysRevLett.87.257904,PhysRevLett.92.207901}
trapped ions quantum simulators are the subject of intensive theoretical and
experimental research. Starting from realization of the simple instances of
quantum magnetism~\cite{friedenauer08}, they have reached quite a maturity
in the recent experimental developments (cf.~\cite{2015arXiv151203559M,
PhysRevA.94.023401,2016arXiv160906429L,PhysRevLett.119.080501,PhysRevLett.118.053001,timecrystal,Zhang2017}). The recent paper by Bollinger's
group~\cite{2017arXiv171107392S}, in addition to the excellent experimental work,
contains also an outstanding analysis of quantum dynamics of the relevant Dicke
model, in which the ions interact essentially with one phononic mode.
Quantum dynamics in general, and in particular for the Dicke-like ion-phonon models, are very challenging for numerical simulations. Exact treatments are possible for small systems only, so that various approximate methods have to be used. One of them is the truncated Wigner approximation, in which both ionic and phononic operators are replaced by complex numbers, the dynamics becomes ``classical'', and only the initial data mimic the "quantumness" of the problem \cite{TWA}. This approach was used in Ref.~\cite{PhysRevA.96.033607}
to study the quantum non-equilibrium dynamics of spin-boson models. More sophisticated ``mean-field'' approaches decorrelate ions from phonons, but treat at least either ions or phonons fully quantum mechanically -- this approach is in particular analyzed in the present paper. Quantum aspects
of the models in question were studied in the series of papers~\cite{PhysRevA.95.013602,Garttner2017,PhysRevA.94.053637}.
We have considered recently the exact quantum dynamics of few ion systems
to demonstrate the robustness of chiral spin currents in a trapped-ion
quantum simulator using Floquet engineering~\cite{PhysRevA.97.010302}. Our
earlier works include studies of dual trapped-ion quantum simulators as
an alternative route towards exotic quantum magnets~\cite{1367-2630-18-3-033011}, and
studies of ion chains with long range interactions forming ``magnetic loops''~\cite{PhysRevA.91.063612}.
Topological edge states in periodically-driven trapped-ion chains~\cite{PhysRevLett.119.210401}, trapped
ion quantum simulators of Rabi lattice models with discrete gauge symmetry~\cite{2015PhRvA..92a3624N}, and hidden frustrated interactions
and quantum annealing in trapped ion spin-phonon chains~\cite{PhysRevA.93.013625} were
also considered recently. Novel ideas for spin-boson models simulated with trapped
ions can be found in Ref.~\cite{2017arXiv170400629L}.
In Ref.~\cite{HBHL}, it has been proposed to use trapped ions for solving difficult optimization problems via quantum annealing. Such scheme, applied to the concrete example of number-partitioning, has come under scrutiny in Ref.~\cite{ncomms11524}. The idea
is to profit from the known mapping between the number partitioning problem and the
ground state of spin Hamiltonians~\cite{0305-4470-15-10-028}. In the interesting domain, where the number partitioning problem is notably
difficult, the system is actually in the spin-glass-like phase, which renders
finding the actual ground state an involved task for classical methods. The annealing
method proposed was found to work well for small number of ions at zero temperature.
In this work we explore in detail a semiclassical approximation to the original problem,
where the quantum correlations between the spins and the phonon bath are neglected.
This, however, allows us to solve the Heisenberg equations of motion in an efficient
way for much larger ion systems. Notably, the approach allows us to explore finite
temperature effects on the annealing protocol.
Our present work is structured as follows. In Sec.~\ref{sec:start} we explain the specific
technical details of the studied model and the calculations. In particular, the
Hamiltonian is explained in~\ref{sec:Ham}, the reduction methods in~\ref{sec:SCdecoupling}
and the annealing protocol~\ref{sec:annprot} along with the basic definitions~\ref{sec:fide}.
In Sec.~\ref{secIII}, we introduce our semiclassical approximation, and benchmark it with a full quantum treatment.
In Sec.~\ref{sec:CalcsAtTfin} we show the results of the semiclassical approach applied to finite
temperature. Here, we set the initial phonon population to non-zero thermal values. A summary
and our main conclusions are provided in Sect.~\ref{sec:concl}. Finally, in the
appendix we provide some further tests to our numerical integration
method (appendix ~\ref{app:Integrationtests}), and a brief study of the optimal bias for
the annealing protocol (appendix ~\ref{sec:optim}).
\section{System}
\label{sec:start}
\subsection{Hamiltonian}
\label{sec:Ham}
We study a chain of $N$ trapped ions interacting by effective spin-spin interactions
subjected to a transverse time-dependent magnetic field. Interactions are generated
by Raman coupling the pseudo-spin degrees of freedom to the phonon modes which
are obtained expanding the Coulomb force between the ions around their equilibrium
positions~\cite{PhysRevLett.92.207901}. The phonon spectrum is defined through its
natural frequencies $\omega_k$ and modes $\xi^{i}_{k}$. The dynamics of the system
is described by a time-dependent Hamiltonian that in the Schr\"odinger picture reads,
\begin{eqnarray}
{\cal H}_{\rm S} \left( t \right)/\hbar &= &
\sum_{k}^{M} {\omega_k \hat{a}^{\dagger}_k \hat{a}_k}
+\sum_{i,k}^{N,M} {\Omega \eta^{(i)}_{k}
\sin{\left(\omega_{\rm L} t\right)} \left(\hat{a}^{\dagger}_k
+ \hat{a}_k\right) \sigma_{x}^{\left( i \right)}} \nonumber\\
&+ & \sum_{i}^{N} {B \left( t \right)
\sigma_{z}^{\left( i \right)}} + \varepsilon \sigma_{x}^{\left( p \right)},
\label{eq:Hamiltonian}
\end{eqnarray}
where $\hat{a}_k$($ \hat{a}^{\dagger}_k$) is the annihilation (creation) operator
of one phonon in the $k$-th mode and $\omega_k$ is the frequency of that mode.
The operators $\sigma_{x}^{\left( i \right)}$, $\sigma_{y}^{\left( i \right)}$, and
$\sigma_{z}^{\left( i \right)}$ are the spin operators in the $i$-th position.
The frequencies $\Omega$ and $\omega_{\rm L}$ are the Rabi frequency and the
beatnote frequency of the laser, respectively. The dimensionless parameters $\eta^{(i)}_{k} $ are
the Lamb-Dicke parameters proportional to the displacement of an ion $i$ in
the vibrational mode $k$, see Ref.~\cite{PhysRevLett.92.207901}. As usual, $t$ is
time, and a time-dependent magnetic field $B(t)$ allows us to perform the quantum
annealing. A small bias term, proportional to $\varepsilon$, has been added in
the $p$-th position to remove the ${\cal Z}_2$ degeneracy. The upper limit of the sum over the ions $i$ is $N$, the number of
ions. The upper limit of the sum in modes $k$ is $M$, the number of modes. The
total number of phonon modes is $3N$, but the Raman beam couples to only $N$ modes,
selected by the wave vector difference of the lasers.
At this point, we may keep our analysis general by making no assumption about the number $M$ of modes.
However, all phonons which are considered are assumed to be coupled to the spin in the same way.
Hereinafter, the upper limits of the sums will be omitted for brevity.
\subsection{Equations of motion}
\label{sec:SCdecoupling}
We compute the Heisenberg equations of motion for the quantum average of every
operator in the Hamiltonian, $\hat{a}_k$, $\hat{a}^{\dagger}_k$, $\sigma_{x}^{\left( i \right)}$,
$\sigma_{y}^{\left( i \right)}$, and $\sigma_{z}^{\left( i \right)}$. Given that these are time
independent operators, the calculation reduces to commutators. Additionally, we
replace $\hat{a}^{\dagger}_k+\hat{a}_k$ by $2 \Re \left[ \hat{a}_k\right]$ and
$\hat{a}_k-\hat{a}^{\dagger}_k$ by $2 {\rm i} \Im \left[ \hat{a}_k\right]$. The
equations of motion (all with real coefficients) read,
\begin{eqnarray}
\frac{d \langle \Re \left[ \hat{a}_k\right] \rangle}{dt} &=&
\omega_k \langle \Im \left[ \hat{a}_k\right] \rangle \label{eem}
\\
\frac{d \langle \Im \left[ \hat{a}_k\right] \rangle}{dt} &=&
-\omega_k \langle \Re \left[ \hat{a}_k\right] \rangle-
\sin{\left(\omega_{\rm L} t\right)} \sum_{j} \Omega \eta^{(j)}_{k}
\langle \sigma_{x}^{\left( j \right)} \rangle,
\nonumber\\
\frac{d \langle \sigma_{x}^{\left( i \right)} \rangle}{dt} &=&
-2 B\left( t \right) \langle\sigma_{y}^{\left( i \right)}\rangle,
\nonumber\\
\frac{d \langle \sigma_{y}^{\left( i \right)} \rangle}{dt} &= & -4
\sum_{l} \Omega \eta^{(i)}_{l} \sin{\left(\omega_{\rm L} t\right)}
\langle \Re{ \left[\hat{a}_l\right]} \sigma_{z}^{\left( i \right)}\rangle
\nonumber\\
&&+2 B\left( t \right) \langle \sigma_{x}^{\left( i \right)}\rangle
-2 \varepsilon \langle \sigma_{z}^{\left( i \right)} \rangle \delta_{p,i},
\nonumber\\
\frac{d \langle \sigma_{z}^{\left( i \right)} \rangle}{dt} &=&4
\sum_{l} \Omega \eta^{(i)}_{l} \sin{\left(\omega_{\rm L} t\right)}
\langle \Re{ \left[\hat{a}_l\right]} \sigma_{y}^{\left( i \right)}\rangle
+2\varepsilon \langle \sigma_{y}^{\left( i \right)} \rangle \delta_{p,i}\nonumber\,.
\end{eqnarray}
\subsection{Annealing protocol}
\label{sec:annprot}
The functional form and value of $\Omega$, $B \left( t \right)$ and $\varepsilon$ determine
the annealing protocol. In these annealing schemes, the initial value of the transverse
magnetic field $B \left( t=0 \right)$ must be sufficiently strong to initialize the system
in the paramagnetic phase, that is, $B$ must be larger than the effective spin-spin interactions
$J \sim \Omega^2 \omega_{\rm rec}/ (\delta \omega_{\rm rad} )$, where $\omega_{\rm rec}$ is the
recoil energy of the photon-ion coupling, $\omega_{\rm rad}$ is the radial trap frequency, and $\delta$ the detuning from the nearest phonon mode.
For typical values, e.g. $\Omega \sim \delta \sim 100$ kHz, $\omega_{\rm rec}\sim 25$ kHz,
$\omega_{\rm rad} \sim 5$ MHz, we obtain effective interactions $J\sim 1$ kHz, so we need an
initial field strength $B(0)\sim 10$ kHz. The annealing scheme proceeds by turning down the
magnetic field according to some functional form in order to adiabatically achieve the ground
state of the Hamiltonian of interest. Given the adiabatic theorem, for a closed system
initiallized in the ground state, the final system is guaranteed
to be in the ground state as long as the system is gapped along the annealing path, and the
variation is slow enough. Generalization to open systems has been proposed in Ref.~\cite{PhysRevA.93.032118}.
We have used a decreasing exponential form for the transverse magnetic field,
$B \left( t\right)=B(0)\,e^{-\frac{t}{\tau}}$ with a decay rate $\tau$. The other parameters,
$\Omega$ and $\varepsilon$, remain constant. An example of the evolution of
the system under this protocol is shown in Fig.~\ref{fig:sigma_x}. Initially,
$\expval{\hat{\sigma}_{x}^{\left( i \right)}}=0$ for all $i$, and the total phononic
population is set to $0$. Within tens of microseconds the phononic modes are
populated. Not surprisingly, the mode next to the resonance becomes the most populated one,
with a population being orders of magnitude larger than the population of the other
modes. In contrast to these rapid changes of the phonon state, the spin dynamics is much
slower. The spin expectations $\expval{\hat{\sigma}_{x}^{\left( i \right)}}$ remain mostly
clustered around zero for hundreds of microseconds. When $B \left( t\right) \simeq \varepsilon$,
the values $\expval{\hat{\sigma}_{x}^{\left( i \right)}}$ start to deviate from zero, and some
acquire positive values, while others become negative. Thus, the spin curves separate
from each other, and we call the time at which this happens the {\it separation time}. At
some point after the separation time, the spin curves saturate, that is, from then on
$\expval{\hat{\sigma}_{x}^{\left( i \right)}}$ remain constant in time. We define the {\it waiting time}
as the time when all $\expval{\hat{\sigma}_{x}^{\left( i \right)}}$ have stopped varying.
At the waiting time, the phononic populations stabilize around certain values, although their
oscillations do never vanish.
The quantum annealer produces final values of $\expval{\hat{\sigma}_{x}^{\left( i \right)}}$
which are not fully polarized , that is, $|\expval{\hat{\sigma}_{x}^{\left( i \right)}}|<1$.
Thus, the final state differs from the classical ground state of the target Hamiltonian, that is,
the Hamiltonian in the absence of a transverse field. Thus, we take as
readout of the annealing protocol the average spin values~\cite{PhysRevE.94.032105,PhysRevA.96.042310,srep22318}.
This is not a problem, as long as
for all spins the sign matches with the one in the classical state. As explained in detail in Ref.~\cite{ncomms11524},
the spin configuration of the target Hamiltonian is determined by the dominant mode,
defined as the one with frequency just below the beatnote frequency $\omega_{\rm L}$.
There are different reasons why the final ground state might show a different spin
pattern: Either, the annealing was too fast, that is, the value of $\tau$ was chosen too
small, or the effective spin model is not valid. This is the case when $\omega_{\rm L}$ is
too close to a resonance $\omega_k$.
\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{Ex_Ev_SigsandOccs.ps}
\caption{(Upper panel) Evolution of $\expval{\hat{\sigma}_{x}^{\left( i \right)}}$
in a system with 4 spins, with initial populations of phononic modes set to $0$.
The biased spin is the pink one, see text for details. (Lower panel) Evolution of
the populations of the phononic modes in a system with 4 spins, with initial
populations of phononic modes set to $0$. $\delta = 1$ MHz.}
\label{fig:sigma_x}
\end{figure}
\subsection{Fidelity of the annealing protocol}
\label{sec:fide}
In order to quantify the success of the annealing protocol, that is, the
ability of the method to identify the target ground state of the spin system,
we will define the following fidelity,
\begin{equation}
F=\begin{cases}
\min\limits_{i} {\left| \expval{\sigma_{x}^{(i)}}\right|} &{\rm if\;}
{\rm sign}{\left[\expval{\sigma_{x}^{(i)}}\right]}={\rm sign}{\left[ \eta_{k_d}^{(i)} \right]}, \forall i\\
0 & {\rm Otherwise}
\end{cases},
\label{eq:Fide_def}
\end{equation}
where $\eta_{i j}$ is the $i$th component of the dominant mode $k_d$ for a fixed value
of the beatnote frequency $\omega_{\rm L}$. That is, the fidelity is zero if the
signs of $\langle \sigma_x^i\rangle$ do not match the signs of $\eta_i$ of the dominant
mode. If the signs are reproduced, the value of the fidelity is defined as the smallest
expectation value of the spins of the ions. Note that with this definition, any non-zero fidelity is good enough for correctly
identifying the ground state pattern, assuming the absence of noise in the system.
\section{Semiclassical approximation}
\label{secIII}
\subsection{Semiclassical equations of motion}
Now we will develop a semiclassical approximation to the exact equations of
motion, Eq.~(\ref{eem}), that will allow us to study larger systems of ions
and the effects of temperature on the annealing protocols.
We make the following approximations:
\begin{eqnarray}
\langle \hat{a}_k \sigma_{\mu}^{\left( i \right)} \rangle &\simeq&
\langle \hat{a}_k \rangle \langle \sigma_{\mu}^{\left( i \right)} \rangle \nonumber\\
\langle \hat{a}^{\dagger}_k \sigma_{\mu}^{\left( i \right)} \rangle &\simeq& \langle \hat{a}^{\dagger}_k \rangle \langle \sigma_{\mu}^{\left( i \right)} \rangle
\end{eqnarray}
with $\mu = \left\{ x,y,z\right\}$. These approximations ignore the quantum
correlations in the coupling between bosonic and spin modes.
Additionally defining the auxiliary variables
$S_k \left( t \right) \equiv \sin{\left(\omega_{\rm L} t\right)}
\sum_{j} \Omega \eta^{j}_{k} \langle \sigma_{x}^{\left( j \right)} \rangle$ and
$J^{\left( i \right)} \left( t \right) \equiv \sin{\left(\omega_{\rm L} t\right)}
\Omega \left(2\sum_{l} \eta^{i}_{l} \langle \Re{ \left[\hat{a}_l\right]}
\rangle \right) +\varepsilon \delta_{p,i}$, we obtain the approximate equations of motion,
\begin{eqnarray}
\frac{d \langle \Re \left[ \hat{a}_k\right] \rangle}{dt} &=&
\omega_k \langle \Im \left[ \hat{a}_k\right] \rangle,
\label{eq:Eqs_motion_simpl}\\
\frac{d \langle \Im \left[ \hat{a}_k\right] \rangle}{dt} &=&
-\omega_k \langle \Re \left[ \hat{a}_k\right] \rangle-S_k \left( t \right),
\nonumber\\
\frac{d \langle \vec{\sigma}^{\left( i \right)} \rangle}{dt} &=& -2
\begin{pmatrix}
0 & B\left( t \right) & 0\\
-B\left( t \right) & 0 & J^{\left( i \right)} \left( t \right)\\
0 & -J^{\left( i \right)} \left( t \right) & 0
\end{pmatrix}
\cdot \langle \vec{\sigma}^{\left( i \right)} \rangle\nonumber
\end{eqnarray}
where a spin vector notation,
$\langle \vec{\sigma}^{\left( i \right)} \rangle =\left( \langle \sigma_{x}^{\left( i \right)} \rangle,
\langle \sigma_{y}^{\left( i \right)} \rangle, \langle \sigma_{z}^{\left( i \right)} \rangle \right)$,
has been used. This is a system of $2\times M+ 3\times N$ non-linear first-order ordinary
differential equations. Hence, it is numerically solved with a first order, ordinary
differential equation solver that uses the Gragg–Bulirsch–Stoer method, stepsize control and
order selection, called ODEX~\cite{ODEXcite}.
\subsection{Comparison of semiclassical approximation to full quantum evolution}
\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{figura.eps}
\caption{\label{fig:fig4} Quantum (a,c) and semiclassical (b,d) evolution
of $\expval{\hat{\sigma}_{x}^{\left( i \right)}}$. The figure is computed with $N=4$ ions
with a detuning $\delta =2900$ kHz. Panels (a) and (b) correspond to $\tau=0.35$ ms
and panels (c) and (d) to $\tau=0.7$ ms. }
\end{figure}
To benchmark the semiclassical method, we have compared it against a full quantum
evolution of the system using Krylov subspaces. The latter is a method to study
the dynamical evolution under time-dependent Hamiltonians that computes a reduced
evolution operator omitting contributions smaller than a certain threshold, that is,
transitions to irrelevant states. Despite this neglection, the Krylov evolution can be
considered an \textit{exact} numerical simulation, as it iteratively determines which part of the
Hilbert space is irrelevant at a given accuracy.
In both cases, semiclassical and full quantum, the evolution of a given initial state
under a time-depending Hamiltonian is calculated with time steps in a recurrent way.
In the fully quantum calculation, the time steps are of the order of $1$ ns, while in
the semiclassical description they are variable, but can be orders of magnitude larger.
In the quantum case, we have to specify a quantum state ---a complex vector in
the joint Fock basis of phononic and spin modes---, containing the amplitudes of every
state of the basis. In the semiclassical case, we only have to supply the initial mean
values of every operator.
It should be noted that the exact quantum evolution requires truncating the maximum
phonon number which in our case was set to two phonons per mode. Such truncation of
the Hilbert space requires sufficiently cool systems. And even with this truncation,
the quantum evolution is restricted to a small number of ions. Considering only one
transverse phonon branch, i.e. $N$ phonon modes, with a maximum population of two phonons
per mode, the Hilbert space dimension is $2^N \times 3^N$, that is, a dimension of 46656
for $N=6$ ions. The semiclassical approach, in contrast, allows us to explore larger systems.
\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{Ex_Ev_ComparExRelErr}
\caption{Evolution of $\expval{\hat{n}_3 \hat{\sigma}_{x}^{\left( i \right)}}$ (a) and of
the relative difference
$(\expval{\hat{n}_3 \hat{\sigma}_{x}^{\left( i \right)}}-\expval{\hat{n}_3} \expval{\hat{\sigma}_{x}^{\left( i \right)}})/\expval{\hat{n}_3 \hat{\sigma}_{x}^{\left( i \right)}} $
for a system of four spins with initial populations of phonons set to zero for a
detuning $\delta=1000$ kHz.
}
\label{fig:diffs}
\end{figure}
\subsubsection{Time evolution of $\expval{\hat{\sigma}_{x}^{\left( i \right)}}$}
The semiclassical model captures well the qualitative behavior of the
evolution of $\expval{\hat{\sigma}_{x}^{\left( i \right)}}$, as exemplified
in Fig.~\ref{fig:fig4}. In the figure we compare the semiclassical evolution
with the exact dynamics for $N=4$ ions for two different decay times. The
discrepancy between semiclassical and exact evolution is smallest for shorter
times ($t\lesssim \tau$), where the semiclassical model is able to correctly
capture the details of the dynamical evolution, most notably little wiggles
in the evolution of the biased ion, $i=1$. Importantly, the semiclassical model
also agrees with the exact evolution regarding general features such as the
separation time, and the sign of each $\expval{\hat{\sigma}_{x}^{\left( i \right)}}$
in the long-time limit. As discussed in more detail in the next paragraph, this
enables a quite accurate prediction of annealing fidelities, despite the fact
that the approximation disregards some quantum properties. Thus it provides
a computationally efficient way to study the behavior of larger systems.
For a better understanding of the errors in the semiclassical approach, we have
exactly calculated the evolution of $\langle \hat n_3 \sigma_x^{(i)} \rangle$, see
Fig.~\ref{fig:diffs}(a), and of $\langle \hat n_3 \rangle \langle \sigma_x^{(i)} \rangle$.
As our semiclassical approximation is based on substituting the former correlator
by the latter one, the discrepancy between both expressions is an indicator for
the quality of the semiclassical approach. In Fig.~\ref{fig:diffs}(b), we plot the
relative difference as a function of time: Initially, the phonon and spin degrees of
freedom are taken as uncorrelated, thus, the semiclassical and the exact description
coincide at $t=0$. On short time scales, both correlators have small absolute values,
but their relative difference becomes large. For times larger than the separation time,
the absolute values of the correlators increase, and the relative errors decrease.
On long time scales, the errors oscillate around mean values of the order 0.1. This
observation suggests that the main errors made in the semiclassical approximation are
introduced at short times, where the transverse magnetic field and its temporal
derivative takes large values.
\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{Fides_na4_diff.eps}
\caption{\label{fig:QFide_na4} Fidelity obtained with the full quantum evolution
$F_{\rm E}$ (a), the semiclassical model $F_{\rm SC}$ (b), and the difference between
the semiclassical model and the full quantum (c). In both cases, we consider a system
of four ions, and we plot the magnitudes as a function of $\omega_{\rm L}$ and
$\tau$. The fidelity is readout after a time $20 \,\tau$.}
\end{figure}
\subsubsection{Fidelity}
\label{secfide}
As we have seen the semiclassical approximation provides a reasonable description
of the dynamics in many configurations. Let us now explore in more detail in which
parameter regions it predicts the correct fidelity for the annealing protocol.
In Fig.~\ref{fig:QFide_na4} we present a comparison of the fidelities obtained from
the exact time evolution and from the semiclassical approach for a system of four ions.
We tune through a broad range of beatnote frequencies $\omega_L$, and vary the decay
time $\tau$ of the magnetic field. The overall agreement is very good: Both methods
predict a small fidelity when the field decays too fast (small $\tau$), or when the
system is too close to one of the phonon resonances. The semiclassical evolution, however,
slightly overestimates the fidelity for small $\tau$, and also slightly below each
phonon resonance, that is, on the ferromagnetic side of the resonance. Notably, the
semiclassical approach works quite well in the glassy regimes above the resonances,
where it estimates correctly the regions in which the annealer fails for any annealing time.
As discussed earlier, the failure of the annealing protocol for small $\tau $ is due to
non-adiabatic behavior in the fast varying field. The failure near the resonance, though,
cannot be fixed by increasing $\tau$, and has its origin in the deviation from of the
Dicke dynamics from the effective spin model. Although such deviations are expected on
both sides of a phonon resonance, the region of zero fidelity is seen only on the glassy
side of each resonance. From that perspective, the size of the spin gap seems to play a
role as well, although in this regime we should not compare it to $\hbar/\tau$, but to
those spin-phonon energy scales which are neglected in the effective spin model, that is,
the first order term in a Magnus expansion, see Ref.~\cite{PhysRevLett.103.120502}.
The main advantage of the semiclassical model is that it can easily be applied to larger
systems. In Fig.~\ref{fig:SCFide_na6}, we consider systems of six and eight ions. Notably,
a broad region of zero fidelity occurs for eight ions between $\omega_5$ and $\omega_6$.
Its origin is unclear to us, and further calculations on the fully quantum evolution would
be needed in order to discriminate whether they are true effects or merely calculation artifacts.
\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{Fide_na6i8}
\caption{\label{fig:SCFide_na6} Fidelity of a system of six ions (a)
and a system of eight ions (b) as a function of $\omega_{\rm L}$ and $\tau$
using the semiclassical approximation. Final times are $20 \tau$.}
\end{figure}
We finish this section by discussing the factor which limits the scalability of the
quantum annealer. As seen above, below a critical detuning from the resonance the
fidelity drops to zero. This sets a limit to the scalability of the quantum annealer,
because, with the number of modes being proportional to the number of ions, the mode
spacing decreases when the system size is increased. However, we note that phonon
spectra are not equidistant, and within the transverse branch, the phonon spacing
is largest at the lower energetic end of the phonon spectrum. Thus, to achieve finite
fidelity in an up-scaled system, one may need to operate in the regime of low-energy
phonons. The semiclassical estimates in Ref.~\cite{ncomms11524} suggest that the
quantum annealing still works in systems with more than twenty ions, presumably large
enough to detect quantum speed-up.
\section{Finite temperature effects on the annealing protocol}
\label{sec:CalcsAtTfin}
Finite temperature effects are expected to reduce the quality of the annealing protocol,
recently, however, thermal effects are shown to aid the annealing protocol
for a 16-qubit problem in a superconducting setup~\cite{ncomms2920}.
In this section we study the robustness of the annealing protocol when
the phonons are initially at finite temperature. To do so, we consider an
initial state with phonon mode populations set as follows: we fix the
temperature $T$ of the phonons, then, the mean values of the number of phonons
for each mode are sampled according to the a bosonic thermal bath probability
distribution at $T$. With these initial conditions, the system is then evolved
semiclassically according to Eqs.~(\ref{eq:Eqs_motion_simpl}). This process
is repeated with different initial values of the population of phonons, sampled
appropriately. After the evolution, the statistical moments are calculated in
order to infer the thermal properties of the system at the final time.
It should be noted that our dynamical model only captures the coherent Hamiltonian
evolution, but no decoherence processes due to interactions with the environment.
Thus, in order to account for all thermal effects, heating events, as they occur
for instance due to trap inhomogeneties, should be taken into account by
considering an increased initial temperature.
\subsection{Classical thermal phonons}
We assume that the initial populations of the phononic modes are determined by
a phonon temperature. In the canonical ensemble, the expected value of the number
operator of the phonons in the $k$-th mode is,
\begin{equation}
\langle \hat{n}_{k} \rangle = \frac{1}{e^{\beta \hbar \omega_k}-1}\,.
\label{eq:n_k}
\end{equation}
The corresponding Hamiltonian of the symmetrized phonon field is
$\hat{\cal H}_{\rm ph} = \hbar \sum_{k} \omega_k\frac{ \hat{a}^{\dagger}_k \hat{a}_k
+ \hat{a}_k \hat{a}^{\dagger}_k}{2}$. The expected value of the annihilation
operator $\alpha_k \equiv \expval{\hat{a}_k}$
is sampled as,
\begin{equation}
P\left(\alpha_k \right) = \frac{1}{\pi \langle \hat{n}_{k} \rangle} e^{-\frac{|\alpha_k|^2}{\langle \hat{n}_{k} \rangle}} \,.
\label{eq:p_alpha_k}
\end{equation}
The complex-valued Gaussian probability distribution function (PDF)
for the random variable $\alpha_k$ is a product of two Normal PDFs ---one real
the other purely imaginary---for the random variables $\Re\left[\alpha_k\right]$
and $\Im\left[\alpha_k\right]$. Both distributions have a mean $\mu=0$ and variance
$\sigma^2=\hat{n}_{k}/2$. We thus use for convenience,
\begin{equation}
P\left(\alpha_k \right) =
N \left(\Re\left[\alpha_k\right];0,\langle \hat{n}_{k} \rangle/2\right)
N \left(\Im\left[\alpha_k\right];0,\langle \hat{n}_{k} \rangle/2\right)\,,
\label{eq:p_reimalpha_k}
\end{equation}
being $N \left(x;\mu,\sigma^2\right)$ the Normal PDF of the random variable $x$ with
mean $\mu$ and variance $\sigma^2$.
\subsection{Effects of temperature on the protocol}
\begin{figure}[t!]
\includegraphics[width=0.5\textwidth]{output_wL_compar_panel.eps}
\caption{\label{fig:ThermalFideandTotPop}
We plot the fidelity (a,b) and the square of the phonon coherences (c,d), as a function
of the initial phonon temperature, as obtained from the semiclassical calculation for
a system of four ions. Different lines correspond to different detunings. Panels (a) and
(c) consider cases where the second phonon resonance is approached from below (where
the magnetic order is ferromagnetic), while panels (b) and (d) consider cases where
the same resonance is approached from above (where a glassy regime occurs near the
resonance). In our system, the second resonance occurs at a frequency $\omega_2=14332.7$
kHz. All calculations were done for $\tau = 10$ ms. Each point is obtained by sampling
over 1,000 runs. The adscribed error is 2 $\sigma$.}
\end{figure}
To evaluate the effects of temperature on the proposed annealing protocol we will consider
different initial temperatures and detunings. In all cases we will fix the decay time
$\tau=10$ ms. Fig.~\ref{fig:ThermalFideandTotPop} shows the fidelity and the total phononic
population per mode in the system. In the figure we compare results obtained with several
values of $\omega_L$ and a broad range of temperatures of the phonons. Quite generally, panels
(a) and (b) show that up to a certain temperature the fidelity is not affected by thermal
phonons, but above this temperature the fidelity drops to zero. The value of this temperature
strongly depends on the detuning, and decreases by several orders of magnitudes when we change
from a far-detuned configuration to a near-resonance scenario. For instance, in the far-detuned
regime at $\omega_{\rm L}=\omega_2-902.41$ kHz (solid squares) the critical temperature is of
the order $0.1$ K, while close to the resonance at $\omega_{\rm L}=\omega_2-35.33$ kHz
(small triangles), the fidelity drop occurs at a temperature of the order $10^{-4}$ K. Such
behavior is seen both when we approach the phonon resonance from above (panel (b))or below
(panel (a)), but we remind that above the resonance there is a finite region in which the
fidelity is zero even at $T=0$, cf. the discussion of Fig.(~\ref{fig:QFide_na4}) in Sec. ~\ref{secfide}.
Regarding the phonon population, we assume that a reasonable estimate of $\langle \hat{n}_k \rangle$
is given by $|\alpha_k|^2$, as would be the case if the phonon field remains coherent.
In Fig.~\ref{fig:ThermalFideandTotPop}(c,d), we may distinguish two different regimes:
At low temperatures, i.e. for $T\lesssim 10^{-5}$ K, the final phonon population is dominated
by those phonons which are produced by the spin-phonon coupling. The number of those phonons is
independent from the temperature, and as shown earlier in Fig. 1(b), such phonons are generated
also at $T=0$. The proximity to the resonance induces a larger population of the dominant mode,
resulting in a larger value for the population as we approach the resonance. In contrast,
for high temperatures, the phonon population is dominated by thermal phonons present already
in the beginning of the evolution. In this case, the phonon population is more or less constant
during the evolution, and the population number strongly depends on the temperature, according
to the initial values from the Boltzmann distribution. As in this case, the time evolution does
not noticeably change the phonon distribution, initial and final distribution are very close,
and so are initial and final temperature. Between the low- and high- temperature phase, there
is a narrow crossover regime, where the number of dynamically generated phonons is similar
to the number of thermal phonons. The temperature at which this happens generally depends on
the detuning, i.e. on $\omega_{\rm L}$. In all cases, the fidelity drop occurs only in the
high-temperature regime, that is, the number of thermal photons must be large compared to the
dynamically generated phonons in order to negatively affect the spin evolution.
\subsection{Thermal tolerance of the Lamb-Dicke regime}
The Hamiltonian Eq.~(\ref{eq:Hamiltonian}) describes the trapped ion system when it is in
the Lamb-Dicke regime, that is, for $k x \ll 1$. In a harmonic oscillator with frequency $\omega$,
we have$\langle x^2 \rangle = \frac{\hbar}{m \omega} \left(\langle n \rangle + \frac{1}{2} \right)$.
With $k=\sqrt{2m\omega_{\rm rec}/\hbar}$, and introducing the Lamb-Dicke parameter
$\eta=\sqrt{\omega_{\rm rec}/\omega}$, we re-write the Lamb-Dicke condition as
\begin{eqnarray}
\eta \sqrt{ 2\langle n \rangle +1 } \ll 1 \,.
\end{eqnarray}
All phonon frequencies $\omega$ are of the order of the radial trap frequency,
$\omega_{\rm rad}=2\pi \times 2,655$ kHz. In our simulation, the recoil frequency is
taken as $\omega_{\rm rec}=2\pi \times 15$ kHz, so we obtain a Lamb-Dicke parameter
$\eta \approx 0.075$. Thus, the Lamb-Dicke regime requires $\sqrt{\langle n \rangle +0.5} \ll 10$,
which is fulfilled by a phonon occupation $\langle n \rangle \lesssim 1$. From that
perspective, we have to disregard those calculations where the phonon population
exceeds this number. From Fig.~\ref{fig:ThermalFideandTotPop}(c,d) we find that, for the
Rabi frequency we have used, phonon numbers above 1 occur in the thermally dominated regime,
independent from the detuning. This regime is characterized by temperatures $>10^{-4}$ K.
\section{Summary and conclusions}
\label{sec:concl}
We have considered a chain of trapped ions with an internal state (``spin'') coupled
to vibrational modes via Raman lasers. The couplings are such that the effective model
describing the ions is a long range spin model with tunable, pseudo-random couplings,
leading to a spin-glass-like phase. The goal of our approach is the adiabatic distillation
of the ground state in the glassy phase starting from a completely paramagnetic state.
To this aim we consider the addition of a time-dependent transverse magnetic field. Our
procedure goes as follows: At the initial time, the magnetic field is strong enough to
ensure the ground state of the spins is a ferromagnetic state, with all spins aligned in
the transverse direction. As time evolves we slowly, ideally adiabatically, remove the
magnetic field such that the final Hamiltonian is our effective long-range spin model in the
spin-glass-like phase.
We have simulated our annealing protocol using the exact evolution by means of a Krylov subspace method
which is feasible for a small number of ions. In order to consider larger systems as well as to
study the effect of temperature on the time evolution we have developed a semiclassical formalism which
ignores the quantum correlations between the ions and the phonons. The quality of this method has been
benchmarked by comparing its predictions with the exact evolution for four ions. The semiclassical
model is found to provide a very accurate qualitative picture of our proposed method, and
allows us to correctly identify the parameter region where the annealing protocol works well.
By means of the semiclassical model we have thus extended our study to larger number of ions, providing
an accurate picture of the ability of the annealing protocol to find the correct ground state depending
on the annealing time.
Finally, the semiclassical model has allowed us to study the robustness of the scheme
for initial phonon states at finite temperature. We find that the effect of temperature
strongly depends on the detuning from a phonon resonance. While in most configurations,
the quantum annealing does not break down within the Lamb-Dicke regime, close to a resonance the situation is different.
Here, the fidelity of the annealing may drop even before the Lamb-Dicke limit is reached, see
Fig.~\ref{fig:ThermalFideandTotPop}. Thus, while state-of-art spin model simulations which
are carried out far off any phonon resonance (e.g. Refs. \cite{PhysRevLett.119.080501,Zhang2017})
require only cooling to the Lamb-Dicke limit, quantum annealing in the interesting glassy regime
requires more cooling. Accordingly, our finding motivates the development of new, more efficient cooling
techniques, as for instance cooling based on electromagnetically-induced
transparency~\cite{PhysRevLett.85.5547,PhysRevLett.110.153002,PhysRevA.93.053401}, which is very well
suited to simultaneously achieve low populations in all radial modes.
\begin{acknowledgments}
This work has been funded by EU grants (EQuaM (FP7-ICT-2013-C No. 323714), OSYRIS (ERC-2013-AdG No. 339106),
SIQS (FP7-ICT-2011-9 No. 600645), and QUIC (H2020-FETPROACT-2014 No. 641122)), Spanish Ministerio de
Econom\'ia y Competitividad grants (Severo Ochoa (SEV-2015-0522), FOQUS (FIS2013-46768-P), FIS2014-54672-P and
FISICATEAMO (FIS2016-79508-P)), Generalitat de Catalunya (2014 SGR 401, 2014 SGR 874, and CERCA program),
and Fundaci\'o Cellex.
TG acknowledges support of the NSF through the PFC@JQI.
\end{acknowledgments}
|
1,108,101,563,290 | arxiv | \section{\label{sec:intro}Introduction}
The detection of nearly 200 different molecular species over the last 50 years demonstrates that the interstellar medium (ISM) is home to a rich chemistry~\cite{McGuire2018}. In star-forming regions a multitude of species has been found and some of them, the so-called prebiotic species, are thought to be involved in the processes leading to the early forms of life.
As gas and dust gradually condense into colder and denser interstellar matter, chemical processes begin to produce molecules from the mainly atomic gas. Although formaldehyde-sized molecules (H$_2$CO) exist in diffuse and translucent clouds~\cite{Liszt2006}, the production of complex species seems to occur in the densest regions, where UV fields are considerably reduced and temperatures are below 15~K~\cite{Caselli2012}.
Models of steady-state or time-dependent gas-phase astrochemistry do not always manage to simulate the observed molecular abundances because physical conditions of interstellar regions can themselves be time dependent, and not all chemical processes are known. Therefore, surface reactions on sub-micrometer-sized dust grains are necessarily invoked for the formation of many organic molecules~\cite{Cazaux2016,Taquet2012}.
The chemistry on the surface of cold grains occurs mostly at low temperature ($\sim$10~K). In the innermost part of the clouds new molecules form in and on H$_{2}$O-dominated dust grain mantles, mainly via reactions of atomic hydrogen and atomic oxygen with molecules and radicals. These processes have been invoked by astronomers for many years now~\citep{Tielens1982}, but it is during the last two decades that laboratory evidence has confirmed that the bulk of interstellar ices is formed in the solid phase through surface reactions (water, methanol, carbon dioxide, formaldehyde, ammonia, formic acid)~\cite{Dulieu2010,Jing2011,Oba2010,Ioppolo2011,Noble2011,Minissale2013,Minissale2015,Hiraoka2002,WatanabeKouchi2002,Fuchs2009,minissale2014c,Ioppolo2014,Minissale2016b,Joshi2012,Zins2012,Pirim2014}.
The solid-state formation of organic material is therefore of considerable interest, as efficient surface reaction
routes also provide a general recipe to form relatively complex (i.e., containing more than six atoms) organic molecules, referred to as COMs, in star- and planet-forming regions~\citep{Charnley2001,Caselli2012,Vasyunin2017}.
One of the most important categories of data that astronomers need from laboratory experiments is to determine the rates for the reactions that contribute to formation and destruction of molecules under interstellar conditions. Two main processes are evoked to explain the chemical diversity observed in space: reactions in the gas phase and reactions in solid state (i.e., at the surface of dust grains).
See Smith (2011)~\cite{Smith2011} for a review on laboratory experiments for gas-phase processes.
For solid-state laboratory astrochemistry, surface science techniques at ultra-high vacuum conditions are used to qualitatively and quantitatively study the chemical physical processes leading to new species. Temperature-controlled desorption, TPD~\cite{Collings2004MRAS}, and the novel temperature-controlled during exposure desorption, TP-DED~\cite{Minissale2016}, techniques are now routinely used to analyze thermal and non-thermal desorption of atoms and molecules, providing the number and nature of newborn species, as well as the binding energies that will be used in the models. On the other hand, Fourier Transform Infrared (FTIR) and UV spectroscopy techniques are used to analyze ice samples, pure and mixed, \textit{in situ}, which allows laboratory astrophysicists to trace the evolution of the sample from the deposition of radicals until the formation of new products.
Until recently, solid state experiments have been carried out using single or double atom/radical sources, hence with a limited number of reactants and restrictions in the design of experiments aimed at simulating the processes leading to COMs. A paper was published this year that describes a triple beam line experiment including a novel atomic carbon source, which extends the possibilities of studying C-atom chemistry and the formation of COMs as this is likely to occur in the low density phase of molecular clouds~\cite{Qasim2020}.
The problem of how to form COMs with high enough efficiency to account successfully for their abundance in the ISM is a major problem in modern astrochemistry and astrobiology.
To date, many of the facts required to fully understand the formation of COMs are yet to be determined. How and under what conditions are these molecules produced?
How far does this chemical complexity can go?
This paper describes the design and performance of VENUS, a new apparatus conceived to study surface reactions and reaction-diffusion processes that lead to the formation of COMs via non-energetic processes under conditions approaching those of the ISM. The experiments take place under ultra-high vacuum (UHV)
conditions (base pressure $1\times10^{-10}$~mbar) in a stainless steel chamber. Through a multi-beam assembly, up to five impacting species can be directed onto the cold target, consisting of polycrystalline gold or of an analog of cosmic dust. The sample temperature can be controlled in the 7 -- 400~K interval; quadrupole mass spectrometry and FTIR spectroscopy are used to characterize the deposited reactants and the final products. In Section~\ref{sec:design} we describe in detail the various components of VENUS, in Section~\ref{sec:performance} we present the experimental performances of the apparatus, and finally in Section~\ref{sec:potential} we discuss the importance of these results with respect to astrochemistry in the solid phase.
\section{\label{sec:design}Instrument design}
\subsection{\label{subsec:overview}Goals and overview}
This article describes the conception and performance of a new experimental facility constructed to investigate reactions between atoms and molecules at low temperatures occurring on substrates of astrochemical relevance.
\begin{figure}[b]
\includegraphics[scale=0.55]{VENUS_figure1.pdf
\caption{A schematic diagram of the experimental setup.
\label{fig:venus1}
}
\end{figure}
The new setup called VENUS -- an acronym of the French phrase \textit{VErs des NoUvelles Synthèses} (Towards new syntheses) -- was designed to serve as an upfront apparatus to study solid-state pathways leading to complex organic molecules in space.
With few exceptions, complex molecules have been mainly detected in the gas phase, but there is strong evidence that they can be formed in ice mantles on interstellar grains~\citep{Herbst_vanDishoeck2009}. The suggested mechanisms leading to solid-state formation of COMs are:
\textit{i)} Hydrogenation (H addition;~\cite{Charnley1995ApJ});
\textit{ii)} Formation after H atom addition/abstraction on grains~\cite{Chuang2016MNRAS};
\textit{iii)} Radical-radical surface reactions (at T$>$30~K~\cite{Garrod_2008});
\textit{iv)} Cosmic-ray (CR) induced radical diffusion~\cite{Reboussin2014MNRAS};
\textit{v)} Impulsive spot heating on grains by CRs~\cite{Ivlev_2015ApJ},
The regions where the formation of COMs occurs generally have temperatures ranging between 10 and 100~K and typical densities of 10$^{4-5}$ cm$^{-3}$ (corresponding to pressures $\leq$~10$^{-12}$~mbar).
While modern technology allows us to easily reach very low temperatures, around 5~K or less if proper shielding is used, the technological limit for the lowest pressure is between 10$^{-11}$ and 10$^{-12}$ mbar, thus well above the "vacuum" conditions of the densest regions of the ISM. However, the operating conditions of ultra-high vacuum (UHV), defined as pressures below $10^{-9}$~mbar, are suitable for conducting experiments under conditions that simulate the solid-state synthesis of molecules in the ISM, because at these pressures the accretion time of typical pollutants (H$_{2}$, H$_{2}$O, CO) is much longer than the average duration of an experiment. At a pressure of 10$^{-10}$~mbar, it takes approximately two and a half hours for a monolayer of H$_{2}$ to form on the surface (assuming a surface density of sites $\sim 10^{15}$ cm$^{-2}$, a collision rate of $\sim 10^{11}$ particles cm$^{-2}$ s$^{-1}$~\cite{woodruffdelchar1994}. This allows time to perform experiments on a surface which is clean and well characterized, with only a few percent of a monolayer contamination per hour.
Therefore, to take into account the mechanisms of formation of COMs that do not involve energetic processes (i.e., mechanisms \textit{i}, \textit{ii}, and \textit{iii}) and to approximate the conditions of the ISM in the laboratory, we had to employ ultra-high vacuum technology and the use of cryogenics. With this in mind, a new setup was designed and constructed for the study of sub-monolayer systems in which two or more (up to four) species could simultaneously react on a cold substrate of astrophysical interest.
VENUS consists of a multi-beam stage and three custom-built stainless steel chambers sealed with Conflat{\tiny$^{^{\circledR}}$} (CF) flanges (Figure~\ref{fig:venus1}). The multi-beam stage is made up of five independent small chambers serving as gas sources, the output of which enters the intermediary HV chamber 1 through a 2 mm diaphragm. The intermediary HV chamber 1 (CH1) and UHV chamber 2 (CH2) are built out of a single stainless steel cylinder and are separated by a S-shaped interior wall. Chambers CH1 and CH1 serve as a compact two-stage differentially-pumped path for the beam lines. The third and final stage is a UHV chamber with a base pressure of 2 $\times$ 10$^{-10}$ mbar, that is the "reaction" chamber and we will refer to it as the \textit{main chamber} henceforth.
The main chamber contains a quadrupole mass spectrometer (QMS) and a gold-coated copper sample holder attached to the cold finger of closed-cycle He cryostat. The gold-coated target temperature can be varied between 7~and 400~K via a computer-controlled resistive heater.
Although the minimum temperature of the surface is 7~K, this is sufficient to model interstellar chemistry, as molecular clouds are estimated to have temperatures in the region of 10~K~\cite{Williams_Herbst2002SurSc}.
At present, VENUS has four operational triply differentially-pumped beam lines having fluxes of about $2 \times 10^{12}$ molecules~cm$^{-2}$~s$^{-1}$, which means that one monolayer (1 ML, corresponding to $10^{15}$ molecules~cm$^{-2}$) is grown on the gold-coated mirror in about 10 minutes. Deposited and newly formed species are monitored \textit{in situ} by means of Fourier transform reflection absorption infrared spectroscopy (FT-RAIRS) using vibrational fingerprint spectra in the mid-infrared (2.5~--~15~$\mu$m) domain.
The VENUS setup has been optimized for the study of very thin layers (sub-monolayers to one monolayer) of adsorbates, although systems of thick ices (pure and mixed) can be investigated as well.
The combination of all these characteristics allows VENUS to perform controlled deposition of very thin layers and to monitor the reaction kinetics in real time, making this experimental facility an ideal tool for studying sub-monolayer systems of several reactants and the formation of complex molecules in the solid phase.
\subsection{\label{subsec:mainch}Intermediary stages and the main chamber}
The apparatus consists of three vacuum chambers as shown in Figure~\ref{fig:chambers}. The first two chambers, CH1 and CH2, are intermediary chambers that serve as two independent stages of differential pumping for the beam lines. CH1 and CH2 are separated by an S-shaped stainless shield (3 mm in thickness) and each chamber is pumped by a turbo-molecular pump (Agilent Technologies, Turbo-V 551 Navigator) having nominal pumping speed of 550~l~s$^{-1}$.
CH1 has a base pressure of $10^{-8}$~mbar monitored by a full-range pirani/cold cathode gauge (Pfeiffer Vacuum, PKR251), and contains a Residual Gas Analizer that is used for the acquisition of mass spectra of the incoming species injected via the beam line ports at left. The intermediary chamber CH2 has a base pressure of $2 \times 10^{-10}$ mbar measured with a cold catode gauge (Pfeiffer Vacuum, IKR270).
\begin{figure}[b]
\includegraphics[scale=0.2]{VENUS_figure2.pdf}
\caption{Schematic top-view (a) and side-view (b) diagrams of the vacuum chambers showing the position of various experimental components. \label{fig:chambers}
}
\end{figure}
A pneumatic gate valve separates the intermediary stages from the UHV chamber via the frontal CF 100 flange. The main chamber, illustrated in Figure~\ref{fig:venus1} and~\ref{fig:chambers}, is a custom-made non-magnetic stainless steel cylinder 60 cm tall, with a diameter of 50 cm, built by {\footnotesize{\copyright}}Kurt J. Lesker Company. A turbo-molecular pump with pumping speed 1000 l s$^{-1}$ (Agilent Technologies, Turbo-V 1001 Navigator) maintains UHV conditions in the chamber, at a base pressure of a few $10^{-10}$ mbar, if an appropriate baking of the chamber is performed at $\sim$70~C$^\circ$ for about 3 days. The Turbo-V 1001 pump is backed via a primary vacuum pipe (<$10^{-3}$ mbar) that runs above the apparatus and is evacuated by a roots pump (Pfeiffer Vacuum, Okta 500), located in the basement, with a nominal pumping speed of 280 to 840~m$^{3}$~h$^{-1}$.
The main chamber is additionally pumped with a three-filament titanium sublimation pump (MECA 2000, PFT.3), which is switched on to achieve optimum UHV conditions whenever the hydrogen load is high.\\
A cold catode gauge (Pfeiffer Vacuum, IKR270) is used to monitor the main chamber pressure.
At the rear CF 100 port of the main chamber is mounted the cryocooler, shown in Figure~\ref{fig:cryocooler}. The sample holder is an OFHC (Oxygen-Free High Conductivity) copper cylinder (9 mm in diameter) mounted at the end of the cold finger of a helium-cooled cryostat (Advanced Research
Systems 4K DE-210SB cryo-cooler coupled to an ARS-10HW
compressor, with a typical cooling power of 0.4~W
at 10~K). The sample holder is mounted on
the third stage of the cryostat, and this whole region is thermally isolated
from the environment by a thermal shield. The protective thermal cover consisting of a nickel-coated copper
(internal) and stainless steel (external surface) cylinder surrounding the whole assemblage is attached to the first stage of the cryostat ($\sim$50~K).
\begin{figure}[b]
\includegraphics[scale=0.3]{VENUS_figure3.pdf}
\caption{Detailed schematic of the cryocooler and the target surface. \label{fig:cryocooler}
}
\end{figure}
Three different sensors are used to measure the temperatures in different parts of the cryocooler: a KP-type thermocouple (Au 0.07\%-Fe/Chromel); a K-type
thermocouple (Nickel-Chromium/Nickel-Aluminium); and a silicone diode (DT 470). The three sensors are connected on the thermal shield, on the second stage of the cryostat, and on the sample holder, respectively. A heating resistor too is attached to the back of the OFHC Cu sample holder. All temperature sensors and the resistor are connected to a Lakeshore 340 temperature controller.
The gold-plated target, that is the sample holder 9~mm circular base facing the beam lines, is optically flat, chemically inert, and ensures optimal mid-infrared light reflectance. The surface temperature is measured with the DT~470 silicon diode and is regulated by the Lakeshore 340 controller to $\pm$0.1 K
with an accuracy of $\pm$2~K in the 8 -- 400~K range, as determined by calibration carried out monitoring the multilayer desorption temperature of CO and N$_2$~\citep{Thanh2018AA}.
The Lakeshore 340 is controlled remotely by a homemade Labview program which allows us to perform the following procedures:
1) regulating heating ramps and final temperature during cooling or heating of the sample;
2) monitoring and recording the temperatures captured by the sensors during an experiment;
3) adjusting the heating power in real time to ensure a linear increase in surface temperature.
The QMS (Hiden 51/3F) is mounted in an upright position to a rotatable CF 40 port soldered off-axis to a differentially pumped rotary CF 100 flange, attached to the bottom port of the main chamber. This configuration allows the mass spectrometer to be rotated around its axis and also to move along a semicircular path around the target (see Fig.~\ref{fig:chambers}). Also, the QMS can be translated vertically, which allows us to align the QMS and the sample with any of the beam lines. The QMS is used for three main purposes: in its low position, to analyse
residual gas in the main chamber as a function of atomic or molecular mass (in the range 1 -- 300~amu); in its high position, facing any of the beam lines, to characterize the chemical nature of the beam as well as the beam intensity, and thus to check the accuracy of the beam alignment;
and again in its high position, facing the target, the mass spectrometer is used to measure the desorption of species from the surface during a TPD. The QMS is controlled via a software program provided by Hiden. Technical settings, such as the energy of the ionizing electrons and the dwell time for each mass, can be controlled via this interface. It also records both the masses being measured and the temperature of the surface, input from the Lakeshore 340 as an external signal on the auxiliary entry.
The target surface is located approximately in the center of the main chamber, at the point where both the four beam lines and the infrared beam converge. The whole cryocooler is mounted on a 3-axis translational stage, which allows translations and fine adjustments of the surface position both in the z (in and out of the main chamber) and the x, y (horizontally and vertically, respectively, relative to the QMS and beam lines) directions.
\subsubsection{\label{subsubsec:ices}Growth of water ice substrates that mimic interstellar icy mantles}
Amorphous solid water (ASW) is reputed to be the most
abundant form of water in the Universe, thanks to its
propensity for forming, molecule after molecule, as a deposit
on interstellar dust particles.
In the laboratory, water ice film are obtained by the condensation of water vapour onto the cold target. The properties of the ice depend upon the incident angle of water molecules, the temperature of the surface both during deposition and
any increase in temperature experienced after deposition~\cite{Kimmel2001JChPh}.
We constructed a water vapour delivery manifold for growing water ice onto the sample surface with control over the ice thickness and deposition rate. One UHV leak valve is mounted to one of the main chamber ports and is connected to a stainless steel gas-handling manifold ending with a 4-mm nozzle.
The nozzle outlet can be positioned at a "high" position to flood the main chamber with water vapour without exposing the sample to the direct flow of molecules emerging directly from the nozzle.
The ice films are grown by admitting water vapor into the main chamber so it freezes out on the gold target. This method, also known as "background deposition",
allows us to control the amount (and thickness) of the deposited ice film. The
water vapor is obtained from deionized water which has been
purified by several freeze-pump-thaw cycles, carried out under vacuum.
The rate at which molecules of water impinge upon the sample will be proportional to the partial pressure of water in the main chamber. By monitoring the partial pressure of the "flooding" species, ice thickness and deposition rate are computer controlled within the accuracy of the pressure gauge reading (i.e., 0.1 monolayers) using a custom-made LabView program.\\
Typically, water vapour is injected into the main chamber until a pressure of 1.2$\times$\(10^{-8}\) mbar is reached, and is kept constant during the deposition process until the desired ice thickness is attained. The purity of the water vapor, that is absence of air contamination, is monitored via the QMS.
The calibration of the water ice thickness was carried out via FT-RAIRS and QMS in a dedicated set of experiments where the amount of water ice deposited was measured as a function of time using the H$_2$O partial pressure in the chamber, since a given pressure corresponds to a measurable flux of molecules impinging on the walls of the chamber. Subsequently, the comparison between the number of deposited water layers (calculated by the LabView program) and the infrared/mass spectra allows definition of the one-monolayer standard expressed in IR-band area units or TPD peak area units, with an accuracy of $\pm 0.2$ monolayers.\\
Three major ice morphologies are used to simulate the icy mantles covering the grains of dust in dark clouds:
-\textit{porous ASW}: mimics a very disordered substrate with a high effective surface area, because of the presence of cracks and pores having themselves an internal surface area. To produce porous ASW, water vapor is dosed while the surface is held at a constant temperature between 10 and 40~K, depending on the required degree of surface disorder.
-\textit{non-porous ASW}: mimics the compact ASW ice which comprises the bulk of
interstellar ice.
To produce non-porous ASW, water is dosed while the surface is held at a constant temperature of 110~K.
-\textit{crystalline water ice}: mimics the crystalline ice seen in some
star-forming regions.
To form crystalline water films,
the surface is held at 120~K during the deposition, then flash heated at 50~K min-1 to 140~K, and finally at 10~K min-1 to 142.5~K. Complete crystallisation of the ice film is checked through verification of the infrared water band profile between 3400 and 3100~cm$^{-1}$, and thus cooled down to 130 K.\\
Once the deposition phase is complete, typically we wait between 30 and 60 minutes (depending on the thickness of the deposited ice) until the pressure in the main chamber has reached its pre-dosing value ($\sim 10^{-10}$~mbar), and only then the sample temperature is cooled further (typically to 10~K) to perform the subsequent experiments.\\
By lowering the water leak manifold to a "low" position, the nozzle is located 1~cm in front of the target, aims at the cold surface and the flow of water vapour can be directed at normal incidence onto the sample. Nevertheless, repeated testing of this configuration proved it was not suited for growing a uniform well characterized layer of water ice. It can however be used to quickly form a thick sublayer of compact ice ($>$100~ML) at a surface temperature of 110~K, on which the subsequent well characterized ice films are grown.
\subsection{\label{subsec:rairs}FT-IR spectrometer}
Composition, thickness, and morphology of the deposited ices and of the newly formed species may be monitored \textit{in situ} through absorption reflection spectroscopy (RAIRS) using a Bruker Vertex 70v spectrometer.
RAIRS was first applied many years
ago, when Pickering and Eckstrom (1959)~\cite{pic59} recorded spectra from carbon monoxide and
hydrogen adsorbed on metal films, using a multiple reflection arrangement in which the infrared beam was at near normal incident to the substrate.
Following the principles laid down by Greenler (1966)~\cite{gre66}, higher sensitivity can be obtained in a reflection experiment if grazing angles of incidence are employed~\cite{hollins94, trenary00, fraser02, bennett04}. FT-RAIRS turns out to be a valuable technique to conduct studies on molecules adsorbed on a surface, and provides information that cannot be inferred from TPD mass spectra. In fact, the combination of TPD and RAIRS provides a better understanding of the system than either technique alone. Besides, IR spectroscopy will allow us to investigate surface-formation processes concerning complex molecules, starting from the recently observed COMs such as methyl formate (HCOOCH$_3$) and dimethyl ether (CH$_3$OCH$_3$) to refractory carbon and nitrogen bearing molecules of astrophysical
and astrobiological importance, that cannot be investigated thoroughly through the TPD technique alone.
\begin{figure}[b]
\includegraphics[scale=0.41]{VENUS_figure4.pdf}
\caption{Detailed schematic of the FT-IR setup showing the infrared beam path and the position of various components located inside the pumped enclosures E1 and E2. \label{fig:rairs}
}
\end{figure}
The Vertex 70v Fourier transform infrared (FT-IR) spectrometer mounted on VENUS possesses airtight lids and windows so it can be kept under vacuum to avoid atmospheric H$_{2}$O and CO$_{2}$ spectral contamination. The FT-IR instrument is set up in a reflection-absorption infrared spectroscopy configuration and the spectra are recorded at a grazing incident angle of 83~$\pm 1^{\circ}$ (Figure~\ref{fig:rairs}). The parallel and unpolarized infrared beam (40~mm in diameter) exiting the FTIR is focused onto the sample using two gold-coated mirrors: a flat and a 90$^{\circ}$ off-axis parabolic (OAP) mirror (effective FL = 400~mm), mounted in a pumped enclosure (E1) adjacent to the main chamber and isolated from it with a ZnSe window. External to the UHV chamber, through an opposite ZnSe window, the infrared beam enters a second pumped enclosure (E2) after being reflected off the sample substrate coated in thin film of poly-crystalline gold that acts as a flat mirror. Inside E2, another 400~mm OAP mirror is used to collect the diverging infrared light coming from the sample, that is finally focused onto the sensor element of a liquid-N$_2$-cooled mercury cadmium telluride (MCT, spectral sensitivity 6000 -- 600~cm$^{-1}$, or 1.6 -- 16~$\mu$m) with a 43~mm OAP. Although copper and silver are also two highly reflective metal surfaces
used commonly for applications in the mid-IR spectral range, with a reflectance percentage comparable to that of gold, only gold-coated mirrors were used on VENUS because they are very broadband, from 0.7 to 10~$\mu$m they reflect more than 98\% of the incident radiation, and this percentage stays in this range up to 25 $\mu$m.
To couple the external enclosures housing the optics and the MCT detector to the main chamber, two custom-designed flanges are employed, whose schematic is shown in Fig~\ref{fig:RAIRSflanges}. Each coupling piece, manufactured by Bruker, is made of two flanges soldered to a stainless steel tube that forms an angle of 7$^{\circ}$ with the flange normal. Such a design allows the infrared beam to enter the main chamber and be incident upon the sample with the desired grazing angle of 83$^{\circ}$. The spectrometer and the two external optical enclosures are pumped with a dry scroll pump (Edwards, model nXDS15i) that allows the pressure to be in the low 10$^{-2}$~mbar.
\begin{figure}[b]
\includegraphics[scale=0.6]{VENUS_figure5.pdf}
\caption{Schematic (a) and 3-D view (b) of the custom-designed double flange used to couple the FT-IR setup to the main chamber. \label{fig:RAIRSflanges}
}
\end{figure}
FT-IR spectra between 4000 and 650 cm$^{-1}$ (2.5 --~15 $\mu$m) may be acquired prior, during, or subsequently to either deposition or TPD experiment, with a typical resolution of 4 cm$^{-1}$. The number of averaged scans per saved infrared spectrum is typically 256, corresponding to an acquisition time of about 2 minutes. In case a shorter acquisition time is needed, such as real time monitoring of reaction kinetics during a TPD, the Vertex 70v spectrometer allows us to use a rapid scan mode at a reduced resolution of 8~and 16~cm$^{-1}$.
The FT-IR spectrometer is controlled via OPUS{\scriptsize$\mathrm{^{^{^{TM}}}}$}, a proprietary software program provided by Bruker. The spectral acquisition settings, such as the resolution, diaphragm aperture, number of scans, spectral range, performance tests, pumping and venting of the spectrometer, etc., can be controlled via this interface. The OPUS \textit{Measurement dialog} also allows us to start recording the background measurement (i.e., spectrum recorded without a sample being on the surface) and the sample spectrum. The result sample spectrum is automatically calculated by dividing the sample spectrum by the background measurement. The software is
used either to record a single spectrum or to start a repeated data collection, that is a series of automatic acquisitions recorded sequentially during a deposition, or during H atom bombardment of the sample, or during a TPD. During the experiments, the standard ceramic mid-infrared (MIR) source and a KBr beamsplitter (spectral range 4800 to 450~cm$^{-1}$) are permanently mounted on the spectrometer. However, a near-infrared (NIR) source and a quartz beamsplitter (15000 to 3300~cm$^{-1}$) are used whenever the system needs alignment of the optics (mirrors, target, and MCT sensor).
\subsection{\label{subsec:beams}Molecular and atomic beam lines}
There are currently four operational beam lines mounted on VENUS, although a fifth source could be implemented if necessary, according to future research programs.
The possibility of using four molecular/atomic beam lines is undoubtedly one of the most important properties of this apparatus, and a unique feature among astrochemistry laboratories throughout the world.
Each beam line is composed of three stages of differentially pumped chambers separated by diaphragms (whose positions are illustrated in Fig.~\ref{fig:venus1}), and was designed to deliver molecular and atomic species onto the sample at a rate low enough to maintain a pressure $< 10^{-9}$~mbar in the main chamber at all times.
The four first stages (or stage 1) of the beam lines compose the multi-beam stage. A frontal view of the apparatus, presented in Figure~\ref{fig:beamports}, shows the five CF 40 ports to which the beam line stage~1 chambers are attached. As mentioned above, only four ports are currently used, that is the central port (central beam, CB), the top port (top beam, TB), the right port (right beam, RB), and the bottom port (bottom beam, BB). The top, right, and bottom ports are soldered as to form an angle of 5$^\circ$ with the central port (normal to the upright axis) so that all the beam lines converge on an ideal point lying onto the central axis of the main chamber.
\begin{figure}[b]
\includegraphics[scale=0.8]{VENUS_figure6.pdf}
\caption{Frontal view of the apparatus showing the five ports to which the first stages the beam lines are attached. \label{fig:beamports}
}
\end{figure}
Stage~1 of each beam line is composed of a quartz tube (the source, 4~mm inner diameter), a vacuum chamber evacuated by a turbo molecular pump (Leybold Vacuum, Turbovac SL80, pumping speed 65~l s$^{-1}$), and a skimmer terminating with a diaphragm of 2 mm in diameter. The pressure of the beam in the quartz tube is typically $\sim$1~mbar, diminishing gradually to pressures of $10^{-4}$ -- $10^{-5}$~mbar within the chamber of the first stage to a value of $10^{-10}$~mbar in the main chamber. Collimators (diaphragms) of 3~mm diameter separate CH1 from CH2, and CH2 from the main chamber, so the beams are angularly resolved to a very small solid angle of $\sim 10^{-5}$ steradian.
The right beam is also equipped with an air-cooled microwave cavity that surrounds the quartz tube (see Figure~\ref{fig:RBstage1}), and with a simple flag shutter. This flag shutter enables us to modulate the beam (let it through or block it) very quickly (within 1 second).
A valve separating each of the four first stages from CH1 can be closed to prevent the penetration of the beam into the main chamber.
\begin{figure}[b]
\includegraphics[scale=0.71]{VENUS_figure7.pdf}
\caption{Schematic of stage 1 of the right beam (RB). Stage 1 of the other beam lines is essentially a replica of the RB first stage without the Surfatron and the flag shutter. \label{fig:RBstage1}
}
\end{figure}
The flux of gaseous species entering TB, CB and BB sources is controlled by an automated regulator (Bronkhorst High-Tech control valve 1~sccm~\footnote{Standard cubic centimetre per minute: 1 sccm = 592\,m$^3$ Pa s$^{-1}$ in SI units.}) set at 10\% of maximal gas flow rate, while H$_{2}$ injected into the RB source is controlled with a 10~sccm regulator typically set at 15\% of its maximal flow rate.\\
The great advantage of the Bronkhorst flux controller over a manual needle valve is that a given molecular flow is accurately reproducible from one experiment to another, even for experiments conducted on different days.
Gas is introduced into the first vacuum chamber through the 4~mm inner diameter tube, producing an effusive beam, and passes through three chambers (stage 1, CH1, CH2) and into the main chamber via three diaphragms. It should be noted that an aligned beam line requires the source (the 25~cm long quartz tube), the three diaphragms, and the target to be perfectly aligned, which is checked beforehand by a laser beam to ensure that all beam lines are aimed at the same spot on the sample. In addition to maximizing the beam flux intensity,
ideally it would be preferable that all the beams be perfectly overlapped on the sample, for a 100\% overlap allows maximum physical and chemical interaction between two or more different species deposited through the beam lines.
With the Bronkhorst control valve by-passed and replaced by a standard leak valve, the central beam is used to inject water vapor or any other substance contained in a glass vial that is liquid at room temperature and has a vapor pressure that can vary from a few mbars (i.e., H$_2$O, CH$_3$OH, HCOOH) to about 1 atm (i.e., CH$_3$CHO). The beams can be molecular or, if the microwave discharge is activated, the RB will be a mixture of atomic and molecular species. Due to recombination of atomic species on surfaces of the beam source, the dissociation efficiency, $\tau$, is never 100\%. $\tau$ is calculated via the formula:
\begin{equation}
\tau = \frac{S_{OFF} - S_{ON}}{S_{OFF}}
\end{equation}
where S is the number of ion counts of the parent molecular species measured with the QMS in front of the target when the discharge is switched off or on. The value of $\tau$ is calculated before every experiment involving atoms. Typically measured dissociation fractions are between 60~and 80\% for H$_{2}$ and D$_{2}$, $\sim$~50\% for O$_{2}$~\cite{Minissale2019AA}, and only $<$5\% for N$_{2}$, due to the greater strength of the nitrogen-nitrogen triple bond.
Air cooling proves to be essential, not only to reduce the discharge temperature, but also to inhibit recombination of hydrogen atoms on the walls of the discharge tube~\cite{wood62}. Microwave-frequency power from a SAIREM's GMS 200\,W 2450\,MHz microwave generator is transmitted to the gas via a Surfatron 2450\,MHz (SAIREM), generating a plasma upon application of a spark. Impedance matching of the microwave cavity is performed using the adjustment screws on the Surfatron to ensure the maximum transmission of forward power (typically, 60\,W are used to dissociate H$_2$/D$_2$) and to reduce the reflected power.
Molecules and atoms delivered from the beam line to the main chamber are at ambient temperature of $\sim$300~K, even if produced in a discharge, due to the transfer of kinetic energy by collisions in the source. Besides, in an effusive source, the molecules in the beam have temperatures close to that of the tube..
Also, we previously determined that atoms are in their ground state~\citep{Amiaud2007, Congiu2009}.
\subsection{\label{subsec:calibration}Calibration procedures}
\subsubsection{\label{subsubsec:beamflux}Beam flux}
One of the first tests we performed was to check the relative intensity of the four beams using carbon dioxide, a highly active mid-infrared species.
Figure~\ref{fig:beams1} shows the intensity of the CO$_{2}$ beam flux expressed in ion counts s$^{-1}$ as a function of source gas supply in sccm, measured with the QMS facing each beam at a time.
The right and central beam have a comparable slope, namely these two beam lines have very similar flux intensities. The saturation value, that is the threshold value where proportionality between the source pressure and the beam line flux is lost, is reached at a source feeding rate of 0.5~sccm for the central beam, and 0.6~sccm for the right beam. The top beam appears to be less intense, which is likely due to a lesser precise alignment between the quartz tube and the diaphragms. The onset of the saturation region is generated by accumulation of molecules within the first stage chambers (see Figure~\ref{fig:RBstage1}), which makes the resulting mean free path of effusing molecules comparable or even shorter than the dimensions of the chamber itself. We thus determined that source feed rates $<0.5$~sccm must be used to avoid flooding of the first stage and blocking the beam path.
In Figure~\ref{fig:beams2} we present the CO$_{2}$ IR band area as a function of deposition time obtained with the four beam lines. Each source was fed using a CO$ _{2}$ flux of 0.2~sccm. By comparing the QMS data shown in Figure~\ref{fig:beams1} and the IR data in Figure~\ref{fig:beams2} we can see that they are consistent. This indicates that the region of the cold target illuminated by IR light is exposed to the four beams, and the difference in IR band area only depends on flux intensity. The amount of deposited molecules grows linearly with time, the right and central beams are the most intense and their fluxes are comparable. The TB appears to have about one third of the flux intensity with respect to CB and RB, while the bottom beam has a very low flux, which is almost certainly due to a poor alignment. For this reason, at the time writing this paper, the bottom beam is so weak that cannot be used for experimental purposes, in that depositions would imply too long exposure times not compatible with a typical experimental sequence. From the integrated band areas of Figure~\ref{fig:beams2} at 5~min exposure, the relative intensities of CB, RB, TB, and BB are 1:1:0.33:0.1, respectively.
\begin{figure}
\scalebox{1}{{
\includegraphics[scale=0.34]{VENUS_figure8.pdf}}
\caption{\label{fig:beams1} Beam intensity as a function of CO$_{2}$ source feed rate in sccm for three of the operational beam sources.}
\end{figure}
\begin{figure}
\scalebox{1}{{
\includegraphics[scale=0.31]{VENUS_figure9.pdf}}
\caption{\label{fig:beams2} Infrared band area as a function of CO$_{2}$ deposition time at a source feed rate of 0.2 sccm measured for the four beam lines.}
\end{figure}
Given a fixed source pressure, absolute beam flux calibration of the molecular beams can be performed by a series of TPD experiments using a fully wetting molecule, such as N$_{2}$, CO, or O$_{2}$, with sticking probability at low temperatures close to unity. Fully wetting species have molecule--substrate bonds stronger than molecule--molecule interactions. Hence, these species exhibit only (sub)monolayer desorption characteristics, and have no multilayer component for TPD spectra corresponding to 1~ML or less. \\
TPD mass spectra \textit{vs} surface temperature used for the flux calibration of the central beam are illustrated in Figure~\ref{fig:13COtpd}. The surface is cooled to 10~K and a timed deposition of $^{13}$CO is performed. The surface is heated and the number of molecules measured with the QMS as they desorb from the surface. The experiment is repeated and the dose is increased until the leading edge keeps shifting to the left, meaning that the surface is being filled with least binding sites being occupied last. From the $^{13}$CO TPD spectra, we notice that the desorption profile of CO stops shifting to the left after a dose of six minutes. From the inset in Fig.~\ref{fig:13COtpd}, showing the logarithm of the desorption rate vs surface temperature, it is clear that TPD profiles for doses $>6$~min increase exponentially from the same multilayer temperature (25~K), and a multilayer peak begins to grow on top of the full monolayer trace corresponding to 6 minutes. Therefore, $\sim$1~ML CO is deposited in 6~min, that is 360 seconds, which gives a CB flux:
$\phi_{CB}=10^{15} \, \mathrm{molecules \, cm}^{-2} / 360 \, s \simeq 3 \times 10^{12} \, \mathrm{molecules \, cm}^{-2} \, s^{-1}$.\\
Because it mesures the amount of species deposited on the target, this method has the advantage of measuring the effective flux on the surface, rather than simply the flux of the beam.
From the values of relatives fluxes we can thus estimate the flux of the other beams:\\
$\phi_{RB} \simeq 3 \times 10^{12}\, \mathrm{molecules \, cm}^{-2} \, s^{-1}$,\\
$\phi_{TB} \simeq 1 \times 10^{12} \, \mathrm{molecules \, cm}^{-2} \, s^{-1}$,\\
$\phi_{BB} \approx 3 \times 10^{11} \, \mathrm{molecules \, cm}^{-2} \, s^{-1}$,\\
with a conservative error of 20\% on $\phi_{CB}$, $\phi_{RB}$, and $\phi_{TB}$, due to the uncertainty associated to the TPD spectra and the integrated infrared band areas.
\begin{figure}[b]
\includegraphics[scale=0.38]{VENUS_figure10.pdf}
\caption{$^{13}$CO TPD series as function of surface temperature used for calibration of the central beam. \textit{Inset:} Logarithm of the desorption rate \textit{vs} temperature showing the exponential behavior of TPD traces corresponding to doses $>$1~ML. \label{fig:13COtpd}
}
\end{figure}
\subsubsection{\label{subsubsec:rairsconstant}Infrared absorption bands and column densities}
In general, for thin layers $\le 20$~ML, the column density $N$ (molecules cm$^{-2}$ ) of deposited or newly formed species can be calculated via a modified Lambert-Beer equation~\citep{bennett04,TEOLIS07}:
\begin{equation}
N = \Lambda \, \frac{\int_{\lambda_1}^{\lambda_2} A(\lambda) \mathrm{d}\lambda}{\mathrm{S}} \label{LambertBeer}
\end{equation}
where $\int A(\lambda) \, \mathrm{d}\lambda$ is the integrated area of the infrared absorption feature (cm$^{-1}$), and S is the corresponding band strength. The constant $\Lambda$ is ${(\ln10 \,\,\, \mathrm{cos} \alpha)}/{2}$,
where the division by 2 corrects for the ingoing and outgoing infrared beam, and $\alpha$ accounts for the angle between the normal of the surface mirror and the infrared beam. Therefore, the determination of the column density of the sample is apparatus dependent and literature values of band strengths ought to be used with care in RAIRS measurements.
The setup dependent value of $\Lambda$ may be measured using two methods:
Method 1): computation of the theoretical value of $\Lambda$ and use of eqn.~\ref{LambertBeer} to derive $N$.
Method 2): a series of TPD experiments is performed to accurately determine the 1~ML dose of a given infrared active molecule. Subsequently, $N$ being known, the integrated absorbance of the infrared band can be measured and the value of $\Lambda$ for a given setup can be calculated from eqn.~\ref{LambertBeer}.
\\
As for the FT-RAIRS setup mounted on VENUS, with a grazing angle of 83$^\circ \pm 1^\circ$, the theoretical value is:
$\Lambda_{theo, VENUS} = 0.14 \pm 0.02$, \\
where the error represents the uncertainty in the angle.
An experimental derivation of $\Lambda$ was done using the $^{13}$CO TPD spectra shown in Section~\ref{subsubsec:beamflux}.
For a coverage of 1~ML (10$^{15}$ molecules~cm$^{-2} \, \pm$ 20\%), the integrated $^{13}$CO band at 2092~cm$^{-1}$ is $(0.006 \pm 0.001)$ cm$^{-1}$; S($^{13}$CO) is $1.3 \times 10^{-17}$ cm~molecules$^{-1}$~\cite{Gerakines1995}; $N$ must be corrected for the actual area of the spot on the sample. The circular spot has a $\sim$3 mm (0.3 cm) diameter, or 0.15~cm radius, which gives an area of $\pi r^2 = (0.07 \pm 0.01)$~cm$^{2}$. Hence, $N$ is 10$^{15}$ molecules cm$^{-2} \times 0.07 = 7 \times 10^{13}$~molecules~cm$^{-2}$. Therefore, by rearranging eqn.~(\ref{LambertBeer}), we obtain:
$
\Lambda_{exp, VENUS} = \frac{N \cdot \mathrm{S}}{\int A(\lambda) \mathrm{d}\lambda} = \frac{7 \times 10^{13} \mathrm{molecules} \, \mathrm{cm}^{-2} \, \cdot \, 1.3 \times 10^{-17} \mathrm{cm \, molecules}^{-1}}{0.006 \, \mathrm{cm}^{-1} } = 0.15 \pm 0.07.
$
We can notice that the two calculated values of $\Lambda$ are consistent, although the theoretical value $\Lambda_{theo, VENUS}$ is affected by a smaller error. Hence, this value seems to be the most reliable for the quantitative measurements of column densities.
In the end, taking into account the value $\Lambda_{theo, VENUS}$ and the scaling factor for the beam size, the Lambert-Beer equation corrected for the calculation of the column density of species X with VENUS is:
\begin{equation}
N_{X} = \frac{0.14}{0.07} \, \frac{\int A(\lambda) \mathrm{d}\lambda}{\mathrm{S_{X}}} = 2 \, \frac{\int A(\lambda) \mathrm{d}\lambda}{\mathrm{S_{X}}}. \label{LambertBeerMOD}
\end{equation}
Considering all sources of error, a 50\% uncertainty on calculated column densities is likely to be conservative enough.
\section{Experimental Performance} \label{sec:performance}
\subsection{Simple reactive systems as probes of beam spot overlap: the NH$_{3}$ + H$_{2}$CO reaction}
In this section we present the first experimental results obtained with the new multi-source apparatus called VENUS. The first representative results concern the thermal reactivity of H$_{2}$CO with NH$_{3}$, which had been previously investigated experimentally and theoretically~\cite{Schutte1993Icar,Bossa2009ApJ,Rimola2010PCCP}. This reaction leads to the formation of aminomethanol, NH$_{2}$CH$_{2}$OH, which occurs via a low barrier nucleophilic addition\footnote{The proposed general mechanism is a nucleophilic attack of the C atom of H$_{2}$CO by the lone pair of the N atom of NH$_{3}$, followed by a proton transfer from the N atom to the O atom~\cite{Courmier2005}.} between ammonia and formaldehyde:
\begin{equation}
\mathrm{NH}_{3} + \mathrm{H}_{2}\mathrm{CO} \longrightarrow \mathrm{NH}_{2}\mathrm{CH}_{2}\mathrm{OH}.
\label{react:aminomethanol}
\end{equation}
In this first experiment we show how we can use the simple chemical system above \mbox{(A+B $\to$ AB)} to quantitatively assess the actual overlap of the beams on the target. NH$_{3}$ was deposited on the surface through the top beam. Formaldehyde is solid at ambient temperature so gaseous H$_{2}$CO was obtained by gently warming (at $\simeq 62^{\circ}$C) paraformaldehyde powder kept in a glass vial~\cite{Minissale2015}. The flux of H$_{2}$CO$_\mathrm{(gas)}$ is adjusted with a precision leak valve and injected into the central beam to be deposited onto the gold surface. Formaldehyde is characterized by two strong bands located at 1730 and 1500~cm$^{-1}$ assigned to the C=O stretching mode and the CH bending mode, respectively.
For each molecule, the characteristic bands are integrated, and then inserted into equation~(\ref{LambertBeerMOD}) with their corresponding band strengths to estimate their column densities.
For NH$_{3}$, we use the value of $1.1 \times 10^{-17}$ cm molecule$^{-1}$ for the band strength of the stretching mode at 3380~cm$^{-1}$~\cite{dHendecourt1986AAS}. For H$_{2}$CO we use the CO stretching mode band at 1730~cm$^{-1}$, which has a band strength of $9.6 \times 10^{-18}$ cm~molecule$^{-1}$~\cite{Schutte1993Icar}.
In Figure~\ref{fig:h2co_ir_tpd}, we show a series of FT-IR spectra (top image) of various timed depositions of formaldehyde at 65~K on the gold-plated target, and the subsequent TPD traces (bottom image) recorded during the linear warmup at 12~K~min$^{-1}$. Both images contain an inset showing the H$_{2}$CO 1730~cm$^{-1}$ band area and the H$_{2}$CO TPD peak area, respectively, as a function of deposition time. We can see that both trends are linear, which indicates that \textbf{a)}~the amount of molecules actually deposited is proportional to the dosing time and \textbf{b)}~qualitatively and quantitatively the two analytical tools implemented on VENUS are consistent (see Fig.\ref{fig:QMS-RAIRS}).
\begin{figure}[b]
\includegraphics[scale=0.35]{VENUS_figure11.pdf}
\caption{FT-IR spectra (top) of various timed depositions of formaldehyde at 65~K on the gold-plated target, and the subsequent TPD traces (bottom) recorded during the linear warmup at 12~K~min$^{-1}$. \textit{Insets}: H$_{2}$CO 1730~cm$^{-1}$ band area and the H$_{2}$CO TPD peak area, respectively, as a function of deposition time. \label{fig:h2co_ir_tpd}
}
\end{figure}
\begin{figure}[b]
\includegraphics[scale=0.38]{VENUS_figure12.pdf}
\caption{Correlation between TPD peak areas and 1730~cm$^{-1}$ band areas of H$_{2}$CO. Each data point corresponds to a given deposition time as per experiments presented in Fig.~\ref{fig:h2co_ir_tpd}. \textit{Inset}: details of the linear fit plotted in red. \label{fig:QMS-RAIRS}
}
\end{figure}
Figure~\ref{fig:IRNH3-H2C0} shows the integrated IR band area of NH$_{3}$ or H$_{2}$CO pure ices for various deposition times, and of various doses of co-deposited NH$_{3}$ and H$_{2}$CO at a surface temperature of 65~K. We can use the integrated band area of the pure ices to estimate the deposited column densities via TB (NH$_{3}$) and CB (H$_{2}$CO). For a dose of 10~min, A$_{NH_{3}}$ is 0.00205~cm$^{-1}$ and A$_{H_{2}CO}$ is 0.00136 cm$^{-1}$ (see Table~\ref{tab:IRNH3-H2C0}). Putting these values in equation~(\ref{LambertBeerMOD}), we obtain N$_{NH_{3}}$ = $3.7 \times 10^{14}$ mol~cm$^{-2}$ and N$_{H_{2}CO}$ = $2.8 \times 10^{14}$ mol~cm$^{-2}$, which gives a relative flux $\Phi_{NH_{3}}$\,:\,$\Phi_{H_{2}CO}$\,$\simeq$\,1\,:\,0.76, that is co-depositions will result in NH$_{3}$\,:\,H$_{2}$CO mixtures with an excess of ammonia.
\begin{figure}[b]
\includegraphics[scale=0.4]{VENUS_figure13.pdf}
\caption{Integrated infrared band area of NH$_{3}$ and H$_{2}$CO in pure ices experiments (filled squares and circles), and in co-depositions of NH$_{3}$ + H$_{2}$CO (empty square and circles) at a surface temperature of 65~K, as a function of deposition time.
\label{fig:IRNH3-H2C0}
}
\end{figure}
\begin{table*}
\caption{\label{tab:IRNH3-H2C0}Integrated IR band areas of NH$_{3}$ and H$_{2}$CO obtained after timed depositions of either pure species or codepositions at a surface temperature of 65~K. The error on all band areas is $\pm$1.5e-4~cm$^{-1}$.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
&\multicolumn{2}{c}{Pure species}&\multicolumn{2}{c}{NH$_{3}$\,+\,H$_{2}$CO experiment}\\
Dose (min)&A$_{NH_{3}}$\,(cm$^{-1}$)&A$_{H_{2}CO}$\,(cm$^{-1}$)&A$_{NH_{3}}$\,(cm$^{-1}$)&A$_{H_{2}CO}$\,(cm$^{-1}$)\\ \hline
2&2.95e-4&2.96e-4&9.83e-5&7.40e-5 \\
4&7.78e-4&5.63e-4&2.59e-4&1.38e-4\\
6&1.23e-3&8.00e-4&4.10e-4&2.05e-4\\
8&1.67e-3&1.09e-3&5.56e-4&2.73e-4\\
10&2.05e-3&1.36e-3&6.83e4&3.25e-4\\
\end{tabular}
\end{ruledtabular}
\end{table*}
From Fig.~\ref{fig:IRNH3-H2C0}, it is also clear that smaller amounts of NH$_{3}$ or H$_{2}$CO are measured on the surface when the two species are co-deposited, which indicates that a reaction takes place and both ammonia and formaldehyde are consumed.
Schutte, Allamandola, and Sandford (1993)~\cite{Schutte1993Icar} have shown that NH$_{2}$CH$_{2}$OH is the dominant product when NH$_{3}$ is in excess with respect to H$_{2}$CO, and we will use this argument to make the following hypotheses:
\textit{i)} working with an excess of NH$_{3}$ impedes formaldehyde polymerisation;
\textit{ii)} formaldehyde is completely consumed if the two beams overlap.\\
A quantitative estimate of the amount of NH$_{3}$ and H$_{2}$CO consumed during the co-deposition experiment, with respect to the pure ice case, is illustrated in Figure~\ref{fig:H2COcomp}.
\begin{figure}[b]
\includegraphics[scale=0.4]{VENUS_figure14.pdf}
\caption{\textit{Top panel:} integrated infrared band area of NH$_{3}$ for deposition of NH$_{3}$ only (filled squares) and co-deposition NH$_{3}$ + H$_{2}$CO (empty squares), as a function of deposition time. \textit{Lower panel:} integrated infrared band area of H$_{2}$CO for deposition of H$_{2}$CO only (filles circles) and co-deposition NH$_{3}$ + H$_{2}$CO (empty circles), as a function of deposition time. Solid and dashed lines represent linear fits of the data. \label{fig:H2COcomp}
}
\end{figure}
$\Delta$A is the difference between the integrated band area of the pure ice and that measured after the NH$_{3}$ + H$_{2}$CO experiment, for a deposition time of 10 minutes. From the values of $\Delta$A$_{NH_{3}}$ and $\Delta$A$_{H_{2}CO}$, using eqn.~(\ref{LambertBeerMOD}), it is straightforward to calculate the amounts of adsobates consumed in the reaction, $\Delta$N(NH$_{3}$) and $\Delta$N(H$_{2}$CO), respectively:
\begin{equation}
\Delta N(\mathrm{NH}_{3}) = 2 \, \frac{\Delta \mathrm{A}(\mathrm{NH}_{3})_{10'}}{S_{\mathrm{NH}{3}}} = (2.5 \pm 0.3) \times 10^{-14} \, \mathrm{molecules \,\, cm}^{-2},
\end{equation}
and
\begin{equation}
\Delta N(\mathrm{H}_{2}\mathrm{CO}) = 2 \, \frac{\Delta \mathrm{A}(\mathrm{H}_{2}\mathrm{CO})_{10'}}{S_{\mathrm{H}{2}CO}} = (2.2 \pm 0.5) \times 10^{-14} \, \mathrm{molecules \,\, cm}^{-2}.
\end{equation}
Since $\Delta$N(NH$_{3}$) $\simeq$ $\Delta$N(H$_{2}$CO) within experimental errors, ammonia and formaldehyde are consumed at the same rate, which confirms that reaction~(\ref{react:aminomethanol}) is at play. Hence, in experiments with an excess of NH$_{3}$ where H$_{2}$CO would react and be completely consumed, we can derive the maximum percentage of overlap of the two beams by calculating the ratio between $\Delta$A$_{H_{2}CO}$ and A$_{H_{2}CO}$. For the 10-min experiment, we found
\begin{equation}
\frac{\Delta\mathrm{A}(\mathrm{H}_{2}\mathrm{CO})_{10'}} {\mathrm{A}(\mathrm{H}_{2}\mathrm{CO})_{10'}}
= \frac{2.2 \times 10^{-14}}{2.8 \times 10^{-14}} = 0.78 \simeq 78\%.
\end{equation}
Extending this analysis to all the experiments done at different doses, we can plot a more general trend and thus reduce the error bar on the estimate of the overlap of beam spots TB and CB. This trend is presented in Figure~\ref{fig:overlap}. We can observe that the fraction of deposited H$_{2}$CO that reacts on the surface, hence the percentage of overlap of the two beams, is $\approx$~75~\%. On the other hand, as one would expect in the case of experiments carried out in excess of ammonia, the fraction of NH$_{3}$ molecules involved in the reaction is smaller, that is $\approx$~67~\%. By scaling the two fractions of consumed molecules of NH$_{3}$ and H$_{2}$CO with the corresponding band strength, it is easy to check that $\Delta$N(NH$_{3}$)$_\mathrm{react}$/$\Delta$N(H$_{2}$CO)$_\mathrm{react} \sim 1$.
\begin{figure}[b]
\includegraphics[scale=0.37]{VENUS_figure15.pdf}
\caption{Fraction of consumed adsorbates ($\Delta$A$_{NH_{3}}$/A$_{NH_{3}}$, blue squares; $\Delta$A$_{H_{3}CO}$/A$_{H_{3}CO}$, red circles) as a function of co-deposition time in the NH$_{3}$ + H$_{2}$CO experiment at 65~K. $\Delta$A$_{H_{3}CO}$/A$_{H_{3}CO} \approx 75\%$ also represents the degree of overlap of the two beam spots. Solid lines are linear fits of the data. \label{fig:overlap}
}
\end{figure}
\subsection{Chemical layer: the NO + H reaction system}
The first original scientific results obtained using VENUS concern the NO~+~H reaction, with the objective to investigate whether H atoms can penetrate into NO ices beyond the first outer monolayer.
In surface reactions and solid-state chemistry, nitric oxide, NO, is thought to be the main precursor species of the NO-bonded molecules~\cite{Halfen_2001} and, together with NH$_{3}$, the precursor molecule of nitrogen-containing organics~\cite{Tsegaw2017,Blagojevic2003MNRAS}. Recently, millimeter-band observations of a molecular cloud uncovered the presence of an essential RNA precursor in space, hydroxylamine (NH$_{2}$OH), whose formation mechanism is predicted to be by chemical models and by laboratory experiments the hydrogenation of NO on dust grain surfaces~\cite{Rivilla2020}.
Figure~\ref{fig:1NO_Hreactivity} shows the integrated area of the NO dimer band at 1775~cm$^{-1}$ as a function of deposition time. The black solid line represents the band area during the deposition of 1~ML (6 min dose) of NO at 10~K on the bare gold substrate, while the subsequent exponential decrease starting at t = 10~min shows the evolution of the NO band area during H bombardment. This experiment confirms that the NO molecule reacts rapidly with H atoms~\cite{Congiu2012ApJ,Congiu2012JChPh} and also shows that the FT-IR spectrometer can probe very thin layers of adsorbates, with a sensitivity of at least 0.2 monolayers.
\begin{figure}[b]
\includegraphics[scale=0.49]{VENUS_figure16.pdf}
\caption{Integrated IR band area of NO (black squares) measured during the deposition phase (black line, 6~min$\simeq$1~ML), and during the H atom exposure time (60 min) fitted with exponential function (blue trace). \label{fig:1NO_Hreactivity}
}
\end{figure}
In Figure~\ref{fig:2NO_Hreactivity} we present the consumption of a NO ice substrate of various initial thickness. A series of depositions (1.2, 2.4, 3.6, 4.8~ML) was performed and each NO ice sample was exposed to H atoms. The initial integrated band area decreases until a plateau is reached. For each of the four experiments, the decrease is well fitted with an exponential function, namely \mbox{Y = Y$_{0}$ + A$_{0}$\,exp(-t/$\tau$)},
whose corresponding reaction rates expressed in minutes/ML are reported in the inset of Fig.~\ref{fig:2NO_Hreactivity}. The correlation between the reactivity and thickness indicates that the thinner the ice the faster the consumption of NO molecules.
\begin{figure}[t]
\includegraphics[scale=0.49]{VENUS_figure17.pdf}
\caption{Evolution of the infrared band area of NO with H atom bombardment for various initial doses of NO. \textit{Inset}: reaction rate indicating the consumption of NO as a function of thickness. \label{fig:2NO_Hreactivity}
}
\end{figure}
\begin{figure}[b]
\includegraphics[scale=0.49]{VENUS_figure18.pdf}
\caption{Comparison between deposited NO (A$_{0}$=A$_\mathrm{R}$~+~A$_1$, black squares) and the amount of NO consumed during H bombardment (A$_1$, red circles) as a function of ice thickness. The green horizontal line, representing the asymptotic limit of the exponential fit of A$_1$, intersects the black line at a thickness value of $\sim$1~ML coverage, which therefore corresponds to the chemical layer (CL).
\label{fig:3NO_Hreactivity}
}
\end{figure}
Also, by subtracting the area reached at t~=~$\infty$ (A$_\mathrm{R}$) from the initial band area, we can calculate the amount of NO that has reacted with H (A$_1$). This result is shown in Figure~\ref{fig:3NO_Hreactivity}. Black squares represent the typical infrared band area (A$_\mathrm{R}$~+~A$_1$) as a function of NO ice initial thickness. Red circles show the amount A$_1$, that is the fraction of NO molecules that have reacted with H atoms. We can see that A$_1$ never surpasses 1~ML thickness although it tends to it asymptotically, as shown by the horizontal green line that indicates the~0.008~cm$^{-1}$ value corresponding to $\sim$1 ML coverage.
These results clearly show that only the outer layer of a slab of NO ice reacts with H atoms coming from the gas phase, meaning that H atoms cannot penetrate beyond the first outer layer of NO. In other terms, the first physical molecular layer of ice coincides with the only accessible chemical layer that is consumed by the reaction NO~+~H to form NH$_{2}$OH, N$_{2}$O, H$_{2}$O~\cite{Thanh2019ECS}. At 10~K these new products are likely to form a barrier shielding the underlying layer from H atoms that cannot penetrate deeper. Moreover, the NO~+~H reaction proceeds faster in the case of thin layers of NO ice, which is likely to be the same conditions found on cosmic dust grains as corroborated by the recent detection of NH$_{2}$OH for the first time, toward a quiescent molecular cloud located in the Galactic Center~\cite{Rivilla2020}.
We obtained similar results with pure ices of formaldehyde exposed to H atoms.
This finding suggests that chemistry at the surface of dust grains only occurs in the first layer of a substrate of NO ice under our experimental conditions, although such results may be more general and apply to other chemical systems of astrophysical interest.
\newpage
\section{Astrophysical potential}\label{sec:potential}
VENUS, the apparatus described in this paper, will be used to study radical-radical reactions and investigate the reaction-diffusion processes that lead to the formation of complex organic molecules on icy grains, that is the formation of the building blocks of life in space. The new experiment has been constructed and successfully commissioned and calibrated. We have already published a Letter investigating the four-species system \{H+H$_2$CO+NO+H$_2$O\}. In Dulieu et al.~\cite{Dulieu2019MNRAS} we demonstrate how formamide (NH$_2$CHO), one of the key molecules of prebiotic chemistry~\cite{Saladino2012}, can be formed by hydrogenation of a mixture of H$_2$CO and NO, with water molecules playing a catalytic role, hence demonstrating the additional potential of an experiment built to study surface reactions using four deposition lines.
VENUS will offer the possibility to direct simultaneously or sequentially atoms and molecular radicals onto an analog of cosmic dust. Its multi-beam configuration will allow us to deposit different ice substrates and investigate at once three-radical systems, while keeping a fully controlled experiment and ensuring maximum purity of the reactants.
{Provided a second microwave cavity is installed,} N-bearing radicals such as NH and NH$_{2}$ can be formed \textit{in situ}, as well as OH and CH$_{3}$ groups. These radicals can be mixed and/or their reactivity and binding energy can be measured on previously deposited pure ices (i.e., H$_{2}$O, H$_{2}$CO, CO, NO) or ice mixtures. Also, by varying some selected experimental parameters (beam fluxes, surfaces temperature, and deposition times), it is possible to distinguish what are the active reactive pathways and estimate reaction rates from the analysis of the number and abundance of the products.
Furthermore, laboratory data on complex reaction systems, as long as they are fully controlled, will be complementary to astrochemical models and the increasingly high-resolution observations of interstellar ices in dark clouds and in circumstellar environments. In particular, with the imminent launch of the James Webb Space Telescope (JWST), a 6.5-meter orbiting infrared observatory scheduled for 2021~\cite{JWST2021}, astronomers will be able to look with greatly improved sensitivity inside dust clouds where stars and planetary systems form, that is environments where COMs are likely to be synthesized on icy grains. This will yield in the near future a wealth of data on the composition of interstellar ices and then it will be important to be able to produce appropriate experimental results, whose inclusion in astrochemical models will give us the most complete understanding possible on the formation of complex molecules in space.
\newpage
\begin{acknowledgments}
This work was supported by the Programme National “Physique et Chimie du Milieu Interstellaire” (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES, by the DIM ACAV+ a funding program of the Region Ile de France, and by the ANR SIRC project (GrantANR-SPV202448 2020-2024). T.N. gratefully acknowledges support from the LabEx MICHEM.
\end{acknowledgments}
\section*{Data availability}
The data that support the findings of this study are available from the corresponding author upon reasonable request.
|
1,108,101,563,291 | arxiv | \section{Introduction}
In the past decade or so, deep learning has contributed to improving the state of the art on many computer vision tasks. Powered by architectures adapted to the image domain, like Convolutional Neural Networks (CNN) \cite{lecun1989backpropagation} or Vision Transformers (ViT) \cite{vit}, deep networks can successfully tackle tasks ranging from classification to dense semantic segmentation. While of clear interest to the design of vision-based robotics systems, deep learning still presents a number of limitations restricting its application to robotics. Firstly, deep learning methods can suffer from sample inefficiency and typically require a large number of annotated images to produce estimators with good generalization capabilities. Deployment in new visual environments are therefore likely to require an expensive data annotation procedure. Secondly, deep vision models have been characterized by a ``depth race'' with architectures of increasing size, which are of limited use for embodied agents requiring high inference performance on resource-constrained hardware.\textcolor{white}{\footnote{Equal contributions.}}
In this work, we consider the problem of learning an instance segmentation model for monocular robot navigation using few annotated images. Our model performs ``coarse'' semantic segmentation by predicting labels at the \(8\times8\) patch level. We address the sample inefficiency issue by using standard data augmentation techniques as well as pretrained weights from a leading self-supervised method \cite{dino}. As for the computational aspect, we show how a ViT can be trained for the the task at hand and yield effective and efficient perceptual backbones for visual servoing.
In summary, we make the following contributions:
\begin{enumerate}
\item Using a ViT pretrained with a label-free self-supervised method, we successfully train a coarse image segmentation model using only 70 training images;
\item We show how the same model can be used to predict labels at different resolutions, allowing a compromise between prediction granularity, inference speed and memory footprint;
\item We perform experimental validation by using the resulting segmentations for visual servoing of a real robot in the Duckietown environment.
\end{enumerate}
Our source code, annotated dataset and additional videos of our experiments can be found at \texttt{https://sachamorin.github.io/dino}.
\section{Background}
\subsection{Vision Transformers}
ViTs~\cite{vit} have recently emerged as a competitor to CNNs~\cite{lecun1989backpropagation} for computer vision tasks~\cite{liu2022convnet}. Echoing the original transformer formulation for natural language processing \cite{transformer}, the ViT architecture decomposes images (``sentences'') into small image patches (``visual words''), typically of size \(8\times8\) or \(16\times16\). ViTs learn vector encodings for each patch via self-attention, that is, encodings are updated with a weighted combination of all other patch encodings. A single transformer layer can therefore learn dependencies between any two patches in the input image.
For the specific task of image segmentation, transformer-based architectures usually rely on some encoder-decoder structure \cite{badrinarayanan2017segnet} where patch encodings are reconstructed or upsampled in some way to predict labels in pixel space \cite{segmenter, zheng2021rethinking, xie2021segformer, ranftl2021vision}.
\subsection{Self-Supervised Learning}
Self-Supervised Learning (SSL) is a subset of unsupervised learning which aims to learn generic and expressive data representations in the absence of labels. Recent successful applications of SSL include transformer-based language models like BERT or RoBERTa \cite{bert, roberta}. A similar trend exists for vision models, where SSL image representations achieve state-of-the-art performance on ImageNet benchmarks \cite{simclr, dino, maske-auto}. An important class of SSL algorithms for vision relies on contrastive learning, where the aim is to learn representations in which samples from the same class (``positive'') are close to one another and samples from distinct classes (``negative'') are separated.
The traditional view of contrastive learning using positive and negative pairs was recently modified in BYOL~\cite{byol}, where the authors managed to train representations on the ImageNet dataset~\cite{imagenet} using only positive pairs and a teacher/student architecture.
A followup to BYOL is DINO~\cite{dino}, which employs a similar architecture with a different loss. Interestingly, the authors show how self-supervised ViTs naturally learn explicit information about the semantic segmentation of an image in their attention masks, a finding that was not replicated in CNNs or in supervised ViTs.
Notwithstanding the undeniable success of SSL, there have been few attempts to extrapolate these ideas to real-world robots where data is abundant but labeling remains a time-consuming task. Some attempts have been made in the Reinforcement Learning (RL) community for navigation tasks~\cite{cc-rig, duckie-transformer}, but there has been little experimental validation of SSL representations for monocular image-based visual servoing \cite{servoing, thesis}. Additionally, standard evaluation procedures for SSL in computer vision tasks~\cite{swin, moco, color}
do not encompass assessment in downstream applications for embodied agents in real-world environments. Our work differs from previous ones as we use pre-trained DINO weights in a limited data setting to navigate a robot.
\subsection{Duckietown}
To assess the performance of our visual-servoing agent, we use the Duckietown platform \cite{duckietown}. Duckietown is an inexpensive open-source platform for autonomous driving education and research. Duckiebots (Fig. \ref{fig:mycroft}) are the main mobile agents and various tiles and accessories are available to design their driving environments.
In this work we exploit the modularity of Duckietown to create two object-rich driving scenes composed of ducks, houses, signs and other Duckiebots (Fig. \ref{fig:lane_following} and \ref{fig:obstacles}). By modifying the layout of the scene after data collection, we can validate the robustness of the ViT segmentation predictions used for visual servoing.
\section{Coarse Semantic Segmentation with Vision Transformers}
\begin{figure}
\centering
\includegraphics[width=.49\textwidth]{pipeline.png}
\caption{\textbf{Coarse semantic segmentation using ViTs.} We encode image patches using a ViT and predict a class label for each patch. We visualize attention heads to assess the effect of training in the transformer encoder.}
\label{fig:pipeline}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.49\textwidth]{figures/controller.png}
\caption{\textbf{Potential-Field based controller for lane following and obstacle avoidance.} The coarse segmentation output is used to compute a left (blue) and right (red) mask which are delivered to a potential-field based controller. The controller receives the mask and maps it as a ``repulsive'' potential to steer away from the half of the image with the most obstacle patches (Equation \ref{eq:loss}).}
\label{fig:controller}
\end{figure}
Our approach is based on the following hypothesis: an agent can successfully and safely navigate an environment with low resolution segmentation masks. Thus, we propose to train a classifier to predict labels for every \(8 \times8 \) patch in an image. Our classifier is a fully-connected network which we apply over ViT patch encodings to predict a coarse segmentation mask (Fig. \ref{fig:pipeline}). For example, given a ViT with \(d\)-dimensional encodings, an 480p image will yield a \(60\times60\times d\) encoding map (\(480/8=60\)). During training, we therefore downsample the original ground truth mask accordingly using nearest neighbor interpolation to derive a single label for each patch. Our simple design avoids bespoke architectural components and allows to reuse pretrained weights from any vanilla ViT architecture, such as those from DINO~\cite{dino}.
\input{tables/resolution}
Our motivation for using ViTs is twofold: first, we hope to leverage the ability of transformers to learn long-range dependencies in an image using a relatively shallow architecture, which is an appealing proposition for real-time control tasks. Second, as with convolutional layers, a trained ViT can run segmentation on images at various resolution, as long as the patch size is the same. This implies that a ViT trained for coarse segmentation of \(8 \times 8\) patches will yield \(30 \times 30\) predictions for 240p images, \( 60 \times 60 \) predictions for 480p images, \(120 \times 120\) predictions for 960p images and so on. One can therefore adjust the granularity of the prediction and the associated computational load by simply downscaling or upscaling input images: a welcome flexibility for deployment on embodied agents with hardware constraints. ViTs can be trained and fine-tuned at different resolutions \cite{vit} and more advanced techniques for improving the performance at different test resolutions is a matter of active research \cite{xie2021segformer}.
\section{Image Segmentation Experiments}
\label{sec:segmentation}
\subsection{Dataset}
Our dataset is composed of RGB images gathered using the on-board camera of our Duckiebot (Fig. \ref{fig:mycroft}). We collected images by teleoperating the robot in the data collection scene presented in Fig. \ref{fig:data_collection}. To increase the diversity in the training scene, various objects were added on the road during the episode. A total of 100 images were sampled from the recordings and labeled with 7 classes: Duckiebot, duckie, white lane marking, yellow lane marking, road sign and human hand. We use a 70-15-15 split for training, validation and testing.
\begin{figure*}[!t]
\centerline{
\subfloat[\textbf{Robot.} Example of the differential drive DB21J Duckiebot model used in our experiments. While other sensors are available, we only used the camera for navigation. \label{fig:mycroft}]{\includegraphics[width=.45\textwidth,height=4.5cm,trim={0 22cm 0 30cm},clip]{duckiebot.jpg}}
\hfil
\subfloat[\textbf{Data collection scene.} We collected 100 images by teleoperating the robot on the scene and occasionally adding various objects on the road. \label{fig:data_collection}]{\includegraphics[width=.45\textwidth,height=4.5cm]{collection.jpg}}
}
\centerline{
\subfloat[\textbf{Lane following scene.} The robot is tasked with completing laps without crossing the yellow line (left) or white line (right). We benchmarked both the outer and inner loops. \label{fig:lane_following}]{\includegraphics[width=.45\textwidth,height=4.5cm]{lane-loop.jpg}}
\hfil
\subfloat[\textbf{Obstacle avoidance scene.} The robot needs to complete laps without colliding with on-road obstacles. For this experiment, the controller ignores the yellow line predictions and the robot can navigate both lanes to avoid obstacles. In total, 4 duckiebots, 2 signs and 4 groups of duckies must be avoided during a loop. \label{fig:obstacles}]{\includegraphics[width=.45\textwidth,height=4.5cm]{obstacles-loop.jpg}}
}
\caption{Experimental setup.}
\label{fig:data_scene}
\end{figure*}
\subsection{Model Training}
We use the ViT architecture and pretrained weights from DINO for our perception backbone. Specifically, the ViT-S/8 architecture consists of 12 transformer blocks---a block includes a self-attention layer followed by 2 fully-connected layers---and takes \(8 \times 8\) patches as input. As with standard ViT architectures, the size of the feature map is constant throughout layers and any number of blocks will output predictions at the same resolution. Each patch is encoded as a 384-dimensional vector which we classify using a fully-connected segmentation head. In Fig. \ref{fig:blocks}, we vary the number of transformer blocks in the backbone as well as the augmentations applied to the training data in order to study segmentation performance. Encouragingly, we find that using only a few transformer blocks is sufficient to achieve good performance. We further observe that metrics plateau or even deteriorate with a backbone deeper than 5 blocks. Moreover, standard data augmentations and finetuning the backbone still appear necessary to maximize performance. The bigger part of the finetuning improvement is observed in the road sign class, which was not heavily featured in our dataset.
For training, we use 480p images, a batch size of 1 image (3600 \(8 \times 8\) patches), and train for 200 epochs of 1000 randomly sampled images. The validation set is used for checkpointing and the model with the best balanced validation accuracy is retained. When training the segmentation head only, we use the Adam optimizer with a learning rate of .001. For finetuning the backbone, we continue training for 200 epochs using the AdamW optimizer with a learning rate of 1e-4. We use standard data augmentations where mentioned: random crops, flips, shifts, scales, rotations, color jittering and Gaussian blur. Reported IoU and Accuracy scores are with respect to the downscaled ground truth masks, e.g., 480p predictions are benchmarked against the 60x60 interpolated masks to reflect patch-level performance.
\begin{figure}
\centerline{
\includegraphics[width=.45\textwidth]{figures/blocks.pdf}
}
\caption{Average class IoU (mIoU) and accuracy (mAcc) of various segmentation models on our Duckietown segmentation test dataset. Metrics are averaged over 3 seeds. The DINO architecture consists of 12 transformer blocks: we therefore probe the intermediary patch representations by training a segmentation head at various depths using a ViT backbone of $n$ blocks. While raw patch representations (\textbf{``No Augmentations''}) perform reasonably well, adding standard image augmentations (\textbf{``Augmentations''}) is hugely beneficial, despite the self-supervised DINO pretraining. In both the ``No Augmentations'' and ``Augmentations'' setup, the ViT backbone parameters are frozen and we only train the segmentation head. As expected, unfreezing the backbone and continuing training (\textbf{``Augmentations + Finetuning''}) increases both performance metrics and is particularly beneficial for the 1-block and 2-block backbones. We could not finetune backbones with more than 4 transformer blocks due to hardware constraints.}
\label{fig:blocks}
\end{figure}
\subsection{Inference}
In Table \ref{tab:inference}, we study the inference speed and quality of the finetuned 1-block and 3-block ViT segmentation models from the previous section. For comparison, we also study backbones built with the first layers of a DINO-pretrained ResNet-50 CNN architecture to perform the same task. We benchmark the same models at different inference resolutions and find that the ViTs perform relatively well even on downscaled images. Importantly, all models can run inference at a reasonable framerate on GPU for 240p or 480p input. CPU inference at 240p would even be conceivable. All benchmarks were run on a \textit{11th Gen Intel Core i7-11800H @ 2.30GHz × 16} CPU and a \textit{NVIDIA GeForce RTX 3060} Laptop GPU. Contrary to the memory efficient CNNs, the ViT 960p resolution could not fit into GPU memory due to the unwieldy amount of patches.
ViT predictions can be further visualized in Fig. \ref{fig:predictions}. Additionally, we compare the attention masks of the pretrained 3-block versus the finetuned one in Fig. \ref{fig:attention} where interestingly, the finetuned model learns to attend mostly to those objects that are relevant to our task.
\begin{figure*}[!t]
\includegraphics[trim={0 13cm 0 0},clip, width=1\textwidth]{figures/predictions.jpg}
\caption{Predictions of the \textbf{same 3-block ViT} at different resolutions. While the model was trained in the 480p regime, it performs well on downscaled (240p) or upscaled (960p) images. The 240p predictions are visually coarse, but accurate for nearby objects or large distant ones. We show in Subsections \ref{subsec:lane} and \ref{subsec:obstacles} how 240p and 480p predictions can be used for navigating a Duckietown environment. The 960p predictions are shown for illustrative purposes and are too slow for real-time navigation.}
\label{fig:predictions}
\end{figure*}
\begin{figure*}[!t]
\centerline{
\begin{tabular}{ccc}
\includegraphics[width=.3\textwidth]{figures/test_1.png} & \includegraphics[width=.3\textwidth]{figures/attn-head-dino1-test1.png} & \includegraphics[width=.3\textwidth]{figures/attn-head-ours1-test1.png}\\
\subfloat[Input image]{\includegraphics[width=.3\textwidth]{figures/test_3.png}} & \subfloat[DINO]{\includegraphics[width=.3\textwidth]{figures/attn-head-dino1_test3.png}} & \subfloat[Finetuned]{\includegraphics[width=.3\textwidth]{figures/attn-head-ours1_test3.png}}
\end{tabular}
}
\caption{Segmentation masks of the DINO 3-block pretrained model comapared with a finetuned version. Results demonstrate how the attention masks of the pretrained model attend to a general variety of objects in the scene like the desk in row two. Interestingly, the attention masks of the finetuned model attend primarily to the objects that belong to one of the classes in our dataset, ignoring objects that are not relevant to our task.}
\label{fig:attention}
\end{figure*}
\section{Navigation Experiments}
\label{sec:navigation}
In this section, we assess the performance of the trained 1-block and 3-block ViT models from Table \ref{tab:inference} on two different monocular visual servoing tasks: lane following and obstacles avoidance. The robot is controlled using a potential-field based controllers which receives as input the segmentation output produced by a trained ViT and outputs steering commands for the robot while maintaining a fixed linear velocity as shown in Fig. \ref{fig:controller}. For the lane-following task the goal is to maintain the vehicle centered in a lane while in the obstacle avoidance scene, the agent can use both lanes to navigate and avoid obstacles.
The scenes for both experiments are modified from the original data collection scene. In Fig. \ref{fig:lane_following}, we show the driving environment used for lane-following where an additional U-turn is added to measure the robustness of the driving agent. The same driving scene is reused for the obstacle avoidance problem but different objects are added on the road (Fig. \ref{fig:obstacles}).
\subsection{Lane following}
\label{subsec:lane}
The objective of this task is to navigate the vehicle on the road (without obstacles) and maintain the vehicle centered between the yellow and white lines. We first obtain segmentation predictions \(\boldsymbol{S}_t \in \mathbb{R}^{n \times n}\) for the current $n\times n$ resized camera frame, then compute a navigation mask \(\boldsymbol{M}_t \in \{0,1\}^{n \times n}\) by extracting the pixel patches corresponding to white and yellow lines. This mask is used to compute a ``repulsive'' potential pushing the agent to steer away from the half of the image with the most line patches. More formally, the steering angle is controlled based on
\begin{equation}
\label{eq:loss}
\phi_{t+1} = \phi_{t} - \gamma \sum_{i, j} (\boldsymbol{P} \odot \boldsymbol{M}_t)_{i, j}.
\end{equation}
where $P$ is a sign mask with $-1$ values on the left and $+1$ values on the right, and $\gamma$ is a weighting factor. This purely reactive controller will reach an equilibrium by keeping the same energy on the right and the left, i.e., by being centered between the white and yellow lines. We present a visualization of the controller in Fig. \ref{fig:controller}.
Each model is evaluated by navigating the agent for five loops (2 outer loops and 3 inner loops) and report the number of minor and major infractions. A minor infraction occurs when the robot steps over either the white or yellow line. A major infraction is defined as any event requiring human intervention to put the agent back on track. Both models are evaluated at 240p and 480p input resolutions and compared against the standard lane following system implemented in Duckietown, which consists of HSV filters alongside a particle filter for state estimation.
The results of the lane following evaluation are reported in Table \ref{table-following}. Both models perform equally well in the outer loop, however,
the inner loop proves more challenging.
The best performing model was with 1-block at 240p with a total of four minor infractions and zero major infractions. We hypothesize this good performance is owed to the high throughput of the model, which allows for better reaction time in the controller. Surprisingly, the high-capacity 3-block model
reports a higher number of infractions. The baseline model is the worst-performing
whose result is a likely consequence of the HSV filters producing false positives for line detection when white or yellow objects are placed in the scene (see Fig. \ref{fig:lane_following}).
\input{tables/following}
These results demonstrate that even though higher capacity models perform better than low capacity ones with respect to test segmentation metrics, more variables should be considered for real-world deployment. Additionally, while the coarse segmentation predictions (see Fig. \ref{fig:predictions} column~2) lower the resolution to increase inference speed, they still appear to hold enough information to accurately navigate the agent within the environment.
\subsection{Obstacle Avoidance}
\label{subsec:obstacles}
The objective for the obstacles avoidance experiment is to navigate the agent through the scene in Fig. \ref{fig:obstacles} while avoiding on-road duckies, signs and Duckiebots. This task was designed to validate the performance of the agent within a more challenging environment, and make use of predictions for all classes. We define the area between both white lines as being drivable, i.e., the agent is allowed to cross the yellow line to avoid obstacles without penalty. Particularly, the controller is the same used in Subsection \ref{subsec:lane} with the difference that the navigation mask now includes all obstacles to be avoided. Therefore, pixels classified as white line, ducks, signs and duckiebots are actively contributing to the ``repulsive'' potential. This potential function encourages the agent to drive on the road while avoiding objects placed on it. The assessment here is similar to that of Subsection \ref{subsec:lane} (same number of loops) with the difference that driving over the yellow line is permitted and small contacts with obstacles are considered minor infractions.
\input{tables/obstacles}
We evaluate the same models as in the previous subsection (1-block, 3-block, at 240p and 480p). For this benchmark, we do not have a particular baseline in the Duckietown stack to compare with. The results are presented in Table \ref{table-obstacles} and are consistent with the ones in Table \ref{table-following}. The best performing model was once again 1-block with an input resolution of 240p, reporting a total of two minor infractions and zero major ones. The high-capacity models did not perform as-well as expected even though the segmentation results produced by those are of superior quality, again suggesting that visual servoing benefits from the higher framerate of the shallow 1-block backbone.
\section{Conclusion}
In this work, we study how embodied agents with vision-based motion can benefit from ViTs pretrained via SSL methods. Specifically, we train a perception model with only 70 images to navigate a real robot in two monocular visual-servoing tasks. Additionally, in contrast to previous SSL literature for general computer vision tasks, our agent appears to benefit more from small high-throughput models rather than large high-capacity ones. We demonstrate how ViT architectures can flexibly adapt their inference resolution based on available resources, and how they can be used in robotic application depending on the precision needed by the embodied agent. Our approach is based on predicting labels for 8x8 image patches, and is not well-suited for predicting high-resolution segmentation masks, in which case an encoder-decoder architecture should be preferred. The low resolution of our predictions does not seem to hinder navigation performance however, and we foresee as an interesting research direction how those high-throughput low-resolution predictions affect safety-critical applications by scaling our method to more challenging scenarios. Moreover, training perception models in an SSL fashion on sensory data from the robot itself rather than generic image datasets (e.g., ImageNet) appears to be a promising research avenue, and is likely to yield visual representations that are better adapted to downstream visual servoing applications.
\section*{Acknowledgment}
The authors would like to thank Gustavo Salazar and Lilibeth Escobar for their help labeling the dataset. Special thanks to Charlie Gauthier for her help setting-up the Duckietown experiments. This research was partially funded by an IVADO (l'Institut de valorisation des donn\'{e}es) MSc. Scholarship and an FRQNT (Fonds de recherche du Québec – Nature et technologies) B1X Scholarship [\emph{S.M.}].
\bibliographystyle{IEEEtran}
|
1,108,101,563,292 | arxiv | \section{Introduction}
\label{sec:introduction}
Quantum phase transitions (QPTs) are often treated in the thermodynamic limit, where the concepts of a critical point, critical exponents, and scaling relations are very clear~\cite{sachdev1999}. However, many physical many-body systems for practical applications these days are finite. For example, nano-devices can be smaller than 100 nm. Then a crystal with a lattice constant of a fraction of a nanometer has just 2 or 3 orders of magnitude to work with in scale. In nuclear physics one has QPTs in finite nuclei which are very far from being in the thermodynamic limit~\cite{caprio2008}. Experiments on trapped ions, recently used to simulate many body Hamiltonians, normally work with small systems~\cite{kimK2009}. Ultracold atoms in optical lattices provide a fourth example, where the number of lattice sites is typically 50 to 100 in each direction, so that in one dimension (1D) they are highly mesoscopic. These systems have been proposed as quantum simulators to solve hard quantum many body problems, including the positive-$U$ Fermi-Hubbard Hamiltonian for general filling~\cite{hofstetter2002,jaksch2004,lewensteinM2007,bloch2008}. Progress towards this goal requires understanding of how finite-size effects change the quantum phase diagram. In this paper, we explore such mesoscopic or finite size effects for both Bose- and Fermi-Hubbard Hamiltonians.
\begin{figure}[t]
\begin{center}
\epsfxsize=8cm \epsfysize=6.5 cm \leavevmode \epsfbox{fig1.eps}
\caption{\label{fig:williams} (color online) \emph{The first Mott lobe of the Bose-Hubbard Hamiltonian}. Quantum depletion for a system of interacting bosons on 51 lattice sites at $T=0$ as a function of chemical potential $\mu$ and hopping parameter $t$, both normalized to interaction strength $U$, obtained with Time-Evolving Block Decimation (TEBD) in imaginary time propagation, as discussed below. The white circular points represent best-converged results of Density Matrix Renormalization Group methods for the boundary of the Mott lobe in the thermodynamic limit~\cite{kuhner2000}.}
\end{center}
\end{figure}
In addition to finite-size effects, ultracold atomic quantum simulators are non-uniform, as ordinarily there is a weak overall harmonic trap keeping atoms from spilling off the edge of the lattice. This non-uniformity gives rise to a ``wedding-cake'' or tiered structure of Mott plateaus separated by superfluid layers for bosons~\cite{batrouniGG2002,gemelke2009}, and similar layered structure for fermions~\cite{schneiderU2008,jordens2008}. The non-uniformity creates even smaller regions of Mott insulator or superfluid, often just 10 or 20 sites in length. For these small regions the quantum phase diagram is substantially different than in the thermodynamic limit: for example, the quantum depletion, (defined precisely below in Sec.~\ref{sec:equations}) shows that the Mott lobe is not closed but instead goes through a very broad crossover into the superfluid regime, as we show in Sec.~\ref{ssec:bose}. Figure~\ref{fig:williams} shows the sharp phase boundary of a Bose Hubbard system in the thermodynamic limit (white circles~\cite{kuhner2000}) and the continuously variable quantum depletion (defined in Sec.~\ref{sec:equations}) obtained in our calculations for 51 sites. As can be seen in the figure, the tip of the Mott lobe is significantly blurred even for this relatively large lattice.
Approximating the effects of a slowly varying trap by using a local chemical potential in the local density approximation, $\mu=\mu(\vec{r})$, is not very useful for cold atoms in 1D. The breakdown of the local density approximation has been explored with quantum Monte Carlo and exact diagonalization elsewhere (see~\cite{rigol2009} and references therein); our work, which isolates finite size effects in small uniform regions, is complementary to previous studies which have explored the harmonic trap structure as a whole~\cite{batrouniGG2002,rigol2004,rigol2009}. The success of quantum simulators relies in part on understanding all possible sources of error~\cite{trotzky2009}, in the same sense as one might do so for a high precision measurement of time-reversal symmetry, for example. Therefore going beyond the local density approximation~\cite{rigol2009} is key to the eventual success of quantum simulators, especially in lower dimensions. Our use of TEBD offers a wide array of quantum measures, many of which are not directly accessible to quantum Monte Carlo~\cite{kuhner1998,sengupta2005,fubini2006} or exact 1D methods~\cite{Lieb1968,gu2004}, including various entanglement entropies and fidelity. Multiple measures provide maximal information, useful for a thorough analysis; in particular, fidelity and fidelity susceptibility make no assumptions about states or symmetries, but simply ask about many body wavefunction overlap in a model-independent way~\cite{venuti2008}.
It has already been shown that the mean-field Gutzwiller approximation, which assumes a product state between sites of the lattice, is not very useful in 1D for Hubbard Hamiltonians~\cite{lewensteinM2007}. But how far does one have to move away from the Gutzwiller approximation in order to get correct results? A high level of effort has been put into alternative methods, such as generalized dynamical mean field theory in two dimensions for Bose-Fermi mixtures~\cite{titvinidze2009}. The convergence parameter in TEBD, which can be couched in terms of the degree of spatial entanglement, allows one to go beyond the Gutzwiller approximation in a methodical manner, and we present a thorough study of this issue. A second question concerning numerical methods is the use of the canonical ensemble vs. the grand canonical ensemble. The two only formally give the same result in the thermodynamic limit. The canonical ensemble is numerically preferred for TEBD and related methods~\cite{daley2004} because the number of Fock states is much smaller and therefore calculations are more efficient. We explicitly explore the effects of ensemble dependence Hubbard Hamiltonian phase diagrams.
This article is outlined as follows. In Sec.~\ref{sec:equations} we present the Bose- and Fermi-Hubbard Hamiltonians, the many quantum measures we use, and a brief description of and motivation for our implementation of TEBD. In Sec.~\ref{sec:bosefermi} we present and discuss our numerical results for mesoscopic effects in ground states of the Bose- and Fermi-Hubbard Hamiltonians, respectively. In Sec.~\ref{sec:compare} we compare canonical to grand canonical ensembles and Gutzwiller to entangled TEBD simulations; these comparisons also serve as a convergence study. Finally, in Sec.~\ref{sec:conclusion}, we conclude.
\section{Models, Measures, and Methods}
\label{sec:equations}
We treat two fundamental Hamiltonians for ultracold quantum gases in optical lattices, the Bose- and Fermi-Hubbard Hamiltonians. In both cases we take the tight-binding lowest-band approximation. The Bose-Hubbard Hamiltonian is
\ba
\hat{H_b}&=& -t\sum_{\langle i,j\rangle}(\hat{b}^{\dagger}_i \hat{b}_j+
\hat{b}_i \hat{b}^{\dagger}_j) +\frac{1}{2}U\sum_i \hat{n}_i^{(b)}(\hat{n}^{(b)}_i-1)\nonumber\\
&&-\mu\sum_i \hat{n}^{(b)}_i\,,\label{eqn:bosehubbard}
\ea
where $t$ is the hopping strength, $U$ is the interaction strength, $\mu$ is the chemical potential, $\hat{b}_i$, $\hat{b}_i^{\dagger}$ are bosonic destruction/creation operators satisfying bosonic commutation relations on-site and commuting between sites, $\hat{n}_i^{(b)}$ is the bosonic number operator, and $\langle i,j\rangle$ denotes a sum over nearest neighbors. The corresponding Fermi-Hubbard Hamiltonian in particle-hole symmetric form is
\ba
\hat{H_f}&=& -t\sum_{\langle i,j\rangle,\sigma}(\hat{f}^{\dagger}_{i\sigma} \hat{f}_{j\sigma}+
\hat{f}^{\dagger}_{j\sigma}\hat{f}_{i\sigma} )\nonumber\\
&&+U\sum_i \left(\hat{n}_{i\downarrow}^{(f)}-\frac{1}{2}\right)
\left(\hat{n}^{(f)}_{i\uparrow}-\frac{1}{2}\right)\nonumber\\
&&-\mu\sum_i \hat{n}^{(f)}_i\,,\label{eqn:fermihubbard}
\ea
where $\hat{f}_{i\sigma},\hat{f}_{i\sigma}^{\dagger}$ are fermionic destruction and creation operators satisfying fermionic anticommutation relations; $t$ and $U$ have the same meaning as in the bosonic case; and we have assumed $s$-wave interactions and spin 1/2, so that $\sigma\in\left\{\uparrow,\downarrow\right\}$. We note that there is an exact mapping under the canonical transformation from the negative-$U$ to positive-$U$ Fermi-Hubbard Hamiltonian at half filling ($\mu=0$). This will appear visually in our phase diagrams in Sec.~\ref{ssec:fermi}.
To find the ground states of these models, we use TEBD~\cite{tebdOpenSource} in 1D and propagate in imaginary time, starting with a state with uniform weights. TEBD has one main convergence parameter, the number of eigenvalues $\chi$ retained in the reduced density matrix. In all cases we have checked convergence by at least doubling $\chi$, and seeing if we obtain the same results in the plots; all plots have at least a $\chi$ of 16, and we have spot-checked a $\chi$ of 32. We emphasize that as we have a very high resolution of ($250\times 100$) pixels in our diagrams, we cannot go to the preferred typical values of $\chi$ in the hundreds, in common use for TEBD calculations; also, grand canonical calculations are inherently more restrictive, due to the much larger state space. We use both canonical and grand canonical ensembles depending on circumstances. The latter is useful to understand the proper interpolation method to be used for the canonical ensemble for smaller numbers of sites. In the canonical ensemble, chemical potential is derived by finite differencing, $\mu \approx E(N+1) - E(N)$; for just a few sites, this would lead to only a few points of resolution on the chemical potential axis without interpolation. That is, a finite difference approximation for the chemical potential is only \emph{directly} useful for large $N$; interpolation can only be justified by comparing to the grand canonical results. We illustrate this point in Sec.~\ref{ssec:ensemble}.
For fermions, there is an additional advantage of grand canonical calculations in that, in practice, we find that imaginary time propagation is less likely to get stuck in a local minimum.
For fermions, the local, or on-site dimension of the Hilbert space is fixed at $d=4$ since we consider only a single band. For bosons $d$ provides a second convergence parameter. In practice we truncate at $d=5+1$, meaning 0 to 5 bosons per site, and we have checked $d=7+1$ for specific points in the phase diagrams and found no visible changes. We implement imaginary time propagation in the second order Suzuki-Trotter scheme to obtain the ground state. Finally, we parallelize over the parameter space $(t/U, \mu/U)$ via MPI for bosons, and $(U/t, \mu/t)$ for fermions. We typically perform our calculations on between $5\times 8$ and $60\times 8$ cores. We note that TEBD becomes exact, aside from the $\mathcal{O}\left(\delta t^2\right)$ error of the Suzuki-Trotter decomposition, for $\chi \geq d^{\lfloor L/2 \rfloor}$.
Our quantum measures can be divided into four classes: moments, correlation, entropies, and fidelity. Although these classes of measures can have significant overlap in the information they provide, nevertheless there are always cases where they are distinct. For example, the spin singlet is maximally entangled but is not correlated. Thus it is useful to consider all classes.
First, for moments, the energy, $E_b$, and filling factor, $\nu_b$, are defined as
\ba
E_b&=&\langle \hat{H}_b\rangle\,,\\
\nu_b&=&\frac{1}{L}\sum_{i=1}^L\langle \hat{n}^{(b)}_i\rangle\,,
\ea
where $L$ is the number of lattice sites. To calculate $E_f$ and $\nu_f$ for fermions, one must also sum over the spin index $\sigma$.
Second, for correlations in bosons we calculated the quantum depletion, $D$, which is a measure of how far away the system is from the Penrose-Onsager definition of superfluidity~\cite{penrose1956}:
\be
D \equiv 1-\frac{\lambda_1}{\sum_{i=1}^{L}\lambda_i}\,,
\ee
where $\{\lambda_i\}$ are the eigenvalues of the single particle density matrix $\langle \hat{b}_i^{\dagger}\hat{b}_j\rangle$ ordered so that $\lambda_1 > \lambda_2 > \cdots > \lambda_L$. For correlations in fermions we use three standard measures~\cite{scalettar1989,varney2009}. The s-wave pairing correlation, $P$, is given by
\ba
P &=& \langle \hat{\Delta}^{\dagger}\hat{\Delta} + \hat{\Delta}\hat{\Delta}^{\dagger} \rangle\,,\\
\hat{\Delta}^{\dagger} &\equiv& \frac{1}{\sqrt{L}}\sum_{i=1}^{L}\hat{f}^{\dagger}_{i\uparrow}\hat{f}^{\dagger}_{i\downarrow}\,.
\label{eqn:pc}
\ea
The charge-density wave correlation, $C$, is given by
\ba
C &=& \langle \hat{\theta}\hat{\theta}^{\dagger}\rangle\,,\\
\hat{\theta}^{\dagger} &\equiv& \frac{1}{\sqrt{L}}\sum_{j=1}^{L}\sum_{\sigma} e^{i\pi j}\hat{n}_{j\sigma}^{(f)}\,.
\label{eqn:cdc}
\ea
The antiferromagnetic structure factor is given by
\ba
A&=&\frac{1}{L}\sum_{i,j=1}^L\left(-1\right)^{i-j}\langle \hat{S}^z_{i}\hat{S}^z_{i}\rangle\, ,\\
\hat{S}^z_{i}&\equiv& \frac{1}{2}\left(\hat{n}_{i\uparrow}^{(f)} - \hat{n}_{i\downarrow}^{(f)}\right)\,.
\ea
Third, for entropies, the average local von Neumann entropy~\cite{kitaev2006}, or average local entropy of entanglement, $\bar{S}$, is
\be
\bar{S}\equiv -\frac{1}{L}\sum_{i=1}^L\mathrm{Tr}\,\left(\hat{\rho}_i\log_d\hat{\rho}_i\right)\in[0,1]
\ee
where the local reduced density matrix, $\hat{\rho}_i$, is defined as
\be
\hat{\rho}_i = \mathrm{Tr}_{j\neq i}\hat{\rho}
\ee
and $\hat{\rho}$ is a pure-state density matrix for the whole system, as we work at zero temperature. Note that the base $d$ of the log normalizes the entropy to lie between zero and unity. We also studied the Q-measure~\cite{meyer2002,brennenGK2003} or average local impurity, $Q$, a special case of generalized entanglement~\cite{barnumH2003,barnumH2004}, which gives similar results to the von Neumann entropy (see Sec.~\ref{sec:bosefermi}):
\begin{equation}
Q\equiv \frac{d}{d-1} \left [1 - \frac{1}{L}\sum_{i=1}^L \mathrm{Tr}\,(\hat{\rho}_i^2) \right
] \in [0, 1]\,. \label{eqn:Qmeasure}
\end{equation}
Fourth, we calculated both fidelity, $f$, and fidelity susceptibility~\cite{buonsante2007,venuti2008,rigol2009b}, $\chi_f$. The fidelity is an overlap measure on the states, and is given by
\be
f\left(\zeta,\zeta+\delta\zeta\right)\equiv\left|\langle
\psi\left(\zeta\right)|\psi\left(\zeta+\delta\zeta\right)\rangle\right|\,,
\label{eqn:fidelity}
\ee
where $\zeta$ is a control parameter in a quantum phase diagram, in our case either chemical potential or hopping. The fidelity susceptibility is a second derivative of the fidelity in the control parameter, and we normalize it to the unit length:
\be
\chi_f\left(\zeta\right)\equiv\frac{1}{L}\frac{2\left(1-f\right)}{\left(\delta
\zeta\right)^2}\,,
\label{eqn:fs}
\ee
This proves to be a more useful quantity than the fidelity, as it is independent of the control parameter step size $\delta\zeta$, and a divergence of $\chi_f$ in the thermodynamic limit signals a quantum phase transition.
Note that $\chi_f$ bears no relation to the convergence parameter for TEBD, $\chi$. In the figures below we take $\zeta$ to be $t/U$ for the Bose-Hubbard Hamiltonian and $U/t$ for the Fermi-Hubbard Hamiltonian.
\section{Quantum Measures on Ground States of Hubbard Hamiltonians}
\label{sec:bosefermi}
\subsection{Bose-Hubbard Hamiltonian}
\label{ssec:bose}
\begin{figure*}
\begin{center}
\epsfxsize=\textwidth \epsfysize=\textwidth \leavevmode \epsfbox{fig2.eps}
\caption{\label{fig:boseAll} (color online) \emph{Quantum Measures for the Bose-Hubbard Hamiltonian}. Shown is the density (first row), average local entropy of entanglement (second row), generalized entanglement or Q-measure (third row), quantum depletion (fourth row), fidelity (fifth row), and fidelity susceptibility (sixth row). The panels show increasing numbers of sites from left to right: 3, 5, 10, 20, and 40.}
\end{center}
\end{figure*}
The Bose-Hubbard Hamiltonian phase diagram has been calculated in the mean field approximation in higher dimensions~\cite{fisher1989} and with both the DMRG method~\cite{monien1998,kuhner2000} and Monte Carlo in one dimension. Trapped Bose systems with an external harmonic~\cite{rigol2009} and linear potential~\cite{carr2009j} have been treated with Monte Carlo and TEBD, respectively. However, these studies did not focus on finite size effects without aiming at the thermodynamic limit.
In Fig.~\ref{fig:boseAll} are illustrated a suite of quantum measures for $L=3,5,10,20,$ and 40 sites, from left to right in column form. The rows represent the different quantum measures as labeled on the right hand side: from top to bottom these are density, entropy, Q-measure, depletion, fidelity, and fidelity susceptibility. The calculation of these measures and their definitions are given in Sec.~\ref{sec:equations}. We emphasize that quantum Monte Carlo cannot be used to obtain entanglement. Only matrix-product-state-based techniques can do so for general local Hamiltonians, including TEBD and DMRG~\cite{schollwock2005}, which lead to an equivalent result, although with different numerical implementations. Bethe-ansatz-based techniques have been used to calculate entanglement in certain limiting cases (near $U=0$) in the Fermi-Hubbard Hamiltonian~\cite{gu2004}, but not generally. The case of $L=3$ is exact diagonalization for $\chi=15$ and $d=6$, but at $L=5$ the result is already approximate, since $6^2=36$ states would be required for exact diagonalization in matrix product state form (see Sec.~\ref{sec:equations}). Each panel of Fig.~\ref{fig:boseAll} is of resolution $80 \times 240$ in $t/U \times \mu/U$. The largest system of $L=40$ took several weeks to complete in the grand canonical ensemble on $20$ nodes with separate memory, each node consisting of 8 cores with shared memory, all on the Golden Energy Computing Organization~\cite{geco} high performance computing cluster ``Ra''. For this reason we do not go to $\chi=32$ for the whole phase diagram, but only make spot checks, since simulations scale as $\chi^3$. We calculated both canonical and grand canonical results, and show the canonical ones in Fig.~\ref{fig:boseAll}.
Turning to the physical interpretation of Fig.~\ref{fig:boseAll}, we begin with number. In the canonical ensemble a definite total number is guaranteed. This can be seen first in the average filling, which proceeds in multiples of $1/L$, as illustrated in the top row of the figure, starting from the vacuum and a filling of zero. The vacuum region is evident in the lower portion of the panels in the top row of where it is shaded a deep blue. The density increases in discrete jumps as the chemical potential is increased; interpolation is used between curves of definite number difference in order to fill in the picture, as described in Sec.~\ref{sec:equations} and Sec.~\ref{ssec:ensemble}.
In the second row we show the quantum depletion. One clearly sees the highly depleted Mott lobes and the lowly depleted outer superfluid region. Note that there are only two Mott lobes displayed in these figures; the apparent third, lowest such lobe is in fact the vacuum, $N=0$, for which quantum depletion is ill-defined.
In keeping with the 1D nature of the problem we consider, we observe that even the superfluid region is still depleted at a level of about 30\,\% far away from the Mott lobes. Again, the smooth nature of the Mott-insulator-superfluid transition near the tip is in strong contrast to the sharper transition in other regions.
In the third and fourth rows of Fig.~\ref{fig:boseAll} we show entropies. For completeness we compare the von Neumann entropy or entropy of entanglement, which is now a standard measure used in quantum information~\cite{nielsenMA2000} and also proposed for the study of quantum phase transitions~\cite{osterlohA2002,kitaev2006,dengS2008}, to the Q-measure, which has been more closely connected to local observables~\cite{barnumH2003,sommaR2004}. We find both measures useful and nearly but not exactly the same; we choose the Q-measure for the rest of our study. Note that the local von Neumann entropy of the Mott insulating regions is low by this measure, which reflects the nearly pure number state on each site in the Mott phase. On the other hand, the superfluid region displays high entropy, because the superfluid state is delocalized across sites, leading to mixed states on each site when measured separately. This contrasts with the usual association of low thermal entropy with a superfluid phase.
In the second through fourth rows one can clearly pick out the Mott-insulating region. It has a sharp phase boundary for small $t/U\lesssim 0.2$. K\"uhner, White and Monien~\cite{kuhner2000} first provided an accurate evaluation of the critical value, $t_c=0.29 \pm 0.1$, for the Kosterlitz-Thouless (KT) transition at the tip of the Mott lobe in the infinite system limit; in our notation this corresponds to a value of $t/U=0.29$. Near $t_c$ the entropy changes more continuously for 3, 5, and 10 sites, while for 20 sites the lobe begins to close. Finally at 40 sites, on the far right of the figure, the region of continuous change is very narrow. We clarify that by ``sharper boundary'' as compared to ``more continuous'' we are speaking in relative terms, specifically of the difference in entropies between states with commensurate and incommensurate filling. There is no truly sharp transition since this is a finite system; all ``quantum phase transitions'' seen here are in fact formally crossovers between regions with different characteristics.
Finally, in the fifth and sixth rows we show the fidelity $f$ and fidelity susceptibility $\chi_f$. In our present implementation we compare the neighboring pixels of the figures in $t/U$ for curves of fixed $N$ and $L$, as described in Sec.~\ref{sec:equations}. This excludes a comparison between changes in total number; such a comparison, which we also performed in the grand canonical ensemble, is not shown, because features in $f$ and $\chi_f$ are dominated by changes in the conserved quantity. In the thermodynamic limit the superfluid region of the phase diagram forms a continuous tensor product space and the fidelity is everywhere zero even for arbitrarily small $\delta\zeta$. This is the well-known \emph{Anderson orthogonality catastrophe}~\cite{anderson1967}. Thus, the fidelity, through its scaling and qualitative behavior, reflects for a finite system the critical behavior of an infinite system. Indeed, the Mott lobes appear clearly in this set of measures, as can be seen in the figure.
In order to provide a more \emph{quantitative} estimate of the tip location for finite systems than a by-eye estimation from Fig.~\ref{fig:boseAll}, we study quantum measures and derivatives thereof for unit filling $N=L$ in the canonical ensemble. Three figures illustrate our results. In Fig.~\ref{fig:estimators} are shown derivatives of quantum measures which could be used to define a crossover; in rows from top to bottom these are derivatives of average local von Neumann entropy, Q-measure, and quantum depletion; and lastly, the fidelity susceptibility. The fidelity susceptibility provides a particularly sharp signature of the Mott phase, a result that was noted previously in the context of perodic boundary conditions and the completely connected graph~\cite{buonsante2007}. Within the $n^{\mathrm{th}}$ Mott lobe, $\chi_f$ assumes values close to those given by second order perturbation theory~\cite{rigol2009b} in the limit that $t/U\to 0^+$:
\be
\chi_f = 2n(n+1)\frac{L-1}{L}
\label{eqn:perturb}
\ee
for open boundary conditions. For periodic boundary conditions the system size cancels and the expression is $\chi_f = 2n(n+1)$. These perturbations represent particle-hole fluctuations, which are weighted more heavily in higher Mott lobes due to Bose stimulation. This causes $\chi_f$ to have a higher value in the upper Mott lobe in Fig.~\ref{fig:boseAll} as compared to the lower lobe.
As the number of sites increases from 3 to 40
the extrema move towards the thermodynamic limit $t_c$. Figure~\ref{fig:tip} summarizes the most useful extrema of these curves in an easy to read single plot showing that quantum depletion and fidelity susceptibility appear to converge most rapidly to their asymptotic values in the thermodynamic limit. However, $\chi_f$ is affected by the boundary conditions, as can be seen in Eq.~\ref{eqn:perturb} as well as in Fig.~\ref{fig:exactDiagonalization}, where we compare periodic and open boundary conditions using both exact diagonalization and TEBD; this was also a significant issue in DMRG and perturbation theory studies, where earlier calculations~\cite{monien1998} obtained a less accurate estimate of the critical point due to the choice of boundary conditions~\cite{kuhner2000}. This figure also serves as a check on our code. We emphasize that open boundary conditions are closer to the conditions of ultracold atom experiments at the present time.
\begin{figure}
\begin{center}
\epsfxsize=8cm \epsfysize=13 cm \leavevmode \epsfbox{fig3.eps}
\caption{\label{fig:estimators} (color online) \emph{Crossover estimators}. First derivative (left column) and second derivative (right column) with respect to $t/U$ of various quantum measures for unit filling in the canonical ensemble. The vertical line corresponds to the known value of $t_c$.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\epsfxsize=8cm \epsfysize=6.5 cm \leavevmode \epsfbox{fig4.eps}
\caption{\label{fig:tip} (color online) \emph{Estimates of the crossover}. Extrema of the first derivative of the depletion (red squares), the fidelity susceptibility (blue triangles), and the second derivative of the Q-measure for unit filling in the canonical ensemble (green circles). All derivatives are with respect to $t/U$. The horizontal line again corresponds to the limit $t_c$. The curves are a guide to the eye.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\epsfxsize=8cm \epsfysize=6.5 cm \leavevmode \epsfbox{fig5.eps}
\caption{\label{fig:exactDiagonalization} (color online) \emph{Open vs. Periodic Boundary Conditions}. Fidelity susceptibility for fixed $N=L$ in a comparative study of open (OBC) and periodic boundary conditions (PBC); the latter can only be obtained via exact diagonalization (ED) in our present code, while OBCs can be compared directly with TEBD. We note that (1) PBC have a substantially different fidelity for smaller systems, with a peak at much smaller $t/U$, and (2) TEBD and exact diagonalization match up to the $\mathcal{O}\left(\delta t^2\right)$ error from the Suzuki-Trotter decomposition for small systems, validating our code.}
\end{center}
\end{figure}
\subsection{Fermi-Hubbard Hamiltonian}
\label{ssec:fermi}
\begin{figure*}[t]
\begin{center}
\epsfxsize=\textwidth \epsfysize=\textwidth \leavevmode \epsfbox{fig6.eps}
\caption{\label{fig:fermiAll} (color online) \emph{Quantum Measures for the Fermi-Hubbard Hamiltonian in Particle-Hole Symmetric Form}. Density (first row), generalized entanglement or Q-measure (second row), charge-density wave correlation (third row), pairing correlation (fourth row), antiferromagnetic structure factor (fifth row), and fidelity susceptibility (sixth row). The panels show increasing numbers of sites from left to right: 3, 5, 9, 10, and 20.}
\end{center}
\end{figure*}
The Fermi-Hubbard Hamiltonian in 1D has an exact solution~\cite{Lieb1968,Lieb2003} via the Bethe ansatz; in principle, one can extract any information from such a solution, including all measures of interest, and the Bethe ansatz has been applied to finite as well as infinite domains. Because of their computational complexity, Bethe ansatz results are not widely used for performing partial traces of the type that we consider here~\cite{gu2004}. Therefore it is useful to consider finite size effects for a suite of measures simultaneously, including measures not previously accessed through exact solution techniques. TEBD or DMRG methods also enable one to ``turn on'' non-mean-field effects, i.e. entanglement, in discrete steps in the form of the parameter $\chi$, as we show in Sec.~\ref{ssec:meanfield}.
Figure~\ref{fig:fermiAll} displays ground state solutions of the Fermi-Hubbard Hamiltonian [Eq.~(\ref{eqn:fermihubbard})] in the grand canonical ensemble for 3, 5, 9, 10, and 20 sites from left to right in columns, following the same pattern as Fig.~\ref{fig:boseAll} but with measures appropriate to fermions. Since we take $\chi=16$, for 3 and 5 sites TEBD is equivalent to exact diagonalization, as $d=4$; in fact, we explicitly checked our results with a separate exact diagonalization routine. The first row shows the density. For positive $U$ we observe that all number states are allowed; as in Sec.~\ref{ssec:bose}, the grand canonical ensemble settles on a total-number-conserving state for these small systems due to the discreteness of the spectrum. For example, for $L=3$ there are seven colored regions for increasing $\mu/t$ and $U/t$ greater than 0, of which all regions are visible up to $U/t \approx 2$; these correspond to 0,1,$\ldots$,6 fermions. For $L=5$ there are eleven colored regions, of which 2 are barely visible due to our choice of scale, and for higher $L$ we observe the same pattern. For odd numbers of sites the exact mapping from positive $U$ to negative $U$ in Eq.~(\ref{eqn:fermihubbard}) at zero chemical potential does not hold, and therefore there is no symmetry around the vertical axis at $U/t=0$ for $L=3,5,9$, while for $L=10,20$ we observe this symmetry. For negative $U$, particles enter only in pairs, excepting at $\mu=0$ for $L$ even, as can be seen from the figure.
In the second row of Fig.~\ref{fig:fermiAll} we show the entanglement entropy. One clearly observes the particle-hole symmetry between negative and positive chemical potential. For odd numbers of sites, the most highly entangled regions for positive $U$ are those in which there is one extra particle or hole above half filling; for even systems it is two extra particles or holes, as can be seen in the stripe-like pattern for $L=20$, for example. States nearer the band insulator, with just a few particles (holes) compared to the vacuum (maximal filling) are less entangled. Entanglement also decreases in the Mott region for increasing $U$ since the system moves closer to a product state. There is a discontinuity in the $Q$-measure between regions of half filling and slightly away from half filling which vanishes at $U$=0. This can be understood as the requirement that the derivatives approaching from above and below are required to satisfy $dQ/dn|_{n=1^+}=-dQ/dn|_{n=1^-}$ by particle-hole symmetry. For negative $U$ the most highly entangled regions are near zero chemical potential and near half-filling. Therefore, in general the pattern for both positive and negative $U$ is that the entanglement is largest near half filling. We show only the Q-measure, having established in Sec.~\ref{ssec:bose} that the average local von Neumann entropy does not show substantial additional information.
In the third row of Fig.~\ref{fig:fermiAll} we show the charge-density correlation (CDC). This measure shows how close the system is to a ``checkerboard'' occupying every other site. For an odd number of sites, as visible in $L=3,5,9$, and negative $U$, the CDC is higher for two pairs than for one pair, by the nature of the measure [see Eq.~(\ref{eqn:cdc})] -- this is due to the fact that there is one extra pair which cannot be placed. For even numbers of sites, as seen in $L=10,20$ it is particle-hole symmetric, as expected, yielding the same result upon flipping all plots around the $\mu=0$ horizontal axis. CDCs do not follow the positive-$U$ to negative-$U$ mapping of the Fermi-Hubbard Hamiltonian. CDCs decrease as $U>0$ is increased, since the system is closer to a Mott product state. In the fourth row of Fig.~\ref{fig:fermiAll} we show the pairing correlations, which are of interest only in the negative-$U$ region where pairing naturally occurs due to the negative interaction.
In the fifth row of Fig.~\ref{fig:fermiAll}, we show the antiferromagnetic structure factor. This measure is the spatial fourier transform of the spin-spin correlation function evaluated at $q=\pi$, and is motivated by the fact that a canonical transformation maps the half-filled positive-$U$ Hubbard model in the strong coupling limit $U/t\gg 1$ onto a spin-1/2 antiferromagnetic Heisenberg model with $J=4 t^2/U$ in an external magnetic field $h=2\mu$
\ba
\hat{H}=J\sum_{\langle i,j\rangle}\mathbf{S}_i\cdot\mathbf{S}_j-h\sum_i \hat{S}^z_{i}\, .
\ea
This measure jumps sharply at half filling where the antiferromagnetic order is expected to exist, grows positively and reaches a maximum for some value of $U$, and then decreases to 0 for large $U$. The symmetry about $\mu$ is preserved because the operators involved are $\hat{S}_z^{\left(i\right)}=\frac{1}{2}\left(\hat{n}_{i \uparrow}^{\left(f\right)}-\hat{n}_{i\downarrow}^{\left(f\right)}\right)$ which vanish for on-site singlet pairs.
In the sixth row of Fig.~\ref{fig:fermiAll} we show the fidelity susceptibility. Due to our use of the grand canonical ensemble, only different total number states have an overlap significantly different from unity for small systems. For larger systems, fidelity susceptibility becomes less dominated by differences in number states as the number of jumps in number approaches the resolution in $(\mu/t,U/t)$, as was also the case for the Bose-Hubbard Hamiltonian in Fig.~\ref{fig:boseAll}.
\section{Key Numerical Comparisons}
\label{sec:compare}
\subsection{Grand Canonical vs. Canonical Ensembles}
\label{ssec:ensemble}
\begin{figure}
\begin{center}
\epsfxsize=8cm \epsfysize=6cm \leavevmode \epsfbox{fig7.eps}
\caption{\label{fig:ensemble} (color online) \emph{Comparison of Canonical Ensemble (CE) and Grand Canonical Ensemble (GCE).} Shown is the Q-measure (generalized entanglement) for the Bose-Hubbard Hamiltonian and $L=20$ sites. The CE is restricted to curves of fixed number difference. To obtain the images in Fig.~\ref{fig:boseAll}, we interpolate the CE by taking the nearest fixed line and thus ``filling in'' the phase diagram. The GCE shows that this method obtains the correct result, thus overcoming the finite differencing error inherent to the CE: $\mu\approx E(N+1)-E(N)$. Small qualitative differences remain between the two ensembles, as can be observed by comparing to the corresponding panel (second row, fourth column) in Fig.~\ref{fig:boseAll}.}
\end{center}
\end{figure}
\begin{figure*}
\begin{center}
\epsfxsize=\textwidth \epsfysize=0.75\textwidth \leavevmode \epsfbox{fig8.eps}
\caption{\label{fig:fullCE} \emph{Quantum Measures for the Bose-Hubbard Hamltonian in the Canonical Ensemble.} This figure shows the actual data underlying Fig.~\ref{fig:boseAll}; for smaller systems, the canonical ensemble provides very poor resolution in chemical potential, as can be observed in the first few columns. As the system size increases, the resolution in chemical potential goes up and the Mott lobe tip appears.}
\end{center}
\end{figure*}
The canonical ensemble is much preferable for large systems because it is more efficient. However, in order to obtain a chemical potential one must approximate the derivative in $\mu \equiv (\partial E / \partial N)|_V$. Our volume is automatically fixed by performing the derivative for a given system size $L$. The derivative is then evaluated by finite differences, either a simple forward difference of $\mu \approx E(N+1)-E(N)$, or a higher order finite difference. At any level of finite difference approximation, for smaller systems there is only a very low resolution for the chemical potential axis. This leads to a lack of information about quantum measures. This issue is highlighted in Fig.~\ref{fig:ensemble} by showing what the actual data is for the canonical ensemble (right panel) as compared to the grand canonical ensemble (left panel) for 20 sites. In the canonical calculation, each horizontal curve represents an allowed value of $N$. The grand canonical ensemble allows us to motivate the interpolation of the canonical ensemble result to go past the finite-difference result and obtain a continuous plot. Our interpolation algorithm is to take the value from the nearest curve according to the standard distance measure $\sqrt{(\mu_2-\mu_1)^2-(t_1-t_2)^2}$. Figure~\ref{fig:fullCE} shows the actual canonical ensemble data used to produce Fig.~\ref{fig:boseAll}.
The initial state chosen for our imaginary time propagation has the same projection onto every Fock state in the Hilbert space allowed by our numerical truncation. In our grand canonical calculations, this includes states which do not have the same total number, $N$. However, for small systems, number states are sufficiently different in energy that imaginary time propagation converges on a state with a definite $N$. On the border between number states one requires a very high $\chi$ to deal with the near degeneracy of states with different $N$. This leads to the regular pattern of bright spots seen in the left panel of Fig.~\ref{fig:ensemble}. The sole effect of higher $\chi$ is to remove bright spots from the entropy and Q-measure. The density of such low-entropy spikes increases with increasing $L$ because a higher $\chi$ is needed for a larger system -- more sites means the possibility for more spatial entanglement. This is another reason why canonical ensemble calculations are preferred for bosons. We have verified that all panels in Fig.~\ref{fig:boseAll} match the grand canonical picture, with the exception of the bright spots, which do not occur in canonical ensemble calculations.
Although we do not illustrate it in Fig.~\ref{fig:ensemble}, the variance in the total number, $\langle \hat{N}^2\rangle - \langle \hat{N}\rangle^2$, where $\hat{N} \equiv \sum_i \hat{n}^{(b)}_i$, is zero to 5 or 6 digits of precision. This is in fact a measure of convergence for our grand canonical simulations; we require all Schmidt values (eigenvalues of the reduced density matrix) be converged to 10 digits in imaginary time propagation, and obtain this total number variance, having started with an initial state with weights on all total numbers allowed by the cut-off $d$, i.e., from 0 atoms to $\left(d-1\right)\times L$ atoms.
\subsection{Gutzwiller Mean Field vs. Entangled Time-Evolving Block Decimation}
\label{ssec:meanfield}
\begin{figure*}
\begin{center}
\epsfxsize=0.8\textwidth \epsfysize=\textwidth \leavevmode \epsfbox{fig9.eps}
\caption{\label{fig:chi} (color online) \emph{Increasing Corrections to Mean Field Gutzwiller Ansatz}. The same quantities as calculated in Fig.~\ref{fig:fermiAll} for the Fermi-Hubbard Hamiltonian, but for 10 sites and in increasing values of $\chi$ from left to right: 1,2,4,8. $\chi=1$ corresponds to the Gutzwiller mean field approximation. The speckle structure observed in the vacuum region and for entanglement in the $\chi=1$ case is simply numerical noise associated with the calculation of small numbers.}
\end{center}
\end{figure*}
Since mean field methods are usually the most tractable approaches to solution of many body problems, identifying their domain of validity is important. For instance, for the three-dimensional Bose-Hubbard Hamiltonian one can obtain the phase diagram via the Hubbard-Stratonovich transformation~\cite{fisher1989}, which is a mean field method, or alternately, via the Gutzwiller mean field approximation, as discussed in Sec.~\ref{sec:introduction}. On the other hand, the Gutzwiller method fails for the one-dimensional Bose-Hubbard Hamiltonian~\cite{lewensteinM2007}. It is natural to ask, how far beyond mean field does one need to go in order to obtain correct results for essential quantum measures?
To address this question, we repeat the calculations of Sec.~\ref{ssec:fermi} for the Fermi-Hubbard Hamiltonian for 10 sites only, starting with the mean field calculation of $\chi=1$ and then doubling $\chi$ successively. Figure~\ref{fig:chi} shows the results: each column corresponds to $\chi=1,2,4,8$ while the rows correspond to the same quantum measures as Fig.~\ref{fig:fermiAll}. Note that $\chi=16$ can be found in the third column of Fig.~\ref{fig:fermiAll}, so we do not repeat it here. We make several remarks concerning this figure. The density is a first moment and only requires lower values of $\chi$, as can be seen in the first row. For $\chi=1$ in the first column, the entanglement shown in the second row is zero to 16 digits of accuracy, reflecting our use of double precision floating point arithmetic in our implementation of TEBD. The entanglement does not converge until $\chi=16$, as can be determined from a comparison of Fig.~\ref{fig:chi} and Fig.~\ref{fig:fermiAll}.
In the third row, for large positive $U/t$ we obtain the correct result for charge density correlations at half-filling even for very low $\chi$ because the Mott phase is nearly a product state. The pairing phase for negative $U/t$, on the other hand, requires higher $\chi$, as can be seen in the fourth row.
This can be understood from the fact that superfluids and superconductors are highly entangled in position space; similarly, in the Bose Hubbard Hamiltonian one must go to high $\chi$ in order to resolve the superfluid region far away from the Mott lobes. By ``spatial entanglement'' we refer to exactly the Schmidt number used in TEBD. The fifth row shows anti-ferromagnetic order.
Finally, in the sixth row we observe that the main features of the fidelity susceptibility appear already for $\chi=2$, because fidelity measures in the grand canonical ensemble are mainly sensitive to total number states in our system; correct total number states are obtained at very low $\chi$. We observe that particle-hole symmetry is maintained even at the mean field level of $\chi=1$, while the positive-$U$ to negative-$U$ mapping at half-filling and zero chemical potential is only good for at least $\chi=2$.
Although Fig.~\ref{fig:chi} shows that $\chi=16$ is sufficient to obtain by-eye converged values of all quantum measures considered in Fig.~\ref{fig:fermiAll}, a more careful quantitative study is desirable. In Fig.~\ref{fig:error} we show this study for both fermions and bosons for 10 sites. Both the average error (left panels) and maximum error (right panels) over all pixels in $(\mu/t,U/t)$ for fermions and $(\mu/t,t/U)$ for bosons are evaluated. The error for any given pixel $p$ is defined as
\be
\varepsilon_p(M) \equiv \left|\frac{M_{p}(2^j)-M_{p}(2^{j-1})}{\frac{1}{2}[M_{p}(2^j)+M_{p}(2^{j-1})]}\right|\,,
\ee
where $M(2^j)$ is the quantum measure of interest evaluated at Schmidt number $\chi=2^j$ and $j$ is an integer, in our case taken to be 1, 2, 3, 4, 5. That is, we evaluate measure $M$ at $\chi$ of 16, 8, 4, 2 and compare it to the evaluation of the same measure at the previous smaller value of $\chi$. The maximum and average errors are given by
\ba
\varepsilon_{\mathrm{max}}(M)&\equiv&\max_p \varepsilon_p(M)\,,\\
\bar{\varepsilon}(M)&\equiv&\frac{\sum_p \varepsilon_p(M)}{\sum_p 1}\,.
\ea
For fermions we calculate these two errors for density, Q-measure, and pairing in the grand canonical ensemble for 10 sites, corresponding to Fig.~\ref{fig:chi}. For brevity, we do not show an analogous figure to Fig.~\ref{fig:chi} for bosons, but calculate maximum and average error for $L=10$ sites and the grand canonical ensemble shown in Fig.~\ref{fig:ensemble} in the third column. As can be seen from the figures, the average error is about 1\,\% for $\chi=16$ for bosons, and the maximum error is about 10\,\%. For fermions it somewhat larger for the same $\chi$. The maximum error appears large, but upon a detailed investigation we find that this error is localized in the vacuum or band insulator regions; this means that the max error is dominated by imaginary time propagation effects. Other regions have an error corresponding to the average error, including the important half-filling Mott region for positive $U$. In general, local measures like density or entanglement converge faster than non-local measures like quantum depletion, as can be seen in the figure. Also, the bosonic measures converge more quickly than do the fermionic measures.
\begin{figure}
\begin{center}
\epsfxsize=8cm \epsfysize=8cm \epsfbox{fig10.eps}
\caption{\label{fig:error} (color online) \emph{Quantitative Error Analysis for increasing entanglement ($\chi$) in TEBD}. Average error (left column) and maximum error (right column) for the Fermi-Hubbard Hamiltonian (top row) and Bose-Hubbard Hamiltonian (bottom row) as a function of increasing powers of $\chi$. For fermions, we show the quantum measures of density (red squares), Q-measure or generalized entanglement (blue triangles), and pairing correlations (green circles); for bosons, we show the error in the Q-measure (red squares) and quantum depletion (blue triangles) in the grand canonical ensemble. Curves are a guide to the eye only.}
\end{center}
\end{figure}
\section{Conclusions}
\label{sec:conclusion}
We explored finite size effects in the Bose- and Fermi-Hubbard Hamiltonians with TEBD and imaginary time propagation, studying a range of quantum measures including moments, correlations, entanglement, and fidelity. We found that entanglement~\cite{nielsenMA2000,weiTC2005} followed the same pattern as more traditional measures for quantum phase transitions, such as depletion for the Bose-Hubbard Hamiltonian and the superfluid-Mott-insulator transition, and pairing, charge-density, and antiferromagnetic structure factor for the Fermi-Hubbard Hamiltonian in the Mott transition; the exception was the tip of the Mott lobe, which was best described by the depletion, in terms of converging to the thermodynamic result more quickly as a function of system size.
Fidelity susceptibility provides a useful alternative measure for characterizing the phase transition, and is a non-local measure.
We compared results computed within the grand canonical ensemble and canonical ensembles: the chemical potential can be approximated in the canonical ensemble by a finite difference, a method only useful for large $N$. However, by first performing a grand canonical calculation and then using it to motivate an interpolation of the canonical ensemble we could subsequently use the more efficient canonical calculations to obtain more highly converged results. The canonical ensemble is also more useful for fidelity susceptibility calculations, which are otherwise dominated by changes in the conserved quantity of total number.
We performed quantitative convergence studies and showed that particle-hole symmetry is maintained in the particle-hole symmetric form of the Fermi-Hubbard Hamiltonian for all values of $\chi$, but the positive-$U$ to negative-$U$ mapping at half-filling and zero chemical potential is correct only for at least $\chi=2$, meaning it is not satisfied in the mean field approximation. We found that for bosons the tip of the Mott lobe is wide-open for smaller system sizes of less than about 20 sites, so that exploring quantum phase transitions by changing chemical potential finds sharp effects while changing hopping find a very broad crossover.
We acknowledge useful discussions with Ippei Danshita, Ludwig Mathey, Ryan Mishmash, Alejandro Muramatsu, and Marcos Rigol. This work was supported by the National Science Foundation under Grant PHY-0547845 as part of the NSF CAREER program (LDC, MLW) and by the Aspen Center for Physics (LDC), by the Summer Undergraduate Research Fellow (SURF) program at the National Institute of Standards and Technology (DGS, RCB), and by the NSF under Physics Frontiers Center grant PHY-0822671 (RCB, CWC).
|
1,108,101,563,293 | arxiv | \section{Introduction}
The complexification of field theories has attracted
recent attention in the context of solving the sign problem.
The complex Langevin (CL) method
\cite{Parisi:1984cs,Klauder:1983sp,Seiler:2012wz}
and the Lefshetz thimble (LT) method
\cite{Pham:1983,Witten:2010cx,Witten:2010zr}
rely on the complexification of dynamical variables of the system.
Although they are hopeful candidates as a solution for
the sign problem, there are still some controversies over their
feasibility or practicality.
One of the difficulties lies in the lack of knowledge of complexified theories,
which we would like to address in this work.
The Langevin method is based on the stochastic quantization \cite{Parisi:1980ys}.
As for real actions,
the Langevin method is ensured to produce
correct results in an infinitely large Langevin-time limit, where the
probability distribution of dynamical variables approaches to
the Boltzmann weight as indicated in the
eigenvalue analysis of the Fokker-Planck
equation, e.g. see Refs.~\cite{Namiki:1992,Damgaard:1987rr}.
As for complex actions \cite{Parisi:1984cs,Klauder:1983sp},
observables in the CL method sometimes converge to wrong results.
Recently, Aarts et al.~showed that the CL method can be justified
if some conditions are satisfied~\cite{Aarts:2009uq,Aarts:2011ax}.
Possible reasons of its breakdown are the spread of CL configurations in the
complexified direction in the complex plane of field
variables~\cite{Aarts:2009uq,Aarts:2011ax}
and singular drift terms of CL equations \cite{Mollgaard:2013qra,Greensite:2014cxa,Nishimura:2015pba}.
A new method, which is referred to as the gauge cooling,
has been proposed to suppress the spread of
configurations in the imaginary direction by using the
complexified gauge invariance~\cite{Seiler:2012wz,Sexty:2013ica,Fodor:2015doa}.
It was shown that the method reproduces correct results of physical
observables in some cases \cite{Seiler:2012wz}.
Later, the gauge cooling method is justified in Ref.~\cite{Nagata:2015uga},
and extended to theories that have a global symmetry with sufficiently large
number of generators~\cite{Nagata:2015ijn}.
In addition, it was found in Ref.~\cite{Mollgaard:2013qra} that the singular drift
problem occurred in a chiral random matrix theory (ChRMT), where complex
eigenvalues of the fermion matrix touch the origin at non-zero chemical potential.
Here the fermion matrix is given as $D+m$ with a Dirac operator $D$.
Zero modes of the fermion matrix cause a singularity in a drift term of the Langevin equation.
Such a singularity originates from the logarithmic term in the effective action,
which is expected to occur for fermionic theories quite in general.
Mollgaard and Splittorff pointed out that
a suitable parameterization of dynamical variables solves this problem in the ChRMT~\cite{Mollgaard:2014mga}.
They showed that the fermion matrix eigenvalues are well
localized, deviating from the origin,
and the simulation converges to the correct result.
It was also shown in Ref.~\cite{Nagata:2015ijn} that the gauge
cooling can be extended to the singular drift problem by using suitable norms.
With regard to the convergence problem,
it is interesting to focus on universal quantities to understand
if the CL method converges correctly.
The CL method can reproduce correct results of the universal
quantities that are defined as holomorphic quantities
if the conditions for justification are satisfied~\cite{Aarts:2009uq,Aarts:2011ax}.
On the other hand, it is unclear if the CL method can reproduce
physical results for non-holomorphic quantities.
However, it may be natural to expect that the original universality
is preserved for the correct convergence cases as in one classical thimble calculation
in the LT method \cite{Cristoforetti:2012su}.
The nearest neighbor spacing (NNS) distribution is one of the universal quantities
which reflect the correlations of the fermion matrix eigenvalues
and the property of the matrix.
The NNS distribution has been investigated in the phase quenched QCD at zero
and non-zero chemical potential by using MC simulations \cite{Markum:1999yr}.
They showed that the NNS distribution obeys some kinds of distributions:~the
Ginibre, Wigner, and Poisson distributions,
depending on the system and the strength of the eigenvalue correlations.
We note that the NNS distribution is obtained from the density of the fermion matrix eigenvalues
and it is not a holomorphic quantity~\cite{Stephanov:1996RM}.
This fact implies that the distribution obtained in the CL method does not
necessarily reproduce the NNS distribution in the original theory~\cite{Nagata:2015ijn}.
However, the Dirac eigenvalues are closely related to the convergence property of
the CL method, and the NNS distribution is a universal quantity obtained from Dirac
eigenvalues.
Therefore, it can be valuable to investigate the universal behavior of the NNS
distribution of the Dirac eigenvalues in the CL method
deeply to understand the complexified theory.
We conjecture that the NNS distribution can reproduce physical
results in the CL method even though it is non-holomorphic.
This conjecture may be natural if the NNS distribution
reflects the properties of the critical points (or saddle points) owing to its universality.
Since the Dirac eigenvalues can depend on the choice of the convergence properties
of the CL method~\cite{Mollgaard:2014mga}, then at least we can expect that
the convergence property of the CL method also affect the NNS distribution.
In the CL simulations, configurations fluctuate
around a classical flow and locate around some critical points.
The different convergence properties indicate that the configurations
locate around different critical points.
Then, we expect that the underlying universality at each critical point
can be different in the two cases with the correct and wrong convergences
in ChRMT \cite{Mollgaard:2013qra,Mollgaard:2014mga}.
In this work, we investigate the NNS distributions of the ChRMT
in the CL method,
where we employ two parameterizations based on
Refs.~\cite{Mollgaard:2013qra,Mollgaard:2014mga}:
one leads to correct convergence and the other leads to wrong one.
Our purposes are following:
(I) analyzing what happens in the wrong and correct convergent cases in the NNS distribution
and (II) comparing the CL result with the previous MC expectations for
the NNS distribution.
This paper is organized as follows.
In the next section, we define the ChRMT and introduce the two parameterizations.
In \secref{Sec:Formalism}, we introduce the CL method.
We also explain the procedure for calculating the
NNS distribution with complex eigenvalues~\cite{Markum:1999yr} in \secref{Sec:NNS}.
In \secref{Sec:Results}, we give results on the NNS distribution and discussions of them.
\secref{Sec:Summary} is devoted to a summary.
\section{Chiral Random Matrix Theory}
\label{Sec:Formalism}
\subsection{Two types of representations}
In this work, we study the ChRMT introduced in
Ref.~\cite{Osborn:2004rf} in
two types of the representation: first one is used in Ref.~\cite{Mollgaard:2013qra},
and the second one is used in Ref.~\cite{Mollgaard:2014mga}.
Those two representations are equivalent under the linear transformation of dynamical variables
before complexification \cite{Bloch:2012bh}.
In both cases, we describe originally complex dynamical variables
in the polar coordinate.
Throughout this work, we consider only the case of zero topological-index and of
two flavor $N_f=2$.
The partition function of the ChRMT is given as
\begin{align}
Z &= \int d\Phi_1 d\Phi_2 (\det (D +m) )^{N_f}
e^{-N \tr (\Phi_1^{\dagger} \Phi_1 + \Phi_2^{\dagger} \Phi_2 )}
\nonumber \\
&= \int d \Phi_1 d \Phi_2 \exp (-S) \ .
\end{align}
$\Phi_{1,2}$ are $N\times N$ complex matrices.
\vspace{3mm}
\underline{Representation (I): Hyperbolic type}
\vspace{3mm}
First, we consider the following case, where the action and the Dirac operator are
\begin{align}
&S = N \mathrm{Tr} [\Phi_1^{\dagger} \Phi_1 + \Phi_2^{\dagger} \Phi_2]
- N_f \mathrm{Tr} \log (G^{-1})
\ , \\
&D (\mu)+m = \left(
\begin{matrix}
0 & X \\
Y & 0
\end{matrix}
\right) +m
\ ,
\end{align}
here $G^{-1} = m^2 -XY$.
The $X$ and $Y$ are complex $N \times N$ matrices,
\begin{align}
X &\equiv i \cosh (\mu) \Phi_1 + \sinh (\mu) \Phi_2^\dagger
\ , \\
Y & \equiv i \cosh (\mu) \Phi_1^\dagger + \sinh (\mu) \Phi_2
\ .
\end{align}
We call this parameterization hyperbolic (Hyp) representation below.
Here, elements of the random matrices, $\Phi_1$ and $\Phi_2$, are originally complex, and
can be parameterized in terms of two real variables.
In the present study, we employ the polar coordinate~\cite{Mollgaard:2014mga}, where
\begin{align}
\PhiM{1}{i j} = \rM{1}{i j} e^{i \tM{1}{i j} }
\ \ ,\
\PhiM{2}{i j} = \rM{2}{i j} e^{i \tM{2}{i j} }
\ ,
\end{align}
for $i,j = 1,\cdots, N$. The matrices $X$ and $Y$ are given as
\begin{align}
X_{ij} &= i \cosh (\mu) \rM{1}{i j} e^{ i \tM{1}{ij} }
+ \sinh (\mu) \rM{2}{j i} e^{-i \tM{2}{j i}}
\ , \\
Y_{i j} & = i \cosh (\mu) \rM{1}{j i} e^{- i \tM{1}{ji}}
+ \sinh (\mu) \rM{2}{i j} e^{i \tM{2}{i j}}
\ .
\end{align}
The action is rewritten as
\begin{align}
S =& - \sum_{i,j} \left( \log (\rM{1}{i j}) + \log (\rM{2}{i j}) \right)
\nonumber \\
& - N_{f} \log \det (G^{-1})
+ N \sum_{i,j} \left( \rM{1}{ij}^2 + \rM{2}{ji}^2 \right)
\ .
\label{Eq:action_HYP}
\end{align}
\underline{Representation (II): Exponential type}
\vspace{3mm}
Next, we consider the other parameterization used in Ref.~\cite{Mollgaard:2014mga} as
\begin{align}
X \equiv & e^\mu \Phi_1 - e^{-\mu} \Phi_2^\dagger \ , \\
Y \equiv & -e^{-\mu} \Phi_1^\dagger + e^\mu \Phi_2\ .
\end{align}
The action is given by
\begin{align}
S = 2 N {\rm Tr} [ \Phi_1^\dagger \Phi_1 + \Phi_2^\dagger \Phi_2] - N_f {\rm Tr} \log (G^{-1}).
\end{align}
This parameterization is referred to as Exponential (Exp) representation below.
We also use the polar coordinate in this representation.
The two representations are equivalent under a linear transformation \cite{Bloch:2012bh,Mollgaard:2014mga}.
Note that the CL method is applied to the ChRMT with the Hyp representation and with the Cartesian
coordinate in Ref.~\cite{Mollgaard:2013qra}, while it is applied to the ChRMT with
the Exp representation and with the polar coordinate in Ref.~\cite{Mollgaard:2014mga}.
In this work, we adopt the polar coordinate both for the two representations.
We will confirm that only the case of the Exp representation reproduces
the correct results if the mass of quark is small at non-zero quark chemical potential.
\subsection{Langevin equations and drift terms}
Now, we apply the Langevin equation to the originally real variables
in the system $\varphi_k \in
\{ r_{1, ij} , r_{2, ij} , \theta_{1,ij}, \theta_{2,ij}|\, i, j = 1, \cdots, N, ( i, j \in \mathbb{Z})\}$.
\begin{align}
\frac{\partial \varphi_k (\tau)}{\partial \tau} = - \frac{\partial S [\varphi]}{\partial \varphi_k} + \eta_k (\tau)
\ ,
\end{align}
where $\tau$ is the Langevin time.
The Gaussian noise, $\eta_k$, is normalized as $\expv{\eta_k (\tau )} = 0$
and $\expv{\eta_k (\tau) \eta_l (\tau')} = 2 \delta_{kl} \delta (\tau -\tau') $.
Since the action is complex at $\mu \neq 0$, those variables are extended to complex
$r_1, r_2, \theta_1, \theta_2 \in \mathbb{R} \to \mathbb{C}$.
In this paper, we use the real Gaussian noise $\eta_k \in \mathbb{R}$
and the adaptive step size method
\cite{Ambjorn:1986fz,Aarts:2009dg,Aarts:2010aq}.
We explicitly show the drift terms in the Hyp
representation in \secref{sec:driftterms}, and those in the Exp representation
were shown in Ref.~\cite{Mollgaard:2014mga}.
In the numerical simulation, we define the discretized Langevin equation
\begin{align}
\varphi_k (\tau+d\tau) = \varphi_k (\tau) + d \tau \biggl(- \frac{\partial S [\varphi]}{\partial \varphi_k}
\biggr) + \sqrt{d\tau}\, \eta_k (\tau).
\end{align}
In the adaptive stepsize method in Ref.~\cite{Aarts:2010aq},
we can adopt an average value of a maximum drift term
at each Langevin time before the thermalization time $\tau_{\rm th}$,
\begin{align}
\left< \mathcal{K}_\mrm{max} \right>_\mrm{th}=
\frac{1}{N_\mrm{th}} \sum_{\tau' =0}^{\tau_\mrm{th}}
\mathcal{K}_\mrm{max} (\tau')
\ ,
\end{align}
where
$\mathcal{K}_\mrm{max} (\tau')
= \max_{k,\varphi=r_1,r_2,\theta_1,\theta_2} \left| \varphi_k (\tau') \right| $
and
$N_\mrm{th} = \tau_\mrm{th}/d\tau$ is the number of configurations
until the thermalization time, $\tau_\mrm{th}$.
After the thermalization time,
we adopt an adaptive stepsize at each Langevin time as
\begin{align}
\epsilon_{\tau} = \min (d\tau, d\tau \left< \mathcal{K}_\mrm{max} \right>_\mathrm{th} / \mathcal{K}_\mrm{max} (\tau) )
\ ,
\end{align}
and then the CL equation is given as
\begin{align}
\varphi_k(\tau + \epsilon_\tau)
= \varphi_k (\tau) + \epsilon_\tau \left(- \frac{\partial S [\varphi]}{\partial \varphi_k} \right)+ \sqrt{\epsilon_\tau} \eta_k (\tau) \ .
\end{align}
\section{Nearest Neighbor spacing distribution} \label{Sec:NNS}
In order to obtain the nearest neighbor spacing (NNS) distributions,
we need a procedure so called unfolding.
We follow the unfolding procedure introduced in Ref.~\cite{Markum:1999yr}.
\subsubsection{Unfolding procedure with complex eigenvalues}
First, we define the density of Dirac eigenvalues as
\begin{align}
\rho (x,y) = \expv{ \sum_k \delta^{(2)} ( z - z_k )} \ , \, ( z = x + i y)
\end{align}
where $\expv{\cdots}$ denotes an ensemble average
and $z_k$ is an eigenvalue of the Dirac operator
$ D(\mu) \ket{z_k} = z_k \ket{z_k},\, (k=1, \cdots 2N)$.
Due to the property of the Dirac operator $\{ \gamma_5, D(\mu)\}=0$,
the eigenvalues appear as $\pm z$ pair \cite{Akemann:2004dr}.
The density of the Dirac eigenvalue $\rho(x,y)$ is non-holomorphic~\cite{Stephanov:1996RM}.
In order to obtain the fluctuation part of the
eigenvalue density, we rewrite $\rho(x,y)$ as
\begin{align}
\rho(x,y) = \rho_{\mrm{ave} } (x,y)+ \rho_{\mrm{fluc}}(x,y) \ ,
\end{align}
where $\rho_{\mrm{ave} }$ and $\rho_{\mrm{fluc}}$ are the average part and the
fluctuation part of the eigenvalue density, respectively.
We consider a map
\begin{align}
z' = x' + i y' = u(x,y) +i v(x,y)
\label{eq:2016feb16eq1}
\ ,
\end{align}
where we impose a condition that the average eigenvalue density is unity for any points on the new coordinate,
namely $\rho_{\mrm{ave} } (x',y')=1$.
It immediately follows from the probability conservation condition that
$\rho_{\mrm{ave} } (x',y')dx'dy'=dx'dy' =\rho_{\mrm{ave}} (x,y)dxdy$ \cite{Markum:1999yr}.
On the other hand, it follows from Eq.~(\ref{eq:2016feb16eq1})
that $dx' dy' = J dx dy,$ where $J=|\partial (x',y')/\partial(x,y)|$ is the Jacobian
of the coordinate transformation. Combining the two relations implies that the
$\rho_{\mrm{ave}} (x,y)$ is nothing but the Jacobian, i.e.,
\begin{align}
\rho_{\mrm{ave}} (x,y) = \left| \partial_x u \partial_y v - \partial_x v \partial_y u \right|
\ .
\label{eq:2016feb16eq2}
\end{align}
$u(x,y)$ and $v(x,y)$ are not determined uniquely only from this condition.
We choose $y'=v(x,y)=y$, which reduces Eq.~(\ref{eq:2016feb16eq2}) to
$\rho_{\mrm{ave}} (x,y) = \partial_x u $ \cite{Markum:1999yr}.
The real part of eigenvalues is expressed as
\begin{align}
x' = u(x,y) = \int_{-\infty}^{x} dt \rho_{\mrm{ave}} (t,y) \equiv N_{\mrm{ave}} (x,y)
\ ,
\end{align}
where $N_{\mrm{ave}}(x,y)$ is the average part of the cumulative
spectral function $N(x,y)$~\cite{Guhr:1997ve}, which is given as
\begin{align}
N (x,y ) = N_{\mrm{ave}} (x,y) + N_{\mrm{fluc}} (x,y)
\ .
\end{align}
\begin{figure}[htbp]
\includegraphics[width=60mm, angle=270]{p000-N20-mu3-m2-Exp-cubic.eps}
\includegraphics[width=60mm, angle=270]{p000-N20-mu3-m2-Exp.eps}
\caption{The cumulative spectral function $N(x,y)$ (histogram) and its
average part $N_{\rm ave}(x,y)$ (solid line) as a function of $x$
at $N=20$, $\tilde{m}=2$, and $\tilde{\mu}=3$ in the Exp representation.
They are obtained from eigenvalues included in a strip around $y=0$.
The top and bottom panels are obtained with a cubic function
and a quintic function, respectively.
\label{fig:Cumu-N20mu3m2}
} \label{fig:Unfolding-N20mu3m2}
\end{figure}
In order to obtain the cumulative spectral function numerically,
we first divide the complex plane in the $y$-direction, where
each strip has a width $dy$.
In the ChRMT, a spacing between two adjacent eigenvalues
is of an order $\mathcal{O}(1/N)$ \cite{Verbaarschot:2000dy}.
We choose the width $dy$ so that it is bigger than this magnitude:
$dy = 0.1$ at $N=20$, for example.
If we adopt too small $dy$, there are few eigenvalues in a given strip.
Then, we calculate $N(x,y)$ for each strip.
In our calculations, we derive the average cumulative spectral function
by fitting $N_{\mrm{ave}}$ with a low order polynomial as in Ref.~\cite{Markum:1999yr}.
Then, we adopt a sufficiently small bin size for $x$
to fit $N_\mrm{ave}$ appropriately.
We use a quintic polynomial throughout this paper, but the
validity of the fitting procedure is checked by comparing results with
the quintic and cubic polynomials.
In \figref{fig:Cumu-N20mu3m2}, we show $N_{\rm ave}(x,y)$ obtained from
the cubic polynomial (top panel), and from the
quintic polynomial (bottom panel).
We find that the NNS distribution is quantitatively insensitive to the choice
of the fitting functions, which will be shown in Appendix \ref{Sec:Fit}.
\begin{figure}[htbp]
\includegraphics[width=30mm, angle=270, bb= 0 0 595 842]{EV-N20-m2-mu3}
\includegraphics[width=30mm, angle=270, bb= 0 0 595 842]{unfolding-N20-m2-mu3}
\caption{
Dirac eigenvalues
at $N=20$, $\tilde{m}=2$, and $\tilde{\mu}=3$
before unfolding procedure (left panel) and after unfolding procedure (right panel).
They are obtained from configurations in late Langevin time.
\label{fig:EV-N20mu3m2}
}
\end{figure}
To illustrate the unfolding procedure, we show Dirac eigenvalues at $N=20$,
$\tilde{m}=2$, and $\tilde{\mu}=3$
obtained with various ensembles at late Langevin times in \figref{fig:EV-N20mu3m2}
where $\tilde{m}=Nm$ and $\tilde{\mu}=\sqrt{N}\mu$.
As in Ref.~\cite{Mollgaard:2014mga},
the width of Dirac eigenvalues in the Exp representation
decreases along the real axis compared with that in the Hyp representation.
The right panel shows the Dirac eigenvalues after the unfolding.
The unfolded distribution is apparently consistent with the
distribution obtained in the phase quenched QCD~\cite{Markum:1999yr}.
\subsubsection{Nearest Neighbor spacing distribution with complex eigenvalues}
In order to derive the NNS distribution from complex eigenvalues,
we need to introduce the NNS, $s$.
There is an ambiguity to define $s$ for the complex
eigenvalues.
We adopt $s_i (\tau) = \min_j \left| z_i' - z_j' \right|$ for
$i = 1, \cdots, 2N$ using unfolded
eigenvalues at Langevin time $\tau$ \cite{Markum:1999yr}.
We construct the distribution $p(s)$ of $s_i (\tau)$ for $(i =1,\cdots,2N)$,
where we use all the configurations
in late Langevin time
with a certain interval.
The NNS distribution $P(s)$ is defined so that it satisfies
two conditions $\int ds P(s) = 1$ and $\int ds s P(s) = 1$.
In order to satisfy these conditions, we use the normalization
and rescaling of $s$ \cite{Markum:1999yr}:
supposing that the first moment of the distribution is $c =
\int_0^{\infty} ds s p(s)$ with $1 = \int_0^{\infty} ds p(s)$,
the new distribution is defined by $P(s) = c p(c s)$.
Once the distribution satisfies the two conditions,
we can compare the distribution obtained in the present work
with some typical distributions.
\subsubsection{Reference NNS distributions}
Here, we explain three typical NNS distributions, which are used as references
to understand numerical results obtained from CL simulations.
At $\mu=0$, the Dirac operator is anti-Hermitian, and its eigenvalues are pure imaginary.
The NNS distribution in the chiral unitary ensemble of RMT follows the Wigner surmise~\cite{Halasz:1995vd,Guhr:1997ve,Markum:1999yr}
\begin{align}
P_W(s) = \frac{32}{\pi^2}s^2 e^{-4s^2/\pi}.
\end{align}
It is expected that the NNS distribution in this work also follows the Wigner surmise.
We will discuss the mass dependence in \secref{Sec:Results}.
On the other hand, at $\mu\neq 0$, the Dirac operator is no more anti-Hermitian,
and its eigenvalues are generally complex.
If the real and imaginary part of eigenvalues have approximately the same average magnitude, then
the system is described by the Ginibre ensemble \cite{Ginibre:1965zz} of non-Hermitian RMT~\cite{Ginibre:1965zz,Markum:1999yr}.
In this case, the NNS distribution is given by
\begin{align}
P_G(s) & = c p(cs) , \\
p(s) & = 2s \lim_{N \to \infty} \left[ \prod_{n=1}^{N-1} e_n(s^2) e^{-s^2} \right] \sum_{n=1}^{N-1} \frac{s^{2n}}{n! e_n(s^2)}
\label{eq:ps-anal},
\end{align}
where $e_n(x) = \sum_{m=0}^n x^m/m!$ and $c=\int_0^\infty ds s p(s) $ \cite{Grobe:1988zz,Markum:1999yr}.
In this paper, we use $N=2000$ in \equref{eq:ps-anal} as a reference distribution.
For uncorrelated eigenvalues, the NNS distribution follows the Poisson distribution.
On the complex plane, the Poisson distribution is given by \cite{Grobe:1988zz}
\begin{align}
P_{P} (s)= \frac{\pi}{2} s e^{-\pi s^2/4}.
\end{align}
\section{Results and discussion}
\label{Sec:Results}
In this section, we show numerical results obtained from the CL simulations.
Our numerical set up is as follows.
We consider the ChRMT with zero topological index $\nu =0$, $N =20$ and $N_{f} =2$.
We utilize the reference step size as $d\tau=1.0\times 10^{-6}$, and
perform the simulation with the adaptive stepsize~\cite{Aarts:2010aq}.
We take the Langevin time $\tau \sim 50$ for thermalization, and $\tau \sim 100$ for measurements.
The measurement is performed by using configurations in late Langevin time, and
the number of typical configurations for the measurement is 2000.
In the following, we show the numerical results for $\tilde{\mu}=1$ and $\tilde{\mu}=3$,
where $\tilde{m}=Nm$ and $\tilde{\mu}=\sqrt{N}\mu$.
\subsection{Results for $N=20$ at $\tilde{\mu} =1$}
In \figref{fig:N20-chiral-mu1}, we show the chiral condensate at $\tilde{\mu}=1$ as a function of $\tilde{m}$.
Here the chiral condensate is given by \cite{Mollgaard:2013qra},
\begin{align}
\frac{1}{N} \expv{\bar{\eta} \eta} &= \frac{1}{N} \expv{ \partial_m \log (Z) }
\nonumber \\
& = \frac{2m N_{f}}{N} \expv{ \mathrm{Tr} \left[ (m^2 - XY)^{-1} \right] }
\ .
\label{Eq:chiral}
\end{align}
There is a small difference between analytical results for the exact \cite{Osborn:2004rf} and phase quenched \cite{Akemann:2004dr} cases for small $\tilde{m}$, which are denoted as ``exact" and ``PQ exact"
in \figref{fig:N20-chiral-mu1}, respectively.
Numerical results with the Exp representation and Hyp representation are
almost consistent with analytical results for the exact and phase quenched cases.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=60mm, angle=270]{chiral-N20-mu1-adaptive.eps}
\end{center}
\caption{The chiral condensate at
$\tilde{\mu}=1$ as a function of $\tilde{m}$.
The solid line shows the exact value in the ChRMT \cite{Osborn:2004rf} and
the dotted line shows the exact value in the phase quenched (PQ)
ChRMT \cite{Akemann:2004dr}.
The results of the Exp and Hyp representations are expressed as squares and circles.
At large $\tilde{m}$, both representations well produce exact results.
Results of the Hyp representation slightly deviate from exact results at small $\tilde{m}$.
Dotted vertical lines show the values of $\tilde{m}$ at which the NNS distribution
is obtained.}
\label{fig:N20-chiral-mu1}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=60mm, angle=270]{Ps-tot-hyp-N20-mass2-mu1.eps}
\end{center}
\caption{
The NNS distribution of
the Hyp notation at $N=20$, $\tilde{m}=2$, and $\tilde{\mu}=1$.}
\label{fig:N20mu1m2H}
\begin{center}
\includegraphics[width=60mm, angle=270]{Ps-tot-hyp-N20-mass30-mu1.eps}
\end{center}
\caption{
The NNS distribution of
the Hyp notation at $N=20$, $\tilde{m}=30$, and
$\tilde{\mu}=1$.}
\label{fig:N20mu1m30H}
\end{figure}
We show the NNS distribution in the Hyp representation
for ($\tilde{\mu}, \tilde{m})=(1,2)$ and (1,30) in
Figs.~\ref{fig:N20mu1m2H} and \ref{fig:N20mu1m30H}, respectively.
The NNS distributions at $\tilde{m}=2$ and $30$ are almost
consistent with the Wigner surmise.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=60mm, angle=270]{Ps-tot-exp-N20-mass2-mu1.eps}
\end{center}
\caption{
The NNS
distribution of
the Exp notation at $N=20$, $\tilde{m}=2$, and $\tilde{\mu}=1$.}
\label{fig:N20mu1m2E}
\begin{center}
\includegraphics[width=60mm, angle=270]{Ps-tot-exp-N20-mass30-mu1.eps}
\end{center}
\caption{
The NNS
distribution of
the Exp notation at $N=20$, $\tilde{m}=30$, and $\tilde{\mu}=1$.}
\label{fig:N20mu1m30E}
\end{figure}
We show the NNS distribution in the Exp representation
for ($\tilde{\mu}, \tilde{m})=(1,2)$ and (1,30) in
Figs.~\ref{fig:N20mu1m2E} and \ref{fig:N20mu1m30E}, respectively.
For $\tilde{m}=30$, the NNS distribution in the Exp representation is almost
the same as that in the Hyp representation.
By comparison, the NNS distribution at $\tilde{m}=2$ is slightly different from
NNS distribution at $\tilde{m}=30$ and is relatively close to the Wigner surmise.
For small $\tilde{m}$, the NNS distributions in the two representations
are slightly different.
The difference increases for $\tilde{\mu}=3$ as we will
show in the next subsection.
\subsection{Results for $N=20$ at $\tilde{\mu} =3$}
\label{Sec:results}
In \figref{fig:N20-chiral},
we show the chiral condensate at $\tilde{\mu} = 3$ as a function of $\tilde{m}$.
The chiral condensate in the Exp representation reproduces the correct result
shown by the solid line, as found in \cite{Mollgaard:2014mga}.
On the other hand, the chiral condensate in the Hyp representation
produces results close to the phase quenched theory
shown by dotted line at low $\tilde{m}$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=60mm, angle=270]{chiral-N20-mu3-adaptive.eps}
\end{center}
\caption{Chiral condensate at $\tilde{\mu}=3$ as a function of $\tilde{m}$.
The solid line shows the exact value in the ChRMT \cite{Osborn:2004rf} and
the dotted line shows the exact value in the phase quenched (PQ)
ChRMT \cite{Akemann:2004dr}.
The results of the Exp and Hyp representations are expressed as
squares and circles. At large $\tilde{m}$, both the representations well produce the exact result.
The result of Hyp representation at small $\tilde{m}$ deviates from the exact result.
Dotted vertical lines show the values of $\tilde{m}$ at which the NNS distribution
is obtained.
}
\label{fig:N20-chiral}
\end{figure}
We show the NNS distributions in the Hyp representation
for ($\tilde{\mu}, \tilde{m})=(3,2)$ and (3,30) in Figs.~\ref{fig:N20mu3m2H} and \ref{fig:N20mu3m30H},
respectively.
The NNS distributions in those cases
are approximately consistent with each other, but slightly smaller than the Ginibre ensemble.
We do not find strong $\tilde{m}$ dependence of the NNS distribution
in the Hyp representation.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=60mm, angle=270]{Ps-tot-hyp-N20-mass2-mu3.eps}
\end{center}
\caption
The NNS distribution of
the Hyp notation
at $N=20$, $\tilde{m}=2$, and $\tilde{\mu}=3$.
}
\label{fig:N20mu3m2H}
\begin{center}
\includegraphics[width=60mm, angle=270]{Ps-tot-hyp-N20-mass30-mu3.eps}
\end{center}
\caption{
The NNS distribution of
the Hyp notation at $N=20$, $\tilde{m}=30$, and $\tilde{\mu}=3$.
}
\label{fig:N20mu3m30H}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=60mm, angle=270]{Ps-tot-exp-N20-mass2-mu3.eps}
\end{center}
\caption{
The NNS distribution of
the Exp notation
at $N=20$, $\tilde{m}=2$, and $\tilde{\mu}=3$.
}
\label{fig:N20mu3m2E}
\begin{center}
\includegraphics[width=60mm, angle=270]{Ps-tot-exp-N20-mass30-mu3.eps}
\end{center}
\caption{
The NNS distribution of
the Exp notation
at $N=20$, $\tilde{m}=30$, and $\tilde{\mu}=3$.
}
\label{fig:N20mu3m30E}
\end{figure}
Next, we show the NNS distribution in the Exp representation
for ($\tilde{\mu}, \tilde{m})=(3,2)$ and (3,30) in Figs.~\ref{fig:N20mu3m2E} and \ref{fig:N20mu3m30E}, respectively.
We find that the NNS distributions depend on $\tilde{m}$.
At small $\tilde{m}$, the NNS distribution is close
to the Wigner surmise.
At large $\tilde{m}$, the NNS distribution approximately follows that in the Ginibre ensemble, and
is almost consistent with that in the Hyp representation.
\subsection{Discussion}
Now, we discuss the interpretation of our results and its implications.
We found that the Ginibre ensemble is
favored at large $\tilde{m}$ in both representations.
This is physically reasonable, because the fermion determinant
is approximated by the mass factor as $\det(\Slash{D}+m) \sim m^{2N}$.
There is no difference between the original theory and phase
quenched theory, and both of them provide the same results.
In this case, the partition function is well described as
the Gaussian form of the complex matrices $\Phi_{1,2}$,
which is nothing but the Ginibre ensemble \cite{Ginibre:1965zz,Verbaarschot:2005rj}.
On the other hand, we found that at small $\tilde{m}$
the Wigner surmise and Ginibre ensemble are favored in the Exp
and Hyp representation, respectively.
The ChRMT used in this work is independent of quark chemical potential $\mu$
~\cite{Osborn:2004rf,Bloch:2012bh}, which implies that the NNS distribution
at finite $\mu$ should be the same with that at zero $\mu$.
As we have explained,
the Wigner surmise is theoretically expected at $\mu = 0$.
Thus, we find that the NNS distribution shows physically expected
behavior even in the CL method, if the Langevin simulation converges
to the correct results.
The Ginibre ensemble in the Hyp representation is caused by
unphysical broadening of the Dirac eigenvalue distribution
due to the failure of the complex Langevin simulation.
Here, we should note that there is a subtlety in the application
of the CL method to the NNS distribution.
The argument to justify the CL method is given for holomorphic observables~\cite{Aarts:2009uq,Aarts:2011ax},
while it is unclear if the CL method can be justified
for non-holomorphic observables.
The NNS distribution is real, and therefore a non-holomorphic quantity.
It is not ensured that the NNS distribution obtained in the CL method agrees with that in the original theory.
Although the Dirac eigenvalue distribution is narrower for the correct
convergence case, the distribution more or less receives broadening caused by
the complexification, namely imaginary parts of the originally real variables.
In this sense, it is non-trivial that the CL method can reproduce the
physical behavior of the NNS distribution.
As we mentioned in the Introduction, we conjecture that the NNS distribution,
a universal quantity defined for the Dirac eigenvalues,
can provide physical behavior even in the complex Langevin simulation.
Although this is highly non-trivial statement due to the non-holomorphy of the
NNS distribution, our numerical results support this conjecture.
Such a conjecture may be also inferred from an analogy between the
CL method and the Lefshetz thimble (LT) method.
In the LT method, dominant critical points are expected to share the symmetry properties with an original integral contour
as shown in the case of one classical thimble contribution \cite{Cristoforetti:2012su}.
In the CL and LT methods, the critical points are located at the same points
on the complex plane, and therefore dominant critical points in the CL method
are also expected to share the symmetry properties of the original theory.
If this conjecture holds, the universal quantities may provide a tool to
understand the convergence properties of the CL simulations, which
will be useful in the study of theories where exact results are not known
such as in QCD.
\section{Summary} \label{Sec:Summary}
In this work, we have performed the first study of the nearest neighbor
spacing (NNS) distributions of Dirac eigenvalues in the chiral random matrix theory (ChRMT)
at non zero quark chemical potential using the complex Langevin (CL) method.
The ChRMT was described in two representations:
the hyperbolic and exponential forms of the chemical potential.
The polar coordinate was adopted in both cases for the description of dynamical variables.
For small quark mass, the hyperbolic case converges to the wrong result
and exponential case converges to the correct result, as shown in
a previous study~\cite{Mollgaard:2014mga}.
We have calculated the NNS distribution for several values of the mass and
chemical potential using the unfolding procedure.
For large mass, the NNS distribution follows the Ginibre ensemble, which
implies that the real and imaginary part of the Dirac eigenvalues
have the same order of magnitude.
For small mass, we found the deviation between two representations.
The NNS distribution follows the Wigner surmise for the correctly converging case,
while it follows the Ginibre ensemble when the simulation converges to the phase quenched result.
The Wigner surmise is physically reasonable according to the chemical potential
independence of the ChRMT.
Thus, the NNS distribution shows the physical behavior even in the
CL method if it converges to the correct result.
There is a subtlety as to whether the NNS distribution, which is
non-holomorphic, can be justified in the CL method.
We speculate from the analogy between the CL and Lefshetz thimble
methods that universal quantities determined from the
properties such as symmetries can be maintained even in the CL method
if the configurations are correctly located around relevant critical points.
If this holds, the universal quantities can be used to test the
convergence properties of the CL method.
At least, our numerical result support this conjecture.
Of course, it is important to consider the theoretical justification
of the conjecture and applications to other universal quantities and to other theories,
which we leave for future studies.
Such a study will deepen our understanding of complexified theories,
and may provide information about a convergence property in the
CL simulation for the study of theories with the sign problem.
\newpage
\acknowledgments
The authors would like to thank Falk Bruckmann, Sayantan Sharma, and Sinji Shimasaki for useful discussions.
T.~I.~thanks Yuta Yoshida for technical help.
T.~I.~ is supported by the Grant-in-Aid for the Japan Society for the Promotion of Science (JSPS) Fellows (No.~25-2059).
K.~K.~ is supported by the Grant-in-Aid for the Japan Society for the Promotion of Science (JSPS) Fellows (No.~26-1717).
K.~N.~ is supported by the JSPS Grants-in-Aid for Scientific Research (Kakenhi) Grants No.~26800154,
and by MEXT SPIRE and JICFuS.
|
1,108,101,563,294 | arxiv | \section{Introduction}
Over the years, cavity quantum electrodynamics (CQED) \citep{berman:cqed, walther2006cavity,miller2005trapped} where atoms interact with quantized electromagnetic fields have emerged as an excellent platform for engineering entanglement \citep{RevModPhys.73.565, PhysRevLett.90.027903}, photon blockade \citep{PhysRevLett.79.1467,Nature2,PhysRevA.95.063842}, generation of quantum light \cite{rempe:prl124:093603:2020} and a variety of quantum correlations \citep{PhysRevLett.80.3948,PhysRevLett.107.023601}. Antibunching and entanglement are a crucial resource for quantum information processing, metrology and quantum teleportation \citep{PhysRevA.98.012121,PhysRevA.99.023828}. In particular, correlated or entangled photons are essential for photon-based quantum metrology and quantum technology \citep{Nature4,PhysRevLett.118.133602,PhysRevA.100.053802,PhysRevLett.94.023601,PhysRevLett.127.123602,PhysRevA.97.023822}. One of the important sources of correlated photons used for a variety of fundamental studies over the years is optical parametric oscillator (OPO) \citep{mandel:prl59:2044:1987,kwait:prl75:4337:1995,RevModPhys.84.777.2012,muller:NaturePhotonics8:224:2014,PhysRevA.103.053710,PhysRevA.62.033802}. However, the generation of correlated photons by an OPO is probabilistic. In recent times, several theoretical proposals for generating nonclassical or entangled radiation fields by coherently driving multilevel atoms inside a cavity have been put forward \cite{xion:prl94:023601:2005, zubairy:pra77:062308:2008,fang:pra81:012323:2010}. Lasing into a single-mode cavity field due to coherent as well as incoherent pumping of multi-level atoms in strong-coupling CQED regime has also been studied \cite{Devi:pra}. Although, with the currently available conventional laser technology, it is possible to produce extremely narrow-band and high-quality single-mode coherent light sources there hardly exists any standard technique to create a laser source that can deliver entangled or non-classically correlated photons on demand.
A one-atom laser using CQED was theoretically conceived three decades ago \cite{muandsavage:pra:1992}. Since then several theoretical as well as experimental studies \cite{ritsch:jmo41:609:1994,walther_PhysRevA.55.3923,kimble_mckeever2004deterministic,khajavikhan2012thresholdless,gsa-sdg:pra42:1737:1990,walther:epl:1997,Nature1,PhysRevA.70.023814} on single-atom CQED laser have been carried out. The steady-state properties of an incoherently pumped single-atom in a single-mode cavity reveal vanishing threshold for lasing action of the cavity mode with interesting atom-photon and photon-photon correlations \cite{gsa-sdg:pra42:1737:1990}. A laser with a single trapped ion inside a high-Q cavity was theoretically proposed by Meyer, Briegel and Walther \cite{walther:epl:1997} about two decades ago. 1980s and 1990s had witnessed a lot of progress in the development of micromaser \cite{PhysRevA.34.3077_filipowicz,Haroche_micromaser_PhysRevA.36.3771,rempe_micromaser_PhysRevLett.64.2783,walther_micromaser_PhysRevLett.82.3795} that uses a mono-energetic beam of Rydberg atoms passing through a high-Q cavity one by one. An optical analog of micromaser, namely microlaser had been realized in the 90s \cite{martini_microlaser_PhysRevA.46.4220,an_microlaser_PhysRevLett.73.3375}. Although a micromaser or microlaser makes use of the strong atom-cavity interactions of single atoms, it is not truly a one-atom maser or laser as it results from the cumulative effect of interactions of a beam of atoms with the cavity.
The recent advancement of trapping and cooling of single atoms and ions opens new prospects for developing a microscopic laser with a single trapped atom inside a high-Q cavity for controlled generation of nonclassical light.
In this context, McKeever {\it et. al.} \citep{Nature1,PhysRevA.70.023814} had experimentally demonstrated lasing with a single trapped atom in the strong-coupling CQED regime in 2003. A one-atom CQED laser is fundamentally different from a standard laser in many respects. First, it can operate at vanishingly small or zero threshold. Second, its frequency depends on the atom-cavity coupling. Third, it can be operated with a single trapped atom at a few photon levels enabling substantial microscopic control over the output signal amplitude and photon statistics. Fourth, as shown in Ref. \cite{Nature1}, it can produce antibunched light which is nonclassical.
Here we propose a model for a single-atom threshold-less bimodal correlated-photon laser (CPL) based on strong-coupling CQED. Our proposed laser system is schematically shown in Fig.\ref{fig 1}. It may consist of a pair of crossed cavities or a bimodal cavity, a single trapped $V$-type three-level atom inside the bimodal cavity or at the symmetry point of the crossed cavities, two incoherent pumps and two lasers to drive the two atomic transitions. The two excited levels of the atom are incoherently pumped by external sources while a pair of coherent drives (lasers) couple the atomic ground state with the two excited states. The incoherent pumping leads to the amplification of the two cavity modes while coherent driving creates entanglement between the two cavity modes. Even without coherent driving but with dual incoherent pumping, the system produces inter-mode photon antibunching without any photon-photon entanglement. This nonclassical correlation can be attributed to an interplay between the coherent Rabi dynamics due to the cavity fields and the dual incoherent pumping. Only dual incoherent pumping of a $V$-type three-level system in free space can not produce any excited-state atomic coherence unless the two transition dipole moments are non-orthogonal \cite{scully_PhysRevA.74.063829}. Since all three levels of a $V$-type three-level atom have different angular momenta, the two transition dipole moments are orthogonal. However, classical coherent fields or quantized cavity fields can produce the excited-state coherence. We interpret our results in terms of the transitions and coherence between the atom-field dressed states. The origin of the photon-photon entanglement lies in the dressing of the cavity-dressed states by the external coherent fields. When the photons are emitted from such doubly dressed states, they carry forward the nonclassical properties of the dressed states in the form of inter-mode photon-photon entanglement. Our results show that when the atom is driven by external coherent drives in the absence of incoherent pumping or in the presence of small incoherent pumping, lasing in the two cavity modes can happen without any inversion due to the atomic coherence. We also show threshold-less lasing behavior with manifestly nonclassical inter-mode correlation. We find that the entanglement is strongly quenched as we slightly increase the strength of the incoherent pump in the presence of fixed coherent drives.
The paper is organised in the following way: we present theoretical formulation of our problem in Sec.~\ref{sec2}. We discuss and analyse our results in Sec.~\ref{sec3}. The paper is concluded in Sec.~\ref{sec4}.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{schematic_paper_fig.eps}
\caption{A schematic diagram showing a single trapped $V$-type three-level atom interacting with two cavity modes either in a crossed cavity (i) or a doubly-resonant cavity (ii) setup. The zoomed view shows the atomic level diagram with $a$, $b$ and $c$ denoting the three atomic levels and the arrows indicate various transition pathways. The cavity modes 1 and 2 couple the lowest level $a$ to the two upper levels $b$ and $c$, respectively. Here $\eta_{ic1}$ and $\eta_{ic2}$ represent two incoherent pumping rates while $\eta_{c1}$ and $\eta_{c2}$ are the coherent coupling rates due to laser fields $L_1$ and $L_2$ that drive the corresponding atomic transitions. $\gamma_1$ and $\gamma_2$ are the atomic decay rates for the transitions $b \rightarrow a $ and $c \rightarrow a$, respectively; while $\kappa_1$ and $\kappa_2$ are the cavity decay rates for the modes 1 and 2, respectively. }
\label{fig 1}
\end{figure}
\section{Theoretical formulation}\label{sec2}
We consider a trapped $V$-type three-level atom either at the intersection of a pair of crossed cavities as schematically shown Fig.~\ref{fig 1} (i) or in a doubly-resonant cavity setup as shown in Fig.~\ref{fig 1} (ii) . Here $a$ denotes the ground state, and $b$ and $c$ represent the two excited states which may be degenerate or quasi-degenerate. The two cavity modes 1 and 2 are tuned to the transitions $a \rightarrow b$ and $a \rightarrow c$, respectively. The two excited levels $b$ and $c$ are pumped by dual incoherent sources while the two atomic transitions may be driven by two classical laser fields as depicted in Fig.~\ref{fig 1}.
\subsection{Hamiltonian}
The Hamiltonian for this system is
\begin{equation}
{\cal H} = {\cal H}_0 + {\cal H}_{int}+ {\cal H}_{drive}~,
\end{equation}
where ${\cal H}_0 = \hbar \omega_b \mid b\rangle \langle b \mid+ \hbar \omega_c \mid c\rangle \langle c \mid + \hbar \omega_1 a^{\dag}_1 a_1 + \hbar \omega_2 a^{\dag}_2 a_2~$ is the free Hamiltonian with $\omega_b$ and $\omega_c$ being the respective atomic transition frequencies. We choose the energy of the lowest level $a$ to be zero. Here $a_i (a_i^{\dagger})$ denotes the annihilation (creation) operator of the $i$-th (=1,2) cavity mode. Under electric-dipole and rotating-wave approximation (RWA), the interaction Hamiltonian ${\cal {H}}_{int}$ can be written as
\begin{eqnarray}
{\cal {H}}_{int} = \hbar \left( g_1 a_1 \sigma^{\dagger}_1 + g_2 a_2 \sigma^{\dagger}_2\right) + {\rm H.c.}~,
\end{eqnarray}
where $g_1$ and $g_2$ are the atom-cavity coupling constants that depend on dipole moment and the respective transition field amplitude. $\sigma^{{\dagger}}_{1}= \mid b \rangle \langle a \mid $ and $\sigma^{{\dagger}}_{2} = \mid c \rangle \langle a \mid$ represent the atomic raising operators and the corresponding lowering operators are $\sigma_{1}= \mid a \rangle \langle b \mid $ and $\sigma_{2} = \mid a \rangle \langle c \mid$, respectively. Here the Hamiltonian corresponding to the driving of the atomic transitions by two classical fields is given by
\begin{equation}
{\cal H}_{drive} = \hbar \eta_{c1} \sigma_1 e^{i \omega_{L_1}t} + \hbar \eta_{c2} \sigma_2 e^{i \omega_{L_2}t} + {\rm H.c.}~,
\end{equation}
where $\eta_{c1}$ and $\eta_{c2}$ are the laser-atom coupling constants. In the rotating reference frame of the laser frequencies, the Hamiltonian reads as
\begin{eqnarray}
{\cal H} &=& \hbar \left (\Delta_b \mid b\rangle \langle b \mid + \Delta_c \mid c\rangle \langle c \mid + \delta_{1L} a^{\dag}_1 a_1 + \delta_{2L} a^{\dag}_2 a_2 \right )+{\cal H}_{int} \nonumber\\
&&+ \hbar \left [ \eta_{c1} \left (\sigma_1 + \sigma^{\dag}_1 \right ) + \eta_{c2} \left (\sigma_2 + \sigma^{\dag}_2 \right ) \right ]~,
\end{eqnarray}
where the detuning parameters are $\Delta_b = \omega_b - \omega_{L_1}$, $\Delta_c = \omega_c - \omega_{L_2}$, $\delta_{1L} = \omega_1 - \omega_{L_1}$ and $\delta_{2L} = \omega_2 - \omega_{L_2}$. Here $\omega_{L_1}$ and $\omega_{L_2}$ are the frequencies of $L_1$ and $L_2$ lasers. The detuning of the two cavity fields are denoted by $\delta_1 = \omega_1 - \omega_b$ and $\delta_2 = \omega_2 - \omega_c$.
\subsection{Dressed-states}
In the absence of all incoherent processes, one can develop a dressed state description of the driven three-level atom interacting with two cavity modes. Dressed state picture of a CQED problem helps to gain insight into the physical processes underlying the relevant quantum dynamics, particularly in the strong-coupling regime.
We use the two-mode Fock basis $\mid n_1, n_2 \rangle$ for the two cavity fields, where $n_1=0, 1, 2, \cdots, \infty $ and $n_2 = 0, 1, 2, \cdots, \infty$ denote the number of photons in the mode 1 and 2, respectively. One can construct three bare basis states $\mid a, n_1, n_2 \rangle$, ~ $\mid b, n_1-1, n_2 \rangle$,~ ${\rm and} \mid c, n_1, n_2-1 \rangle$ which will be involved in the closed coherent quantum dynamics in the absence of all incoherent processes and couplings to the classical derives.
Defining the excitation number operators $N_1 = a^{\dag}_1 a_1 + \mid b \rangle \langle b \mid$ and $N_2 = a^{\dag}_2 a_2 + \mid c \rangle \langle c \mid$ and applying the unitary transformation
$\hat{U}(t) = {\rm exp} \left[-i \left(\omega_1 N_1 + \omega_2 N_2\right)t\right]~$\cite{barnett2002methods}, one can obtain the following effective Hamiltonian in the absence of the lasers
\begin{eqnarray}
\tilde{H} = -\delta_1 \mid b \rangle \langle b \mid - \delta_2 \mid c \rangle \langle c \mid + {\cal H}_{int}~.
\label{Heff}
\end{eqnarray}
If the two cavity fields are tuned to the two-photon resonance $\delta_{1} = \delta_{2} = \delta$, then the eigenvalues of $\tilde{H}$ are $\tilde{E}_0 = - \delta $ and $\tilde{E}_{\pm} = - \frac{\delta}{2} \pm \frac{1}{2} \sqrt{ \delta^2 + 4 \left ( n_1 g_1^2 + n_2 g_2^2 \right )}$. For limited Fock states, the characteristic eigenvalue and eigenstate analysis can be found elsewhere \citep{Pal2019PhotonphotonCW}.
The eigenstate $\mid \psi_0(n_1,n_2) \rangle$ corresponding to the eigenvalue $\tilde{E}_0$ is given by
\begin{eqnarray}
\mid \psi_0(n_1, n_2) \rangle &=& \frac{1}{\sqrt{g_2^2n_2+g_1^2n_1}} \left[ g_1\sqrt{n_1}\mid c,n_1,n_2-1\rangle - g_2\sqrt n_2\mid b,n_1-1,n_2\rangle \right]~.
\end{eqnarray}
which is a coherent superposition of the two atom-field joint states involving only the two atomic excited states. It results from the destructive quantum interference between the two cavity field-induced transition pathways $ \mid b,n_1-1,n_2\rangle \longleftrightarrow \mid a, n_1, n_2 \rangle $ and $ \mid c,n_1,n_2 - 1 \rangle \longleftrightarrow \mid a, n_1, n_2 \rangle $ connecting
the two joint atom-field excited states to the joint atom-field state with the atom being in the ground state. This is an excited dark state due to the quantized cavity fields. The dark-state resonance and associated phenomena such as coherent population trapping and electromagnetically induced transparency (EIT) are well-known for a Lambda-type three level system interacting with two classical coherent fields. In a recent work, Rempe's group \cite{rempe:prl124:093603:2020} has experimentally demonstrated
the generation of single-mode quantum light utilizing the dark state of a coherently driven $\Lambda$-type three-level atom inside a single-mode cavity.
However, for a $V$-type three-level system, the dark state resonance and associated effects are qualitatively different \citep{Nature3}. The excited-state coherence in $V$-type three-level atoms driven by classical coherent fields may lead to transient coherent population trapping \citep{ARIMONDO1996257}, quantum beat and correlated two-mode lasers as shown by Scully's group \cite{scully:prl55:2802:1985,scully:prl60:1988,scullu:zubairy:pra35:752:1987,scully:pra38:1988}.
The other two dressed states corresponding to the eigenvalues $E_{\pm}$ are coherent superposition of all three bare states. For $\delta = 0$, these two dressed states can be expressed as
\begin{eqnarray}
\mid \psi_{\pm}(n_1, n_2) \rangle &=& \frac{1}{\sqrt{2}} \mid a,n_1,n_2\rangle \pm \frac{1}{\sqrt{2(g_2^2n_2+g_1^2n_1)}} \nonumber \\
&\times& \left [ g_2\sqrt{n_2}\mid c,n_1,n_2-1\rangle
+ g_1\sqrt n_1\mid b,n_1-1,n_2\rangle \right ]~.
\label{eqn psipm}
\end{eqnarray}
We call these three dressed states, namely, $\mid \psi_{0}(n_1, n_2)$, $\mid \psi_{\pm}(n_1, n_2) \rangle $ as belonging to the $(n_1, n_2)$-photon sector. Thus the two-mode strong-coupling CQED with a $V$-type three-level atom enables the creation of tripartite superposition states involving the atom and the two quantized fields. The incoherent pumping and other incoherent processes such as spontaneous emission as well as classical drives will induce transitions to the states outside these three dressed states, thus coupling all other photon sectors. While dual incoherent pumping can lead to the enhancement of the excited-state population and the consequent lasing into the cavity modes, the coherent atomic drives can generate further coherence between the cavity-dressed states. This coherence plays an essential role in creating entanglement between two cavity modes. Furthermore, the coherent drives can lead to the amplification of the cavity modes without population inversion due to the coherence.
To keep the dressed-state analysis simple, we assume that $\delta = 0 $ and $g_1 = g_2 = g$, $\eta_{ic1} = \eta_{ic2} = \eta_{ic} $, $\kappa_1 = \kappa_2 = \kappa$ and $\gamma_1 = \gamma_2 = \gamma$. The (0,0)-photon sector, that is when both the cavity fields are in vacuum, there exists only one joint state $\mid a, 0, 0 \rangle$. Each of the $(n_1,0)$- and $(0,n_2)$-photon sectors have two dressed states
\begin{eqnarray}
\mid \psi_{\pm}(n_1,0) \rangle &=& \frac{1}{\sqrt{2}} \left [ \mid a, n_1, 0 \rangle \pm \mid b, n_1-1,0 \rangle \right ]~,
\nonumber \\
\mid \psi_{\pm}(0,n_2) \rangle &=& \frac{1}{\sqrt{2}} \left [ \mid a, 0, n_2 \rangle \pm \mid c,0,n_2-1 \rangle \right ].
\label{eqn psibc}
\end{eqnarray}
The corresponding eigenvalues are $\pm g$. These are the familiar dressed states of two-state Jaynes-Cummings model. If the fields contain zero photon in one mode and a finite number of photons in the other mode, then in the absence of incoherent processes and any classical drive, the system effectively reduces to a two-level system. However, atomic drives and incoherent processes can connect $\psi_{\pm}(n_1,0)$ and $\psi_{\pm}(0,n_2)$ to $\mid a, n_1 - 1,0 \rangle$ and $\mid a, 0,n_2 - 1\rangle$, respectively. As we will show in the next section, when dual incoherent pumps are weak, that is, $\eta_{ic} < (\kappa + \gamma) $, the lasing actions and the nature of two-photon correlations between the generated photons can be interpreted in terms of an interplay between the coherent and incoherent transitions involving some low-lying bare or dressed states. For strong incoherent pumping $\eta_{ic} \ge (\kappa + \gamma) $, higher photon sectors get involved in the dynamics. The possible transitions between the bare states as well as between the dressed states belonging to a few low-lying photon sectors are schematically shown in Fig.\ref{fig 2}
\begin{figure*}
\includegraphics[width=0.9\linewidth]{dressed_state_coherent_3.eps}
\caption{{ (i) Bare-state level diagram with various transition pathways due to different incoherent processes and
coherent atomic drives. (ii) and (iii) A few low lying dressed states with transitions due the incoherent processes only.
Here $g_1= g_2 = g$ (atom-field coupling), $\eta_{ic1} = \eta_{ic2} = \eta_{ic}$ (incoherent pumps) and $\eta_{c1} = \eta_{c2} = \eta_c$ (coherent drives). In (i), the green and orange- double headed arrows represent the coherent couplings due to the cavity and driving fields, respectively. The curvy upward and downward arrows indicate incoherent pumping and atomic or cavity decay processes, respectively. In the dressed-state picture, the frequency spacing between the dressed states belonging to the same photon sector are marked. Note that the dressed-state diagram involving couplings among (0,0) $\Longleftrightarrow$ (0,1) $\Longleftrightarrow$ (0,2) is the same as in (ii) with `(1,0)' and `(2,0)' being replaced by `(0,1)' and `(0,2)', respectively. Similarly, the dressed-state diagram involving (0,1)$\Longleftrightarrow$ (1,1), photon sectors will be same as in (iii) with `(1,0)' being replaced by `(0,1)'. The spacing between different levels is not to be scaled.} }
\label{fig 2}
\end{figure*}
\subsection{ Incoherent pumping and damping}
To include incoherent pumping and various damping processes in the system, we resort to the density matrix equation in Lindblad form
\begin{eqnarray}
\frac{\partial{\hat\rho_{sf}}}{\partial t} = - \frac{i}{\hbar}\left[ {\cal{H}}(\tilde{H}),\hat\rho_{sf}\right] + \sum_{i=1,2}\frac{\kappa_i}{2}\mathcal{L}_{a_i} + \sum_{j=1,2}\frac{\gamma_{j}}{2}\mathcal{L}_{\sigma_j} + \sum_{j=1,2}\frac{\eta_{icj}}{2}\mathcal{L}_{\sigma_j^{\dagger}} ~,
\label{eq1}
\end{eqnarray}
Where $\rho_{sf}$ is the density matrix of joint system, comprising of the sub-system atom (denoted by `$s$' and the cavity fields (denoted by `$f$'). $\cal L_{\rm{x}}$ is a Liouville super operator where $x$ stands for any of the three operators $a_i,\sigma^{\dagger}_j,\sigma_j$. $\kappa_i$ denotes the decay rate of the cavity field mode $i$ (i = 1, 2). Here the Lindbladian superoperators
\begin{align*}
\mathcal{L}_{a_i}&=\left[2a_i\rho_{sf}a_i^\dagger- \left \{ a_i^\dagger a_i, \rho_{sf} \right \}\right]~, \\
\mathcal{L}_{\sigma_j}&= \left[ 2\sigma_j\rho_{sf}\sigma_j^{\dagger} - \left \{ \sigma_j^{\dagger} \sigma_j, \rho_{sf} \right \} \right ]~, \nonumber
\end{align*}
describe the cavity and atomic damping, respectively; while
\begin{equation}
\mathcal{L}_{\sigma_j^{\dagger}}= \left[ 2\sigma_j^{\dagger}\rho_{sf}\sigma_j - \left \{ \sigma_j \sigma_j^{\dagger}, \rho_{sf} \right \} \right ]~,
\end{equation}
describes incoherent pumping to the two upper levels of the atom. Here $\{ \alpha, \beta \} $ implies anticommutation between $\alpha$ and $\beta$. The Liouvillian in Eq. (\ref{eq1}) can also be expressed in a different form using the complex Hamiltonian
\begin{eqnarray}
H_c = {\cal H} + i \hbar \sum_{i =1,2} \frac{\kappa_i}{2} a_i^\dagger a_i +
i \hbar \sum_{j =1,2} \frac{\gamma_j}{2} \sigma_j^{\dagger} \sigma_j +
i \hbar \sum_{j =1,2} \frac{\eta_{ic j}}{2} \sigma_j \sigma_j^{\dagger}~.
\end{eqnarray}
Explicitly, this form is
\begin{eqnarray}
\frac{\partial{\hat\rho_{sf}}}{\partial t} = - \frac{i}{\hbar}\left[ {H_c \hat\rho_{sf} - \hat{\rho}_{sf} H_c^{\dagger}}\right] + \sum_{i=1,2}\kappa_i a_i\rho_{sf}a_i^\dagger + \sum_{j=1,2} \left [ \gamma_{j} \sigma_j\rho_{sf}\sigma_j^{\dagger}
+ \eta_{ic j} \sigma_j^{\dagger}\rho_{sf}\sigma_j \right ] ~,
\end{eqnarray}
To numerically solve \citep{JOHANSSON20131234} the density matrix equation we use joint atom-field basis states $\mid \alpha, n_1, n_2\rangle$ where $\alpha$ denotes any of the three atomic states, $n_1$ and $n_2$ are the photon numbers. For a given set of system parameters, we first verify the convergence of our numerical results against the variation of the photon numbers in the two modes. Once we ensure the convergence, we truncate the Fock basis of the two fields for the chosen parameter set. We numerically obtain the steady-state solution of the joint atom-field density matrix.
\begin{figure}[h]
\includegraphics[width=0.7\linewidth]{saturation_with_N_1.eps}
\caption{The steady-state average photon numbers $\bar n_1=\bar n_2=\bar n$, as a function of the maximum Fock-state number $(N_{max})$ in each mode considered for our calculations. The scaled atom-cavity coupling parameter is ${g}_1={g}_2 =$ 10, the cavity damping rates are $\kappa_1=\kappa_2 = 1$ MHz, incoherent pump strengths are $ \eta_{ic1}=\eta_{ic2} = 2$ MHz and both modes are at resonance, i.e., $\delta_1 = \delta_2 = 0$. }
\label{fig 3}
\end{figure}
\subsection{Second-order photon-photon correlations and entanglement}\label{D}
We calculate the expectation value of an observable operator $\hat{O}$ of interest in the steady-state by
\begin{eqnarray}
\langle\hat O\rangle\nonumber={\rm Tr}\left[\hat O \hat\rho_{sf}^{ss}\right]&=&\sum_{\alpha}\sum_{n}\sum_{m}\langle \alpha,n,m|\hat O \hat\rho_{sf}|\alpha, n,m\rangle~.
\label{eq12}
\end{eqnarray}
where $\alpha$ stands for an atomic state, $n$ and $m$ represent photon number states for mode 1 and 2, respectively. Here $\hat\rho_{sf}^{ss} = \hat\rho_{sf}(t \rightarrow \infty)$ stands for the steady-state joint density matrix. If the operator $\hat{O}$ describes only the atomic properties such as atomic population inversion or coherence, then one can use the reduced atomic density matrix to calculate the expectation value. The reduced atomic density matrix is given
\begin{equation}
\hat{\rho}_s = {\rm Tr}_{\rm field} [\hat \rho_{sf}]~.
\end{equation}
where ${\rm Tr}_{\rm field}$ implies tracing over the two cavity field modes. The expectation value is then given by $\langle\hat O\rangle\nonumber={\rm Tr}_{\rm atom} \left[\hat O \hat\rho_{s} \right]$, where ${\rm Tr}_{\rm atom}$ stands for tracing over the atomic states.
On the other hand, if $\hat O $ refers to a field property only then we can use reduced two-mode field density matrix given by
\begin{equation}
\hat{\rho}_{f} = {\rm Tr}_{\rm atom } [\hat \rho_{sf}]~.
\end{equation}
Then the expectation value is given by $\langle\hat O\rangle\nonumber={\rm Tr}_{\rm field} \left[\hat O \hat\rho_{f} \right]$. We calculate the two-time Hanbury Brown-Twiss (HBT) second-order correlation function \cite{JURCZAK1995480} as
\begin{eqnarray}
g_{ij}^{(2)}(\tau) &=& \frac{\langle a^{\dag}_i(t) a^{\dag}_j(t+\tau)a_j(t+\tau)a_i(t)\rangle}{\langle a^{\dag}_i(t) a_i (t)\rangle \langle a^{\dag}_j(t+\tau) a_j (t+\tau)\rangle}~.
\label{twotime}
\end{eqnarray}
where $i=1,2$ and $j=1,2$ denote the mode indexes. If $i=j$ then we have intra-mode correlation, otherwise we obtain inter-mode correlation. Here $\langle a^{\dag}_i a_i\rangle$ is the mean photon number of the $i^{th}$ cavity mode. Using quantum regression theorem \cite{PhysRev.129.2342,PhysRev.157.213,carmichael1999statistical}, HBT function can be calculated for fluctuations around the steady state. Then we have
\begin{eqnarray}
g_{ij}^{(2)}(\tau)= \frac{{\rm Tr}_{\rm field} \left[a^{\dag}_i a^{\dag}_j a_j a_i \rho_{f}(t\rightarrow\infty)\right]}{{\rm Tr}_{\rm field} \left[ a^{\dag}_i a_i\rho_{f}(t\rightarrow \infty)\right] {\rm Tr}_{\rm field} \left[ a^{\dag}_j a_j\rho_{f}(t\rightarrow \infty)\right]}~.
\label{g20}
\end{eqnarray}
For $\tau \sim 0$, the system's response manifests into photonic correlation ($g_{ij}^{(2)}(0) > 1$) or anticorrelation ($g_{ij}^{(2)}(0) < 1$). When $g_{ij}^{(2)}(0) \gg 1$, the photons are highly correlated in bunched form. i.e. bunching occurs. In contrast, when $g_{ij}^{(2)}(0) < 1$, there occurs photon antibunching and as $g_{ij}^{(2)}(0)\rightarrow 0$, the perfect photon blockade \citep{PhysRevLett.79.1467} takes place. Now, we define the frequency domain of correlation by Fourier transforming $g_{ij}^{(2)}(\tau)$
\begin{equation}
\mathcal{F}_{ij}(\omega)=\int_{-\infty}^{\infty} [{g_{ij}^{(2)}(\tau)-g_{ij}^{(2)}(\infty)}]e^{(i\omega\tau)} \,d\tau ~.
\end{equation}
Depending on the coherence in the system, $g_{ij}^{(2)}(\tau)$ as a function of $\tau$ may exhibit oscillatory behavior. In the limit $\tau \rightarrow \infty$, $g_{ij}^{(2)}(\tau) \rightarrow 1$. To know the frequencies of the oscillations and other frequency-domain features of HBT correlations \cite{kingshuk_PhysRevA.103.033310}, it is important to study not only the time dependence of $g_{ij}^{(2)}(\tau)$ but also the frequency dependence of $\mathcal{F}_{ij}(\omega)$. In the present context, the HBT correlations as a function of $\tau$ will decay to the steady value of unity with a time constant $\tau_d \sim (\kappa + \gamma)^{-1}$ while the oscillations will bear the signature of quantized Rabi dynamics.
Next, to explore the entanglement between the photons of the two cavity modes,
we take partial transpose of Peres and Horodecki \cite{PhysRevLett.77.1413} over the mode 2 (or 1) of the two-mode reduced density matrix. The resulting matrix is denoted as $\rho^{{\rm PT,2}}_f$ ($\rho^{{\rm PT,1}}_f$). Partial transpose over field mode 2 implies that the basis operators of the bipartite density matrix change as $\mid mn\rangle \langle \mu \nu \mid \rightarrow \mid m \nu \rangle \langle \mu n\mid$. The negativity of at least one eigenvalue of $\rho^{{\rm PT,2}}_f$ or ($\rho^{{\rm PT,1}}_f$) implies that the two field modes are entangled.
\section{Results and Discussions}\label{sec3}
To present and discuss the results in a systematic way, we first present the results in the absence of classical drives, and analyze the
effects of the dual incoherent pumps. We then discuss the effects of coherent drives in the presence as well as in absence of incoherent pumps.
Thus we clearly discern the effects of coherent drives from the incoherent pumps and vice versa.
For all our numerical results, we consider $\gamma_1 = \gamma_2 = \gamma = 1$, implying that we scale all the frequency quantities by $\gamma$.
\subsection{The effects of incoherent pump only}
\begin{figure}
\centering
\vspace{.25in}
\begin{tabular}{@{}cc@{}}
\includegraphics[width=0.9\linewidth]{n1n2_g_1.eps}\\
\includegraphics[width=0.4\linewidth]{n1_g_p_1.eps}\hfill
\includegraphics[width=0.41\linewidth]{n2_g_p_1.eps}\\
\end{tabular}
\caption{(a) The average number of photons $\bar n_1=\bar n_2=\bar n$ are plotted as a function of scaled atom-cavity coupling parameter ${g}_1={ g}_2 = g= 10$. The other parameters are: $\delta_1 = \delta_2 =0$, $\kappa_1=\kappa_2=1$, $ \eta_{ic1}=\eta_{ic2} = 2, 1.5, 1, 0.5$ . The surface plots (b) and (c) depict the variation of $\bar n_1$ (b) and $\bar n_2$ (c) as a function of coupling parameters ${ g}_1$ and ${g}_2$, keeping $ \eta_{ic1}=\eta_{ic2} = 2$ with all other parameters remaining fixed. }
\label{fig 4}
\end{figure}
\begin{figure}
\centering
\vspace{.25in}
\begin{tabular}{@{}cc@{}}
\includegraphics[width=0.7\linewidth]{intrag210_vs_g.eps}\\
\includegraphics[width=0.71\linewidth]{intra.eps}
\end{tabular}
\caption{(a) Equal-time intra-mode HBT correlation functions $g^{(2)}_{11(22)}(0)$ and (b) inter-mode $g^{(2)}_{12}(0)$ are plotted as a function of $ g$ for different incoherent pump strengths $\eta_{ic}= 0.5$ (solid), $\eta_{ic}=1$ (dashed-dotted), $\eta_{ic}=1.5$ (dotted) and $\eta_{ic}=2$ (dashed). All other parameters are same as in Fig.~\ref{fig 4} .}
\label{fig 5}
\end{figure}
\begin{figure}
\centering
\vspace{.25in}
\begin{tabular}{@{}cc@{}}
\includegraphics[width=0.7\linewidth]{nb1_nb2_detuning_p_1.eps}\\
\includegraphics[width=0.7\linewidth]{surface_all.eps}
\end{tabular}
\caption{(a) $ \bar n_1$ is plotted as a function of $\delta_1$ for $\eta_{ic}=2$ (dashed), $\eta_{ic}=1.5$ (dotted), $\eta_{ic}=1$ (dashed-dotted), $\eta_{ic}=0.5$ (solid), for $\delta_2 = 0$, $g_1=g_2= 10$ and $\kappa_1 = \kappa_2 = 1$. The surface plots (b) and (c) exhibit the variation of $ \bar n_1$ and $ \bar n_2$, respectively, as a function of both $\delta_1$ and $\delta_2$ for $\eta_c=0.5$. The subplots (d) and (e) display the same as in (b) and (c), respectively, but for $\eta_c=2$. }
\label{fig 6}
\end{figure}
\begin{figure}
\centering
\vspace{.25in}
\begin{tabular}{@{}cc@{}}
\includegraphics[width=0.5\linewidth]{g2_t.eps}\hfill
\includegraphics[width=0.5\linewidth]{P_all_furierg12.eps}
\end{tabular}
\caption{(a) The inter-mode HBT correlation function $g^{(2)}_{12}(\tau)$ is plotted against time-delay $\tau$ and (b) the spectrum $\mathcal{F}_{12}(\omega)$ of the correlation is plotted against frequency $\omega$ for $\eta_{ic}=0.5$ (dashed), $\eta_{ic}=1$ (dotted), $\eta_{ic}=1.5$ (dashed-dotted), $\eta_{ic}=2$ (solid) keeping the light-matter coupling $g_1=g_2=10$ fixed. The rest of the parameters asre same as in Fig. \ref{fig 4}. }
\label{fig 7}
\end{figure}
\begin{figure}
\centering
\vspace{.25in}
\begin{tabular}{@{}cc@{}}
\includegraphics[width=0.6\linewidth]{sig_all.eps}\\
\includegraphics[width=0.58\linewidth]{P_all_furierg22_1.eps}
\end{tabular}
\caption{(a) The intra-mode HBT functions $g^{(2)}_{11}(\tau)=g^{(2)}_{22}(\tau)$ are plotted against time delay $\tau$ and (b) the spectrum ${\mathcal F}_{jj}(\omega)$ is plotted against frequency $\omega$, for $\eta_{ic}=0.5$ (dashed), $\eta_{ic}=1$ (dotted), $\eta_{ic}=1.5$ (dashed-dotted), $\eta_{ic}=2$ (solid) with $g_1= g_2 = 10.$ The rest of the parameters asre same as in Fig. \ref{fig 4}. }
\label{fig 8}
\end{figure}
\begin{figure}
\centering
\vspace{.25in}
\begin{tabular}{@{}cc@{}}
\includegraphics[width=\linewidth]{dressed_state_coherent_2.eps}\hfill
\end{tabular}
\caption{Dressed-state level diagram showing couplings due to the coherent drives only for $\eta_{c1} = \eta_{c2} = \eta_{c}$ and $\delta = 0$. The drives connect the dressed states from different photon sectors and the transition amplitude for each of the transitions is mentioned.}
\label{fig 9}
\end{figure}
\begin{figure}
\centering
\vspace{.25in}
\begin{tabular}{@{}cc@{}}
\includegraphics[width=0.5\linewidth]{d_all_p_1.eps}\\
\includegraphics[width=0.5\linewidth]{n2_p_all_1.eps}
\end{tabular}
\caption{(a) The smallest eigenvalue of $\rho_{f}^{PT,2}$ is plotted against $\delta_{2L}$, for $\eta_{ic}=0.01$ (dotted), $\eta_{ic}=0.1$ (dashed). It shows the death of entanglement with the increase of incoherent pump strength $\eta_{ic}$. (b) average photon-number $\bar n_2$ and $\bar n_1$ vs. $\delta_{2L}$ is plotted for $\eta_{ic}=0.01$ (solid), $\eta_{ic}=0.1$ (dashed-dotted). The rest of the parameters are $g=10, \kappa_1 = \kappa_2 = 1, \eta_{c1}=\eta_{c2}=\eta_{c}=2$ and $\delta_{1L}=0$.}
\label{fig 10}
\end{figure}
\begin{figure}
\centering
\vspace{.25in}
\begin{tabular}{@{}cc@{}}
\includegraphics[width=0.5\linewidth]{d_e_all_same.eps}\\
\includegraphics[width=0.5\linewidth]{n_e_all_same.eps}\\
\includegraphics[width=0.5\linewidth]{g12_e_all_same.eps}
\end{tabular}
\caption{ The smallest eigenvalue of $\rho_{f}^{1,2^T}$ (a), $\bar n_2=\bar n_1=\bar n$ (b) and $g^{(2)}_{12}(0)$ (c) are plotted against $\delta_{1L}=\delta_{2L}=\delta_{L}$ for $\eta_c=0.1$ (solid), $\eta_c=0.5$ (dashed), $\eta_c=1$ (dotted), $\eta_c=1.5$ (dashed-dotted) and $\eta_c=2$ (diamond marker). The rest of the parameters are same as in Fig. \ref{fig 10} with $\eta_{ic}=0$.}
\label{fig 11}
\end{figure}
First we consider the case when the three-level atom inside the cavity is pumped incoherently with no coherent drive being present. The level diagram with only incoherent couplings between the dressed states is schematically shown in Fig. \ref{fig 2} (ii) and (iii). Fig.~\ref{fig 3} shows the convergence of the values of the average photon numbers $\bar n_1=\bar n_2=\bar n$, against the variation of the Fock basis size $N_{max}$ for a chosen set of equal parameters for the two cavity modes. We find that the results converge for $N_{max} \ge 5$ for the chosen parameter set as mentioned in the figure caption. For the strong-coupling regime limited by $g_1 \le 10$ and $g_2 \le 10$; and $\eta_{ic1} \le 2$ and $\eta_{ic2} \le 2$, we can safely restrict Fock states per cavity mode at $N_{max} = 6$. Throughout our analysis, we set $N_{max} = 6$ for the two modes and do not take the value of $g_1$ or $g_2$ grater than 10 and that of $\eta_{ic1}$ or $\eta_{ic2}$ greater than 2.
Fig.~\ref{fig 4} demonstrates that the dual incoherent pumping leads to the thresholdless two-mode lasing action. From Fig.~\ref{fig 4}(a), one can notice that as the atom-cavity coupling strengths $ g_1= g_2= g$ increase for equal and fixed incoherent pump strengths $\eta_{ic1}=\eta_{ic2}=\eta_{ic}$ the average photon numbers $\bar n_1=\bar n_2 = \bar n$ first start growing and then saturate. Larger the value of $\eta_{ic}$, the larger is the saturation limit revealing that the dual incoherent pumping leads to amplification in both the modes. The surface plots in Fig.~\ref{fig 4}(b) and \ref{fig 4}(c) show the variation of $\bar n_1$ and $\bar n_2$ against both $g_1$ and $g_2$. These plots suggest that one can control the relative photon numbers in the two modes by changing the two atom-cavity coupling parameters. Below the saturation limit of $g_i$ ($i=1,2$), the average photon number in the $i$th mode is directly proportional to $g_i$. If we switch off one the incoherent pumps, then the system reduces to an effective two-level atom pumped incoherently inside a single-mode cavity, leading to single-mode CQED laser.
In Fig.~\ref{fig 5}(a) equal-time Hanbury Brown-Twiss (HBT) intra-mode correlation function $g_{11(22)}^{(2)}(0)$ is plotted against coupling parameter ${g_1 = g_2 = g}$, for four different values of $\eta_{ic}$: 0.5 (solid), 1.0 (dashed), 1.5 (dotted), 2 (dashed-dotted). If all other parameters corresponding to both the modes are equal, then $g_{11}^{(2)}(0) = g_{22}^{(2)}(0)$. Fig.~\ref{fig 5}(b) shows equal- time inter-mode HBT correlation function $g_{12}^{(2)}(0)$ for the same parameters . For small values of $\eta_{ic}$ and atom-cavity coupling constants being not very high, both inter and intra-mode HBT correlations show nonclassical behavior characterized by $g_{11(22)}^{(2)}(0)<1, g_{12}^{(2)}(0)<1$. As $\eta_{ic}$ increases for fixed values of $g$, the inter-mode correlation remains nonclassical ($g_{12}^{(2)}(0)<1$), while the intra-mode correlation approaches thermal limit, i.e., $1 < g_{11(22)}^{(2)}(0) < 2$. It is interesting to note that, the cavity photons of the same mode are most likely to appear in pairs, particularly for strong incoherent pumping, In contrast, if a photon in one mode is created in the case of weak incoherent pumping, a photon in the other mode can not be created at the same time. This feature may be explained from the point of view of an interplay between the coherent cavity-field Rabi dynamics and the incoherent processes as schematically depicted in the level diagram of Fig.\ref{fig 2}. When the atom is initially in the ground state $\mid a, 0, 0 \rangle$ with the two cavity fields in the vacuum, the incoherent pumping can raise the atom to either of the excited states $\mid b, 0, 0 \rangle$ or $\mid c, 0, 0 \rangle$. Suppose, the atom is excited to $\mid b, 0, 0 \rangle$, Then the atom can either emit a photon in cavity mode 1 or into the vacuum by atomic decay. If it emits a photon into the cavity mode 1, then it will find itself in state $\mid a, 1, 0 \rangle$ until the photon leaks out of the cavity with the characteristic time scale $\kappa^{-1}$. In both the cases of atomic decay and cavity decay, the system will be reset to its initial state $\mid a, 0, 0 \rangle$. Now, if the incoherent pumping strength is weak, then it is most likely that the system will be reset to its initial state rather than being promoted to the next higher photon sector. In the strong-coupling CQED regime with one photon in mode 1 and no photon in mode 2, there will Rabi oscillations between the two atomic states $\mid a \rangle$ and $\mid b \rangle$ before the atom or the photon decays. Thus, for weak incoherent pumping, until and unless the atom is excited to $\mid c, 0, 0 \rangle$ from the reset state $\mid a, 0, 0 \rangle$ by incoherent processes, a photon in the mode 2 will not appear. This is the physical origin of the inter-mode photon-photon anti-bunching.
In Fig.~\ref{fig 6}(a), we plot $\bar{n}_1$ vs. $\delta_1$ for $\delta_2 = 0$ for four different values of $\eta_{ic}$ as mentioned in the legend of the figure. It is important to note that, as we increase $\eta_{ic}$ above a certain value ($\eta_{ic}\ge 1.5$), a single-peak structure is split into a double peak one. Figures~\ref{fig 6}(b) and \ref{fig 6}(c) display the variation of $\bar n_1$ and $\bar n_2$, respectively, as functions of both $\delta_1$ and $\delta_2$ for $\eta_{ic}= 0.5$ while
Fig.~\ref{fig 6}(d) and \ref{fig 6}(e) show the same, respectively, for $\eta_{ic}= 2$. Figures~\ref{fig 6}(b) and \ref{fig 6}(c) exhibit only one peak structure while Fig.~\ref{fig 6}(d) and \ref{fig 6}(e) clearly show the two-peak structures. From these $(\delta_1, \delta_2)$ plots, the positions of the two peaks for $\bar{n}_1$ are detected to be $(g/2, -\sqrt{2} g)$ and $(-g/2, g)$ while those for $\bar{n}_2$ are $(\sqrt{2} g, -g/2)$ and $(-g, g/2)$, indicating that probably the lasing in the two modes occur due to stimulated transitions between the dressed state. For weak incoherent pumping, the amplification or lasing occurs primarily due to the transitions in the lower photon sectors, namely, (0,0)-, (1,0)- and (0,1)-sectors. In these sectors, the cavity-field stimulated transitions $\mid b, 0, 0 \rangle \rightarrow \mid a, 1, 0 \rangle $ and $\mid c, 0, 0 \rangle \rightarrow \mid a, 0, 1 \rangle $ to the two modes occur independently in two non-coupled two-level systems. The intensity of the emitted radiation is broadened due to strong CQED coupling giving the linewidth $\sim 2 g + \gamma + \kappa$ which is 22 for the parameters chosen. When incoherent pumping strength is increased beyond 1.5, the higher photon sectors (0,2),(2,0) becomes involved in the stimulated transition process. As shown in the bare- and dressed-state level diagram of Fig.\ref{fig 2}, all three bare- or dressed-states become involved in the process for higher photon-sectors, leading to two-peak structure in the radiation intensity.
Fig.~\ref{fig 7}(a) shows the variation of $g_{12}^{(2)}(\tau)$ against $\tau$ while Fig.~\ref{fig 7}(b) exhibits the spectrum $\mathcal{F}_{12}(\omega)$ as a function of frequency $\omega$ for four different values of $\eta_{ic}$ as mentioned in the legend. As $\tau\rightarrow \infty$, $g_{12}^{(2)}(\tau)$ reaches the value of unity implying that the two modes become uncorrelated in the long time delay limit. For low values of $\eta_{ic}$, $g_{12}^{(2)}(\tau)$ exhibits oscillatory decay from nonclassical domain $(g_{12}^{(2)}(\tau) < 1)$ towards coherent limit $(g_{12}^{(2)}(\tau) \rightarrow 1)$. As we increase $\eta_{ic}$, the amplitude of the oscillations gradually diminish. These oscillations are reminiscent of quantum beats and indicative of atomic coherence. The Fourier transform of $g_{12}^{(2)}(\tau) - 1$, that is, ${ \cal F}_{12}(\omega)$ as a function of $\omega$ as shown in Fig.\ref{fig 7}(b) reveals that there is one dominant frequency of oscillations which is $2g$ in this case. We recall the the vacuum-field Rabi splitting of two-level Jaynes-Cummings model is $2g$. This reveals that for low incoherent pumping, the photons are generated due to stimulated transitions in the (1,0)- and (0,1)-photon sectors. Remarkably, the anti-correlation between the two field modes $g_{12}^{(2)}(\tau)- 1 < 0$ is also reflected in the ${\cal F}_{12}(\omega)$ vs. $\omega$ plots. This can be ascertained from the observation that the central peak of ${\cal F}_{12}(\omega)$ at $\omega = 0$ is negative. We have checked that as we gradually increase the strength of the incoherent pump, a new smaller dip appears at $\omega = \sqrt{2}g$ in the ${\cal F}_{12}(\omega)$ vs. $\omega$ plot. The dressed-state energies of (1,1)-photon sector for $\delta_1 = \delta_2 =0$ are $-\sqrt{2} g$, 0 and $\sqrt{2}g$ and for (2,0) and (0,2) photon sectors it is $\pm \sqrt{2}g$. But (1,1) photon sector will not take part in this process as discussed in Eq. (\ref{eq 19}). This indicates that for relatively higher incoherent pump strength, the oscillations in the photon-photon correlations correspond to Rabi oscillations in these two-states CQED dynamics.
We display the variation of $g_{11(22)}^{(2)}(\tau)$ as a function of $\tau $ against in Fig.~\ref{fig 8} (a) for four different values of $\eta_{ic}$. The correspond spectrum is shown in Fig.~\ref{fig 8} (b). As in the case of inter-mode correlations, the intra-mode correlation too exhibits oscillatory decay. However, in contrast to inter-mode case, the intra-mode correlations become predominantly classical $(g_{12}^{(2)}(\tau) \ge 1 )$ for $\eta_{ic} > 1$ while they are mostly nonclassical for $\eta_{ic} \le 1$. These correlation properties are also manifested in the spectral profiles in that the central peak of ${\cal F}_{ii}(\omega) $ at $\omega = 0$ is positive or negative for the classical or nonclassical correlations, respectively.
\subsection{The effects of coherent drives: Photon-Photon entanglement}
Here we calculate the lowest eigenvalue of partially-transposed matrix of the reduced two-mode field density matrix as discussed in the subsection \ref{D} of the previous section to verify the existence of photon-photon entanglement. The negativity of the eigenvalue is a clear signature of the entanglement. For our chosen parameter regime and truncated Fock space, the reduced two-mode field density matrix is of 36$\times$36 dimension.
For simplicity, as before, we consider the two cavity fields are in two-photon resonance with the atomic transitions, that is, $\delta_1 = \delta_2= \delta$.
Under this condition, in a frame rotating with the driving laser frequencies,
the eigenvalues of the three dressed states are given by
\begin{align*}
E_0/{\hbar} &= n_1 \delta_{1L}+ n_2\delta_{2L} - \delta~, \\
E_{\pm}/{\hbar} &= n_1 \delta_{1L} + n_2 \delta_{2L} - \frac{\delta}{2} \pm \frac{1}{2} \sqrt{ \delta^2 + 4 \left ( n_1 g_1^2 + n_2 g_2^2 \right ) }~,
\end{align*}
where $\delta_{1L} = \omega_1 - \omega_{1L} = \delta_1 + \Delta_{b}$ and $\delta_{2L} = \omega_2 - \omega_{2L} = \delta_2 + \Delta_c$ are two cavity-drive detuning parameters. The dressed-state level diagram depicting the transitions due to the coherent drives that connect different photon sectors is shown in Fig.~\ref{fig 9} for $\eta_{c1} = \eta_{c2} = \eta_{c}$.
The amplitudes of the transitions are mentioned in the diagram. These transition amplitudes are given by
\begin{eqnarray}
\langle \psi_{\alpha}(n_1, n_2) \mid \hbar \eta_c \left (\sigma_1 + \sigma_2 + {\rm H.c.} \right ) \mid \psi_{\beta}(n_1', n_2') \rangle~.
\end{eqnarray}
where $\alpha, \beta = 0, \pm$, $\mid \psi_{\alpha}(n_1, n_2) \rangle$ and $\mid \psi_{\beta}(n_1', n_2') \rangle$ refer to the dressed states as given
by Eqs. \ref{eqn psipm} and \ref{eqn psibc}. In the absence of incoherent pumping, as $\eta_c$ increases from zero the coherent drives will couple from lower to higher dressed states successively as a consequence of the competition between the driving and the damping processes. The drives make superposition of the cavity-dressed states. The superposition state is an atom-field entangled state. Since the field is a two-mode one, if the atom is measured in a particular state, the atom-field entangled state will be projected into a two-mode entangled field state.
For numerical analysis of the effects of coherent drives, we set $\kappa_1 = \kappa_2 = \kappa = 1$, $\delta_1 = \delta_2 = 0$, $g_1=g_2 = 10$. We plot
the smallest eigenvalue of $\rho_{f}^{{\rm PT,2}}$ against $\delta_{2L}$ in Fig.~\ref{fig 10}(a) for $\eta_{c1}=\eta_{c2} = \eta_c = 2$, $\delta_{1L} = 0 $ for two values of $\eta_{ic}$: $\eta_{ic}=0.01$ (dashed) and $\eta_{ic}=0.1$ (solid). Both the eigenvalues are negative though the magnitude is quite small ($\sim 10^{-3})$. We notice that, for $\eta_{ic} = 0.001$ the eigenvalue becomes appreciably negative while for $\eta_{ic} = 0.1$ the magnitude of the eigenvalue is negligibly small ($\sim 0$) compared to that for the former one. This implies that even a small increase of incoherent pump can destroy the entanglement. One can further notice that the eigenvalue for $\eta_{ic} = 0.001$ has four dips: two at $\delta_{2L} = \pm \sqrt{2} g $ and the other two at $\delta_{2L} = g/\sqrt{2} $. These dips correspond to the entangled states of the two-mode cavity fields. According to the dressed-state level diagram of Fig.~\ref{fig 9}, $\delta_{2L} = \pm \sqrt{2} g $ corresponds to the shifts of the two dressed states $\mid \psi^{\pm}(1,1) \rangle$ in (1,1)-photon sector while $\delta_{2L} = \pm g/\sqrt{2} $ corresponds to the shifts of the two dressed states in (0,2)-photon sector. Since the value of $\eta_{c1} = \eta_{c2} = 2$ is greater than the effective decay rate $(\kappa + \gamma)/2 = 1$, the atom is being relatively strongly driven in this case leading to the excitation of these higher level dressed states.
To know the average photon numbers in the two cavity modes when they are entangled, we plot both $\bar{n}_2$ and $\bar{n}_1$ as a function of $\delta_{2L}$ in Fig.~\ref{fig 10}(b).
The peaks in $\bar{n}_2$ vs. $\delta_{2L}$ plot at $\delta_{2L} = \pm g$ correspond to vacuum-field Rabi splittings in the transitions between (0,0) and (0,1) photon sectors, while the two other peaks at $\delta_{2L} = \pm g/\sqrt{2}$ correspond to one-photon Rabi splittings in the transitions between (0,1) and (0,2) photon sectors. We find that at $\delta_{2L} = \pm \sqrt{2} g $, $\bar{n}_1 \simeq \bar{n}_2 \simeq 0.05$. In contrast at $\delta_{2L} = \pm g/\sqrt{2} $, $\bar{n}_1 \simeq 0.05 $ and $\bar{n}_2 \simeq 0.38$. This means that the entangled states corresponding to $\delta_{2L} = \pm g/\sqrt{2}$ have more number of photons in the second cavity mode due to resonant enhancement of the dressed states belonging to (0,2)- photon sector while the first cavity mode contains less number of photons as $\delta_{1L}=0$ corresponds to off-resonance due to the large shifts $g/\sqrt{2} \simeq 7.1$. The reason for the small number of photons in both the modes in the entangled states corresponding to $\delta_{2L} = \pm \sqrt{2} g$ is the existence of the dark state $ \mid \psi_0(1,1) \rangle$ in the (1,1)-photon sector. The probability of photon emission into the two cavity modes is proportional to the quantity
\begin{eqnarray}
\mid \langle \psi_{\alpha} (n_1,n_2) \mid \left [ a_1^{\dagger} \sigma_1 + a_2^{\dagger}\sigma_2 \right ] \mid \psi_{\beta}(n_1,n_2) \rangle \mid^2 ~.
\label{eq 19}
\end{eqnarray}
For any two dressed states belonging to (1,1)-photon sector under two-photon resonance condition for the two cavity modes, the above quantity vanishes
while it is finite for (1,0)-, (2,0)-, (0,1)- and (0,2)-photon sectors. This explains why there is small number of photons in the entangled states corresponding to $\delta_{2L} = \pm \sqrt{2} g$. Moreover, $ \mid \psi_0(1,1) \rangle$ can not be populated by the the drives due to destructive interference between the transition pathways as schematically shown in Fig. \ref{fig 9}.
Next we study the system driven by the dual coherent drives in the absence of any incoherent pump. The smallest eigenvalue of $\rho_{f}^{{\rm PT,2}}$ as a function $\delta_{2L}=\delta_{1L}=\delta_{L}$ is plotted in
Fig.~\ref{fig 11} (a) . This figure shows that the eigenvalue becomes more negative with the increase of the coherent drive strength. Fig.~\ref{fig 11}(b) in $\bar{n}_2,\bar{n}_1$ vs. $\delta_{L}$ plots, the peak at $\delta_{L}= g = 10$ corresponds to the emissions from (0,1),(1,0) photon sector. An additional peak appears at $\delta_{L} = g/\sqrt{2} \simeq 7.1 $ with the increase of $\eta_c$ as a signature of the emission from (0,2),(2,0)-photon sector. Again, an emergence of another peak is observed at $\delta_{L} = g/\sqrt{3} \simeq 6 $ as an evidence of probing (2,1) and (1,2) photon sectors. Since both $\bar n_1$ and $\bar n_2$ are equal and appreciably enhanced, it is meaningful to analyse the statistical properties in this parameter regime. In Fig. \ref{fig 11} most of the dips in the negativity of the eigenvalue correspond to the peaks in the photon number populations in both the modes as can be inferred by comparing Figs. \ref{fig 11} (a) with \ref{fig 11}(b). In Fig. \ref{fig 11}(c), $g_{12}^{(2)}(0)$ is plotted as a function of $\delta_L$. It is to be noted that the inter-mode antibuncing does not always necessarily mean inter-mode entanglement.
As mentioned before, the origin of the entanglement can be traced to the dressing of the cavity-dressed states by the coherent drives. The possibility of generating atom-photon entangled states, Schroedinger cat states of cavity fields and two-mode entangled field states by driving two-level atoms with a strong coherent field inside a single-mode cavity had been discussed by Solano, Agarwal and Walther \cite{PhysRevLett.90.027903} about two decades ago. They considered the dressed states of atoms due to the strong classical drive and then these atomic dressed states are then superposed by the cavity field, producing atom-cavity dressed states and thus leading to the entanglement. In our case, the two strong cavity fields first dress the three-level atom, leading to bimodal atom-cavity dressed states which are then further dressed by the coherent drives. Thus tripartite entangled states involving the atom and the two cavity modes are produced. Now, if the $V$-type three-level atom is measured in the ground state, the tripartite entangled state will be projected into a two-mode entangled field state. In fact, generation of two-mode entangled fields in the microwave frequency domain using a bimodal superconducting cavity and circular Rydberg atoms had been experimentally demonstrated by Heroche's group in 2001 \cite{haroche:pra64:1050:2001}. In that experiment, the cavity was driven by excited Rydberg atoms that had passed through the cavity one by one. In 2005, Xiong, Scully and Zubairy \cite{xion:prl94:023601:2005} had theoretically shown that a "Correlated-spontaneous emission laser" using an ensemble of cascade or $\Lambda$-type three-level atoms driven by two classical coherent fields inside a doubly resonant cavity can produce two-mode entangled radiation with large number of photons in each mode.
\section{Conclusions}\label{sec4}
In conclusion, we have proposed a single-atom CQED laser that can generate nonclassically correlated and entangled photon pairs. We have shown that when the two cavity modes are incoherently pumped without any coherent drive, the laser produces amplified signals for both the modes with inter-mode nonclassical photon-photon correlation but with no photon-photon entanglement. The nonclassical behavior is manifested though inter-mode photon-photon anti-bunching which is shown to result from an interplay between the incoherent processes and cavity-field induced coherent processes. When coherent atomic drives are used with small or no incoherent pumping, the laser generates inter-mode entangled photon pairs. We have demonstrated that the entanglement arises due to the dressing of the cavity-dressed states by the coherent drives. Our results show that increasing the incoherent pumping rates even by a small amount destroys the entanglement. As elaborated in our paper, not only the intensities of the generated fields, but also the photon-photon correlations and entanglement can be controlled at a microscopic level by varying the incoherent pumping rates, coherent drive strengths and detuning parameters, thus providing a wide range of functionality of our proposed laser system.
Our proposal may be implemented experimentally by combining the currently available technologies of cavity QED and atom traps. For example, an atom trap containing a single bosonic atom may be loaded inside a bimodal cavity. The three levels of the atom may be chosen in the following way: Let the atom in the ground electronic $S$-state be prepared in the hyperfine spin state $\mid F, m_F = 0 \rangle$, where $F$ is the hyperfine spin quantum number and $m_F $ denotes the magnetic hyperfine spin quantum number. The two excited ($P$) states be represented by $\mid F', m_F' = - 1\rangle$ and $\mid F', m_F' = 1\rangle$, where the excited hyperfine spin $F' \ge 1$. Now, if the two cavity modes are chosen to be left and right circularly polarized fields then our proposed one-atom laser will produce polarization-entangled photon pairs with antibunching. This may be realized with a doubly resonant cavity for which the two cavity axes for the two modes may coincide as shown in Fig.\ref{fig 1} (ii). As regards the implementation of incoherent optical pumping, Kimble's group at Cal-tech has already experimentally demonstrated such pumping in CQED using incoherent Raman transitions in a single-atom far-off resonance trap (FORT) loaded inside a cavity \cite{thesis_boozer2005raman}. In the single-mode CQED laser with a single trapped Cs atom reported in Ref. \cite{Nature1}, the laser output was available for a short duration limited by the lifetime of the trap which was 50 ms. This time limitation may be overcome by using single-ion trap instead of a single-atom trap in CQED regime \cite{walther:epl:1997, blatt_PhysRevLett.114.023602}. Because. the lifetime of an ion-trap is quite large. Furthermore, an ion trap may offer some advantages as far as coherent or incoherent optical pumping is concerned as the stability of an ion-trap does not depend on the internal electronic states of the ion unlike that of an atom-trap. Therefore, ion-trap CQED may be an interesting research direction for generating nonclassical and entangled photon pairs for practical applications.
\bibliographystyle{apsrev4-2.bst}
|
1,108,101,563,295 | arxiv | \section{Introduction}
Despite intensive efforts in the last two decades understanding the nature of the pseudogap phase in cuprate high temperature
superconductors remains an open problem \cite{Norman:2005}. At zero hole doping, the insulating parent compounds of these systems are three-dimensional (3D) antiferromagnetic insulators.
As holes are introduced, the superconducting transition temperature ($T_c$) is finite above a critical doping, reaching a maximum at a hole concentration known as optimal doping. The regime with hole doping less than optimal -- known as underdoped regime -- hosts the remarkable pseudogap phase. In this regime, the normal (non-superconducting) phase above $T_c$ has an anisotropic spectral gap of unknown origin (`pseudo'-gap), and behaves strikingly differently from a Fermi liquid. It is generally understood that the crux of the problem of $d$-wave pairing resides in the pseudogap phase, from which superconductivity develops at lower temperatures.
The idea that in the underdoped regime of the cuprates various charge, spin, or current ordered states compete with superconductivity has been apparent in the past few years~\cite{Varma:1997,Chakravarty:2001,Kivelson:2003,Hosur:2012,Orenstein:2012,Chakravarty:2013,Wu:2011,Wu:2013}.
A compelling evidence is provided in the class of materials YBa$_2$Cu$_3$O$_{6+x}$ (YBCO), in which, upon suppression of superconductivity by a strong magnetic field, exquisite quantum oscillations of various electronic properties with the applied field point to the existence of Fermi surface pockets \cite{Doiron-Leyraud:2007,Sebastian:2008}. Since in the overdoped regime the Fermi surface is large and hole-like, the existence of Fermi pockets in the underdoped regime indicates a Fermi surface reconstruction near optical doping. A change in the topology of the Fermi surface from hole-like at higher doping to electron-like at low doping is also indicated by measurements of low temperature Hall and Seebeck coefficients which turn negative in the underdoped regime \cite{LeBoeuf:2007,Chang:2010,Laliberte:2011}.
Although various charge, spin, and current ordered states have been proposed to account for the Fermi surface reconstruction in YBCO \cite{Chakravarty-Kee,Millis-Norman}, none had so far been observed in bulk-sensitive probes until recently.
In recent X-ray diffraction experiments, two groups have independently discovered \cite{Chang:2012,Ghiringhelli:2012,Kivelson,Tranquada} strong evidence for an
incipient charge density wave (CDW) instability (with scattering peaks at wave vectors $Q_1=(q_1,0,0.5)$ and $Q_2=(0,q_2,0.5)$, with $q_1 \sim q_2 \sim 0.31$) competing with superconductivity in a range of doping in the underdoped regime of YBCO. Although not conclusively known if the peaks derive from equal distributions of domains with only uni-axial CDW correlations (i.e., with modulations given by only $Q_1=(q_1,0,0.5)$ or $Q_2=(0,q_2,0.5)$) or a bi-axial CDW with both wave vectors co-existing, the lack of anisotropy in the scattering signals (intensities, widths) indicates a coupling between $Q_1$ and $Q_2$ and a bi-axial CDW order. The temperature ($T$) dependent correlation length above $T_c$ and only short ranged correlations perpendicular to the CuO$_2$ planes indicate that the observed charge order is only quasi-static. However, the near divergence of the correlation length as $T \rightarrow T_c$ (and that the scattering signals significantly increase on application of magnetic field below $T_c$) indicate that a true thermodynamic CDW transition at some critical temperature ($T_{CDW}<T_c$) is preempted by the superconducting transition at $T_c$. Evidence for a similar charge density wave transition in the underdoped regime has also been found in other recent experiments~\cite{LeBoeuf:2013,Hinton:2013,sebastian:2014,Fujita:2014,Comin:2014}.
In this paper we treat a two-dimensional (2D) bi-axial $Q_1=(2\pi/3,0)$ and $Q_2=(0,2\pi/3)$ CDW state in mean field theory (valid for temperatures $T<T_{CDW}$ and magnetic fields high enough to eliminate the superconductivity) and investigate the reconstructed Fermi surface and experimentally measured transport coefficients as functions of temperature and hole doping appropriate for the underdoped regime of the cuprates. Although the experimental evidence is that for a slight incommensuration in the CDW scattering vectors (i.e., $q_1 \sim q_2 \sim 0.31$) in this paper we work with a commensurate CDW for simplicity (i.e. we take $q_1=q_2=0.33$, corresponding to charge modulations with periodicity of three lattice vectors). We find that, below the CDW transition temperature and in the appropriate regime of hole doping, the Fermi surface topology changes from a large hole-like Fermi surface at higher doping to a few
electron and hole-like Fermi pockets in the CDW state.
A similar fermi surface reconstruction in terms of a different CDW state was recently assumed to explain the low frequency of quantum oscillations in the pseudogap phase of the cuprates~\cite{sebastian}.
An explanation of the Fermi pockets observed in the recent quantum oscillation experiment in Ref.~\cite{sebastian:2014} in terms of the bi-axial CDW state considered in the present work is left for future study.
We calculate the CDW Hall and Seebeck coefficients using the Kubo formula which reduces to the Boltzmann expressions in the experimentally appropriate limits. We show that, in the appropriate doping range where the low temperature phase (in the absence of superconductivity) is a bi-axial CDW, the Hall and Seebeck coefficients turn negative for $T<T_{CDW}$, as seen in the experiments. Our calculations establish that the recent observations of strong incipient CDW instability in YBCO \cite{Chang:2012,Ghiringhelli:2012} are broadly consistent with the other recent experimental surprises in underdoped regime, namely, the existence of Fermi surface pockets \cite{Doiron-Leyraud:2007,Sebastian:2008} and negative low temperature transport coefficients indicating Fermi surface reconstruction \cite{LeBoeuf:2007,Chang:2010,Laliberte:2011}.
The paper is organized as follows. In Sec.~\ref{secii}, we calculate the Fermi surface in the presence of a bi-axial $Q_1$ and $Q_2$ CDW order parameter in mean-field theory. We present in Sec.~\ref{seciii} the temperature and doping dependent crossover in sign of the Hall and Seebeck coefficients using the Kubo formula. In Sec.~\ref{seciv}, we show that such sign changes in the coefficients does not occur in a uni-axial CDW state. We summarize and conclude in Sec.~\ref{secv}.
\begin{figure}[t]
\includegraphics[width = 1\linewidth]{fermi_surface_cdw.eps}
\caption{
(color online)
{(a) The CDW band structure from the eigenvalues $E_n( \mathbf{k} )$ of the matrix $H( \mathbf{k} )$, Eq.~(\ref{h_matrix}). The chemical potential, corresponding to hole doping $p=0.125$, is noted by a dashed line. Here, the CDW order parameter $W = 0.069$ eV at $T=0$.
(b) The Fermi surface for $p=0.125$ in the first Brillouin zone. The green dashed square in the middle represents the reduced Brillouin zone (RBZ) boundary determined by the CDW momenta $Q_1$ and $Q_2$. Note that the Fermi surface in the CDW state consists of two electron pockets in RBZ: one large electron pockets at the center $(0,0)$ and the corners $(\pi/3,\pi/3)$ of the RBZ (red) and two hole pockets at the boundary $(\pi/3,0)$ and $(0,\pi/3)$ (red).
}
}
\label{fermi_surface_cdw}
\end{figure}
\section{Mean-field Hamiltonian and CDW Fermi surface\label{secii}}
To calculate the Fermi surface and the normal-state quasiparticle Hall and Seebeck coefficients, we consider electrons on a two-dimensional square lattice of unit lattice constant, with a dispersion given by
\begin{eqnarray}
\label{dispersion}
\varepsilon_ \mathbf{k}
&=&
-2t \left( \cos k_x + \cos k_y \right) +
4 t^\prime \cos k_x \cos k_y
\nonumber\\
&&-2t^{\prime\prime} \left( \cos 2 k_x + \cos 2 k_y \right),
\end{eqnarray}
where $t$ is a nearest-neighbor, $t^\prime$ is a next-nearest-neighbor, and $t^{\prime\prime}$ is a next-to-next-nearest-neighbor hopping integrals. For the Fermi surface of YBCO measured in photoemission experiments, we choose the parameters $t=0.3$ eV, $t^\prime = 0.32 t$, and $t^{\prime\prime} = 0.1 t^\prime$.
Within a mean-field picture, the Hamiltonian for the CDW state is described by the quasiparticles subject to a periodic modulation with wave vector $Q_1=(2\pi/3,0)$ and $Q_2=(0,2\pi/3)$,
\begin{equation}
H_{MF}
=\sum_{ \mathbf{k} ,\sigma} \left( \varepsilon_ \mathbf{k} - \mu \right) c_{ \mathbf{k} \sigma}^\dagger c_{ \mathbf{k} \sigma}+
\sum_{ \mathbf{k} ,Q_i,\sigma} W_{Q_i} c_{ \mathbf{k} +Q,\sigma}^\dagger c_{ \mathbf{k} \sigma} + H.c.,
\end{equation}
where $c_{ \mathbf{k} \sigma}$ is the annihilation operator for an electron of spin $\sigma$ and momentum $ \mathbf{k} $, and
$
W_{Q_i} = U\sum_{ \mathbf{k} } \langle c_{ \mathbf{k} +Q_i, \sigma}^\dagger c_{ \mathbf{k} , \sigma} \rangle
$ is a CDW order parameter with on-site Coulomb repulsion $U$.
By taking $W_{Q_1} = W_{Q_2} = W$, we can express the Hamiltonian in terms of a nine-component quasiparticle spinor $\psi_\sigma( \mathbf{k} )$ as
\begin{equation}
H_{MF} =
\sum_{ \mathbf{k} \in \text{RBZ},\sigma}
\psi_\sigma^\dagger( \mathbf{k} ) H( \mathbf{k} ) \psi_\sigma( \mathbf{k} ),
\end{equation}
where the reduced Brillouin zone (RBZ) is given by $|k_x|, |k_y|\le \pi / 3$, and $H( \mathbf{k} )$ is $9\times 9$ matrix
\begin{widetext}
\begin{equation}
\label{h_matrix}
H( \mathbf{k} ) =
\begin{pmatrix}
\xi_{ \mathbf{k} } & W & W & W & 0 & 0 & W & 0 & 0\\
W & \xi_{ \mathbf{k} +Q_1} & W & 0 & W & 0 & 0 & W & 0\\
W & W & \xi_{ \mathbf{k} -Q_1} & 0 & 0 & W & 0 & 0 & W\\
W & 0 & 0 & \xi_{ \mathbf{k} +Q_2} & W & W & W & 0 & 0\\
0 & W & 0 & W & \xi_{ \mathbf{k} +Q_1+Q_2} & W & 0 & W & 0 \\
0 & 0 & W & W & W & \xi_{ \mathbf{k} -Q_1+Q_2} & 0 & 0 & W \\
W & 0 & 0 & W & 0 & 0 & \xi_{ \mathbf{k} -Q_2} & W & W \\
0 & W & 0 & 0 & W & 0 & W & \xi_{ \mathbf{k} +Q_1-Q_2} & W \\
0 & 0 & W & 0 & 0 & W & W & W & \xi_{ \mathbf{k} -Q_1-Q_2}
\end{pmatrix}.
\end{equation}
\end{widetext}
The energy spectrum $E_n( \mathbf{k} )$ and corresponding eigenoperators $\chi_n( \mathbf{k} )$ can be obtained by diagonalizing the matrix $H( \mathbf{k} )$.
Note that the chemical potential has to be adjusted for the value of $W$ to preserve the doping $p$.
The number of electrons per unit cell equals $2N$ due to spin degeneracy, and the doping of the cuprates is counted from half-filling, hence $p = 1 - 2N\,$, where $N$ is the occupied part of the Brillouin zone
\begin{equation}
N = \sum_{ \mathbf{k} \in \text{RBZ},n} f(E_n( \mathbf{k} ))\,,
\end{equation}
where $f(x) = 1/(1+\exp (x/T)$ is a Fermi distribution function. We find that at $T=0$, $\mu = -0.6989 t$ corresponds to half-filled ($p=0$) and $\mu = -1.0027 t$ corresponds to $p = 0.125$. In Fig.~\ref{fermi_surface_cdw}, we plot the band structure and the Fermi surface corresponding to the CDW state from the eigenvalues $E_n( \mathbf{k} )$ for hole doping $p=0.125$ in the first Brillouin zone. We find that the Fermi surface consists of electron pockets as well as hole pockets in the reduced Brillouin zone given by the green dashed line. Note that there are two electron pockets in RBZ: one electron pocket at the center $(0,0)$ and the other at the corners $(\pi/3,\pi/3)$ of the RBZ (blue and red), and two hole pockets at the boundary $(\pi/3,0)$ and $(0,\pi/3)$ (red). Since for higher doping, the free electron Fermi surface in the absence of CDW (given by $\varepsilon_{\bf k}$ in Eq.~(\ref{dispersion}) set to the chemical potential) is large and hole-like, the electron pockets in Fig.~\ref{fermi_surface_cdw} at lower doping indicate a
doping-induced Fermi surface reconstruction.
\section{Hall and Seebeck Coefficients\label{seciii}}
In linear-response theory, three conductivity tensors $\hat \sigma$, $\hat \alpha$, and $\hat \kappa$ are related with charge current $\vec J$ and thermal current $\vec Q$ such as
\begin{equation}
\begin{pmatrix}
\vec J \\ \vec Q
\end{pmatrix}
=
\begin{pmatrix}
\hat \sigma & \hat \alpha \\
T \hat \alpha & \hat \kappa
\end{pmatrix}
\begin{pmatrix}
\vec E \\ - \vec \nabla T
\end{pmatrix}.
\end{equation}
The electric field induced by a thermal gradient in the absence of an electric current $\vec J = 0$ is given by $\vec E = \hat \sigma^{-1} \hat \alpha \vec \nabla T$. Therefore, the Seebeck coefficient $S$, defined as the $E_x$ generated by a thermal gradient $\nabla_x T$, reads as, $S = \alpha_{xx}/\sigma_{xx}$.
The Hall coefficient, defined as the transverse $E_{y}$ generated by a magnetic field $B$, is given by $R_H = \sigma_{xy}/\sigma_{xx}^2$.
\begin{figure}[t]
\includegraphics[width = \linewidth]{temp_dep.eps}
\caption{
Temperature dependence of the Seebeck and Hall coefficients in the CDW state.
(a) Normal-state Seebeck coefficients $S/T$ at various doping concentrations $p$ are plotted as a function of temperature $T$.
The negative Seebeck coefficients at low temperature for $p=0.092$ and $0.125$ are ascribed to the electron pockets in the Fermi surface due to the CDW, while $S/T$ at $p=0.08$ ($W=0$) remains positive.
(b) Hall coefficients $R_H$ in arbitrary units for various $p$ are plotted as a function of $T$.
The sign of the Hall coefficient changes at lower temperature for $p=0.092$ and 0.125, while it remains positive at all temperatures for $p=0.08$.
}
\label{fig-temp_dep}
\end{figure}
We calculate the dc transport properties using the Kubo formula by applying perturbative electric and magnetic fields: $\vec E = E_0 \hat x \cos \omega t$, $\vec B = q \hat z A_0 \sin q y$.
Further, we assume that the scattering time $\tau(E, \mathbf{k} )$ is independent of the energy, but momentum dependent: $\tau(E, \mathbf{k} ) = \tau( \mathbf{k} )$.
Phenomenologically, we retain a possible momentum dependence of the scattering time because, as has been shown before, an assumption of a momentum independent $\tau$ results in a negative Seebeck coefficient in the normal state~\cite{hildebrand, kontani, storey} (i.e., above the CDW transition temperature), inconsistent with experiments. In this work we assume the scattering time $\tau( \mathbf{k} ) = ( 1 + \alpha ( \cos k_x + \cos k_y ) )^2$ with $\alpha=0.4$. This form of $\tau(k)$ is purely phenomenological and designed to enhance the contribution of the free fermion hole pockets at high temperatures (i.e., above the CDW transition temperature) located near $(\pi,\pi)$ and symmetry related points in the first Brillouin zone. By enhancing the contribution of the hole pockets the high temperature ($W=0$) Seebeck coefficient is positive, in accordance with the experiments~\cite{Chang:2012,Ghiringhelli:2012}. The precise functional
form of $\tau(k)$ is unimportant, however, and any other momentum dependence of the scattering time that enhances the contribution of the high temperature hole pockets to the Seebeck coefficient works just as well. Let us also emphasize that the drop in the Seebeck and Hall coefficients with decreasing $T$ and the eventual negative sign at low temperatures (Fig.~\ref{fig-temp_dep}) are both robust features of the bi-axial CDW state and are immune to variations in the precise functional forms of the scattering time. In particular they continue to hold even with a $k$-independent $\tau$.
To lowest order in the applied fields $\vec{E},\vec{B}$,
the conductivity $\sigma_{xx}$ is given by the Fourier transform of the imaginary time-ordered current-current correlation function, $$\sigma_{xx} = \lim_{q\to 0} \frac{1}{\omega} \text{Im}\int^{1/T} d\tau e^{i(\omega+i\delta)\tau} \langle T_\tau J_x(q,\tau)J_x(q,\tau)\rangle,$$ and
$\sigma_{xy}$ is given by,
\begin{eqnarray}
\sigma_{xy} &=& \lim_{q\to 0}\frac{B}{\omega q} \text{Re} \int_0^{1/T} d\tau d\tau^\prime e^{i(\omega+i\delta)\tau} \nonumber\\\nonumber
&\times &
\langle
T_\tau J_y(q,\tau) J_x(0,0) J_y(-q,\tau^\prime)
\rangle,
\end{eqnarray}
where $J_i = e \sum_{ \mathbf{k} ,n} \chi_n^\dagger( \mathbf{k} ) \frac{\partial E_n( \mathbf{k} )}{\partial k_i} \chi_n( \mathbf{k} )$ is the electric current operator.
The thermal conductivity coefficient $\alpha_{xx}$ is given by the appropriate correlation function of the thermal current $\vec Q$ and the electric current $\vec{J}$, $$\alpha_{xx} = \lim_{q\to 0} \frac{1}{\omega} \text{Im}\int^{1/T} d\tau e^{i(\omega+i\delta)\tau} \langle T_\tau Q_x(q,\tau)j_x(q,\tau)\rangle,$$ where $Q_i = \sum_{ \mathbf{k} ,n} \chi_n^\dagger( \mathbf{k} ) E_n( \mathbf{k} )\frac{\partial E_n( \mathbf{k} )}{\partial k_i} \chi_n( \mathbf{k} )$.
By taking $q \to 0$, $\omega \to 0$ and ignoring the inter-band contributions, we have the following results that are same as those obtained from the Boltzmann equation:
\begin{eqnarray}
\label{conds}
&&
\alpha_{xx} =
-\frac{2e}{T}\sum_{ \mathbf{k} ,n}
\tau( \mathbf{k} )
(v_n^x)^2 E_n( \mathbf{k} ) \left( -\frac{\partial f(E_n( \mathbf{k} ))}{\partial E_n( \mathbf{k} )}\right),
\\
&&
\sigma_{xx} =
2e^2\sum_{ \mathbf{k} ,n}
\tau( \mathbf{k} )
(v_n^x)^2 \left( -\frac{\partial f(E_n( \mathbf{k} ))}{\partial E_n( \mathbf{k} )}\right),
\\
&&
\sigma_{xy} =
\frac{2e^3B}{\hbar c}\sum_{ \mathbf{k} ,n}
\tau^2( \mathbf{k} )
[ v_{n}^{x} v_{n}^{y} v_{n}^{xy} - (v_{n}^{x})^2 v_{n}^{yy}]\left( -\frac{\partial f(E_n( \mathbf{k} ))}{\partial E_n( \mathbf{k} )}\right),
\end{eqnarray}
where $v_{n}^x = \frac{\partial E_n( \mathbf{k} )}{\partial k_x}$ and
$v_{n}^{xy} = \frac{\partial^2 E_n( \mathbf{k} )}{\partial k_x \partial k_y}$.
For a given doping concentration $p$, the temperature dependence of the CDW order parameter $W(p,T)$ is assumed to scale with a mean-field exponent below the critical temperature $T_{CDW}(p)$ such that $W(p,T) = W(p) \left\vert 1 - T/T_{CDW}(p) \right\vert^{1/2}$ for $T<T_{CDW}(p)$, whereas $W(p,T)=0$ for $T>T_{CDW}(p)$.
Here, we used the phenomenological form for $T_{CDW}(p) = 142.6 (\text{K}) \times (p-p_{CDW})^{0.3}$ (where $p_{CDW} = 0.085$), which, although arbitrary, gives a best fit to the experiment~\cite{Laliberte:2011}.
Further, we assume the doping dependence of the CDW order parameter $W(p)$ below the critical temperature $T_{CDW}(p)$ such that it disappears below (above) a lower (upper) critical doping, producing a CDW dome consistent with the experiments. On the lower doping side we take $W(p) = W \left\vert (p-p_{CDW})/(p_\text{max}-p_{CDW}) \right\vert^{1/2}$ for $p> p_{CDW}$, and $W(p) = 0$ for $p<p_{CDW}$, and similarly for the higher doping side. Here, we set $p_\text{max} = 0.125$, and $W=0.069$ eV.
Although we use these values in our numerical calculations, we emphasize that our results are robust to reasonable variations of these parameters in the underdoped regime.
We notice that as CDW disappears, the Fermi surface becomes the contour of the dispersion $\varepsilon_ \mathbf{k} - \mu = 0$, which is closed around the corners of the Brillouin zone (the $M$ points), giving rise to the usual hole-like Fermi pockets.
With the conductivities given in Eqs.~(7,8,9), and the $T$ and $p$ dependences of CDW in hand, we calculate the Seebeck and Hall coefficients ($S = \alpha_{xx}/\sigma_{xx}$ and $R_H = \sigma_{xy}/\sigma_{xx}^2$, respectively) in the underdoped regime.
Fig.~\ref{fig-temp_dep} shows the Seebeck coefficient normalized by temperature, $S/T$, and the Hall coefficient $R_H$ as a function of $T$ for various doping concentrations $p$. As the temperature lowers, in the presence of CDW, the signs change from positive to negative in both Seebeck and Hall coefficients, while those for $p=0.08$ ($p<p_{CDW}$) remain positive. This indicates that the emergence of the electron-like pockets due to presence of the CDW gives rise to the negative Seebeck and Hall coefficients at low temperatures.
\begin{figure}[t]
\includegraphics[width = \linewidth]{dop_dep.eps}
\caption{
Doping dependences of Seebeck and Hall coefficients in the bi-axial CDW state with $Q_1=(2\pi/3,0)$ and $Q_2=(0,2\pi/3)$.
(a) With increasing doping concentration $p$, Seebeck coefficients $S/T$ for $T=20$ K and $30$ K change the sign from positive to negative, while $S/T$ at $T=60$ K ($W=0$) remains positive throughout.
(b) Hall coefficients $R_H$ vs. $p$ show the similar behavior as the doping dependence of Seebeck coefficient $S/T$.}
\label{fig-dop_dep}
\end{figure}
In Fig.~\ref{fig-dop_dep} we show the doping dependence of the Seebeck and Hall coefficients near the lower critical doping in the CDW phase (analogous results, not shown here, are found also near the higher critical doping).
Fig.~\ref{fig-dop_dep}a (\ref{fig-dop_dep}b) shows $S/T$ ($R_H$) as a function of doping concentration $p$ for various temperatures $T$. For a given temperature $T < T_{CDW}$, the positive Seebeck and Hall coefficients for $p<p_{CDW}$ ($W(p)=0$ for $p$ below the lower critical doping) becomes negative on crossing the critical doping concentration $p_{CDW}$, where the electron-like pockets emerge due to the CDW. When temperature is above the CDW critical temperature $T_{CDW}$, the Seebeck and Hall coefficients remain positive due to the hole-like Fermi surface around the $M$ points in the first Brillouin zone. Such temperature and doping dependences in the Seebeck and Hall coefficients (specifically, the sign changes at low temperatures in the presence of a bi-axial CDW) are qualitatively consistent with the experiments.
\section{Uni-axial CDW with $Q=(2\pi/3,0)$ or $(0,2\pi/3)$\label{seciv}}
\begin{figure}[b]
\includegraphics[width = 1\linewidth]{fermi_surface_stripe.eps}
\caption{
(color online)
(a) The CDW band structure from the eigenvalues $E_n( \mathbf{k} )$ of the $3\times 3$ matrix in Eq.~(\ref{ham_stripe}). The chemical potential, corresponding to hole doping $p=0.125$, is displayed by a dashed line ($\mu = -1.0027 t$ for $W = 0.2267 t$ at $T=0$).
(b) The Fermi surface in the first Brillouin zone is depicted.
In this case, the reduced Brillouin zone (RBZ) boundary is determined by the CDW momentum $Q=(2\pi/3,0)$. Note that the Fermi surface in the RBZ consists of a hole pocket centered at $(\pi/3,\pi)$ depicted in red.
}
\label{fermi_surface_stripe}
\end{figure}
In this section, we show that a uni-axial CDW state with $Q=(2\pi/3,0)$ or $(0,2\pi/3)$ cannot produce such temperature or doping dependent crossover in sign of the Seebeck and Hall coefficients. This result is a consequence of the fact that reconstructed Fermi surface of uni-axial CDW with $Q=(2\pi/3,0)$ or $(0,2\pi/3)$ consists of only hole pockets and thus the Hall and Seebeck coefficients remain positive even at low temperatures.
To calculate the Fermi surface and the normal-state quasiparticle Hall and Seebeck coefficients in a uni-axial CDW with $ \mathbf{Q} =(2\pi/3,0)$ or $(0,2\pi/3)$, we use a 2D square lattice tight-binding dispersion, Eq.~(\ref{dispersion}).
\begin{figure}[t]
\includegraphics[width = \linewidth]{coeffs_stripe.eps}
\caption{
Temperature and doping dependences of the Seebeck and Hall coefficients in the uni-directional $Q = (2\pi/3,0)$ CDW state.
(a) and (b) the temperature dependence for $p=0.08$, 0.092, and 0.125.
(c) and (d) the doping dependence for $T=20$ K, 30 K, and 60 K.
}
\label{fig:coeff}
\end{figure}
Then, a uni-axial CDW with $ \mathbf{Q} =(2\pi/3,0)$ or $(0,2\pi/3)$ is given by a periodic density modulation $\delta \rho (\mathbf{r}) = W\cos (\mathbf{Q}\cdot \mathbf{r})$, where $\mathbf{r} = (x,y)$.
In the reciprocal momentum space the Fourier transform $W_ \mathbf{q} $ can be expressed as
$
W_ \mathbf{q} = W \delta_{ \mathbf{q} ,\mathbf{Q}}.
\label{h_stripe}
$
The mean-field Hamiltonian can be written as
\begin{eqnarray}
H_{MF}
&=&
\sum_{ \mathbf{k} ,\sigma} \left( \varepsilon_ \mathbf{k} - \mu \right) c_{ \mathbf{k} \sigma}^\dagger c_{ \mathbf{k} \sigma}+
W \sum_{ \mathbf{k} ,\sigma} c_{ \mathbf{k} + \mathbf{Q} ,\sigma}^\dagger c_{ \mathbf{k} \sigma} + H.c.
\nonumber \\
&=&\frac{1}{2}
\sum_{ \mathbf{k} \in RBZ}
\psi_ \mathbf{k} ^\dagger
\begin{pmatrix}
\xi_{ \mathbf{k} -\mathbf{Q}} & W & W \\
W & \xi_{ \mathbf{k} } & W \\
W & W & \xi_{ \mathbf{k} +\mathbf{Q}}
\end{pmatrix}
\psi_{ \mathbf{k} },
\label{ham_stripe}
\end{eqnarray}
where $RBZ$ is $|k_x|\le \frac{\pi}{3}, |k_y|\le \pi$ and $\psi_{ \mathbf{k} }^\dagger = [ c_{ \mathbf{k} -\mathbf{Q}} , c_{ \mathbf{k} }, c_{ \mathbf{k} +\mathbf{Q}} ]$.
Fig.~\ref{fermi_surface_stripe} illustrates (a) band structure and (b) Fermi surface in the 1st Brillouin zone at $p=0.125$ and $T=0$.
We calculate the Hall and Seebeck coefficients in the presence of a uni-axial CDW with $ \mathbf{Q} =(2\pi/3,0)$ as functions of temperature $T$ and doping $p$ in the underdoped regime (Fig.~\ref{fig:coeff}) with the same parameters as in the previous sections:$p_{CDW}=0.085$, $p_{\text{max}}=0.125$, $W=0.069$ eV, and $T_{CDW}(p) = 142.6 (\text{K}) \times (p-p_{CDW})^{0.3}$.
Due to the unidirectional feature of stripe CDW, the Seebeck $S$ and Hall coefficients $R_H$ depend upon the directions of the applied thermal gradient and transverse electric field generated by the magnetic field: When parallel to $Q=(2\pi/3,0)$, or perpendicular to $Q$.
Contrary to the bi-axial with $Q_1$ and $Q_2$ CDW state, in both cases, the sign change with onset of CDW order parameter in the coefficients $S$ and $R_H$ does not occur. Positive $S$ and $R_H$ at high temperatures or $p<p_{CDW}$ ($W=0$) remain positive even in the presence of this uni-axial CDW order parameter. It can be explained by the presence of the hole pockets in the Fermi surface in the uni-axial CDW state with $ \mathbf{Q} =(2\pi/3,0)$ or $(0,2\pi/3)$.
\section{Conclusion\label{secv}}
Motivated by strong experimental evidence of an incipient charge density wave instability in the underdoped regime of high-$T_c$ cuprates, we calculate the Fermi surface topology and the resulting Hall and Seebeck coefficients for a mean field $Q_1=(2\pi/3,0)$ and $Q_2=(0,2\pi/3)$ bi-axial CDW state. We show that, in the appropriate doping ranges in which the low temperature state (in the absence of superconductivity) is a bi-axial CDW, the Fermi surface consists of electron and hole pockets, resulting in the Hall and Seebeck coefficients becoming negative at low temperatures. Our calculated temperature and doping dependences of the transport coefficients, specifically a change of sign at low temperatures in a restricted range of hole-doping, are qualitatively consistent with experiments.
In addition to explaining the temperature dependent crossover of the magneto-transport coefficients in the underdoped regime of cuprates in terms of the recently observed CDW state, our results also show that the uni-axial CDW state is inconsistent with the phenomenology of transport coefficients in the cuprates at high magnetic fields.
\begin{acknowledgments}
{\it This work is supported by
NSF-PHY (1104527), and AFOSR (FA9550-13-1-0045).}
\end{acknowledgments}
|
1,108,101,563,296 | arxiv | \section{Introduction}
Acoustic analogue of a black hole has been a lot studied in the literature
as a concrete laboratory model for probe several aspects of curved space
quantum field theory.
In 1981, Unruh\cite{1,2} showed that if a fluid is barotropic and inviscid,
and the flow of the fluid is irrotational, the equation of motion that
fluctuation of the velocity potential of acoustic disturbance obeys, is
identical to that of a minimally coupled massless scalar field propagating
in an effective curved spacetime Loretizian geometry, which can simulate an
artificial black hole\cite{1,2}.
This paper is organized as follows. In section 2, we obtain the effective
acoustic geometry. In section 3, we will show the Klein-Gordon equation in
the sonic black hole scenario. In section 4, we describe the acoustic black
hole in the presence of a disclination and the amplification sound wave.
Section 5 is devoted to present our conclusions.
\section{Effective acoustic geometry}
In the absence of chemical reactions, the number of particles entering and
leaving a collision in a fluid will be conserved. For non relativistic
process, the total mass of the particles involved in the collision process
will also be conserved. As a result, if we consider a volume element of the
fluid, $dV(t)$ (with a given set of fluid particles), which moves with the
fluid, the amount of mass inside this volume element must remain constant.
Let be $\rho =\rho (\vec{r},t)$ the mass density (mass per unit volume) and
let $M$ denote the total mass in the volume, $V(t)$. Then%
\begin{equation}
\frac{dM}{dt}=\frac{d}{dt}\dint\limits_{V(t)}\rho
dV=\dint\limits_{V(t)}\left( \frac{d\rho }{dt}+\rho \vec{\nabla}_{r}\cdot
\vec{v}\right) dV=0, \label{1}
\end{equation}%
where $\vec{v}=\vec{v}(\vec{r},t)$ is the average velocity of the fluid at
point $\vec{r}$ and time $t$. Since the volume \ element, $dV(t)$, is
arbitrary, the integrand must be zero and we find%
\begin{equation}
\frac{d\rho }{dt}+\rho \vec{\nabla}_{r}\cdot \vec{v}=0. \label{2}
\end{equation}%
If we that the convective derivative is given by $d/dt=\partial /\partial t+$
$\vec{v}\cdot \vec{\nabla}_{r}$, then we can also write%
\begin{equation}
\frac{\partial \rho }{\partial t}+\vec{\nabla}_{r}\cdot \left( \rho \vec{v}%
\right) =0, \label{3}
\end{equation}%
where $\rho \vec{v}$ is the mass flux. The eq. (\ref{3}) is a continuity
equation and it is a direct consequence of the conservation of mass in the
fluid. The Euler equation is given by%
\begin{equation}
\rho \frac{d\vec{v}}{dt}\equiv \rho \lbrack \frac{\partial \vec{v}}{\partial
t}+(\vec{v}\cdot \vec{\nabla})\vec{v}]=\vec{f}+\vec{F}, \label{4}
\end{equation}%
where $f$ is a force per unit volume acting on the walls of the volume
element and $F$ is an external force per unit volume which couples to the
particles inside the volume element (it could be an electric or magnetic
field for example).
We consider a fluid been inviscid (zero viscosity), with the only forces
present being those due to pressure $p$, i.e., $\vec{f}=-\vec{\nabla}p$. In
this case, $\vec{F}$ equal to zero. In the sense, we consider that the fluid
is locally irrotational (free vortex), that is, $\vec{v}=-\vec{\nabla}\phi ,$
and the fluid is barotropic, i.e., the density $\rho $ is a function of
pressure $p$ only. In this case, we can define the enthalpy $h$ as%
\begin{equation}
h(p)=\dint\limits_{0}^{p}\frac{dp^{\prime }}{\rho (p^{\prime })} \label{5}
\end{equation}%
or%
\begin{equation}
\vec{\nabla}h=\frac{\vec{\nabla}p}{\rho }. \label{6}
\end{equation}%
The equation (\ref{4}) now reduce to
\begin{equation}
-\frac{\partial \phi }{\partial t}+h+\frac{1}{2}\left( \vec{\nabla}\phi
\right) ^{2}=0. \label{7}
\end{equation}
We will follow to study sound wave, the usual procedure and linearize the
continuity and Euler 's equations around some background flow, by setting $%
\rho =\rho _{0}+\epsilon \rho _{1},$ $p=p_{0}+\epsilon p_{1},$ $\phi =\phi
_{0}+\epsilon \phi _{1}$, and discarding all terms of order $\epsilon ^{2}$
or higher.
Then, the continuity equation leads to%
\begin{equation}
\frac{\partial \rho _{0}}{\partial t}+\vec{\nabla}\cdot (\rho _{0}\vec{v}%
_{0})=0. \label{8}
\end{equation}%
and%
\begin{equation}
\frac{\partial \rho _{1}}{\partial t}+\vec{\nabla}\cdot (\rho _{1}\vec{v}%
_{0}+\rho _{0}\vec{v})=0, \label{9}
\end{equation}
Expanding $h(p)$ as $h(p_{0}+\epsilon p_{1})\simeq h(p_{0})+\epsilon \frac{dh%
}{dp}|_{p=p_{0}}=h_{0}+\epsilon \frac{p_{1}}{\rho _{0}}=h_{0}+\epsilon
h_{1}, $ the eq. (\ref{4}) becomes%
\begin{equation}
-\frac{\partial \phi _{0}}{\partial t}+h_{0}+\frac{1}{2}(\vec{\nabla}\phi
_{0})^{2}=0, \label{10}
\end{equation}%
and%
\[
-\frac{\partial \phi _{1}}{\partial t}+\frac{p_{1}}{\rho _{0}}-\vec{v}%
_{0}\cdot \vec{\nabla}\phi _{1}=0,
\]%
that is%
\begin{equation}
p_{1}=\rho _{0}\left( \frac{\partial \phi _{1}}{\partial t}+\vec{v}_{0}\cdot
\vec{\nabla}\phi _{1}\right) . \label{11}
\end{equation}%
Then, since the fluid is barotropic we have%
\begin{equation}
\rho _{1}=\frac{\partial \rho }{\partial p}p_{1}. \label{12}
\end{equation}%
Substituting eq. (\ref{11}) into (\ref{12}) we get%
\begin{equation}
\rho _{1}=\frac{\partial \rho }{\partial p}\rho _{0}\left( \frac{\partial
\phi _{1}}{\partial t}+\vec{v}_{0}\cdot \vec{\nabla}\phi _{1}\right) .
\label{13}
\end{equation}%
Now, substituting eq. (\ref{13}) into eq. (\ref{9}) we obtain
\begin{widetext}
\begin{eqnarray}
\label{14}
-\frac{\partial }{\partial t}\left[ \frac{\partial \rho }{\partial p}\rho
_{0}\left( \frac{\partial \phi _{1}}{\partial t}+\vec{v}_{0}\cdot \vec{\nabla%
}\phi _{1}\right) \right] +\vec{\nabla}\cdot \left[ \rho _{0}\vec{\nabla}%
\phi _{1}-\frac{\partial \rho }{\partial p}\rho _{0}\vec{v}_{0}\left( \frac{%
\partial \phi _{1}}{\partial t}+\vec{v}_{0}\cdot \vec{\nabla}\phi
_{1}\right) \right] =0.
\end{eqnarray}
\end{widetext}The eq. (\ref{14}) describes the propagation of the linearized
scalar potential $\phi _{1},$ if $\phi _{1}$ is determined, eq. (\ref{11})
determines $\rho _{1}$. Thus, this wave equation completely determines the
propagation of acoustic disturbances, where the local speed of sound is
defined by%
\begin{equation}
c^{-2}\equiv \frac{\partial \rho }{\partial p}. \label{15}
\end{equation}
Thus, it can now be shown that the eq. (\ref{14}) can also be obtained from
the usual curved space Klein Gordon equation\cite{wisser}%
\begin{equation}
\frac{1}{\sqrt{-g}}\partial _{\mu }\left( \sqrt{-g}g^{\mu \nu }\partial
_{\nu }\right) \phi =0, \label{16}
\end{equation}%
where $g_{\mu \nu }$ is a metric tensor (with Lorentizian signature), not of
spacetime itself. but an acoustic $\ ^{\prime \prime }$analog spacetime$%
^{\prime \prime }$.
\section{Klein-Gordon equation in the sonic black hole scenario - Draining
bathtub flow model}
In this model, the velocity potential, in polar coordinates is given by\cite%
{wisser}%
\begin{equation}
\phi (r,\theta )=A\log r+B\theta , \label{17}
\end{equation}%
where $A$ and $B$ are real constants and $\phi $ present a sink in the
origin. This leads to the velocity profile%
\begin{equation}
\vec{v}=\frac{A}{r}\hat{r}+\frac{B}{r}\hat{\theta}, \label{18}
\end{equation}%
then, the metric in the exterior region, i.e., outside of core at $r=0$,
turns out
\begin{widetext}
\begin{eqnarray}
ds^{2}=-\left( c^{2}-\frac{A^{2}+B^{2}}{r^{2}}\right) dt^{2}-\frac{2A}{r}%
drdt-2Bd\theta dt+dr^{2}+r^{2}d\theta ^{2}+dz^{2}, \label{19}
\end{eqnarray}
\end{widetext}where $c$ is the velocity of sound.
Defining\cite{sou}%
\[
dt\rightarrow dt+\frac{\left\vert A\right\vert r}{\left(
r^{2}c^{2}-A^{2}\right) }dr\text{ ; \ \ }d\theta \rightarrow d\theta +\frac{%
B\left\vert A\right\vert r}{r\left( r^{2}c^{2}-A^{2}\right) }dr\text{\ \ \ \
\ \ }
\]%
we obtain, after a rescaling of the time coordinate by $c$
\begin{widetext}
\begin{eqnarray}
ds^{2}=-\left( 1-\frac{A^{2}+B^{2}}{c^{2}r^{2}}\right) dt^{2}-\left( 1-\frac{%
A^{2}}{c^{2}r^{2}}\right) ^{-1}dr^{2}-2\frac{B}{c}d\theta dt+r^{2}d\theta
^{2}+dz^{2}, \label{20}
\end{eqnarray}
\end{widetext}where , the radius of the ergosphere is given by the vanishing
of $g_{00}$, i.e., $r_{e}=\left( A^{2}+B^{2}\right) ^{1/2}/c$, and it has a
singularity at $r_{h}=\left\vert A\right\vert /c$, which signifies the event
horizon. We observe on eq. (\ref{18}) that for $A>0$ we are dealing with a
past event horizon, i.e., acoustic white hole and for $A<0$ we dealing with
a future acoustic horizon, i.e., acoustic black hole.
\section{Acoustic black hole in the presence of a disclination and
amplification sound wave}
In a recent paper\cite{F} we have discussed the phenomenon of sound
amplification in the acoustic black hole analogue.
Thus, in this paper we propose the analyze the influence of an acoustic
black hole analogue in the presence of a disclination in the sound wave
amplification. In the geometric approach, the medium with a disclination has
the line element given by
\begin{widetext}
\begin{eqnarray}
ds^{2}=-\left( 1-\frac{A^{2}+B^{2}}{c^{2}r^{2}}\right) dt^{2}-\left( 1-\frac{%
A^{2}}{c^{2}r^{2}}\right) ^{-1}dr^{2}-2\frac{B}{c}\alpha d\theta dt+r^{2}\alpha^2 d\theta
^{2}+dz^{2}, \label{21}
\end{eqnarray}
\end{widetext}in cylindrical coordinates. This metric is equivalent to the
boundary condition with periodicity of $2\pi \alpha $ instead of $2\pi $
around the $z-$axis. In the Volterra process\cite{Kle} of disclination
creation, this corresponds to remove $(0<\alpha \leq 1)$ or insert $(2\pi
>\alpha \geq 1)$ a wedge of material of dihedral angle $\lambda =2\pi
(\alpha -1)$\cite{G}.
But, for the velocity potential given by eq. (\ref{18}), the analogue black
hole metric is basically a $(2+1)$ dimensional flow with a sink at the
origin. The metric given by (\ref{21}) reduce to
\begin{widetext}
\begin{eqnarray}
ds^{2}=-\left( 1-\frac{A^{2}+B^{2}}{c^{2}r^{2}}\right) dt^{2}-\left( 1-\frac{%
A^{2}}{c^{2}r^{2}}\right) ^{-1}dr^{2}-2\frac{B}{c}\alpha d\theta dt+r^{2}\alpha^2 d\theta
^{2}. \label{22}
\end{eqnarray}
\end{widetext}
Now, we write the Klein-Gordon equation (\ref{16}) in the background metric (%
\ref{22}) and we can separate variables by the substitution%
\[
\phi (t,r,\theta )=\exp i(\omega t-m\theta )R(r),
\]%
where $m$ is an integer, we assume that $\omega >0$, then, the radial
function satisfies the equation given by
\begin{widetext}
\begin{eqnarray}
\frac{1}{r}\left( 1-\frac{A^{2}}{c^{2}r^{2}}\right) \frac{d}{dr}\left[
r\left( 1-\frac{A^{2}}{c^{2}r^{2}}\right) \frac{d}{dr}\right] R(r)+\left[
\omega ^{2}-\frac{2Bm\omega }{\alpha cr^{2}}-\frac{m^{2}}{\alpha ^{2}r^{2}}%
\left( 1-\frac{A^{2}+B^{2}}{c^{2}r^{2}}\right) \right] R(r)=0.
\label{24}
\end{eqnarray}
\end{widetext}
Introducing the tortoise coordinate $r^{\ast }$ such that%
\begin{equation}
\frac{d}{dr^{\ast }}=\left( 1-\frac{A^{2}}{r^{2}c^{2}}\right) \frac{d}{dr}
\label{25}
\end{equation}%
which implies that%
\begin{equation}
r^{\ast }=r+\frac{|A|}{2c}\log \left\vert \frac{r-\frac{|A|}{c}}{r+\frac{|A|%
}{c}}\right\vert . \label{26}
\end{equation}%
Observe that the horizon $r=\frac{|A|}{c}$ maps to $r^{\ast }\rightarrow
-\infty $ and while $r\rightarrow \infty $ corresponds to $r^{\ast
}\rightarrow +\infty $. Now, introducing a new radial function $g(r^{\ast
})\equiv r^{1/2}R(r),$ we obtain the equation%
\begin{equation}
\frac{d^{2}g(r^{\ast })}{dr^{\ast 2}}+\left[ q(r)-\frac{1}{2r^{2}}\left(
\frac{dr}{dr^{\ast }}\right) ^{2}-\left( \frac{A^{2}}{r^{4}c^{2}}-\frac{3}{%
4r^{2}}\right) \frac{dr}{dr^{\ast }}\right] g(r^{\ast })=0, \label{27}
\end{equation}%
where%
\begin{equation}
q(r)=\frac{A^{2}m^{2}+Bm^{2}-c^{2}m^{2}r^{2}-2B\alpha mr^{2}\omega +\alpha
^{2}r^{4}\omega ^{2}}{c^{2}\alpha ^{2}r^{4}}. \label{28}
\end{equation}%
Now, analyzing eq. (\ref{27}) when $r\rightarrow \infty $, we obtain%
\begin{equation}
\frac{d^{2}g(r^{\ast })}{dr^{\ast 2}}+\omega ^{2}g(r^{\ast })=0, \label{29}
\end{equation}%
whose solution is given by%
\begin{equation}
g(r^{\ast })=\exp \left( i\omega r^{\ast }\right) +\mathcal{R}\exp (-i\omega
r^{\ast }). \label{30}
\end{equation}%
The first term of eq. (\ref{30}) corresponds to an ingoing wave and the
second term corresponds to de reflected wave, where $\mathcal{R}$ is the
reflection coefficient in the sense of potential scattering. Now using this
solution of the differential equation together with it complex conjugate, we
calculate the Wronskian of the solutions (\ref{30}) given by%
\begin{equation}
\mathcal{W}(+\infty )=-2i\omega \left( 1-|\mathcal{R}|^{2}\right) .
\label{31}
\end{equation}
Thus, we considering the solution near the horizon, is that, $r^{\ast
}\rightarrow -\infty ,$ the eq. (\ref{27}) becomes%
\begin{equation}
\frac{d^{2}g(r^{\ast })}{dr^{\ast 2}}+\left( \omega -m\Omega _{H,\alpha
}\right) ^{2}g(r^{\ast })=0 \label{32}
\end{equation}%
where $\Omega _{H,\alpha }\equiv \frac{Bc}{\alpha A^{2}}$ is the angular
velocity of the acoustic black hole in the presence of a disclination. Near
the horizon, we suppose that just the solution identified by ingoing wave is
physical, is that%
\begin{equation}
g(r^{\ast })=\mathcal{T}\exp [i\left( \omega -m\Omega _{H,\alpha }\right)
r^{\ast }], \label{33}
\end{equation}%
where $\mathcal{T}$ is the transmission coefficient. Once again, we
calculate the Wronskian of the solutions (\ref{33})
\begin{equation}
\mathcal{W}(-\infty )=-2i\left( \omega -m\Omega _{H,\alpha }\right)
\left\vert \mathcal{T}\right\vert ^{2}. \label{34}
\end{equation}%
Thus, remind that two linearly independent solutions of the same
differential equation must lead to a constant Wronskian, so of eqs. (\ref{31}%
) and (\ref{34}) we obtain%
\begin{equation}
\left\vert \mathcal{R}\right\vert ^{2}=1-\left( 1-\frac{m}{\omega }\Omega
_{H,\alpha }\right) \left\vert \mathcal{T}\right\vert ^{2}. \label{35}
\end{equation}%
We can observe in eq. (\ref{35}) that, for frequencies in the range $%
0<\omega <$ $m\Omega _{H,\alpha }$, the reflection coefficient has a
magnitude larger than unity whose imply the amplification relation of the
ingoing sound wave near horizon regions. This imply that the ingoing wave
removes mass (energy) of the acoustic black hole\cite{F}. As $\Omega
_{H,\alpha }$ depends on the disclination, then, the same affects the
quantity of removed energy of the hole. When $0<\alpha \leq 1$ whose
corresponds to remove a wedge of material it is possible to accentuate the
quantity of retired energy of the acoustic black hole, in other words,
larger amplification of the ingoing sound wave and when $2\pi >\alpha \geq 1$
whose corresponds to insert a wedge of material represent to the possibility
to attenuate the quantity of removed energy of the acoustic black hole.
\section{Conclusions}
In this paper we shown that the presence of the disclination modify the
quantity of removed energy of the acoustic black hole and that, it is
possible to accentuate or to attenuate the amplification of the removed
energy of the acoustic black hole and still exists the possibility to cancel
the superradiance effect to $\alpha $ equal to $m\Omega /\omega $ where $%
\Omega $ $\equiv \frac{Bc}{A^{2}}$ is the angular velocity of the acoustic
black hole in the absence of the disclination, in this case, the reflection
coefficient is equal to unity. Those aspects perhaps can be proven in future
experimental realizations.
\vskip2.0 cm
{\large \textbf{Acknowledgments:}} I would like thank to MCT/CNPq and
FAPESQ-PB for the partial financial support.
|
1,108,101,563,297 | arxiv | \section{Introduction}
The sharing of resources in animals can be defined as an active or passive transfer of food from one individual to another \cite{stevens2004conceptual, jaeggi2011evolution, jaeggi2013natural}.
Some examples are food division in primates \cite{brown1978food, feistner1989food, boesch1989hunting, de1989food, mitani2006reciprocal}, blood sharing in vampire bats \cite{wilkinson1984reciprocal, wilkinson1990food, wilkinson2016non}, and cooperative breeding in birds and fishes \cite{koenig2004ecology, wong2011evolution}.
From an external human observer, this behavior really resembles an act of \quotes{generosity}.
Indeed, once an individual has acquired some resources, it chooses to equalize them among the members of the community (or with a partner) instead of being \quotes{selfish} and having a larger income.
Resource division can be viewed as an instance of animal cooperation and biological altruism, subjects that have a long tradition in evolutionary biology \cite{dugatkin1997cooperation, kappeler2006cooperation, okasa2013biological, barta2016individual}.
Darwin himself already noticed that altruism seems to be in contrast with the natural selection theory, since the altruistic act consists in a decrease of the donor fitness \cite{darwin1859origin}.
Many episodes of food sharing and altruistic behaviors occur between relatives, such as parent-offspring sharing \cite{feistner1989food}, and can be explained by kin-selection \cite{hamilton1964genetical, taylor1996make, foster2006kin}.
However, several other instances do not involve kin-related individuals, and require alternative explanations.
Some classical proposals are reciprocity \cite{trivers1971evolution}, repression of competition \cite{frank2003repression}, or ideas based on group selection \cite{wilson1994reintroducing}.
A concept tightly related to resource sharing is inequity aversion, defined as the tendency to negatively react in response to unequal subdivision of resources between individuals \cite{brosnan2014evolution, oberliessen2019social}.
One widespread behavior is the protest of the individual getting less than the partner, called first-order inequity aversion.
Animals that show this behavior are, for example, monkeys \cite{brosnan2003monkeys, takimoto2010capuchin}, dogs \cite{range2009absence} and crows \cite{wascher2013behavioral}.
In fewer cases, it is the individual receiving more than its fair share that protests, possibly acting in a way to equalize the resources.
This is dubbed second-order inequity aversion.
It seems that only chimpanzees show experimental evidence of this apparent \quotes{generous} behavior, by protesting if the partner receives less \cite{brosnan2010mechanisms}, or by choosing a fair resource division \cite{proctor2013chimpanzees}.
Importantly, inequity-averse behaviours are all inefficient for the short-term income: the individual that protests typically lowers its resource gain.
A widely used theoretical framework for describing inequity-aversion is the ultimatum game \cite{guth1982experimental, camerer2003behavioral}.
It is played by a \textit{proposer} who has to divide a certain amount of resource with a \textit{receiver}.
This second player has the possibility to accept the proposal or to reject it, in the latter case neither party receives anything.
Note that a fair choice of the proposer (i.e. dividing the resource in half) corresponds to the $2^{nd}$ order inequity-averse behavior, while the rejection of an unfair proposal to the $1^{st}$ order one.
A lot of theoretical work has been done to understand the emergence of fairness in the ultimatum game \cite{debove2016models}, since a one-shot game would always lead to the receiver's acceptance of the division with the minimum possible amount of resource for the receiver \footnote{This is the only evolutionary stable strategy.}.
Some mechanisms that can lead to the emergence of fairness are based on the knowledge of the players' reputation \cite{nowak2000fairness, andre2011social}, noise in the decision rule (with a larger noise for the receiver) \cite{gale1995learning}, spite \cite{huck1999indirect, forber2014evolution}, and spatial population structure \cite{page2000spatial, alexander2007structural, iranzo2011spatial}.
Most of these models as well as the ones for the emergence of cooperation are based on the evolutionary point of view. They address the question on how natural selection has led to a successful \quotes{cooperative/fair phenotype}, i.e. having a higher fitness than the one of defectors.
The natural framework to describe this kind of processes is evolutionary game theory, where one looks for the conditions in which the cooperation is stable against the emergence of defectors \cite{axelrod1981evolution, nowak2006five, debove2016models}.
The present work is based on a different approach: we assume that animals have the ability to learn and change their strategies in the most efficient way for their survival in a given environment.
This allows us to move the focus from the time-scale of evolution of a large population to the scale of an individual life span, who interacts with relatively few other individuals within its community.
In particular, we adopt Markov decision processes, \cite{sutton1998introduction}, which, in the case of more than one agent, are called Markov games \cite{littman1994markov, hu1998multiagent, claus1998dynamics}.
In short, in this framework agents know the environmental state in which they are, and, depending on this state, can choose actions according to a policy.
As a consequence of these actions, the agents receive rewards (here represented by resources), and the game moves to a new environmental state which depends on the player actions and the previous state.
Each agent wants to find the policy that maximizes its resource income within a given time horizon.
There are few works that study non-kin cooperation using reinforcement learning instead of evolution \cite{sandholm1996multiagent, claus1998dynamics, ezaki2016reinforcement}.
They focus on the emergence of optimal strategies mainly in the context of the iterated prisoner dilemma.
Differently, the Markov model introduced in the following takes inspiration from resource-gathering and sharing tasks between two animals, and, in particular, from the blood sharing in vampire bats, as described in the next section.
Generous resource sharing, or a $2^{nd}$ order inequity-aversion behavior, emerges if the optimal policy for the player is to divide a certain amount of acquired resources in equal parts instead of keeping a larger fraction.
The model allows for analytical solution and, albeit its simplicity, exhibits a rich phenomenology of the optimal strategy, i.e. being generous or selfish, depending on few parameters.
These key parameters have a straightforward biological interpretation: the abundance of resource in the environment, the player asymmetry in the ability of performing the tasks (typically overlooked in the traditional literature of cooperation), the number of games that a player expects to play, i.e. the time horizon, and the fraction of resources kept in the selfish choice.
Specifically, three different phases, corresponding to generous, exploitative and selfish behaviour, can be discerned.
It is important to stress that individuals are \quotes{greedy}, meaning that they are interested in obtaining the largest possible personal gain, without considering the resource income of their partners.
This clearly implies that, if one were looking at the immediate reward only, it would be always optimal to be selfish.
However, since individuals are interested in the return within a certain time horizon, generosity can provide advantages in the future, becoming, possibly, the most efficient strategy even for \quotes{greedy} players.
\section{A relevant example: food sharing in vampire bats}
\begin{figure}
\includegraphics[width=0.6\textwidth]{bats.png}
\centering
\caption{Schematic illustration of the blood gathering and sharing shown by vampire bat communities.}
\label{fig:bats}
\end{figure}
The model that we will introduce is mainly inspired by the well-studied \quotes{generous} behaviour shown by blood-feeding vampire bats \cite{wilkinson1984reciprocal, wilkinson1990food, wilkinson2016non}.
After gathering blood during a given night, these animals frequently share it to other individuals when they come back to the roost.
This behaviour is shown most of the times by mothers feeding their own offspring, and can be explained by kin selection.
However a relevant fraction of blood sharing occurs among unrelated adults.
The key factors that seems to be at the basis of this behaviour are, first, the fact that after two consecutive nights without blood a bat can reach starvation, and, second, that there is around $18\%$ of probability that an individual fails to obtain blood during the night \cite{wilkinson1984reciprocal}.
In the light of this, food sharing seems to be necessary to prevent starvation, and, on the long run, the collapse of the community.
Therefore, even though the individual fitness decreases in the short-term period, being generous may provide a significant long-term advantage, since it guarantees the survival of the community.
Inspired by these findings, in this paper we will build a minimal model of resource sharing that retains the fundamental ingredients leading to this behaviour.
The game is between two individuals that can interact repeatedly in time. This requires that the animals are able to discriminate between different individuals within the community, since, in principle, different strategies can be learned depending on the partner.
This seems to be true in bats: specific pairs can establish long-term relationships by helping each other \cite{wilkinson1990food}.
It is important to stress that the only information that the bat needs about the parter is its health state. They do not need to remember, for example, its actions at the previous stage, as a Tit-for-Tat strategy would require \cite{axelrod1981evolution}, which is the classical mechanism through which reciprocal cooperation is obtained.
One stage of the game corresponds to one night, and describes the situation in which one individual gathers some blood, and the second fails in performing this task or stays in the roost, as shown in the upper part of Figure \ref{fig:bats}.
At subsequent stages of the game the two roles can be inverted, and this depends on a given probability that can account for a possible \quotes{asymmetry} between the players: one can be the gatherer more frequently than the other one.
Actually, this is also observed in field studies regarding chimpanzees, where, for example, the degree of dominance within the community, the age, and the sex determine the frequency at which animals perform resource-gathering tasks \cite{boesch1989hunting}.
After having obtained some food, the gatherer comes back to the roost and chooses whether to share it with individuals that have failed the task, or to keep most of it for itself (bottom right part of Figure \ref{fig:bats}).
As a result of this division, the two individuals feed themselves, and their health changes accordingly to the amount of food received (bottom left part of Figure \ref{fig:bats}).
As in real bats, if the individual that failed the gathering task does not receive enough food, it may starve and, as a consequence, it will not be able to gather resources anymore.
\section{A minimal decision-making model for resource gathering and sharing}
\label{sec:model}
\begin{figure}
\includegraphics[width=0.6\textwidth]{model2.pdf}
\centering
\caption{Illustration of one temporal step of the resource-gathering-sharing model. Box (a): one of the two players is chosen to be the \textit{proposer}, labelled with $p$, with a certain probability depending on the first-player specialization $s \in \left[ -1, 1 \right]$.
The other individual becomes the \textit{bystander}, $b$.
Box (b): the proposer performs the resource gathering task, acquiring a given amount of resource $R$ depending on its internal variable $h_p$, which can be thought as its health, and the resource abundance $a$.
Box (c): playing a dictator game, the proposer chooses how to divide the resource with the bystander.
It can either split $R$ in equal parts, $r_p = r_b = R/2$, or be selfish and keep a larger fraction of resource, $r_p=R/2(1+\delta)$, where $\delta \in (0,1]$.
The proposer strategy is determined by its policy $\pi_p$, i.e. the probability of playing the generous action and split the resource in equal parts.
Box (d): finally, each player updates its health, $h_i$, as a function of the received resource fraction $r_i$.}
\label{fig:model}
\end{figure}
Here we describe the model in abstract mathematical terms.
Even though it is inspired by the behaviour of vampire bats, it can be used as a backbone in whatever context where generous resource sharing emerges.
It is performed by two animals/players, whose only purpose is to acquire as many resources as possible for themselves.
At each iteration of the game (which we are going to call also \quotes{episode}), one of the two players is selected to be the \textit{proposer}, $p$, box (a) of Figure \ref{fig:model}: it will be the active individual who carries out a certain task to acquire resources.
The other player, called the \textit{bystander}, $b$, remains passive until the end of the iteration.
The model features a degree of specialization of the players, such that at every iteration, one individual can be consistently more likely to play as proposer than the other individual.
Specifically, the first player can be selected with probability $(1+s)/2$.
The parameter $s \in [-1,1]$ can be interpreted as the \textit{specialization} of the first player with respect to the second one for the given task: $s=1$ means that the first player will always perform the task, while, $s=0$ defines a symmetric case where both can be chosen with the same probability.
After being selected, the proposer performs the task, Figure \ref{fig:model}b, which consists in acquiring a certain amount of resource $R = a \cdot h_p$, depending on the abundance of the resource in the environment $a$, and an internal variable of the proposer $h_p$.
The variable $h$ can be identified with the player health.
In the interest of simplicity, we consider $h$ to be a binary variable: $h \in \lbrace 0,1 \rbrace$.
The value $h=0$ defines an exhausted agent, who will not gather any resource, while $h=1$ a healthy one.
Typically, individuals that get food are also the ones that have more decision power in the resource sharing \cite{wilkinson1984reciprocal, boesch1989hunting}.
Therefore we let the same player that has performed the task also to choose how to share the resource with the bystander.
\\
The resource-division stage, Figure \ref{fig:model}c, consists in a mini-dictator game \cite{camerer2003behavioral}, a simplified version of the ultimatum game \cite{guth1982experimental}.
It is worth pointing out that using the more complex ultimatum game would not change the optimal strategies, as shown in \ref{sec:UG}.
Specifically, the game consists in two actions for the proposer: splitting the resource $R$ in equal parts, i.e. the \quotes{generous} action, such that he obtains $r_p = R/2$, or being \quotes{selfish} by keeping a larger fraction $r_p = R/2(1+\delta)$, with $\delta \in (0,1]$.
The bystander is passive and will receive resources according to the proposer choice: $r_b = R/2$ for the equal action, and $r_b = R/2(1-\delta)$ for the unequal.
The exogenous parameter $\delta \in (0,1]$, called \textit{inequity strength}, indicates how large is the selfish-choice profit for the proposer at the expense of the bystander.
It can be interpreted as a constraint on the maximum amount of resource that a player can keep or control.
For example, the case $\delta=1$ defines a game in which the proposer is able to consume all of the resource or, alternatively, it is able to protect it from the attempted theft of the bystander.
The decision of a player, i.e. being \quotes{generous} or \quotes{selfish}, is encoded by the policy $\pi$, which defines the probability of playing the equal action.
This is the parameter that each individual can control in order to maximize the amount of obtained resource.
The final step consists in updating the health according to the received fraction of resource $r$, Figure \ref{fig:model}d.
One simple choice to express the health dependency on $r$ is the Heaviside step function $f(r) = H(r-\theta)$, which is $1$ if $r \ge \theta$, or $0$ if $r < \theta$.
In order to be healthy, the player has to receive an amount of resource larger than the threshold $\theta$, which represents the minimal food ration that a player needs to be in good shape.
\begin{table
\centering
\caption{Key parameters of the model.}
\begin{tabular}{rlrl}
$s$ & First-player specialization & $\delta$ & Inequity strength \\
$a$ & Resource abundance & $\theta$ & Food threshold \\
$\gamma$ & Discount factor & $(1-\gamma)^{-1}$ & Time horizon \\
\end{tabular}
\label{tab:params}
\end{table}
The objective function to be maximized by player $i$ is the exponentially discounted return:
\begin{equation}
G_i (\pi_i, \pi_{-i}) = \mathbb{E}_{\pi_i, \pi_{-i}} \left[ \sum_{t=1}^\infty r_i^{(t)} \gamma^{t-1} \right] ,
\label{eq:return}
\end{equation}
which is averaged over the stochastic game evolution, the player policy $\pi_i$, and the other player policy $\pi_{-i}$.
The random variable $r_i^{(t)}$ indicates the fraction of resource obtained at the iteration $t$ of the model, and $\gamma \in [0,1)$, the so-called \textit{discount factor}, determines the value of future rewards, such that increasing $\gamma$, the player gives more importance to what it is going to earn at future time steps.
Since $1 - \gamma$ can be interpreted as the probability that the game ends at each iteration, it sets the average length of a game to $(1-\gamma)^{-1}$, the so-called \textit{time horizon}, defining the number of episodes over which players want to maximize the return.
Each individual can control its own policy $\pi_i$ affecting the outcome of resource fraction $r_i^{(t)}$, both for himself and the other individual.
Here, solving the problem for the player $i$ means finding the policy $\pi_i$ which maximizes its own profit $G_i$ by knowing that the other player $j$ is trying to optimize at the same time $G_{j}$.
The obtained optimal policies belong to a Nash equilibrium: any unilateral change of strategy by a single player would reduce the payoff.
Here we also want to stress that the Markov property of the model implies that players take decisions only on the basis of the present state, and independently on previous decisions or game history.
This problem can be formalized using Bellman optimality equation as shown in the Methods.
It turns out that the model admits an analytical solution, and we were able to find the optimal strategies $\pi^*_i$ depending on the parameters of the model, as shown below.
Before presenting the solution, we would like to point out the general guiding lines that have driven the model construction.
The first important observation is that the optimal strategy of a simple dictator game is always to play selfishly (see \ref{sec:generalization}).
Therefore, we had to introduce an additional reasonable mechanism that allows generosity to emerge.
By taking inspiration from animal behavior, we paired the resource sharing with the gathering of that resource, using the health as the variable that couples the two tasks.
Specifically, the gathering efficiency depends on the health, which, in turn, is affected by how the sharing of the resources is performed.
This coupling is what we assumed to be the basic mechanism to obtain equal resource sharing.
Starting from this idea, we aimed at building a Markov game with a \quotes{minimal mathematical complexity}.
Therefore, we chose the most simple possible sharing task, i.e. a dictator game, a gathering task where the obtained resource is proportional to the health, and a health which is a binary variable updated with a step function.
In general, the purpose of keeping the complexity low is to allow mathematical techniques to provide more insights about the model solution.
In our case, the chosen structure is simple enough to explicitly derive an analytical solution.
The fact that there are four free parameters seems to go against the purpose of keeping the complexity low, however, since the model remains solvable, one can have a full control on how the solution depends on them.
Moreover, since all of them have a reasonable biological interpretation, the dependence of the solution on those parameters allows one to identify general rules that govern the generous-selfish dilemma, as discussed later.
\section{Results}
\subsection{Inequity aversion is a matter of time horizon and player specialization}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{ph_diag.pdf}
\caption{Optimal strategies for the maximization of the exponentially discounted return (\ref{eq:return}) as a function of the three model parameters: the player specialization $s$, the discount factor $\gamma$, and the inequity strength of the selfish choice $\delta$.
Looking at the panel (a), starting from the left plot, the red volume is defined by Equation (\ref{eq:selfish_ph}), and represents the region of parameters such that the selfish strategy is optimal: $\pi_1^* = 0, \pi_2^* = 0$.
The blue volume shown in the central diagram, which rests on the selfish block, defines the \quotes{exploitation} phase: $\pi_1^* = 0, \pi_2^* = 1$, Equation (\ref{eq:expl_ph}).
Above these two regions, the optimal strategy is the generous one, $\pi_1^* = 1, \pi_2^* = 1$, represented by the green volume in the third plot, Equation (\ref{eq:generous_ph}).}
\label{fig:ph_diag}
\end{figure}
Even though each player wants to greedily maximize its own profit, the generous strategy, i.e. $\pi_i^* = 1$, turns out to be the most efficient one in a broad region of the parameter space.
To intuitively understand why, one must note that an equal resource division can result in a higher long-term reward for the proposer.
Indeed, the generous choice provides more food to the other player, who has a higher chance to be \quotes{healthy} at the end of the episode (this can or cannot happen depending on the model parameters).
As a consequence, if it will play as proposer at future episodes, it could gather a larger amount of resource which will be shared with the former proposer.
This benefit typically disappears for the selfish choice: the other player gets few resources becoming \quotes{exhausted}, which, in turn, implies that the amount of food that it could get at future gathering tasks will be zero.
Therefore the optimal strategy must be a balance between this long-term benefit that a generous strategy can provide, and the immediate higher profit given by a selfish division.
Before describing the optimal solution, it is important to note a useful symmetry of the model: the game is invariant by changing the sign of the player-specialization $s$ and exchanging the first with the second player.
This allows us to consider only half of the player-specialization domain $s \in [0,1]$: all the results at negative $s$ can be immediately recovered by taking the result for the positive $s$ and inverting the players.
Figure \ref{fig:ph_diag} shows which strategy maximizes the objective function (\ref{eq:return}) varying the three free parameters $s$, $\delta$, $\gamma$, and considering only half of the specialization interval for the reason described above.
The role of the resource abundance, $a$, and the minimal food threshold, $\theta$, are discussed in the next section, and here fixed at $a = 1$, and $\theta = 1/2$.
Three distinct phases emerge as optimal strategies: \textit{selfishness}, where both players choose to keep a larger fraction of resource, \textit{exploitation}, where one player is selfish and the other generous, and the case of two individuals that split the resource in equal parts, \textit{generosity}.
As derived in \ref{sec:solution}, the range of parameters in which being selfish is optimal, $\pi_1^* = 0, \pi_2^* = 0$, is given by the following inequality:
\begin{equation}
\text{Selfishness: } \;\; \gamma < \frac{2 \delta}{1 + |s| + 2 \max{(0, \delta - |s|)}} ,
\label{eq:selfish_ph}
\end{equation}
which defines the red volume in the Figure \ref{fig:ph_diag}a.
As the intuition suggests, the selfish behaviour emerges when players give much more importance to the short-term profit, i.e. having a sufficiently small discount factor $\gamma$.
Consistently, in the extreme case of $\gamma = 0$, where each player considers only the immediate reward, the selfish behaviour is always optimal.
A different scenario appears in the following region of parameters, represented in the second plot of Figure \ref{fig:ph_diag}b in blue:
\begin{equation}
\text{Exploitation: } \;\; \frac{2 \delta}{1 + s} < \gamma < \frac{2 \delta}{1 - s + 2 \delta} .
\label{eq:expl_ph}
\end{equation}
Here what is optimal for the first player is being selfish, while for the second one is being generous, $\pi_1^* = 0, \pi_2^* = 1$.
The two inequalities above also imply that $s > \delta$ (by verifying that the left-hand term is less than the right-hand one).
Therefore, if there is a strong asymmetry between the individuals, i.e. the first is much more specialized, playing as proposer with probability $p > (1+s)/2$, a dominance relationship of one player over the other can emerge.
Taking advantage of the model symmetry, and substituting $s \rightarrow -s$ in Equation (\ref{eq:expl_ph}), one can immediately recover the case in which the two roles are inverted: $\pi_1^* = 1, \pi_2^* = 0$, that exists for $s < -\delta$, i.e. a second player more specialized than the first one.
Finally, for a sufficiently large discount factor, the generous behaviour can be the more rewarding strategy, $\pi_1^* = 1, \pi_2^* = 1$.
Specifically, for:
\begin{equation}
\text{Generosity: } \;\; \gamma > \frac{2 \delta}{1 - |s| + 2 \delta} .
\label{eq:generous_ph}
\end{equation}
This region is shown in green in the third plot of Figure \ref{fig:ph_diag}c, which in the limit $\gamma \rightarrow 1$ becomes the optimal strategy for any choice of the other parameters (except $|s|=1$).
\begin{figure}[t]
\centering
\includegraphics[width=0.6\textwidth]{ph_diag_section.pdf}
\caption{Section of the three-dimensional phase diagram at two values of the inequity strength: $\delta = 1/2$ and $\delta = 1$.
The right axis refers to the time horizon $(1-\gamma)^{-1}$.}
\label{fig:ph_diag_section}
\end{figure}
The panel (a) of Figure \ref{fig:ph_diag_section} is useful to summarize the results discussed so far.
It represents a section of the three-dimensional phase diagram at fixed inequity strength $\delta=1/2$, therefore describing all the games in which the selfish resource division is $3/4 R$ for the proposer and $1/4 R$ for the bystander.
It is clear that the emergence of inequity aversion, i.e. the generous strategy, is favoured by a high discount factor, that is when players give more importance to the future rewards that a healthy mate can provide.
However, if the player specialization is sufficiently large, even for high $\gamma$, the game can enter the blue area.
In such a case, the first agent plays as proposer much more frequently than the second one, taking control of the game: it can be selfish because the potential future profit provided by the other player becomes low.
At the same time, the second player is going to be the bystander with very high probability, and therefore it has to play generously (in the unlikely case of being selected as proposer) in order to make the first one in a healthy state and receive at least $1/4$ at the next resource divisions.
Finally, when the discount factor goes below the line defined by Equation (\ref{eq:selfish_ph}), both the players start to be selfish, having an interest in maximizing the short-term profit.
Note that here the selfish strategy is always optimal if $\gamma < 1/2$, or, equivalently, the number of episodes that the players expect to play, i.e. the time horizon $(1-\gamma)^{-1}$, is less that $2$.
The limiting case of $\delta = 1$ is shown in panel (b) of Figure \ref{fig:ph_diag_section}.
Here the choice is between an equal division and keeping all the resource.
Since the gain of the selfish division increases with respect the previous case, the region associated with generosity reduces, but does not disappear.
Differently, exploitation does not longer represent an optimal behaviour, because the bystander of a selfish division does not receive anything, and therefore it is not interested in keeping the other player healthy.
\subsection{The role of resource abundance}
Environmental factors affect the amount of resources in the habitat, resulting in different outcomes of a gathering task, which are independent of the animal \quotes{health} or its ability to fulfil the task.
Moreover, changes of the available food amount can have strong effects on the sharing behavior, as shown, for instance, by chimpanzees in the Ta\"i National Park, which are more generous when the prey is large, typically an adult of a Colobus monkeys, with respect when they capture infants \cite{boesch1989hunting}.
In our minimal model, $a$ is the free parameter that can describe the quantity of resource provided by the environment.
The role that it plays in determining the optimal strategy is coupled with $\theta$, i.e. the minimal quantity of food that a player needs to be healthy (both the two parameters are kept fixed in the previous section, $a=1, \theta=1/2$).
Indeed, they determine how the health is updated: for example, high abundance and low food threshold should lead more easily to healthy players, and, in turn, this can change the optimal policies.
As shown in \ref{sec:ab}, the optimal solution is controlled by the ratio between the two parameters through the following inequalities:
\begin{equation}
\text{Necessary conditions for generosity: } \; \; 1 - \delta < \frac{2 \theta}{a} \le 1 .
\label{eq:abundance_ineq}
\end{equation}
If they are satisfied, the optimal behaviour is exactly described by the relations shown in the previous section: (\ref{eq:selfish_ph}), (\ref{eq:expl_ph}), (\ref{eq:generous_ph}).
Otherwise, if one of the two inequalities is violated, the selfish behaviour ($\pi^*_i=0$) is the most efficient one for each parameter choice.
Intuitively, this happens because, if (\ref{eq:abundance_ineq}) is true, the generous resource division leads to a healthy partner (the health-update function, Figure \ref{fig:model}d, reads $H[a/2 - \theta] = 1$), while the selfish one drives him to exhaustion, $H[a/2(1-\delta) - \theta] = 0$.
This provides the long-term benefit for the equal division (and not for the unequal one) described at the beginning of the previous section, which is the key to have generosity.
On the contrary, if the environment provides too few resources such that (\ref{eq:abundance_ineq}) is broken, $a < 2 \theta$, then there is not enough food to make the other player healthy: the long term benefit disappears, implying that it is always convenient to keep as much food as possible through the unequal division.
Also in the case of a very rich environment, i.e. $a \ge 2 \theta / (1-\delta)$, the optimal policy is always being selfish.
In this case, there is enough food to make the bystander always healthy, even as a recipient of the selfish division.
As a consequence, the long term benefit of having a healthy partner is now given not only by the equal, but also by the unequal division.
Since this latter sharing additionally provides a larger amount of immediate resource, being selfish is always the most advantageous strategy.
\section{Discussion}
The presented results show that generosity can emerge as the optimal strategy of a model for resource gathering and sharing tasks in animals.
Importantly, it emerges even though the agents are interested in maximizing only their personal income, i.e. they are \quotes{greedy}.
However, an equal resource division can be more rewarding than the selfish one in the long run, specifically, when the inequality (\ref{eq:generous_ph}) is satisfied.
The underlying mechanism is that a generous player provides more food to the partner, increasing its health and, thus, making it more efficient at gathering resources in the future, that can be potentially shared with the player.
It is worth noting that the crucial ingredient that makes the equal resource sharing optimal is the coupling of the sharing task with the resource gathering through the animal health, as mentioned in Section \ref{sec:model}.
To better understand this observation, one can gradually decouple the two tasks by introducing a random component in the health-update function.
For example, with probability $\eta$, the player health is chosen with equal probability to be $0$ or $1$, while, with probability $1-\eta$, the step function used above is employed.
As shown in \ref{sec:generalization}, as $\eta$ increases, the effect that the sharing task has on the resource gathering becomes weaker, and the selfish strategy tends to be the optimal strategy for a larger region of parameters, and becomes the only solution for $\eta=1$.
The analytical solution of the Bellman optimality equation allows us to identify general rules that govern the fair-selfish dilemma.
First, increasing the discount factor $\gamma$, and therefore putting more weights on the rewards obtained in the future, favours generosity.
This parameters can also be interpreted as the probability that the game repeats at the next step, implying that $(1-\gamma)^{-1}$ is the expected number of episodes to be played, i.e. the time horizon.
Therefore, the longer an agent expects to play, the larger is the parameter space associated to inequity aversion.
The second crucial quantity is the symmetry between the players, described by the specialization parameter $s$.
It can have two interpretations.
The first one is how much more frequently a player performs the task with respect to the other, for example because of dominance ranks within the community.
The alternative interpretation is about the ability in performing the task, in particular the difference in the probability of success of gathering resource between the two players.
It appears that a strong player asymmetry disfavours generosity, possibly leading to the dominance of one player over the other, if (\ref{eq:expl_ph}) is satisfied.
Finally, a crucial role is played by the resource abundance: the generous behaviour is possible only if the environment provides an amount of resources within the window defined by (\ref{eq:abundance_ineq}).
The transition from selfish to generous behavior as a consequence of different resource availability is observed, for example, in the chimpanzees in the Ta\"i National Park.
They are more generous when the prey is large, typically an adult of a Colobus monkeys, with respect when they capture infants \cite{boesch1989hunting}.
Moreover, it seems that chimpanzees in the rain forest show much more resource sharing than the ones in savanna-woodlands \cite{boesch1989hunting}, and, again, this can be due to a change in the resource availability.
An important consideration is in order about the simple rule discussed above: generosity is favored by a long time horizon. This closely resembles the condition for cooperation in reciprocity \cite{trivers1971evolution}.
For example, in the classical setting of reciprocity, which identifies Tit for Tat as the cooperative strategy in an iterated prisoner dilemma, cooperation emerges if the probability of interacting again with the same partner (the counterpart of $\gamma$) is sufficiently large \cite{axelrod1981evolution, stephens1996modelling, nowak2006five, whitlock2007costs, buettner2009cooperation}.
This analogy can be traced back to a similar way of thinking about generosity: it emerges from greedy players who want to maximize their personal fitness/return, without considering, for example, the fitness of relatives as in kin selection. Moreover, in both models, generosity can be chosen because it provides an advantage in the long run, while being inefficient at the present time.
However, there are substantial differences with our approach here.
First, the fundamental mechanism that induces cooperation is different: here it is based on the feedback that the resource sharing has on the next gathering task through the health variable.
In reciprocity the key is to recognize the partner and to remember the game outcome at the previous step, allowing agents to play strategies such as tit-for-tat or win-stay lose-shift \cite{nowak1993strategy}.
We want to stress that our Markov game does not require to remember the previous outcome of the game, but the agent must be able to recognize the health state of his partner, which is more biologically plausible.
Second, here we focus on a specific instance of cooperation: the choice of dividing acquired resources in equal parts, i.e. second order inequity aversion.
To better understand how this is related with classical theories of cooperation, it is useful to introduce the cost in fitness for the generous action, $c$, and the benefit $b$ that this action provides to the recipient.
If one computes these two quantities for inequity aversion, one finds $c = R\delta/2$, which is the difference between what a player can potentially acquire by being selfish and what it actually gains by the equal sharing, and $b = R\delta/2$, the recipient receives exactly what the donor looses.
This condition, $c = b$, typically does not allow the emergence of cooperation in evolutionary games.
For example, in the seminal work of Martin Nowak \cite{nowak2006five}, five mechanism for cooperation are introduced and, for each of them, an inequality states when the strategy is evolutionary stable.
In reciprocity, for example, the inequality reads $\gamma > c/b$, where $\gamma$ is the probability that there will be another interaction with the same player.
If $c=b$ cooperation is no longer stable neither for reciprocity nor for all the other four mechanisms.
The last important difference is that, as already stressed above, instead of looking at generosity as an evolutionary outcome, the present work recovers generosity as the best strategy of a decision-making problem that animals can learn through trial and error.
A crucial advantage of the latter approach is that learning works on much faster time-scales than evolution.
This allows individuals to modify strategies during their life time in response to varying environmental conditions, such as abundances of resources in their habitat or changes in the social ranks within a community.
The model is clearly oversimplified to address real situations and many additions are conceivable.
Obviously, the drawback of increasing the model complexity is that the system of equations becomes analytically unsolvable. Nonetheless, it can be approached with numerical methods, such as dynamic programming techniques \cite{sutton1998introduction}.
Just to mention some interesting generalizations, more than two agents can be considered, each with a private policy about the fraction of resource to share with the other players.
Also, the health space can be expanded including intermediate states between the fully healthy and the exhausted player, and more than two choices for the resource sharing can be added.
In this direction, recent works from the DeepMind lab \cite{leibo2017multi, hughes2018inequity, Wang2018EvolvingIM} study dynamics of cooperation-competition in Markov games inspired by real systems, such as a pair of wolves that has to catch a prey in a grid-world environment, or several agents who have to harvest resources whose spawning rate drops to zero if all the resources have been gathered.
This goes in the direction of adding more realistic details to the game: the system complexity increases, and analytical approaches are no longer possible.
However, there are extremely powerful tools and algorithms to efficiently find rewarding strategies.
Most of them are based on (deep) reinforcement learning \cite{mnih2015human, li2017deep}.
The problem of generosity and cooperation is just one among many systems in which Markov games, together with reinforcement learning techniques, can find application.
Notably, this framework is also at the basis of several recent successes in \quotes{artificial intelligence}, e.g. \cite{silver2016mastering, moravvcik2017deepstack}.
The concept of reinforcement learning and its application in Markov decision processes allows us to introduce another important remark.
These kind of algorithms are, generally speaking, grounded on the idea of trial and error.
Importantly, a lot of behavioral and neuroscientific evidence claims that animals can learn using very similar processes, in particular temporal difference algorithms \cite{dayan2008reinforcement, niv2009reinforcement}.
This leads to the biologically reasonable assumption that animals usually learn efficient strategies for the daily-tasks performed in nature.
This can include how to acquire and divide resources, implying that the generous (or the selfish) sharing can be viewed as the optimal policy of a decision-making problem.
Clearly, whether resource sharing and, in general, cooperation between animals are shaped by evolution or by a learning process is an extremely difficult question to answer \cite{lehmann2008social}, with a strong specificity on the species and the behavior under study.
Classical approaches rely mainly on the evolutionary mechanism \cite{nowak2006five}.
Alternatively, our model shows that a subset of these behaviors, i.e. generous resource sharing, can emerge as a result of the learning of a simple Markov game without requiring a population of players and evolutionary time scales.
A reasonable possibility is that what really happens in animals is a combination of the two mechanisms.
Interestingly, recent successes in obtaining cooperation between AI agents are based on the combination of evolution and learning \cite{Wang2018EvolvingIM}.
The algorithm works on two different time scales.
On the faster time scale, the agent learns with reward function composed of the actual environmental rewards, plus a part of \quotes{intrinsic motivations}.
On the slower evolutionary time scale, a population of agents can evolve the shape of those intrinsic motivations after having completed a learning task, trying to maximize a given fitness function.
As a result, the intrinsic motivations are not connected to real sources of rewards, but greatly help the algorithm to learn efficient strategies \cite{singh2009rewards, singh2010intrinsically}.
If one assumes that the behavior is mainly driven by learning, then our game can be potentially used to describe and predict dynamics that really happen in animals.
The case of vampire bats would be the most natural one for testing predictions, since their behavior closely resembles our game.
Moreover, it can be reasonable to assume that they learn, because individuals select other specific individuals as sharing partners \cite{wilkinson1990food}, and partner selection can be carried out only within the time scales of learning.
A fascinating possibility would be testing the emergence of different optimal strategies by controlling, for example, the specialization parameter $s$.
To this end, one could force an individual to stay in the roost, therefore introducing asymmetry between the animals in performing the gathering task.
Does this affect the amount of blood transferred during the sharing, leading to some sort of exploitative behavior?
Another possible test could involve the quantity of resource available, and, in particular, if overabundance or scarcity of food would reduce the generous sharing.
As a final remark, we want to mention a further context in which generalizations of our model can be potentially applied.
Let us express in other words the key mechanism for the emergence of generosity: a selfish action would lead to an exhausted partner, who will get no resource the next time it will play as proposer, making also the first player exhausted and unable to get resource any longer.
Therefore generosity can be seen as an \quotes{insurance} against the community collapse (i.e. the two players exhausted).
This really resembles the mechanism proposed for the emergence of egalitarianism in Bushmen indigenous groups, where there is a strong social pressure towards economic equality and sharing of resources \footnote{We thank an anonymous referee for pointing out this parallelism.}. Anthropologists suggest that this social convention allows the group to survive in adverse environmental conditions \cite{cashdan1980egalitarianism}.
Differently, other communities which have the possibility to accumulate resources show social stratification and a stronger unbalance of ownership of wealth items. Qualitatively, one can imagine that, in the latter case, inequality (\ref{eq:abundance_ineq}) is violated and selfishness emerges. Clearly, these kind of phenomena require further modelling efforts that go beyond the purpose of the present paper.
However, these qualitative agreements tempt us to speculate that complex features of the primitive human society share a similar backbone mechanism as the one proposed here.
\section*{Acknowledgements}
We are grateful to Agnese Seminara, Jacopo Grilli, Alberto Pezzotta, Matteo Adorisio, Claudio Leone, and Mihir Durve for useful discussions.
\section{Methods}
\subsection{Bellman optimality equation}
The model can be described as an extension of a Markov Decision Process \cite{howard1960dynamic, sutton1998introduction} for more than one agent: a Markov game \cite{littman1994markov, claus1998dynamics}.
It is defined by a set of states $\mathcal{S}$, and a set of actions that each player can take $\mathcal{A} = \mathcal{A}_1 \otimes \mathcal{A}_2$ (for the two-player case).
From each state $s \in \mathcal{S}$ and set of actions $\vec{a} = (a_1, a_2) \in \mathcal{A}$ the game jumps to a new state $s'$ according to the transition probabilities $p(s'|s,\vec{a}) \in PD(\mathcal{S})$, and each player gets a reward according to the reward function $\vec{r}(s, \vec{a}) \in \mathbb{R}^2$.
The player $i$ chooses which action to take from a given state with a probability given by its policy: $\pi_i(a_i|s) \in PD(\mathcal{A}_i)$.
In Table \ref{tab:mdp} it is shown how the model can be cast into this framework (see \ref{sec:MDP} for a more detailed explanation.)
To compute the optimal strategy, the key quantity to consider is the quality function $Q_i(s,a_i,\pi_i,\pi_{-i}) = \mathbb{E}_{\pi_i,\pi_{-i}}\left[ \sum_{t=1}^\infty r_i^{(t)} \gamma^{t-1} | S^{(1)}=s, A^{(1)}_i=a_i \right]$, which is the expected return (\ref{eq:return}) of the player $i$ starting from the state $S^{(1)} = s$, choosing the action $A^{(1)}_i = a_i$, by playing with a policy is $\pi_i$, and by knowing the policy of the other player $\pi_{-i}$.
The optimization problem of maximizing the return (\ref{eq:return}) by knowing that also the other player is optimizing its return simultaneously, can be solved through the following Bellman optimality equation (derived in \ref{sec:solution}):
\begin{equation}
\begin{cases}
\pi^*_i(a_i|s) = \delta (a_i - \underset{b}{\text{argmax}} Q^*_i(s, b) )
\\
Q^*_i(s, a_i) = \mathbb{E}_{p, \pi^*_{-i}} \left[ r_i(s, a_i, a_{-i}) + \gamma\; \underset{b}{\max}Q^*_i(s, b) \right]
\end{cases}
\label{eq:bellman}
\end{equation}
where the optimal policy from the state $s$ is deterministic and consists in choosing the action that maximizes the best quality $Q^*_i(s, a_i)$.
\begin{table
\centering
\caption{States, actions, rewards and transitions probabilities of the gathering-sharing model.}
\begin{tabular}{cccc}
State, $s$ & Action, $\vec{a}$ & Reward, $\vec{r}$ & Transition probability, $p(s'|s,\vec{a})$ \\
\midrule
$1h$ & $(e,\emptyset)$ & $(\frac{1}{2}, \frac{1}{2})$ & $q \;$ to $1h$, $\;\;$ $1-q \;$ to $2h$ \\
$1h$ & $(u,\emptyset)$ & $\frac{1}{2}(1+\delta,1-\delta)$ & $q \;$ to $1h$, $\;\;$ $1-q \;$ to $2t$ \\
$1t$ & $(e,\emptyset)$ & $(0,0)$ & $q \;$ to $1t$, $\;\;$ $1-q \;$ to $2t$ \\
$1t$ & $(u,\emptyset)$ & $(0,0)$ & $q \;$ to $1t$, $\;\;$ $1-q \;$ to $2t$ \\
$2h$ & $(\emptyset,e)$ & $(\frac{1}{2}, \frac{1}{2})$ & $q \;$ to $1h$, $\;\;$ $1-q \;$ to $2h$ \\
$2h$ & $(\emptyset,u)$ & $\frac{1}{2}(1-\delta,1+\delta)$ & $q \;$ to $1t$, $\;\;$ $1-q \;$ to $2h$ \\
$2t$ & $(\emptyset,e)$ & $(0,0)$ & $q \;$ to $1t$, $\;\;$ $1-q \;$ to $2t$ \\
$2t$ & $(\emptyset,u)$ & $(0,0)$ & $q \;$ to $1t$, $\;\;$ $1-q \;$ to $2t$ \\
\bottomrule
\end{tabular}
\caption{The model is composed of four states identified by the individual playing as proposer, $1$ or $2$, and its health, $h$ for a healthy player, $t$ for an exhausted one.
For each state, the proposer can choose an equal, $e$, or an unequal sharing $u$, while the bystander can only play a fictitious action $\emptyset$.
For each state-action pair the rewards are shown in the third column, and the transition probabilities in the fourth one, where $q = (1+s)/2$ is the probability to choose the first player as proposer.
For more details, see \ref{sec:MDP}}
\label{tab:mdp}
\end{table}
|
1,108,101,563,298 | arxiv |
\section{Introduction}
In the digital age, portrait and self-portrait photographs are extremely popular and have spread to every corner of the world.
Although these images are casual in nature, many users would like to retouch them before sharing online.
The general purpose photo editing tools, such as Adobe Photoshop, provide professional retouching results, however they target only highly skilled users.
In research community, the methods for portrait editing can be roughly grouped into two categories: transfer-based approaches \citep{fu2020poisson,Shu_2017_CVPR,nguyen2021lipstick,Cho_2019_CVPR} and user-guided approaches \citep{xiao2019interactive,zhang2017real}. The transfer-based methods allow the user to transfer colors, lights and shadows from a user-specified reference image to target images. Although these methods can generate visually appealing results, they do not support fine-grained control, since the transfer process is often designed in a fully automated manner. Moreover, it may also be challenging to find suitable reference images to achieve the desired transfer effect.
The user-guided approaches allow the user to control the retouching results by using color strokes \citep{sangkloy2017scribbler}, color and texture patches \citep{xiao2019interactive,Xian_2018_CVPR,zhang2017real}, tags \citep{Kim_2019_ICCV} or even texts \citep{Bahng_2018_ECCV}. The color strokes or patches based methods require moderate user interaction, such as sketching strokes or copying patches to the region of interest. Those methods are intuitive and easy to use, and ideal for the scenarios in which the user only retouches a few images. However, applying them to a large amount of photos is tedious and time consuming. Besides, very sparse user inputs sometimes confuse the neural network that may produce unpredictable colorization results \citep{you2019pi}. The tag/text based methods support batch processing naturally, but their way of control is not intuitive and they cannot support fine tuning.
In this paper, we propose an effective framework that adopts conditional GAN \citep{mirza2014conditional,isola2017image} for fine-grained geometry, color, light and shadow editing (see Fig.~\ref{fig:teaser}). Given the training images, We first extract features (e.g., edge maps, color, etc.) as conditional information for controllable image generation. For better visualization to the end users, the features correspond to different editing operations are separated and saved in the form of images. Then we feed the concatenated conditional information into the generator and discriminator of our models, and train the generator to reconstruct the original images. Different from the previous conditional GAN based approaches where the generators and discriminators are usually fed with the same conditional information, we convert generator inputs to the form that are noise robust and easy for hand editing, and convert discriminator inputs to the form that can guide controllable image generation more efficiently. After training, the model can take unseen conditional inputs for image synthesis or editing. Moreover, it is flexible to extend our framework for more editing operations by concatenating additional conditional information with those discussed in this paper.
Our main contributions can be summarized as follows:
\begin{itemize}
\item We design an all-in-one model for portrait geometry, color, light and shadow editing without compromising performance, which generates photo realistic images and reduces the complexity of deployment and maintenance.
\item We demonstrate novel and convenient ways for fine-grained portrait image editing, including color editing using palette and slider, and light/shadow editing using masks. These methods are easy to use for both professional and non-professional users.
\item We propose an asymmetric conditional GAN architecture to improve the performance of fine-grained color editing using palette, and enhance the robustness of geometry editing guided by edge maps.
\item We design a region-weighted discriminator that enables explicitly assigning higher weight to the more important regions of the generated images, like eyes and face, when computing loss.
\item Our palette and slider based color editing enables batch image processing, which is not supported in the previous fine-grained editing methods.
\end{itemize}
\begin{figure*}[t!]
\centering
\includegraphics[width=\linewidth]{figs/teaser2.pdf}
\caption{Our method supports fine-grained editing of geometries, colors, lights and shadows using a single neural network model. It allows the user to edit colors via a color palette and/or slider, geometries via edge maps, and lights and shadows via masks. It can also generate images from hand-drawn sketches. The percentages in (d) indicate the yellow ratio in the first row of palette, which controls hair color.}
\label{fig:teaser}
\end{figure*}
\section{Related Work}
\myparagraph{Conditional GAN.}
Unconditional GANs, proposed by \citet{goodfellow2014generative}, take only a random noise vector as input, thereby is unable to support user control. In contrast, conditional GANs \citep{mirza2014conditional}
train on a labeled data set and allow the user to specify the label to control each generated instance.
Conditional GANs demonstrate promising performance on various image synthesis tasks \citep{isola2017image,gu2019mask,park2019semantic,shamsolmoali2020image}, where a widely used form for image editing is image-to-image translation \citep{pang2021image}.
However, the existing variants of conditional GANs usually feed the same label/conditional information into generator and discriminator, which is sub-optimal since these two components are designed for different purpose.
Therefore, our proposed asymmetric conditional GAN feeds different conditional information into generator and discriminator to better fit the application needs.
\myparagraph{Portrait Image Editing.}
Portrait image editing has attracted considerable interest due to its wide application, which consists of a long list of tasks, including geometry editing \citep{wu2021coarse}, colorization \citep{zhang2017real}, relighting and shadow manipulation \citep{zhou2019deep,wang2020single,zhang2020portrait}, stylization \citep{li2018closed, yi2019apdrawinggan}, pose and expression control \citep{Tewari_2020_CVPR} and so on. Recently, there are also a few attempts to enable the control of multiple attributes using a single model. For example, MichiGAN \citep{tan2020michigan} is designed for disentangled hair manipulation of structure, shape and so on. However, each attribute in MichiGAN requires a separate condition module, which makes it inflexible to extend. In this work, we conduct extensive experiments to show that our model can be extended for a wide range of tasks by simply concatenating more conditional information to the inputs.
\myparagraph{Colorization and Color Editing.} The existing image colorization and color editing methods can be categorized into two groups, user-guided approach and transfer-based approach. The user guided approaches usually control the results using sparse user inputs, like strokes \citep{levin2004colorization,sangkloy2017scribbler}, points/patches \citep{zhang2017real}, and so on. These methods often do not support batch editing. The transfer-based approach allows automatic color transfer from the reference images \citep{reinhard2001color,he2018deep,afifi2021histogan}, but do not allow fine-grained control. Similar to our approach, the transfer-based image recoloring method proposed by \citet{chang2015palette} also takes palette as input. However, they do not incorporate spatial information into the algorithm, so their method is inflexible for fine-grained editing.
\myparagraph{Light and Shadow Editing.} There is also a long line of research on portrait image relighting or light/shadow editing. Most of the recent approaches \citep{zhang2020portrait,NestmeyerLML20,SunLBXR21,HouZSBT021} use target light for control. For example, \citet{NestmeyerLML20} use physics-based relight, but find noise increases when there is little light. \citet{ShuHSSPS18} also propose method to transfer light from reference image, however, they notice the method fails when adding/removing no-diffuse effects. We adopt the mask-based method, which is more controllable during fine-grained light and shadow editing.
\section{Method}
\subsection{Overview}
We first extract portrait edge maps, colors, and light/shadow masks from the training images. Then the conditional GAN is trained to reconstruct the original images with the extracted constraint features as input. After training, we can modify the model inputs to achieve fine-grained portrait image editing. To ease reading, we list the main notations in Table~\ref{tab:notations}.
\begin{table}[t]
\centering
\caption{Summary of the main notations.}
\begin{tabular}{cl}
\toprule
Notation & Description \\
\midrule
${\bm{I}}_{E}$ & edge map\\
${\bm{I}}_{CP}$ & color palette\\
${\bm{I}}_L$ &light mask\\
${\bm{I}}_S$ & shadow mask\\
${\bm{I}}_E$ & original edge map\\
${\bm{I}}_C$ & color map\\
$n({\bm{I}}_{E})$ & noisy edge map \\
\hline
$G$ & the generator\\
${\bm{I}}_G$ & the input of $G$\\
${\bm{P}}_F$ & the output of $G$\\
\hline
$m({\bm{P}}_F)$ & applying the skin and eye masks to ${\bm{P}}_F$\\
$D_L$ & the multi-scale discriminator for local facial components \\
${\bm{I}}_{D-L}$ & the input of $D_L$\\
$D_G$ & the multi-scale discriminator for global appearance\\
${\bm{I}}_{D-G}$ & the input of $D_G$\\
$\mathcal{L}_{MD-L}$ & the loss function for $D_L$\\
$\mathcal{L}_{MD-G}$ & the loss function for $D_G$\\
\bottomrule
\end{tabular}
\label{tab:notations}
\end{table}
\subsection{Model Inputs}
\label{sec:model_input_sec}
\subsubsection{Edge Maps}
\label{sec:edge_map_extract}
Edge maps, which provide salient structural information for images, are often used to control the geometry of the generated images. The DexiNed model~\citep{poma2020dense} shows encouraging edge detection performance on outdoor images. To adapt DexiNed for portrait images, we annotate 27 portrait images manually and then combine them with the Barcelona Images for Perceptual Edge Detection (BIPED) dataset \citep{poma2020dense} that contains 250 hand-annotated outdoor images for model training. Given an input image, the trained model outputs a probability matrix of the same dimension as the input, where each value in the matrix represents the likelihood of the corresponding pixel to be part of an edge. Since the generated edge maps are noisy, we apply the following steps to further reduce noise: 1) apply the Gaussian filter to the input edge map and then set the pixels below a pre-defined threshold $\beta$ to 0 ($\beta=0.35$ in our implementation); 2) move a $5 \times 5$ sliding window through the edge map with stride size 1, and filter out the bottom 20\% values in each window; 3) apply the Gaussian filter in step 1) to generate the final edge map. Comparing with the classic edge detection algorithms \citep{canny1986computational,xie2015holistically,yi2019apdrawinggan}, we observe the portrait-tailored DexiNed model produces visually clean results that facilitate editing.
\subsubsection{Color Palette}
\label{sec:color_palette}
\begin{figure}[t!]
\centering
{\footnotesize
\includegraphics[width=0.19\columnwidth]{figs/29993_r.png}
\includegraphics[width=0.19\columnwidth]{figs/29993_r_c.png}
\includegraphics[width=0.19\columnwidth]{figs/29993_r_c2.png}\\
\makebox[0.19\columnwidth]{(a)}
\makebox[0.19\columnwidth]{(b)}
\makebox[0.19\columnwidth]{(c)}\\
}
\caption{Color palette extraction. Given the input image (a), we compute a semantic segmentation, consisting of hair, face skin, eyebrows, eyes, nose, mouth, neck and background. Then the average color of each segment is computed to generate the color map (b). Finally, (b) is transformed to a color palette (c), by taking the average colors of hair, face, eye, lip and background (from top to bottom).
}
\label{fig:color_palette}
\end{figure}
Fine-grained color editing is often a tedious work and usually requires expertise. To help get rid of the tedious work, we design our framework to support automated color editing without compromise of fine grained control. Our framework achieves this objective with the help of color palettes.
We use the CelebAMask-HQ dataset \citep{DBLPLee0W020} for model training, which contains high-resolution portrait images and manually annotated facial segmentation masks. If there is any additional portrait images from the other sources, the pretrained face parsing model released by \citet{DBLPLee0W020} can also be used for annotation. Then we compute the average RGB pixel value of each facial component as the corresponding color map. An example is shown in Fig.\ref{fig:color_palette}(b). However, editing image with color map still requires users to decide the boundary of each facial component. To further simply the color editing work, we take colors of the 5 most important components from the color map, namely average colors of hair, face, eye, lip and background, to create a color palette for each image. An example is shown in Fig.\ref{fig:color_palette}(c). Order of these components in the color palettes are fixed, if any component is not found we use black as the default color. It is possible to add more facial components for finer grained color control. However, exposing non-professional users to too many choices may increase the editing complexity, so we let the model to help decide colors of the other components based on the context.
\subsubsection{Light and Shadow Masks}
We apply a simple yet effective light extraction algorithm \citep{shen2013real} to extract facial light from the input images, and facial shadow extraction can be achieved by applying the same algorithm on the inverted image.
Then the extracted light and shadow are binarized with the threshold $0.15$ after being normalized to the range of $[0, 1]$.
Finally, the binary images are processed using a 2D median filter with window size $7\times7$ to generate the light and shadow masks.
Fig.~\ref{fig:light_removal} shows an example of light and shadow mask extraction.
\begin{figure}[t!]
\centering
{\footnotesize
\includegraphics[width=0.19\columnwidth]{figs/119.jpg}
\includegraphics[width=0.19\columnwidth]{figs/119_hl.jpg}
\includegraphics[width=0.19\columnwidth]{figs/119_hl_mask.jpg}
\includegraphics[width=0.19\columnwidth]{figs/119_sd.jpg}
\includegraphics[width=0.19\columnwidth]{figs/119_sd_mask_r.jpg}\\
\makebox[0.19\columnwidth]{Original}
\makebox[0.19\columnwidth]{Light}
\makebox[0.19\columnwidth]{Light Mask}
\makebox[0.19\columnwidth]{Shadow}
\makebox[0.19\columnwidth]{Shadow Mask}\\
}
\caption{Light and shadow mask extraction.}
\label{fig:light_removal}
\end{figure}
\subsection{Edge Map Noising Methods}
We apply three noising methods, namely random removal, random shift and random lines, to the edge maps to improve model robustness. This allows even beginners to edit extracted edge maps and obtain high quality results. We use $\mathcal{N} = \{n_{RR}(\cdot), n_{RS}(\cdot), n_{RL}(\cdot), n_{N}(\cdot)\}$ to denote the set of noising functions, where $n_{RR}(\cdot), n_{RS}(\cdot), n_{RL}(\cdot)$ denotes random remove, random shift and random line, respectively. And $n_{N}(\cdot)$ denotes \textbf{not} to apply any of the noising functions, i.e., simply returning the original input. Given an edge map ${\bm{I}}_E$, we apply a random noising function $n(\cdot)$ uniformly selected from $\mathcal{N}$ to generate the noisy edge map $n({\bm{I}}_E)$, then $n({\bm{I}}_E)$ is used as part of the inputs during model training. During test, we do not apply any noising function to ${\bm{I}}_E$, that is, we directly feed ${\bm{I}}_E$ into the model for inference.
\begin{figure}[t!]
\centering
{\footnotesize
\includegraphics[width=0.19\columnwidth]{figs/noise_9_origin.png}
\includegraphics[width=0.19\columnwidth]{figs/noise_rr_9_m.png}
\includegraphics[width=0.19\columnwidth]{figs/noise_rs_9_m.png}
\includegraphics[width=0.19\columnwidth]{figs/noise_rl_9_m.png}\\
\makebox[0.19\columnwidth]{(a)}
\makebox[0.19\columnwidth]{(b)}
\makebox[0.19\columnwidth]{(c)}
\makebox[0.19\columnwidth]{(d)}\\
}
\caption{Examples of the edge map noising methods. (a) Original edge map; (b) Random removal; (c) Random shift; (d) Random lines.}
\label{fig:edge_noise}
\end{figure}
\ \\
\noindent\textbf{Random removal.} We remove a randomly selected region from the input in this noising method, which is similar to the masks used in \citep{yang2020deep}, except that we apply it to the edge maps instead of the original images. Let $h$ and $w$ be the height and width of the input image. We select a random rectangle ${\bm{R}}$ of height $h^\prime$ and width $w^\prime$ from the edge map to remove, where $h^\prime \in (0, \alpha \times h)$ and $w^\prime \in (0, \alpha \times w)$ are randomly selected from the intervals, and $0 < \alpha < 1$ is a hyper-parameter determines the max dimensions of ${\bm{R}}$. We set $\alpha=0.33$ in our implementation. See Fig.~\ref{fig:edge_noise}(b) for an example of random removal. This noising method improves the robustness of the model so that it can deal with missing lines in edge maps.
\ \\
\noindent\textbf{Random shift.} We first select a random rectangle ${\bm{R}}$ of height $h^\prime$ and width $w^\prime$ from the edge map following the same steps in random removal, and then shift the lines in ${\bm{R}}$ by a random displacement. Fig.~\ref{fig:edge_noise}(c) shows an output generated by this noising method, where the lines on face are shifted towards the right hand side. The introduced noises during training helps model to learn to generate more coherent images when the edge map contains inaccurate lines drawn by user.
\ \\
\noindent\textbf{Random lines.} Adding random lines to edge map can be achieved via similar operations as random shift. Assume ${\bm{R}}$ is the random rectangle selected from the original edge map, and ${\bm{R}}^{\prime}$ is the target area that lines in ${\bm{R}}$ will be copied to, where ${\bm{R}}^{\prime}$ is of the same dimension as ${\bm{R}}$. In random shift, the lines in ${\bm{R}}^{\prime}$ are overwritten by white color, which looks like being shifted from ${\bm{R}}$ to ${\bm{R}}^{\prime}$. However, the random lines function, Fig.~\ref{fig:edge_noise}(d) for example, keeps the original lines in the target area ${\bm{R}}^{\prime}$, which looks adding extra lines to the edge map. This function is designed to improve model's robustness to extra noisy lines caused by edge map extraction or hand editing.
\subsection{Asymmetric Conditional GAN}
We build our model upon pix2pixHD \citep{wang2018high}, a conditional GAN based model designed for high-resolution photorealistic image-to-image translation.
In the previous conditional GAN based models, both the generators and the discriminators take the same conditional information as input during model training.
To facilitate fine-grained control, we propose a novel asymmetric conditional GAN, where the generator and the discriminators take relevant but different conditional information as input. This feature distinguishes our method and the other conditional GAN based image synthesis methods~\citep{dash2021high,wang2018high,chen2018sketchygan}.
We show the network architecture in Fig.~\ref{fig:model_arch}.
\begin{figure*}[t!]
\centering
{\footnotesize
\includegraphics[width=1.0\linewidth]{figs/model_arch.pdf}
}
\vspace{-1em}
\caption{Model architecture. In contrast to the existing conditional GAN models, we feed the generator and the discriminators relevant but different types of conditional information. The input to the generator ${\bm{I}}_G = \mathrm{concat}( n({\bm{I}}_{E}), {\bm{I}}_{CP}, {\bm{I}}_{L}, {\bm{I}}_{S})$, consisting of noisy edge map $n({\bm{I}}_E)$, color palette ${\bm{I}}_{CP}$, light mask ${\bm{I}}_L$ and shadow mask ${\bm{I}}_S$, can be directly edited by the user. There are two multi-scale discriminators $D_G$ and $D_L$. $D_G$, which takes ${\bm{I}}_{D-G} = \mathrm{concat}( {\bm{I}}_{E}, {\bm{I}}_{C}, {\bm{I}}_{L}, {\bm{I}}_{S}, {\bm{P}})$ as input, judges global visual appearance. $D_L$, on the other hand, takes ${\bm{I}}_{D-L} = \mathrm{concat}( {\bm{I}}_{E}, {\bm{I}}_{C}, {\bm{I}}_{L}, {\bm{I}}_{S}, m({\bm{P}}))$ as input and provides feedback of local facial components to $G$.}
\label{fig:model_arch}
\end{figure*}
\subsubsection{Generator}
The original pix2pixHD model generator, composed of a global generator network and an additional local enhancer network, is able to produce images of resolution $2048 \times 1024$ or even higher. It is observed that even without the local enhancer network the quality of generated images of resolution $1024 \times 512$ or lower are still promising, so we only keep the global generator network in our implementation to reduce computation resource usage. To generate higher resolution images, one could add the local enhancer network back.
The generator $G$ consists of three main components, a convolutional front-end, 9 residual blocks, and a transposed convolutional back-end. We concatenate noisy edges $n({\bm{I}}_{E})$, color palette ${\bm{I}}_{CP}$, light mask ${\bm{I}}_{L}$, shadow mask ${\bm{I}}_{S}$ as conditional information to generate the output image. The concatenated input $${\bm{I}}_G = \mathrm{concat}( n({\bm{I}}_{E}), {\bm{I}}_{CP}, {\bm{I}}_{L}, {\bm{I}}_{S})$$ is passed through the three generator components sequentially, where they are down-sampled in the convolutional front-end and up-sampled in the transposed convolutional back-end. Finally, the generator yields a fake image ${\bm{P}}_F = G({\bm{I}}_G)$.
\subsubsection{Discriminators}
\label{sec:discriminators}
We design multiple discriminators to provide feedback to the generator $G$ based on more explicit conditions. In the existing conditional GAN models~\citep{dash2021high,wang2018high,chen2018sketchygan}, the generators and the discriminators are fed with the same conditional input. However, such design either constrains the convenience of the control of the generators, or hinders the discriminators from providing better feedback for the generator outputs. Thus, we propose an \textbf{asymmetric conditional GAN}, that allows generators and discriminators to use conditional information in \textit{different} forms, which are more relevant to their objective.
Different from the original multi-scale discriminators used \citet{wang2018high}, we also propose a \textbf{region-weighted discriminator}, which consists of two multi-scale discriminators, namely $D_G$ for the global appearance and $D_L$ for local facial components. As shown in Fig.~\ref{fig:model_arch}, the architectures of these two multi-scale discriminators are similar, but they take different inputs and their parameters are not shared.
Compared with hair and background, humans are usually more sensitive to the quality of generated faces and eyes, so it is intuitive to give their corresponding regions higher weights when computing loss. Therefore, we use semantic segmentation model to extract face and eye masks from the real portrait images ${\bm{P}}_R$, and apply masks to ${\bm{P}}$ to get $m({\bm{P}})$, where ${\bm{P}}$ denotes ${\bm{P}}_R$ or ${\bm{P}}_F$, and $m({\bm{P}})$ denotes the concatenation of face and eye regions extracted from ${\bm{P}}$.
We feed
$${\bm{I}}_{D-L} = \mathrm{concat}( {\bm{I}}_{E}, {\bm{I}}_{C}, {\bm{I}}_{L}, {\bm{I}}_{S}, m({\bm{P}}))$$
into this region-weighted discriminator to compute loss $\mathcal{L}_{MD-L} = D({\bm{I}}_{D-L})$.
To ensure the coherence of the whole image, we train another multi-scale discriminator with loss function $\mathcal{L}_{MD-G} = D_{L}({\bm{I}}_{D-G})$, where
$${\bm{I}}_{D-G} = \mathrm{concat}( {\bm{I}}_{E}, {\bm{I}}_{C}, {\bm{I}}_{L}, {\bm{I}}_{S}, {\bm{P}}).$$
We can observe the difference between the generator and discriminator inputs. For geometry control, $n({\bm{I}}_{E})$ is used in the generator inputs to simulate the noisy edges, while ${\bm{I}}_{E}$ is used in the discriminators' inputs, which allows the discriminators to find the possible regions caused by noises easily and thus provide better instructions to the generator on how to handle the noises. Similarly, for color control we use ${\bm{I}}_{CP}$ in the generator inputs and ${\bm{I}}_{C}$ in the discriminators' inputs. ${\bm{I}}_{CP}$ is easier for users to edit, and ${\bm{I}}_{C}$ is easier for discriminators to learn since it also contains positional information of the color pixels.
Each multi-scale discriminator is also composed of two different discriminator networks. Both of the two discriminator networks take the same input to predict a fake/real label. Then the losses are computed based on their predictions. In multi-scale discriminator $D_{G}$, the first discriminator network passes inputs through several convolutional layers to compute the loss $\mathcal{L}_{O-G}$. The second discriminator down-samples the inputs with an average pool layer first, and then processes the down-sampled inputs with a similar network as the first one to compute the loss $\mathcal{L}_{DS-G}$. The final loss for $D_G$ is computed with $$\mathcal{L}_{MD-G} = \mathcal{L}_{O-G} + \mathcal{L}_{DS-G}.$$
Note that the parameters of the two networks are not shared. Loss $\mathcal{L}_{MD-L}$ is computed in a similar way for $D_{L}$.
\subsubsection{Loss Functions}
As described above, we use $G$ to generate the fake portrait image ${\bm{P}}_F = G({\bm{I}}_G)$. The other conditional information can be concatenated with ${\bm{P}}_R$ or ${\bm{P}}_F$ to get the inputs for the two discriminators, ${\bm{I}}_{D-G}$ for $D_G$ and ${\bm{I}}_{D-L}$ for $D_L$. In order to differentiate the inputs when the different images ${\bm{P}}={\bm{P}}_R$ and ${\bm{P}}={\bm{P}}_F$ are used, we add additional subscripts $R$ and $F$ to ${\bm{I}}_{D-G}$ and ${\bm{I}}_{D-L}$
$$
\mathcal{L}_{MD-G} = \mathbb{E}\left[\log D_G({\bm{I}}_{D,R-G})\right] + \mathbb{E}\left[\log (1- D_G({\bm{I}}_{D,F-G}))\right],
$$
and
$$
\mathcal{L}_{MD-L} = \mathbb{E}\left[\log D_L({\bm{I}}_{D,R-L})\right] + \mathbb{E}\left[\log (1- D_L({\bm{I}}_{D,F-L}))\right].
$$
We define the GAN loss as
\begin{equation}
\min_{G} \max_{D_G,D_L} \mathcal{L}_{GAN} = \min_{G} \max_{D_G,D_L} \left(\mathcal{L}_{MD-G} + \mathcal{L}_{MD-L}\right).
\label{eq:loss_gan}
\end{equation}
In addition to $\mathcal{L}_{GAN}$, we also compute the perceptual loss \citep{johnson2016perceptual} and the feature matching loss \citep{wang2018high} to further improve the quality of the generated images. These two loss functions have proven effective in many image synthesis tasks~\citep{yang2020deep,pang2018visual,dosovitskiy2016generating}. To leverage the knowledge learned by the pretrained models, we feed real images ${\bm{P}}_R$ and and fake images ${\bm{P}}_F$ into the pretrained VGGNet \citep{simonyan2014very} to compute the perceptual loss
\begin{gather}
\mathcal{L}_{VGG} = \mathbb{E}\left[ \sum_{i,j} \lambda_i\left\|\Phi_{i,j}({\bm{P}}_R) - \Phi_{i,j}({\bm{P}}_F)\right\|_1\right],
\label{eq:loss_perc}
\end{gather}
where $\Phi_{i}(\cdot)$ is the feature map at the $i$-th VGGNet layer, $\Phi_{i,j}(\cdot)$ is the $j$-th element in the feature map, and $\lambda_{i}$ is the weight for the $i$-th layer. We compute the feature matching loss in a similar way, except that we compute the distance between the feature maps extracted from $D_G$
\begin{gather}
\mathcal{L}_{FM} = \mathbb{E}\left[\sum_{n,i,j} \lambda^{\prime}_i\left\|\Phi^{\prime}_{n,i,j}({\bm{I}}^{\prime}_{D,R}) - \Phi^{\prime}_{n,i,j}({\bm{I}}^{\prime}_{D,F})\right\|_1\right],
\label{eq:loss_fm}
\end{gather}
where $\Phi^{\prime}_{n,j}(\cdot)$ is the feature map extracted from the $i$-th layer of the $n$-th discriminator of $D_G$, since $D_G$ contains two discriminator networks. $\lambda^{\prime}_i$ is the weight of the $i$-th layer. So in addition to Eq.~\ref{eq:loss_gan}, we also adjust $G$ parameters with
\begin{gather}
\min_{G} (\mathcal{L}_{VGG} + \mathcal{L}_{FM}).
\end{gather}
\section{Experiments}
In this section, we first describe the implementation details of our model. Then, we present the experimental results of various fine-grained editing operations supported by our model. Finally, we perform ablation studies to quantitatively verify the effectiveness of the proposed model.
\subsection{Implementation Details}
\textbf{Dataset.} We use the CelebAMask-HQ dataset \citep{DBLPLee0W020} in our experiments, which contains 30,000 high-resolution portrait images. Each image has a manually-annotated segmentation mask of facial attributes in 19 pre-defined classes. We rescale the images to 512 $\times$ 512, and use 29,490 randomly selected images for model training, and the remaining 510 images for testing.
\ \\
\noindent\textbf{Model training.} We use the Adam optimizer \citep{KingmaB14} to train our models for 60 epochs with batch size 64. We set a constant learning rate 0.0002 for the first 30 epochs, and then linearly decrease it to zero for the next 30 epochs. We use VGG19 \citep{simonyan2014very} when computing the perceptual loss.
\subsection{Fine-Grained Editing}
\subsubsection{Geometry}
\begin{figure}[t!]
\centering
{\footnotesize
\includegraphics[width=0.16\columnwidth]{figs/edge_12367_r_real.jpg}
\includegraphics[width=0.16\columnwidth]{figs/edge_12367_r_edge_o.jpg}
\includegraphics[width=0.16\columnwidth]{figs/edge_12367_r_edge_s.jpg}
\includegraphics[width=0.16\columnwidth]{figs/edge_12367_r_fake.jpg} \\
\includegraphics[width=0.16\columnwidth]{figs/edge_12434_r_real.jpg}
\includegraphics[width=0.16\columnwidth]{figs/edge_12434_r_edge_o.jpg}
\includegraphics[width=0.16\columnwidth]{figs/edge_12434_r_edge_s.jpg}
\includegraphics[width=0.16\columnwidth]{figs/edge_12434_r_fake.jpg} \\
\includegraphics[width=0.16\columnwidth]{figs/edge_13643_r_real.jpg}
\includegraphics[width=0.16\columnwidth]{figs/edge_13643_r_edge_o.jpg}
\includegraphics[width=0.16\columnwidth]{figs/edge_13643_r_edge_s.jpg}
\includegraphics[width=0.16\columnwidth]{figs/edge_13643_r_fake.jpg} \\
\includegraphics[width=0.16\columnwidth]{figs/edge_13683_r_real.jpg}
\includegraphics[width=0.16\columnwidth]{figs/edge_13683_r_edge_o.jpg}
\includegraphics[width=0.16\columnwidth]{figs/edge_13683_r_edge_s.jpg}
\includegraphics[width=0.16\columnwidth]{figs/edge_13683_r_fake.jpg} \\
\makebox[0.16\columnwidth]{Original}
\makebox[0.16\columnwidth]{Original edges}
\makebox[0.16\columnwidth]{Edited edges}
\makebox[0.16\columnwidth]{Generated images}\\
}
\caption{Our model allows the user to edit geometry by changing the edge maps. The changes of edges are highlighted in red boxes.}
\label{fig:exp_edge_edit}
\end{figure}
Our proposed framework allows users to perform the common fine-grained portrait image editing operations using a single trained model, which supports geometry, color, light, and shadow editing. It is also flexible to extend the model to support more editing operations, e.g., freckle editing. In the experiments below, we modify the inputs in $\{{\bm{I}}_{E}, {\bm{I}}_{CP}, {\bm{I}}_{L}, {\bm{I}}_{S}\}$ to demonstrate how they control the synthesised images.
\noindent\textbf{Edge maps.}
The edge maps extracted from the original portrait images provide the most important structural information for image reconstruction, so it is more convenient to directly modify the extracted edge maps for minor geometry editing. As the examples shown in Fig.~\ref{fig:exp_edge_edit}, we modify ${\bm{I}}_{E}$ and keep the other constraint inputs ${\bm{I}}_{CP}, {\bm{I}}_{L}, {\bm{I}}_{S}$ unchanged to demonstrate many interesting applications, including add or remove accessories like eyeglasses and jewelry, hair style design, adjust facial components, and so on. With the help of our model, the users can conveniently generate photo-realistic images with just a few stroke editing steps, while these editing operations may be very complex and time consuming with the traditional photo editing software.
\begin{figure}[h!]
\centering
{\footnotesize
\makebox[0.16\columnwidth]{}
\makebox[0.16\columnwidth]{Scribbler \citep{sangkloy2017scribbler}}
\makebox[0.16\columnwidth]{}
\makebox[0.16\columnwidth]{}
\vspace{2pt} \\
\includegraphics[width=0.16\columnwidth]{figs/hd11_r_real.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd11_base.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd11_r_fake.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd11_r_fake_concat.jpg} \\
\includegraphics[width=0.16\columnwidth]{figs/hd10_r_real.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd10_base.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd10_r_fake.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd10_r_fake_concat.jpg} \\
\hdashrule{\columnwidth}{1pt}{1pt} \\
\makebox[0.16\columnwidth]{}
\makebox[0.16\columnwidth]{DeepFaceEditing \citep{chen2021DeepFaceEditing}}
\makebox[0.16\columnwidth]{}
\makebox[0.16\columnwidth]{}
\vspace{2pt}
\\
\includegraphics[width=0.16\columnwidth]{figs/hd4_r_real.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd4_base.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd4_r_fake.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd4_r_fake_concat.jpg} \\
\includegraphics[width=0.16\columnwidth]{figs/hd5_r_real.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd5_base.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd5_r_fake.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd5_r_fake_concat.jpg} \\
\includegraphics[width=0.16\columnwidth]{figs/hd1_r_real.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd1_base.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd1_r_fake.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd1_r_fake_concat.jpg} \\
\hdashrule{\columnwidth}{1pt}{1pt} \\
\makebox[0.16\columnwidth]{}
\makebox[0.16\columnwidth]{DeepPlasticSurgery \citep{yang2020deep}}
\makebox[0.16\columnwidth]{}
\makebox[0.16\columnwidth]{}
\vspace{2pt}
\\
\includegraphics[width=0.16\columnwidth]{figs/hd6_r_real.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd6_base.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd6_r_fake.jpg}
\includegraphics[width=0.16\columnwidth]{figs/hd6_r_fake_concat.jpg} \\
\makebox[0.16\columnwidth]{Sketches}
\makebox[0.16\columnwidth]{Baseline}
\makebox[0.16\columnwidth]{Ours}
\makebox[0.16\columnwidth]{More Colors (Ours)} \\
}
\caption{Though our method is not designed for sketches-to-image translation, it can generate visually pleasing images from hand-drawn sketches of various styles.}
\label{fig:exp_sketch2img}
\end{figure}
\noindent\textbf{Hand-drawn sketches.}
Although our framework is not designed to handle hand-drawn sketches on purpose, we find it can generate compelling results for hand-drawn sketch to image translation. In Fig.~\ref{fig:exp_sketch2img}, we compare our results with Scribbler \citep{sangkloy2017scribbler}, DeepFaceEditing \citep{chen2021DeepFaceEditing} and DeepPlasticSurgery \citep{yang2020deep}. In addition, we also test on an example drawn by ourselves. We use the model trained without light and shadow masks in this experiment for fair comparison with the baseline methods. As we can see from Fig.~\ref{fig:exp_sketch2img}, the style of these sketches are very different from the edge maps used to train the model, so we apply additional edge processing steps: 1) apply the steps described in \S\ref{sec:edge_map_extract} to extract edges from the Scribbler \citep{sangkloy2017scribbler} sketches, and keep the other sketches unchanged; 2) concatenate the edges and random color palettes as inputs to generate the intermediate images; 3) extract edges (\S\ref{sec:edge_map_extract}) from the intermediate images, and repeat step 2 to generate the final results. Scribbler \citep{sangkloy2017scribbler} is also designed for fine-grained colorization, but compared with their results the color of our method is more photo-realistic. Besides, compared with DeepFaceEditing \citep{chen2021DeepFaceEditing}, our results preserves more identity information about the sketches.
\subsubsection{Colors}
\begin{figure*}[t!]
\centering
{\footnotesize
\includegraphics[width=0.09\columnwidth]{figs/color_12361_r_real.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12361_r_lip.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12361_r_lip2.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12361_r_hair.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12361_r_hair2.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12361_r_skin3.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12361_r_skin2.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12361_r_eye.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12361_r_all.jpg}
\\
\includegraphics[width=0.09\columnwidth]{figs/color_11283_r_real.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_11283_r_lip.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_11283_r_lip2.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_11283_r_hair3.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_11283_r_hair2.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_11283_r_skin3.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_11283_r_skin2.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_11283_r_eye.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_11283_r_all.jpg}
\\
\includegraphics[width=0.09\columnwidth]{figs/color_10363_r_real.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_10363_r_lip.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_10363_r_lip2.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_10363_r_hair.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_10363_r_hair2.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_10363_r_skin3.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_10363_r_skin2.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_10363_r_eye.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_10363_r_all.jpg}
\\
\includegraphics[width=0.09\columnwidth]{figs/color_12340_r_real.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12340_r_lip.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12340_r_lip2.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12340_r_hair.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12340_r_hair2.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12340_r_skin3.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12340_r_skin2.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12340_r_eye.jpg}
\includegraphics[width=0.09\columnwidth]{figs/color_12340_r_all.jpg}
\\
\makebox[0.09\columnwidth]{Original}
\makebox[0.09\columnwidth]{Lip 1}
\makebox[0.09\columnwidth]{Lip 2}
\makebox[0.09\columnwidth]{Hair 1}
\makebox[0.09\columnwidth]{Hair 2}
\makebox[0.09\columnwidth]{Face Skin 1}
\makebox[0.09\columnwidth]{Face Skin 2}
\makebox[0.09\columnwidth]{Eye}
\makebox[0.09\columnwidth]{All}
\\
}
\caption{Fine-grained color editing. The color palettes are shown in the right bottom corner of the corresponding images. The same edge maps and shadow/light masks extracted from the original images are used as inputs without modification, which are not shown in the figure. Colors from top to bottom in the palette controls hair, skin, eyes, lips and background color, respectively.
}
\label{fig:exp_color_edit}
\end{figure*}
\begin{figure}[t!]
\centering
{\footnotesize
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12203_r_real.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12203_r_1.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12203_r_0.25.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12203_r_0.5.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12203_r_0.75.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12203_r_0.jpg}
\\
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12235_r_real.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12235_r_1.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12235_r_0.25.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12235_r_0.5.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12235_r_0.75.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12235_r_0.jpg}
\\
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_11234_r_real.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_11234_r_1.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_11234_r_0.25.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_11234_r_0.5.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_11234_r_0.75.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_11234_r_0.jpg}
\\
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12273_r_real.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12273_r_1.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12273_r_0.25.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12273_r_0.5.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12273_r_0.75.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_ctr_12273_r_0.jpg}
\\
\makebox[0.12\columnwidth]{Original}
\makebox[0.12\columnwidth]{0\%}
\makebox[0.12\columnwidth]{25\%}
\makebox[0.12\columnwidth]{50\%}
\makebox[0.12\columnwidth]{75\%}
\makebox[0.12\columnwidth]{100\%}
\\
}
\caption{Color editing using sliders. The user blends two colors, brown and blonde by controlling the mixing ratio. See the first row in the color palettes. }
\label{fig:exp_color_edit_ctr}
\end{figure}
\begin{figure}[t!]
\centering
{\footnotesize
\includegraphics[width=0.12\columnwidth]{figs/color_chg_h_625.png}
\includegraphics[width=0.12\columnwidth]{figs/color_chg_h_650.png}
\includegraphics[width=0.12\columnwidth]{figs/color_chg_h_675.png}
\includegraphics[width=0.12\columnwidth]{figs/color_chg_h_700.png}
\\
\includegraphics[width=0.12\columnwidth]{figs/color_chg_v_825.png}
\includegraphics[width=0.12\columnwidth]{figs/color_chg_v_850.png}
\includegraphics[width=0.12\columnwidth]{figs/color_chg_v_875.png}
\includegraphics[width=0.12\columnwidth]{figs/color_chg_v_900.png}
\\
\makebox[0.12\columnwidth]{25\%}
\makebox[0.12\columnwidth]{50\%}
\makebox[0.12\columnwidth]{75\%}
\makebox[0.12\columnwidth]{100\%}
\\
}
\caption{Horizontal (top) and vertical (bottom) color sliders for eye color editing. The percentage denotes the ratio of blue in the slider. See also the accompanying video demonstration.}
\label{fig:exp_color_slider_direction}
\end{figure}
In this section, we evaluate the proposed framework for fine-grained color editing. As the examples presented in Fig.~\ref{fig:exp_color_edit}, users only need to edit the color palettes ${\bm{I}}_{CP}$ shown in the right bottom corner of the images to control the color of each facial components. Compared with the recent work \citep{afifi2021histogan,chen2021DeepFaceEditing}, our approach does not require any reference image, and gives users more freedom to specify any preferred color to edit the target facial components only, while keeping color of the other facial components unchanged. Compared with the other fine-grained color editing approaches \citep{sangkloy2017scribbler,zhang2017real,xiao2019interactive}, which usually require user to specify color points or strokes on the target components, our approach is more convenient for batch editing. In the experiments, we only demonstrated editing the color of hair, skin, eyes, lips and background, but it is convenient to extend the model to support more facial components.
Our model has been proven to support color editing using the combination of colors, although it is trained with a single color for each facial component. In Fig.~\ref{fig:exp_color_edit_ctr}, we show how to perform hair color editing using a slider to control the combination of two colors. As we adjust the ratio of the two colors by move the slider from right to left, the hair color in generated image also changes gradually. While most of the facial components are not very sensitive to the left/right position of the two colors when using slider, like the hair color editing examples shown in Fig.~\ref{fig:exp_color_edit_ctr}, so the horizontal sliders can be used to change their color. However, we find color of eyes changes in a different way. As the examples shown in the first row of Fig.~\ref{fig:exp_color_slider_direction}, color of the left eye changes first when we adjust the slider horizontally from left to right. It is possibly because the eyes occupy two separate segments in the images, while the other facial components, such as hair and face skin, only occupy one single segment. Therefore, we can use a vertical slider for more natural control of the eye colors as shown in the second row of Fig.~\ref{fig:exp_color_slider_direction}.
Besides, we also test the color combination methods. As presented in Fig.~\ref{fig:exp_color_edit_layout}, we compare slider and vertical stripe pattern for hair color control. When we adjust the ratio of the two colors to $1:1$, the generated hair colors are very close. Moreover, the results do not demonstrate obvious pattern correlated with the combination method. For example, we do not observe stripe in the generated image when using vertical stripe pattern. Compared with the vertical stripe pattern method, slider is more intuitive and convenient to use for end users.
\begin{figure}[t!]
\centering
{\footnotesize
\makebox[0.12\columnwidth][c]{Original}
\makebox[0.12\columnwidth]{}
\makebox[0.12\columnwidth]{}
\makebox[0.12\columnwidth]{}
\makebox[0.12\columnwidth]{}
\vspace{2pt}
\\
\includegraphics[width=0.12\columnwidth]{figs/color_l_12033_r_real.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_l_12423_r_real.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_l_12138_r_real.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_l_12136_r_real.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_l_12369_r_real.jpg}
\\
\makebox[0.12\columnwidth][c]{Editing}
\makebox[0.12\columnwidth]{}
\makebox[0.12\columnwidth]{}
\makebox[0.12\columnwidth]{}
\makebox[0.12\columnwidth]{}
\vspace{2pt}
\\
\includegraphics[width=0.12\columnwidth]{figs/color_l_12033_r_1.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_l_12423_r_1.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_l_12138_r_1.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_l_12136_r_1.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_l_12369_r_1.jpg}
\\
\includegraphics[width=0.12\columnwidth]{figs/color_l_12033_r_2.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_l_12423_r_2.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_l_12138_r_2.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_l_12136_r_2.jpg}
\includegraphics[width=0.12\columnwidth]{figs/color_l_12369_r_2.jpg}
\\
}
\caption{Blending colors using sliders (top) and vertical stripe patterns (bottom). Both control can yield similar results.
}
\label{fig:exp_color_edit_layout}
\end{figure}
\begin{figure}[t!]
\centering
{\footnotesize
\makebox[0.12\columnwidth]{}
\includegraphics[width=0.12\columnwidth]{figs/dist_10383_r_real.jpg}
\includegraphics[width=0.12\columnwidth]{figs/dist_10383_r_color.jpg}
\includegraphics[width=0.12\columnwidth]{figs/dist_10383_r_fake.jpg}
\\
\includegraphics[width=0.12\columnwidth]{figs/dist_12361_r_real.jpg}
\includegraphics[width=0.12\columnwidth]{figs/dist_12031_r_real.jpg}
\includegraphics[width=0.12\columnwidth]{figs/dist_12031_r_color.jpg}
\includegraphics[width=0.12\columnwidth]{figs/dist_12031_r_fake.jpg}
\\
\makebox[0.12\columnwidth]{}
\includegraphics[width=0.12\columnwidth]{figs/dist_12937_r_real.jpg}
\includegraphics[width=0.12\columnwidth]{figs/dist_12937_r_color.jpg}
\includegraphics[width=0.12\columnwidth]{figs/dist_12937_r_fake.jpg}
\\
\makebox[0.12\columnwidth]{Original}
\makebox[0.12\columnwidth]{Reference}
\makebox[0.12\columnwidth]{Color Palette}
\makebox[0.12\columnwidth]{Result}
\\
}
\caption{Color transfer via modifying the color palette. See the main text for details.}
\label{fig:exp_color_transfer_dist}
\end{figure}
In Fig.~\ref{fig:exp_color_transfer_dist}, we demonstrate that our model can also be trained for color transfer. In stead of computing the average color as described in \S\ref{sec:color_palette}, we use color palette to represent the color distributions of each facial component from the reference images. More specifically, we sort the color pixels extracted from each facial component, and uniformly sample colors to create the new color palette, where each 1-pixel-width column represent one sampled color. Then we use the new color palette as conditional information for color transfer after model training. As demonstrated in Fig.~\ref{fig:exp_color_transfer_dist}, our method ensures color transfer from the same facial component of the reference image (i.e., prevents transfer from background to hair, skin, etc.), while most of the previous transfer based methods do not have such explicit restriction and hence are less controllable.
\begin{figure*}[t!]
\centering
{\footnotesize
\includegraphics[width=0.10\columnwidth]{figs/hl_11237_r_real_c.jpg}
\hspace{2pt}
\includegraphics[width=0.10\columnwidth]{figs/hl_11237_r_fake_c.jpg} \hspace{-5pt}
\includegraphics[width=0.10\columnwidth]{figs/hl_11237_r_fake2_c.jpg} \hspace{-5pt}
\includegraphics[width=0.10\columnwidth]{figs/hl_11237_r_fake3_c.jpg}
\hspace{2pt}
\includegraphics[width=0.10\columnwidth]{figs/sd_11237_r_fake_c.jpg} \hspace{-5pt}
\includegraphics[width=0.10\columnwidth]{figs/sd_11237_r_fake2_c.jpg}
\hspace{2pt}
\includegraphics[width=0.10\columnwidth]{figs/ls_11237_r_fake_c.jpg} \\
\includegraphics[width=0.10\columnwidth]{figs/hl_13953_r_real_c.jpg}
\hspace{2pt}
\includegraphics[width=0.10\columnwidth]{figs/hl_13953_r_fake_c.jpg} \hspace{-5pt}
\includegraphics[width=0.10\columnwidth]{figs/hl_13953_r_fake2_c.jpg} \hspace{-5pt}
\includegraphics[width=0.10\columnwidth]{figs/hl_13953_r_fake3_c.jpg}
\hspace{2pt}
\includegraphics[width=0.10\columnwidth]{figs/sd_13953_r_fake_c.jpg} \hspace{-5pt}
\includegraphics[width=0.10\columnwidth]{figs/sd_13953_r_fake2_c.jpg}
\hspace{2pt}
\includegraphics[width=0.10\columnwidth]{figs/ls_13953_r_fake_c.jpg} \\
\includegraphics[width=0.10\columnwidth]{figs/hl_13443_r_real_c.jpg}
\hspace{2pt}
\includegraphics[width=0.10\columnwidth]{figs/hl_13443_r_fake_c.jpg} \hspace{-5pt}
\includegraphics[width=0.10\columnwidth]{figs/hl_13443_r_fake2_c.jpg} \hspace{-5pt}
\includegraphics[width=0.10\columnwidth]{figs/hl_13443_r_fake3_c.jpg}
\hspace{2pt}
\includegraphics[width=0.10\columnwidth]{figs/sd_13443_r_fake_c.jpg} \hspace{-5pt}
\includegraphics[width=0.10\columnwidth]{figs/sd_13443_r_fake2_c.jpg}
\hspace{2pt}
\includegraphics[width=0.10\columnwidth]{figs/ls_13443_r_fake_c.jpg} \\
\includegraphics[width=0.10\columnwidth]{figs/hl_12573_r_real_c.jpg}
\hspace{2pt}
\includegraphics[width=0.10\columnwidth]{figs/hl_12573_r_fake_c.jpg} \hspace{-5pt}
\includegraphics[width=0.10\columnwidth]{figs/hl_12573_r_fake2_c.jpg} \hspace{-5pt}
\includegraphics[width=0.10\columnwidth]{figs/hl_12573_r_fake3_c.jpg}
\hspace{2pt}
\includegraphics[width=0.10\columnwidth]{figs/sd_12573_r_fake_c.jpg} \hspace{-5pt}
\includegraphics[width=0.10\columnwidth]{figs/sd_12573_r_fake2_c.jpg}
\hspace{2pt}
\includegraphics[width=0.10\columnwidth]{figs/ls_12573_r_fake_c.jpg} \\
\includegraphics[width=0.10\columnwidth]{figs/hl_12193_r_real_c.jpg}
\hspace{2pt}
\includegraphics[width=0.10\columnwidth]{figs/hl_12193_r_fake_c.jpg} \hspace{-5pt}
\includegraphics[width=0.10\columnwidth]{figs/hl_12193_r_fake2_c.jpg} \hspace{-5pt}
\includegraphics[width=0.10\columnwidth]{figs/hl_12193_r_fake3_c.jpg}
\hspace{2pt}
\includegraphics[width=0.10\columnwidth]{figs/sd_12193_r_fake_c.jpg} \hspace{-5pt}
\includegraphics[width=0.10\columnwidth]{figs/sd_12193_r_fake2_c.jpg}
\hspace{2pt}
\includegraphics[width=0.10\columnwidth]{figs/ls_12193_r_fake_c.jpg} \\
\makebox[0.10\columnwidth]{Original}
\hspace{2pt}
\makebox[0.10\columnwidth]{Light \#1} \hspace{-5pt}
\makebox[0.10\columnwidth]{Light \#2} \hspace{-5pt}
\makebox[0.10\columnwidth]{Light \#3}
\hspace{2pt}
\makebox[0.10\columnwidth]{Shadow \#1} \hspace{-5pt}
\makebox[0.10\columnwidth]{Shadow \#2}
\hspace{2pt}
\makebox[0.10\columnwidth]{Light + shadow}\\
}
\caption{Light and shadow editing via binary masks. We show the light and shadow masks in the bottom- and top-right corners, respectively, and highlight the modified masks with red boundaries.}
\label{fig:exp_light}
\end{figure*}
\subsubsection{Lights \& Shadows}
Our model can also be used for fine-grained light and shadow editing through the modification of the binary light and shadow masks. As the examples show in Fig.~\ref{fig:exp_light}, it is convenient to add or remove shadow/light to the desired area with simple mask modification operations. Our model is feed forward, so it can be easily implemented for interactive editing. In Fig.~\ref{fig:exp_light_compare}, we also compare the images generated using our model and a few existing portrait relighting methods. Our method automatically generates shadow and light masks (second-to-last column), which are binary images, and feed them into our neural network model. We observe that our method can yield visually pleasing results comparable to the previous methods, while providing users with a direct way to edit light and shadow.
In our implementation, we apply a simple method to extract light and shadow masks from the user-specified light image (second column in Fig.~\ref{fig:exp_light_compare}) and the original portrait images, and then generate the final masks through the \textit{AND}/\textit{OR} Boolean operations. Note that it is also possible to build 3D face models \citep{Tran_2018_CVPR,Wu_2019_CVPR} from which high-quality light and shadow masks can be generated. We leave it as a future work.
\begin{figure*}[htbp!]
\centering
{\footnotesize
\includegraphics[width=0.12\columnwidth]{figs/ffhq1.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq1_sl.jpg}
\hspace{2pt}
\includegraphics[width=0.12\columnwidth]{figs/ffhq1_a.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq1_b.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq1_c.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq1_d.jpg}
\hspace{2pt}
\includegraphics[width=0.06\columnwidth]{figs/ffhq1_merge.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq1_fake.jpg}
\\
\includegraphics[width=0.12\columnwidth]{figs/ffhq2.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq2_sl.jpg}
\hspace{2pt}
\includegraphics[width=0.12\columnwidth]{figs/ffhq2_a.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq2_b.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq2_c.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq2_d.jpg}
\hspace{2pt}
\includegraphics[width=0.06\columnwidth]{figs/ffhq2_merge.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq2_fake.jpg}
\\
\includegraphics[width=0.12\columnwidth]{figs/ffhq4.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq4_sl.jpg}
\hspace{2pt}
\includegraphics[width=0.12\columnwidth]{figs/ffhq4_a.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq4_b.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq4_c.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq4_d.jpg}
\hspace{2pt}
\includegraphics[width=0.06\columnwidth]{figs/ffhq4_merge.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq4_fake.jpg}
\\
\includegraphics[width=0.12\columnwidth]{figs/ffhq5.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq5_sl.jpg}
\hspace{2pt}
\includegraphics[width=0.12\columnwidth]{figs/ffhq5_a.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq5_b.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq5_c.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq5_d.jpg}
\hspace{2pt}
\includegraphics[width=0.06\columnwidth]{figs/ffhq5_merge.jpg}
\includegraphics[width=0.12\columnwidth]{figs/ffhq5_fake.jpg}
\\
\makebox[0.12\columnwidth]{Original}
\makebox[0.12\columnwidth]{Target}
\hspace{2pt}
\makebox[0.12\columnwidth]{\citet{HouZSBT021}}
\makebox[0.12\columnwidth]{\citet{zhou2019deep}}
\makebox[0.12\columnwidth]{\citet{sun2019single}}
\makebox[0.12\columnwidth]{\citet{sengupta2018sfsnet}}
\hspace{2pt}
\makebox[0.06\columnwidth]{Masks}
\makebox[0.12\columnwidth]{Ours}
\\
}
\caption{Comparison of light and shadow editing with previous methods. The masks used in our method are computed automatically from the user-specified light directions. }
\label{fig:exp_light_compare}
\end{figure*}
\subsubsection{Portrait Artworks}
Though the main objective of our method is for portrait photo editing, we observe that our method can also be applied to various types of portrait artworks. As illustrated in Fig.~\ref{fig:artwork}, our model can improve photo-realism of oil paintings, pencil sketches, sculptures and black-and-white photos. We generate the results in two simple steps: 1) extracting edge maps, colors and shadow/light masks from the original image (\S\ref{sec:model_input_sec}); and 2) editing colors via a palette.
\begin{figure}[t!]
\centering
{\footnotesize
\includegraphics[width=0.192\columnwidth]{figs/g10.jpg}
\includegraphics[width=0.192\columnwidth]{figs/bb.jpg}
\includegraphics[width=0.192\columnwidth]{figs/s2.jpg}
\includegraphics[width=0.192\columnwidth]{figs/d2.jpg}
\includegraphics[width=0.192\columnwidth]{figs/a2.jpg}\\
\vspace{2pt}
\includegraphics[width=0.192\columnwidth]{figs/g10_fake.jpg}
\includegraphics[width=0.192\columnwidth]{figs/bb_fake.jpg}
\includegraphics[width=0.192\columnwidth]{figs/s2_fake.jpg}
\includegraphics[width=0.192\columnwidth]{figs/d2_fake.jpg}
\includegraphics[width=0.192\columnwidth]{figs/a2_fake.jpg}\\
\makebox[0.192\columnwidth]{(a)}
\makebox[0.192\columnwidth]{(b)}
\makebox[0.192\columnwidth]{(c)}
\makebox[0.192\columnwidth]{(d)}
\makebox[0.192\columnwidth]{(e)}\\
}
\caption{Our method can improve photo-realism of various types of portrait artworks. The upper row shows the input images and the bottom row is the generated results. (a) and (b) oil paintings; (c) pencil sketch; (d) sculpture; (e) colorization of black-and-white photo.}
\label{fig:artwork}
\end{figure}
\subsection{Ablation Study}
\subsubsection{Asymmetric Conditional GAN}
\textbf{Fine-grained color control.} We first conduct experiments to quantitatively analyze the performance of the proposed asymmetric conditional GAN for fine-grained color control. We denote by \textbf{AC-GAN} our model trained with the asymmetric conditional GAN, which takes color palette ${\bm{I}}_{CP}$ as the generator input, and average color map ${\bm{I}}_{C}$ as the discriminators input. Denote by \textbf{C-GAN} our model trained with the original conditional GAN, which takes ${\bm{I}}_{CP}$ as both generator and discriminator inputs.
The test split is used for evaluation to compare the images generated by \textbf{AC-GAN} and \textbf{C-GAN}. Instead of the original color palettes, we use the color palette of a randomly selected image ${\bm{P}}_{rand}$ as ${\bm{I}}_{CP}$ to control image colors. Besides, we also use the manually annotated segmentation masks to generate color palette ${\bm{O}}_{CP}$ for each synthesised image, following the steps described in \S\ref{sec:color_palette}. For each facial component, we compute the Euclidean distance between their corresponding average colors in ${\bm{I}}_{CP}$ and ${\bm{O}}_{CP}$. Besides, we also compare color distribution between the corresponding facial components in each synthesised image and ${\bm{P}}_{rand}$. Similar to \citet{afifi2021histogan}, we adopt the average of KL divergence to measure the difference between RGB color histograms in our experiments.
From the average color distances shown in Table~\ref{tb:ablation_color}, we can see that colors of the images generated by \textbf{AC-GAN} are closer than those of \textbf{C-GAN} to the desired ones on the color palette for most of the facial components. Especially for lip, hair and eyes, users are usually more interested in editing their color. The face colors generated by \textbf{C-GAN} is slightly closer to the input for color control, which is probably because of the higher color variance of faces, so the value for face may be less representative. Since KL divergence treat distant and neighbouring columns in the RGB color histograms indifferently, which does not directly measure the color distance, so we use it as a reference to compare color distributions. The results in Table~\ref{tb:ablation_color} show that the color distributions generated by \textbf{AC-GAN} are closer to that of the real images for 3 out of the 5 components.
\begin{table}[ht!]
\centering
\caption{Color control using the original and asymmetric conditional GAN.}
\begin{small}
\begin{tabular}{lccccc}
\toprule
& \textbf{Hair} & \textbf{Skin} & \textbf{Eyes} & \textbf{Lip} & \textbf{BG.} \\
\midrule
\textbf{Avg. Color Dist. } & & & \\
\textbf{C-GAN} & 41.92 & \textbf{41.24} & 50.22 & 50.12 & 42.71 \\
\textbf{AC-GAN} & \textbf{41.20} & 41.72 & \textbf{49.94} & \textbf{47.88} & \textbf{40.44} \\
\midrule
\textbf{Color Hist. Dist.} & & & \\
\textbf{C-GAN} & 1.1103 & \textbf{0.7294} & 0.5773 & \textbf{0.8494} & 2.3353 \\
\textbf{AC-GAN} & \textbf{1.0933} & 0.7309 & \textbf{0.5763} & 0.8711 & \textbf{2.3039} \\
\bottomrule
\end{tabular}
\end{small}
\label{tb:ablation_color}
\end{table}
\noindent\textbf{Robustness to noisy edge maps.} The asymmetric conditional GAN is also possibly useful to improve the model robustness to noisy edge maps. To verify this, we apply the proposed edge map noising methods $n(\cdot)$ to the edge maps in the training split, the use the asymmetric and original conditional GAN to train the model. Similar to the experiments above, the models are denoted with \textbf{AC-GAN} and \textbf{C-GAN}, respective. \textbf{AC-GAN} takes $n({\bm{I}}_{E})$ as generator input and ${\bm{I}}_{E}$ as discriminators input, while \textbf{C-GAN} takes $n({\bm{I}}_{E})$ as both generator and discriminators input. In addition, we also compare with the model trained with the original edge maps ${\bm{I}}_{E}$ only, i.e., without applying the proposed edge map noising methods, which is denoted with \textbf{O-Edge}.
For the test split, we apply the different noising methods $n(\cdot)$ to the edge maps first, and then feed the same set of noisy edge maps into the models during inference. The average of the Structural Similarity Index (SSIM) \citep{wang2004image} between the original image and the synthesised images are computed to compare their robustness to the noises.
As shown in Table~\ref{tb:ablation_edge}, \textbf{AC-GAN} consistently outperforms \textbf{O-Edge} and \textbf{C-GAN} in all evaluation scenarios, which effectively helps improve model robustness to the common edge map noises introduced by hand editing and edge map extraction models, including random remove, random shift and random lines. This again demonstrates the effectiveness of our proposed method in various editing tasks.
\begin{table}[ht!]
\centering
\caption{Impact of the noisy edge maps on the model trained with the conditional GAN and the proposed asymmetric conditional GAN. We use SSIM to measure the similarity between real and fake images. \textbf{O}, \textbf{RR}, \textbf{RS} and \textbf{RL} represent original edge map, random removal, random shift and random lines, respectively.}
\begin{tabular}{lcccc}
\toprule
& \textbf{O} & \textbf{RR} & \textbf{RS} & \textbf{RL} \\
\midrule
\textbf{O-Edge} & 0.5930 & 0.5751 & 0.5895 & 0.5875 \\
\textbf{C-GAN} & 0.5974 & 0.5857 & 0.5939 & 0.5939 \\
\textbf{AC-GAN} & \textbf{0.6006} & \textbf{0.5896} & \textbf{0.5973} & \textbf{0.5974} \\
\bottomrule
\end{tabular}
\label{tb:ablation_edge}
\end{table}
\subsubsection{Region-Weighted Discriminators}
We also compare the quality of the generated images with and without the region-weighted discriminator $D$ introduced in \S\ref{sec:discriminators}. We use \textbf{MD} to denote the model trained with the original multi-scale discriminator ($D_G$ only), and use \textbf{RW-MD} to denote the proposed model that use region-weighted multi-scale discriminators ($D_G$ and $D_L$). Same as above, the training set is used to train \textbf{MD} and \textbf{RW-MD}, and the test set is used for evaluation.
The results are shown in Figure~\ref{fig:rw_md}. Since the face and eye masks are used to give these regions higher weight when training \textbf{RW-MD}, we use the face parsing model trained by \citep{DBLPLee0W020} to annotate the fake images by identifying the corresponding segments and classifying their labels, which are then compared with the manual annotations. The average of semantic segmentation F1 is used to quantify the reconstruction quality. From the results we can see, the average F1 scores of \textbf{RW-MD} are higher than \textbf{MD} for both face and eyes, where the improvements of eyes are more obvious.
\begin{figure}[htbp!]
\centering
{\footnotesize
\includegraphics[width=0.45\columnwidth]{figs/rw_md.png}
}
\caption{Semantic segmentation F1 of the edited images generated with MD and RW-MD.}
\label{fig:rw_md}
\end{figure}
\begin{figure}[htbp!]
\centering
{\footnotesize
\makebox[0.12\columnwidth][c]{Original}
\makebox[0.12\columnwidth]{}
\makebox[0.12\columnwidth]{}
\makebox[0.12\columnwidth]{}
\makebox[0.12\columnwidth]{}
\vspace{2pt}
\\
\includegraphics[width=0.12\columnwidth]{figs/fail_12310_r_real.jpg}
\includegraphics[width=0.12\columnwidth]{figs/fail_12903_r_real.jpg}
\includegraphics[width=0.12\columnwidth]{figs/fail_12830_r_real.jpg}
\includegraphics[width=0.12\columnwidth]{figs/fail_10323_r_real.jpg}
\makebox[0.12\columnwidth]{}
\\
\makebox[0.12\columnwidth][c]{Editing}
\makebox[0.12\columnwidth]{}
\makebox[0.12\columnwidth]{}
\makebox[0.12\columnwidth]{}
\makebox[0.12\columnwidth]{}
\vspace{2pt}
\\
\includegraphics[width=0.12\columnwidth]{figs/fail_12310_r_fake.jpg}
\includegraphics[width=0.12\columnwidth]{figs/fail_12903_r_fake.jpg}
\includegraphics[width=0.12\columnwidth]{figs/fail_12830_r_fake.jpg}
\includegraphics[width=0.12\columnwidth]{figs/fail_10323_r_fake.jpg}
\includegraphics[width=0.12\columnwidth]{figs/fail_10323_r_fake_u.jpg}
\\
\makebox[0.12\columnwidth]{(a)}
\makebox[0.12\columnwidth]{(b)}
\makebox[0.12\columnwidth]{(c)}
\makebox[0.12\columnwidth]{(d)}
\makebox[0.12\columnwidth]{(e)}
}
\caption{Limitations. (a) Some fine details, such as freckles, are missing; (b)-(c) Background are changed. (d) missing jewelry; (e) generated image after erasing the jewelry in edge maps for (d), i.e., geometry editing.}
\label{fig:limitations}
\end{figure}
\section{Discussions \& Future Work}
We presented a new method for portrait image editing. Using a single neural network model, our method allows the user to edit colors, lights, shadows as well as geometries in an easy and intuitive manner. Our method can also generate visually pleasing images from hand-drawn sketches.
While we have demonstrated high quality editing results, our approach is still subject to a few limitations that can be addressed in follow-up work.
First, we observe our neural network model often removes small facial features, such as freckles in the results (Fig~\ref{fig:limitations}(a)), since our edge map extraction method usually ignores them. One possible solution is to provide an additional mask and annotate training data to model those small features. Second, our model focuses on foreground objects, thereby has difficulty in colorizing background patterns (Fig~\ref{fig:limitations}(b),(c)). This is because of the large variety in background patterns, which are more difficult for the model to learn. Inspired by \citet{yang2020deep}, one can add foreground masks so that the model generates portraits only, while keeping the original background unchanged. Third, our model cannot generate jewelry and headgear
that are not available in the training data. For example, the jewelry in Fig~\ref{fig:limitations}(d) is missing in the generated image. This problem can be solved by providing more relevant training data. Fourth, with the help of color palette, our method enables fine-grained color editing in batches. However, when applying our model to videos, we observe that the generated results often have flickering artifacts due to lack of consideration of temporal coherence. It is interesting to extend our method for fine-grained video color editing with video generation models.
Though we focus on portrait image editing in the paper, we believe the proposed asymmetric conditional GAN (AC-GAN) is a general framework that can be applied to other controllable generation tasks. For example, AC-GAN can deal with the situations when the controlling signal is over simplified or abstracted for model to learn (e.g., auto-encoder encoded representation of edges), or when there is too few training data. We will explore along this direction in the near future.
|
1,108,101,563,299 | arxiv | \section{Introduction.}
Discrete differential geometry studies discrete equivalents of the geometric notions and methods of classical differential
geometry, such as notions of curvature and integrability for polyhedral surfaces. In this connection, discrete surfaces have been studied one after another with
strong ties to mathematics physics and great potential for computer analysis, architecture, numerics. Progress in this field is to a
large extent stimulated by its relevance for computer graphics and mathematical physics\cite{Bobenko-2,Bobenko-4}.
\par Recently, the expansion of computer graphics and applications in mathematical physics have given a great impulse to the issue of giving discrete equivalents of affine differential geometric objects\cite{Bobenko-1,Craizer-1,Craizer-2}. In \cite{Bobenko-3} a consistent definition of discrete affine spheres is proposed, both for definite and indefinite metrics, and in \cite{Matsuura} a similar construction is done in the context of improper affine spheres.
\par Following the ideas of Klein, presented in his famous lecture at Erlangen, several geometers in the early 20th century proposed the study
of curves and surfaces with respect to different transformation groups. In geometry, an affine transformation, affine map or an affinity is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence.
\par A centroaffine transformation is nothing but a general linear transformation $\R^n\ni x\mapsto Ax\in\R^n$, where $A\in GL(n,\R)$.
In 1907 Tzitz$\mathrm{\acute{e}}$ica found that for a surface in Euclidean 3-space the property that the ratio of the Gauss curvature to the fourth power of
the distance of the tangent plane from the origin is constant is invariant under a centroaffine transformation. The surfaces with this property
turn out to be what are now called Tzitz$\mathrm{\acute{e}}$ica surfaces, or proper affine spheres with center at the origin. In centroaffine differential geometry, the theory of hypersurfaces has a long history. The notion of centroaffine minimal
hypersurfaces was introduced by Wang \cite{Wang} as extremals for the area integral of the centroaffine metric. See also \cite{Y-Y-L,Yu-Y-L} for the classification results about centroaffine translation surfaces and centroaffine ruled surfaces in $\R^3$.
\par Smooth geometric objects and their transformations should belong to the same geometry. In particular discretizations should be invariant with respect to the same transformation group as the smooth objects are(projective, affine, m\"obius etc). This paper is concerned with some invariant properties of the discrete centroaffine indefinite surface, which is organized as follows: Basic concepts of classical centroaffine differential geometry are presented in Section 2.
In Section 3 we define the discrete centroaffine indefinite surface, and then obtain the structure equations, compatibility conditions and some centroaffine invariants.
In section 4, the Laplacian operator is defined as the gradient of the Dirichlet energy for a discrete centroaffine indefinite surface. The discrete centroaffine indefinite surface with constant coefficients is considered in section 5. Section 6 deals with the convexity of a discrete centroaffine indefinite surface.
\section{Centroaffine hypersurfaces.}
Prior to the introduction of a discrete centroaffine indifinite surfaces theory, in this section, we recall some fundamental notions for centroaffine
hypersurfaces in $\R^{n+1}$. For details we refer to
\cite{L-S-W}, \cite{Liu-W-1}, \cite{S-S-V} or \cite{Wang}. Let $x: \M \rightarrow
\R^{n+1}$ be a hypersurface immersion and $[\dots]$ the standard
determinant in $\R^{n+1}$. $x$ is said to be a centroaffine
hypersurface if the position vector of $x$, denoted also by $x$, is
always transversal to the tangent space $ x_*({\mathbf{T}\M})$ at
each point of $\M$ in $\R^{n+1}$. We define a symmetric bilinear
form $G$ on ${\mathbf T}\M$ by
\begin{equation}\label{metric}
G=-\sum_{i,j=1}^n{\frac{[e_1(x),e_2(x),\dots,e_n(x),e_ie_j(x)]}{[e_1(x),e_2(x),\dots,e_n(x),x]}}\
\theta ^{i}\otimes \theta ^j,
\end{equation}
where $\{ e_1,e_2,\dots,e_n \}$ is a local basis of $\mathbf T\M$ with
the dual basis $\{\theta^1,\theta^2,\dots,\theta^n\}$. Note that $G$ is
globally defined. A centroaffine hypersurface $x$ is said to be non-degenerate if $G$ is non-degenerate.
We call $G$ the centroaffine metric of $x$.
We say that a hypersurface is definite (or indefinite) if $G$ is definite (or indefinite).
\begin{rem}
Geometrically, a hypersurface $x$ with positive (resp. negative) definite centroaffine metric $G$ is the
locally strongly convex hypersurface in
$\R^{n+1}$ and such hypersurface is called hyperbolic type (respectively, elliptic type) in \cite{L-L-S}.
In particular, $G$ is definite if $x$ is locally
strongly convex in $\R^{n+1}$.
\end{rem}
\par
Let $x: \M \rightarrow \R^{n+1}$ be a non-degenerate centroaffine
surface. Then $x$ induces a centroaffinely invariant metric $G$ and
a so-called induced connection $\nabla$. The difference of the
Levi-Civita connection $\widehat{\nabla}$ of $G$ and the induced
connection $\nabla$ is a $(1,2)-$tensor $C$ on $M$ with the property
that its associate cubic form $\widehat{C}$, defined by
\begin{equation}
\widehat{C}(u,v,w)=G(C(u,v),w), u,v,w\in TM,
\end{equation}
which is totally symmetric. The so-called Tchebychev form is defined by
\begin{equation}\label{Tch-form}
\widehat{T}=\frac{1}{n}\mathrm{trace}_{G}(\widehat{C}).
\end{equation}
Let $T$ be the Tchebychev vector field on $M$ defined by the equation
\begin{equation}\label{Tch-vector}
G(T,v)=\widehat{T}(v), v\in TM.
\end{equation}
It is proved by Wang in \cite{Wang} that a centroaffine surface $x:\M^n \rightarrow \R^{n+1}$ is called centroaffine minimal if $\mathrm{trace}_{G}(\widehat{\nabla}T)=0$,
and the centroaffine mean curvature is defined by
\begin{equation}\label{H-def}
H=\frac{1}{n}\mathrm{trace}_{G}(\widehat{\nabla}T).
\end{equation}
The Gauss equation of $x$ can be written as(in the following, we use
the Einstein summation convention and the range of indices is $1\leq
i,j,k,\dots\leq n$)
\begin{equation}\label{gauss-eqn}
\frac{\pab^2x}{\pab x^i\pab x^j}=\Gamma_{ij}^ke_k(x)-G_{ij}x.
\end{equation}
\par Then the Riemannian curvature tensor is given by
\begin{equation}
\widehat{R}^{l}_{ijk}=\frac{\pab\widehat{\Gamma}^{l}_{ij}}{\pab u^k}-\frac{\pab\widehat{\Gamma}^{l}_{ik}}{\pab u^j}+\widehat{\Gamma}^{p}_{ij}\widehat{\Gamma}^{l}_{pk}
-\widehat{\Gamma}^{p}_{ik}\widehat{\Gamma}^{l}_{pj} ,
\end{equation}
and
\begin{equation}
\widehat{R}_{mijk}=G_{ml}\widehat{R}^{l}_{ijk},
\end{equation}
where $\widehat{\Gamma}^{k}_{ij}$ is the Levi-Civita connection of $G$.
\par If $n=2$, the Gauss curvature of $x$ is defined by
\begin{equation}\label{gauss-cuv-def}
\kappa=\frac{-\widehat{R}_{1212}}{\det(G_{ij})}.
\end{equation}
\par Let $x:M\rightarrow \mathbb{R}^3$ be an centroaffine indefinite surface. We introduce local asymptotic coordinates $(u, v)$ of $G$ such that
\begin{equation}
G=h(\rd u\otimes\rd v+\rd v\otimes\rd u)
\end{equation}
for some local function $h>0$. Using appropriate functions $\lambda,\mu,\varphi,\psi$ we define $1$-forms
\begin{equation}
\Lambda:=\lambda\rd u:=\frac{[x_{uu},x_u,x]}{[x_u,x_v,x]}\rd u,\quad \mathcal{M}:=\mu\rd v=\frac{[x_{uv},x_v,x]}{[x_u,x_v,x]}\rd v
\end{equation}
and cubic forms
\begin{equation}
\Phi:=\varphi\rd u^3:=h\frac{[x_u,x_{uu},x]}{[x_u,x_v,x]}\rd u^3,\quad \Psi:=\psi\rd v^3=-h\frac{[x_{u},x_{vv},x]}{[x_u,x_v,x]}\rd v^3.
\end{equation}
Then we have the following structure equations
\begin{eqnarray}
x_{uu} &=& (\frac{h_u}{h}+\lambda)x_u+\frac{\varphi}{h}x_v, \\
x_{uv} &=& \mu x_u+\lambda x_v-hx, \\
x_{vv} &=& \frac{\psi}{h}x_u+(\frac{h_v}{h}+\mu)x_v
\end{eqnarray}
and the integrability conditions
\begin{eqnarray}
&(\ln h)_{uv}+\frac{\varphi\psi}{h^2}-\lambda\mu+h = 0 \\
&\varphi_v=h(\lambda_u-\frac{h_u}{h}\lambda),\quad \psi_u=h(\mu_v-\frac{h_v}{h}\mu), \\
&\lambda_v=\mu_u.
\end{eqnarray}
We will have
\begin{eqnarray}
&C^1_{11}=\frac{h_u}{2}+\lambda,\quad C^2_{11}=\frac{\varphi}{h},\\
&C^1_{12}=\mu,\quad C^2_{12}=\lambda,\\
&C^1_{22}=\frac{\psi}{h},\quad C^2_{22}=\frac{h_v}{h}+\mu,\\
&T^1=\frac{\mu}{h},\quad T^2=\frac{\lambda}{h}.
\end{eqnarray}
Let $\ll,\gg$ be the inner product of the forms on $M$ induced by the centroaffine metric $G$. Then, by the definition of the centroaffine metric, the Tchebychev form, and the forms $\Lambda, \mathcal{M}, \Phi$ and $\Psi$, we have
\begin{eqnarray}
&\ll \Lambda, \mathcal{M}\gg=\frac{\lambda\mu}{h} =\frac{1}{2}||T||^2,\\
&\ll \Phi, \Psi\gg=\frac{\varphi\psi}{h^3}=J-\frac{3}{2}||T||^2=\tilde{J}, \\
& \kappa=-\frac{(\ln h)_{uv}}{h},\\
& \kappa-1=\ll \Phi, \Psi\gg-\ll \Lambda, \mathcal{M}\gg,
\end{eqnarray}
where, $J$ is the Pick invariant and $\kappa$ the Gauss curvature of $G$. There are the
following propositions:
\begin{prop}(i) $\Lambda=\mathcal{M}=0$ if and only if x is a proper equiaffine sphere centered at
origin $O\in\mathbb{R}^3$; (ii) $\Phi=\Psi=0$ if and only if x is a quadric (\cite{Liu-W-1}).
\end{prop}
\begin{prop}
(i) An centroaffine indefinite surface $x$ in $\R^3$ is a centroaffine extremal
(minimal) surface if and only if $\Lambda$ and $\mathcal{M}$ satisfy $ \lambda_u= \mu_v = 0$; (ii) an indefinite centroaffine
surface $x$ in $\R^3$ is a centroaffine Tchebychev surface if and only if $\Phi$ and $\Psi$ satisfy $\varphi_v=\psi_u=0 $(\cite{Liu-W-1}).
\end{prop}
\section{Discrete centroaffine indefinite immersions.}
Here, we define discrete analogues of centroaffine immersions in a purely geometric manner. These constitute particular `discrete surface' which are maps
\begin{equation}\label{dis-map}
\vec{r}:\mathbb{Z}^2\rightarrow \mathbb{R}^3,\qquad (n_1,n_2)\mapsto \vec{r}(n_1,n_2).
\end{equation}
In the following, we suppress the arguments of functions of $n_1$ and $n_2$ and denote increments of the discrete variables by subscripts, for example,
$$\vec{r}=\vec{r}(n_1,n_2),\vec{r}_1=\vec{r}(n_1+1,n_2),\vec{r}_2=\vec{r}(n_1,n_2+1).$$
Moreover, decrements are indicated by overbars, that is,
$$\vec{r}_{\bar{1}}=\vec{r}(n_1-1,n_2),\vec{r}_{\bar{2}}=\vec{r}(n_1,n_2-1).$$
The following notation for difference operators is adopted:
$$\Delta_i\vec{r}=\vec{r}_i-\vec{r},\quad \Delta_{12}=\vec{r}_{12}-\vec{r}_1-\vec{r}_2+\vec{r}.$$
Now we will give a definition for the discrete centroaffine indefinite surface. Especially, in the following, the point $\vec{r}(n_1,n_2)$ indicates the terminal point of the vector $\vec{r}(n_1,n_2)$ with its starting point at the origin $O$.
\begin{defn}\label{dics}(Discrete centroaffine indefinite surface) A two-dimensional lattice (net) in three-dimensional affine space
\begin{equation}
\vec{r}:\mathbb{Z}^2\rightarrow \mathbb{R}^3
\end{equation}
is called a discrete centroaffine indefinite surface if it has the following properties:
\begin{enumerate}
\item[(a)] Any point $\vec{r}(n_1,n_2)$ and its neighbours $\vec{r}_{\bar{1}},\vec{r}_{\bar{2}},\vec{r}_1,\vec{r}_2$ lie on a plane $\pi$.
\item[(b)] The origin $O$ is not in the plane $\pi$.
\item[(c)] The three points $\vec{r}(n_1,n_2), \vec{r}(n_1+\epsilon_1,n_2), \vec{r}(n_1,n_2+\epsilon_2)$ are nonlinear, where $\epsilon_1,\epsilon_2\in\{1,-1\}$.
\end{enumerate}
\end{defn}
In analytical terms, Condition (a) can be translated into
\begin{equation}
\det[\vec{r}_1-\vec{r},\vec{r}_2-\vec{r},\vec{r}-\vec{r}_{\bar{1}}]=0,\quad \det[\vec{r}_1-\vec{r},\vec{r}_2-\vec{r},\vec{r}-\vec{r}_{\bar{2}}]=0.
\end{equation}
Condition (b) and Condition (c) imply
\begin{equation}\label{ind-vec}
\det[r_1,r_2,r]\neq0,\quad \det[r_{\bar{1}},r_{\bar{2}},r]\neq0,\quad \det[r_1,r_{\bar{2}},r]\neq0,\quad \det[r_{\bar{1}},r_2,r]\neq0.
\end{equation}
\par So that the position vector of the discrete surface considered here obeys the discrete `Gauss equation'
\begin{eqnarray}
\vec{r}_{11}-\vec{r}_1 &=& \alpha(\vec{r}_1-\vec{r})+\beta(\vec{r}_{12}-\vec{r}_1), \label{Stru-c-1}\\
\vec{r}_{12} &=& a\vec{r}+b(\vec{r}_1-\vec{r})+c(\vec{r}_2-\vec{r}), \label{Stru-c-2}\\
\vec{r}_{22}-\vec{r}_2 &=& \gamma(\vec{r}_2-\vec{r})+\delta(\vec{r}_{12}-\vec{r}_2),\label{Stru-c-3}
\end{eqnarray}
where $\alpha,\beta,a,b,c,\gamma$ and $\delta$ are discrete functions from $\mathbb{Z}^2$ to $\mathbb{R}$.
\par The following proposition describes some centroaffine invariants included in the above structure equations. Exactly, all coefficients in Eqs. (\ref{Stru-c-1})-(\ref{Stru-c-3}) are centroaffine invariant.
\begin{prop}\label{Equ-pro}If two discrete centroaffine indefinite surface are centroaffinely equivalent, they have same discrete functions $a,b,c,\alpha,\beta,\gamma, \delta$.
\end{prop}
{\bf Proof.}
If two discrete centroaffine indefinite surface $\vec{r}(i,j)$ and $\vec{\bar{r}}(i,j)$ are centroaffinely equivalent, there exists a non-degenerate matrix $P$ satisfying that
$\vec{r}(i,j)=P\vec{\bar{r}}(i,j).$ According to Eqs. (\ref{Stru-c-1})-(\ref{Stru-c-3}) and (\ref{ind-vec}), we obtain
\begin{gather}
\alpha=\frac{\det[\vec{r}_1,\vec{r}_{11},\vec{r}_{12}]}{\det[\vec{r},\vec{r}_1,\vec{r}_{12}]}, \qquad
\beta=\frac{\det[\vec{r},\vec{r}_{1},\vec{r}_{11}]}{\det[\vec{r},\vec{r}_1,\vec{r}_{12}]}, \\
\gamma=\frac{\det[\vec{r}_2,\vec{r}_{22},\vec{r}_{12}]}{\det[\vec{r},\vec{r}_2,\vec{r}_{12}]}, \qquad
\delta=\frac{\det[\vec{r},\vec{r}_{2},\vec{r}_{22}]}{\det[\vec{r},\vec{r}_2,\vec{r}_{12}]}, \\
a=\frac{\det[\vec{r}_{12},\vec{r}_{1},\vec{r}_{2}]}{\det[\vec{r},\vec{r}_1,\vec{r}_{2}]},\quad
b=\frac{\det[\vec{r},\vec{r}_{12},\vec{r}_{2}]}{\det[\vec{r},\vec{r}_1,\vec{r}_{2}]}, \quad
c =\frac{\det[\vec{r},\vec{r}_{1},\vec{r}_{12}]}{\det[\vec{r},\vec{r}_1,\vec{r}_{2}]}.
\end{gather}
It can be easily seen that $$\bar{\alpha}=\alpha, \bar{\beta}=\beta, \bar{\gamma}=\gamma,\bar{\delta}=\delta, \bar{a}=a,\bar{b}=b,\bar{c}=c,$$
which show the discrete functions $a,b,c,\alpha,\beta,\gamma, \delta$ are invariant under centroaffine transformation.
\\ \rightline{$\Box$}
By Definition \ref{dics}, there are some limitation to the coefficients in Eqs. (\ref{Stru-c-1})-(\ref{Stru-c-3}).
\begin{rem}
\begin{itemize}
\item[(1)] From Eq. (\ref{ind-vec}), that is, Condition (b) and Condition (c), we can derive that $a-b-c\neq0,\quad b\neq0,\quad c\neq0.$
\item[(2)] $a=1$ means $\vec{r}_{12}-\vec{r}, \vec{r}_2-\vec{r}, \vec{r}_1-\vec{r}$ are coplanar, which implies the discrete centroaffine surface is a plane locally.
\end{itemize}
\end{rem}
Thus, in this paper we assume
\begin{equation}\label{assume}
a-b-c\neq0,\quad bc\neq0,\quad a\neq1.
\end{equation}
Hence, the compatibility conditions of Eqs. (\ref{Stru-c-1})-(\ref{Stru-c-3}) yield
\begin{eqnarray}
\alpha &=& \frac{(1-a_1)(a-b-c)}{(a-1)b_1}, \label{Dis-Con-F} \\
\gamma &=& \frac{(1-a_2)(a-b-c)}{(a-1)c_2}, \label{Dis-Con-S} \\
\frac{c}{b} &=& \frac{c_{12}(a_2-b_2-c_2)}{b_{12}(a_1-b_1-c_1)},\\
\frac{1-a_{12}}{b_{12}}&=& \frac{(a_2-1)(a_1-1)c}{(a-1)(a_2-b_2-c_2)}(1-K),
\end{eqnarray}
and
\begin{eqnarray}
(a_2-1)b\beta_2+(a-1)(a_1-c_1) &=& (a-1)b_1\beta+(a_1-1)(a-c), \\
(a_1-1)c\delta_1+(a-1)(a_2-b_2) &=& (a-1)c_2\delta+(a_2-1)(a-b), \label{Dis-Con-L}
\end{eqnarray}
where $ K=\frac{[(a-1)b_1\beta+(a_1-c_1)(1-a)-(1-a_1)(a-c)][c_2\delta(a-1)+(a_2-1)(a-b)-(a-1)(a_2-b_2)]}{bc(a_2-1)(a_1-1)}.$
\begin{rem} In view of $a-b-c\neq0, a\neq1,$ Eqs. (\ref{Dis-Con-F}) and (\ref{Dis-Con-S}), it is obvious that
\begin{equation}
\alpha\neq 0,\gamma\neq 0.
\end{equation}
\end{rem}
\par Integrable discrete versions of indefinite affine spheres have been constructed in \cite{Bobenko-3,Bo-Pin-99} by the following equations, which is shown that
the underlying discrete Gauss-Codazzi equations reduce to an integrable discrete Tzitzeica system.
\begin{thm}(The discrete Tzitzeica system). Discrete affine sphere are governed by the discrete Gauss equations
\begin{eqnarray}
\vec{r}_{11}-\vec{r}_1 &=& \frac{H_1-1}{H_1(H-1)}(\vec{r}_1-r)+\frac{A}{H-1}(\vec{r}_{12}-\vec{r}_1),\label{AS-Con-F} \\
\vec{r}_{12}+\vec{r} &=& H(\vec{r}_1+\vec{r}_2), \label{AS-Con-M}\\
\vec{r}_{22}-\vec{r}_2 &=& \frac{H_2-1}{H_2(H-1)}(\vec{r}_2-\vec{r})+\frac{B}{H-1}(\vec{r}_{12}-\vec{r}_2). \label{AS-Con-L}
\end{eqnarray}
They are compatible modulo
\begin{gather}
A_2=\frac{H_1}{H}A,\quad B_1=\frac{H_2}{H}B,\\
H_{12}=\frac{H(H-1)}{H^2(H_1+H_2-H_1H_2)-H+ABH_1H_2},
\end{gather}
which is termed the discrete Tzitzeica system\cite{Bo-Pin-99}.
\end{thm}
By comparing Eqs. (\ref{Stru-c-1})-(\ref{Stru-c-3}) and Eqs. (\ref{AS-Con-F})-(\ref{AS-Con-L}), it is easy to see
\begin{prop}\label{BeqC} For a discrete centroaffine indefinite surface, if the coefficients satisfy that $b=c, a-b-c=-1$, it is a discrete affine sphere.
\end{prop}
Using Eqs. (\ref{Stru-c-1})-(\ref{Stru-c-3}), we get the following chain rule
\begin{align}
\left[\vec{r}(m+1,n),\vec{r}(m+2,n),\vec{r}(m+1,n+1)\right]&\nonumber\\
=&\left[\vec{r}(m,n),\vec{r}(m+1,n),\vec{r}(m,n+1)\right]A(m,n), \label{Tran-1}\\
\left[\vec{r}(m,n+1),\vec{r}(m+1,n+1),\vec{r}(m,n+2)\right]& \nonumber\\
=& \left[\vec{r}(m,n),\vec{r}(m+1,n),\vec{r}(m,n+1)\right]B(m,n)\label{Tran-2},
\end{align}
where
\begin{eqnarray}
A(m,n) &=& \left(
\begin{array}{ccc}
0 & \beta_{mn}(a_{mn}-b_{mn}-c_{mn})-\alpha_{mn} & a_{mn}-b_{mn}-c_{mn} \\
1 & 1+\alpha_{mn}+b_{mn}\beta_{mn}-\beta_{mn} & b_{mn} \\
0 & c_{mn}\beta_{mn} & c_{mn} \\
\end{array}
\right),
\\
B(m,n) &=& \left(
\begin{array}{ccc}
0 & a_{mn}-b_{mn}-c_{mn} & \delta_{mn}(a_{mn}-b_{mn}-c_{mn})-\gamma_{mn}\\
0 & b_{mn} & b_{mn}\delta_{mn}\\
1 & c_{mn} & 1+\gamma_{mn}+c_{mn}\delta_{mn}-\delta_{mn}\\
\end{array}
\right).
\end{eqnarray}
By a direct computation, we have
\begin{equation}\label{detA-B}
|A|=c_{mn}\alpha_{mn},|B|=b_{mn}\gamma_{mn},
\end{equation}
where $|A|$ indicates the determinant of $A$.
\par Eqs. (\ref{Tran-1}) and (\ref{Tran-2}) give
\begin{align*}
\left[\vec{r}(m+1,n+1),\vec{r}(m+2,n+1),\vec{r}(m+1,n+2)\right]& \\
=\left[\vec{r}(m,n),\vec{r}(m+1,n),\vec{r}(m,n+1)\right]&A(m,n)B(m+1,n) \\
=\left[\vec{r}(m,n),\vec{r}(m+1,n),\vec{r}(m,n+1)\right]&B(m,n)A(m,n+1),
\end{align*}
which leads to
\begin{equation}\label{com-con}
A(m,n)B(m+1,n) = B(m,n)A(m,n+1).
\end{equation}
\par Obviously, the result of this chain rule is accordance with Eqs. (\ref{Stru-c-1})-(\ref{Stru-c-3}), so they have same compatibility conditions. Thus, we conclude
\begin{prop}The equation $$ A(m,n)B(m+1,n) = B(m,n)A(m,n+1)$$ is equivalent to the compatibility conditions (\ref{Dis-Con-F})-(\ref{Dis-Con-L}).
\end{prop}
Given the initial three points $\vec{r}(0,0),\vec{r}(1,0),\vec{r}(0,1)$ with the discrete functions $a,b,c,\alpha,\beta,\gamma, \delta,$ all point groups in the discrete centroaffine surface can be generated according to the chain rule shown in Figure \ref{fig-gpoint} and Figure \ref{fig-gpoint1}. Especially, under a centroaffine transformation, the discrete functions $a,b,c,\alpha,\beta,\gamma, \delta,$ are invariant. On the other hand, there exists a centroaffine transformation which changes $\vec{r}(0,0),\vec{r}(1,0),\vec{r}(0,1)$ to $(1,0,0)^{\mathrm{Tran}}, (0,1,0)^{\mathrm{Tran}},(0,0,1)^{\mathrm{Tran}}.$ Hence we can always choose $$\vec{r}(0,0)=(1,0,0)^{\mathrm{Tran}}, \vec{r}(1,0)=(0,1,0)^{\mathrm{Tran}},\vec{r}(0,1)=(0,0,1)^{\mathrm{Tran}}.$$
\begin{figure}[hbtp]
\centering
\includegraphics[width=.6\textwidth]{gpoint.eps}
\caption{The chain graph of the points in proper order. }
\label{fig-gpoint}
\end{figure}
\begin{figure}[hbtp]
\centering
\includegraphics[width=.8\textwidth]{gpoint1.eps}
\caption{The chain graph of the points in reverse order. }
\label{fig-gpoint1}
\end{figure}
\par Now, we can directly obtain the point group $\vec{r}(m,n),\vec{r}(m+1,n),\vec{r}(m,n+1)$ by the transition matrices, and it is similar for the point group $\vec{r}(-m,-n),\vec{r}(-m+1,-n),\vec{r}(-m,-n+1)$. By Figure \ref{fig-gpoint} and Figure \ref{fig-gpoint1},
\begin{align}
\left[\vec{r}(m,n),\vec{r}(m+1,n),\vec{r}(m,n+1)\right]& \nonumber\\
=\left[\vec{r}(0,0),\vec{r}(1,0),\vec{r}(0,1)\right]A(0,0)&A(1,0)\cdots A(m-1,0)B(m,0)B(m,1)\cdots B(m,n-1)\label{Transp}
\end{align}
and
\begin{align}
\left[\vec{r}(-m,-n),\vec{r}(-m+1,-n),\vec{r}(-m,-n+1)\right]& \nonumber\\
=\left[\vec{r}(0,0),\vec{r}(1,0),\vec{r}(0,1)\right]A^{-1}(-1,0)\cdots &A^{-1}(-m,0)B^{-1}(-m,-1)\cdots B^{-1}(-m,-n),\label{Transr}
\end{align}
where $m>0,n>0$ and $m,n\in\mathbb{Z}$.
\par Under a centroaffine transformation, the points $\vec{r}(0,0), \vec{r}(1,0),\vec{r}(0,1)$ can be changed to
$$\vec{r}(0,0)=(1,0,0)^{\mathrm{Tran}}, \vec{r}(1,0)=(0,1,0)^{\mathrm{Tran}},\vec{r}(0,1)=(0,0,1)^{\mathrm{Tran}}.$$
and the discrete functions $a,b,c,\alpha,\beta,\gamma, \delta$ are invariant. Thus the following proposition is obvious.
\begin{prop}\label{Exi-pro}Given the discrete function $a,b,c,\alpha,\beta,\gamma, \delta$ satisfying the relations (\ref{Dis-Con-F})-(\ref{Dis-Con-L}), there exists an unique discrete centroaffine indefinite surface under a centroaffine transformation.
\end{prop}
Then by Proposition \ref{Equ-pro} and Proposition \ref{Exi-pro}, we obtain
\begin{cor}\label{Lem-same}If two discrete centroaffine indefinite surface $\vec{r}(i,j)$ and $\vec{\bar{r}}(i,j)$ have same coefficients, that is, $$\bar{\alpha}(i,j)=\alpha(i,j), \bar{\beta}(i,j)=\beta(i,j), \bar{\gamma}(i,j)=\gamma(i,j),\bar{\delta}(i,j)=\delta(i,j),$$ $$\bar{a}(i,j)=a(i,j),\bar{b}(i,j)=b(i,j),\bar{c}(i,j)=c(i,j),$$ they are centroaffine equivalent.
\end{cor}
\par Given all point group $\{\vec{r}(i,j)\}$, we can generate the discrete surface by the following mode on the left side of Figure \ref{fig-DS}, and every triangle represents a face in the discrete centroaffine indefinite surface. Moreover, the plane $\pi$ can be regarded as {\it the tangent plane} at the point $\vec{r}(i,j)$ on the right side of Figure \ref{fig-DS}.
\begin{figure}[hbtp]
\centering
\begin{tabular}{cc}
\includegraphics[width=.45\textwidth]{DS.eps} & \includegraphics[width=.45\textwidth]{TP.eps}
\end{tabular}
\caption{{\it Left} : Each vertex neighborhood of a triangle mesh looks topologically like an oriented piece of the
plane. {\it Right} : The tangent plane of the point $\vec{r}(i,j)$. }
\label{fig-DS}
\end{figure}
\par A discrete centroaffine indefinite surface $M=(V,E,F)$ is a simplicial 2-complex consisting of
\begin{itemize}
\item distinct vertices $$V=\{\vec{r}(i,j), i,j\in\mathbb{Z}\};$$
\item oriented edges
\begin{align*}
E=&\{[\vec{r}(i,j-1),\vec{r}(i,j)],[\vec{r}(i,j),\vec{r}(i-1,j)],[\vec{r}(i-1,j+1),\vec{r}(i,j)],\\
&[\vec{r}(i,j),\vec{r}(i,j+1)],[\vec{r}(i+1,j),\vec{r}(i,j)],[\vec{r}(i,j),\vec{r}(i+1,j-1)],i,j\in\mathbb{Z}\};
\end{align*}
\item oriented faces $$F=\{(\vec{r}(i,j),\vec{r}(i,j+1),\vec{r}(i+1,j)),(\vec{r}(i,j-1),\vec{r}(i+1,j-1),\vec{r}(i,j)),i,j\in\mathbb{Z}\}.$$
\end{itemize}
Let star($\vec{r}(i,j)$) denote the triangles of $M$ that contain $\vec{r}(i,j)$ as a vertex. For an edge $\vec{e}$, let star($\vec{e}$) denote the (at most two) triangles of $M$ that contain $\vec{e}$ as an edge. Let $T=(\vec{p},\vec{q},\vec{r})$ denote an oriented triangle of $M$ with vertices $p,q,r\in M$. The volume of an oriented surface $M$ is the oriented volume enclosed by the cone of the surface over the origin in $\R^3$:
\begin{equation}
\textbf{Vol}\{M\}:=\frac{1}{6}\sum_{T=(\vec{p},~\vec{q},~\vec{r})\in M}\det[\vec{p},\vec{q},\vec{r}].
\end{equation}
By simplification, we take the following notations:
$$d(i,j)=a(i,j)-b(i,j)-c(i,j),\quad \textbf{V}(i,j)=\textbf{Vol}\{(\vec{r}(i,j),\vec{r}(i+1,j),\vec{r}(i,j+1))\}.$$
Thus
\begin{align}
\textbf{Vol}\{\mathrm{star}([\vec{r}(i,j),\vec{r}(i-1,j+1)])\}&=[1-d(i-1,j)]\textbf{V}(i-1,j).\\
\textbf{Vol}\{(\vec{r}(i-1,j),\vec{r}(i,j),\vec{r}(i,j+1))\}&=c(i-1,j)\textbf{V}(i-1,j).
\end{align}
We also have
\begin{align}
\textbf{Vol}\{\mathrm{star}([\vec{r}(i,j),\vec{r}(i+1,j-1)])\}&=[1-d(i,j-1)]\textbf{V}(i,j-1).\\
\textbf{Vol}\{(\vec{r}(i,j-1),\vec{r}(i+1,j),\vec{r}(i,j))\}&=b(i,j-1)\textbf{V}(i,j-1).
\end{align}
On the other hand
\begin{align}
\textbf{Vol}\{(\vec{r}(i,j-1),\vec{r}(i,j),\vec{r}(i-1,j))\}=-d(i-1,j-1)\textbf{V}(i-1,j-1).
\end{align}
Finally we get
\begin{align}
\textbf{Vol}\{\mathrm{star}(\vec{r}(i,j))\}&=-d(i-1,j-1)\textbf{V}(i-1,j-1)+[1-d(i-1,j)]\textbf{V}(i-1,j)\nonumber\\
&+[1-d(i,j-1)]\textbf{V}(i,j-1)+\textbf{V}(i,j)\nonumber\\
&=[-d_{i-1,j-1}+(1-d_{i-1,j})|B_{i-1,j-1}|+(1-d_{i,j-1})|A_{i-1,j-1}|\nonumber\\
&+|A_{i-1,j-1}B_{i,j-1}|~]\textbf{V}_{i-1,j-1}\nonumber\\
&=[-d_{i-1,j-1}+(1-d_{i-1,j})|B_{i-1,j-1}|+(1-d_{i,j-1})|A_{i-1,j-1}|\nonumber\\
&+|A_{i-1,j-1}B_{i,j-1}|~]\times|A(0,0)A(1,0)\cdots A(i-2,0)|\nonumber\\
&\times|B(i-1,0)\cdots B(i-1,j-2)|V(0,0).\label{Vol-rij}
\end{align}
In the tangent plane,
\begin{align}
\textbf{Vol}_1\{\mathrm{star}(\vec{r}(i,j))\}&=-d(i-1,j-1)\textbf{V}(i-1,j-1)+c(i-1,j)\textbf{V}(i-1,j)\nonumber\\
&+b(i,j-1)\textbf{V}(i,j-1)+\textbf{V}(i,j)\nonumber\\
&=[-d_{i-1,j-1}+c_{i-1,j}|B_{i-1,j-1}|+b_{i,j-1}|A_{i-1,j-1}|\nonumber\\
&+|A_{i-1,j-1}B_{i,j-1}|~]\textbf{V}_{i-1,j-1}\nonumber\\
&=[-d_{i-1,j-1}+c_{i-1,j}|B_{i-1,j-1}|+b_{i,j-1}|A_{i-1,j-1}|\nonumber\\
&+|A_{i-1,j-1}B_{i,j-1}|~]\times|A(0,0)A(1,0)\cdots A(i-2,0)|\nonumber\\
&\times|B(i-1,0)\cdots B(i-1,j-2)|V(0,0).\label{tVol-rij}
\end{align}
\section{Laplacian operator on centroaffine simplicial surface.}
Let $f:\mathcal{S}\rightarrow\R^3$ be a simplicial surface $S:=f(\mathcal{S})$. Then the half square edge energy of the simplicial surface $S$ is given by
$$E(S)=\frac{1}{2}\sum_{(i,j)\in E}||f(i)-f(j)||^2,$$
where $$||f(i)-f(j)||^2=\sum_{k=1}^3|f_k(i)-f_k(j)|^2$$
and $f_k$ is the $k-$th coordinate function. Then
$$E(S)=\sum_{k=1}^3E(f_k),$$
where $E(f_k)$ is the Dirichlet energy of the $k-$th coordinate function.
Using the similar method as in \cite{Bo-Pin-07}, the gradient of $E(S)$ at the vertex $f(i)$ is equal to
$$\nabla_{f(i)}E(S)=\sum_{j:(i,j)\in E}(f(i)-f(j)).$$
In analogy to the smooth case, the Laplacian operator is defined as the
gradient of the Dirichlet energy. So we obtain the {\it combinatorial Laplacian operator}
\begin{equation}\label{Laplacian}
(\Delta f)(i)=\sum_{j:(i,j)\in E}(f(i)-f(j)).
\end{equation}
This Laplacian operator is very useful due to its simple definition using only the connectivity of the mesh. Besides that matrix is symmetric, and so it has real eigenvalues and orthogonal eigenvectors. On the other hand, from the following proposition, it is closely related to centroaffine transformation.
\begin{prop}
$\Delta f=0$ and $\Delta f= sf$ is centroaffine invariant, where $s$ is a centroaffine invariant.
\end{prop}
{\bf Proof.} Under a centroaffine transformation $\tilde{f}=Pf$, where $P$ is a non-degenerate matrix, from Eq. (\ref{Laplacian}) we can show that
\begin{eqnarray*}
(\Delta \tilde{f})(i)&=&\sum_{j:(i,j)\in E}(\tilde{f}(i)-\tilde{f}(j)) \\
&=& \sum_{j:(i,j)\in E}(Pf(i)-Pf(j)) \\
&=& P\sum_{j:(i,j)\in E}(f(i)-f(j)).
\end{eqnarray*}
Hence, $\Delta f=0$ and $\Delta f= sf$ can generate $\Delta \tilde{f}=0$ and $\Delta \tilde{f}= s\tilde{f}$ respectively.
\\ \rightline{$\Box$}
For a discrete centroaffine indefinite surface $S=(V,E,F)$, by Eq.(\ref{Laplacian}) and Figure \ref{fig-DS}, we have
\begin{equation}\label{Lap1}
\Delta \vec{r}(i,j)=6\vec{r}(i,j)-\vec{r}(i-1,j)-
\vec{r}(i-1,j+1)-\vec{r}(i,j-1)-\vec{r}(i,j+1)-\vec{r}(i+1,j-1)-\vec{r}(i+1,j).
\end{equation}
\begin{defn}
A discrete centroaffine indefinite surface is called harmonic if $\Delta \vec{r}=0.$
\end{defn}
A discrete centroaffine indefinite surface $S=(V,E,F)$ is harmonic if and only if the point $\vec{r}(i,j)$ lies the center of gravity of its immediate neighbors $\vec{r}(i-1,j)$,
$\vec{r}(i-1,j+1)$,$\vec{r}(i,j-1)$,$\vec{r}(i,j+1)$,$\vec{r}(i+1,j-1)$ and $\vec{r}(i+1,j)$, distributing the vertices over the space in a good way. Now, the following two corollaries are obvious.
\begin{cor}
A discrete centroaffine indefinite surface is harmonic if and only if
$$\vec{r}(i,j)=\frac{\vec{r}(i-1,j)+
\vec{r}(i-1,j+1)+\vec{r}(i,j-1)+\vec{r}(i,j+1)+\vec{r}(i+1,j-1)+\vec{r}(i+1,j)}{6}.$$
\end{cor}
\begin{cor}A discrete centroaffine indefinite surface satisfies that $\Delta \vec{r}= s\vec{r}$ if and only if
$(6+s)\vec{r}(i,j)= \vec{r}(i-1,j)+
\vec{r}(i-1,j+1)+\vec{r}(i,j-1)+\vec{r}(i,j+1)+\vec{r}(i+1,j-1)+\vec{r}(i+1,j).$
\end{cor}
\section{Discrete centroaffine indefinite surface with constant coefficients.}
In this section, we will consider the discrete centroaffine indefinite surface with constant coefficients in structure equations (\ref{Stru-c-1})-(\ref{Stru-c-3}).
If all the coefficients are constant, the compatibility conditions (\ref{Dis-Con-F})-(\ref{Dis-Con-L}) may be written as
\begin{gather}\label{com-con-c}
-\alpha b=-\gamma c=a-b-c=bc(\beta\delta-1).
\end{gather}
The transition equations (\ref{Transp}) and (\ref{Transr}) are changed to
\begin{gather}
\left[\vec{r}(m,n),\vec{r}(m+1,n),\vec{r}(m,n+1)\right]=\left[\vec{r}(0,0),\vec{r}(1,0),\vec{r}(0,1)\right]A^mB^n,\label{Ite}\\
\left[\vec{r}(-m,-n),\vec{r}(-m+1,-n),\vec{r}(-m,-n+1)\right]=\left[\vec{r}(0,0),\vec{r}(1,0),\vec{r}(0,1)\right](A^{-1})^m(B^{-1})^n,
\end{gather}
where
\begin{eqnarray}
A &=& \left(
\begin{array}{ccc}
0 & \beta(a-b-c)-\alpha & a-b-c \\
1 & 1+\alpha+b\beta-\beta & b \\
0 & c\beta & c \\
\end{array}
\right),
\\
B &=& \left(
\begin{array}{ccc}
0 & a-b-c & \delta(a-b-c)-\gamma\\
0 & b & b\delta\\
1 & c & 1+\gamma+c\delta-\delta\\
\end{array}
\right),
\end{eqnarray}
$m>0,n>0$ and $m,n\in\mathbb{Z}$. As in Eqs. (\ref{detA-B}) and (\ref{com-con}), we have
\begin{equation}
|A|=c\alpha,\quad|B|=b\gamma,\quad AB=BA.
\end{equation}
In Proposition \ref{BeqC}, we obtain the relation between a discrete centroaffine indefinite surface and a discrete affine sphere. The following proposition gives another result.
\begin{prop}\label{Pro-Aff-sph}A discrete centroaffine indefinite surface with constant coefficients is a discrete affine sphere if and only if $|A|=|B|=1$.
\end{prop}
{\bf Proof.} Firstly, if $|A|=|B|=1$, which implies $\alpha=\frac{1}{c}, \gamma=\frac{1}{b}$, Eq. (\ref{com-con-c}) generates
$\frac{b}{c}=\frac{c}{b}.$ Then we have $b=-c$ or $b=c$. If $b=-c$, it is easy to see $\alpha=-\frac{1}{b}$ and $a=a-b-c=-\alpha b=1$, which is contradictory to the assumption $a\neq1$ in Eq. (\ref{assume}). If $b=c$, we get $\alpha=\frac{1}{b}$ and $a-b-c=-\alpha b=-1$, which satisfies the condition of a discrete affine sphere in Proposition \ref{BeqC}.
\par On the other hand, if $b=c, a-b-c=-1$, Eq. (\ref{com-con-c}) gives $\alpha=\gamma=\frac{1}{b}=\frac{1}{c}$. It immediately shows that $|A|=c\alpha=1,\quad|B|=b\gamma=1$.
\\ \rightline{$\Box$}
The following proposition shows an interest property of a discrete centroaffine indefinite surface with the constant coefficients.
\begin{prop}\label{Prop-locally}
A discrete centroaffine indefinite surface with the constant coefficients is self-equivalent locally, that is, the patch $$\vec{r}(i,j),\cdots,\vec{r}(i+m,j+n)$$ is centroaffine equivalent to the patch $$\vec{r}(k,l),\cdots,\vec{r}(k+m,l+n),$$ where $i,j,k,l,m,n\in\mathbb{Z}$
\end{prop}
{\bf Proof.}
For the vector groups $\{\vec{r}(i,j),\vec{r}(i+1,j),\vec{r}(i,j+1)\}$ and $\{\vec{\bar{r}}(k,l),\vec{\bar{r}}(k+1,l),\vec{\bar{r}}(k,l+1)\}$, there exists a non-degenerate matrix $P$ satisfying that $$\{\vec{r}(i,j),\vec{r}(i+1,j),\vec{r}(i,j+1)\}=P\{\vec{\bar{r}}(k,l),\vec{\bar{r}}(k+1,l),\vec{\bar{r}}(k,l+1)\}.$$
Since the functions $\alpha,\beta,a,b,c,\gamma,\delta$ are constant, by Eqs. (\ref{Stru-c-1})-(\ref{Stru-c-3}), we get
$$\{\vec{r}(i,j),\cdots,\vec{r}(i+m,j+n)\}=P\{\vec{r}(k,l),\cdots,\vec{r}(k+m,l+n)\},$$
which completes the proof.
\\ \rightline{$\Box$}
\begin{rem}From the proposition \ref{Prop-locally}, locally, we can arbitrarily choose a patch to display a discrete centroaffine indefinite surface with constant coefficients.
\end{rem}
Since the coefficients $a,b,c,\alpha,\beta,\gamma,\delta$ and $\Delta\vec{r}=0$ are invariant under centroaffine transformation, we can choose $$\vec{r}(0,0)=(1,0,0)^{\mathrm{Tran}}, \vec{r}(1,0)=(0,1,0)^{\mathrm{Tran}},\vec{r}(0,1)=(0,0,1)^{\mathrm{Tran}}.$$
to obtain the following proposition by a direct computation.
\begin{prop}
A discrete centroaffine indefinite surface $S=(V,E,F)$ with constant coefficients is harmonic if and only if it satisfies that
\begin{gather}
(1+\alpha-\beta)(\frac{1}{\alpha}+\frac{b}{c})+(1+\gamma-\delta)(\frac{1}{\gamma}+\frac{c}{b})-\frac{c}{b}-\frac{b}{c}=6,\label{Harmonic-F} \\
\frac{\delta}{\gamma}(1+\alpha)=\frac{1}{\alpha}(1+\gamma)-\frac{1}{b}-1,\\
\frac{\beta}{\alpha}(1+\gamma)=\frac{1}{\gamma}(1+\alpha)-\frac{1}{c}-1.\label{Harmonic-L}
\end{gather}
\end{prop}
\par Finally, according to above results, we give some examples for the discrete centroaffine indefinite surface with constant coefficients.
\par {\bf Example 1.} It is not hard to check that $\beta=\delta=0, \alpha=\gamma=b=c=0.5$ and $a=0.75$ satisfy Eq. (\ref{com-con-c}), and we get a discrete surface like saddle surface shown in Figure \ref{figure-saddle}.
\begin{figure}[hbtp]
\centering
\begin{tabular}{c}
\includegraphics[width=.4\textwidth]{saddle.eps}
\end{tabular}
\caption{A discrete centroaffine indifinite surface analogy of saddle surface. }
\label{figure-saddle}
\end{figure}
\par {\bf Example 2.} From the compatibility conditions (\ref{com-con-c}) and Eqs. (\ref{Harmonic-F})-(\ref{Harmonic-L}), by assuming $\alpha=1$, we get a harmonic discrete centroaffine indefinite surface with
$$a=-b=c=-\frac{1}{3}, \beta=\delta=2, \gamma=-1.$$
In particular, we obtain the coordinates of the corresponding points
\begin{gather*}
\vec{r}(0,0)=(1,0,0)^{\mathrm{Tran}},\vec{r}(1,0)=(0,1,0)^{\mathrm{Tran}},\vec{r}(0,1)=(0,0,1)^{\mathrm{Tran}},\\
\vec{r}(-1,1)=(1,1,-5)^{\mathrm{Tran}},\vec{r}(0,-1)=(2,-2,1)^{\mathrm{Tran}},\vec{r}(1,-1)=(3,1,1)^{\mathrm{Tran}},\\
\vec{r}(-1,0)=(0,-1,2)^{\mathrm{Tran}},\vec{r}(-1,-1)=(-1,-1,-1)^{\mathrm{Tran}}.
\end{gather*}
By these points, we obtain a local graph of a harmonic discrete centroaffine indefinite surface in the left of Figure \ref{figure-exp1}. With the iterative formula (\ref{Ite}), we generate 16 points and display it as a graph of a harmonic discrete centroaffine indefinite surface in the right of Figure \ref{figure-exp1}.
\begin{figure}[hbtp]
\centering
\begin{tabular}{cc}
\includegraphics[width=.25\textwidth]{local.eps} & \includegraphics[width=.3\textwidth]{global.eps}
\end{tabular}
\caption{Graph of a harmonic discrete centroaffine indefinite surface. }
\label{figure-exp1}
\end{figure}
\par {\bf Example 3.} In the smooth case, a centroaffine surface with $\Delta_gx=-2x$ is a centroaffine minimal affine sphere. Here we consider a discrete affine sphere of constant coefficients with $\Delta \vec{r}=s\vec{r}$, where $s$ is constant. Using Eqs. (\ref{Lap1}), (\ref{com-con-c}) and Proposition \ref{Pro-Aff-sph}, by a simple calculation, we get $s=8$. Furthermore, we can obtain $a=-3,b=c=\alpha=\gamma=-1,\beta\delta=0$.
Especially, if $\beta=\delta=0$, the discrete affine sphere is the face of a tetrahedron with $\vec{r}_{11}=\vec{r}_{22}=\vec{r}$ and $\vec{r}_{12}=-\vec{r}-\vec{r}_1-\vec{r}_2$, as shown in the left of Figure \ref{figure-AS}.
If $\beta=0,\delta\neq0$, the structure equations of the discrete affine sphere are
$$\vec{r}_{11}=\vec{r},\quad \vec{r}_{12}=-\vec{r}-\vec{r}_1-\vec{r}_2,\quad \vec{r}_{22}=(1-\delta)\vec{r}-\delta\vec{r}_1-2\delta\vec{r}_2.$$ Using $\delta=1$ and $\delta=-1$, we get the graphs shown with 9 vertices in the middle of Figure \ref{figure-AS} and in the right of Figure \ref{figure-AS}, respectively. In fact, there are only 6 vertices according to $\vec{r}_{11}=\vec{r}$.
\begin{figure}[hbtp]
\centering
\begin{tabular}{ccc}
\includegraphics[width=.2\textwidth]{affinesphere0.eps}&\includegraphics[width=.23\textwidth]{affinesphere1.eps}& \includegraphics[width=.13\textwidth]{affinesphere-1.eps}
\end{tabular}
\caption{Discrete affine sphere with $\Delta\vec{r}=8\vec{r}$. }
\label{figure-AS}
\end{figure}
\par From Proposition \ref{BeqC}, Proposition \ref{Pro-Aff-sph}, Eqs. (\ref{Vol-rij}) and (\ref{tVol-rij}) we have
\begin{cor} For a discrete affine sphere with constant coefficients we have
$$\textbf{V}(i,j)=\textbf{V}(0,0),\quad\frac{\textbf{Vol}\{\mathrm{star}(\vec{r}(i,j))\}}{\textbf{V}(0,0)}=6,\quad\frac{\textbf{Vol}_1\{\mathrm{star}(\vec{r}(i,j))\}}{\textbf{V}(0,0)}=2+b+c.$$
\end{cor}
\section{Locally convexity of a discrete centroaffine indefinite surface.}
Obviously, a discrete centroaffine indefinite surface is convex at the point $\vec{r}(n_1,n_2)$ if these points
$$\vec{r}_{\bar{1}\bar{1}}, \vec{r}_{\bar{1}\bar{2}}, \vec{r}_{\bar{2}\bar{2}}, \vec{r}_{\bar{1}2}, \vec{r}_{1\bar{2}}, \vec{r}_{11}, \vec{r}_{12}, \vec{r}_{22}$$ lie on the same side of the tangent plane $\pi$ of $\vec{r}(n_1,n_2)$, which implies the following determinants have the same sign.
\begin{gather}
\begin{split}
&[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{1}\bar{1}}-\vec{r}],\quad [\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{1}\bar{2}}-\vec{r}], \quad [\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{2}\bar{2}}-\vec{r}], \\
&[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{1\bar{2}}-\vec{r}], \quad[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{1}2}-\vec{r}],\quad
[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{11}-\vec{r}], \\
&[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{12}-\vec{r}], \quad [\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{22}-\vec{r}].
\end{split}\label{Det-all}
\end{gather}
Firstly, from Eq. (\ref{Stru-c-2}), it is easy to check
\begin{equation}
\begin{split}
[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{12}-\vec{r}]&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},(a-1)\vec{r}]\\
&=(a-1)[\vec{r},\vec{r}_1,\vec{r}_2].
\end{split}\label{Det1}
\end{equation}
On the other hand, by Eq. (\ref{Stru-c-3}) we have
\begin{equation}
\begin{split}
[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{22}-\vec{r}]&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{22}-\vec{r}_2+\vec{r}_2-\vec{r}]\\
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\delta(\vec{r}_{12}-\vec{r}_2)]\\
&=(a-1)\delta[\vec{r},\vec{r}_1,\vec{r}_2].
\end{split}\label{Det2}
\end{equation}
Since $a-1\neq 0$ by the assumption (\ref{assume}) , comparison of Eq. (\ref{Det1}) and Eq. (\ref{Det2}) shows that
\begin{equation}
\delta\geq 0.
\end{equation}
By Eq. (\ref{Stru-c-1}), we may examine
\begin{equation}
\begin{split}
[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{11}-\vec{r}]&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{11}-\vec{r}_1+\vec{r}_1-\vec{r}]\\
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\beta(\vec{r}_{12}-\vec{r}_2)]\\
&=(a-1)\beta[\vec{r},\vec{r}_1,\vec{r}_2].
\end{split}
\end{equation}
Similarly, it is convenient to conclude
\begin{equation}
\beta\geq0.
\end{equation}
Again, from Eqs. (\ref{Stru-c-1})-(\ref{Stru-c-3}), it is clear that
\begin{equation}
\begin{split}
[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{1}2}-\vec{r}]&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{1}2}-\vec{r}_{\bar{1}}+\vec{r}_{\bar{1}}-\vec{r}]\\
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\frac{1}{c_{\bar{1}}}(\vec{r}_2-a_{\bar{1}}\vec{r}_{\bar{1}})]\\
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\frac{1}{c_{\bar{1}}}(\vec{r}_2-a_{\bar{1}}(\vec{r}_{\bar{1}}-\vec{r}+\vec{r}))]\\
&=-\frac{1}{c_{\bar{1}}}(a_{\bar{1}}-1)[\vec{r},\vec{r}_1,\vec{r}_2].
\end{split}
\end{equation}
By the assumption (\ref{assume}), comparing the above equation with Eq. (\ref{Det1}) leads to
\begin{equation}
-\frac{1}{c_{\bar{1}}}(a_{\bar{1}}-1)(a-1)>0.
\end{equation}
In like manner we shall derive
\begin{equation}
\begin{split}
[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{2}1}-\vec{r}]&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{2}1}-\vec{r}_{\bar{2}}+\vec{r}_{\bar{2}}-\vec{r}]\\
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\frac{1}{b_{\bar{2}}}(\vec{r}_1-a_{\bar{2}}\vec{r}_{\bar{2}})]\\
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\frac{1}{b_{\bar{1}}}(\vec{r}_1-a_{\bar{2}}(\vec{r}_{\bar{2}}-\vec{r}+\vec{r}))]\\
&=-\frac{1}{b_{\bar{1}}}(a_{\bar{2}}-1)[\vec{r},\vec{r}_1,\vec{r}_2],
\end{split}
\end{equation}
and
\begin{equation}
-\frac{1}{b_{\bar{2}}}(a_{\bar{2}}-1)(a-1)>0.
\end{equation}
Also, we have
\begin{equation}
\begin{split}
[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{2}\bar{2}}-\vec{r}]&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{2}\bar{2}}-\vec{r}_{\bar{2}}+\vec{r}_{\bar{2}}-\vec{r}]\\
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\frac{\delta_{\bar{2}\bar{2}}}{\gamma_{\bar{2}\bar{2}}}(\vec{r}_{\bar{2}1}-\vec{r}_{\bar{2}})]\\
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\frac{\delta_{\bar{2}\bar{2}}}{\gamma_{\bar{2}\bar{2}}}(\vec{r}_{\bar{2}1}-\vec{r}+\vec{r}-\vec{r}_{\bar{2}})]\\
&=-\frac{1}{b_{\bar{1}}}(a_{\bar{2}}-1)\frac{\delta_{\bar{2}\bar{2}}}{\gamma_{\bar{2}\bar{2}}}[\vec{r},\vec{r}_1,\vec{r}_2],
\end{split}
\end{equation}
and
\begin{equation}
\frac{\delta_{\bar{2}\bar{2}}}{\gamma_{\bar{2}\bar{2}}}\geq0.
\end{equation}
Similarly,
\begin{equation}
\begin{split}
[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{1}\bar{2}}-\vec{r}]
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{1}\bar{2}}-\vec{r}_{\bar{1}}+\vec{r}_{\bar{1}}-\vec{r}]\\
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},-\frac{1}{\gamma_{\bar{1}\bar{2}}}(\vec{r}_{\bar{1}2}-\vec{r}_{\bar{1}})]\\
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},-\frac{1}{\gamma_{\bar{1}\bar{2}}}(\vec{r}_{\bar{1}2}-\vec{r}+\vec{r}-\vec{r}_{\bar{1}})]\\
&=-\frac{1}{c_{\bar{1}}}(a_{\bar{1}}-1)\frac{-1}{\gamma_{\bar{1}\bar{2}}}[\vec{r},\vec{r}_1,\vec{r}_2],
\end{split}
\end{equation}
and
\begin{equation}
\gamma_{\bar{1}\bar{2}}<0.
\end{equation}
Noticing that
\begin{equation}
\begin{split}
[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{1}\bar{2}}-\vec{r}]
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{1}\bar{2}}-\vec{r}_{\bar{2}}+\vec{r}_{\bar{2}}-\vec{r}]\\
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},-\frac{1}{\alpha_{\bar{1}\bar{2}}}(\vec{r}_{\bar{2}1}-\vec{r}_{\bar{2}})]\\
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},-\frac{1}{\alpha_{\bar{1}\bar{2}}}(\vec{r}_{\bar{1}2}-\vec{r}+\vec{r}-\vec{r}_{\bar{2}})]\\
&=-\frac{1}{b_{\bar{1}}}(a_{\bar{2}}-1)\frac{-1}{\alpha_{\bar{1}\bar{2}}}[\vec{r},\vec{r}_1,\vec{r}_2],
\end{split}
\end{equation}
and from it we obtain
\begin{equation}
\alpha_{\bar{1}\bar{2}}<0.
\end{equation}
Finally, from
\begin{equation}
\begin{split}
[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{1}\bar{1}}-\vec{r}]
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{1}\bar{1}}-\vec{r}_{\bar{1}}+\vec{r}_{\bar{1}}-\vec{r}]\\
&=[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\frac{\beta_{\bar{1}\bar{1}}}{\alpha_{\bar{1}\bar{1}}}(\vec{r}_{\bar{1}2}-\vec{r}_{\bar{1}})]\\
&=\frac{\beta_{\bar{1}\bar{1}}}{\alpha_{\bar{1}\bar{1}}}[\vec{r}_1-\vec{r}, \vec{r}_2-\vec{r},\vec{r}_{\bar{1}2}-\vec{r}+\vec{r}-\vec{r}_{\bar{1}}]\\
&=-\frac{1}{c_{\bar{1}}}(a_{\bar{1}}-1)\frac{\beta_{\bar{1}\bar{1}}}{\alpha_{\bar{1}\bar{1}}}[\vec{r},\vec{r}_1,\vec{r}_2],
\end{split}
\end{equation}
we find
\begin{equation}
\frac{\beta_{\bar{1}\bar{1}}}{\alpha_{\bar{1}\bar{1}}}\geq0.
\end{equation}
Then it is easy to conclude
\begin{prop}\label{pro-convex}
A discrete surface is convex at the point $\vec{r}(n_1,n_2)$ if and only if
\begin{equation}
\begin{split}
\delta\geq 0,\quad \beta\geq0,\quad c_{\bar{1}}(a_{\bar{1}}-1)(a-1)<0,\quad b_{\bar{2}}(a_{\bar{2}}-1)(a-1)<0,
\end{split}
\end{equation}
$$$$
$$ \frac{\delta_{\bar{2}\bar{2}}}{\gamma_{\bar{2}\bar{2}}}\geq0,\quad \frac{\beta_{\bar{1}\bar{1}}}{\alpha_{\bar{1}\bar{1}}}\geq0,\quad \gamma_{\bar{1}\bar{2}}<0,\quad \alpha_{\bar{1}\bar{2}}<0.$$
\end{prop}
It is immediate to see from this proposition
\begin{cor}If a discrete centroaffine indefinite surface is locally convex everywhere, then $\delta=\beta=0,\gamma<0,\alpha<0$.
\end{cor}
By the Eq. (\ref{com-con-c}) and Proposition \ref{pro-convex}, we obtain
\begin{cor}A discrete centroaffine indefinite surface with constant coefficients is locally convex everywhere if and only if
\begin{equation}\label{C-Convex}
\alpha=c<0, \gamma=b<0, \beta=\delta=0, a=b+c-bc.
\end{equation}
\end{cor}
\begin{rem}The discrete affine sphere shown in the left of Figure \ref{figure-AS} is a closed locally convex everywhere.
\end{rem}
We give another example which is local convex discrete centroaffine indefinite surface with constant coeffients.
\par {\bf Example.} It is convenient to check $\alpha=c=-2, \gamma=b=-1, a=-5$ and $\beta=\delta=0$ satisfy Eq. (\ref{C-Convex}). Thus, we can obtain a discrete surface shown in Figure \ref{figure-convex} with 25 points. In fact, there are only 10 points because $\vec{r}_{22}=\vec{r}$.
\begin{figure}[hbtp]
\centering
\begin{tabular}{cc}
\includegraphics[width=.25\textwidth]{convex.eps} & \includegraphics[width=.25\textwidth]{convex-1.eps}
\end{tabular}
\caption{Locally convex discrete centroaffine indefinite surface. }
\label{figure-convex}
\end{figure}
\par If a discrete centroaffine indefinite surface is locally strongly convex, all determinants in Eq. (\ref{Det-all}) can not be zero. Hence we have
\begin{cor}There is not a discrete centroaffine indefinite surface which is locally strongly convex everywhere.
\end{cor}
{\bf Proof.} If a discrete surface is locally strongly convex everywhere, from the above calculations we have
$$\delta> 0,\quad \beta>0,\quad c_{\bar{1}}(a_{\bar{1}}-1)(a-1)<0,\quad b_{\bar{2}}(a_{\bar{2}}-1)(a-1)<0,$$
$$ \frac{\delta_{\bar{2}\bar{2}}}{\gamma_{\bar{2}\bar{2}}}>0,\quad \frac{\beta_{\bar{1}\bar{1}}}{\alpha_{\bar{1}\bar{1}}}>0,\quad \gamma_{\bar{1}\bar{2}}<0,\quad \alpha_{\bar{1}\bar{2}}<0,$$
which imply $\gamma_{\bar{2}\bar{2}}>0, \alpha_{\bar{1}\bar{1}}>0.$ This is inconsistent with $\gamma_{\bar{1}\bar{2}}<0,\quad \alpha_{\bar{1}\bar{2}}<0.$
\\ \rightline{$\Box$}
\renewcommand{\refname}{\bf\fontsize{12}{12}\selectfont References}
\bibliographystyle{amsplain}
|
1,108,101,563,300 | arxiv | \section{Introduction \label{s1}}
Flavor-changing neutral current (FCNC) processes present as
stringest tests of the Standart Model (SM) as well as looking for
new physics beyond the SM. In SM, FCNC processes take place only
at the loop level and therefore these decays are suppressed and
for this reason experimentally their measurement is difficult.
Recently, B factory experiments such as Belle and BaBar have
measured FCNC decays due to the $b \rightarrow s \ell^+ \ell^-$
transition at inclusive and exclusive level [1,2,3,4].
Theoretically, the $b \rightarrow s \ell \ell$ transition is described by
the effective Hamiltonian and the FCNC's are represented by QCD
penguin and electroweak penguin (EW) operators. The EW operators
appear in the photon and Z mediated diagrams. Although it is
experimentally very difficult to measure, $B \rightarrow X \,\nu
\bar{\nu}$ decay is an extremely good and "pure" channel to study
Z mediated EW penguin contribution within the SM and its beyond
[5,6,7,8,9].
Another exclusive decay which is described at inclusive level
by the $b \rightarrow s \ell^+ \ell^-$ transition is the baryonic
$\Lambda_b \rightarrow \Lambda \ell \ell$ decay. Unlike the mesonic
decays, the baryonic decay could maintain the helicity structure
of the effective Hamiltonian for the $b \rightarrow s $ transition [10].
In this work, we study the $\Lambda_b \rightarrow \Lambda \,\nu
\bar{\nu}$ decay in the leptophobic Z$'$ Model, as a possible
candidate of new physics in the EW penguin sector. The leptophobic
Z$'$ gauge bosons appear naturally in Grand Unified Theories and
string inspired $E_{6}$ models. Here we would like to note that a
similar mesonic $B \rightarrow K(K^{*}) \,\nu \bar{\nu}$ decay and
$\Delta m_{B_{s}}$ mass difference of $B_{s}$ meson in the context
of leptophobic Z$'$ model are studied in [11] and [12],
respectively. In section 2, we briefly consider the leptophobic
model based on $E_{6}$ model. In section 3, we present the
analytical expressions for the branching ratio and the
longitudinal, transverse and normal polarizations of $\Lambda$
baryon, as well as polarization of $\Lambda_b$. In Section 4, we
give numerical analysis and our concluding remarks.
\section{Theoretical background on leptophobic Z$'$ model \label{s2}}
In this section we briefly review the leptophobic Z$'$ model. From
GUT or string-inspired point of view, the $E_{6}$ model [13] is a
very natural extention of the SM. In this model $U(1)^{'}$ gauge
group remains after the symmetry breaking of the $E_{6}$ group. We
assume that the $E_{6}$ group is broken through the following way
$E_{6} \rightarrow SO(10)\times U(1)_{\psi} \rightarrow SU(5)\times
U(1)_{\chi}\times U(1)_{\psi} \rightarrow SU(2)_{L}\times U(1)\times
U(1)^{'}$, where $U(1)^{'}$ is a linear combination of two
additional $U(1)$ gauge groups with $Q^{'} = Q_{\psi}
\cos\theta-Q_{\chi}\sin\theta$, where $\theta$ is the mixing
angle. When all couplings are GUT normalized, the interaction,
Lagrangian of fermions with Z$'$ gauge boson can be written as
\begin{eqnarray}{\cal L}_{int} &=& -\lambda
\frac{g_{2}}{\cos\theta_{w}}\sqrt{\frac{5
\sin^{2}\theta_{w}}{3}}\bar{\psi}\gamma_{\mu}\Bigg(Q^{'}+\sqrt{\frac{3}{5}}
\delta Y_{SM}\Bigg)\psi Z_{\mu}^{'}~, \end{eqnarray} where $\lambda =
\frac{g_{Q^{'}}}{g_{y}}$ is the ratio of gauge couplings $\delta =
- \tan\chi/\lambda$ [14]. In the general case fermion--Z$'$ gauge
boson couplings contain two arbitrary free parameters $\tan\theta$
and $\delta$ [15]. From Eq.(1), it follows that the Z$'$ gauge
boson can be leptophobic when $Q^{'}+\sqrt{\frac{3}{5}} \delta
Y_{SM} = 0$ for the lepton doublet and $e^c$, simultaneously. In
the conventional embedding, the Z$'$ boson can be made leptophobic
with $\delta = -1/3$ and $\tan\theta = \sqrt{\frac{3}{5}}$. In the
Leptophobic Z$'$ model, FCNC's can arise through the mixing
between SM fermions and exotic fermions. In principle, the mixing
of the left handed fermions can lead to the large Z-mediated
FCNC's. In order to forbid this large Z-mediated FCNC's, we
consider the case when mixing take place only between the
right-handed fermions. So, the mixing between right-handed
ordinary and exotic fermions can induce the FCNC's when their Z$'$
charges are different [16]. In the Leptophobic Z$'$ model, the
Lagrangian for $b \rightarrow q (q=s,d)$ transition containing FCNC at
tree level can be written as \begin{eqnarray}{\cal L}^{Z^{'}} &=&
-\frac{g_{2}}{2\cos\theta_{w}}U_{qb}^{Z^{'}} \bar
q_{R}\gamma^{\mu}b_{R}Z_{\mu}^{'}~.\end{eqnarray} Using upper experimental
bounds on branching ratio for the $B \rightarrow K \,\nu \bar{\nu}$ [17]
and $B \rightarrow \pi \,\nu \bar{\nu}$ [18] in [11] for model parameters
$U_{sb}^{Z^{'}}$ and $U_{db}^{Z^{'}}$ following bounds are
obtained : $|U_{sb}^{Z^{'}}|\leq0.29$ and
$|U_{db}^{Z^{'}}|\leq0.61$. Analysis of $\Delta m_{B_{s}}$ leads
to the more stringest bound for the $U_{sb}^{Z^{'}}$ [12]:
$|U_{sb}^{Z^{'}}|\leq0.036$ at $m_{Z'}= 700 GeV$ and
$|U_{db}^{Z^{'}}|\leq0.051$ at $m_{Z'}=1 TeV$. These constraints
on $U_{qb}^{Z^{'}}$ we will use in our numerical calculations.
\section{Matrix Elements for the $\Lambda_b \rightarrow \Lambda \,\nu \bar{\nu}$ Decay \label{s3}}
The $\Lambda_b \rightarrow \Lambda \,\nu \bar{\nu}$ decay at quark level
is described by $b \rightarrow s \,\nu \bar{\nu}$ transition. In the SM,
effective Hamiltonian responsible for the $b \rightarrow s \,\nu
\bar{\nu}$ transition is given by
\begin{eqnarray} \label{effH} {\cal H}_{eff}^{SM} &=&
\frac{G_{F}\,\alpha}{2\pi\sqrt{2}} V_{tb} V_{ts}^* C_{10} \,\bar
s\gamma_{\mu}(1-\gamma_5)b\,\bar \nu\gamma_\mu(1-\gamma_5)\nu~, \end{eqnarray}
where $G_{F}$ and $\alpha$ are the Fermi and fine structure
constant respectively, $V_{tb}$ and $V_{ts}$ are elements of the
Cabibbo-Kobayashi-Maskawa (CKM) matrix, $C_{10}$ is the Wilson
coefficient whose explicit form can be found in [17,18,19]. As we have
already mentioned, in the leptophobic model, $U(1)^{'}$ charge
is zero for all the ordinary left and right handed lepton fields
within the SM. However, 27 representation of $E_{6}$ contain new
right handed neutrino which is absent in SM and it can give
additional contribution to the processes due to the $b \rightarrow s
\,\nu \bar{\nu}$ transition. Then the effective Hamiltonian
describing $b \rightarrow s \,\nu_{R} \bar{\nu_{R}}$ transition can be
written as \begin{eqnarray} \label{effH} {\cal H}_{eff}^{new} &=&
\frac{\pi\,\alpha}{\sin^{2}2\theta_{w}M_{Z^{'}}^{2}}
U_{sb}^{Z^{'}} Q_{\nu_{R}}^{Z^{'}} \,\bar
s\gamma_{\mu}(1+\gamma_5)b\,\bar \nu\gamma_\mu(1+\gamma_5)\nu~, \end{eqnarray}
where $Q^{Z'}_{\nu_{R}}=\frac{1}{2}
x\sqrt{\frac{5\sin^{2}\theta_{w}}{3}}$. $U^{Z'}_{qb}$ and $x$ are
model dependent parameters and we take $x=1$ for simplicity. Having
effective Hamiltonian for $b \rightarrow s \,\nu \bar{\nu}$ transition,
our next problem is the derivation of the decay amplitude for the
$\Lambda_b \rightarrow \Lambda \,\nu \bar{\nu}$ decay, which can be
obtained by calculating the matrix element of the effective
Hamiltonian for $b \rightarrow s \,\nu \bar{\nu}$ transition between
initial $\Lambda_b$ and final $\Lambda$ baryon states. It follows
that the matrix elements $\left< \Lambda \left| \bar s \gamma_\mu
(1\mp\gamma_{5}) b \right| \Lambda_b \right>$ are needed for calculating
the decay amplitude for the $\Lambda_b \rightarrow \Lambda \,\nu
\bar{\nu}$ decay. These matrix elements parametrized in terms of
the form factors are as follows ([20],[21])
\begin{eqnarray} \label{mel1} \left<
\Lambda \left| \bar s \gamma_\mu b \right| \Lambda_b \right> &=& \bar
u_\Lambda \Bigg[f_1 \gamma_\mu + i\,f_2 \sigma_{\mu\nu} q^{\nu} +
f_3 q_{\mu}\bar \Bigg]u_{\Lambda_{b}}~, \end{eqnarray}
\begin{eqnarray} \label{mel2} \left< \Lambda \left| \bar s \gamma_\mu\gamma_5 b
\right| \Lambda_b \right> &=& \bar u_\Lambda\Bigg[g_1 \gamma_\mu\gamma_5
+ i\,g_2 \sigma_{\mu\nu} q^{\nu}\gamma_5 + g_3 q_{\mu}\gamma_5
\Bigg] u_{\Lambda_{b}} \end{eqnarray} where $q=p_{\Lambda_{b}}-p_{\Lambda}$
is the momentum transfer and $f_i$ and $g_i$ $(i=1,2,3)$ are the
form factors. Note that form factors $f_3$ and $g_3$ do not give
contribution to the considered decay since neutrinos are
massless. Using Eqs. (3)-(4)-(6) and (7) for the decay amplitude of
the $\Lambda_b \rightarrow \Lambda \,\nu \bar{\nu}$ decay in leptophobic
model, we have \begin{eqnarray} \label{ib} {\cal M}&=& \frac{\alpha G_F}{2
\sqrt{2} \, \pi} V_{tb} V_{ts}^* C_{10}^{\nu} \Bigg\{\bar
\nu\gamma_\mu(1-\gamma_5)\nu \times \bar u_\Lambda \Bigg[A_1
\gamma_\mu(1+\gamma_5)
+ B_1 \gamma_\mu(1-\gamma_5) \nonumber \\
&+&i\,\sigma_{\mu\nu}q^{\nu}A_2(1+\gamma_5)+ B_2(1-\gamma_5))
\Bigg]u_{\Lambda_{b}}\nonumber
\\&+&C_{RR}\bar \nu\gamma_\mu(1+\gamma_5)\nu \times \bar u_\Lambda
\Bigg[B_1\gamma_\mu(1+\gamma_5)
+A_1\gamma_\mu(1-\gamma_5) \nonumber \\
&+&i\,\sigma_{\mu\nu}q^{\nu}B_2(1+\gamma_5)+
A_2(1-\gamma_5))\Bigg]u_{\Lambda_{b}}\Bigg\}~, \end{eqnarray} where
$C_{RR}=\frac{\pi Q U_{qb}^{Z'}}{\alpha V_{tb} V_{ts}^*
C_{10}^{\nu}}\frac{m_{Z}^{2}}{m_{Z'}^{2}}$.
It is shown in [21] that when HQET is applied the number of
independent form factors are reduced to two ($F_{1}$ and $F_{2}$),
irrelevant of the Dirac structure of the corresponding operators,
i.e., $\left<\Lambda(p_{\Lambda})\left|\bar s\Gamma b\right|
\Lambda(p_{\Lambda_{b}})\right> = \bar u_\Lambda\Bigg[F_{1}(q^2) +
{\not\!\upsilon}F_{2}(q^2)\Bigg]\Gamma u_{\Lambda_{b}}$, where
$\Gamma$ is an arbitrary Dirac matrix and
$\upsilon^{\mu}=p^{\mu}/m_{\Lambda_{b}}$ is the four velocity of
$\Lambda_{b}$. Using this result we get \begin{eqnarray}
f_1&=&g_1=F_1+\sqrt{r}F_2,\nonumber \\
f_2&=&g_2=\frac{F_2}{m_{\Lambda_{b}}}~,\end{eqnarray} where $r =
m_{\Lambda}^{2}/m_{\Lambda_{b}}^{2}$.
Our next problem is the calculation of $\Lambda$ baryon
polarizations using the matrix element in Eq.(7). In the rest
frame of $\Lambda$, the unit vectors along the longitudinal,
normal and transversal components of $\Lambda$ are chosen as \begin{eqnarray}
s_L^\mu &=& \left( 0,\vec{e}_L \right) = \left( 0, \frac{\vec{p}_\Lambda}
{\left| \vec{p}_\Lambda \right|} \right)~, \nonumber \\
s_T^\mu &=& \left( 0,\vec{e}_T \right) = \left( 0, \frac{\vec{e}_L \times
\vec{\xi}_{\Lambda_b}}{\left| \vec{e}_L \times \vec{\xi}_{\Lambda_b}
\right|} \right)~, \nonumber \\
s_N^\mu &=& \left( 0,\vec{e}_N \right) = \left( 0,\vec{e}_L \times \vec{e}_T
\right)~.
\end{eqnarray}
The longitudinal component of the $\Lambda$ baryon polarization is boosted
to the center of mass frame of the nuetrino--antineutrino pair by Lorentz
transformation, yielding
\begin{eqnarray}
\left( s_L^\mu \right)_{CM} = \left( \frac{\left| \vec{p}_\Lambda \right|}{m_\Lambda},
\frac{E_\Lambda \vec{p}_\Lambda }{m_\Lambda \left| \vec{p}_\Lambda \right|} \right)~,
\end{eqnarray}
where $E_\Lambda$ and $m_\Lambda$ are the energy and mass of the $\Lambda$
baryon in the CM frame of nuetrino--antineutrino pair. The remaining two
unit vectors $s_T^\mu$ and $s_N^\mu$ are unchanged under Lorentz
transformation.
After integration over neutrino and antineutrino momentum the
differential decay rate of the $\Lambda_b \rightarrow \Lambda \,\nu
\bar{\nu}$ decay for any spin direction $\vec{\xi}$ along the
$\Lambda$ baryon can be written as (in the rest frame of
$\Lambda_b$) \begin{eqnarray} d\Gamma=\frac{1}{4}\,\,d\Gamma^{0}
\Bigg[1+\frac{I_{1}}{I_{0}}\vec{e}_{L}\cdot\vec{\xi}_{\Lambda_{b}}\Bigg]
\Bigg[1+\vec{P}_{\Lambda}\cdot \vec{\xi_{\Lambda}}\Bigg].\end{eqnarray}
\begin{eqnarray}\vec{P}_{\Lambda}=\frac{1}{1+\frac{I_{1}}{I_{0}}\hat{e}_{L}\cdot
\hat\xi_{\Lambda_{b}}}\Bigg[\Bigg(\frac{I_2}{I_0}+\frac{I_3}{I_0}\hat{e}_{L}\cdot
\hat\xi_{\Lambda_{b}}\Bigg)\hat{e}_{L}+\frac{I_4}{I_0}\hat{e}_{N}
+\frac{I_5}{I_0}\hat{e}_{T}\Bigg]\end{eqnarray} \begin{eqnarray}
d\Gamma^{0}=\frac{3G_{F}^{2}\alpha_{em}^{2}|V_{tb} V_{ts}^*|^2
|C_{10}|^2}{32.32.4\pi^{7}m_{\Lambda_{b}}}|\vec{p_{\Lambda}}|I_0
dE_{\Lambda} d\Omega_{\Lambda}\end{eqnarray} The functions $I_{i}$ in
Eq.(13) have following expressions: \begin{eqnarray} \label{Delta} I_{0}&=&
\frac{32}{3}\pi\Bigg\{ (q^2p_{\Lambda}\cdot
p_{\Lambda_{b}}+2p_{\Lambda}\cdot q\,p_{\Lambda_{b}}\cdot q)
(|A_1|^2+|B_1|^2+|C_{RR}|^2(|B_1|^2+|A_1|^2))\nonumber\\&+&
(q^2p_{\Lambda}\cdot p_{\Lambda_{b}}-4p_{\Lambda}\cdot
q\,p_{\Lambda_{b}}\cdot q)q^2
(|A_2|^2+|B_2|^2+|C_{RR}|^2(|B_2|^2+|A_2|^2))\nonumber\\&+&
6m_{\Lambda}q^2 p_{\Lambda_{b}}\cdot q
\Bigg[Re[A_1A_2^{\ast}]+Re[B_1B_2^{\ast}]+|C_{RR}|^2Re[B_1B_2^{\ast}]
+|C_{RR}|^2Re[A_1A_2^{\ast}]\Bigg] \nonumber\\&-&6m_{\Lambda_{b}}q^2
p_{\Lambda}\cdot q
\Bigg[Re[A_2B_1^{\ast}]+Re[A_1B_2^{\ast}]+|C_{RR}|^2Re[B_2A_1^{\ast}]
+|C_{RR}|^2Re[B_1A_2^{\ast}]\Bigg]
\nonumber\\&-&6m_{\Lambda_{b}}m_{\Lambda}q^2
\Bigg[Re[A_1B_1^{\ast}]+q^2Re[A_2B_2^{\ast}]+|C_{RR}|^2Re[B_1A_1^{\ast}]\nonumber\\&+&
|C_{RR}|^2q^2Re[B_2A_2^{\ast}]\Bigg] \Bigg\}\end{eqnarray}
\begin{eqnarray} \label{PLI1} I_{1}&=&\frac{32}{3I_{0}}\pi p_{\Lambda}\Bigg\{
(-m_{\Lambda_{b}}) (q^2-2p_{\Lambda}\cdot
q)\Bigg[(|A_1|^2-|B_1|^2)+|C_{RR}|^2(|B_1|^2-|A_1|^2)\Bigg]\nonumber\\&+&
m_{\Lambda_{b}}q^2(q^2+4p_{\Lambda}\cdot
q)\Bigg[(|A_2|^2-|B_2|^2)+|C_{RR}|^2(|B_2|^2-|A_2|^2)\Bigg]\nonumber\\&+&
6m_{\Lambda_{b}}m_{\Lambda}q^2
\Bigg[Re[A_1A_2^{\ast}]-Re[B_1B_2^{\ast}]+|C_{RR}|^2Re[B_1B_2^{\ast}]
-|C_{RR}|^2Re[A_1A_2^{\ast}]\Bigg]\nonumber\\&+&
2q^2(p_{\Lambda_{b}}\cdot p_{\Lambda}+p_{\Lambda_{b}}\cdot
q)\Bigg[Re[A_1B_2^{\ast}]-Re[A_2B_1^{\ast}]-|C_{RR}|^2Re[B_2A_1^{\ast}]\nonumber\\&+&
|C_{RR}|^2Re[B_1A_2^{\ast}]\Bigg]\Bigg\}\end{eqnarray}
\begin{eqnarray} \label{PLI2} I_{2}&=&\frac{32}{3I_{0}}\pi
p_{\Lambda}m_{\Lambda_{b}}\Bigg\{2m_{\Lambda}p_{\Lambda}(|A_1|^2+|B_1|^2)+(q^2+
p_{\Lambda_{b}}\cdot q)(|A_1|^2-|B_1|^2)
\nonumber\\&+&2q^2(3m_{\Lambda_{b}}+p_{\Lambda})
\Bigg[Re[A_2B_1^{\ast}]+|C_{RR}|^2Re[B_2A_1^{\ast}]\Bigg]\nonumber\\&-&
2q^2(3m_{\Lambda_{b}}-p_{\Lambda})
\Bigg[Re[A_1B_2^{\ast}]+|C_{RR}|^2Re[B_1A_2^{\ast}]\Bigg]\nonumber\\&-&
\frac{2}{m_{\Lambda}}p_{\Lambda}q^4\Bigg[Re[A_2B_2^{\ast}]+|C_{RR}|^2Re[B_2A_2^{\ast}]\Bigg]
\nonumber\\&+&\frac{2}{m_{\Lambda}}q^2(m_{\Lambda}p_{\Lambda}+p_{\Lambda_{b}}\cdot
p_{\Lambda}+p_{\Lambda}\cdot
q)\Bigg[Re[B_1B_2^{\ast}]+|C_{RR}|^2Re[A_1A_2^{\ast}]\Bigg]
\nonumber\\&+&\frac{2}{m_{\Lambda}}q^2(m_{\Lambda}p_{\Lambda}-p_{\Lambda_{b}}\cdot
p_{\Lambda}+p_{\Lambda}\cdot
q)\Bigg[Re[A_1A_2^{\ast}]+|C_{RR}|^2Re[B_1B_2^{\ast}]\Bigg]
\nonumber\\&+&\frac{2p_{\Lambda}}{m_{\Lambda}}(q^2-2p_{\Lambda_{b}}\cdot
p_{\Lambda}+2p_{\Lambda}\cdot q-2p_{\Lambda_{b}}\cdot q)
\Bigg[Re[A_1B_1^{\ast}]+|C_{RR}|^2Re[B_1A_1^{\ast}]\Bigg]\nonumber\\&+&
|C_{RR}|^2(2m_{\Lambda_{b}}p_{\Lambda}-q^2-2p_{\Lambda_{b}}\cdot
q)\Bigg[|E_{1}|^2+|D_{1}|^2\Bigg]\nonumber\\&-& q^2
(4m_{\Lambda_{b}}p_{\Lambda}+q^2-4p_{\Lambda_{b}}\cdot
q)\Bigg[|A_{2}|^2+|B_{2}|^2\Bigg] \nonumber\\&-& q^2|C_{RR}|^2
(4m_{\Lambda_{b}}p_{\Lambda}+q^2-4p_{\Lambda_{b}}\cdot
q)\Bigg[|D_{2}|^2+|E_{2}|^2\Bigg] \Bigg\}\end{eqnarray}
\begin{eqnarray} \label{PLI3} I_{3}&=&\frac{64\pi}{3I_{0}}
\Bigg\{\frac{E_{\Lambda}}{m_{\Lambda}}\Bigg[q^4p_{\Lambda}p_{\Lambda_{b}}\Bigg(
Re[A_2B_2^{\ast}]-|C_{RR}|^2(Re[B_1A_1^{\ast}]-Re[B_2A_2^{\ast}])\Bigg)
\nonumber\\&-&m_{\Lambda_{b}}q^2p_{\Lambda}\cdot
q\Bigg[Re[A_1A_2^{\ast}]+Re[B_1B_2^{\ast}]-|C_{RR}|^2(Re[B_1B_2^{\ast}]+
Re[A_1A_2^{\ast}])\Bigg]\Bigg]\nonumber\\&+&
m_{\Lambda}q^2p_{\Lambda_{b}}\cdot
q\Bigg[Re[A_2B_1^{\ast}]+Re[A_1B_2^{\ast}]+|C_{RR}|^2(Re[B_2A_1^{\ast}]+
Re[B_1A_2^{\ast}])\Bigg]\nonumber\\&+&|C_{RR}|^2p_{\Lambda_{b}}\cdot q
p_{\Lambda}\cdot q Re[B_1A_1^{\ast}]-(q^2p_{\Lambda_{b}}\cdot
p_{\Lambda}-2p_{\Lambda_{b}}\cdot q p_{\Lambda}\cdot
q)Re[A_1B_1^{\ast}]\Bigg]\nonumber\\&+&\frac{m_{\Lambda_{b}}}
{m_{\Lambda}}p_{\Lambda}^{2}\Bigg[m_{\Lambda_{b}}q^2\Bigg(
Re[A_1A_2^{\ast}]+Re[B_1B_2^{\ast}]+|C_{RR}|^2
(Re[B_1B_2^{\ast}]+Re[A_1A_2^{\ast}])\Bigg)\nonumber\\&+&m_{\Lambda}q^2\Bigg(
Re[A_2B_1^{\ast}]+Re[A_1B_2^{\ast}]+|C_{RR}|^2
(Re[B_2A_1^{\ast}]+Re[B_1A_2^{\ast}])\Bigg)\Bigg] \Bigg\}\end{eqnarray}
\begin{eqnarray} \label{PN} I_{4}&=&\frac{16\pi}{3I_{0}}\Bigg\{
-m_{\Lambda}m_{\Lambda_{b}}q^2\Bigg[(|A_1|^2+|B_1|^2)+q^2(|A_2|^2+|B_2|^2)
+|C_{RR}|^2(|B_1|^2+|A_1|^2)\nonumber\\&+&|C_{RR}|^2q^2(|B_2|^2+|A_2|^2)\Bigg]+
2q^4p_{\Lambda}\cdot p_{\Lambda_{b}}
\Bigg[Re[A_2B_2^{\ast}]+|C_{RR}|^2Re[B_2A_2^{\ast}]\Bigg]\nonumber\\&-&
2m_{\Lambda_{b}}q^2p_{\Lambda}\cdot q
\Bigg[Re[A_1A_2^{\ast}]+Re[B_1B_2^{\ast}]+|C_{RR}|^2(Re[B_1B_2^{\ast}]
+Re[A_1A_2^{\ast}])\Bigg]\nonumber\\&+& 2m_{\Lambda}q^2p_{\Lambda}\cdot
q\Bigg[Re[A_2B_1^{\ast}]+Re[A_1B_2^{\ast}]+|C_{RR}|^2(Re[B_2A_1^{\ast}]
+Re[B_1A_2^{\ast}])\Bigg]\nonumber\\&-& 2(q^2p_{\Lambda}\cdot
p_{\Lambda_{b}}-2p_{\Lambda}\cdot q\,p_{\Lambda_{b}}\cdot q)
\Bigg[Re[A_1B_1^{\ast}]+|C_{RR}|^2Re[B_1A_1^{\ast}]\Bigg]
\Bigg\}\end{eqnarray}
\begin{eqnarray} \label{PT}
I_{5}&=&\frac{16\pi m_{\Lambda_{b}}p_{\Lambda}}{3I_{0}}\Bigg\{
2m_{\Lambda_{b}}q^2\Bigg[Im[A_1A_2^{\ast}]+Im[B_1B_2^{\ast}]+
|C_{RR}|^2(Im[B_1B_2^{\ast}]+Im[A_1A_2^{\ast}])\Bigg]\nonumber\\&+&
2m_{\Lambda}q^2\Bigg[Im[A_2B_1^{\ast}]+Im[A_1B_2^{\ast}]+
|C_{RR}|^2(Im[B_2A_1^{\ast}]+Im[B_1A_2^{\ast}])\Bigg]\nonumber\\&-&
2q^4\Bigg[Im[A_2B_2^{\ast}]+|C_{RR}|^2Im[B_2A_2^{\ast}]\Bigg]\nonumber\\&+&
2(2m_{\Lambda_{b}}^{2}-q^2-2p_{\Lambda}\cdot p_{\Lambda_{b}})
\Bigg[Im[A_1B_1^{\ast}]+|C_{RR}|^2Im[B_1A_1^{\ast}]\Bigg]
\Bigg\}\end{eqnarray}
Here the kinematics and the relationships for the form factors are
given as follows
\begin{eqnarray}
q^2&=&m_{\Lambda_{b}}^{2}+m_{\Lambda}^{2}-2m_{\Lambda_{b}}E_{\Lambda}\nonumber
\\p_{\Lambda}\cdot p_{\Lambda_{b}}&=&m_{\Lambda_{b}}E_{\Lambda}\nonumber
\\p_{\Lambda_{b}}\cdot
q&=&m_{\Lambda_{b}}^{2}-m_{\Lambda_{b}}E_{\Lambda}\nonumber
\\p_{\Lambda}\cdot q&=&m_{\Lambda_{b}}E_{\Lambda}-m_{\Lambda}^{2}
\end{eqnarray}
and \begin{eqnarray}
A_1&=&(f_1-g_1)\,\,\, , A_2=(f_2-g_2)\,\,\, ,\nonumber
\\B_1&=&(f_1+g_1)\,\,\, , B_2=(f_2+g_2)\,\,\,.\end{eqnarray}
\section{Numerical analysis and discussion \label{s4}}
In this section, we present our numerical analysis on the
branching ratio and $P_L,P_N,P_T$ polarizations of $\Lambda$
baryon. The value of input parameters, we use in our calculations
are
\begin{eqnarray} & &
m_{\Lambda_{b}} =5.62\, GeV \, , \, m_{\Lambda} =1.116 \, GeV \, , \nonumber \\
& & f_B=0.2 \, GeV \, , \, \,\, |V_{tb} V^*_{ts}|=0.04 \, \, , \,
\, \alpha^{-1}=137\, , \nonumber \\ & & |C_{10}^{\nu}|=4.6 \, ,\,\,
G_F=1.17 \times 10^{-5}\, GeV^{-2} \, ,\,\, \tau_{\Lambda_{b}}=1.24 \times 10^{-12} \, s \,\, , \nonumber \\
& & |U_{sb}^{Z'}|\leq0.29\,\, , \,\,\,\,|U_{db}^{Z'}|\leq0.61.
\end{eqnarray} We have assumed that all the neutrinos are
massless. From the expressions of branching ratio and $\Lambda$
baryon polarizations it follows that the form factors are the main
input parameters. The exact calculation for the all form factors
which appear in $\Lambda_{b}\rightarrow \Lambda$ transition does not
exist at present. For the form factors, we will use results from
QCD sum rules method in corporation with heavy quark effective
theory [23,24]. The $q^2$ dependence of the two form factors are
given as follows :\begin{eqnarray} F(q^2)=\frac{F(0)}{1+a_{F}
(q^2/m_{\Lambda_{b}}^2) + b_{F} (q^2/m_{\Lambda_{b}}^2)^2}~.\end{eqnarray} The
values of the form factors parameters are given listed in Table1
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{1}{|c|}{ $ $} &
\multicolumn{1}{|c|}{ $F(0)$} &
\multicolumn{1}{|c|}{ $a_F$} &
\multicolumn{1}{|c|}{ $b_F$} \\
\hline
$F_1$ & $0.462$ & $-0.0182$ & $-0.000176$ \\
\hline
$F_2$ & $-0.077$ & $-0.0685$ & $0.00146$ \\
\hline
\end{tabular}
\end{center}
\caption{ Form factors for $\Lambda_b \rightarrow \Lambda \,\nu
\bar{\nu}$ decay in a three parameters fit.\label{table1}}
\end{table}.
Now, let us examine the new effects to the branching ratio of the
$\Lambda_b \rightarrow \Lambda \, \nu \bar{\nu}$ decay and polarization
effects of the $\Lambda$ baryon in the leptophobic Z$'$ boson. In
the Leptophobic Z$'$ model,there are two new parameters, namely,
the effective FCNC coupling constant $|U_{sb}^{Z'}|$ and the mass
for the Z$'$ boson. Although the mass of Z$'$ boson range is given
$365\,GeV\,\leq M_{Z'}\leq615\,GeV$ in the D0 experiment, we take
$M_{Z'}=700GeV$ and $|U_{sb}^{Z'}|\leq0.29$ [22], which we will
use in our calculations. In Fig. 1, we present the dependence of
the differential decay branching ratio (BR), for $\Lambda_b \rightarrow
\Lambda \,\nu \bar{\nu}$ decay, as function of the momentum
square,$q^2$. We observe from this figure that the differential
branching ratio at low $q^2$ region is one order larger compared
to that of the SM prediction. In Fig.2, we show our prediction for
the branching ratio as a function of the effective coupling
constant $|U_{sb}|$ in the Leptophobic Z$'$ Model. We see from
this figure with increasing $|U_{sb}|$, BR also increases. We see
that difference between two models is clear when $|U_{sb}|\geq
0.4$. In Fig.3, the dependence of the longitudinal polarization
$P_L$ for $\Lambda_b \rightarrow \Lambda \,\nu \bar{\nu}$ decay is
presented. For completeness, in this figure we also present the
prediction of SM for $P_L$. From this figure we see that the
magnitude and sign of $P_L$ are different in these models.
Therefore a measurement of the magnitude and sign of $P_L$ can
provide valuable information about the new physics effects. With
increasing $q^2$ the difference between two models decreases. At
least up to $q^2=10 GeV^{2}$ there is noticeable difference
between these models and this can serve as a good test for
discrimination of these theories. The transverse component $P_T$
of the $\Lambda$ polarization is a T-odd quantity. A nonzero value
of $P_T$ could indicate CP violation. In the SM, there is no CP
violating phase in the CKM element of $V_{tb} V^*_{ts}$ and since
parametrization of the form factors are real, they can not induce
$P_T$ in the $\Lambda_b \rightarrow \Lambda \,\nu \bar{\nu}$ decay.
Therefore if transverse $\Lambda$ polarization is measured in the
experiments to be nonzero, it is an indication of the existence of
new CP violating source and new types of interactions. The
parameter $U_{sb}^{Z'}$ in leptophobic Z$'$ model, in principle,
can have imaginary part and therefore it can induce CP violation.
But in the considered model, there is no interference terms
between SM and leptophobic Z$'$ model contributions and terms
proportional to $U_{sb}^{Z'}$ involves to the branching ratio and
polarization effects in the form $|U_{sb}^{Z'}|^2$. Therefore it
cannot induce CP violation. Depicted in Fig.4 is the dependence of
the normal polarization $P_N$ for $\Lambda_b \rightarrow \Lambda \,\nu
\bar{\nu}$ decay. The difference between the two models in
prediction of $P_N$ becomes considerable at $q^2\geq10 GeV^{2}$.
Therefore analysis of $P_N$ in this region can be informative for
discriminating these models.
When $\Lambda$ is not polarized, summing over the $\Lambda$ spin in Eq.(11),
we get \begin{eqnarray}
d\Gamma=\frac{d\Gamma^{0}}{2}\Bigg(1+
\alpha_{\Lambda_{b}}\hat{e}_{L}\cdot \hat
\xi_{\Lambda_{b}}\Bigg)~,\end{eqnarray} where
$\alpha_{\Lambda_{b}}=\frac{I_1}{I_0}$. So, the polarization of
$\Lambda_{b}$ is $P_{\Lambda_{b}}\equiv \alpha_{\Lambda_{b}}$. In
Fig.5 we present the dependence of polarization of $\Lambda_{b}$
baryon $\alpha_{\Lambda_{b}}$ on $q^2$. We see that up to $q^2=13
GeV^2$ the sign of $\alpha_{\Lambda_{b}}$ in leptophobic model is
positive and negative in SM. When $q^2>13 GeV^2$ situation becomes
vice versa. For this reason measurement of the sign and magnitude of
$\alpha_{\Lambda_{b}}$ at different $q^2$ can provide us essential
information about existence of new physics. For unpolarized
$\Lambda_{b}$ ; i.e. $\hat \xi_{\Lambda_{b}}=0$, we have
\begin{eqnarray}\vec{P_{\Lambda}}=\alpha_{\Lambda}\hat{e}_{L}~,\end{eqnarray} with
$\alpha_{\Lambda}=\frac{I_2}{I_0}$ which leads to the result that the $\Lambda$
polarization is purely longitudinal, i.e. $P_{\Lambda}=
\alpha_{\Lambda}$ and $P_N=P_T=0$.
In conclusion, we have studied the Semileptonic $\Lambda_b \rightarrow
\Lambda \,\nu \bar{\nu}$ decay in the Leptophobic Z$'$ Model. We
have analysed the longitudinal, transverse and normal polarization
of the $\Lambda$ baryon on $q^2$ dependence. The sensitivity of
the branching ratio on $|U_{qb}^{Z'}|$ is investigated and the
dependence of the polarization parameter of $\Lambda_{b}$ baryon
on $q^2$ is also investigated. We find that all physical
observables are very sensitive to the existence of new physics
beyond SM.
\vspace{2cm}
We would like to thank T. M. Aliev and M. Savc{\i} for useful
discussions.
\newpage
\renewcommand{\topfraction}{.99}
\renewcommand{\bottomfraction}{.99}
\renewcommand{\textfraction}{.01}
\renewcommand{\floatpagefraction}{.99}
\begin{figure}
\centering
\includegraphics[width=5in]{dBRdEL}
\caption{The dependence of the differential decay branching
ratio(BR),for $\Lambda_b \rightarrow \Lambda \,\nu \bar{\nu}$ decay, as
function of the momentum transfer,$q^2$ \label{f1}.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=5in]{Usb}
\caption{Integrated branching ratio(BR)as function of the
$|U_{sb}|$, effective coupling constant in the Leptophobic Z$'$
Model. \label{f2}.}
\end{figure}
\clearpage
\begin{figure}
\centering
\includegraphics[width=5in]{BRPLL}
\caption{The dependence of the longitudinal polarization $P_L$
for $\Lambda_b \rightarrow \Lambda \,\nu \bar{\nu}$ decay.\label{f3}}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=5in]{BRPNN}
\caption{The dependence of the normal polarization $P_N$ for
$\Lambda_b \rightarrow \Lambda \,\nu \bar{\nu}$ decay. \label{f4}}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=5in]{alfaLBB}
\caption{The distribution of $\alpha_{\Lambda_{b}}$ as a function
of $q^2$. \label{f5}}
\end{figure}
\clearpage
\newpage
|
1,108,101,563,301 | arxiv | \section{Introduction}
In this article, we pursue the representation-theoretic approach to
Fomin-Zelevinsky's cluster algebras \cite{FominZelevinsky02}
\cite{FominZelevinsky03} \cite{BerensteinFominZelevinsky05}
\cite{FominZelevinsky07} developed by Marsh-Reineke-Zelevinsky
\cite{MarshReinekeZelevinsky03}, Buan-Marsh-Reineke-Reiten-Todorov
\cite{BuanMarshReinekeReitenTodorov06}
\cite{BuanMarshReitenTodorov07}, Caldero-Chapoton and Caldero-Keller
\cite{CalderoChapoton06} \cite{CalderoKeller06}
\cite{CalderoKeller08}, Geiss-Leclerc-Schr\"oer
\cite{GeissLeclercSchroeer05} \cite{GeissLeclercSchroeer06}
\cite{GeissLeclercSchroeer07b}, \cite{BuanIyamaReitenScott07} and
many others, {\em cf.}\ the surveys \cite{BuanMarsh06} \cite{Keller08c}
\cite{Reiten06} \cite{Ringel07} .
Our emphasis here is on cluster algebras {\em with coefficients}.
More precisely, we investigate certain symmetric cluster algebras of
geometric type with coefficients. Coefficients are of great
importance both in geometric examples of cluster algebras
\cite{GekhtmanShapiroVainshtein03}
\cite{GekhtmanShapiroVainshtein05}
\cite{BerensteinFominZelevinsky05}
\cite{Scott06} \cite{GeissLeclercSchroeer07b} and in the study of
duality phenomena \cite{FockGoncharov03} as shown in
\cite{FominZelevinsky07}. Following \cite{BuanIyamaReitenScott07},
we consider two types of categories which allow us to incorporate
coefficients into the representation-theoretic model:
\begin{itemize}
\item[1)] $2$-Calabi-Yau Frobenius categories;
\item[2)] $2$-Calabi-Yau `subtriangulated' categories, {\em i.e.} full
subcategories of the form $^\perp\!(\Sigma {\mathcal D})$ of a
$2$-Calabi-Yau triangulated category ${\mathcal C}$, where
${\mathcal D}$ is a rigid functorially finite subcategory of
${\mathcal C}$ and $\Sigma$ the suspension functor of ${\mathcal
C}$.
\end{itemize}
In both cases, we establish the link between the category and its associated
cluster algebra using (variants of) cluster characters in the sense of
Palu \cite{Palu06}. For subtriangulated categories, we use the
restriction of the cluster characters constructed in \cite{Palu06}.
For Frobenius categories, we construct a suitable variant in section~\ref{CC2FC}
(Theorem~\ref{thm1}).
The work of Geiss-Leclerc-Schr\"oer \cite{GeissLeclercSchroeer06}
\cite{GeissLeclercSchroeer07b} and Buan-Iyama-Reiten-Scott
\cite{BuanIyamaReitenScott07} provides us with large classes of
$2$-Calabi-Yau Frobenius categories and of $2$-Calabi-Yau
subtriangulated categories which admit cluster structures in the
sense of \cite{BuanIyamaReitenScott07}. Our general results imply
that for these classes, the $2$-Calabi-Yau categories do yield
`categorifications' of the corresponding cluster algebras with
coefficients (Theorems~\ref{thm2} and ~\ref{thm3}). As an
application, we show that Conjectures 7.2, 7.10 and 7.12 of
\cite{FominZelevinsky07} hold for these cluster algebras
(Proposition~\ref{prop5} and Theorem~\ref{thm3}). Let us recall the
statements of these conjectures:
\begin{itemize}
\item[7.2] cluster monomials are linearly independent;
\item[7.10] different cluster monomials have different
$\mathbf{g}$-vectors and the $\mathbf{g}$-vectors of the cluster
variables in any given cluster form a basis of the ambient lattice;
\item[7.12] the $\mathbf{g}$-vectors of a cluster variable with respect to
two neighbouring clusters are related by a certain piecewise linear
transformation (so that the $\mathbf{g}$-vectors equal the $\mathbf{g}^\dagger$-vectors
of \cite{DehyKeller08}).
\end{itemize}
In the case of cluster algebras with principal coefficients admitting
a categorification by a 2-Calabi-Yau subtriangulated category, we
obtain a representation-theoretic interpretation of the $F$-polynomial
defined in section~3 of \cite{FominZelevinsky07}, {\em cf.}\ Theorem~\ref{thm4}. This interpretation implies in
particular that Conjecture~5.4 of \cite{FominZelevinsky07} holds in
this case: The constant coefficient of the $F$-polynomial equals $1$.
We also deduce that the multidegree of the $F$-polynomial associated
with a rigid indecomposable equals the dimension vector of the
corresponding module (Proposition~\ref{prop7}). By combining this with recent work by
Buan-Marsh-Reiten \cite{BuanMarshReiten08a}, {\em cf.}\ also \cite{Dupont07}, we obtain a counterexample to
Conjecture~7.17 (and 6.11) of \cite{FominZelevinsky07}. We point out
that the corresponding computations were already present in
G.~Cerulli's work \cite{Cerulli08}. Following a suggestion by A.~Zelevinsky,
we show that, by assuming the existence of suitable categorifications,
instead of the equality claimed in Conjecture~7.17, one does have an
inequality: The multidegree of the $F$-polynomial is greater or equal
to the denominator vector (Proposition~\ref{prop8}). We also show in certain cases that the transformation rule for
$\mathbf{g}$-vectors predicted by Conjecture~6.10 of
\cite{FominZelevinsky07} does hold (Proposition~\ref{prop9}).
Let us emphasize that our proofs for certain cluster algebras
of some of the conjectures
of \cite{FominZelevinsky07} depend on the existence of suitable
Hom-finite $2$-Calabi-Yau categories with a cluster-tilting object.
This hypothesis imposes a finiteness condition on the corresponding
cluster algebra (to the best of our knowledge, it is not known how
to express this condition in combinatorial terms). The construction
of such $2$-Calabi-Yau categories is a non trivial problem for which
we rely on \cite{BuanMarshReinekeReitenTodorov06} in the acyclic
case and on \cite{GeissLeclercSchroeer06}
\cite{GeissLeclercSchroeer07b} \cite{BuanIyamaReitenScott07} and
\cite{Amiot08a} in the non acyclic case. As A.~Zelevinsky has kindly
informed us, many of the conjectures of \cite{FominZelevinsky07}
will be proved in \cite{DerksenWeymanZelevinsky09} in full
generality building on \cite{MarshReinekeZelevinsky03} and
\cite{DerksenWeymanZelevinsky07}.
{\bf Acknowledgments.} We thank A.~Zelevinsky for stimulating
discussions and for informing us about the ongoing work in
\cite{DerksenWeymanZelevinsky09} and \cite{Cerulli08}. We are grateful to
A.~Buan and I.~Reiten for sharing the results on dimension vectors
obtained in \cite{BuanMarshReiten08a}. We thank J.~Schr\"oer and O.~Iyama for
their interest and for motivating discussions. The first-named
author gratefully acknowledges a 5-month fellowship of the Liegrits
network (MRTN-CT 2003-505078) during which this research was carried
out. Both authors thank the referee for his great help in making
this article more readable.
\section{Recollections}
\subsection{Cluster algebras} \label{ss:ClusterAlgebras}
In this section, we recall
the construction of cluster algebras of geometric type with
coefficients from \cite{FominZelevinsky07}. For an integer $x$,
we use the notations
\[
[x]_+=\max(x,0)
\]
and
\[
\opname{sgn}(x)= \left\{ \begin{array}{ll} -1 & \mbox{if $x<0$;} \\ 0 & \mbox{if $x=0$;} \\ -1 & \mbox{if $x<0$.}
\end{array} \right.
\]
The {\em tropical semifield} on a finite family of variables $u_j$,
$j\in J$, is the abelian group (written multiplicatively) freely
generated by the $u_j$, $j\in J$, endowed with the {\em addition}
$\oplus$ defined by
\[
\prod_j u_j^{a_j} \oplus \prod_j u_j^{b_j} = \prod_j u_j^{\min(a_j,
b_j)}.
\]
Let $1\leq r\leq n$ be integers. Let $\P$ be the tropical semifield
on the indeterminates $x_{r+1}, \ldots, x_n$. Let $\mathbb{Q}\P$ be the
group algebra on the abelian group $\P$. It identifies with the
algebra of Laurent polynomials with rational coefficients in the
variables $x_{r+1},\ldots , x_n$. Let ${\mathcal F}$ be the field of
fractions of the ring of polynomials with coefficients in $\mathbb{Q}\P$ in
$r$ indeterminates. A {\em seed} in ${\mathcal F}$ is a pair
$(\tilde{B},\mathbf{x})$ formed by
\begin{itemize}
\item an $n\times r$-matrix $\tilde{B}$ with integer entries
whose {\em principal part} $B$ (the submatrix formed by the first
$r$ rows) is antisymmetric;
\item a free generating set $\mathbf{x}=\{x_1, \ldots, x_r\}$
of the field ${\mathcal F}$.
\end{itemize}
The matrix $\tilde{B}$ is called the {\em exchange matrix}
and the set $\mathbf{x}$ the {\em cluster} of the
seed $(\tilde{B}, \mathbf{x})$.
Let $1\leq s\leq r$ be an integer. The {\em seed mutation in the direction $s$}
transforms the seed $(\tilde{B},\mathbf{x})$ into the seed
$\mu_s(\tilde{B},\mathbf{x})=(\tilde{B}', \mathbf{x}')$, where
\begin{itemize}
\item the entries $b'_{ij}$ of $\tilde{B}'$ are given by
\[
b'_{ij} = \left\{ \begin{array}{ll} -b_{ij} & \mbox{if $i=s$ or
$j=s$; } \\
b_{ij}+ \opname{sgn}(b_{is}) [b_{is} b_{sj}]_+ & \mbox{otherwise.}
\end{array} \right.
\]
\item The cluster $\mathbf{x'}=\{x'_1, \ldots x'_r\}$ is
given by $x'_j=x_j$ for $j\neq s$ whereas $x'_s\in{\mathcal F}$ is determined
by the {\em exchange relation}
\[
x'_s x_s = \prod_{i=1}^n x_i^{[b_{is}]_+} + \prod_{i=1}^n
x_i^{[-b_{is}]_+}.
\]
\end{itemize}
Mutation in a fixed direction is an involution.
Let $\mathbb{T}_r$ be the {\em $r$-regular tree}, whose edges are labeled by
the numbers $1,\ldots, r$ so that the $r$ edges emanating from each
vertex carry different labels. A {\em cluster pattern} is the assignment
of a seed $(\tilde{B}_t, \mathbf{x}_t)$ to each vertex $t$ of $\mathbb{T}_r$ such
that the seeds assigned to vertices $t$ and $t'$ linked by
an edge labeled $s$ are obtained from each other by the seed
mutation $\mu_s$.
Fix a vertex $t_0$ of the $r$-regular tree $\mathbb{T}_r$. Clearly, a
cluster pattern is uniquely determined by the {\em initial seed}
$(\tilde{B}_{t_0}, x_{t_0})$, which can be chosen arbitrarily.
Fix a seed $(\tilde{B}, \mathbf{x})$ and let
$(\tilde{B}_t, \mathbf{x}_t)$, $t\in \mathbb{T}_r$
be the unique cluster pattern with initial seed $(\tilde{B}, \mathbf{x})$.
The {\em clusters} associated with $(\tilde{B}, \mathbf{x})$ are
the sets $\mathbf{x}_t$, $t\in\mathbb{T}_r$. The {\em cluster variables}
are the elements of the clusters. The {\em cluster algebra}
${\mathcal A}(\tilde{B}) = {\mathcal A}(\tilde{B},\mathbf{x})$ is the $\mathbb{Z}\P$-subalgebra of ${\mathcal F}$
generated by the cluster variables. Its {\em ring of coefficients} is
$\mathbb{Z}\P$. It is a `cluster algebra without coefficients' if $r=n$ and
thus $\mathbb{Z}\P=\mathbb{Z}$.
\subsection{Cluster algebras from ice quivers} \label{ss:IceQuivers}
As we have seen in the
previous subsection, our cluster
algebras are given by certain integer matrices $\tilde{B}$. Such
matrices can also be encoded by `ice quivers': A {\em quiver} is a
quadruple $Q=(Q_0, Q_1, s,t)$, where $Q_0$ is a set (the set of {\em
vertices}), $Q_1$ is a set (the set of {\em arrows}) and $s$ and $t$
are two maps $Q_1 \to Q_0$ (taking an arrow to its {\em source}
respectively to its {\em target}). An {\em ice quiver} is a pair
$(Q,F)$ consisting of a quiver $Q$ and a subset $F$ of its set of
vertices ($F$ is the set of {\em frozen vertices}).
Let $(Q,F)$ be an ice quiver such that the set $Q_0$ is the set of natural
numbers from $1$ to $n$, the set $Q_1$ is finite and the set $F$ is the set of
natural numbers from $r+1$ to $n$ for some $1\leq r\leq n$.
The associated integer matrix $\tilde{B}(Q,F)$ is the $n\times r$
matrix whose entry $b_{ij}$ equals the
number of arrows from $i$ to $j$ minus the number of arrows from $j$ to $i$.
The {\em cluster algebra with coefficients ${\mathcal A}(Q,F)$} is defined as the cluster
algebra ${\mathcal A}(\tilde{B}(Q,F))$.
The matrix $\tilde{B}(Q,F)$ determines the ice quiver $(Q,F)$ if
\begin{itemize}
\item[1)] $Q$ does not have loops (arrows from a vertex to itself) and
\item[2)] $Q$ does not have $2$-cycles (pairs of distinct arrows $\alpha$, $\beta$ such that
$s(\alpha)=t(\beta)$ and $t(\alpha)=s(\beta)$) and
\item[3)] there are no arrows between any vertices of $F$.
\end{itemize}
Given integers $1 \leq r\leq n$ each integer matrix $\tilde{B}$ with
antisymmetric principal part $B$ (formed by the first $r$ rows of
$\tilde{B})$, is obtained as the matrix associated with a unique ice
quiver satisfying these properties. The {\em mutation of
ice quivers} satisfying conditions 1)-3) is defined via
the mutation of the corresponding integer matrices recalled
in section~\ref{ss:ClusterAlgebras}.
\subsection{Krull-Schmidt categories} \label{ss:KrullSchmidtCategories}
An additive category {\em has split idempotents}
if each idempotent endomorphism $e$ of an object $X$ gives rise to a direct sum decomposition
$Y\oplus Z\stackrel{_\sim}{\rightarrow} X$ such that $Y$ is a kernel for $e$. A {\em Krull-Schmidt category} is an
additive category where the endomorphism rings of indecomposable objects are
local and each object decomposes into a finite direct sum of indecomposable
objects (which are then unique up to isomorphism and permutation).
Each Krull-Schmidt category has split idempotents.
We {\em write $\mathsf{indec}({\mathcal C})$} for the set of isomorphism classes
of indecomposable objects of a Krull-Schmidt category ${\mathcal C}$.
Let ${\mathcal C}$ be a Krull-Schmidt category.
An object $X$ of ${\mathcal C}$ is {\em basic} if every indecomposable of ${\mathcal C}$
occurs with multiplicity $\leq 1$ in $X$. In this case, $X$ is fully
determined by the full additive {\em subcategory $\opname{add}\nolimits(X)$} whose objects are the
direct factors of finite direct sums of copies of $X$. The map $X \mapsto \opname{add}\nolimits(X)$
yields a bijection between the isomorphism classes of basic objects and
the full additive subcategories of ${\mathcal C}$ which are stable under
taking direct factors and only contain finitely many indecomposables
up to ismorphism.
Let $k$ be an algebraically closed field. A {\em $k$-category} is
a category whose morphism sets are endowed with structures of
$k$-vector spaces such that the composition maps are bilinear. A
$k$-category is {\em $\opname{Hom}$-finite} if all of its morphism spaces
are finite-dimensional. A {\em $k$-linear category} is a $k$-category
which is additive. Let ${\mathcal C}$ be a $k$-linear $\opname{Hom}$-finite
category with split idempotents. Then ${\mathcal C}$ is a Krull-Schmidt category.
Let ${\mathcal T}$ be an additive subcategory of ${\mathcal C}$ stable under taking direct
factors. The {\em quiver $Q=Q({\mathcal T})$ of ${\mathcal T}$}
is defined as follows: The vertices of $Q$ are the isomorphism classes of indecomposable
objects of ${\mathcal T}$ and the number of arrows from the isoclass of
$T_1$ to that of $T_2$ equals the dimension of the space of irreducible
morphisms
\[
\mathsf{irr}(T_1, T_2) = \mathsf{rad}(T_1, T_2)/\mathsf{rad}^2(T_1,T_2) \: , \;
\]
where $\mathsf{rad}$ denotes the {\em radical of ${\mathcal T}$}, {\em i.e.}\ the ideal such
that $\mathsf{rad}(T_1,T_2)$ is formed by all non isomorphisms from $T_1$ to $T_2$.
\subsection{$2$-Calabi-Yau triangulated categories} \label{ss:CalabiYauCategories}
Let $k$ be an algebraically closed field. Let ${\mathcal C}$ be a $k$-linear
triangulated category with suspension functor $\Sigma$. We assume
that ${\mathcal C}$ is {\em $\opname{Hom}$-finite} and has split idempotents.
Thus, it is a Krull-Schmidt category. For objects
$X$, $Y$ of ${\mathcal C}$ and an integer $i$, we define
\[
\opname{Ext}^i(X,Y)={\mathcal C}(X,\Sigma^i Y).
\]
An object $X$ of ${\mathcal C}$ is {\em rigid} if $\opname{Ext}^1(X,X)=0$.
Let $d$ be an integer. Following \cite{VandenBergh08}, {\em cf.}\ ~also
\cite{Keller08b}, we define the category ${\mathcal C}$ to be {\em
$d$-Calabi-Yau} if there exists a family of linear forms
\[
t_X : {\mathcal C}(X,\Sigma^d X) \to k \: , \; X\in\opname{obj}\nolimits({\mathcal C})\: , \;
\]
such that the bilinear forms
\[
\langle, \rangle: {\mathcal C}(Y, \Sigma^d X) \times {\mathcal C}(X,Y) \to k \: , \; (f,g) \mapsto t_X(f\circ g)
\]
are non degenerate and satisfy
\[
\langle \Sigma^p f, g\rangle = (-1)^{pq} \langle \Sigma^q g, f\rangle
\]
for all $f$ in ${\mathcal C}(Y, \Sigma^q X)$ and all $g\in {\mathcal C}(X, \Sigma^p Y)$,
where $p+q=d$.
Let us assume that ${\mathcal C}$ is $2$-Calabi-Yau. A {\em cluster-tilting
subcategory of ${\mathcal C}$} is a full additive subcategory ${\mathcal T}\subset
{\mathcal C}$ which is stable under taking direct factors and such that
\begin{itemize}
\item for each object $X$ of ${\mathcal C}$, the functors
${\mathcal C}(X,?):{\mathcal T}\to \mod k$ and ${\mathcal C}(?,X):{\mathcal T}^{op}\to\mod k$
are finitely generated;
\item an object $X$ of ${\mathcal C}$ belongs to ${\mathcal T}$ iff we have
$\opname{Ext}^1(T,X)=0$ for all objects $T$ of ${\mathcal T}$.
\end{itemize}
A {\em cluster-tilting object} is a basic object $T$ of ${\mathcal C}$ such that
$\opname{add}\nolimits(T)$ is a cluster-tilting subcategory. Equivalently, an object $T$
is cluster-tilting if it is rigid and if each object $X$ satisfying
$\opname{Ext}^1(T,X)=0$ belongs to $\opname{add}\nolimits(T)$. The following definition is
taken from section~1 of \cite{BuanIyamaReitenScott07}. Recall that ${\mathcal C}$ is
a $\opname{Hom}$-finite $k$-linear triangulated category with split
idempotents which is $2$-Calabi-Yau.
\begin{definition}[\cite{BuanIyamaReitenScott07}]
\label{def:TriangClusterStructure}
The cluster-tilting subcategories of ${\mathcal C}$ {\em determine a cluster
structure on ${\mathcal C}$} if the following hold:
\begin{itemize}
\item[0)] There is at least one cluster-tilting subcategory in ${\mathcal C}$.
\item[1)] For each cluster-tilting subcategory ${\mathcal T}'$ of
${\mathcal C}$ and each indecomposable $M$ of ${\mathcal T}'$, there is a unique (up
to isomorphism) indecomposable $M^*$ not isomorphic to $M$ and such
that the additive subcategory ${\mathcal T}''=\mu_M({\mathcal T}')$ of ${\mathcal C}$ with
set of indecomposables
\[
\mathsf{indec}({\mathcal T}'')=\mathsf{indec}({\mathcal T}')\setminus\{M\} \cup
\{M^*\}
\]
is a cluster-tilting subcategory;
\item[2)] In the situation of 1), there are triangles
\[
\xymatrix{M^* \ar[r]^f & E \ar[r]^g & M \ar[r] & \Sigma M^*}
\mbox{ and }
\xymatrix{M \ar[r]^s & E' \ar[r]^t & M^* \ar[r] & \Sigma M^*} \: , \;
\]
where $g$ and $t$ are minimal right ${\mathcal T}'\cap{\mathcal T}''$-approximations
and $f$ and $s$ are minimal left ${\mathcal T}'\cap{\mathcal T}''$-approximations.
\item[3)] For any cluster-tilting subcategory ${\mathcal T}'$, the quiver
$Q({\mathcal T}')$ does not have loops nor $2$-cycles.
\item[4)] We have $Q(\mu_M({\mathcal T}'))=\mu_M(Q({\mathcal T}'))$ for each cluster-tilting
subcategory ${\mathcal T}'$ of ${\mathcal C}$ and each indecomposable $M$ of ${\mathcal T}'$.
\end{itemize}
The cluster tilting subcategory ${\mathcal T}''=\mu_M({\mathcal T}')$ of 1) is
the {\em mutation of ${\mathcal T}'$ at the indecomposable object $M$}.
The {\em mutation of a cluster-tilting object $T$} is defined via
the mutation of the cluster-tilting subcategory $\opname{add}\nolimits(T)$.
\end{definition}
\begin{lemma} \label{lemma:Multiplicities} Suppose that the
cluster-tilting subcategories determine a cluster structure on
${\mathcal C}$. Then, in the situation of condition~2) of the above
definition~\ref{def:TriangClusterStructure}, the following hold:
\begin{itemize}
\item[a)] The space $\opname{Ext}^1(M,M^*)$ is one-dimensional (hence, by the $2$-Calabi-Yau
property, so is the space $\opname{Ext}^1(M^*,M)$) and the triangles of 2) are non split.
\item[3)] The multiplicity
of an indecomposable $U$ of ${\mathcal T}'\cap{\mathcal T}''$ in $E$ equals the number
of arrows from $U$ to $M$ in the quiver $Q({\mathcal T}')$ and that from $M^*$ to $U$
in $Q({\mathcal T}'')$; the multiplicity of $U$ in $E'$ equals the number of
arrows from $M$ to $U$ in $Q({\mathcal T}')$ and that from $U$ to $M^*$ in
$Q({\mathcal T}'')$;
\end{itemize}
\end{lemma}
\begin{proof} a) The first triangle yields an exact sequence
\[
{\mathcal C}(M,E) \to {\mathcal C}(M,M) \to \opname{Ext}^1(M,M^*) \to 0.
\]
By the absence of loops required in condition 3),
each radical morphism from $M$ to $M$ factors through $E$. Since $k$
is algebraically closed, the radical is of codimension $1$ in the
local algebra ${\mathcal C}(M,M)$. Thus, the space $\opname{Ext}^1(M,M^*)$ is one-dimensional.
The minimality of the approximations implies that the triangles are non split.
b) This follows if we combine the definition of the quivers $Q({\mathcal T}')$ and $Q({\mathcal T}'')$,
with the approximation properties of $f$, $g$, $s$ and $t$.
\end{proof}
We refer to section~1, page~11 of
\cite{BuanIyamaReitenScott07} for a list of classes of examples
where this assumption holds. In particular, this list contains
the cluster categories associated with finite quivers without oriented
cycles and the stable categories of preprojective algebras of Dynkin quivers.
We refer the reader to the surveys \cite{BuanMarsh06} \cite{Reiten06} \cite{Keller08}
\cite{Keller08c}
for more information on cluster categories and to
the survey \cite{GeissLeclercSchroeer08}
for more information on stable categories of Dynkin quivers.
\subsection{Cluster characters} \label{ss:ClusterCharacters}
The notion of cluster character is due to Palu \cite{Palu08}.
Under suitable assumptions, cluster characters allow one to pass
from $2$-Calabi-Yau categories to cluster algebras. We recall
the definition and the main construction from \cite{Palu08}.
Let $k$ be an algebraically closed field and
${\mathcal C}$ a $k$-linear $\opname{Hom}$-finite triangulated category with
split idempotents which is $2$-Calabi-Yau.
Let $R$ be a commutative ring. A {\em cluster character} on ${\mathcal C}$
with values in $R$ is a map $\zeta: \opname{obj}\nolimits({\mathcal C}) \to R$ such that
\begin{itemize}
\item[1)] we have $\zeta(L)=\zeta(L')$ if and $L$ and $L'$ are isomorphic,
\item[2)] we have $\zeta(L\oplus M) = \zeta(L)\zeta(M)$ for all objects $L$ and $M$ and
\item[3)] if $L$ and $M$ are objects such that $\opname{Ext}^1(L,M)$ is one-dimensional
(and hence $\opname{Ext}^1(M,L)$ is one-dimensional) and
\[
L \to E \to M \to \Sigma L \mbox{ and } M \to E' \to L \to \Sigma M
\]
are non-split triangles, then we have
\begin{equation} \label{eq:ClusterCharacterExchangeEquation}
\zeta(L) \zeta(M) = \zeta(E) + \zeta(E').
\end{equation}
\end{itemize}
Assume that ${\mathcal C}$ has a cluster-tilting object $T$ which is the
direct sum of $r$ pairwise non isomorphic indecomposable summands $T_1$,
\ldots $T_r$. In a vast generalization of Caldero-Chapoton's work
\cite{CalderoChapoton06}, Palu has shown in \cite{Palu08} that there
is a canonical cluster-character
\[
X^T_?: \opname{obj}\nolimits({\mathcal C}) \to \mathbb{Z}[x_1, \ldots, x_r] \: , \; M \mapsto X^T_M
\]
such that $X^T_{\Sigma T_i}=x_i$ for $1\leq i \leq r$. Let us recall
Palu's construction. First we need to introduce some more notation.
Let $C$ be the endomorphism algebra of $T$. Let $\mod C$ denote the
category of $k$-finite-dimensional right $C$-modules. For each
$1\leq i \leq r$, the morphism space ${\mathcal C}(T,T_i)$ becomes an
indecomposable projective right $C$-module denoted by $P_i$. Its
simple top is denoted by $S_i$. For $L$ and $M$ in $\mod C$, define
\[
\langle L, M\rangle = \dim \opname{Hom}_C(L,M) - \dim \opname{Ext}^1_C(L,M)
\]
and put
\[
\langle L, M\rangle_a = \langle L,M \rangle - \langle M,L \rangle.
\]
By Theorem~11 of \cite{Palu08}, the map $(L,M) \mapsto \langle
L,M\rangle_a$ induces a well-defined bilinear form on the
Grothendieck group $K_0(\mod C)$. By \cite{KellerReiten07}, for any
$X\in {\mathcal C}$, we have triangles
\[
T^X_1\to T^X_0\to X\to \Sigma T^X_1 \mbox{ and }
X\to \Sigma^2 T^0_X\to \Sigma^2 T^1_X\to \Sigma X,
\]
where $T^X_1$, $T^X_0$, $T_X^0$ and $T_X^1$ belong to $\opname{add}\nolimits(T)$.
The {\em index} and {\em coindex} of $X$ with respect to $T$ are defined to
be the classes in $K_0(\opname{add}\nolimits T)$
\[
\opname{ind}_T (X)=[T^X_0]-[T^X_1] \mbox { and }
\opname{coind}_T (X)= [T_X^0]-[T_X^1].
\]
For an object $M$ of ${\mathcal C}$, one defines
\[
X^T_M = \prod_{i=1}^r x_i^{-[\operatorname{coind}_{T}(M): T_i]}
\sum_e \chi(Gr_e({\mathcal C}(T,M)) \, \prod_{i=1}^r x_i^{\langle S_i,e\rangle_a},
\]
where $e$ runs through the positive elements of the Grothendieck
group $K_0(\mod C)$ and $Gr_e({\mathcal C}(T,M))$ denotes the variety
of submodules $U$ of the right $C$-module ${\mathcal C}(T,M)$ such
that the class of $U$ is $e$ and $\chi$ is the Euler characteristic
(of the underlying topological space if $k=\mathbb{C}$ or of $l$-adic
cohomology if $k$ is arbitrary).
\subsection{From $2$-CY categories to cluster algebras without coefficients}
In this section, we show how certain $2$-Calabi-Yau triangulated categories can be linked to
cluster algebras without coefficients via cluster characters. All we need to do is to combine the results
recalled in sections \ref{ss:CalabiYauCategories} and \ref{ss:ClusterCharacters}. In
the main part of the paper, we will concentrate on the case where our
cluster algebras do have coefficients.
Let $k$ be an algebraically closed field and ${\mathcal C}$ a $\opname{Hom}$-finite
$k$-linear $2$-Calabi-Yau triangulated category with split idempotents as defined
in section~\ref{ss:CalabiYauCategories}. Let $T$ be a
cluster-tilting object in ${\mathcal C}$. Assume that $T$ is the direct sum
of $r$ pairwise non isomorphic indecomposable objects $T_1$, \ldots,
$T_r$. Let
\[
\zeta_T : \opname{obj}\nolimits({\mathcal C}) \to \mathbb{Q}(x_1, \ldots, x_r)
\]
be Palu's cluster character associated with $T$ as recalled in
section~\ref{ss:ClusterCharacters}. In particular, we have
\begin{equation} \label{eq:ClusterCharInitialValues}
\zeta_T(\Sigma T_i)=x_i \mbox{ for } 1\leq i\leq r.
\end{equation}
Now assume that the cluster-tilting subcategories define a cluster structure
on ${\mathcal C}$ ({\em cf.}\ section~\ref{ss:CalabiYauCategories}). A cluster-tilting
object $T'$ is {\em reachable from $T$} if $\opname{add}\nolimits(T')$ is obtained
from $\opname{add}\nolimits(T)$ be a finite sequence of mutations as defined in
\ref{ss:CalabiYauCategories}. A rigid object $M$ is {\em reachable
from $T$} if it lies in $\opname{add}\nolimits(T')$ for a cluster-tilting object $T'$
reachable from $T$. Let $Q$ be the quiver of the endomorphism
algebra $C$ of $T$, or equivalently, the quiver of the category
$\opname{add}\nolimits(T)$. We consider $Q$ as an ice quiver with empty set of frozen
vertices and denote by ${\mathcal A}(Q)$ the associated cluster algebra
without coefficients (defined by specializing the construction of
\ref{ss:IceQuivers} to the case where the set of frozen vertices is
empty). It is the subalgebra of $\mathbb{Q}(x_1, \ldots, x_r)$ generated by
the cluster variables.
\begin{proposition} \label{prop:ClusterCharacterProperties}
Assume that the above assumptions hold and
in particular that the cluster-tilting subcategories define
a cluster-structure on ${\mathcal C}$ ({\em cf.}\
section~\ref{ss:CalabiYauCategories}).
\begin{itemize}
\item[a)] The map $M \mapsto \zeta_T(\Sigma M)$ induces a surjection from
the set of rigid objects reachable from $T$ onto the set of cluster
variables of the cluster algebra ${\mathcal A}(Q)$.
\item[b)] The surjection of a) induces a
surjection from the set of cluster-tilting objects reachable from
$T$ onto the set of clusters of ${\mathcal A}(Q)$.
\end{itemize}
\end{proposition}
\begin{proof} Clearly, part a) follows from part b). Let us prove part b).
Let $\mathbb{T}_r$ be the $r$-regular tree and let $t_0$ be a fixed vertex of
$\mathbb{T}_r$. Let $B$ be the antisymmetric matrix associated with the
quiver $Q$ and let $\mathbf{x}$ be the initial cluster $x_1, \ldots, x_r$.
Let $(B_t, \mathbf{x}_t)$, $t\in \mathbb{T}_r$, be the unique cluster pattern
with initial seed $(B_{t_0}, \mathbf{x}_{t_0})=(B,\mathbf{x})$
({\em cf.}\ section~\ref{ss:ClusterAlgebras}). To each
vertex $t$ of $\mathbb{T}$, we assign a cluster-tilting object $T_t$ with
indecomposable direct summands $T_{t,1}$, \ldots, $T_{t,r}$ such that
\begin{itemize}
\item[1)] we have $T_{t_0}=T$ and
\item[2)] if $t$ is linked to $t'$ by an edge labeled $s$, then $T_{t'}$
is obtained from $T_t$ by mutating at the summand $T_{t,s}$.
\end{itemize}
It follows from point 1) of the definition of a cluster structure that
$T_t$ is well-defined for each vertex $t$ of $\mathbb{T}$. Moreover, it follows
from point 4) of the same definition that for each vertex $t$ of $\mathbb{T}$,
the antisymmetric matrix $B_t$ corresponds to the quiver of the category
$\opname{add}\nolimits(T_t)$ under the bijection of section~\ref{ss:IceQuivers}. We
claim that for each vertex $t$ of $\mathbb{T}$, the cluster character takes
the shift $\Sigma T_{t,i}$ of the indecomposable direct summand $T_{t,i}$
of $T_t$ to the cluster variable $x_{t,i}$, $1 \leq i\leq r$. Indeed, this
holds for $t=t_0$ by equation~(\ref{eq:ClusterCharInitialValues}).
Now assume that it holds for some vertex $t$ and that $t$ is linked
to a vertex $t'$ by an edge labeled $s$. We know that the indecomposable summands
of $T_{t'}$ are the $T_{t',i}=T_{t,i}$ for $i\neq s$ and a new summand $T'_{t,s}$
which is not isomorphic to $T_{t,s}$. By part a) of lemma~\ref{lemma:Multiplicities},
the extension space between $T_{t,s}$ and $T_{t',s}$
is one-dimensional and we have non split triangles
\[
T_{t',s} \to E \to T_{t,s} \to \Sigma T_{t',s} \mbox{ and }
T_{t,s} \to E' \to T_{t',s} \to \Sigma T_{t,s}.
\]
Here, the middle terms are sums of copies of the $T_{t,i}$, $i\neq s$,
and the multiplicities are determined by the quivers of the
endomorphism algebras of $T$ and $T'$, as indicated in part b)
of lemma~\ref{lemma:Multiplicities}. More precisely, if $b_{ij}^t$ denotes the $(i,j)$-entry of
the exchange matrix, the summand $T_{t,i}$ occurs in $E$ with multiplicity
$[b_{is}^t]_+$ and in $E'$ with multiplicity $[b_{si}^t]_+=[-b_{is}^t]_+$. Now
if we use points 2) and 3) of the definition
of a cluster character, we see that the induction hypothesis
and equation~(\ref{eq:ClusterCharacterExchangeEquation})
yield the exchange relation
\[
x_{t,s} \zeta_T(\Sigma T_{t',s}) =\prod_{i=1}^n x_i^{[b^t_{ik}]_+} + \prod_{i=1}^n
x_i^{[-b^t_{ik}]_+}.
\]
Thus, we have $\zeta_T(\Sigma T_{t',s}) = x_{t',s}$ as required.
\end{proof}
\subsection{Frobenius categories} \label{ss:FrobeniusCategories}
A {\em Frobenius category} is an exact category
in the sense of Quillen \cite{Quillen73} which has enough projectives and enough injectives
and where an object is projective iff it is injective. By definition,
such a category is endowed with a class of admissible exact sequences
\[
0 \to L \to M \to N \to 0.
\]
Following \cite{GabrielRoiter92} we will call the left morphism
$L\to M$ of such a sequence an {\em inflation}, the right morphism a
{\em deflation} and, sometimes, the whole sequence a {\em
conflation}. Let ${\mathcal E}$ be a Frobenius category. Its associated
stable category $\ul{{\mathcal E}}$ is the quotient of ${\mathcal E}$ by the ideal of
morphisms factoring through a projective-injective object. It was
shown by Happel \cite{Happel87} that $\ul{{\mathcal E}}$ has a canonical
structure of triangulated category. We have
\[
\opname{Ext}^i_{\mathcal E}(L,M) \stackrel{_\sim}{\rightarrow} \opname{Ext}^i_{\ul{{\mathcal E}}}(L,M)
\]
for all objects $L$ and $M$ of ${\mathcal E}$ and all integers $i\geq 1$.
An object $M$ of ${\mathcal E}$ is {\em rigid} if $\opname{Ext}^1_{\mathcal E}(M,M)=0$.
Let $k$ be an algebraically closed field and ${\mathcal E}$ a $\opname{Hom}$-finite
Frobenius category with split idempotents.
Suppose that ${\mathcal E}$ is a {\em $2$-Calabi-Yau Frobenius category}, {\em i.e.}\ its associated stable
category ${\mathcal C}=\ul{{\mathcal E}}$ is $2$-Calabi-Yau in the sense of section~\ref{ss:CalabiYauCategories}.
A {\em cluster-tilting subcategory of ${\mathcal E}$} is a full additive subcategory ${\mathcal T}\subset
{\mathcal E}$ which is stable under taking direct factors and such that
\begin{itemize}
\item for each object $X$ of ${\mathcal E}$, the functors
${\mathcal E}(X,?):{\mathcal T}\to \mod k$ and ${\mathcal E}(?,X):{\mathcal T}^{op}\to\mod k$
are finitely generated;
\item an object $X$ of ${\mathcal E}$ belongs to ${\mathcal T}$ iff we have
$\opname{Ext}^1_{\mathcal E}(T,X)=0$ for all objects $T$ of ${\mathcal T}$.
\end{itemize}
Clearly if these conditions hold, each projective-injective
object of ${\mathcal E}$ belongs to ${\mathcal T}$.
A {\em cluster-tilting object} is a basic object $T$ of ${\mathcal E}$ such that
$\opname{add}\nolimits(T)$ is a cluster-tilting subcategory. Equivalently, an object $T$
is cluster-tilting if it is rigid and if each object $X$ satisfying
$\opname{Ext}^1_{\mathcal E}(T,X)=0$ belongs to $\opname{add}\nolimits(T)$. The following definition
is taken from section~1 of \cite{BuanIyamaReitenScott07}. Recall
that ${\mathcal E}$ is a $k$-linear $\opname{Hom}$-finite Frobenius category with
split idempotents such that the associated stable category
${\mathcal C}=\ul{{\mathcal E}}$ is $2$-Calabi-Yau.
\begin{definition}[\cite{BuanIyamaReitenScott07}]
\label{def:ExactClusterStructure}
The cluster-tilting subcategories of ${\mathcal E}$ {\em determine a cluster
structure on ${\mathcal E}$} if the following hold:
\begin{itemize}
\item[0)] There is at least one cluster-tilting subcategory in ${\mathcal E}$.
\item[1)] For each cluster-tilting subcategory ${\mathcal T}'$ of ${\mathcal E}$ and
each non projective indecomposable $M$ of ${\mathcal T}'$, there is a unique
(up to isomorphism) non projective indecomposable $M^*$ not
isomorphic to $M$ and such that the additive subcategory
${\mathcal T}''=\mu_M({\mathcal T}')$ of ${\mathcal E}$ with set of indecomposables
\[
\mathsf{indec}({\mathcal T}'')=\mathsf{indec}({\mathcal T}')\setminus\{M\} \cup
\{M^*\}
\]
is a cluster-tilting subcategory;
\item[2)] In the situation of 1), there are conflations
\[
\xymatrix{0 \ar[r] & M^* \ar[r]^f & E \ar[r]^g & M \ar[r] & 0}
\mbox{ and }
\xymatrix{0 \ar[r] & M \ar[r]^s & E' \ar[r]^t & M^* \ar[r] & 0} \: , \;
\]
where $g$ and $t$ are minimal right ${\mathcal T}'\cap{\mathcal T}''$-approximations
and $f$ and $s$ are minimal left ${\mathcal T}'\cap{\mathcal T}''$-approximations.
\item[3)] For any cluster-tilting subcategory ${\mathcal T}'$, the quiver
$Q({\mathcal T}')$ does not have loops nor $2$-cycles.
\item[4)] We have $Q^\circ(\mu_M({\mathcal T}'))=\mu_M(Q^\circ({\mathcal T}'))$ for each
cluster-tilting subcategory ${\mathcal T}'$ of ${\mathcal E}$ and each non projective
indecomposable $M$ of ${\mathcal T}'$, where $Q^\circ({\mathcal T}')$ denotes the
quiver obtained from $Q({\mathcal T}')$ be removing all arrows between
projective vertices.
\end{itemize}
The cluster tilting subcategory ${\mathcal T}''=\mu_M({\mathcal T}')$ of 1) is
the {\em mutation of ${\mathcal T}'$ at the non projective indecomposable object $M$}.
The {\em mutation of a cluster-tilting object $T$} is defined via
the mutation of the cluster-tilting subcategory $\opname{add}\nolimits(T)$.
\end{definition}
\begin{lemma} \label{lemma:ExactMultiplicities} Suppose that the
cluster-tilting subcategories determine a cluster structure on
${\mathcal E}$. Then, in the situation of condition~2) of the above
definition~\ref{def:ExactClusterStructure}, the following hold:
\begin{itemize}
\item[a)] The space $\opname{Ext}^1(M,M^*)$ is one-dimensional (hence, by the $2$-Calabi-Yau
property, so is the space $\opname{Ext}^1(M^*,M)$) and the conflations of 2) are non split.
\item[b)] The multiplicity
of an indecomposable $U$ of ${\mathcal T}'\cap{\mathcal T}''$ in $E$ equals the number
of arrows from $U$ to $M$ in the quiver $Q({\mathcal T}')$ and that from $M^*$ to $U$
in $Q({\mathcal T}'')$; the multiplicity of $U$ in $E'$ equals the number of
arrows from $M$ to $U$ in $Q({\mathcal T}')$ and that from $U$ to $M^*$ in
$Q({\mathcal T}'')$;
\end{itemize}
\end{lemma}
We omit the proof of the lemma since it is entirely parallel to that
of lemma~\ref{lemma:Multiplicities}. Large classes of examples of
Frobenius categories where the cluster-tilting objects define a
cluster-structure are obtained in \cite{GeissLeclercSchroeer07b} and
\cite{BuanIyamaReitenSmith08}, {\em cf.}\ ~the survey
\cite{GeissLeclercSchroeer08} and
example~\ref{example:Frobenius2CYRealization} below. For an
extension of the theory from the antisymmetric to the
antisymmetrizable case, we refer to \cite{Demonet08}.
\section{Cluster characters for 2-Calabi-Yau Frobenius categories}
\label{CC2FC}
Let $k$ be an algebraically closed field and ${\mathcal E}$ a $k$-linear
Frobenius category with split idempotents. We assume that ${\mathcal E}$ is
$\opname{Hom}$-finite and that the stable category ${\mathcal C}=\ul{{\mathcal E}}$ is $2$-Calabi-Yau
({\em cf.}\ section~\ref{ss:CalabiYauCategories}).
\begin{definition} \label{def:ExactClusterCharacter}
A {\em cluster character on ${\mathcal E}$
with values in a commutative ring $R$} is a map $\zeta: \opname{obj}\nolimits({\mathcal E}) \to R$ such that
\begin{itemize}
\item[1)] we have $\zeta(L)=\zeta(L')$ if and $L$ and $L'$ are isomorphic,
\item[2)] we have $\zeta(L\oplus M) = \zeta(L)\zeta(M)$ for all objects $L$ and $M$ and
\item[3)] if $L$ and $M$ are objects such that $\opname{Ext}^1_{\mathcal E}(L,M)$ is one-dimensional
(and hence $\opname{Ext}^1_{\mathcal E}(M,L)$ is one-dimensional) and
\[
0\to L \to E \to M \to 0 \mbox{ and } 0\to M \to E' \to L \to 0
\]
are non-split triangles, then we have
\begin{equation} \label{eq:ExactClusterCharacterExchangeEquation}
\zeta(L) \zeta(M) = \zeta(E) + \zeta(E').
\end{equation}
\end{itemize}
\end{definition}
From now on, we assume in addition that ${\mathcal E}$ contains a cluster-tilting object $T$.
Using $T$ we will construct a cluster character on ${\mathcal E}$ and link it
to Palu's cluster character associated with the image of $T$ in the
triangulated category ${\mathcal C}=\ul{{\mathcal E}}$ ({\em cf.}\ section~\ref{ss:ClusterCharacters}).
Let $C$ be the endomorphism algebra of $T$ (in ${\mathcal E}$) and $\ul{C}=\opname{End}_{{\mathcal C}}(T)$. Let
\[
F=\opname{Hom}_{\mathcal E}(T,?) : {\mathcal E} \to \mod C,
\]
\[
G=\opname{Hom}_{\mathcal C}(T,?) : {\mathcal C} \to \mod \ul{C}.
\]
Let $T_i$, $1\leq i\leq n$, be the pairwise non isomorphic
indecomposable direct summands of $T$. We choose the numbering of
the $T_i$ so that $T_i$ is projective exactly for $r<i\leq n$ for
some integer $1\leq r\leq n$. For $1\leq i\leq n$, let $S_i$ be the
top of the indecomposable projective $P_i=FT_i$. Note that $C$ and
$\ul{C}$ are finite dimensional $k$-algebras, so finitely presented
modules are the same as finitely generated modules. As in section~4
of \cite{KellerReiten07}, we identify $\opname{Mod}\nolimits \ul{C}$ with the full
subcategory of $\opname{Mod}\nolimits C$ formed by the modules without composition
factors isomorphic to one of the $S_i$, $r< i\leq n$. Let ${\mathcal D} C$ be
the unbounded derived category of the abelian category $\opname{Mod}\nolimits C$ of
all right $C$-modules. Let $\opname{per}\nolimits C$ be the perfect derived category
of $C$, {\em i.e.}\ the full subcategory of ${\mathcal D} C$ whose objects are all
the complexes quasi-isomorphic to bounded complexes of finitely
generated projective $C$-modules. Let ${\mathcal D}^b(\mod C)$ the bounded
derived category of $\mod C$ identified with the full subcategory of
${\mathcal D} C$ whose objects are all complexes whose total homology is
finite-dimensional over $k$. As shown in section~4 of
\cite{KellerReiten07}, we have the following embeddings
\[
\mod \ul{C} \hookrightarrow \opname{per}\nolimits C \hookrightarrow {\mathcal D}^b(\mod C).
\]
We have a bilinear form
\[
\langle \ , \ \rangle : K_0(\opname{per}\nolimits C) \times K_0({\mathcal D}^b(\mod C))\longrightarrow \mathbb{Z}
\]
defined by
\[
\langle [P], [X]\rangle = \sum (-1)^i \mbox{dim}\ \opname{Hom}_{{\mathcal D}^b(\mod C)}(P, \Sigma^i X),
\]
where $K_0(\opname{per}\nolimits C)$ (resp. $K_0({\mathcal D}^b(\mod C))$) is the Grothendieck group of $\opname{per}\nolimits C$
(resp. ${\mathcal D}^b(\mod C) $) and $\Sigma$ is the shift functor of ${\mathcal D}^b(\mod C)$.
For arbitrary finitely generated $C$-modules $L$ and $N$, put
\[
[L,N]= \, ^0[L,N]=\mbox{dim}_k\opname{Hom}_C(L,N)\ \ \mbox{and} \ \
^i[L,N]=\mbox{dim}_k\opname{Ext}_C^i(L,N) \ \mbox{for}\ \ i\geq 1.
\]
Let
\[
\langle L, N\rangle_{\tau} = [L,N]- \, ^1[L,N]\ \mbox{and} \ \langle L,N\rangle_3 = \sum_{i=0}^3\
(-1)^i \ ^i[L,N]
\]
be the truncated Euler forms on the split Grothendieck group $K_0^{sp}(\mod C)$.
By the proposition below, if $L$ is a $\ul{C}$-module, then $\langle L, N\rangle_3$
only depends on the dimension vector $\underline{\dim}\, \ L$ in $K_0(\mod C)$. We put
\[
\langle \underline{\dim}\, \ L, N\rangle_3 = \langle L, N \rangle_3.
\]
\begin{proposition}
\label{prop1}
\begin{itemize}
\item [a)] The restriction of the map
\[
K_0(\opname{per}\nolimits C) \longrightarrow K_0(D^b(\mod C))=K_0(\mod C)
\]
induced by the inclusion of $\opname{per}\nolimits C$ into ${\mathcal D}^b(\mod C)$
to the subgroup generated by the $[S_i]$, $1\leq i\leq r$, is injective.
\item [b)] If $L$, $N$ are two $\ul{C}$-modules such that $\underline{\dim}\, \ L = \underline{\dim}\, \ N$ in $K_0(\mod C)$, then
\[
\langle L, Y\rangle_3 = \langle N, Y\rangle_3
\]
for each finitely generated $C$-module $Y$.
\end{itemize}
\end{proposition}
\begin{proof} a) We need to show that for arbitrary finitely generated
$\ul{C}$-modules $L$, $N$
with $\ul{\mbox{dim}}\ L = \ul{\mbox{dim}}\ N$, we have $[L]=[N]$ in $K_0(\opname{per}\nolimits C)$.
Let
\[
0=L_s\subset L_{s-1}\subset \cdots \subset L_0=L
\]
and
\[
0=N_s\subset N_{s-1}\subset \cdots \subset N_0=N
\]
be composition series of $L$ and $N$ respectively. By
\cite{KellerReiten07}, we know that every $\ul{C}$-module has
projective dimension at most $3$ in $\mod C$. Assume for simplicity
that $L_{s-1}=S_1$, $L_{s-2}/L_{s-1}=S_2$. Denote by $P_i^{*}$ a
minimal projective resolution of $S_i$. Then we have the following
commutative diagram
\[
\xymatrix{0 \ar[r] & P_1^3 \ar[r] \ar[d] & P_1^2 \ar[r] \ar[d] & P_1^1 \ar[r] \ar[d] & P_1^0 \ar[r] \ar[d] & L_{s-1} \ar[r] \ar[d] & 0\\
0\ar[r] & P_1^3\oplus P_2^3 \ar[r] \ar[d] & P_1^2\oplus P_2^2 \ar[r] \ar[d] & P_1^1\oplus P_2^1 \ar[r] \ar[d]& P_1^0\oplus P_2^0 \ar[r] \ar[d] & L_{s-2} \ar[r] \ar[d] & 0\\
0\ar[r] & P_2^3 \ar[r] & P_2^2 \ar[r] & P_2^1 \ar[r] & P_2^0 \ar[r] & L_{s-2}/L_{s-1} \ar[r] & 0}
\]
where the middle term is a projective resolution of $L_{s-2}$.
In this way, we inductively construct projective resolutions for $L$ and $N$.
If $m_i$ is the multiplicity of $S_i$ in the composition factors of $L$ and $N$,
then we obtain projective resolutions of $L$ and $N$ of the form
\[
0\to \bigoplus_{i=1}^r(P_i^3)^{m_i}\xrightarrow{f_3}\bigoplus_{i=1}^r(P_i^2)^{m_i}
\xrightarrow{f_2}\bigoplus_{i=1}^r(P_i^1)^{m_i}\xrightarrow{f_1}\bigoplus_{i=1}^r(P_i^0)^{m_i}\to L\to 0,
\]
\[
0\to \bigoplus_{i=1}^r(P_i^3)^{m_i}\xrightarrow{g_3}\bigoplus_{i=1}^r(P_i^2)^{m_i}
\xrightarrow{g_2}\bigoplus_{i=1}^r(P_i^1)^{m_i}\xrightarrow{g_1}\bigoplus_{i=1}^r(P_i^0)^{m_i}\to N\to 0.
\]
Let $P^L$ ({\em resp.} $P^N$) be the projective resolution complex of $L$ ({\em resp.} $N$).
We have $L\cong P^L$ and $N\cong P^N$ in $\opname{per}\nolimits C$, which implies $[L]=[P^L]=[P^N]=[N]$
in $K_0(\opname{per}\nolimits C)$.
b) We have
\[
\langle L, Y\rangle_3 = \langle P^L, Y \rangle = \langle [P^L], [Y]\rangle,
\]
\[
\langle N, Y\rangle_3 = \langle P^N, Y \rangle = \langle [P^N], [Y]\rangle.
\]
By a), we have $[P^L]=[P^N]$ in $K_0(\opname{per}\nolimits C)$, which implies the equality.
\end{proof}
One should note that the truncated Euler form $\langle \ , \ \rangle_3$
does not descend to the Grothendieck group $K_0(\mod C)$ in general
(except if the global dimension of $C$ is not greater than 3),
{\em cf.}\ remark~\ref{remark:gldim3}.
\vspace{0.5cm}
Using the bilinear forms introduced so far,
for $M\in{\mathcal E}$, we define the Laurent polynomial
\[
X'_M = \prod_{i=1}^n x_i^{\langle FM, S_i\rangle_{\tau}}
\sum_e \chi(Gr_e(\opname{Ext}^1_{\mathcal E}(T,M))) \, \prod_{i=1}^n x_i^{-\langle e,S_i\rangle_3}.
\]
Here we consider $\opname{Ext}^1_{\mathcal E}(T,M)$ as a right $C$-module via the natural
action of $C=\opname{End}_{\mathcal E}(T)$ on the first argument; the sum ranges over all
the elements of the Grothendieck group; for a $C$-module $L$, the notation $Gr_e(L)$ denotes the
projective variety of submodules of $L$ whose class in the Grothendieck group is $e$; for
an algebraic variety $V$, the notation $\chi(V)$ denotes the Euler characteristic
(of the underlying topological space of $V$ if $k=\mathbb{C}$ and of $l$-adic
cohomology if $k$ is arbitrary).
Since ${\mathcal C}$ is 2-Calabi-Yau, the object $\ul{T}=\oplus_{i=1}^r T_i$ is a
cluster tilting object of ${\mathcal C}$. For an object $M$ of ${\mathcal C}$, put
\[
X_M = X^{\ul{T}}_M \: , \;
\]
where $M \mapsto X^{\ul{T}}_M$ is Palu's cluster character associated
with the cluster-tilting object $\ul{T}$, {\em cf.}\ section~\ref{ss:ClusterCharacters}.
The following theorem shows that $M\mapsto X'_M$ is a cluster character
on ${\mathcal E}$ and that, if we specialize the `coefficients' $x_{r+1}, \ldots, x_n$
to $1$, it specializes to the composition of Palu's cluster character
$M \mapsto X_M$ with the suspension functor $M \mapsto \Sigma M$.
Notice that this theorem does not involve cluster algebras (but paves
the way for establishing a link with cluster algebras when ${\mathcal E}$
admits a cluster structure, {\em cf.}\ ~Theorem~\ref{thm2} below).
\begin{theorem}\label{thm1}
As above, let $k$ be an algebraically closed field and ${\mathcal E}$ a $k$-linear
Frobenius category with split idempotents such that
${\mathcal E}$ is $\opname{Hom}$-finite, the stable category ${\mathcal C}=\ul{{\mathcal E}}$ is $2$-Calabi-Yau
and ${\mathcal E}$ contains a cluster-tilting object $T$. For an
object $M$ of ${\mathcal E}$, let $X'_M$ and $X_M$ be the Laurent
polynomials defined above.
\begin{itemize}
\item[a)] We have $X'_{T_i}=x_i$ for $1\leq i \leq n$.
\item[b)] The specialization of $X'_M$ at $x_{r+1}=x_{r+2}= \ldots = 1$
is $X_{\Sigma M}$, where $\Sigma$ is the suspension of ${\mathcal C}$.
\item[c)] For any two objects $L$ and $M$ of ${\mathcal E}$, we have
$X'_{L\oplus M} = X'_L X'_M$.
\item[d)] If $L$ and $M$ are objects of ${\mathcal E}$ such that $\opname{Ext}_{\mathcal E}^1(L,M)$
is one-dimensional and we have non split conflations
\[
0 \to L \to E \to M \to 0 \mbox{ and } 0 \to M \to E' \to L \to 0 \: , \;
\]
then we have
\[
X'_L X'_M = X'_E + X'_{E'}.
\]
\end{itemize}
\end{theorem}
\begin{proof}
a) is straightforward.
b) We have
\[
X_{\Sigma M}=\prod_{i=1}^r x_i^{-[\operatorname{coind}_{\ul{T}} (\Sigma M): T_i]}
\sum_e \chi(Gr_e(G\Sigma M)) \, \prod_{i=1}^r x_i^{\langle S_i,e\rangle_a}.
\]
Now by the definition, we have
\[
G\Sigma M = \opname{Hom}_{{\mathcal C}}(T,\Sigma M) = \opname{Ext}_{{\mathcal E}}(T,M).
\]
Therefore, we only need to show that the exponents of $x_i$, $1\leq i\leq r$, in the corresponding terms of $X_{\Sigma M}$ and $X_M'$ are equal.
There exists a triangle in ${\mathcal C}$
\[
T_M^1\to T_M^0 \to M\to \Sigma T_1
\]
with $T_M^0$ and $T_M^1$ in $\opname{add}\nolimits{\ul{T}}$. We may and will assume that this triangle is minimal, {\em i.e.} does not admit a non zero direct factor of the form
\[
T'\to T'\to 0 \to \Sigma T'.
\]
Since ${\mathcal E}$ is Frobenius, we can lift this triangle to a short exact sequence in ${\mathcal E}$
\[
0\to T_M^1\to T_M^0\oplus P\to M\to 0,
\]
where $P$ is a projective of ${\mathcal E}$. Applying the functor $F$ to this short exact sequence, we get a
projective resolution of $FM$ as a $C$-module,
\[
0\to FT_M^1\to F(T_M^0\oplus P) \to FM \to 0.
\]
Therefore, we have
\[
\langle FM,S_i\rangle_{\tau} = [FT_M^0\oplus FP, S_i]-[FT_M^1, S_i] = [FT_M^0,S_i]-[FT_M^1,S_i]
\]
for $1\leq i\leq r$.
On the other side, we have the following minimal triangle
\[
\Sigma M\to \Sigma^2 T_M^1\to \Sigma^2 T_M^0\to \Sigma^2 M.
\]
By the definition of the coindex, we get
\[
- [\mbox{coind}_{\ul{T}}\ (\Sigma M):T_i] = - [T_M^1 - T_M^0 :T_i] = \langle FM,S_i\rangle_{\tau},\ \mbox{for}\ \, 1\leq i\leq r.
\]
Next we will show that $\langle S_i, e\rangle_{a}=-\langle e,
S_i\rangle_3.$ Let $N$ be a $C$-module such that $\mbox{\ul{dim}}\
N=e$. Note that $N$ and the $S_i, \ 1\leq i\leq r$, are $\ul{C}$-modules
and that all of them are finitely presented $C$-modules. Therefore, they
lie in the perfect derived category $\opname{per}\nolimits (C)$. Thus, we can use the
relative 3-Calabi-Yau property of $\opname{per}\nolimits (C)$ ({\em {\em cf.}\
\cite{KellerReiten07}}) to deduce that $\langle S_i,
e\rangle_a=-\langle e, S_i\rangle_3$. We have
\[
\opname{Ext}_C^2(N,S_i) =\opname{Ext}_C(S_i,N)=\opname{Ext}_{\ul{C}}(S_i,N),
\]
\[
\opname{Ext}_C^3(N,S_i) =\opname{Hom}_C(S_i,N)=\opname{Hom}_{\ul{C}}(S_i,N),
\]
for $1\leq i\leq r$.
By the definition of $\langle S_i, N\rangle_{a}$, we have
\begin{eqnarray*}
\langle S_i, N\rangle_{a} &=&\mbox{dim}_k\opname{Hom}_{\ul{C}}(S_i,N)-\mbox{dim}_k\opname{Ext}_{\ul{C}}(S_i,N)+
\mbox{dim}_k\opname{Ext}_{\ul{C}}(N,S_i)-\mbox{dim}_k\opname{Hom}_{\ul{C}}(N,S_i)\\
&=&\mbox{dim}_k\opname{Hom}_{C}(S_i,N)-\mbox{dim}_k\opname{Ext}_{C}(S_i,N)+\mbox{dim}_k\opname{Ext}_C(N,S_i)-\mbox{dim}_k\opname{Hom}_C(N,S_i)\\
&=&^3[N,S_i]-\,^2[N,S_i]+\,^1[N,S_i]-[N,S_i]\\
&=&-\langle N, S_i\rangle_3.
\end{eqnarray*}
c) is proved in exactly the same way as Corollary~3.7 in \cite{CalderoChapoton06}.
d) Let
\[
0 \to L \xrightarrow{i} E \xrightarrow{p} M \to 0 \ \,\mbox{ and }\ \, 0 \to M \xrightarrow{i'} E' \xrightarrow{p'} L \to 0 \: , \;
\]
be the non-split conflations in ${\mathcal E}$, and
\[\Sigma L\xrightarrow{G\Sigma i} \Sigma E\xrightarrow{G\Sigma p} \Sigma M \to \Sigma^2 L
\]
\[
\Sigma M\xrightarrow{G\Sigma i'} \Sigma E'\xrightarrow{G\Sigma p'} \Sigma L \to \Sigma^2 N
\]
the associated triangles in ${\mathcal C}$. For any classes $e$, $f$, $g$ in the Grothendieck group
$K_0(\mod \ul{C})$, let $X_{e,f}$ be the variety whose points are the $\ul{C}$-submodules
$E\subset G\Sigma E$ such that the dimension vector of $(G\Sigma i)^{-1}E$
equals $e$ and the dimension vector of $(G\Sigma p)E$ equals $f$. Similarly,
let $Y_{f,e}$ be the variety whose points are the $\ul{C}$-submodules
$E\subset G\Sigma E'$ such that the dimension vector of $(G\Sigma i')^{-1}E$
equals $f$ and the dimension vector of $(G\Sigma p')E$ equals $e$. Put
\[
X_{e,f}^g=X_{e,f}\cap Gr_g(G\Sigma E),
\]
\[
Y_{f,e}^g=Y_{f,e}\cap Gr_g(G\Sigma E').
\]
Since ${\mathcal C}$ is a $2$-CY triangulated category, by section 5.1 of
\cite{Palu08} we also have
\[
\chi (Gr_e(G\Sigma L)\times Gr_f(G\Sigma M))=\sum_g\chi(X_{e,f}^g)+\chi(Y_{f,e}^g).
\]
Therefore, part d) is a consequence of the following lemma.
\begin{lemma}
\label{lem1}
If $X_{e,f}^g\neq \emptyset$, then we have the following equality
\[
-\langle g, S_i\rangle_3 +\langle FE, S_i\rangle_{\tau} = -\langle e+f , S_i\rangle_3 +\langle FL, S_i\rangle_{\tau} +\langle FM, S_i\rangle_{\tau}, 1\leq i\leq n.
\]
\end{lemma}
{\it Proof.} We have the following commutative diagram as in section
4 of \cite{Palu08}
\[
\xymatrix{(G\Sigma i)^{-1}E \ar[r]^-{\alpha} \ar[d]_i & E \ar[r]^-{\beta} \ar[d]_j & (G\Sigma p)E \ar[r] \ar[d]_k & 0\\
G\Sigma L \ar[r]^-{G\Sigma i} & G\Sigma E \ar[r]^-{G\Sigma p} &
G\Sigma M \ar[r] & G\Sigma^2 L}
\]
where $i$, $j$, $k$ are monomorphisms, $\beta$ is an epimorphism and $[E]=g$,
$[G\Sigma i)^{-1}E]=e$, $[G\Sigma p)E]=f$ in $K_0(\mod C)$.
One can easily show that $\ker G\Sigma i = \ker \alpha$. We have an exact sequence
\[
0\to \ker \alpha \to (G\Sigma i)^{-1}E \to E \to (G\Sigma p)E\to 0.
\]
If we apply $F=\opname{Hom}_{{\mathcal E}}(T,?)$ to the short exact sequence
\[
0\to L\to E\to M\to 0,
\]
we get the long exact sequences of $C$-modules
\[
0\to FL\to FE\to FM\to G\Sigma L\xrightarrow{G\Sigma i}\ G \Sigma E\to \ldots,
\]
and
\[
0\to FL\xrightarrow{Fi} FE\xrightarrow{Fp} FM\to \ker \alpha\to 0.
\]
Since $\ker \alpha$, $(G\Sigma i)^{-1}E$, $E$, $(G\Sigma p)E$ are $\ul{C}$-modules,
and the projective dimensions of $FL$, $FE$, $FM$ are not greater than 1, we can
use the method of Proposition ~\ref{prop1} to construct the projective resolutions
and compute the truncated Euler forms. We get that
\[
\langle e,S_i\rangle_3 +\langle f, S_i\rangle_3 = \langle g,S_i\rangle_3 +\langle \ker \alpha, S_i\rangle_3,
\]
and
\[
\langle FL,S_i\rangle_3+\langle FM,S_i\rangle_3=\langle FE,S_i\rangle_3+\langle \ker \alpha,S_i\rangle_3.
\]
Note that $\langle FL,S_i\rangle_3=\langle FL,S_i\rangle_{\tau}$, $\langle FM,S_i\rangle_3=\langle FM,S_i\rangle_{\tau}$ and $\langle FE,S_i\rangle_3=\langle FE,S_i\rangle_{\tau}$, which implies
\[
\langle FL,S_i\rangle_{\tau}+\langle FM,S_i\rangle_{\tau}-
\langle e+f,S_i\rangle_3=\langle FE,S_i\rangle_{\tau}-\langle g,S_i\rangle_3.
\]
\end{proof}
\begin{remark} \label{remark:gldim3}
If $C$ has finite global dimension, the Grothendieck group $K_0(\mod C)$ has
the Euler form $\langle \ , \ \rangle$. We can then define a Laurent polynomial
$X_M^f$ as follows
\[
X_M^f=\prod_{i=1}^n x_i^{\langle FM, S_i\rangle}
\sum_e \chi(Gr_e(\opname{Ext}^1_{\mathcal E}(T,M))) \, \prod_{i=1}^n x_i^{\langle S_i,e\rangle}.
\]
One can show that in this case $X_M'=X_M^f$.
In fact, if $\mbox{gldim} C <\infty$, then the perfect derived category $\opname{per}\nolimits (C)$
equals $D^b(\mod C)$, and $S_i$ belongs to $\opname{per}\nolimits (C)$ for all $i$.
Thus, we have
\[
\langle S_i,e\rangle =\sum_{i=0}^3(-1)^i \mbox{dim}\opname{Ext}^i_C(S_i,e)=-\langle e,S_i\rangle_3
\]
and $\langle FM, S_i\rangle_{\tau}=\langle FM, S_i\rangle$. The assumption that $C$ is of finite
global dimension holds for the examples constructed in \cite{BuanIyamaReitenScott07} by Proposition~I.2.5~b)
of [loc. cit.] and for the examples constructed in \cite{GeissLeclercSchroeer06} by
Proposition~11.5 of [loc.cit.].
\end{remark}
\section{Index and $\mathbf{g}$-vector}
\label{IG}
\subsection{Index} \label{ss:Index}
As in section~\ref{CC2FC}, we let $k$ be an algebraically closed
field and ${\mathcal E}$ a $k$-linear Frobenius category with split
idempotents. We assume that ${\mathcal E}$ is $\opname{Hom}$-finite and that the
stable category ${\mathcal C}=\ul{{\mathcal E}}$ is $2$-Calabi-Yau ({\em cf.}\
section~\ref{ss:CalabiYauCategories}). Moreover, we assume that
${\mathcal E}$ admits a cluster-tilting object $T$ and we write
$C=\opname{End}_{\mathcal E}(T)$ and $\ul{C}=\opname{End}_{\mathcal C}(T)$.
Let ${\mathcal D}(\opname{Mod}\nolimits C)$ be the derived category of $C$-modules, ${\mathcal D}^-(\mod C)$
the right bounded derived category of $\mod C$, ${\mathcal H}^-({\mathcal P})$ the right
bounded homotopy category of finitely generated projective $C$-modules.
It is well known that there is an equivalence
\[
{\mathcal H}^-({\mathcal P})\stackrel{\sim}{\longrightarrow}{\mathcal D}^-(\mod C).
\]
\begin{proposition}\label{prop2}
For an arbitrary $\ul{C}$-module $Z$ which is also a finitely presented $C$-module we have a canonical isomorphism
\[
D \opname{Hom}_{{\mathcal D}^-(\mod C)}(Z,?) \stackrel{\sim}{\longrightarrow} \opname{Hom}_{{\mathcal D}^-(\mod C)}(?,Z[3]).
\]
\end{proposition}
\begin{proof} For arbitrary $X\in {\mathcal D}^-(\mod C)$, by the equivalence,
we have a $P_X\in {\mathcal H}^-({\mathcal P})$ such that $X\cong P_X$ in ${\mathcal D}^-(\mod C)$.
Assume that $P_X$ has the following form
\[
\ldots \to P_m\to P_{m+1}\to \ldots \to P_{n-1}\to P_n\to 0\to 0\ldots.
\]
Put
\begin{eqnarray*}
X_0 &=& \ldots \to 0 \to 0\to P_n\to 0\ldots,\\
X_i &=&\ldots \to 0\to P_{n-i} \to \ldots \to P_n \to 0 \ldots, \
\mbox{for}\ i>0.
\end{eqnarray*}
Clearly, the complex $P_X$ is the direct limit of the complexes
$X_i$. We write $\mbox{hocolim}$ for the total left derived functor
of the functor of taking the direct limit. Since taking direct
limits over filtered systems is an exact functor, the functor
$\mbox{hocolim}$ is simply induced by the direct limit functor.
Thus, we have $P_X \cong \mbox{hocolim} \ X_i$ in ${\mathcal D}(\opname{Mod}\nolimits C)$.
Note that by Proposition 4 of \cite{KellerReiten07}, $Z$ belongs to
$\opname{per}\nolimits C$, {\em i.e.}\ $Z$ is compact in ${\mathcal D}(\opname{Mod}\nolimits C)$. So we have
\begin{eqnarray*}
\opname{Hom}_{{\mathcal D}(\opname{Mod}\nolimits C)}(Z,X) &\cong& \opname{Hom}_{{\mathcal D}(\opname{Mod}\nolimits C)}(Z, P_X)\\
&\cong& \opname{Hom}_{{\mathcal D}(\opname{Mod}\nolimits C)}(Z, \mbox{hocolim} X_i)\\
&\cong& \mbox{colim} \opname{Hom}_{{\mathcal D}(\opname{Mod}\nolimits C)}(Z,X_i).
\end{eqnarray*}
By the definition of $X_i$, we know that $X_i \in \opname{per}\nolimits C$.
Since $\opname{per}\nolimits C$ is a full subcategory of ${\mathcal D}(\opname{Mod}\nolimits C)$,
by the relative 3-Calabi-Yau property of $\opname{per}\nolimits C$, we have the following
\[
\mbox{colim} \opname{Hom}_{{\mathcal D}(\opname{Mod}\nolimits C)}(Z,X_i) \cong \mbox{colim} D\opname{Hom}_{{\mathcal D}(\opname{Mod}\nolimits C)}(X_i, Z[3]).
\]
It is easy to see that this colimit is a stationary system,
{\em i.e.} $\exists\ N$ such that for $i>N$, we have
\[
D\opname{Hom}_{{\mathcal D}(\opname{Mod}\nolimits C)}(X_i, Z[3])\cong D\opname{Hom}_{{\mathcal D}(\opname{Mod}\nolimits C)}(X_{i+1}, Z[3]).
\]
Thus, we have
\begin{eqnarray*}
\mbox{colim} D\opname{Hom}_{{\mathcal D}(\opname{Mod}\nolimits C)}(X_i, Z[3])&\cong & D \ \mbox{lim} \opname{Hom}_{{\mathcal D}(\opname{Mod}\nolimits C)}(X_i, Z[3])\\
&\cong & D\opname{Hom}_{{\mathcal D}(\opname{Mod}\nolimits C)}(\mbox{hocolim} X_i, Z[3])\\
&\cong & D\opname{Hom}_{{\mathcal D}(\opname{Mod}\nolimits C)}(P_X, Z[3]).
\end{eqnarray*}
Note that since ${\mathcal D}^-(\mod C)$ is a full subcategory of ${\mathcal D}(\opname{Mod}\nolimits C)$, we get the isomorphism
\[
D \opname{Hom}_{{\mathcal D}^-(\mod C)}(Z,X) \stackrel{\sim}{\longrightarrow} \opname{Hom}_{{\mathcal D}^-(\mod C)}(X,Z[3]).
\]
\end{proof}
For each $X\in {\mathcal E}$, there is a unique minimal conflation (up to isomorphism)
\[
0\to T_X^1\to T_X^0\to X\to 0
\]
with $T_X^0,T_X^1\in \opname{add}\nolimits T$. As in \cite{Palu08}, put
\[
\mbox{ind}_T(X)=[T_X^0]-[T_X^1] \ \ \mbox{in} \ K_0(\opname{add}\nolimits T).
\]
By the proof of Theorem~\ref{thm1}, we have
\[
\mbox{ind}_T(X) = \sum_{i=1}^n \langle FX, S_i\rangle_{\tau} [T_i].
\]
The following result is easily deduced from Theorem~2.3 of \cite{DehyKeller08}.
\begin{lemma}
\label{lem2}
If $X$ is a rigid object of ${\mathcal E}$, then $X$ is determined up to isomorphism by
$\mbox{ind}_T(X)$, i.e. if \ $Y$ is rigid and $\mbox{ind}_T(X) = \mbox{ind}_T(Y)$,
then $X$ is isomorphic to $Y$.
\end{lemma}
\begin{proof}
Since $\mbox{ind}_T(X)=\mbox{ind}_T(Y)$, we have $\mbox{ind}_{\ul{T}}(X)=\mbox{ind}_{\ul{T}}(Y)$ in the stable
category $\ul{{\mathcal E}}$. By Theorem~2.3 of \cite{DehyKeller08}, we have $X\cong Y$ in $\ul{{\mathcal E}}$.
Thus, there are ${\mathcal E}$-projectives $P_X$ and $P_Y$ such that $X\oplus P_X\cong Y\oplus P_Y$ in ${\mathcal E}$.
Consider the minimal right $T$-approximation of $X\oplus P_X$
\[
0\to T^1\to T^0\to X\oplus P_X\to 0,
\]
we have $\mbox{ind}_T(X\oplus P_X) =\mbox{ind}_T(Y\oplus P_Y)=[T^0]-[T^1]$. Note that
\[
\mbox{ind}_T(X)=\mbox{ind}_T(X\oplus P_X)-[P_X]=\mbox{ind}_T(Y\oplus P_Y)-[P_Y]=\mbox{ind}_T(Y),
\]
which implies $[P_X]=[P_Y]$ in $K_0(\opname{add}\nolimits T)$. Thus, we have $P_X\cong P_Y$ and $X\cong Y$ in ${\mathcal E}$.
\end{proof}
\subsection{$\mathbf{g}$-vector} \label{ss:gvector}
Let us recall the definition of $\mathbf{g}$-vectors from section~7
of \cite{FominZelevinsky07}. Let $1< r\leq n$ be integers. Let
$\tilde{B}=(\tilde{b}_{ij})$ be an $n\times r$ matrix with integer
entries, whose principal part $B$ ({\em i.e.}\ the submatrix formed by the
first $r$ rows) is antisymmetric. Let ${\mathcal A}(\tilde{B})$ be the
cluster algebra with coefficients associated with $\tilde{B}$, {\em cf.}\
the end of section~\ref{ss:ClusterAlgebras}. Let $z$ be an element
of ${\mathcal A}(\tilde{B})$. Suppose that we can write $z$ as
\[
z = R(\hat{y}_1,\ldots, \hat{y}_r)\prod_{i=1}^nx_i^{g_i},
\mbox{where} \ \hat{y}_j=\prod_{i=1}^nx_i^{\tilde{b}_{ij}},
\]
where $R(\hat{y}_1,\ldots, \hat{y}_r)$ is a primitive rational
polynomial. If $\mbox{rank}\, \tilde{B}=r$, then the
$\mathbf{g}$-vector of $z$ is defined by
\[
g(z)=(g_1,\ldots, g_r).
\]
Note that $\mbox{rank}\,\tilde{B}=r$ implies that the $\mathbf{g}$-vector is well-defined.
As in the previous section, we let $k$ be an algebraically closed
field and ${\mathcal E}$ a $k$-linear Frobenius category with split
idempotents. We assume that ${\mathcal E}$ is $\opname{Hom}$-finite and that the
stable category ${\mathcal C}=\ul{{\mathcal E}}$ is $2$-Calabi-Yau ({\em cf.}\
section~\ref{ss:CalabiYauCategories}). Moreover, we assume that
${\mathcal E}$ admits a cluster-tilting object $T$ and we write
$C=\opname{End}_{\mathcal E}(T)$ and $\ul{C}=\opname{End}_{\mathcal C}(T)$. Let $T_1$, $T_2$,
$\ldots$, $T_n$ be the pairwise non isomorphic indecomposable direct
summands of $T$ numbered in such a way that $T_i$ is projective iff
$r< i\leq n$. We define $B(T)=(b_{ij})_{n\times n}$ to be the
antisymmetric matrix associated with the quiver of the endomorphism
algebra of $T$. Let $B(T)^0$ be the submatrix formed by the first
$r$ columns of $B(T)$. We suppose that we have $\opname{rank}\nolimits\, B(T)^0=r$.
In analogy with the definition of $\mathbf{g}$-vectors in a cluster
algebra, for $M\in {\mathcal E}$, if we can write $X_M'$ as
\[
X_M'=R(\hat{y}_1,\ldots, \hat{y}_r)\prod_{i=1}^nx_i^{g_i},
\mbox{where} \ \hat{y}_j=\prod_{i=1}^nx_i^{b_{ij}},
\]
where $R(\hat{y}_1,\ldots, \hat{y}_r)$ is a primitive rational
polynomial, then we define the $\mathbf{g}$-vector $g_T(X_M')$ of
$M$ with respect to $T$ to be
\[
g_T(X_M')=(g_1,\ldots, g_r).
\]
As in the cluster algebra case, this is well-defined since $\opname{rank}\nolimits\, B(T)^0=r$.
\begin{proposition}\label{prop3}
Assume that $\opname{rank}\nolimits\, B(T)^0 = r$.
For arbitrary $M\in {\mathcal E}$, the $\mathbf{g}$-vector $g_T(X_M')$ is well-defined and its $i$-th coordinate is given by
\[
g_T(X_M')(i)=[\mbox{ind}_T(M):T_i],\ 1\leq i\leq r.
\]
\end{proposition}
\begin{proof} By the relative 3-Calabi-Yau property of ${\mathcal D}^-(\mod C)$, for $1\leq i\leq n$, $1\leq j\leq r$, we have
\begin{eqnarray*}
\langle S_i, S_j\rangle_3 &=&[S_i, S_j]-\,^1\,\![S_i,S_j]+\,^2\,\![S_i,S_j]-\,^3\,\![S_i,S_j]\\
&=&[S_i, S_j]-\,^1\,\![S_i,S_j]+\,^1\,\![S_j,S_i]-\,[S_j,S_i]\\
&=&^1\,\![S_j,S_i]-\,^1\,\![S_i,S_j]\\
&=&b_{ij},
\end{eqnarray*}
where the last equality follows from the definition of $B(T)$.
Recall the definition of $X_M'$
\[
X_M'=\prod_{i=1}^n x_i^{\langle FM, S_i\rangle_{\tau}}
\sum_e \chi(Gr_e(\opname{Ext}^1_{\mathcal E}(T,M))) \ \prod_{i=1}^n x_i^{- \langle e, S_i\rangle_3}.
\]
Let $e$ be the dimension vector of a $C$-submodule of
$\opname{Ext}_{{\mathcal E}}^1(T,M)$ and $e_j$ its $j$-th coordinate in the basis of
the $S_i$, $1\leq i\leq n$. Then we have
\[
- \langle e, S_i\rangle_3 = - \sum_{j=1}^r e_j\langle S_j,S_i\rangle_3 = \sum_{j=1}^rb_{ij}e_j.
\]
Therefore, we get
\[
\prod_{i=1}^n x_i^{- \langle e,S_i\rangle_3}=\prod_{i=1}^nx_i^{\sum_{j=1}^r b_{ij}e_j}=\prod_{j=1}^r\hat{y}_j\,^{e_j}.
\]
Thus, we can write
\[
X_M'=\prod_{i=1}^n x_i^{\langle FM, S_i\rangle_{\tau}}
(\sum_e \chi(Gr_e(\opname{Ext}^1_{\mathcal E}(T,M))) \ \prod_{j=1}^r \hat{y}_j\,^{e_j}).
\]
The polynomial
\[
R(\hat{y}_1, \ldots, \hat{y}_r) = \sum_e \chi(Gr_e(\opname{Ext}_{{\mathcal E}}^1(T,M))) \prod_{j=1}^r \hat{y}_j\,^{e_j}
\]
is primitive since it has constant term $1$. Thus, by definition we have $g_T(X_M')(i)=\langle FM, S_i\rangle_{\tau} = [\mbox{ind}_T(M): T_i]$.
\end{proof}
\begin{corollary}\label{cor1} As above, let ${\mathcal E}$ be a $\opname{Hom}$-finite
$k$-linear Frobenius category such that its stable category
${\mathcal C}=\ul{{\mathcal E}}$ is $2$-Calabi-Yau and assume that
\begin{itemize}
\item ${\mathcal E}$ admits a cluster-tilting object $T$ with indecomposable direct summands
$T_1$, \ldots, $T_n$ numbered in such a way that $T_i$ is projective
iff $r<i\leq n$, where $1<r\leq n$ is an integer;
\item the first $r$ columns of the antisymmetric matrix $B(T)$ associated with
the quiver of the algebra $C=\opname{End}_{\mathcal E}(T)$ are linearly independent.
\end{itemize}
Then the following hold.
\begin{itemize}
\item[a)] The map $M \mapsto X'_M$ induces an injection from the set
of isomorphism classes of non projective rigid indecomposables of
${\mathcal E}$ into the set $\mathbb{Q}(x_1, \ldots, x_n)$.
\item[b)] Let $I$ be a
finite set and $T^i$, $i\in I$, cluster tilting objects of ${\mathcal E}$.
Suppose that for each $i\in I$, we are given an object $M_i$ which
belongs to $\opname{add}\nolimits T^i$ and does not have non zero projective direct
factors. If the $M_i$ are pairwise non isomorphic, then the
$X_{M_i}'$ are linearly independent.
\end{itemize}
\end{corollary}
\begin{proof} a) clearly follows from b). Let us prove b).
First, we will show that we can assign a degree to each $x_i$ such
that for every $1 \leq i \leq r$ the degree of $\hat{y}_i$ is 1.
Indeed, it suffices to put $\mbox{deg}(x_i)=k_i$, where the $k_i$ are rationals such that we have
\[
(k_1, k_2, \ldots, k_n)B(T)^0=(1,1,\ldots, 1).
\]
Since $\opname{rank}\nolimits\, B(T)^0 =r$, this equation does admit a solution.
Thus, the term of strictly minimal total degree in $X_{M_j}'$ is
\[
\prod_{i=1}^nx_i^{[\operatorname{ind}_T(M_j):T_i]}.
\]
Suppose that the $X_{M_i}'$ are linearly dependent, {\em i.e.} there
is a non-empty subset $I'$ of $I$ and rationals $c_i$, $i\in I'$,
which are all non zero such that
\[
\sum_{i\in I'} c_i X_{M_i}'=0.
\]
If we consider the terms of minimal total degree of the polynomial above, we find
\[
\sum_{j\in I''} c_j \prod_{i=1}^n x_i^{[\operatorname{ind}_T(M_j):T_i]}=0
\]
for some non-empty subset $I''$ of $I$. Since the $M_j$ are all
pairwise non isomorphic, Lemma~\ref{lem2} implies that the indices
$\mbox{ind}_T(M_j)$ are all distinct. Thus, the monomials
$\prod_{i=1}^n x_i^{[\operatorname{ind}_T(M_j):T_i]}$ are linearly
independent. Contradiction.
\end{proof}
\begin{remark} If the algebra $C$ has finite global dimension, then
the condition $\opname{rank}\nolimits B(T)^0=r$ is superfluous. Indeed, let $A$ be
the Cartan matrix of $C$. Then $B(T)^0$ is the submatrix formed by
the first $r$ columns of the invertible matrix $A^{-t}$.
\end{remark}
\vspace{0.2cm}
Next we will investigate the relation between the indices of an exchange pair.
Recall that $F$ is the functor $\opname{Hom}_{{\mathcal E}}(T,?): {\mathcal E} \to \mod C$. A
conflation of ${\mathcal E}$
\[
0\to X\to Y\to Z\to 0
\]
is $F$-{\em exact} if
\[
0\to FX\to FY\to FZ\to 0
\]
is exact in $\mod C$. The $F$-exact sequences define a new exact
structure on the additive category ${\mathcal E}$. For each $X$, we have an
$F$-exact conflation
\[
0\to T_1\to T_0\to X\to 0.
\]
This shows that ${\mathcal E}$ endowed with the $F$-exact sequences has
enough projectives and that its subcategory of projectives is $\opname{add}\nolimits
T$. Moreover, if we denote by $\opname{Ext}_F^i(X,Z)$ the $i$-th extension
groups of the category ${\mathcal E}$ endowed with the $F$-exact sequences,
then $\opname{Ext}_F^1(X,Z)$ is the cohomology at $\opname{Hom}_{{\mathcal E}}(T_1,Z)$ of the
complex
\[
0\to \opname{Hom}_{{\mathcal E}}(X,Z)\to \opname{Hom}_{{\mathcal E}}(T_0, Z)\to \opname{Hom}_{{\mathcal E}}(T_1,Z)\to 0\to \ldots.
\]
\begin{lemma}\label{lem3}
For $X, Z\in {\mathcal E}$, there is a functorial isomorphism
\[
\opname{Ext}_F^i(X,Z) \stackrel{\sim}{\longrightarrow} \opname{Ext}_C^i(FX, FZ).
\]
\end{lemma}
\begin{proof}
Clearly, the derived functor
\[
\mathbb{L} F: {\mathcal D}^b({\mathcal E})\to {\mathcal D}^b(\mod C)
\]
is fully faithful.
Thus, $\opname{Ext}_F^i(X,Z) \stackrel{\sim}{\longrightarrow} \opname{Ext}_C^i(FX,
FZ)$.
\end{proof}
Now Proposition~15.4 of \cite{GeissLeclercSchroeer07b} still holds
in our general setting.
\begin{proposition}\label{prop4}
Let $T$ and $R$ be cluster tilting objects of ${\mathcal E}$. Let
\[
\eta':0\to R_k\to R'\to R_k^*\to 0, \ \ \eta'':0\to R_k^*\to R''\to R_k\to 0
\]
be the two exchange sequences associated to an indecomposable direct
summand $R_k$ of $R$ which is not ${\mathcal E}$-projective. Then exactly one
of $\eta'$ and $\eta''$ is $F$-exact. Moreover, we have
\[
\underline{\dim}\, \opname{Hom}_{{\mathcal E}}(T,R_k)+\underline{\dim}\, \opname{Hom}_{{\mathcal E}}(T,R_k^*) =
\max \{\underline{\dim}\, \opname{Hom}_{{\mathcal E}}(T,R'),\underline{\dim}\, \opname{Hom}_{{\mathcal E}}(T,R'')\}.
\]
\end{proposition}
\begin{proof}
Using Lemma~\ref{lem3}, the proof is the same as
that of proposition 15.4 in \cite{GeissLeclercSchroeer07b}.
\end{proof}
\begin{corollary}
Under the assumptions of the above proposition, put
\[
I'=\mbox{ind}_T(R')-\mbox{ind}_T(R_k),
\]
\[
I''=\mbox{ind}_T(R'')-\mbox{ind}_T(R_k).
\]
Then we have
\begin{eqnarray*}
\mbox{ind}_T(R_k^*)=\left\{\begin{array}{ll} I',\, \mbox{if}\ \underline{\dim}\, FI'\geq \underline{\dim}\, FI'', \\
I'', \,\mbox{if}\ \underline{\dim}\, FI'\leq \underline{\dim}\, FI''.
\end{array}
\right.
\end{eqnarray*}
and exactly one of these cases occurs.
Let $h(i) = [\mbox{ind}_T(R')-\mbox{ind}_T(R''):T_i]$, for $1\leq i \leq n$. Then $h$ is a linear combination of the columns of $B(T)^0$.
\end{corollary}
\begin{proof} The first part follows from Proposition ~\ref{prop4} directly, because the index is additive on $F$-exact sequences.
Since $(R_k, R_k^*)$ is an exchange pair, we have
\[
X_{R_k}'X_{R_k^*}' = X_{R'}'+X_{R''}'.
\]
For simplicity, we write
\[
H_M = \sum_e\chi(Gr_e(\opname{Ext}_{{\mathcal E}}(T, M))) \prod_{i=1}^r\hat{y}_i\, ^{e_i}
\]
for $X_M'$.
By Proposition~\ref{prop3}, we have
\begin{eqnarray*}
\prod_{i=1}^nx_i^{[\operatorname{ind}_T(R_k)+\operatorname{ind}_T(R_k^*) : T_i]} H_{R_k}H_{R_k^*}
= \prod_{i=1}^nx_i^{[\operatorname{ind}_T(R'): T_i]}H_{R'} +\prod_{i=1}^nx_i^{[\operatorname{ind}_T(R''): T_i]}H_{R''}.
\end{eqnarray*}
Assume that $\mbox{ind}_T(R_k^*) = \mbox{ind}_T(R') - \mbox{ind}_T(R_k)$. We have
\[
H_{R_k}H_{R_k^*} -H_{R'} = \prod_{i=1}^nx_i^{[\operatorname{ind}_T(R'')-\operatorname{ind}_T(R'): T_i]}H_{R''}.
\]
By comparing the minimal total degree we get that
$\prod_{i=1}^nx_i^{[\operatorname{ind}_T(R'')-\operatorname{ind}_T(R'):
T_i]}$ is a monomial in $\hat{y}_i$, $1\leq i\leq r$, which implies
the result.
\end{proof}
\section{Frobenius 2-Calabi-Yau realizations}
\label{F2R}
Recall the bijection defined in section~\ref{ss:IceQuivers} between
antisymmetric integer $n\times n$-matrices and finite quivers
without loops nor 2-cycles with vertex set $\{1,2, \ldots, n\}$: The
quiver $Q$ corresponds to the matrix $B$ iff $b_{ij}>0$ exactly when
there are arrows from $i$ to $j$ in $Q$ and in this case their
number is $b_{ij}$.
We call an $n\times n$ antisymmetric integer matrix $B$ {\em
acyclic} if the corresponding quiver $Q$ does not have oriented
cycles. Two matrices $B$ and $B'$ are called {\em mutation
equivalent} if we can obtain $B'$ from $B$ by a series of matrix
mutations followed by conjugation with a permutation matrix.
Let $0\leq r<n$ be positive integers and let $(Q,F)$ be an ice
quiver ({\em cf.}\ ~section~\ref{ss:IceQuivers}) with vertex set
$Q_0=\{1,\ldots, n\}$ and set of frozen vertices $F=\{r+1, \ldots,
n\}$. We define $\tilde{B}$ to be the $n\times r$ matrix formed by
the first $r$ columns of the skew-symmetric matrix associated with
$Q$ and we let ${\mathcal A}(Q,F)={\mathcal A}(\tilde{B})$ be the cluster algebra with
coefficients associated with $\tilde{B}$, {\em cf.}\
sections~\ref{ss:ClusterAlgebras} and \ref{ss:IceQuivers}.
\begin{definition} \label{def:Frobenius2CYRealization}
A {\em Frobenius 2-Calabi-Yau realization} of the cluster
algebra ${\mathcal A}(\tilde{B})$ is a Frobenius category ${\mathcal E}$ with a
cluster tilting object $T$ as in section~\ref{CC2FC} such that
\begin{itemize}
\item[1)] ${\mathcal E}$ has a cluster structure in the sense of
\cite{BuanIyamaReitenScott07}, {\em cf.}\
section~\ref{ss:FrobeniusCategories};
\item[2)] $T$ has exactly $n$ indecomposable pairwise non isomorphic summands
$T_1$, $T_2$, $\ldots$, $T_n$ and among these, precisely $T_{r+1}$, $\ldots$,
$T_n$ are projectives;
\item[3)] The matrix $\tilde{B}$ equals the matrix formed by the first
$r$ columns of the antisymmetric matrix associated with the quiver
of the endomorphism algebra of $T$ in ${\mathcal E}$.
\end{itemize}
\end{definition}
\begin{remark} \label{remark:No2CYFrobeniusRealization}
Suppose we have a Frobenius $2$-CY realization of a cluster algebra
${\mathcal A}(Q,F)$ as above. Let $1\leq s\leq r$. Then by
Lemma~\ref{lemma:ExactMultiplicities} b), we have conflations
\[
0\to T_s^*\to E\to T_s\to 0,
\]
\[
0\to T_s\to E'\to T_s^*\to 0.
\]
Here the middle terms are the sums
\[
E = \bigoplus_{b_{is}>0} T_i^{b_{is}} \: , \; E' = \bigoplus_{b_{is}<0} T_i^{-b_{is}}.
\]
Therefore, none of the first $r$ vertices of $Q$ can be a source or a sink.\\
\end{remark}
\begin{example} \label{example:Frobenius2CYRealization}
All quivers obtained from Theorem~2.3 of \cite{GeissLeclercSchroeer07b}
and, more generally, from Theorem~II.4.1 of \cite{BuanIyamaReitenScott07} admit Frobenius
$2$-Calabi-Yau realizations.
We illustrate this on the following specific case taken
from section~II.4 of [loc.~cit.]. Let $\Delta$ be the graph
\[
\xymatrix@=0.4cm{ &2\ar@{-}[dr]\\ 1\ar@{-}[ur]\ar@{-}[rr] &&3 \ .}
\]
Let $\Lambda$ be the completion of the preprojective algebra of
$\Delta$ and $W$ the Weyl group associated with $\Delta$. Let
$w$ be the element of $W$ given by the reduced word
$s_2s_1s_2s_3s_2$. Let $e_i$,
$i=1, 2, 3$, be the primitive idempotents corresponding to the
vertices of $\Delta$. Let $I_i=\Lambda(1-e_i)\Lambda$.
By Theorem~II.2.8 of \cite{BuanIyamaReitenScott07}, the category $\mbox{Sub}\,\Lambda/I_w$
formed by all $\Lambda$-submodules of finite direct sums of
copies of $\Lambda/I_w$ is a Frobenius category whose associated
stable category is $2$-Calabi-Yau; moreover, it contains
the cluster-tilting object
\[
T=\Lambda/I_2\oplus \Lambda/I_2I_1\oplus
\Lambda/I_2I_1I_2\oplus \Lambda/I_2I_1I_2I_3 \oplus
\Lambda/I_w.
\]
According to Proposition~II.1.11 of [loc.~cit.], in this
decomposition, each direct factor differs from the preceding one by
one indecomposable direct summand $T_i$, $1\leq i\leq 5$, and among
these, exactly $T_3$, $T_4$ and $T_5$ are projective-injective.
Moreover, by Theorem~II.4.1 of [loc.~cit.], the quiver of the
cluster-tilting object is
\[\xymatrix@=0.5cm{& T_1\ar[dr] \\
T_2\ar[ur]\ar[d] && T_3\ar[d]\ar[dll]\\
T_4\ar[rr]&&T_5\ar[ull] \ .}
\]
Using Theorem~I.1.6 of \cite{BuanIyamaReitenScott07}, one can easily show that the
category $\mbox{Sub}\,\Lambda/I_w$ is a Frobenius 2-Calabi-Yau
realization of the cluster algebra ${\mathcal A}(\tilde{B})$ given
by the matrix
\begin{eqnarray*}
\tilde{B}=\left(\begin{array}{cc} 0&-1\\1&0\\-1&0\\0&-1\\0&1
\end{array}
\right).
\end{eqnarray*}
\end{example}
We return to the general setup. Following \cite{DehyKeller08} we define a
cluster-tilting object $T'$ of ${\mathcal E}$ to be {\em reachable from $T$}
if it is obtained from $T$ by a finite sequence of mutations. We
define an indecomposable rigid objects $M$ to be {\em reachable from
$T$} if it occurs as a direct factor of a cluster-tilting object
reachable from $T$.
\begin{theorem}\label{thm2}
Let $1<r\leq n$ be integers and ${\mathcal A}(\tilde{B})$ the cluster
algebras with coefficients associated with an initial $n\times
r$-matrix $\tilde{B}$ of maximal rank. Suppose that ${\mathcal A}(\tilde{B})$
admits a Frobenius $2$-CY realization ${\mathcal E}$ with cluster tilting
object $T$.
\begin{itemize}
\item [a)] The map $M\mapsto X_M'$ induces a bijection from the set
of isomorphism classes of indecomposable rigid non projective
objects of ${\mathcal E}$ reachable from $T$ onto the set of cluster
variables of ${\mathcal A}(\tilde{B})$. Under this bijection, the
cluster-tilting objects reachable from $T$ correspond to the
clusters of ${\mathcal A}(\tilde{B})$.
\item [b)] The map $M\mapsto \mbox{ind}_T(M)$ is a bijection from
the set of isomorphism classes of indecomposable rigid non
projective objects of ${\mathcal E}$ reachable from $T$ onto the set of
$\mathbf{g}$-vectors of cluster variables of ${\mathcal A}(\tilde{B})$.
\end{itemize}
\end{theorem}
\begin{proof}
a) It follows from Theorem~\ref{thm1} c) that $X_M'$ is a cluster variable
for each indecomposable rigid $M$ reachable from $T$ and from the existence
of a cluster structure on ${\mathcal E}$ that the map $M\mapsto X_M'$
is a surjection onto the set of cluster variables. The injectivity of the map
$M\mapsto X_M'$ follows from Lemma~\ref{lem2} and Proposition~\ref{prop3}.
The second statement follows from the first one and the fact that
${\mathcal E}$ has a cluster structure.
b) The map is injective by Lemma~\ref{lem2}. It is surjective thanks
to part a) and Proposition~\ref{prop3}.
\end{proof}
\begin{theorem}\label{prop5}
Let $1<r\leq n$ be integers and ${\mathcal A}(\tilde{B})$ the cluster
algebras with coefficients associated with an initial $n\times
r$-matrix $\tilde{B}$ of maximal rank. Suppose that ${\mathcal A}(\tilde{B})$
admits a Frobenius $2$-CY realization ${\mathcal E}$ with cluster tilting
object $T$.
\begin{itemize}
\item [a)]Conjecture 7.2 of \cite{FominZelevinsky07} holds for ${\mathcal A}$,
{\em i.e.} cluster monomials are linearly independent.
\item [b)]Conjecture 7.10 of \cite{FominZelevinsky07} holds for ${\mathcal A}$, {\em i.e.} \\
1) Different cluster monomials have different $\mathbf{g}$-vectors with respect to a given initial seed.\\
2) The $\mathbf{g}$-vectors of the cluster variables in any given
cluster form a $\mathbb{Z}$-basis of the lattice $\mathbb{Z}^r$.
\item [c)] Conjecture 7.12 of \cite{FominZelevinsky07} holds for
${\mathcal A}$, {\em i.e.} if $(g_1,\ldots, g_r)$ and $(g_1',\ldots,g_r')$
are the $\mathbf{g}$-vectors of one and the same cluster variable
with respect to two clusters $t$ and $t'$ related by the mutation at
$l$, then we have
\begin{eqnarray*}
g_j'=\left\{\begin{array}{ll} -g_l\ \ \ \ \ \ \ \ \ \ \ \ \ \ \, \quad \ \ \ \ \ \quad \quad \quad \mbox{if}\ \, j=l\\
g_j+[b_{jl}]_+g_l-b_{jl}\min(g_l,0)\ \ \ \mbox{if}\ \, j\neq l
\end{array}
\right.
\end{eqnarray*}
where the $b_{ij}$ are the entries of the $r\times r$-matrix $B$
associated with $t$ and we write $[x]_+$ for $\mbox{max}(x,0)$ for any
integer $x$.
\end{itemize}
\end{theorem}
\begin{proof} a) By Theorem~\ref{thm2} a), each cluster monomial $m$ is
the image $X'_M$ of a rigid object $M$ of ${\mathcal E}$, where $M$ does not
have any non zero projective direct factor. Moreover, such an object
$M$ is unique up to isomorphism. Thus, given a set $m_1$, \ldots,
$m_N$ of pairwise distinct cluster monomials, we obtain a set $M_1$,
\ldots $M_N$ of pairwise non isomorphic rigid objects without
projective direct factors such that $X'_{M_i}=m_i$ for $1\leq i\leq
N$. Thus, by Corollary~\ref{cor1} b), the images $X'_{M_i}=m_i$ of
the $M_i$ are not only pairwise distinct but in fact linearly
independent.
b) Let us prove 1). Let $m$ and $m'$ be two distinct cluster
monomials. We would like to compare their $\mathbf{g}$-vectors with
respect to a given initial cluster. By theorem~\ref{thm2} a), we may
assume that this given cluster consists of the images under
$M\mapsto X'_M$ of the indecomposable direct factors of $T$. Still
by theorem~\ref{thm2} a), the monomials $m$ and $m'$ are the images
$X'_M$ and $X'_{M'}$ of two non isomorphic rigid objects $M$ and
$M'$ of ${\mathcal E}$ without non zero projective direct factors. Thus $M$
and $M'$ are still non isomorphic in the stable category
${\mathcal C}=\ul{{\mathcal E}}$. But by theorem~2.3 of \cite{DehyKeller08}, non
isomorphic rigid objects have distinct indices $\opname{ind}_T(M)$ and
$\opname{ind}_T(M')$. Therefore, they have distinct $\mathbf{g}$-vectors by
Proposition~\ref{prop3}. Now let us prove 2). Let a cluster
$\mathbf{x}'$ be given. By theorem~\ref{thm2} a), the variables
$x'_i$ in $\mathbf{x}'$ are the images under $M\mapsto X'_M$ of the
indecomposable non projective direct summands $T'_i$ of a
cluster-tilting object $T'$ reachable from $T$. By
Proposition~\ref{prop3}, the $\mathbf{g}$-vector of each $x'_i$ is
the index of $T'_i$. Now by Theorem~2.6 of \cite{DehyKeller08}, the
indices of the indecomposable direct factors of a cluster-tilting object form a
basis of the lattice $K_0(\opname{add}\nolimits \ul{T})$, where $\ul{T}$ is the image
of $T$ in ${\mathcal C}$. Thus the $\mathbf{g}$-vectors of the $x'_i$ form a
basis of the lattice $\mathbb{Z}^r$.
c) By Theorem~\ref{thm2} a), we may assume that under the maps
$M \mapsto X'_M$, the clusters $t$ and
$t'$ correspond to the cluster-tilting object
$T$ and another cluster-tilting object $T'$ obtained from $T$ by
mutation at the non projective indecomposable direct factor $T_l$.
Moreover, the given cluster variable $x$ corresponds to some non
projective rigid indecomposable object $X$. By
Proposition~\ref{prop3}, the $\mathbf{g}$-vectors of $x$ with
respect to $t$ and $t'$ are given by the components of the indices
$\opname{ind}_T(X)$ and $\opname{ind}_{T'}(X)$ in the bases formed by the
$\opname{ind}_T(T_i)$, $1\leq i\leq r$, respectively the $\opname{ind}_{T'}(T'_i)$,
$1\leq i\leq r$, where the $T_i$ and the $T'_i$ are the non
projective indecomposable direct factors of $T$ respectively $T'$.
Now Theorem~3.1 of \cite{DehyKeller08} tells us exactly how
$\opname{ind}_T(X)$ and $\opname{ind}_{T'}(X)$ are related: Let
\[
\xymatrix{T_l \ar[r] & E' \ar[r] & T_l^* \ar[r] & \Sigma T_l}
\mbox{ and }
\xymatrix{T_l^* \ar[r] & E \ar[r] & T_l \ar[r] & \Sigma T_l^*}
\]
be the exchange triangles associated with the mutation from $T$ to $T'$.
Let
\[
\phi_+ : K_0(\opname{add}\nolimits T) \to K_0 (\opname{add}\nolimits T') \mbox{ and }
\phi_- : K_0(\opname{add}\nolimits T) \to K_0 (\opname{add}\nolimits T')
\]
be the linear maps which send the classes $[T_i]$, $i\neq l$,
to themselves and send $[T_l]$ to
\[
\phi_+([T_l]) = [E]-[T_l^*] \mbox{ respectively }
\phi_-([T_l]) = [E'] - [T_l^*].
\]
Then by Theorem~3.1 of \cite{DehyKeller08}, we have
\[
\opname{ind}_{T'}(X) = \left\{ \begin{array}{ll}
\phi_+(\opname{ind}_T(X)) & \mbox{ if } [\opname{ind}_T(X): T_l]\geq 0 \\
\phi_-(\opname{ind}_T(X)) & \mbox{ if } [\opname{ind}_T(X): T_l]\leq 0.
\end{array} \right.
\]
We leave it to the reader to check that this yields exactly
the rule given in the assertion.
\end{proof}
Let $\tilde{B}$ be a $2r\times r$ matrix whose principal ($i.e.$ top
$r\times r$) part $B_0$ is mutation equivalent to an acyclic matrix,
and whose complementary ($i.e.$ bottom) part is the $r\times r$
identity matrix. Let ${\mathcal A}(\tilde{B})$ be the cluster algebra with the
initial seed $({\bf x}, \tilde{B})$.
\begin{theorem}
With the above notation, the cluster algebra ${\mathcal A}(\tilde{B})$ does
not admit a Frobenius $2$-CY realization.
\end{theorem}
\begin{proof}
Suppose that ${\mathcal A}(\tilde{B})$ has a Frobenius $2$-CY realization
${\mathcal E}$. Then there is a cluster tilting object $T$ of ${\mathcal E}$ with
$2r$ indecomposable direct summands. Then we have $B(T)^0 =
\tilde{B}$. Since $B_0$ is mutation equivalent to an acyclic matrix
$B_c$ by a series of mutations, we have a cluster tilting object
$T'$ such that the quiver of the stable endomorphism algebra of $T'$
corresponds to $B_c$. Let $A$ be the stable endomorphism algebra of
$T'$. By the main theorem of \cite{KellerReiten06}, we have a triangle
equivalence $\ul{{\mathcal E}}\simeq {\mathcal C}_{A}$, where ${\mathcal C}_{A}$ is the cluster
category of $ A$. Thus the cluster tilting graph of ${\mathcal E}$ is
connected and every rigid object of ${\mathcal E}$ can be extended to a
cluster tilting object of ${\mathcal E}$.
Let $F = \opname{Hom}_{{\mathcal E}}(T,?)$. Let $S_i$, $1\leq i\leq 2r$, be the
simple modules of $\opname{End}_{{\mathcal E}}(T)$. For each object $M$ of ${\mathcal E}$, we
have the Laurent polynomial
\[
X_M'=\prod_{i=1}^{2r}x_i^{\langle FM, S_i\rangle_{\tau}} \sum_e \chi(Gr_e (\opname{Ext}^1_{{\mathcal E}}(T,M))) \prod_{i=1}^{2r} x_i^{- \langle e, S_i\rangle_3}.
\]
Let
\[
y_j=\prod_{i=1}^{2r} x_i^{b_{ij}}, 1\leq j\leq r.
\]
As in Proposition~\ref{prop3}, we can rewrite $X_M'$ as
\[
X_M' = \prod_{i=1}^{2r}x_i^{\langle FM, S_i\rangle_{\tau}} (1+ \sum_{e\neq 0} \chi(Gr_e(\opname{Ext}_{{\mathcal E}}^1(T,M))) \prod_{i=1}^r y_j^{e_j}),
\]
where $e_j$ is the $j$-th coordinate of $e$ in the basis of the $S_i$, $1\leq i\leq 2r$.
If the indecomposable object $M$ is rigid and not isomorphic to $T_i$ for $r< i\leq 2r$, then $X_M'$ is a cluster variable of ${\mathcal A}(\tilde{B})$. By the definition of the rational function $\mathcal{F}_{l,t}$ associated with the cluster variable $x_{l,t}$ in \cite{FominZelevinsky07},
we have
\begin{eqnarray*}
\mathcal{F}_M &=& X_M'(x_1=x_2=\ldots =x_r=1)\\
&=& \prod_{i=r+1}^{2r}x_i^{\langle FM, S_i\rangle_{\tau}} (1+\sum_{e\neq 0} \chi(Gr_e(\opname{Ext}_{{\mathcal E}}^1(T,M))) \prod_{j=r+1}^{2r} x_j^{e_{j-r}}).
\end{eqnarray*}
Put
\[
G_M= 1+\sum_{e\neq 0} \chi(Gr_e\opname{Ext}_{{\mathcal E}}^1(T,M)) \prod_{j=r+1}^{2r} x_j^{e_{j-r}}.
\]
Note that $G_M$ is always a polynomial of $x_i$, $r+1\leq i\leq 2r$, with constant term 1.
By Proposition 5.2 in \cite{FominZelevinsky07}, we know that the polynomial $\mathcal{F}_M$ is not divisible by $x_i$, $r+1\leq i\leq 2r$.
Now for $i> r$, we have $\langle FM, S_i\rangle_{\tau}\geq 0$ in general, which implies that $\langle FM, S_i\rangle_{\tau}=0$.
In particular, $\langle FM, S_i\rangle_{\tau} = [\mbox{ind}_T(M) : T_i] = 0$, for $r+1\leq i \leq 2r$.
Consider $M=\Sigma T_1$, which is rigid and indecomposable, so $X_M'$ is a cluster variable of the cluster algebra ${\mathcal A}(\tilde{B})$. But in the Frobenius category ${\mathcal E}$ we have the conflation
\[
0\to T_1\to P\to \Sigma T_1\to 0,
\]
where $P$ is an injective hull of $T_1$,
which implies
\[
\mbox{ind}_T(M)=[P]-[T_1].
\]
Thus there is always some $r+1 \leq i\leq 2r$ such that $[\mbox{ind}_T(M) : T_i] \neq 0$. Contradiction.
\end{proof}
\begin{remark} In the above notation, if $B_0$ is acyclic, then it is easy to
deduce that the cluster algebra ${\mathcal A}(\tilde{B})$ does not have a
Frobenius $2$-CY realization. Indeed in this case, one of the first
$r$ vertices of $Q$ which corresponds to $\tilde{B}$ is always a
sink. This is incompatible with the existence of a Frobenius $2$-CY
realization by remark~\ref{remark:No2CYFrobeniusRealization}.
\end{remark}
\section{Triangulated 2-Calabi-Yau realizations}
\label{T2R}
\subsection{Definitions.}
Let $B=(b_{ij})_{n\times n}$ be an antisymmetric integer matrix and
${\mathcal A}(B)$ the associated cluster algebra. A $2$-Calabi-Yau
triangulated category ${\mathcal C}$ is called a {\em triangulated
2-Calabi-Yau realization} of the matrix $B$ if ${\mathcal C}$ admits a
cluster tilting object $T$ such that
\begin{itemize}
\item ${\mathcal C}$ has a cluster structure in the sense
\cite{BuanIyamaReitenScott07}, {\em cf.}\ ~section~\ref{ss:CalabiYauCategories};
\item $T$ has exactly $n$ non isomorphic indecomposable direct summands $T_1$, $\ldots$, $T_n$;
\item The antisymmetric matrix $B(T)$ associated with the quiver of the
endomorphism algebra of $T$ equals $B$.
\end{itemize}
We denote a triangulated $2$-CY realization of $B$ by ${\mathcal C}\supset
\opname{add}\nolimits T$.
Let $n_1$ and $n_2$ be positive integers. Let $B_1$ and $B_2$ be
antisymmetric integer $n_1\times n_1$ resp. $n_2\times
n_2$-matrices. Let $B_{21}$ be an integer $n_2\times n_1$-matrix
with non negative entries. Let ${\mathcal C}_i\supset {\mathcal T}_i$ be a
triangulated $2$-CY realization of $B_i$, $i=1,2$. Let $B$ be the
matrix
\begin{eqnarray*}
\left(\begin{array}{ll} B_1&-B_{21}^t\\B_{21}&\quad B_2
\end{array}
\right).
\end{eqnarray*}
A {\em gluing} of ${\mathcal C}_1\supset {\mathcal T}_1$ with ${\mathcal C}_2\supset {\mathcal T}_2$
with respect to $B$ is a triangulated $2$-CY realization
${\mathcal C}\supset {\mathcal T}$ of $B$ endowed with full additive subcategories
${\mathcal T}_1'$ and ${\mathcal T}_2'$ such that
\begin{itemize}
\item $\opname{Hom}_{{\mathcal C}}({\mathcal T}_1',{\mathcal T}_2')=0$;
\item The set $\mbox{indec}({\mathcal T})$ is the disjoint
union of $\mbox{indec}({\mathcal T}_1')$ with $\mbox{indec}({\mathcal T}_2')$;
\item There is a triangle equivalence
\[
^{\perp}(\Sigma {\mathcal T}_1')/({\mathcal T}_1')\stackrel{\sim}{\longrightarrow} {\mathcal C}_2
\]
inducing an equivalence ${\mathcal T}_2'\stackrel{\sim}{\longrightarrow}{\mathcal T}_2$;
\item There is a triangle equivalence
\[
^{\perp}(\Sigma {\mathcal T}_2')/({\mathcal T}_2')\stackrel{\sim}{\longrightarrow}{\mathcal C}_1
\]
inducing an equivalence ${\mathcal T}_1'\stackrel{\sim}{\longrightarrow} {\mathcal T}_1$.
\end{itemize}
A {\em principal gluing} of ${\mathcal C}_1\supset {\mathcal T}_1$ is a gluing of
${\mathcal C}_1\supset {\mathcal T}_1$ with ${\mathcal C}_2\supset {\mathcal T}_2$ with respect to
\begin{eqnarray*}
\left(\begin{array}{ll} B_1&-I_{n_1} \\ I_{n_1}&\quad 0 \end{array} \right),
\end{eqnarray*}
where ${\mathcal C}_2$ is the cluster category of $(A_1)^{n_1}$ and ${\mathcal T}_2$
the image of the subcategory of finitely generated projective modules.
It is well known that each acyclic matrix $B$ admits a triangulated
$2$-CY realization ${\mathcal C}_{Q_B}$, where ${\mathcal C}_{Q_B}$ is the cluster
category of the quiver $Q_B$ corresponding to $B$. In the last
subsection, we will see that ${\mathcal C}_{Q_B}$ does admit a principal
gluing.
\begin{conjecture} \label{conj:ExistenceOfGlueings}
If ${\mathcal C}_1$ and ${\mathcal C}_2$ are algebraic, a gluing exists for any matrix
$B_{21}$ with non negative entries.
\end{conjecture}
Amiot's work \cite{Amiot08a} provides some evidence for the
conjecture: Indeed, if ${\mathcal C}_1$ and ${\mathcal C}_2$ are generalized cluster
categories \cite{Amiot08a} associated with Jacobi-finite quivers
with potential \cite{DerksenWeymanZelevinsky07}, it is easy to
construct a quiver with potential which provides a gluing as
required by the conjecture.
\subsection{Cluster algebras with coefficients.}
Let $B$ be an antisymmetric integer $n\times n$-matrix. Suppose that
the matrix $B$ admits a triangulated $2$-CY realization ${\mathcal C}$ with
the
cluster tilting subcategory ${\mathcal T}=\opname{add}\nolimits T$. Let $T_i$, $1\leq i\leq
n$, be the non isomorphic indecomposable direct summands of $T$. By
the definition, we have $B(T)=B$. The mutations of the matrix $B$
correspond to the mutations of the cluster tilting object $T$. Fix
an integer $0< r\leq n$ and consider the submatrix $B^0$ of $B$
formed by the first $r$ columns of $B$. If $l\leq r$, then we have
\[
\mu_l(B^0)=(\mu_l(B))^0,
\]
where $\mu_l$ is the mutation in the direction $l$. Thus we can
view the cluster algebra ${\mathcal A}(B^0)$ with coefficients as a
sub-cluster algebra of ${\mathcal A}(B)$, {\em cf.}\ Ch.~III of \cite{BuanIyamaReitenScott07}.
Denote by ${\mathcal P}$ the full subcategory of ${\mathcal C}$ whose objects are the
finite direct sums of copies of $T_{r+1},\ldots, T_n$. We define a
subcategory of ${\mathcal C}$
\[
{\mathcal U}=\,^\perp\!(\Sigma{\mathcal P})=\{X\in {\mathcal C}| \opname{Ext}^1_{{\mathcal C}}(T_i,X)=0\mbox{ for }r<i\leq n\}.
\]
By Theorem~I.2.1 of \cite{BuanIyamaReitenScott07}, the quotient
category ${\mathcal U}/{\mathcal P}$ is a 2-Calabi-Yau triangulated category and the
projection ${\mathcal U} \to {\mathcal U}/{\mathcal P}$ induces a bijection between the cluster
tilting subcategories of ${\mathcal C}$ containing ${\mathcal P}$ and the cluster
tilting subcategories of ${\mathcal U}/{\mathcal P}$. Thus, a mutation of a cluster
tilting object in ${\mathcal U}/{\mathcal P}$ can be viewed as a mutation of a cluster
tilting object in ${\mathcal U}\subset{\mathcal C}$ which does not affect the direct
summands $T_i$, $r<i\leq n$. This exactly corresponds to a mutation
of the matrix $B$ in one of the first $r$ directions. In particular,
a mutation of the cluster algebra ${\mathcal A}(B^0)$ corresponds to a
mutation of a cluster tilting object in ${\mathcal U}$.
Recall from section~\ref{ss:ClusterCharacters} that on ${\mathcal C}$, we
have Palu's cluster character associated with $T$, which is given by
the formula
\[
X_M=X^T_M=\prod_{i=1}^{n}x_i^{-[\operatorname{coind}_T M: T_i]}
\sum_e\chi(Gr_e(\opname{Hom}_{{\mathcal C}}(T, M)))\prod_{i=1}^{n}x_i^{\langle S_i,
e\rangle_a}.
\]
We consider the composition of this map with the shift:
\[
X_M'=X_{\Sigma M} =\prod_{i=1}^{n}x_i^{[\operatorname{ind}_T M: T_i]}
\sum_e\chi(Gr_e(\opname{Hom}_{{\mathcal C}}(T, \Sigma M)))\prod_{i=1}^{n}x_i^{\langle
S_i, e\rangle_a}.
\]
We consider the restriction of the map $M \mapsto X_M' $ to the
subcategory ${\mathcal U}$. It follows from
Proposition~\ref{prop:ClusterCharacterProperties} that if $M$ is an
indecomposable rigid object reachable from $T$ in ${\mathcal U}$, then $X_M'$
is a cluster variable of ${\mathcal A}(B^0)$. We will rewrite this variable
so as to express its $\mathbf{g}$-vector (if it is defined) in terms
of the index of $M$: Let $M$ be an object of ${\mathcal U}$. Then
$\opname{Hom}_{{\mathcal C}}(T, \Sigma M)$ is an $\opname{End}_{{\mathcal C}}(T)$-module which
vanishes at each vertex $r< i\leq n$. Let $e$ be the image
of $\opname{Hom}_{{\mathcal C}}(T, \Sigma M)$ in the
Grothendieck group of $\mod \opname{End}_{{\mathcal C}}(T)$. Let $e_j$ be the $j$-th
coordinate of $e$ with respect to the basis $S_i$, $1\leq i\leq n$.
We have
\begin{eqnarray*}
\langle S_i, e\rangle_a&=&\langle S_i, e\rangle_{\tau} -\langle e, S_i\rangle_{\tau}\\
&=& \sum_{j=1}^r e_j(\langle S_i, S_j\rangle_{\tau} - \langle S_j, S_i\rangle_{\tau}) \\
&=& \sum_{j=1}^r e_j (\opname{Ext}^1_{\opname{End}_{{\mathcal C}}(T)}(S_j,S_i)-\opname{Ext}^1_{\opname{End}_{{\mathcal C}}(T)}(S_i,S_j))\\
&=& \sum_{j=1}^r b_{ij}e_j.
\end{eqnarray*}
As in section~\ref{IG}, put
\[
y_j=\prod_{i=1}^{n}x_i^{b_{ij}}\, , \mbox{for} \ 1\leq j\leq r.
\]
Then $X_M'$ can be rewritten as
\[
X_M'=\prod_{i=1}^{n}x_i^{[\operatorname{ind}_T ( M):
T_i]}(1+\sum_{e\neq 0} \chi(Gr_e(\opname{Hom}_{{\mathcal C}}(T, \Sigma
M)))\prod_{j=1}^ry_j^{e_j}).
\]
As in section~\ref{IG}, when $\opname{rank}\nolimits\, B^0=r$, we can define the
$\mathbf{g}$-vector of $M\in {\mathcal U}$ with respect to a cluster tilting
object $T$. Thus we have proved part a) of the following
proposition. We leave the easy proof of part b) to the reader.
\begin{proposition}\label{prop6} Suppose that $\opname{rank}\nolimits\,B^0=r$. Let
$M$ be an object of ${\mathcal U}$.
\begin{itemize}
\item[a)] The $\mathbf{g}$-vector of $X_M'$ with respect to the
initial cluster is given by
\[
g_T(X_M')(i)= [\mbox{ind}_T\, (M): T_i],\,\mbox{for}\, 1\leq i\leq r.
\]
\item[b)] The index of the image of $M$ in ${\mathcal U}/{\mathcal P}$ with respect to
the image of $T$ is
\[
\sum_{i=1}^r g_T(X_M')(i)[T_i].
\]
\end{itemize}
\end{proposition}
In analogy with the definition in section~\ref{F2R}, we define a
cluster tilting object $T'$ of ${\mathcal U}$ to be {\em reachable} from $T$
if it is obtained from $T$ by a sequence of mutations at
indecomposable rigid objects of ${\mathcal U}$ not in ${\mathcal P}$. We define an
indecomposable rigid object of ${\mathcal U}$ to be {\em reachable} from $T$
if it is a direct factor of a cluster tilting object reachable from
$T$.
\begin{theorem}\label{thm3}
Let $B$ be an antisymmetric integer $n\times n$-matrix and $1\leq
r\leq n$ an integer such that the submatrix $B^0$ of $B$ formed by
the first $r$ columns has rank $r$. Let ${\mathcal A}={\mathcal A}(B^0)$ be the
associated cluster algebra with coefficients. Assume that the matrix
$B$ admits a triangulated $2$-CY realization given by a triangulated
category ${\mathcal C}$ with a cluster tilting object $T$ which is the sum of
$n$ indecomposable direct factors $T_1$, \ldots, $T_n$. Denote by
${\mathcal P}$ the full subcategory of ${\mathcal C}$ whose objects are the finite
direct sums of copies of $T_{r+1},\ldots, T_n$ and define the
subcategory ${\mathcal U}$ of ${\mathcal C}$ by
\[
{\mathcal U}=\,^\perp\!(\Sigma{\mathcal P})=\{X\in {\mathcal C}| \opname{Ext}^1_{{\mathcal C}}(T_i,X)=0\mbox{ for } r<i\leq n\}.
\]
For $M\in{\mathcal C}$, define ({\em cf.}\ section~\ref{ss:ClusterCharacters})
\[
X_M'=X^T_{\Sigma M} =\prod_{i=1}^{n}x_i^{[\operatorname{ind}_T M:
T_i]} \sum_e\chi(Gr_e(\opname{Hom}_{{\mathcal C}}(T, \Sigma
M)))\prod_{i=1}^{n}x_i^{\langle
S_i, e\rangle_a}.
\]
Then the following hold.
\begin{itemize}
\item[a)] The map $M \mapsto X'_M$ induces a bijection from the set
of isomorphism classes of indecomposable rigid objects of ${\mathcal U}$ not
belonging to ${\mathcal P}$ and reachable from $T$ onto the set of cluster
variables of ${\mathcal A}(B^0)$. Under this bijection, the cluster-tilting
objects of ${\mathcal U}$ reachable from $T$ correspond to the clusters of
${\mathcal A}(B^0)$.
\item[b)] The map $M \mapsto [\opname{ind}_T(M):T_i]_{1\leq i \leq r}$ is
a bijection from the set of indecomposable rigid objects of ${\mathcal U}$
not belonging to ${\mathcal P}$ and reachable from $T$ onto the set of
$\mathbf{g}$-vectors of cluster variables of ${\mathcal A}(B^0)$.
\item [c)]Conjecture 7.2 of \cite{FominZelevinsky07} holds for ${\mathcal A}$,
{\em i.e.} the cluster monomials are linearly independent over $\mathbb{Z}$. Moreover,
the cluster monomials form a basis of the $\mathbb{Z}[x_{r+1}, \ldots, x_n]$-submodule
of ${\mathcal A}(B^0)$ which they generate.
\item [d)]Conjecture 7.10 of \cite{FominZelevinsky07} holds for ${\mathcal A}$, {\em i.e.} \\
1) Different cluster monomials have different $\mathbf{g}$-vectors with respect to a given initial seed.\\
2) The $\mathbf{g}$-vectors of the cluster variables in any given
cluster form a $\mathbb{Z}$-basis of the lattice $\mathbb{Z}^r$.
\item [e)] Conjecture 7.12 of \cite{FominZelevinsky07} holds for
${\mathcal A}$, {\em i.e.} if $(g_1,\ldots, g_r)$ and $(g_1',\ldots,g_r')$
are the $\mathbf{g}$-vectors of one and the same cluster variable
with respect to two clusters $t$ and $t'$ related by the mutation at
$l$, then we have
\begin{eqnarray*}
g_j'=\left\{\begin{array}{ll} -g_l\ \ \ \ \ \ \ \ \ \ \ \ \ \ \, \quad \ \ \ \ \ \quad \quad \quad \mbox{if}\ \, j=l\\
g_j+[b_{jl}]_+g_l-b_{jl}\min(g_l,0)\ \ \ \mbox{if}\ \, j\neq l
\end{array}
\right.
\end{eqnarray*}
where the $b_{ij}$ are the entries of the $r\times r$-matrix $B$
associated with $t$ and we write $[x]_+$ for $\mbox{max}(x,0)$ for
any integer $x$.
\end{itemize}
\end{theorem}
\begin{proof}
It follows from Proposition~\ref{prop:ClusterCharacterProperties}
that the map $M\mapsto X'_M$ is well-defined and surjective onto the
set of cluster variables of ${\mathcal A}(B^0)$. It is injective by
Proposition \ref{prop6} b) because rigid objects of
${\mathcal U}/{\mathcal P}$ are determined by their indices and the map taking a rigid object $M$
of ${\mathcal U}$ without non zero direct factors in ${\mathcal P}$ to its image
in ${\mathcal U}/{\mathcal P}$ is injective (up to isomorphism). This also implies part
b). The same proof as for Corollary \ref{cor1} b) yields the linear
independence of the cluster monomials in c). Let us prove that
the cluster monomials form a basis of the $\mathbb{Z}[x_{r+1}, \ldots, x_n]$-submodule
of ${\mathcal A}(B^0)$ which they generate. Indeed, over $\mathbb{Z}$, this
submodule is spanned by the images $X'_M$ of all rigid objects
of ${\mathcal U}$ obtained as direct sums of objects of ${\mathcal P}$ and indecomposable rigid objects
reachable from $T$ not belonging to ${\mathcal P}$. Such objects $M$ are in
particular rigid in ${\mathcal T}$ and they can be distinguished (up to
isomorphism) by their indices. Now again, the same proof
as for Corollary~\ref{cor1} b) shows that these $X'_M$ are linearly
independent over $\mathbb{Z}$. Clearly this implies that the cluster monomials
form a basis of the $\mathbb{Z}[x_{r+1}, \ldots, x_n]$-submodule
of ${\mathcal A}(B^0)$ which they generate. As in the proof of Theorem~\ref{prop5} b),
the assertions in part d) follow from the interpretation of the $\mathbf{g}$-vector
given in \ref{prop6} b) and the facts that
\begin{itemize}
\item[1)] rigid objects of ${\mathcal U}/{\mathcal P}$ are determined by their indices
(Theorem~2.3 of \cite{DehyKeller08}) and
\item[2)] the indices of the indecomposable direct factors of a
cluster-tilting subcategory ${\mathcal T}$ of ${\mathcal U}/{\mathcal P}$ form a basis of
$K_0({\mathcal T})$ (Theorem~2.6 of \cite{DehyKeller08}).
\end{itemize}
Part e) is proved in exactly the same way as the corresponding
statement for cluster algebras with a $2$-CY Frobenius realization
in Theorem \ref{thm2} c).
\end{proof}
\begin{example}
Let $A_4$ be the quiver $3\to 1\to 2\leftarrow 4$, ${\mathcal C}_{Q}$ the
corresponding cluster category. The following is the AR quiver of
${\mathcal C}_Q$, where $P_i$, $1\leq i\leq 4$, are the indecomposable
projective $kQ$-modules.
\[\xymatrix@=0.4cm{
& P_4\ar[dr] & & 123\ar[dr]& & \Sigma P_3\ar[dr]\\
\ldots & & *+<0.1cm>[F]{\ \, P_2}\ar[ur]\ar[dr]& & 12\ar[ur]\ar[dr]& & *+<0.1cm>[F]{\Sigma P_1}\ar[dr] &&\ldots\\
\ldots & *+<0.1cm>[F]{\ \, P_1}\ar[ur]\ar[dr] & & 124\ar[ur]\ar[dr] & & 2\ar[ur]\ar[dr] & &*+<0.1cm>[F]{\Sigma P_2}&\ldots\\
P_3\ar[ur] & & S_1\ar[ur]& & *+<0.1cm>[F]{\ \, M}\ar[ur]& & \Sigma
P_4\ar[ur]}
\]
Let $T=P_1\oplus P_2\oplus P_3\oplus P_4$ be the canonical cluster
tilting object in ${\mathcal C}_Q$, ${\mathcal P}=\opname{add}\nolimits (P_3\oplus P_4)$. It is easy to
see that the indecomposable objects in ${\mathcal U}/{\mathcal P}\cong {\mathcal C}_{A_2}$ are
exactly $P_1, P_2, M, \Sigma P_1, \Sigma P_2$. In this case, the
matrix $B(T)^0$ is
\begin{eqnarray*}
\left(\begin{array}{ll} 0&1\\-1&0\\1&0\\0&1
\end{array}
\right).
\end{eqnarray*}
We have $\mbox{rank}\,B(T)^0=2$. Moreover, the cluster algebra ${\mathcal A}(B(T)^0)$ has principal coefficients.
\end{example}
\subsection{Cluster algebras with principal coefficients.}
In this subsection, we suppose that $2r=n$ and that the
complementary part of $B^0$ is the $r\times r$ identity matrix. Thus
the cluster algebra ${\mathcal A}(B^0)$ has principal coefficients. Recall
that for the matrix $B$, we have a triangulated $2$-CY realization
${\mathcal C}\supset \opname{add}\nolimits T$ and we have fixed ${\mathcal P}=\opname{add}\nolimits (T_{r+1}\oplus
\ldots\oplus T_{2r})$. Let ${\mathcal Q}=\opname{add}\nolimits(T_1\oplus\ldots\oplus T_r)$.
Let ${\mathcal C}_1={\mathcal U}/{\mathcal P}$ and ${\mathcal C}_2=\,^\perp(\Sigma{\mathcal Q})/{\mathcal Q}$ be the quotient
categories, ${\mathcal T}_1=\opname{add}\nolimits(\pi_1(T_1\oplus\ldots\oplus T_r))$ and
${\mathcal T}_2=\opname{add}\nolimits(\pi_2(T_{r+1}\oplus \ldots\oplus T_{2r}))$ the
corresponding cluster tilting subcategories, where $\pi_1$ and
$\pi_2$ are the respective projection functors. Then ${\mathcal C}$ is a
gluing of ${\mathcal C}_1\supset {\mathcal T}_1$ with ${\mathcal C}_2\supset {\mathcal T}_2$ with
respect to the matrix $B$.
As in section~\ref{F2R}, for a cluster variable $x_{l,t}$ of the
cluster algebra ${\mathcal A}(B^0)$ which corresponds to an indecomposable
rigid object $M\in {\mathcal U}$ and not in ${\mathcal P}$, we denote the rational
function $\mathcal{F}_{l,t}$ defined in section 3 of
\cite{FominZelevinsky07} by $\mathcal{F}_M$. Since $x_{l,t}=X_M'$,
we have
\begin{eqnarray*}
\mathcal{F}_M&=&X_M'(x_1=\ldots=x_r=1)\\
&=& \prod_{i=r+1}^{2r} x_i^{[\operatorname{ind}_T(M):T_i]}(1+\sum_{e\neq 0} \chi(Gr_e(\opname{Hom}_{{\mathcal C}}(T, \Sigma M)))\prod_{j=r+1}^{2r}x_j^{e_{j-r}}).
\end{eqnarray*}
The following result is now a consequence of Proposition 3.6 and 5.2
in \cite{FominZelevinsky07}. We give a proof based on representation
theory. Note that conjecture 5.4 of \cite{FominZelevinsky07} will be
proved in full generality in \cite{DerksenWeymanZelevinsky09}.
\begin{theorem}\label{thm4}
Conjecture 5.4 of \cite{FominZelevinsky07} holds for ${\mathcal A}(B^0)$, {\em i.e.}
the polynomial $\mathcal{F}_M$ has constant term 1. Thus we have
\[
\mathcal{F}_M=1+\sum_{e\neq 0} \chi(Gr_e(\opname{Hom}_{{\mathcal C}}(T, \Sigma M)))\prod_{j=r+1}^{2r}x_j^{e_{j-r}}.
\]
\end{theorem}
\begin{proof}
We need to show that for each $i>r$, $[\mbox{ind}_T(M):T_i]$ is zero.
Since $X_M'$ is a cluster variable and $M$ is indecomposable,
we have the following two cases:\\
{\em Case 1:} $M\cong \Sigma T_j$ for some $j\leq r$.
We have $\mbox{ind}_T(M)=-[T_j]$, which implies that
$[\mbox{ind}_T(M):T_i]=0$.\\
{\em Case 2:} $M$ is not isomorphic to $\Sigma T_j$ for any $j\leq
r$. Recall that by assumption, $M$ is not isomorphic to $T_j$ for
any $j>r$. We have the following minimal triangle
\[
T_M^1\to T_M^0\to M\to \Sigma T_M^1
\]
with $T_M^0$, $T_M^1$ in $\opname{add}\nolimits T$ and
$\mbox{ind}_T(M)=[T_M^0]-[T_M^1]$. Since $M$ belongs to ${\mathcal U}$, for
each $i>r$ we have $\opname{Hom}_{{\mathcal C}}(M, \Sigma T_i)=0$. If we had $[T_M^1:
T_i]\neq 0$ for some $i>r$, then the above minimal triangle would
have a non zero direct factor
\[
T_i\to T_i\to 0\to \Sigma T_i.
\]
Suppose that we have $[T_M^0: T_i]\neq 0$ for some $i>r$. Applying
the functor $F=\opname{Hom}_{{\mathcal C}}(T,?)$ to the triangle, we get a minimal
projective resolution of $FM$ as an $\opname{End}_{{\mathcal C}}(T)$-module. Note
that for $i>r$, the projective module $FT_i$ is also a simple
module, which implies that $FM$ is decomposable. Contradiction.
\end{proof}
Suppose that the indecomposable rigid object $M$ of ${\mathcal C}$
is reachable from $T$ and consider the polynomial
$\mathcal{F}_M$ of Theorem~\ref{thm4}. We define the {\em
$f$-vector} $f_T(M)=(f_1, \ldots, f_r)$ of $M$ with respect to $T$
by
\[
\mathcal{F}_M|_{Trop(u_1,\ldots,u_r)}(u_1^{-1},\ldots,u_r^{-1})=u_1^{-f_1}\ldots u_r^{-f_r} \: , \;
\]
where $\mbox{Trop}(u_1, \ldots, u_r)$ is the tropical
semifield defined in section~\ref{ss:ClusterAlgebras}.
\begin{proposition}\label{prop7}
Suppose that $M$ is not isomorphic to $ T_i$ for $1\leq i\leq 2r$,
and let $\underline{\dim}\,\, \opname{Hom}_{{\mathcal C}}(T, \Sigma M)=(d_1, \ldots, d_r)$. Then
we have
\[
d_i=f_i, \,\, 1\leq i\leq r.
\]
\end{proposition}
\begin{proof}
By Theorem~\ref{thm4}, we have
\[
\mathcal{F}_M
= 1+\sum_{e\neq 0} \chi(Gr_e(\opname{Hom}_{{\mathcal C}}(T, \Sigma M)))\prod_{j=r+1}^{2r}x_j^{e_{j-r}}.
\]
Therefore, we obtain
\begin{eqnarray*}
\mathcal{ F}_M|_{Trop(u_1,\ldots, u_r)}(u_1^{-1},\ldots,
u_r^{-1})&=&1 \oplus \,
\bigoplus_{e\neq 0} \chi(Gr_e(\opname{Hom}_{{\mathcal C}}(T, \Sigma M)))\prod_{j=1}^{r}u_j^{-e_j}\\
&=& u_1^{-d_1}\ldots u_n^{-d_r}.
\end{eqnarray*}
\end{proof}
Under the assumptions above, we have proved that the dimension
vector of $\opname{Hom}_{{\mathcal C}}(T, \Sigma M)$ equals the $f$-vector $f_T(M)$.
Conjecture 7.17 of \cite{FominZelevinsky07} states that the
$f$-vectors coincide with the denominator vectors in general. But by
recent work of A. Buan, R. Marsh and I. Reiten
\cite{BuanMarshReiten08a}, the dimension vectors do not always
coincide with the denominator vectors. In fact, as shown in
\cite{BuanMarshReiten08a}, for a quiver $Q$ whose underlying graph
is an affine Dynkin diagram, the vector $\dim \opname{Hom}_{{\mathcal C}_Q}(T,M)$ is
different from the denominator vector of $X^T_M$ if $M=R$ and $R$
is a direct factor of $T$, where $R$ is a rigid regular
indecomposable of maximal quasi-length. This leads to the following
minimal counterexample to Conjecture 7.17 in
\cite{FominZelevinsky07}. Let us point out that the corresponding
computations already appear in \cite{Cerulli08}. In subsection~{5.5}
below, we will show that in many cases, the $f$-vector is greater or
equal to the denominator vector.
\subsection{A counterexample.}
\begin{example}
Let $Q$ be the following quiver
\[
\xymatrix@=0.4cm{& 3\ar[dl]\\ 1\ar@2[rr]& & 2\ar[ul]\ .}
\]
Let ${\mathcal A}(Q)$ be the cluster algebra associated with the initial seed
given by $Q$ and ${\bf x}=(x_1,x_2,x_3)$. Consider the mutations
at $3,2,1$. Let ${\bf x}^{t_3}$ be the corresponding cluster. We have
\[
x_1^{t_3}=\frac{x_1^2+2x_1x_2+x_2^2+x_3}{x_1x_2x_3}
\]
and the corresponding $F$-polynomial is
\[
F_{x_1^{t_3}}=1+(1+y_1+y_1y_2)y_3+y_1y_2y_3^2.
\]
Then the $f$-vector of $x_1^{t_3}$ does not coincide with the denominator vector.
\end{example}
Let us interpret this counterexample in terms of representation theory.
Let $A_{2,1}$ be the quiver
\[\xymatrix@=0.4cm{
& 3\ar[dr]\\
1\ar[ur]\ar[rr] & &2\ .}
\]
Consider the cluster category ${\mathcal C}_{A_{2,1}}$ of $kA_{2,1}$. Let
$P_i$, $1\leq i\leq 3$, be the indecomposable projective modules
and $S_i$ the corresponding simple modules. Then
\[
T=P_1\oplus P_2\oplus \tau S_3,
\]
is a cluster tilting object of ${\mathcal C}_{A_{2,1}}$,
where $\tau$ is the Auslander-Reiten translation functor. The quiver $Q_T$ of $T$ looks like
\[
\xymatrix@=0.4cm{& \tau S_3\ar[dl]\\
P_1\ar@2[rr]& &P_2\ar[ul]\ .}
\]
We will show that the cluster category ${\mathcal C}_{A_{2,1}}\supset \opname{add}\nolimits T$
admits a principal gluing. For this, consider the following quiver
$Q_1:$
\[
\xymatrix@=0.5cm{& 6\ar[r] &3\ar[d]\\
4\ar[r]&1\ar[r]\ar[u]&2&5\ar[l]\ .}
\]
It admits a cluster category ${\mathcal C}_{Q_1}$. Let $T_{Q_1}=kQ_1$ be the
canonical cluster tilting object in ${\mathcal C}_{Q_1}$. Let
$T'=\mu_3(\mu_6(T_{Q_1}))$ be the cluster tilting object obtained by
mutations from $T_{Q_1}$. Denote the non isomorphic indecomposable
direct summands of $T'$ by $T'_i$, $1\leq i\leq 6$. Then the quiver
of $Q_{T'}$ is
\[
\xymatrix@=0.4cm{& & T'_6\ar[d]\\
& & T'_3\ar[dl]\\
T'_4\ar[r]&T'_1\ar@ 2[rr]& &T'_2\ar[ul]&T'_5\ar[l]\ .}
\]
Let ${\mathcal P}= \opname{add}\nolimits (T'_4\oplus T'_5\oplus T'_6)$. Then ${\mathcal U}/{\mathcal P}$ is a
2-Calabi-Yau triangulated category and admits a cluster tilting
object with the quiver $Q_T$. By the main theorem of
\cite{KellerReiten06}, we know that there is a triangle equivalence
${\mathcal U}/{\mathcal P}\simeq {\mathcal C}_{A_{2,1}}$. Thus, we see that the matrix $B(T')$
admits a triangulated $2$-CY realization ${\mathcal C}_{Q_1}$ which is the
required principal gluing of ${\mathcal C}_{A_{2,1}}\supset \opname{add}\nolimits T$. We may
assume that the images of $T'_1, T_2', T_3'$ coincide with $P_1,
P_2, \tau S_3$ in ${\mathcal C}_{A_{2,1}}$ respectively. Denote the shift
functor in ${\mathcal C}_{Q_1}$ (resp. ${\mathcal C}_{A_{2,1}}$) by $\Sigma$ (resp.
$[1]$).
Let $N$ be the preimage of $S_3$ in ${\mathcal C}_{Q_1}$. Then one can easily
compute
\[
\underline{\dim}\, \opname{Hom}_{{\mathcal C}_{Q_1}}(T',\Sigma N)=\underline{\dim}\, \opname{Hom}_{{\mathcal C}_{A_{2,1}}}(T, \tau S_3)=(1,1,2).
\]
Note that the denominator vector of $X_N'$ equals the denominator
vector of $X_{\tau S_3}^T$. Now the result follows from the
Proposition above.
\subsection{An inequality.}
Let ${\mathcal T}$ be a 2-Calabi-Yau triangulated category with
cluster-tilting object $T$. Recall that we have the generalized
Caldero-Chapoton map
\[
X_M^T = \prod_{i=1}^n x_i^{-[\operatorname{coind}_{T}(M): T_i]}
\sum_e \chi(Gr_e(GM)) \, \prod_{i=1}^n x_i^{\langle S_i,e\rangle_a},
\]
where $G$ is the functor $\opname{Hom}_{{\mathcal C}}(T,?): {\mathcal C}\to \mod
\opname{End}_{{\mathcal C}}(T)$. The following proposition is proved in greater
generality in \cite{DerksenWeymanZelevinsky09}.
\begin{proposition}\label{prop8}
For each $M$ in ${\mathcal T}$, let $\underline{\dim}\,\, GM=(m_1,\ldots, m_n)$ and let $1\leq i\leq n$. We have
\[
-[\mbox{coind}_T(M):T_i] + \langle S_i, e\rangle_a \geq -m_i,
\]
for each submodule $N$ of $GM$ with $\underline{\dim}\, N=e$. Thus the exponent
of $x_i$ in the denominator of $X_M$ is less or equal to $m_i$.
\end{proposition}
\begin{proof}
This result holds for the case $M\cong \Sigma T',\,T'\in \opname{add}\nolimits T$
obviously. We assume that $M$ is indecomposable and not isomorphic
to any $\Sigma T'$. The case where $M$ is decomposable is a
consequence of the multiplication theorem for $X_?$. Now by Lemma~7
of \cite{Palu08}, we have $$-[\mbox{coind}_T(M):T_i]=-\langle S_i,
GM\rangle_{\tau}.$$ Note that we have the short exact sequence of
$\opname{End}_{{\mathcal C}}(T)$-modules
\[
0\to N\to GM\to GM/N\to 0.
\]
By applying the functor $\opname{Hom}(S_i, ?)$, we get
\[
\langle S_i, N\rangle_{\tau}+\langle S_i, GM/N\rangle_{\tau}-\langle S_i, GM\rangle_{\tau}+ \mbox{dim}\, \opname{Ext}^2(S_i, N)\geq 0.
\]
By the stable 3-Calabi-Yau property of $\mod \opname{End}_{{\mathcal C}}(T)$ proved
in \cite{KellerReiten07}, we have $\mbox{dim}\opname{Ext}^2(S_i,N)\leq
\mbox{dim} \opname{Ext}^1(N,S_i)$. Therefore, we have
\begin{eqnarray*}
-[\mbox{coind}_T(M):T_i] + \langle S_i, e\rangle_a &\geq& -\langle N,S_i\rangle_{\tau}-\langle S_i, GM/N\rangle_{\tau}-\mbox{dim}\opname{Ext}^2(S_i,N)\\
&\geq&-[N,S_i]- [S_i, GM/N]+ \,^1[S_i, GM/N]\\
&\geq& - m_i.
\end{eqnarray*}
\end{proof}
\subsection{Behaviour of the $\mathbf{g}$-vectors under mutation.}
Let $B=(b_{ij})$ be an antisymmetric integer $r\times r$-matrix. Let
${\mathcal C}\supset \opname{add}\nolimits T$ be a triangulated $2$-CY realization of $B$. Let
$T_1, \ldots, T_r$ be the non isomorphic indecomposable factors of
$T$. Let $1\leq l\leq r$ be an integer and $T'=\mu_l(T)$ the
mutation of $T$ at $T_l$. Thus, the non isomorphic indecomposable
factors of $T'$ are $T_1,\ldots, T_l^*,\ldots, T_r$. Let ${\mathcal C}_1$ be
a principal gluing of ${\mathcal C}\supset \opname{add}\nolimits T$ and ${\mathcal C}_2$ a principal
gluing of ${\mathcal C}\supset \opname{add}\nolimits T'$ (we assume such gluings exist).
For each indecomposable object
$M\in {\mathcal C}$ reachable from $T$, we denote by $\mathcal{F}_M^{T}$ and
$\mathcal{F}_M^{T'}$ the $\mathcal{F}$-polynomials of $M$ with
respect to ${\mathcal C}_1$ and ${\mathcal C}_2$ respectively. Following
\cite{FominZelevinsky07}, we define the integers $h_l$ and $h_l'$ by
\begin{eqnarray*}
u^{h_l}=\mathcal{F}_M^{T}|_{Trop(u)}(u^{[-b_{k1}]_{+}},\ldots, u^{-1},\ldots, u^{[-b_{kn}]_{+}}),\\
u^{h_l'}=\mathcal{F}_M^{T'}|_{Trop(u)}(u^{[b_{k1}]_{+}},\ldots,
u^{-1},\ldots, u^{[b_{kn}]_{+}}),
\end{eqnarray*}
where $u^{-1}$ is in the $l$-th position.
The following proposition shows that if the gluings ${\mathcal C}_1$ and
${\mathcal C}_2$ exist (for example if ${\mathcal C}$ is algebraic and
Conjecture~\ref{conj:ExistenceOfGlueings} holds), then Conjecture~6.10
of \cite{FominZelevinsky07} holds for the cluster
algebra with principal coefficients associated with $B$.
\begin{proposition}\label{prop9}
In the above notation, we have
\[
h_l'=-[[\mbox{ind}_{T}(M):T_l]]_{+}, \ \ h_l=\mbox{min}(0,
[\mbox{ind}_{T}(M):T_l]).
\]
\end{proposition}
\begin{proof}
Let $S_i$, $1\leq i\leq r$, be the top of the indecomposable right
projective $\opname{End}_{{\mathcal C}}(T')$-module $\opname{Hom}_{{\mathcal C}}(T', T'_i)$. First we
will show that $g_l=[\mbox{ind}_T(M):T_l]> 0$ iff $S_l$ occurs as a
submodule of the module $\opname{Hom}_{{\mathcal C}}(T', \Sigma M)$ and that the
multiplicity of $S_l$ in the socle of $\opname{Hom}_{{\mathcal C}}(T', \Sigma M)$
equals $[\mbox{ind}_T(M):T_l]$.
Suppose that $g_l> 0$. Then we have the following triangle
\[
T^1_M\to T^{0'}_M\oplus (T_l)^{g_l}\to M\to \Sigma T^1_M
\]
with $T^1_M$, $T^{0'}_M$ in $\opname{add}\nolimits T$ and $[T^{0'}_M:T_l]=0$, where
$(T_l)^{g_l}$ is the sum of $g_{l}$ copies of $T_l$. Applying the
functor $\opname{Hom}_{{\mathcal C}}(T', ?)$ to the shift of the above triangle, we
get the exact sequence
\[
0\to \opname{Hom}_{{\mathcal C}}(T', \Sigma (T_l)^{g_l})\to \opname{Hom}_{{\mathcal C}}(T',
\Sigma M)\to \opname{Hom}_{{\mathcal C}}(T', \Sigma^2 T^1_M) \to \ldots
\]
Note that $\opname{Hom}_{{\mathcal C}}(T',\Sigma (T_l)^g)\cong(S_l)^{g_l}$, {\em
i.e.} $S_l$ occurs with multiplicity $\geq g_l$ in the socle of
$\opname{Hom}_{{\mathcal C}}(T', \Sigma M)$. If the multiplicity of $S_l$ in the
socle of $\opname{Hom}_{{\mathcal C}}(T',\Sigma M)$ was $>g_l$, then $S_l$ would
occur in the socle of $\opname{Hom}_{{\mathcal C}}(T',\Sigma^2T_M^1)$. This is not
the case since $\opname{Hom}_{{\mathcal C}}(T',\Sigma^2T_M^1)$ is the sum of
injective indecomposables not isomorphic to the injective hull
$\opname{Hom}_{{\mathcal C}}(T',\Sigma^2T_l)$ of $S_l$. Conversely, if $S_l$ occurs
in the socle of $\opname{Hom}_{{\mathcal C}}(T', \Sigma M)$, thanks to the split
idempotents property of ${\mathcal C}$, we have an irreducible morphism
$\alpha: \Sigma T_l\to \Sigma M$ in ${\mathcal C}$. Thus, by the definition
of the index, we get $g_l>0$. Moreover, the multiplicity of $S_l$
equals $g_l$ by the same argument as before.
Assume that $g_l>0$. For an arbitrary submodule $U$ of
$\opname{Hom}_{{\mathcal C}}(T', \Sigma M)$, let $\mbox{dim}\, U=(e_1, \ldots, e_n)$.
We will show that
\[
e_l\leq g_l +\sum_i[b_{il}]_{+}e_i.
\]
Indeed, consider the projective resolution of the simple module $S_l$
\[
\ldots \to \oplus P_i^{b_{il}}\to P_l\to S_l\to 0.
\]
Applying the functor $\opname{Hom}_{\opname{End}_{{\mathcal C}}(T')}(?,U)$, we get the exact sequence
\[
0\to \opname{Hom}(S_l, U)\to \opname{Hom}(P_l, U)\to \opname{Hom}(\oplus P_i^{b_{il}}, U)\to \ldots,
\]
which implies the inequality because the dimension of $\opname{Hom}(S_l,U)$
is less or equal to the multiplicity of $S_l$ in the socle of
$\opname{Hom}_{{\mathcal C}}(T',\Sigma M)$, which equals $g_l$. By
Theorem~\ref{thm4}, we have
\begin{eqnarray*}
u^{h_l'}&=&\mathcal{F}_M^{T'}|_{Trop(u)}(u^{[b_{k1}]_{+}},\ldots, u^{-1},\ldots, u^{[b_{kn}]_{+}})\\
&=& 1\oplus \bigoplus_{e}\chi(Gr_e(\opname{Hom}_{{\mathcal C}}(T',\Sigma M)))u^{-e_l}\prod_{i\neq l}(u^{[b_{ki}]_{+}})^{e_i}.
\end{eqnarray*}
We have just shown that for each $e$, we have
\[
-e_l+\sum_i[b_{il}]_{+}e_i\geq g_l,
\]
and the equality occurs if $e$ is the dimension vector of the
submodule $(S_l)^{g_l}$. We conclude that we have
$h_l'=-[\mbox{ind}_{T}(M):T_l]$. If $g_l\leq 0$, then $S_l$ does not
occur in the socle of $\opname{Hom}_{{\mathcal C}}(T',\Sigma M)$ and it is easy to
see that $h_l'=0$. Dually, we have the equality $h_l=\mbox{min}(0,
[\mbox{ind}_{T}(M):T_l])$.
\end{proof}
\subsection{Acyclic cluster algebras with principal coefficients.}
Let $B$ be an antisymmetric integer $r\times r$-matrix. Assume that
$B$ is acyclic. Let $Q$ be the corresponding quiver of $B$ with set
of vertices $Q_0=\{1,\ldots, r\}$ and with set of arrows $Q_1$. Let
${\mathcal C}_Q$ be the cluster category of $Q$, $T=kQ$ the canonical cluster
tilting object of ${\mathcal C}_Q$. We claim that the cluster category
${\mathcal C}_Q\supset \opname{add}\nolimits T$ admits a principal gluing.
Indeed, we define a new quiver $\tilde{Q} =
Q\overleftarrow{\coprod}Q_0$ associated with $Q$: Its set of
vertices is $\{1,\ldots, 2r\}$, and its arrows are those of $Q$ and
new arrows from $r+i$ to $i$ for each vertex $i$ of $Q$. Since $Q$
is acyclic, so is $\tilde{Q}$, hence $k\tilde{Q}$ is
finite-dimensional and hereditary. Thus, we have the cluster
category ${\mathcal C}_{\tilde{Q}}$ which is a triangulated $2$-CY
realization of the matrix
\begin{eqnarray*}
\left(\begin{array}{ll} B&-I_r\\I_r&\quad 0
\end{array}
\right).
\end{eqnarray*}
In particular, ${\mathcal C}_{\tilde{Q}}\supset \opname{add}\nolimits k\tilde{Q}$ is a
principal gluing for ${\mathcal C}_Q\supset \opname{add}\nolimits T$. Thus,
Proposition~\ref{prop6}, Theorem~\ref{thm3}, Theorem~\ref{thm4} and
Proposition~\ref{prop7} hold for acyclic cluster algebras with
principal coefficients.
Let $P_i$, $1\leq i\leq 2r$, be the non isomorphic indecomposable
projective right modules of $k\tilde{Q}$. Let ${\mathcal P}=\opname{add}\nolimits
(P_{r+1}\oplus \ldots \oplus P_{2r})$. We have a triangle
equivalence
\[
^\perp(\Sigma{\mathcal P})/{\mathcal P}\stackrel{\sim}{\longrightarrow} {\mathcal C}_Q.
\]
Recall that there is a partial order on $\mathbb{Z}^r$ defined by
\[
\alpha \leq \beta \,\,\mbox{iff}\ \, \alpha(i)\leq \beta(i), \
\mbox{for} \, 1\leq i\leq r, \, \mbox{where} \, \alpha,\, \beta \in
\mathbb{Z}^r.
\]
\begin{proposition}
Let $B$ be a $2r\times r$ integer matrix, whose principal part is
antisymmetric and acyclic and whose complementary part is the
identity matrix. Let $\sigma$ be a sequence $k_1$, $\ldots$, $k_m$
with $1\leq k_i\leq r$. Denote by $B_{\sigma}$ the matrix
\[
\mu_{k_1}\circ\mu_{k_2}\ldots \circ\mu_{k_m}(B)=(b_{ij}^{\sigma}).
\]
Let $E_{\sigma}=(e_1,e_2,\ldots, e_r)$ be the complementary part of
$B_{\sigma}$, where $e_i\in \mathbb{Z}^r$, $1\leq i\leq r$. Then for each
$i$, we have $e_i\leq 0$ or $e_i\geq 0$.
\end{proposition}
\begin{proof}
Suppose that there is some $k$ such that $e_k\nleq 0$ and $e_k\ngeq
0$. For simplicity, assume that $k=1$, {\em i.e.} there are $\ r<
i,\,j \leq 2r$ such that $b_{i1}^{\sigma}>0$ and
$b_{j1}^{\sigma}<0$.
Let $Q$ be the quiver corresponding to the principal part of $B$ and
$\tilde{Q}$ as constructed above. By the argument above, there is a
cluster tilting object $T'$ of ${\mathcal C}_{k\tilde{Q}}$ such that
$B(T')^0= B_{\sigma}$. We have arrows $P_i\to T_1'$ and $T_1'\to
P_j$, where $T_1'$ is the indecomposable direct summand of $T'$
corresponding to the first column of $B_{\sigma}$. Now if we
consider the mutation in direction $1$ of $T'$, we will have an
arrow $P_i\to P_j$ in $Q_{\mu_1(T')}$. But this is impossible, since
for $r< l\leq 2r$, the $P_l$ are simple pairwise non isomorphic
modules so we have
\[
\opname{Hom}_{{\mathcal C}_{k\tilde{Q}}}(P_i,P_j) = \opname{Hom}_{k\tilde{Q}}(P_i, P_j) =0.
\]
\end{proof}
\def$'${$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
|
1,108,101,563,302 | arxiv | \section{Introduction}
White dwarfs offer a unique view into the properties of planetary systems.
A star can retain much of its planetary system as it leaves the main sequence
and evolves through the red giant stage.
Although the inner $\sim 5$ AU of a planetary system may
evaporate within the red giant atmosphere and planets
that were in marginally stable orbits may escape during stellar mass loss,
most outer planets and even much of
the Oort-cloud-like extremities of planetary systems should persist
around white dwarfs \citep{debes02}.
The white dwarf Giclas 29-38 (WD 2326+049; G29-38 hereafter)
has garnered intense interest since \citet{ZB87}
discovered infrared emission from this object far in excess of the
photosphere. The excess emission, confirmed
both by ground-based \citep{tokunaga} and space-based \citep{chary} observations,
begins to deviate from a pure photosphere
in the near-infrared. While it was initially speculated that a Jupiter-sized companion could
supply all the infrared excess, follow-up observations
(e.g. Kuchner et al. 1998, Graham et al. 1990) appear to
have ruled this out. The colors measured with ISOCAM
suggested the infrared excess is due to particulate matter rather than
a brown dwarf companion \citep{chary}.
Besides strong hydrogen absorption lines,
G29-38 has some photospheric metal lines, including \ion{Ca}{2}
which should rapidly ($< 10^4$ yr) sink out of the atmosphere due to
gravitational sedimentation
\citep{alcock80,koester97,zuckreid}.
Possible explanations for the atmospheric metals
and the infrared excess include interstellar medium (ISM) accretion
\citep{dupuis,koester97,zuckreid},
cometary breakup \citep{alcock}, and asteroid disruption \citep{jura}.
We turned the powerful instruments of the
{\it Spitzer} Space Telescope \citep{werner} towards G29-38 to
measure its brightness from 4.5--24 $\mu$m. These observations probably
probe a planetary system in a late stage of evolution.
G29-38's main sequence progenitor was probably an A star with mass
approximately 3.1 $M_\odot$ \citep{weidemann}, similar to many stars
known to have debris disks and potential planetary systems \citep{riekeAstar}.
The post-AGB phase age is estimated to be $\sim 5\times 10^8$ yr
\citep{postAGBage}.
\def\extra{
infrared excess 2--5 $\mu$m, far above photosphere, Tcolor=1200 K,
title: "an orbiting white dwarf?"
tauPR for particle radius 0.5 micron --> 10 yr
minimum mass of grains 3e17 g
"accumulating this much orbital material in such close proximity
to G29-38 seems very unlikely given the rapid depletion due to the Poynting-Robertson effect
and the absence of any spectral peculiarities in its photospheric spectrum"
adopt distance 14 pc \citep{ZB87}
}
\begin{figure}[th]
\plotone{f1.eps}
\caption[f1.eps]{
Spectral energy distribution of the white dwarf G29-38
from the ultraviolet through infrared. The infrared observations
are described in \S2 and the ultraviolet through
near-infrared observations and model are described in \S3.
The solid line through the infrared data is a modified blackbody
fit to the continuum.
\label{wdspec}}
\end{figure}
\section{Observations\label{obssec}}
Figure~\ref{wdspec} shows the spectral energy distribution of
G29-38.
Observations with the Infrared Array Camera (IRAC) \citep{fazio}
were performed on 2004 Nov 26 10:54 UT.
At each of five dithers
in a Gaussian spatial distribution, a 30 sec frame was taken
with the 4.5 and 8 $\mu$m arrays.
We extracted our photometry using the method \citep{reachcal} used
for the IRAC calibration stars. The quoted 4.5 and 8 $\mu$m fluxes, measured
from the basic calibrated data,
are $8.8\pm 0.3$ and $8.2\pm 0.3$ mJy, respectively.
We calculated color corrections of
0.995 and 1.17 in the 4.5 and 8 $\mu$m channels, making the estimated
fluxes 9.2 and 7.0 mJy at the IRAC nominal
wavelengths of 4.493 and 7.782 $\mu$m.
Observations with the Multiband Imaging Photometer for Spitzer
(MIPS) \citep{rieke} were performed on 2004 Dec 2 04:45 UT.
Three cycles of small-scale photometry dithers were taken
with 10 sec frames (total exposure time 420 sec); the
flux at 24 $\mu$m is 2.4 mJy.
Observations with the Infrared Spectrograph (IRS) \citep{houck}
were performed on 2004 Dec 8 3:26 UT.
The source was first observed on the blue-filter portion of the
peak-up array;
the flux measured from the 16 $\mu$m peak-up image is 3.7 mJy.
The spectrum from 5.2 to 14.4 $\mu$m was assembled by differencing
observations at two nods on each of the 4 spectral orders.
The flux derived from the
IRS spectrum convolved with the IRAC 8 $\mu$m bandpass
is consistent with the IRAC flux.
The IRS spectrum shows continuum emission and a strong 9--11 $\mu$m
feature characteristic of silicate minerals. It shows none of the
bands at 6.2, 7.7, 8.6, 11.3, or 12.6 $\mu$m characteristic of
polycyclic aromatic hydrocarbons that normally dominate
mid-infrared spectra of the ISM.
The excess emission above photospheric is approximated by
two modified blackbodies with temperatures 890 K and 290 K
and dilution factors (at 10 $\mu$m wavelength)
$4.2\times 10^{-16}$ and $6.3\times 10^{-15}$,
and emissivity proportional to $\nu^{0.5}$.
These functions are only intended as mathematical
approximations of the continuum shape.
\def\extra{
If the emission were optically thick
at 10 $\mu$m, the dilution factors would imply,
for the two temperature components, emitting regions
of size 5 and 18 mas, and distances from the star
of 7 and 27 $R_\odot$, respectively. These are lower
limits to the size of the emitting region, which in fact
must be optically thin at 10 $\mu$m in order to present
such a high-contrast silicate emission feature.
}
\section{Modeling\label{modsec}}
We adopt the
following basic parameters for G29-38. It is a
DA4 type white dwarf, with atmospheric spectrum
dominated by H lines \citep{green}.
It has an assumed mass of 0.69 $M_\odot$,
surface gravity $\log g=8.14$, and
temperature $11,800$ K \citep{bergeron95}.
The parallax is
$\pi=0.071''+/-0.004''$ \citep{usno}, implying a distance of 14 pc.
We assembled the spectral energy distribution
from the IUE low-dispersion spectrum
covering 0.115--0.3148 $\mu$m \citep{holberg}
and a model atmosphere for $T=12,000$ K
and $\log g=8$ covering 0.35--60 $\mu$m (courtesy D. Koester).
We used the 2MASS photometry ($J=13.132\pm 0.029$, $H=13.075\pm 0.022$,
$K_s=12.689\pm 0.029$) and optical spectrophotometry
from Palomar \citep{greenstein} for normalization, giving
extra weight to the 2MASS J-band photometry,
the longest-wavelength where we think the photosphere dominates.
The integrated luminosity of the star is $2\times 10^{-3} L_\odot$.
\begin{figure}[th]
\epsscale{.80}
\plotone{f2.eps}
\caption[f2.eps]{
Continuum-subtracted spectrum of the silicate feature from G29-39,
compared to similarly subtracted silicate features of the diffuse
ISM, the O-rich star Mira,
comet Hale-Bopp, and the zodiacal light.
\label{wdspec_sil}}
\end{figure}
The 9--11 $\mu$m emission feature indicates silicates.
Figure~\ref{wdspec_sil} compares the shape of the G29-38 silicate feature
to that of interstellar dust \citep{kemper},
Solar System zodiacal light, the O-rich mass-loss from the star Mira \citep{sloan},
and comet Hale-Bopp \citep{crovisier}.
The G29-38 silicate feature is redder than that of the ISM,
which has a single, rounded peak at 9.7 $\mu$m and nearly linear slopes
on either side. This suggests that the material around G29-38 is not
accreted interstellar matter but rather is indigenous to the star.
The G29-38 silicate feature is more compact than the zodiacal light
silicate feature as well as the exozodiacal light around $\beta$ Pic feature \citep{reachzody}.
The Hale-Bopp spectrum has a prominent
11.3 $\mu$m peak that is not seen in the G29-38 spectrum.
The Mira silicate feature is somewhat redder and wider than the ISM feature, roughly
intermediate between the ISM and G29-38 feature shape.
The G29-38 silicate feature shape is most similar to (and approximately
intermediate between) those of Mira and the zodiacal light, but the line-to-continuum
ratio of G29-38 is much greater than that of the zodiacal light (125\% versus 6\%)
suggesting a very different particle size distribution.
We computed theoretical emission spectra for grains
(size $a=$0.01--1000 $\mu$m, $dn/da\propto a^{-3.5}$)
of various compositions (amorphous carbon, crystalline enstatite,
amorphous pyroxene [50/50 Mg/Fe], amorphous
olivine [50/50] and crystalline forsterite)
around G29-38 to further elucidate the properties of
the dust around that star.
The temperatures and emissivities over a
range of distances (0.003--3.5 AU = 0.6--750 $R_\odot$)
from the white dwarf were integrated over a spherical,
presumed-optically-thin shell with a radial profile
$n\propto r^{-\alpha}$ and an inner cutoff at $R_{min}$.
The temperatures of 0.5 $\mu$m radius grains of
amorphous carbon (olivine) are
600 K (920 K) at 3 $R_\odot$ and
130 K (110 K) at 1 AU from the white dwarf.
\def\extra{
The temperature
can be approximated by $T=T_1 r^{-\delta}$, with $\delta\sim 0.4$--0.5
and $T_1=212$ K for small carbon grains, 65 K
for large olivine or pyroxene grains, and 33 K for pure crystalline
enstatite.
Comparing to the observed spectral shape, we require the
region emitting at 10 $\mu$m to be $<10 R_\odot$, and
the region emitting at 24 $\mu$m to be $<100 R_\odot$.
}
Grains hotter than 1500 K are presumed sublimated and are
not included in the emissivity.
\def\extra{
The observed color temperature (from the continuum shape
near 10 $\mu$m wavelength)
is $> 700$ K, which requires that the bulk of the observed emission
arises within $10 R_\odot$ of the star, even for submicron grains.
The 24 $\mu$m color temperature $\sim 290$ K can be produced
by small grains at 30--100 $R_\odot$ or large grains at 9 $R_\odot$
from the white dwarf.
The emitting region sizes inferred from the color
temperatures are consistent with the lower limits from the
dilution factors. Thus we shall model the emission from a cloud
extending from an inner radius of 0.2 $R_\odot$ to an outer
radius of 1 AU, and consider the compositions and radial density
profile consistent with the observed emission. WHAT DOES THIS MEAN?
}
Since we assume the cloud is optically thin, we cannot further
constrain the geometry of the emitting region, which could
range from a spherical shell to a flattened disk.
The G29-38 silicate feature resembles the
amorphous olivine model, with excess on the red side
that can be accounted for by mixing in forsterite.
Pyroxene can not contribute significantly to the observed spectrum;
its 9--11 $\mu$m feature is too blue.
Amorphous olivine and fosterite alone cannot explain the spectrum,
however, because it cannot emit at 3--6 $\mu$m
without also producing a 1.6 $\mu$m feature that exceeds
the near-infrared H-band flux.
The near-infrared continuum shape at these
wavelengths matches the amorphous carbon model.
The contrast of the silicate feature demands a
particle size distribution that favors small particles, unlike that of
interplanetary dust grains \citep{gruen85}.
On the assumption that all the particles participate in the same
collisional evolution, we assume that silicate and carbon particles
share the same size distribution.
\begin{figure}[th]
\epsscale{1}
\plotone{f3.eps}
\caption[f3.eps]{The infrared excess of G29-38 is compared to
the best-fit model (smooth solid line) consisting of three compositions:
amorphous carbon (dotted line), amorphous olivine (long-dashed line),
and forsterite (short-long dashed line).
\label{fplot}}
\end{figure}
The distance of the emitting region from the star is constrained as follows.
In order to produce significant
emission at 5 $\mu$m, we require $R_{min}<5 R_\odot$.
The best-fitting models all require small $R_{min}$, implying
that the dust exists nearly up to its sublimation temperature.
The radial profile shape requires high $\alpha$, suggesting the
emitting region extends from 1--10 $R_\odot$ from the star.
Figure~\ref{fplot} shows the observed spectrum and the best-fit model.
Experiments with other geometries yielded similar results:
a face-on slab required $\alpha\sim 3$, $R_{min}\sim 2.5 R_\odot$.
A dust cloud this small is consistent with the measurements of
\citet{kuchner98} that showed G29-38 to be unresolved in K-band
at 55 mas (160 $R_\odot$) resolution.
\section{Conclusions}
A cloud of small grains 1--10 $R_\odot$ from G29-38 creates its
mid-infrared emission.
The luminosity of the infrared excess is 3\% of the luminosity of the
star, high by the standards of debris disks, indicating nontrivial
disk opacity in the ultraviolet-visible range.
Our models suggest a dust grain abundance ratio (by number)
of olivine:carbon:forsterite of 5:12:2.
Considering only the atoms in these grains,
assuming material densities of 2.2 and 2.5 g~cm$^{-3}$
for the amorphous carbon and silicates,
the abundance ratio (by number) of C:O is 3:1.
The total dust mass required to generate the observed mid-infrared
flux is of order $10^{18}$ g; a larger mass could be present if
there are larger, cooler grains that do not contribute to the observed flux.
This inferred mass corresponds to that of a $\sim$10 km
diameter asteroid or comet, slightly less than
that of the interplanetary dust responsible
for the zodiacal light in the Solar System \citep{fixsendwek}.
Where could this dust cloud have come from? It was not
generated by the white dwarf (by any known mechanism), nor
is it likely to be planetary system material that was at
this location before the star was a red giant, whose
atmosphere extended much further than the current dust cloud.
The silicate feature is suggestive of dust formed in O-rich
mass-loss during the red giant or AGB phase, but the presence of
significant and comparable amounts of silicate and carbonaceous dust
(and no PAH) does not seem compatible with such dust, which
in any event could not have survived at 10 $R_\odot$ from the star.
The material must have been transported inward.
If the material originates from the planetary system of the progenitor,
then gravitational perturbations would occasionally send smaller
bodies close to the star.
If the small body is collisionally fragmented, or loses mass
by cometary sublimation, or breaks apart by thermal or gravitational stress,
the Poynting-Robertson effect will cause particles of radius $a$ ($\mu$m) and
distance $r$ ($R_\odot$) to spiral into the star on timescales
of $4 r^2 a$ yr, i.e. a few years for
material contributing to the mid-infrared emission.
The particles would continue toward
the star until they sublimate, suffer mutual collisions,
disrupt, with part of the material landing in the
photosphere, part being blown out of the
system by radiation pressure, and part remaining on bound orbits.
If the entire mid-infrared-emitting
mass (i.e. particles out to $\sim 10 R_\odot$)
were lost in the Poynting-Robertson timescale, the accretion
rate is of order $10^{15}$ g~yr$^{-1}$.
This flow could be supplied by collisional comminution of debris from
an asteroidal and cometary cloud like the one in the Solar System
\cite{sykes86}.
This mass loss rate is comparable to the `high' ISM accretion
scenario calculated by \citep{dupuis}, which appears inconsistent
with G29-38's present location in a low density ISM environment.
A plausible model for the infrared excess around G29-38 involves
the tidal disruption of an asteroid near the white dwarf \citep{graham,jura}.
Since short-period comets are composed largely of refractory material,
we generalize this model to include the tidal disruption
of a comet. Comets are commonly observed to
pass near the Sun, so by analogy a surviving reservoir of comets
(and possibly a surviving massive outer planet) would inevitably
lead to star-grazing cometary passages. With its low
luminosity, the white dwarf would not drive a high sublimation
rate for star-grazing comets, reducing the related stresses that appear to
cause Sun-grazing comets to split. Comets may then travel rather
close to the white dwarf, where they could be disrupted by
tidal forces. Our new observations support this model, since the
emitting region lies close to the star, where a large solid body
would be tidally disrupted.
Based on the high C/O ratio in the dust, it is also conceivable that
the `comet' was condensed out of the AGB mass-loss \citep{kuchnersaeger}.
The details of the tidal disruption model presented here are
different from previous papers which assume larger grains and
an optically thick disk.
If the emitting region is a disk, it is either optically thin
or it is viewed relatively face-on, with the line of sight
from the white dwarf to the Sun relatively free of dust.
For a thin disk to absorb 3\% of the luminosity of the
star it would have to be opaque in the ultraviolet,
where there is no evidence of significant extinction and no
2175 \AA~ absorption bump in the IUE spectrum.
An alternate and long-standing model for both the photospheric metals and the
infrared excess is dredge-up and mass loss.
There are already difficulties with dredge-up models associated with the need
for vertical velocities that are at odds with G29-38's g-mode pulsations
that are overwhelmingly horizontal. Further, standard white dwarf models are layered
from prior nuclear burning;
hypothetical mixing from the pulsations would have to reach many orders of
magnitude below the convection zone boundary, well below even the degeneracy
boundary (about $10^{-6} M_{\odot}$) to dredge up material from the C/O layer
(presumably around $10^{-3}M_{\odot}$ or deeper).
Further, the dredge-up hypothesis cannot explain
the abundant silicates;
invoking these materials just below the He-layer is
very difficult to accommodate within our current understanding of nuclear
burning of He in post-main sequence stars appropriate to white dwarf stars in
the mass range of G29-38.
Our observations therefore argue that the observed
material was not formed in the white dwarf; it formed well outside its current
position.
The presence of a dust cloud around G29-38 is likely related to
the anomalous presence of metals in its photosphere.
As discussed by \citet{zuckerman}, the photospheric metals could
be accreted from interstellar gas, but then the
star would need to be within a dense cloud (because
the lifetime of metals in the photosphere [$< 10^4$ yr]
is smaller than the crossing time of a dense cloud [$\ga 10^4$ yr]),
which is not true for G29-38.
The other major model for white dwarf photospheric metals is
cometary impact \citep{zuckreid,debes02}. Based on the presence of
a strong silicate feature and the overall mid-infrared
spectral energy distribution, we believe this latter model
prevails for G29-38.
The abundance of photospheric metals for G29-38
is among the highest of any white dwarf, and the infrared excess
is also the highest of any known white dwarf.
Another white dwarf, WD 1337+705 (G 238-44), with even higher
atmospheric Ca abundance \citep{zuckerman} does not show infrared
excess at the level seen in G29-38
in our {\it Spitzer} photometric survey of white dwarfs
at 4.5 and 8 $\mu$m.
It is possible that a recent disruption of an asteroid or comet has
occurred in G29-38, re-populating the star's dust cloud
at a level much higher than the long-term steady state, as indeed
occurs in debris disk around main sequence stars \citep{riekeAstar} and
the Solar System \citep{sykes86}.
Are there planets around G39-38?
The 9--11 $\mu$m spectral feature proves there is a cloud of small
silicate grains. We ascribed the 3--6 $\mu$m continuum to amorphous carbon dust,
and the carbon-silicate mix matches the spectral energy distribution,
but there is no spectral signature for carbon other than its color temperature.
If we ascribe the 890 K continuum to a planet, then it must be
not only very hot but also very large, with radius
0.2 $R_\odot$---an extremely unusual object.
If the dust cloud is indeed due to an asteroid or comet, perhaps
other, cooler planets await discovery around
the fascinating star G29-38.
\acknowledgements
This work is based in part on observations made with the {\it Spitzer Space
Telescope}, which is operated by the Jet Propulsion Laboratory, California
Institute of Technology under NASA contract 1407. Support for this work was
provided by NASA through award Project NBR: 1269551
issued by JPL/Caltech to the University of Texas.
|
1,108,101,563,303 | arxiv | \section{Introduction}
The goal of modeling the observed ``random'' fluctuations of the velocity field and the intermittent character of the small scale dissipated energy in fully developed turbulent flows has played a critical role in the development of mathematical concepts around multifractal processes. In particular,
random multiplicative cascades first considered by the Russian
school \cite{kol62,ob62,NovSt64} and subsequently developed by B. Mandelbrot \cite{Man74a,Man74b} in order to mimic
the energy transfer from large to small scales \cite{Fri95}
represent the paradigm of multifractal random distributions.
They are the basis of a lot of mathematical work
and have led to a wide number of applications
far beyond the field of turbulence.
Mandelbrot cascades ($\cal M$-cascades) mainly consist
in building a random measure by using a multiplicative iterative rule. One starts with
some large interval with a constant measure density $W_0$ and splits
this interval in two equal parts. The measure density in the
left and right half parts are obtained by multiplying $W_0$ by respectively two independent, identically distributed
positive factors (say $W_1^L$ and $W_1^R$). This operation is then repeated
independently on the two sub-intervals and so on, {\em ad infinitum}, in
order to converge towards a singular measure which properties
have been studied extensively (see e.g. \cite{Man74b,KaPe76,Gui87,Bar04}).
The main disadvantage of $\cal M$-cascades is that they involve a
specific scale ratio ($s=2$ in general) and are limited to live in the starting interval.
This lack of stationarity and continuous scale invariance
is obviously not suited to accounting for natural phenomena.
In order to circumvent these problems, continuous extensions of Mandelbrot cascades were proposed by first Barral and Mandelbrot \cite{BaMan02} and later by Bacry and Muzy \cite{MuBa02,BaMu03}. The idea under these
constructions is to replace the discrete multiplicative density
$\prod_{i=1}^n W_i = e^{\sum_{i=1}^n \ln W_i}$ by $e^{\omega_\ell(t)}$ where $\omega_{\ell}(t)$ is an infinitely divisible noise chosen with a logarithmic correlation function designed to mimic the tree-like (in general a dyadic tree) structure underlying $\cal M$-cascades \cite{ArnBaMaMu98,MuDeBa00,sm01}.
The above definitions of random cascade measures led to a large number of extensions notably in order to construct stochastic processes with some predefined regularity properties.
The most popular approach was initiated by Mandelbrot (see e.g. \cite{ManFisCal97}) and consists in compounding a self-similar stochastic process like the (fractional-) Brownian motion $B_H(t)$ with a ``multifractal time"
$M(t)$, a multifractal measure of the interval $[0,t]$ as provided by a continuous cascade.
Such a compounded process $B_H(M(t))$ inherits the multifractal
scaling properties of $M(t)$ with a "main" regularity that corresponds to the self-similar process $B_H(t)$.
An alternative but related approach, was initially proposed
with the construction of ``the Multifractal Random Walk" (MRW) in \cite{MuDeBa00,BaDeMu01} and that consists of interpreting $M(dt)$ as the local variance of a fractional Brownian motion $B_H(t)$. As
emphasized in \cite{MuBa02,lud08,accp09}, this amounts to constructing a multifractal process as (the limit of) a stochastic integral like $\int e^{\omega_{\ell}(t)} dB_H(t)$.
This point of view is inspired from classical stochastic volatility models of financial markets that aim at accounting for the observed bursts in price fluctuations by the random nature of the variance of an underlying Gaussian law. It can also be understood as a formalization of the so-called "Kolmogorov refined similarity hypothesis" \cite{Fri95} according to which the velocity fluctuation $\delta v$ and
the local rate of dissipated energy $\varepsilon$ are related as,
in its Lagrangian version \cite{rshLag09}, $\delta_\tau v \sim \varepsilon_\tau^{1/2} dW_\tau$, $dW_\tau$ being a noise such that $\EE{dW_\tau^2} = \tau$. Accordingly, the multifractal properties
of the velocity field directly come from those of the local
dissipation field $\varepsilon$, a multifractal cascade interpreted as a ``local variance''.
The main advantage of continuous cascades and their associated random processes is that they provide a large class of parsimonious models with stationary increments and continuous stochastic scale invariance properties. From a practical point of view, they are easy to calibrate from data and can provide simple analytical or numerical solutions to many statistical problems. However, the aforementioned methods to derive a multifractal process from a multifractal measure lack of flexibility and in particular hardly allow one to describe
processes that are not invariant by time reversal. This means that
the increment distribution skewness (i.e. non vanishing third order moments of the increments at different scales) as observed in turbulence or the leverage effect (i.e. some causal, asymmetric relationship between increment signs and increment amplitudes) as observed in financial time series cannot be captured easily within this framework. All former attempts to address this issue were based on studying a random noise like $e^{\omega_\ell(t)} \Delta B_H(t)$ for a specific cross-covariance between $\omega_\ell$ and $\Delta B_H$ \cite{PocBou02,BaDuMu12}. However, as discussed in \cite{BaDuMu12}, this approach leads to a non degenerated limiting process only in some restricted range $H > 1/2$ and therefore cannot be used as a model for financial markets ($H \simeq 1/2$) or turbulence ($H \simeq 1/3$).
More recently, Chevillard et al. \cite{CheGaRhoVa17} introduced a new model which obeys to multifractal scaling with a (signed) third order structure function that behaves as a linear function, like in turbulence.
This model is defined within the context of Gaussian Multiplicative Chaos (i.e. a Gaussian version of $\omega_{\ell}(t)$) and mainly consists in considering a fractional integration of a product like $\omega' e^{\omega} dB$ where $\omega'$ represents a peculiar intermittent version of $\omega$, independent of $\omega$ but correlated with the white noise $dB$. The authors have shown that
their construction leads to a properly defined intermittent skewed random process. However, we can point out that the model of Chevillard et al. is far from being simple, involves cumbersome computations and is not characterized by self-similar properties, it follows a multifractal scaling only in the limit of small scales.
In this paper, we propose a different path to solve
the challenging issue of mixing multiscaling properties and non-invariance by time reversal.
Beyond this question, our construction also offers an appealing alternative way to build a large class of multifractal functions
with well controlled scaling (or regularity) properties and that are
characterized by new features as compared to former constructions.
Our approach is directly inspired from the random cascade
picture originally proposed by Richardson \cite{Rich22} under which, in turbulent flows, large eddies are stretched and broken into smaller ones to which they transfer a fraction of their energy and so on, up to the dissipation scale. Instead of trying to capture the velocity field intermittency from the final state of such a cascade, i.e., the dissipation field (which is a multifractal measure), the previous scenario suggests that one could describe the overall flow as a {\em superposition} of coexisting structures at all scales correlated with each other by a cascading intensity (energy).
This viewpoint of decomposing a function as a weighted sum of waveforms (the "eddies") at different scales precisely corresponds to a wavelet transform representation \cite{Mallatbook}. The construction of discrete multiplicative cascades along the wavelet tree associated with orthogonal wavelet basis has been already
proposed two decades ago by Arneodo, Bacry and Muzy \cite{ArnBacMuz98}.
These "$\cal W$-cascades" have proven to be an appealing alternative to $\cal M$-cascade based constructions in order to directly build multifractal stochastic processes \cite{ArnBaMaMu98,BarSeu05}. They also provided a suitable approach to extent the framework of stochastic self-similarity and build new random functions with a non trivial spectrum of oscillating singularities \cite{ABJM97,ABJM98}.
Our goal in this paper is thus to construct an extension of
$\cal W$-cascades in order to get rid of the restraining grid structure of wavelet orthogonal basis that prevent, very much like grid-bounded $\cal M$-cascades, $\cal W$-cascades from being able to simply account for stationarity and continuous scale invariance. For that purpose, we just consider a continuous sum over space $t$ and scales $\ell$ of ``sythetizing wavelets'' weighted by a factor that is precisely given by the stochastic density involved in continuous cascades, $e^{\omega_{\ell}(t)}$. We show that such a construction allows us to obtain well defined processes with a large flexibility on their scaling and regularity properties. Moreover it generically leads to skewed processes and is able to reproduce the leverage effect for specific shapes of synthetizing wavelets. In that respect, they can naturally be invoked as models for synthetic turbulence or calibrated to account for asset fluctuations in financial markets.
Various numerical examples are provided throughout the paper in order to illustrate our
purpose and notably to show the model ability to reproduce many of the
observed features of the longitudinal
velocity field fully developed turbulence experiments.
The paper is structured as follows: in section \ref{sec:wcasc} we recall the main lines of continuous
cascades (Sec. \ref{sec:c-casc}) and $\cal W$-cascades (Sec. \ref{sec:w-casc}) constructions.
This notably allows us to introduce the process $\omega_{\ell}(t)$ and review its main statistical properties. Our new constrution of continuous cascades in the wavelet plane
($\cal C W$-cascades) are introduced in section \ref{sec:wc-cascades}.
After the definition and the statement of a weak convergence result,
we provide some numerical examples (Sec. \ref{sec:wc-cascade_def}). In Sec. \ref{sec:wc-cascades_wavelets}, we study their wavelet transform and establish the almost sure minimum regularity of their paths. Scaling and self-similarity
properties of this new class of processes are studied
in section \ref{sec:scaling}. We prove that their structure functions are characterized by a power-law behavior with some non-linear $\zeta_q$ spectrum of scaling exponents (Sec. \ref{sec:wc-cascades_ss}) and
study the correlation functions of the absolute increments (Sec. \ref{sec:wc-cascades_corr}). We finally
study the properties of $\cal CW$-cascades as respect to time reversal, notably the behavior of the skewness and the leverage function (Sec. \ref{sec:cw-cascades_skew}).
Concluding remarks and prospects for future research are given in section \ref{sec:conclusion}
while technical material and proofs are provided in appendices.
\section{Continuous cascades and $\cal W$-cascades}
\label{sec:wcasc}
This section contains a brief review of the notions
of ``continuous'' and ``wavelet'' multiplicative
cascades. The first ones were introduced as stationary, self-similar singular measures with log-infinitely divisible multifractal properties while
$\cal W$-cascades are the wavelet transform counterparts
of Mandelbrot discrete multiplicative cascades. As
explained in Sec. \ref{sec:wc-cascades},
both constructions will be mixed in order to build
continuous wavelet cascade models.
\subsection{Log-infinitely divisible continuous cascades}
\label{sec:c-casc}
Continuous random cascades are stochastic measures $M(t)$ introduced few years ago (\cite{BaDeMu01,SchMar01,BaMan02,MuBa02,BaMu03}) in order to extend, within a stationary and grid-free framework,
the Mandelbrot discrete multiplicative cascades.
Such a measure $M(t) = \int_0^t dM_u$
can have exact multifractal scaling properties in the sense that it satisfies, for a given $T >0$, $\forall t \leq T$:
\begin{equation}
\label{eq:scaling_cascades}
\EE{M(t)^q} = K_q \left(\frac{t}{T} \right)^{\zeta_q}
\end{equation}
where $\zeta_q$ is a non-linear
concave function called the multifractal spectrum (or the spectrum of structure functions scaling exponents in the context of turbulence \cite{Fri95,Ar_Turb96}) and $K_q$ is a (eventually infinite) prefactor
corresponding the the $q$-order moment at scale $t=T$.
In \cite{MuBa02, BaMu03} (see also \cite{BaMan02}) $dM(t)$ is obtained as the (weak) limit, when $\ell \to 0$ of the measure with density:
\begin{equation}
\label{eq:def_M}
dM_\ell(t) = e^{\omega_{\ell}(t)} dt
\end{equation}
where $e^{\omega_{\ell}(t)}$ is a stationary log-infinitely divisible process
representing the continuous cascade from scale $T$ to scale $\ell$. Its precise definition
and main properties, reviewed in Appendix \ref{app:cont_casc}, notably imply that it satisfies
(thanks to Eqs. \eqref{rhoexact} and \eqref{charf}):
\begin{equation}
\label{eq:def_phi}
\EE{e^{q \omega_{\ell}(t)}} = \left(\frac{T}{\ell}\right)^{\phi(q)}
\end{equation}
where $\phi(q)$ is the cumulant generating function associated with
an infinitely divisible law as provided by the celebrated Levy-Khintchine Theorem \cite{Fel71}. Moreover, $\omega_{\ell}(t)$ verifies, $\forall s < 1, u \in [0,T]$,
\begin{equation}
\label{eq:ss-omega}
\omega_{s \ell}(s u) \stackrel{\cal L}{=} \Omega_s + \omega_\ell(u)
\end{equation}
where $\stackrel{\cal L}{=}$ means that the two processes have the same
finite dimensional distributions of any order and $\Omega_s$ is a random variable independent of the process $\omega_\ell(t)$ with the same distribution as $\omega_{s T}(t)$.
The multifractal scaling \eqref{eq:scaling_cascades} follows since
Eq. \eqref{eq:ss-omega} entails
\begin{equation}
M(st_0) = \lim_{\ell \to 0} \int_0^{s t_0} e^{\omega_{s \ell}(u)} du \stackrel{\cal L}{=} se^{\Omega_s} \lim_{\ell \to 0} \int_0^{t_0} e^{\omega_{\ell}(u)} du = se^{\Omega_s} M(t_0)
\end{equation}
and thus by choosing $t_0=T$ and $s = t/T$, we have
$$
\EE{M(t)^q} = s^q \EE{e^{q\Omega_{s}}} \EE{M(T)^q}
$$
that leads, using Eq. \eqref{eq:def_phi}, to Eq. \eqref{eq:scaling_cascades} with
$$
\zeta_q = q-\phi(q) \; .
$$
As discussed in the introduction, a large class of multifractal processes $X(t)$ with stationary increments can be obtained from a multifractal measure $M(t)$. One can, as suggested by Mandelbrot \cite{ManFisCal97,Man99}, compound a self-similar process $B_H(t)$ (e.g., a fractional Brownian motion) by the non-decreasing function $M(t)$ so that:
$$
X(t) = B_H(M(t)) \; .
$$
Within this approach $M(t)$ is referred to as the "multifractal time".
Since $B_H(st) \stackrel{\cal L}{=} s^H B_H(t)$, one has
$$
X(st) \stackrel{\cal L}{=} s^H e^{H \Omega_s} X(t)
$$
that entails, given the increment stationarity of $X(t)$, the multiscaling of the structure functions:
$$
\EE{|X(t+\tau)-X(t)|^q} = C_q \tau^{\zeta_q}
$$
with $\zeta_q = qH-\phi(qH)$. Another approach initiated in \cite{MuDeBa00,BaDeMu01} (see
also \cite{MuBa02}) is to consider the limit $\ell \to 0$
of the sequence $X_\ell(t) =\sum_{k=1}^{t/\ell} \ell^{\alpha} e^{\omega_\ell(k)} dW_k$ where $dW_k$ is a fractional
Gaussian noise and $\alpha$ an appropriate constant chosen to
ensure the convergence (in some specific sense) of the series.
This amounts to interpreting $e^{\omega_\ell(k)}$ as a stochastic
variance or, in the field of mathematical finance, a "stochastic volatility". Related
constructions consist in considering stochastic integrals like $\int e^{\omega_{\ell}(t)} dB_H(t)$,
where $B_H(t)$ is a fractional Brownian motion \cite{MuBa02,lud08,accp09}.
All these approaches were extensively used and studied
in the literature as paradigms of multifractal
processes satisfying exact stochastic scale-invariance properties and also considered
as toy models for applications like turbulence or financial time series.
However, as recalled in the introduction, they mainly involve separately the construction
of a multifractal measure and a self-similar process and do not consist in directly
building the random process $X(t)$ with some specific properties. It then results
a lack of flexibility in the obtained features, in particular,
as emphasized previously and discussed in \cite{PocBou02,BaDuMu12,CheGaRhoVa17}, skewed statistics and "leverage effect" cannot be obtained in a fully satisfactory way through these approaches. As reviewed below, wavelet cascades offer an interesting alternative in the
sense that they do not rely on any preset self-similar process and consists in directly
building $X(t)$ with a control of its scaling or regularity properties.
\subsection{$\cal W$-cascades}
\label{sec:w-casc}
In Ref. \cite{ArnBacMuz98}, the authors introduced the so-called
$\cal W$-cascades as the natural
transposition of Mandelbrot $\cal M$-cascades within the framework
a orthogonal wavelet transform. This allows one to construct multifractal processes or distributions with a precise control of their regularity properties.
The idea is to build a new class of (multi-)fractal functions $Z(t)$ from their explicit representation over
a wavelet basis:
\begin{equation}
\label{eq:wcascade1}
Z(t) = \sum_{j=0}^\infty \sum_{k=0}^{2^{j-1}} c_{j,k} \psi_{j,k}(t)
\end{equation}
where $\psi_{j,k}(t) = 2^{j} \psi(2^j t-k)$ and $\{ \psi_{j,k}(t) \}_{j,k \in \mathbb Z}$ constitutes an orthogonal
wavelet basis of the interval. The wavelet coefficients $c_{j,k} = \int Z(u) \psi_{j,k}(u) du $ are chosen according to the multiplicative cascade rule \cite{ArnBacMuz98}:
\begin{equation}
\label{eq:wcascade2}
c_{j,2k} = W_0 \; c_{j-1,k}, \; \; c_{j,2k+1} = W_1 \; c_{j-1,k}
\end{equation}
with $W_0$ and $W_1$ are i.i.d. copies of a real valued random variable $W$.
We see that, if $W$ is a positive random variable, the law of the wavelet coefficient $c_{j,k}$ is precisely
given by the law of the density of a $\cal M$-cascade at construction step $j$:
\begin{equation}
\label{eq:wcascade3}
c_{j,k} = \prod_{m=0}^{j} W_m
\end{equation}
It has been shown in \cite{ArnBacMuz98} that, under some mild condition on the law of $W$, $Z(t)$ is a well
defined multifractal process with almost surely Lipschitz regular paths.
Such a construction of multifractal functions associated with specific
random wavelet series has been extended recently
by Barral and Seuret \cite{BarSeu05}. If $M$ stands for a Mandelbrot
multifractal measure (a $\cal M$-cascade)
constructed from an iterative random multiplicative rule as described in the introduction, these authors considered a wavelet random series like \eqref{eq:wcascade1} but where $c_{j,k}$ is given by the measure of the associated dyadic interval $I_{j,k}=[k2^{-j},(k+1)2^{-j}]$:
$$
c_{j,k} = 2^{-j \alpha} M(I_{j,k}) \; .
$$
Barral and Seuret proved that the scaling and regularity
properties of $Z(t)$ are directly inherited from those
of $M(t)$. Moreover, they have shown that, under specific conditions, replacing $\prod_{i=1}^j W_i$ of $\cal W$-cascades by the limit measure of the associated dyadic interval, $M(I_{j,k})$ does not change the multifractal
properties (see the Sec. \ref{sec:wc-cascades_ss} for more details).
Unlike the constructions of multifractal processes
based on multifractal measures, $\cal W$-cascades allow one to directly build multifractal processes without the need for any additional self-similar process.
However, very much like $\cal M$-cascades, they do involve a dyadic tree and a finite time interval
that can hardly be used to fit most of experimental situations.
For that reason, in the same manner as $\cal M$-cascades have been extended to log-infinitely divisible
continuous cascades, we aim at defining a continuous version of $\cal W$-cascades.
\section{Continuous $\cal W$-cascades}
\label{sec:wc-cascades}
In this section, we introduce the new class of models we consider in the paper.
Our goal is to extend the previously described $\cal W$-cascades
to a grid-free background. The main idea is to replace the discrete sum \eqref{eq:wcascade1}
by a continuous sum over space and scales and the discrete product in Eq. \eqref{eq:wcascade3} at
scale $\ell = 2^{-j}$ by its ``continuous" (i.e. stationary and "grid free") counterpart $e^{\omega_\ell(t)}$
described in Sec. \ref{sec:c-casc} and Appendix \ref{app:cont_casc}.
\subsection{Definition and numerical illustrations}
\label{sec:wc-cascade_def}
\subsubsection{Definition and convergence}
Let us define a stochastic process $X(t)$ as a (continuous)
sum of localized waveforms (i.e. "wavelets") of size $\ell$ and which
intensity is given by the class of stationary, log-infinitely divisible
process $e^{\omega_{\ell}(t)}$ used in the definition of continuous cascades of Section \ref{sec:c-casc}.
Let $H>0$, $\ell >0$ and consider the following (well defined) integral:
\begin{equation}
\label{eq:def_X}
X_\ell(t) = \int_{\ell}^T \! s^{H-2} ds \! \int_{-\infty}^{+\infty} e^{\omega_{s}(b)} \varphi \left(\frac{t-b}{s} \right) db
\end{equation}
where $\omega_{s}(u)$ is the infinitely divisible process defined
in Section \ref{sec:c-casc} and $\varphi(t)$ is a wavelet
that can be chosen as a square integrable smooth function with compact support (e.g, the interval $[-{1 \over 2},{1 \over 2}]$)
and which is sufficiently oscillating so that its
first $N$ moments vanish. Hereafter, we will referred to this wavelet as the ``synthetizing wavelet".
As emphasized below, this amounts, in some sense, to interpreting $s^H e^{\omega_{\ell}(t)}$
as the continuous wavelet transform of $X(t)$ at time $t$
and scale $\ell$, Eq. \eqref{eq:def_X} corresponding to the continous
wavelet reconstruction formula.
Let us remark that, if $\phi(1) < \infty$, one can, without loss of generality (since it simply consists in redefining the parameter $H$), always assume
that in Eq. \eqref{eq:def_X}, $\omega_s(t)$ is chosen such that
\begin{equation}
\label{eq:phi1}
\EE{e^{\omega_s(t)}} = 1 \Leftrightarrow \phi(1) = 0 \; .
\end{equation}
In Appendix \ref{App:conv_C0}, we establish the weak convergence
of $X_{\ell}(t)$ in the space of continuous functions when $\ell \to 0$. Namely, we show that the weak limit
\begin{equation}
\label{eq:lim_X}
\lim_{\ell \to 0} X_\ell(t) = X(t)
\end{equation}
exists as a continuous function provided:
\begin{equation}
\label{eq:conv-condition}
H > \frac{\phi(2)}{2} \; .
\end{equation}
We will call such a limit $X(t)$ a ``continuous wavelet cascade'' process or a ${\cal CW}$-cascade.
Let us notice that the condition \eqref{eq:conv-condition} is precisely
the analog of the condition for $L^2$ convergence of $\cal W$-cascades established by Arneodo et al. (condition $H1$ of Proposition 1 in \cite{ArnBacMuz98}).
\begin{center}
\begin{figure}[h]
\includegraphics[width=0.9\textwidth]{Fig_wavelets.pdf}
\vspace{-3cm}
\caption{Examples of synthetizing wavelets $\varphi(t)$. (a) The compactly supported
odd function $h^{(2)}(t)$ (Eq. \eqref{eq:smH}). (b) The "Mexican hat", i.e. the even wavelet $g^{(2)}(t)$ (Eq. \eqref{eq:gauss_deriv}). (c) A non symmetric wavelet obtained by smoothing an asymmetric version
of the Haar wavelet.}
\label{fig_wavelets}
\end{figure}
\end{center}
\begin{center}
\begin{figure}[h]
\includegraphics[width=0.7\textwidth]{Fig_path1.pdf}
\caption{Sample paths of continuous log-normal $\cal W$-cascades $X(t)$ corresponding to
3 different values of the parameter
$H$. In all cases, the intermittency coefficient and the integral scale were chosen
to be respectively $\lambda^2 = 0.025$ and $T = 1024$ and the synthetizing wavelet the $C^2$ version
of the Haar wavelet, $h^{(3)}(t)$.}
\label{fig_paths}
\end{figure}
\end{center}
\subsubsection{Numerical examples}
In numerical experiments, one has to choose the infinitely divisible law of the process $\omega_{\ell}(t)$,
the regularity parameter $H$, the integral scale $T$ and the synthetizing wavelet $\varphi(t)$.
The simulation procedure simply consists in a discretization of
Eq. \eqref{eq:def_X}. Its main lines are described in Appendix \ref{app:simus}.
All the examples provided in the paper are involving log-normal cascades which are the simplest
ones to handle and that involve a single variance parameter $\lambda^2$. This means that $\omega_{\ell}(t)$
is a Gaussian process with a covariance function given by expression \eqref{rhoexact}.
In that respect, thanks to condition \eqref{eq:phi1}, the function $\phi(q)$ simply reads:
\begin{equation}
\label{eq:phiq_ln}
\phi(q) = \frac{\lambda^2}{2}q(q-1) \; .
\end{equation}
and the condition \eqref{eq:conv-condition} simply becomes
$\lambda^2 < 2 H$.
Let us remark that this condition appears to be
less restrictive than the condition of existence of a MRW $\lambda^2 < 1/2$ \cite{BaDeMu01,DelourPHD} and it is a priori possible to build ${\cal CW}$-cascades with a large intermittency coefficient.
Among the choices we made for the synthetizing wavelets $\varphi$
there is the class of smooth variants of the Haar wavelet:
\begin{equation}
\label{eq:smH}
h^{(n)}(t) = \left(\mathbb{I} \underset{n}{\star \ldots \star} \mathbb{I} \star h \right)(t)
\end{equation}
where $\mathbb{I}(t)$ is the indicator function of the interval $[0,1/2)$, $h(t) = \mathbb{I}(t+1/2)- \mathbb{I}(t)$ is the Haar wavelet and $\star$ stands for the convolution product. Notice that $h^{(n)}(t)$
has one vanishing moment and is of class $C^{n-1}(\mathbb{R})$.
We also use wavelets in the class of the derivatives of the Gaussian function:
\begin{equation}
\label{eq:gauss_deriv}
g^{(n)}(t) = \frac{d^n}{dt^n} e^{-\frac{x^2}{2}} \; .
\end{equation}
Some examples among these wavelet classes are plotted in Fig. \ref{fig_wavelets} (dilation and normalization factors are arbitrary).
Note that one can also consider asymmetric wavelets
as the one illustrated in the right panel of Fig. \ref{fig_wavelets} that is contructed by asymmetrizing and smoothing the Haar wavelet
(see Sec. \ref{sec:cw-cascades_skew} for an usage of asymmetric wavelets).
Some examples of sample paths of $X(t)$ are plotted in Figs. \ref{fig_paths}.
All processes were generated using a Gaussian process $\omega_{\ell}(t)$
with the intermittency coefficient $\lambda^2 = 0.025$ and the integral scale $T = 2048$.
The synthetizing function was chosen to be $\varphi(t) = h^{(3)}(t)$.
The scaling parameter $H$ has been chosen to be respectively $H=0.36$, $H=0.51$ and $H = 0.76$ from top to bottom.
One can see that $H$ directly controls
the global regularity of the paths, the larger $H$, the more regular the path of $X(t)$ is. This is the analog of the parameter $H$ of the fractional Brownian motion \cite{Taqqu}. It is noteworthy that despite $X(t)$ appears to be a process with
zero mean stationary increments, unlike all constructions proposed so far for mulifractal stationary processes, it does not involve any supplementary "white" or "colored" noise,
like the Gaussian white noise, since it is only based on the "intensity" process $e^{\omega_{\ell}(t)}$ as a source of randomness (see the remark at the end of the next section).
\subsection{Wavelet transform, global regularity and reconstruction formula}
\label{sec:wc-cascades_wavelets}
In order to study and characterize the limiting process $X(t)$,
one can compute its wavelet transform \cite{Mallatbook}.
If $\psi(t)$ stands for some analyzing wavelet,
let us introduce the kernel $K_{\varphi,\psi}(x,s)$ defined as:
\begin{equation}
K_{\varphi,\psi}(x,s) = s^{-1} \int \varphi \left(t \right) \psi \left(\frac{t-x}{s} \right) dt \; .
\end{equation}
The wavelet transform of $X(t)$ can then be simply expressed as:
\begin{equation}
\label{eq:wtX}
W(x,a) = a^{-1} \int X(t) \psi \left( \frac{t-x}{a} \right) dt = \int_0^T s^{H-2} ds \! \! \int_{-\infty}^{+\infty} db \; e^{\omega_{s}(b)} K_{\varphi,\psi}\left(\frac{x-b}{s},\frac{a}{s} \right)
\end{equation}
It is easy to show that if $\psi$ has more than $N$ vanishing moments one
has $K_{\varphi,\psi}(x,s) \sim s^{N}$ when $s \ll 1$ while if $\varphi$
has also at least $N$ vanishing moments, then $K_{\varphi,\psi}(x,s) \sim s^{-(N+1)}$ when $s \gg 1$.
In that respect, for each $x$, $K_{\varphi,\psi}(x,s)$ is maximum around $s = 1$.
Moreover, if $\psi$ is also supported by $[-1/2,1/2]$, then $K(x,s) = 0$ if $|x|\geq s$.
In the sequel, for the sake of simplicity and to avoid cumbersome considerations about
the tails of $K(x,s)$ for large and small $s$, we will suppose that the kernel $K(x,s)$ is
non vanishing only in the time-scale interval $s \in [\kappa s,1]$, $x \in [-s,s]$ for some $\kappa < 1$. Under this assumption, the wavelet transform of $X(t)$ can be approximated as:
\begin{equation}
\label{eq:wt-approx}
W(x,a) \simeq \left\{
\begin{array}{ll}
& \!\!\! \int\limits_{
\kappa a}^{\min(T,a)} s^{H-2} ds \int\limits_{x-a}^{x+a} e^{\omega_{s}(b)} K_{\varphi,\psi}\left(\frac{x-b}{s},\frac{a}{s} \right) db, \; \; \mbox{if} \; \kappa a \leq T \\
& 0 \; \; \mbox{otherwise}
\end{array}
\right.
\end{equation}
Let us remark that in the case one chooses the analyzing wavelet $\psi(t) = \delta^{(1)}(t)$ with
\begin{equation}
\label{eq:def_delta1}
\delta^{(1)}(t) = \delta(t+1)-\delta(t),
\end{equation}
where $\delta(t) dt$
stands for the Dirac distribution, then $W(t,a)$ is nothing
but the increment of $X(t)$ at scale $a$ \cite{MuBaAr93PhysRevE}:
\begin{equation}
\label{eq:poor_man_wavelet}
W(t,a) = \delta_a X(t) = X(t+a)-X(t)
\end{equation}
Despite this "poor man's wavelet" does not possess the requested regularity and oscillating
properties (in particular approximation \eqref{eq:wt-approx} is not supposed to hold), in the sequel, we will often consider that the increments statistics can be deduced from the
wavelet transform statistics as a particular case (see e.g. \cite{MuBaAr93PhysRevE} for
a discussion on this specific topic).
Along the same line, if $\psi(t)= \delta^{(2)}(t)$,
with
\begin{equation}
\label{eq:def_delta2}
\delta^{(2)}(t) = \delta(t+2)+\delta(t)-2 \delta(t+1)
\end{equation}
then $W(t,a)$ corresponds to the increments of second order of $X(t)$: $W(t,a) = X(t+2a)+X(t)-2X(t+a)$.
Hereafter, we will refer to increments of first or second order when it is necessary to distinguish
these two specific types of wavelet transforms.
Let us define the spectrum of structure function scaling exponents
\begin{equation}
\label{eq:def_zeta}
\zeta_q = qH - \phi(q)
\end{equation}
and consider its Legendre transform:
\begin{equation}
\label{eq:singspectr-wc}
F(h) = 1+\inf_q(qh-\zeta_q) \; .
\end{equation}
Let us suppose that $\exists \eta > 0$ such that $F(h) < 0$ for all $0 \leq z < \eta$.
In Appendix \ref{App:conv_C0}, we use expression \eqref{eq:wt-approx} to show that, for any $L>0$, almost surely,
the paths of $X(t)$ have a uniform Lipschitz regularity $\alpha$ on $[0,L]$
for all $0 < \alpha < h_{\min}$ with
$$
h_{\min} = \arg \max \{F(h) < 0\}_{h < H-\phi'(0)} \; .
$$
This result about the almost sure regularity of the paths $X(t)$ extends to continuous $\cal W$-cascades
a similar property proven by Arneodo et al. in the case of discrete $\cal W$-cascades \cite{ArnBacMuz98}.
For example, in the log-normal case, $\phi(q)$ is provided in Eq. \eqref{eq:phiq_ln} and therefore $h_0 = H-\phi'(0) = H+\frac{\lambda^2}{2}$. Since $F(h)= 1-\frac{(h-h_0)^2}{2 \lambda^2}$, we obtain, provided $H > \sqrt{2 \lambda^2} -\frac{\lambda^2}{2}$, i.e., $h_{\min} = h_0-\sqrt{2 \lambda^2} > 0$.
Let us notice that, unlike discrete $\cal W$-cascades, the weighting
process $s^H e^{\omega_\ell(t)}$ involved in the continuous version
is always positive. Nevertheless, let us remark that wavelet analysis is also helpful to show that, in the construction \eqref{eq:def_X}, instead of choosing the (positive) weights $s^{H} e^{\omega_s(b)}$, one could equivalently build $X(t)$ from a zero mean weighting process that has the same law as $W(b,s)$, the wavelet transform of $X(t)$. Indeed, let us consider the
following process:
\begin{equation}
\label{wt-recons}
X'(t) = \lim_{\ell \to 0} \lim_{R \to \infty} \int_{\ell}^{R} ds \; s^{-2} \! \int_{-\infty}^{+\infty} W(b,s) \varphi \left(\frac{t-b}{s} \right) db
\end{equation}
where $W(b,s)$ is the wavelet transform of $X(t)$ as defined in \eqref{eq:wtX}.
According to the wavelet point-wise reconstruction formula (e.g. as introduced in \cite{HT91}),
if the analyzing wavelet
$\psi$ satisfies some mild conditions as respect to the synthetizing wavelet $\varphi$, one expects that $X'(t)=X(t)$ at every point of continuity of $X(t)$. It thus results that
\eqref{wt-recons} provides a whole family of versions of
continuous $\cal W$-cascades contruction were the weights are chosen to
be a zero mean process as simply obtained from any wavelet transform of the original $X(t)$ function.
\section{Self-similarity and scaling properties}
\label{sec:scaling}
In this section we study the scaling and self-similarity properties of $X(t)$
as defined in Eqs. \eqref{eq:def_X} and \eqref{eq:lim_X}.
\subsection{Stochastic self-similarity of the wavelet transform. Multifractal Scaling}
\label{sec:wc-cascades_ss}
Let us first point out that, from the construction
of $\omega_{\ell}$ as recalled in Appendix \ref{app:cont_casc},
Eq. \eqref{charf} can be naturally generalized as:
\begin{equation}
\label{charf_2}
\EE{e^{\sum_{m=1}^q i p_m \omega_{\ell_m}(x_m)}} = e^{\sum_{j=1}^q \sum_{k=1}^{j}
\alpha(j,k) \rho_{\max(\ell_k,\ell_j)}(x_k-x_j)}
\end{equation}
for any sequence $0, \ell_1,\ldots,\ell_q < T$. From the expression
\eqref{rhoexact} of $\rho_\ell(t)$,
it results that Eq. \eqref{eq:ss-omega} can be extended as equality in
law for processes of both space and scale variables: one has, $\forall r < 1$, $0 < \ell \leq T$ and $u \in [0,T]$,
\begin{equation}
\label{eq:ss-omega2}
\omega_{r \ell}(s u) \stackrel{\cal L'}{=} \Omega_r + \omega_\ell(u)
\end{equation}
where where $\stackrel{\cal L'}{=}$ means that the two processes have the same
finite dimensional distributions of any order as processes in the half-plane
$(\ell,u)$ and $\Omega_r$ is a random variable of same law as $\omega_{rT}(u)$ independent of the process $\omega_\ell(t)$.
Let $r < 1$ and $a \ll T$. From Eq. \eqref{eq:wt-approx}, the rescaled version
of the wavelet transform of $X(t)$ reads:
\begin{eqnarray*}
W(rx,ra) & \simeq & \int\limits_{
\kappa r a}^{ra} s^{H-2} ds \int\limits_{rx-ra}^{rx+ra} e^{\omega_{s}(b)} K_{\varphi,\psi}\left(\frac{rx-b}{s},\frac{ra}{s} \right) db \\
& = & r^{H} \int\limits_{
\kappa a}^{a} s^{H-2} ds \int\limits_{x-a}^{x+a} e^{\omega_{rs}(rb)} K_{\varphi,\psi}\left(\frac{x-b}{s},\frac{a}{s} \right) db \; .
\end{eqnarray*}
Thanks to Eq. \eqref{eq:ss-omega2}, we can thus establish the self-similarity of the
wavelet transform of $X(t)$:
\begin{equation}
\label{eq:wt-ss}
W(rx,ra) \stackrel{{\cal L'}}{=} r^{H} e^{\Omega_r} W(x,a) \; .
\end{equation}
\begin{center}
\begin{figure}[h]
\vspace{-0.5cm}
\includegraphics[width=0.6\textwidth]{Fig_tauq.pdf}
\caption{Power-law scaling of structure functions of $X(t)$. In the top panel
the structure function $S(q,a)$ of $X(t,a)$ (with $H=0.36$) are plotted in double logarithmic scale for $q=1,2,3,4$. The scaling behavior holds up
to $a \simeq 2. 10^3$ which corresponds to the integral scale $T$. In the bottom panel, the estimated $\zeta_q$ ($\circ$) are compared to the expected log-normal expressions as given by Eq. \eqref{eq:zetaln} (dashed lines) for $H=0.36$, $0.52$ and $0.77$ and $\lambda^2 = 0.025$.}
\label{fig_sf}
\end{figure}
\end{center}
From Eq. \eqref{eq:wt-ss}, the definition of $\Omega_r$ , Eqs. \eqref{eq:def_phi} and \eqref{eq:def_zeta}
it results that, for all $a \leq T$, the wavelet structure functions are characterized by the
multifractal scaling:
\begin{equation}
\label{eq:wt-sf}
S(q,a) = \EE{|W(x,a)|^q} = C_q \left(\frac{a}{T}\right)^{qH-\phi(q)} = C_q a^{\zeta_q} \; ,
\end{equation}
where $C_q = \EE{|W(x,T)|^q}$.
$X(t)$ has therefore multifractal properties in the sense
that the absolute moments of its wavelet transform behave has a power-law
with a non-linear concave multifractal spectrum $\zeta_q$ \cite{MuzBacArn91,MuBaAr93PhysRevE}.
In particular, by considering Eq. \eqref{eq:poor_man_wavelet}, we deduce that
the moments of absolute increments behave has power-laws with the multifractal
spectrum $\zeta_q$:
\begin{equation}
\label{eq:wt-sf}
\EE{|X(t+a)-X(t)|^q} = C_q a^{\zeta_q} \; .
\end{equation}
This result is illustrated in Fig. \ref{fig_sf} where the estimated structure functions
\begin{equation}
\label{eq:def_sf}
S(q,a) = \sum_k |X(k+a)-X(k)|^q
\end{equation}
are computed on realizations
of log-normal versions of $X(t)$ with $H=0.36$, $0.51$ and $0.76$ respectively and
$\lambda^2 = 0.025$. In each case, the integral scale has been set to $T= 2048$ and the overall sample length corresponds to 16 integral scales. In the top panel of Fig. \ref{fig_sf} we have plotted $\log_{10}S(q,a)$ as a function of $\log_{10} a$ in the case $H=0.36$. We see that a power-law behavior extends from the smallest scale up to the integral scale. The estimated $\zeta_q$ spectrum, as obtained from a linear fit of these
log-log plots, are reported in the bottom panel (symbols $\circ$). In the three cases, we obtain, within a good precision, the expected scaling exponents:
\begin{equation}
\label{eq:zetaln}
\zeta_q = \left(H + \frac{\lambda^2}{2}\right) q - \frac{\lambda^2}{2}q^2
\end{equation}
represented by the dashed lines.
Let us notice that in the case $H = 0.5125$, one has $\zeta_2 = 2$
and therefore the successive increments of $X(t)$ are uncorrelated as in the
Brownian motion (see next Section). When $H=0.36$ (precisely $H = 0.3585$), one has $\zeta_3 = 1$ as expected for the increments of the longitudinal velocity in fully deve\-loped turbulence \cite{Fri95}.
\begin{center}
\begin{figure}[t]
\includegraphics[width=1\textwidth]{Fig_Corr1.pdf}
\caption{Estimated correlation function of the increments of $X(t)$ for
$H = 0.36$ (top panels) and $H=0.76$ (bottom panels). In both cases $\omega_\ell(t)$ is a Gaussian process of intermittency coefficient $\lambda^2 = 0.025$. The synthetizing wavelet $\varphi(t)$ is a "mexican hat" wavelet, $g^{(2)}(t)$ defined in Eq. \eqref{eq:gauss_deriv}. As expected when $H$ is small enough increments are anti-correlated
while for $H$ large enough. The expected power-law behavior \eqref{eq:inc_cf} are represented by the solid lines in left panels. }
\label{fig_corr}
\end{figure}
\end{center}
It is well known that the spectrum of scaling exponents
$\zeta_q$ can be, for a large class of functions, related
to the singularity spectrum, i.e., the fractal (Haussdorf) dimension of the sets of iso-H\"older regularity \cite{ParFri85,Jaf97a,Jaf97b,BM04}. This
is the so-called multifractal formalism.
At this stage, it is tempting to conjecture that the result proven by Barral and Seuret for random wavelet series \cite{BarSeu05} is also valid in the
framework introduced here and that the multifractal formalism holds for continuous wavelet cascades. In our context, the Barral-Seuret result would say that $D(h)$ the singularity spectrum of $X(t)$,
can be simply obtained from $D_M(\alpha)$, the singularity spectrum of the underlying log-infinitely divisible cascade $dM_t = \lim_{\ell \to 0} e^{\omega_{\ell}(t)} dt$.
More precisely, if $D_M(\alpha)$ is the singularity
spectrum of the continuous cascade $M(t)$ as provided
by the multifractal formalism \cite{BM04}, i.e., in the range where $1+\min_q (q\alpha -\phi(q)) >0 $, one has
$$
D_M(\alpha) = 1+\inf_q(q(\alpha-1)+\phi(q))
$$
and thus, from definition \eqref{eq:singspectr-wc},
\begin{equation}
\label{eq:d=F}
D_M(\alpha) = F(\alpha+H-1) \; .
\end{equation}
Then the analog of Barral-Seuret result (Theorem 1.1 of \cite{BarSeu05}) result would assert that
the singularity spectrum $D(h)$ of $\cal W$-cascades is
provided by $D(h) = D_M(h+1-H)$ which according to
\eqref{eq:d=F}, would simply mean that the singularity spectrum of $X(t)$ is $D(h) = F(h)$
in the range where $F(h)$, as given by \eqref{eq:singspectr-wc} is positive. Since $F(h) = 1+\inf_q(qh-\zeta_q)$ is the Legendre transform of the spectrum of scaling exponents of wavelet transform structure functions, this would imply that
the multifractal formalism holds for continuous $\cal W$-cascades.
\subsection{Increment correlation functions and magnitude covariance}
\label{sec:wc-cascades_corr}
In this section, we study the behavior of various correlation functions associated with the increments (or wavelet coefficients) or the powers of their absolute values.
Let us first define the increment correlation function
\begin{equation}
\label{eq:def_incf}
\rho (a,\tau) = \Cov{\delta_a X(t)}{\delta_a X(t+\tau)} = \EE{\delta_a X(t) \delta_a X(t+\tau)}
\end{equation}
It is easy to show that, when $a \ll \tau$, one has \cite{RamBaMu18}
$$
\rho(\tau) \simeq \frac{\partial }{\partial \tau} \tau^{-1} \EE{\delta_\tau X(t)^2}
$$
and therefore, from Eq. \eqref{eq:wt-sf}, one has:
\begin{equation}
\label{eq:inc_cf}
\rho(\tau) \simeq A \tau^{\zeta_2-2} \; .
\end{equation}
Let us remark that if $\zeta_2 < 1$, then the prefactor $A$ is negative and
the increments are anti-correlated while if $\zeta_2 > 1$, the correlations
are positive. This is reminiscent of the correlations of a fractional Brownian motion \cite{ManVan68,Taqqu} increments where $\zeta_2 = 2H$ and for which the value $H = 1/2$ separates the regimes of negatively and positively correlated fluctuations.
The behavior \eqref{eq:inc_cf} is illustrated empirically using two samples of $X(t)$
with respectively $H=0.36$ and $H = 0.76$ while $\lambda^2 =0.025$ and $T=2048$ in both
cases. The empirical increments correlation function have been estimated at scale $\tau = 1$ from samples of length 16 integral scales. As expected, one can observe, in the right top and right bottom panels, a power-law behavior in both cases
and that $\zeta_2-2$ provides a good fit (solid lines). In the left panels, we see that the process with
$H = 0.36$, corresponding to $\zeta_2 = 0.695$ is characterized by anti-correlated increments while the one with $H=0.76$, corresponding to $\zeta_2 = 1.495$, has positively
correlated increments.
The behavior correlation function of the absolute power of the increments can
be inferred from the self-similarity properties of the wavelet transform $W(x,a)$.
The same kind of scaling argument as previously used to establish the scaling
of structure functions can be used. Indeed, from Eq. \eqref{eq:wt-ss}, one has, for $a \leq \tau \leq T-\kappa a$:
\begin{equation}
\EE{|W(x_0,ra_0)|^q|W(x_0+r\tau_0,ra_0)|^p} = r^{\zeta_{q+p}} \EE{|W(x_0,a_0)|^q|W(x_0+\tau_0,a_0)|^p}
\end{equation}
Let us now choose $\varepsilon \ll 1$ and set $T' = \frac{T}{1+\varepsilon}$.
Let us consider $r a_0 = a$, $r \tau_0 = \tau$ and
$a = \varepsilon \tau$.
The previous equation can be rewritten as:
\begin{equation}
\EE{|W(x_0,a)|^q|W(x_0+\tau,a)|^p} = \left(\frac{\tau}{T'} \right)^{\zeta_{q+p}} \EE{|W(x_0,\varepsilon T')|^q|W(x_0+T',\varepsilon T')|^p}
\end{equation}
If one considers $\varepsilon \to 0$ (i.e. one chooses $a$ very small), then $T' \simeq T$ and $W(x_0,\varepsilon T')$ becomes
independent from $W(x_0+T',\varepsilon T')$ and one has:
\begin{equation}
\EE{|W(x_0,\varepsilon T')|^q|W(x_0+T',\varepsilon T')|^p} \simeq
\EE{|W(x_0,\varepsilon T')|^q}\EE{|W(x_0+T',\varepsilon T')|^p} \sim K_{p,q}(a) \left(\frac{\tau}{T}\right)^{-\zeta_q-\zeta_q}
\end{equation}
where $K_{p,q}(a)$ is a constant that depends on $a$.
Given the scaling \eqref{eq:wt-sf}, this entails, for a fixed small value of $a$:
\begin{equation}
\label{eq:corr-scaling}
\EE{|W(x_0,a)|^q|W(x_0+\tau,a)|^p} \sim K_{p,q}(a) \left(\frac{\tau}{T}\right)^{\zeta_{p+q}-\zeta_q-\zeta_p}
\end{equation}
showing that correlation functions of the powers of the wavelet transform absolute value, behave, at a given scale, as a power-law as a function of the time
lag $\tau$ with a scaling exponent $\zeta_{p+q}-\zeta_p-\zeta_q$.
It is important to notice that this exponent does not depend on $H$ and is only provided by the $\phi(q)$ the non-linear part of $\zeta_q$.
\begin{center}
\begin{figure}[h]
\includegraphics[width=0.8\textwidth]{Fig_CorrAbs1.pdf}
\caption{Estimated correlation function of the absolute
increments of $X_{LN}(t)$ for $H = 0.37$ ($\circ$) and $H=0.77$ (solid lines). As expected they both behave as $\tau^{-\lambda^2}$ independently of $H$ (represented by the dashed line in the bottom panel).}
\label{fig_corrabs}
\end{figure}
\end{center}
This behavior is illustrated in Fig. \ref{fig_corrabs} where we have plotted the estimation of absolute increments (i.e. in the case $p=q=1$) for the two log-normal processes with $H=0.36$ and $H=0.77$ considered previously. As expected both behave, independently of $H$, as $\tau^{\zeta_{2}-2\zeta_1}$ that corresponds, according to Eq.
\eqref{eq:zetaln} to the power-law $\tau^{-\lambda^2}$ (represented by the dashed line in the bottom panel).
Eq. \eqref{eq:corr-scaling} can be used to compute the behavior
of magnitude covariance as defined in \cite{ArnBaMaMu98,MuDeBa00}. Indeed, since:
\begin{equation}
\Cov{\ln|W(x_0,a)|}{\ln|W(x_0+\tau,a)|} = \frac{\partial_q \partial_p \ln \EE{|W(x_0,a)|^q|W(x_0+\tau,a)|^p}}{\partial p \partial q} \bigg\rvert_{q,p=0} \;,
\end{equation}
we obtain the well known logarithmic magnitude covariance for multifractal
processes \cite{ArnBaMaMu98,MuDeBa00}:
\begin{equation}
\Cov{\ln|W(x_0,a)|}{\ln|W(x_0+\tau,a)|} \simeq \zeta''(0) \ln\left( \frac{\tau}{T} \right) = -\lambda^2 \ln\left( \frac{\tau}{T} \right)
\end{equation}
where we defined the intermittency coefficient as $\lambda^2 = -\zeta''(0)$.
\section{Skewness and leverage effect: applications to turbulence and stock market data}
\label{sec:cw-cascades_skew}
\subsection{Skewness of increment pdf at all scales}
Because of the self-similarity relationship \eqref{eq:wt-ss}, one can
also expects a scaling of odd moments of the wavelet transform, i.e.,
$\forall k \in \mathbb{N}$, $\forall a \leq T$,
\begin{equation}
\label{eq:sf_odd}
S^\star(2k+1,a) = \EE{W(x,a)^{2k+1}} = V_{2k+1} \; a^{\zeta_{2k+1}}
\end{equation}
This shows in particular that generically, the skewness
$$
\frac{ \EE{W(x,a)^{3}}}{ \EE{W(x,a)^{2}}^{3/2}} \sim a^{\zeta_3-\frac{3}{2} \zeta_2 }
$$
increases (in absolute value) when $a \to 0$.
The computation of the constant $V_{2k+1}$ for a given waveform $\varphi$ is tedious
and can be only written as an intricate multiple integral but it can
be shown it is non-zero unless $\varphi$ satisfies very specific conditions (see Fig. \ref{fig_sk}).
This shows that our approach, unlike constructions bases
on classical random measures, generically
leads to skewed multifractal processes with a skewness that, like
the flatness, increases as one goes from coarse to fine scales.
\begin{center}
\begin{figure}[h]
\includegraphics[width=0.8\textwidth]{figPdf.pdf}
\caption{Estimated ``standardized" probability density function (pdf) of the ``wavelet coefficients" at scale $a$, $p_a(w)$. At each scale, $a=T$,$T \over 8$,$T \over 64$,$T \over 512$, $T \over 4096$,
where $T$ stands for the integral scale, we have displayed $\ln p_a(w)$ up to an additive constant for the sake of clarity: pdf at small scales are displayed above pdf at coarser scales. On the top left and right panels the considered $\cal W$-cascade $X(t)$ is a log-normal process with a synthetizing wavelet $\varphi$ that is a odd function ($g^{(1)}(t)$, the first derivative of the Gaussian function). We can check that, due to the intermittency, the flatness increases from large to small scales. On the left panel the analyzing wavelet is odd since it is $\psi(t)= \delta^{(1)}(t)$ (the one that corresponds to the increments of $X(t)$) while on the right panel one uses $\psi(t)=\delta^{(2)}(t)$ and thus considers the second order increments. As expected, one clearly observes, like in turbulence, skewed distributions of increments at all scales. The second order increments are distributed with a symmetric law. On the bottom panels, the opposite effect is observed since the process $X(t)$ as been constructed using a symmetric wavelet, namely $g^{(2)}(t)$, the second order derivative of the Gaussian function.}
\label{fig_pdf}
\end{figure}
\end{center}
\begin{center}
\begin{figure}[h]
\vspace{-1.5cm}
\includegraphics[width=0.8\textwidth]{Fig_s3lin.pdf}
\vspace{-1cm}
\caption{Signed third order structure function $S^\ast(3,a)$ as a function of the scale $a$. On the left
panel the mean order 3 moment of the increments is computed on $X(t)$ corresponding to the anti-symmetric
synthetizing wavelet $g^{(1)}(t)$ (Eq. \eqref{eq:gauss_deriv}). This confirms the (negative) skewness already illustrated in Fig. \ref{fig_pdf} and that, as expected for turbulence, $S^\ast(3,a)$ behaves as a linear function. In the right panel, $X(t)$ is built using the non-symmetric wavelet depicted in Fig. \ref{fig_wavelets}.
The third order structure function function behaves as a linear function and skewness can be observed for both first ($\circ$) and second order increments (solid line).}
\label{fig_sk}
\end{figure}
\end{center}
\vspace{-1.5cm}
The scaling relationship \eqref{eq:sf_odd} indicates that
the skewness will be zero
at all scales if it vanishes at the largest scale $T$. It is noteworthy that this may be the case if the synthetizing wavelet $\varphi$ in Eq. \eqref{eq:def_X} is an even or odd function. Indeed, if the wavelet $\varphi(t)$ is a symmetric function, because $\omega_{\ell}(t) \stackrel{{\cal L}}{=} \omega_{\ell}(-t)$, we see from definition \eqref{eq:def_X} that $X_\ell(t)$ is invariant by
time reversal, i.e., one has $X_\ell(-t) \stackrel{{\cal L}}{=} X_\ell(t)$. It thus results that if the analyzing wavelet an odd function
(as e.g. for the increments $\delta_{\tau} X(t)$) the wavelet transform of $X(t)$ will have a symmetric
law implying that all odd moments are zero. In order to observe some skewness in the increment law, it is thus necessary to consider non-symmetric synthetizing wavelets $\varphi$. Along the same line, if the synthetizing wavelet $\varphi(t)$ is anti-symmetric,
then $X(t)$ is odd by time reversal, i.e.,
$ X_\ell(-t) \stackrel{{\cal L}}{=} -X_\ell(t)$
and if the analyzing wavelet is even (as e.g. when one computes
the second order increments of $X(t)$, $\delta^{(2)}_\tau X(t) = X(t+2 \tau)+X(t)-2 X(\tau)$), the wavelet transform will have
a symmetric law. This means that if a skewness is observed in both first and second order increments, the wavelet $\varphi(t)$ is neither an even nor an odd function.
\begin{center}
\begin{figure}[h]
\includegraphics[width=1\textwidth]{FigPath_exp2.pdf}
\vspace*{-1cm}
\caption{Paths of turbulence velocity field and stock market data as compared to their $\cal CW$-cascade models. In the top left panel are represented the longitudinal velocity field space variations (in $m/s$) obtained in a high Reynolds number turbulence experiment ($R_\lambda \simeq 700$) realized by Castaing et al. \cite{CasHe94}. The top right panel corresponds to the cascade model obtained by fitting the turbulence multiscaling but also the skewness of law of first order increments. In the left bottom panel is plotted the detrended daily variations of the logarithm of the Dow-Jones index over the more than 4 decades (from 1939 to 1985). The right bottom panel represents its $\cal CW$-cascade model where we mainly account for the skewness of second order increments. We can see that the "ramp" like behavior of turbulence data and "arch" like patterns of stock market data are remarkably well reproduced by ${\cal CW}$-cascades.}
\label{fig_paths2}
\end{figure}
\end{center}
These behaviors are illustrated in Fig. \ref{fig_pdf} where whe have displayed the probability density functions (pdf)
of the first and second increments at different scales
of two versions of log-normal continuous wavelet cascades $X(t)$ that are respectively antisymmetric and symmetric by time reversal. In the first case, we used $\varphi = g^{(1)}$ while in the second case $\varphi = g^{(2)}$ (see Eq. \eqref{eq:gauss_deriv}). In both cases, we chose $\omega_{\ell}$ as a log-normal cascade with $\lambda^2 = 0.025$, an integral scale $T = 2^{12}$ and $H=0.355$. In that respect, according to \eqref{eq:zetaln}, $\zeta_3 = 1$ and $X(t)$ was designed to mimic the main features of velocity records in experiments of fully developed turbulence. All the estimations have been performed on a sample of length $128$ integral scales. All the probability density functions reported in Fig. \ref{fig_pdf} are standardized, i.e., represent the distribution of increments normalized by their root-mean-square. The are displayed in semi-log scales and shifted so that large scale distribution are below fine scale ones. In that way, one can clearly observe the
intermittency as an increasing of the flatness from large to small scales. We can check in the top-right (resp. bottom-left) panel of Fig. \ref{fig_pdf} that if $\varphi$ is odd (resp. even), the second (resp. first) order increments are symmetrically distributed. We can see the in top-left panel that, at all scales the increments of the anti-symmetric version of $X(t)$ are negatively skewed. The shapes of these skewed pdf, with an increasing flatness are strikingly similar to distribution of longitudinal velocity increments in fully developed turbulence \cite{CaGaHo90,CasHe94}.
In the symmetric version of $X(t)$, the skewness is only observed on its second order increments (bottom-right panel).
The fact that an anti-symmetric synthezing wavelet allows one to reproduce the observed skewness behavior of the velocity field
in turbulence is directly illustrated in left panel of Fig. \ref{fig_sk}: we have plotted, in linear scale,
the signed third order structure function $S^\star(3,a)$ as a function of the scale $a$ estimated for a log-normal $\cal CW$-cascade
calibrated precisely to model the spatial fluctuations of the longitudinal velocity in turbulence
(i.e., one sets $\zeta_3 = 1$ and $\lambda^2 = 0.025$).
It can be checked that for such a process we have $ S^\star(3,a) = -K a$.
In full analogy with turbulence, one can
wonder if $K$ could not be chosen such that, as in Kolmogorov theory, $K = \frac{4}{5} \varepsilon$ where $\varepsilon$ is the mean dissipation rate $\varepsilon \sim \EE{ a^{-1} \nu \int_0^a dt (\frac{\partial}{\partial t} X(t))^2} \;$ \cite{Fri95}.
Since $X(t)$ is not differentiable, in order for this to be meaningful, we show,
in Appendix \ref{App:diss_anomaly}, that the so-called ``dissipative anomaly'' \cite{EySree06} property of the velocity field
can be reproduced within our framework: if one chooses a ``viscosity" $\nu(\ell)$ such that $\nu(\ell) \sim \ell^{4/3-\lambda^2}$, then:
\begin{equation}
\label{eq:diss_an}
\lim_{\ell \to 0} \nu(\ell) \; \EE{ \left(\frac{\partial X_\ell(t)}{\partial t }\right)^2 } = \varepsilon \; ,
\end{equation}
for some $0 < \epsilon < \infty$.
Let us point out that it is likely that, from the definition Eq. \eqref{eq:def_X}, one could establish the following
generalization of Eq. \eqref{eq:diss_an}:
$$
\nu(\ell) \left( \frac{\partial X_\ell}{\partial t} \right)^2 dt \underset{\ell \rightarrow 0}{\longrightarrow} \varepsilon(dt)
$$
where $\varepsilon(dt)$ represents a singular measure corresponding to the multifractal dissipation \cite{menesree91} in the limit of vanishing viscosity and the convergence being interpreted in a weak sense.
Since the (negative) skewness of turbulence fields appears mainly on first order increments (or on odd analyzing wavelets),
as one can see in the top panels of Fig. \ref{fig_paths2}, it can be directly visualized as ``ramp" like (i.e. a slow increase followed by a rapid fall) patterns on the
velocity profile and its model. This behavior is very different from the "arch" like shapes that can be associated with the negative skewness of the second order increments. Such a feature is obviously present in the fluctuations of market prices as illustrated in the example of the Dow-Jones index the bottom left panel of Fig. \ref{fig_paths2}. In the right panels, we plotted sample paths of the corresponding log-normal models for turbulence ($H=0.355$, $\lambda^2 =0.025$ and $\varphi = g^{(1)}$) and stock market data
($H=0.51$, $\lambda^2=0.025$ and $\varphi = g^{(2)}$).
A close inspection of the Dow-Jones evolution in the bottom-left panel of Fig. \ref{fig_paths2} reveals
that this process has not only skewed second order increments by also display skewed first order increments.
Indeed, it is well known that financial return variations are characterized by
rapid large drops that are followed by slower upward moves. It thus results that, in order to model
the dynamics of market prices, a $\cal CW$-cascade should involve a wavelet that is neither odd nor even.
In the left panel of Fig. \ref{fig_sk}, we have computed the behavior of $S^\star(3,a)$ as a function of $a$
for both first and second order increments when the wavelet is the non-symmetric wavelet displayed
in Fig. \ref{fig_wavelets}(c) (the other parameters are those chosen for turbulence).
We see that both types of increments are characterized by a skewness of comparable magnitude since one observes third order moments a behaves as similar linear functions of the scale.
As discussed below, asymmetric synthetizing wavelets allow one to account for another feature observed on stock market data, namely the leverage effect.
\subsection{Leverage effect}
The fact that the sythetizing wavelet $\varphi(t)$ has no particular symmetry can be reflected
by different statistical quantities. As discussed notably by Pommeau \cite{Pom82}, there exists a wide variety of correlation functions
that allow one to reveal the lack of time reversal symmetry of a process $X(t)$.
The ``leverage'' function is a particular example of such a measure. It consists in computing the correlation between
the increments at some scale and their "amplitude" (e.g. their absolute value and their squared value) after or before some time lag $\tau$ \cite{BouMatPot01,PocBou02,BaDuMu12}:
\begin{equation}
\label{eq:def_lev}
{\cal L}_q(\tau) = Z_{q,\ell}^{-1} \EE{\delta_\ell X(t) |\delta_{\ell} X(t+\tau)|^q}
\end{equation}
where $Z_{q,\ell}^{-1}$ is a properly chosen normalization constant: for example, in \cite{BouMatPot01},
the authors studied ${\cal L}_2$ with $Z_{2,\ell} = \EE{\delta X_\ell^2}^2$.
In general one will consider
the behavior in the small scale regime, i.e., the limit $\ell \to 0$ with a wel chosen normalization.
The so-called ``leverage effect'' is a property that has been empirically observed on stock prices and stock market indices
indicating that past returns are (negatively)
correlated with future volatility (i.e. ${\cal L}_q(\tau) < 0$ for $\tau \geq 0$) while the reverse is not true (${\cal L}_q(\tau) \simeq 0$ for $\tau < 0$).
In that context "returns" means increments of log-price while "volatility" stands for squared or absolute returns. The leverage effect can be intuitively explained by the fact that, after a large price drop, some kind of panic takes place with a large uncertainty
and thus a large volatility results. The symmetric situation, i.e a large upward
price move, does not induce any crisis and therefore does not trigger any noticeable volatility variation.
\begin{center}
\begin{figure}[h]
\includegraphics[width=0.7 \textwidth]{Fig_LevLogLog.pdf}
\caption{Scaling of the leverage function. Estimated curves $|{\cal L}_1(\tau)|$ are plotted as a function of $|\tau|$ is double logarithmic scale (symbols ($\circ$)). The estimations have been performed on samples of lenght $L = 2^{17}$ of log-normal ${\cal CW}$-cascades with $\lambda^2 = 0.025$, $T = 2^{10}$ and, from top to bottom, $H=0.7, 0.51, 0.33$ and $0.1$.
The solid lines represent the corresponding theoretical power-laws of exponent $H-1-\phi(2)$.}
\label{fig_Levloglog}
\end{figure}
\end{center}
\begin{center}
\begin{figure}[h]
\includegraphics[width=0.6 \textwidth]{Fig_levFunction.pdf}
\caption{Leverage functions ${\cal L}_1(\tau)$ estimated for
a log-normal $\cal{CW}$-cascade with $H=0.515$ and $\lambda^2 = 0.025$ (solid lines). The various curves correspond to different asymmetry factors $\alpha$ in the synthetizing wavelet defined in Eq. \eqref{eq:asym_wavelet}, namely $\alpha = \frac{1}{8}$, $\frac{1}{3}$, $\frac{1}{2}$ and $1$. One sees that the behavior at lags $\tau >0$ does not depend on $\alpha$ whereas the leverage function at negative lags becomes smaller and smaller as the asymmetry increases. For comparison purpose, the estimated leverage function from daily Dow-Jones
index data are also reported (grey curve).}
\label{fig_LevFunction}
\end{figure}
\end{center}
Different attempts to account for this effect within the standard class of econometric models have been proposed \cite{BouMatPot01,PerMas03}. Since the class of Multifractal Random Walk as described in Sec. \ref{sec:c-casc} remarkably accounts for many
of ``stylized facts" of asset fluctuations, some authors considered different variants of these models
that break the time reversal symmetry
by introducing specific correlations between cascade and noise terms \cite{PocBou02,BaDuMu12}. But as mentionned in the introductory
section, such approaches cannot lead to well defined continuous time limits unless the noise has long-range correlations \cite{Poc03,BaDuMu12,Che17}.
Since continuous $\cal W$-cascades are generically not invariant by time reversal one expects that it should be possible
to account for the leverage effect by a specific choice of the synthetizing wavelet $\varphi$.
According to the definitions \eqref{eq:def_lev} and \eqref{eq:def_X}, ${\cal L}_q(\tau)$ can be expressed as
an intricate integral.
In Appendix \ref{App:Leverage}, we show, using some heuristic approximations, that, when $q=1$, the leverage function
behaves as:
\begin{eqnarray*}
{\cal L}_1(\tau) & \underset{\tau \gg \ell}{\sim} & C_{+} |\tau|^{H-1-\phi(2)} \\
{\cal L}_1(\tau) & \underset{\tau \ll -\ell}{\sim} & C_{-} |\tau|^{H-1-\phi(2)}
\end{eqnarray*}
where both scaling laws are valid in the range $|\tau| \ll T$.
We have shown that the prefactors $C_{+}$ and $C_{-}$ depend on the synthetizing wavelet $\varphi$ as:
\begin{eqnarray*}
C_{+} & = & -\int_{0}^{+\infty} \! \! s^{H-1} ds \int \! du \varphi \left(u \right) C_s'(su+1) \\
C_{-} & = & -\int_{0}^{+\infty} \! \! s^{H-1} ds \int \! du \varphi \left(u \right) C_s'(su-1) \\
\end{eqnarray*}
with $C_s(z) = \EE{e^{\omega_s(t)+\omega_s(t+z)}}$.
This scaling law is illustrated in Fig. \ref{fig_Levloglog} where
we have checked, using estimations from simulated data, it holds for different values of $H$ in the case of a log-normal cascade built with an anti-symmetric wavelet.
Notice that from the previous expression, $C_-$ and $C_+$ are
not necessarily equal and the leverage effect could therefore be observed when the leverage ratio:
\begin{equation}
\label{eq:leverageratio}
\kappa = \frac{C_+}{C_-}
\end{equation}
is very large (i.e. $\kappa \gg 1$).
\begin{center}
\begin{figure}[h]
\includegraphics[width=0.7 \textwidth]{Fig_LevRatio.pdf}
\caption{Evolution of the leverage ratio $\kappa$ (Eq. \eqref{eq:leverageratio}) as a function of the coefficient $\alpha$ characterizing the asymmetry of the synthezing wavelets defined in \eqref{eq:asym_wavelet}. The numerical integrations have been performed in the log-normal case with $\lambda^2 = 0.025$ and $H = 1/2+\lambda^2$. The symbols ($\circ$) represent the value
of $\kappa$ as estimated from the empirical leverage function obtained from simulated data of length 128 $T$ with $T = 2^{13}$.}
\label{fig_ratio}
\end{figure}
\end{center}
For illustration purpose and in order to handle reasonable expressions, let us consider the ``simple''
case when $\varphi(t)$ is a piece-wise constant function:
\begin{equation}
\label{eq:asym_wavelet}
\varphi(t) = \alpha^{-1}\mathbb{I}_{[0,\alpha)}(t)-\mathbb{I}_{[-1,0)}(t)
\end{equation}
where $0<\alpha<1$ controls the wavelet asymmetry.
In Fig. \ref{fig_LevFunction} we have plotted the leverage function
estimated from numerical simulations of
log-normal ${\cal CW}$-cascades with a synthezing wavelet corresponding to Eq. \eqref{eq:asym_wavelet} with different asymmetry factors $\alpha= 1/8,1/3,1/2$ and $1$. We chose $T = 2^{13}$, $\lambda^2 = 0.025$ and $H=0.51$ so that the increments are uncorrelated ($\zeta_2 = 1$). We first clearly see that the behavior for positive lags appears to not depend on $\alpha$ while the situation is very different for negative lags: as $\alpha$ decreases one sees that the leverage function lessens more and more. One goes from a symmetric leverage function in the case $\alpha=1$ to a situation where it is almost zero at negative lags. This latter case is very interesting to reproduce the empirical features observed on stock market: for comparion we have plotted the leverage function estimated from the Dow-Jones index daily returns over a period extending from 1939 to 2018 (grey line in Fig. \ref{fig_LevFunction}).
These empirical findings show that the leverage
ratio strongly depends $\alpha$ and becomes arbitrary large
when $\alpha$ is small. Even if it is possible to show that
the constants $C_\pm$ are bounded in the case when $\varphi$ is defined by Eq. \eqref{eq:asym_wavelet}, their exact value can however be hardly obtained or approximated under a closed form. We then estimated them by means of a numerical integration.
The so computed leverage ratios are reported in Fig. \ref{fig_ratio}.
In the case when $\alpha = 1$, we naturally recover the
fact that the leverage effect is not present while we observe that
$\kappa$ strongly increases for small $\alpha$.
Notice that our numerical estimations confirm that both $C_+$ and $C_-$ are negative in the range $0.1 \leq \alpha \leq 1$. We also observed that $C_+$ is almost independent of $\alpha$ while $C_{-}$ becomes arbitrary small as $\alpha \to 0.048$ where it changes its sign.
It thus seems possible, within this model, to obtain a "perfect" leverage effect with an infinite leverage ratio corresponding to a vanishing leverage function in the domain $\tau <0$.
These numerical computations of the leverage ratio have been checked
using empirical estimation from numerical simulations (symbols $\circ$).
\section{Summary and prospects}
\label{sec:conclusion}
In this paper we have proposed a new way to build random
multifractal functions with stationary increments, exact scaling and self-similarity properties.
Our model just consists in extending former
$\cal W$-cascades by replacing the framework of orthogonal
wavelet basis by that of continuous wavelet transform
and the discrete multiplicative weights by their log-infinitely divisible counterpart, i.e. the process $e^{\omega_{\ell}(t)}$
involved in the construction of continuous cascade measures. We have shown that our construction provides almost surely Liphscitz regular paths and studied its self-similarity and scaling properties. As emphasized in Sec.\ref{sec:cw-cascades_skew}, $\cal CW$-cascades are in general skewed multifractal processes and are characterized by
a non-symmetric correlation between increment signs and amplitudes (the so-called "leverage effect").
As far as applications to turbulence are concerned, since our framework can easily reproduce the skewed intermittency phenomenon observed for the velocity fluctuations, together with the dissipative anomaly, it provides undoubtedly
a promising way to account for many stochastic aspects of fluid dynamics in regimes of fully developed turbulence.
It that respect, vector and 3-dimensional extensions of $\cal CW$-cascades with the possibility of introducing well
known dynamical properties like incompressibility (as e.g. in \cite{CheRobVar10,ChePeGa16}) could be interesting in order to get a more realistic model.
Beyond applications to specific contexts and the previously mentioned problems,
a fundamental question is to know which features of the sythetizing wavelet remain observable through the associated continuous cascade.
Is there a analog of the famous black holes "no-hair theorem" \cite{MTW73} in the present framework ? This "inverse problem" is interesting since we already known that some important properties like
skewness, the behavior of the leverage function, the prefactor values in scaling relationships may depend on the specific wavelet shape but the issue is to precisely know in what respect and also which properties of $\varphi$ (like e.g. the number of vanishing moments, the moment values,...) can be recovered from empirical data.
Finally, let us mention that, by simply replacing
$e^{\omega_{\ell}(t)}$ by its lacunary version introduced in \cite{MuBai16}, our framework may also allow one to build random functions that are almost everywhere smooth and singulular on random Cantor sets. One could also slightly extend the definition of ${\cal CW}$-cascades in order to build a stationary variant of the class of lacunary wavelet series that possess oscillating singularities defined
and studied by Arneodo {\em et al.} \cite{ABJM97,ABJM98}. Such processes
are not self-similar but possess a self-similar wavelet transform.
All these prospects will be considered in a future research.
|
1,108,101,563,304 | arxiv | \section{Introduction}
\label{sec:Introduction}
In \emph{The Outline of History}, H.~G.~\citet[p.~1100]{HG-Wells-21:history} declared that ``Human history becomes more and more a race between education and catastrophe.'' Although Wells wasn't talking about Autonomous Vehicle (AV) technology, he was talking about economic disasters, and AVs are a prime example of a phenomenon with significant risks to society.
AVs have much to offer but they also bring forth potential catastrophes by affecting the lives, livelihoods, and wellbeing of humans. We focus on the important ethical concerns of human autonomy and dignity, justice, and equity \citep{EGE-18:ethics} concerning AVs.
We posit that since AV deployments involve people and machines, they should be viewed as sociotechnical systems \citep{AIES-18:ethics}. Previous work on sociotechnical systems in AI pursues challenges in representation and reasoning as a way to provide a computational basis for user requirements. In contrast, here we tackle the problem of understanding stakeholder perceptions that underlie any such requirements. Our approach thus exemplifies ``social-systems analysis'' as an interdisciplinary assessment of the human impact of AI that ``thinks through all the possible effects of AI systems on all parties'' \citep[p.~313]{Crawford+Calo-16:AI-blind-spot}.
Previous works largely consider the opinions of ``experts'' (whether technologists, decision makers, or ethicists). However, we take the stance that to develop a suitable foundation for an ethical response, we must understand the concerns of stakeholders most likely affected by the introduction of AVs.
Accordingly, we study the perceptions and concerns of professional and semi-professional drivers recognizing them as a source of ground knowledge about potential harms and benefits of AVs. We aim to turn the tables by using their perceptions and concerns to advise the experts. This process respects the reasonable pluralism in democratic societies in which ``every one of us has different life experiences that inform our values'' \cite[p.~35]{Himmelreich-20:ethics-technology}.
To elicit and understand the perceptions and values of stakeholders requires in-depth conversations in which they are able to articulate and explain their positions. To this end, we applied a qualitative research methodology based on semi-structured interviews with drivers. The resulting understanding brings out the ethical ramifications of deploying AVs as perceived by drivers, and can be a useful basis for decision makers formulating policies that will govern AV deployments.
\paragraph{Organization}
The remainder of the paper proceeds as follows.
Section~\ref{sec:Justice} explains how we approach the social
ramifications of AVs. Section~\ref{sec:Study} explains the motivation and design of our study.
Section~\ref{sec:Results} presents and analyzes the results of our study.
Section~\ref{sec:Conclusions}
presents concluding remarks.
\section{Justice and Social Ramifications}
\label{sec:Justice}
We distinguish two levels of ethical concern in relation to AI and other technologies: micro-and macro-ethics, reflecting a distinction made by \citet{AIES-18:ethics}. Micro-ethics is concerned with morally charged interactions between relatively small numbers of individuals, as represented by classic ethical theories such as deontology \citep{Kant-1785:morals}, utilitarianism \citep{Mill-1863:utilitarianism}, and virtue theory \citep{Aristotle-350BCE:ethics}. Macro-ethics is concerned with the ethics of systems. Through there may be no sharp cutoff between micro-ethics and macro-ethics, clear examples of each exist.
Much of AI research focuses on micro-ethics. The classic trolley problem scenario \citep{Foot-67:trolley} in research on AVs exemplifies what \citet{Dubljevic+Bauer-20:AVs-society} call ``ethics on the road.'' \citet{Bergmann-18:AV-socio-political} used the trolley problem in a virtual reality study concerned with how people react in a variety of vehicular dilemmas while balancing concern for their own safety against the safety of others.
Macro-ethics focuses on difficult questions like distributive justice \citep{Rawls-99:justice,Sen-09:justice}: How should we structure the basic institutions of society, as well as distribute goods and opportunities? What kinds of economic incentives will improve society?
Focusing on the macro issues---the socioeconomic impact of AVs---we posit that fairness requires decision makers and relevant leaders to take account of stakeholder perspectives in designing policies to mitigate the negative impacts and enhance the positive impacts of AV technology.
\citet{Rawls-99:justice} presents an influential theoretical model for analyzing the fairness of socioeconomic institutions and outcomes. His ``original position'' thought experiment provides an objective and illuminating way to think about the basic structure of society and what members of societies owe each other. In the original position---conceptualized as a hypothetical device, not an actual historical condition or ``state of nature''---people are behind a ``veil of ignorance'' about their individual characteristics; thus, they do not know who they are.
\emph{Reflective equilibrium} \citep{Rawls-99:justice} is a prominent method to test the normative and empirical adequacy of ethical principles. It implies a coherence procedure of applying general principles from theories to specific moral intuitions or considered moral judgments of stakeholders. Specifically, self-interested interlocutors propose principles of justice and test them against considered judgments on particular examples.
We see two reasons to adopt a Rawlsian perspective. First, ``justice as fairness'' highlights the value of equity when talking about the basic structure of society and, we presume, the public shares a significant interest in creating fair results for drivers as AVs become more integrated into our transportation systems. Second, the original position involves all relevant stakeholders so that policies are chosen fairly. Specifically, in the later stages of the original position, stakeholders would refine the abstract principles to concrete policies based on relevant new sociopolitical data.
\section{Study Design and Methodology}
\label{sec:Study}
In developing policies concerning the deployment of AVs, decision makers may accept that drivers need protection from emerging technological advances or that they are part of the problem to be solved to successfully deploy AV technology. Either way, public policies should be informed by the experiences and worldviews of drivers since they would be affected most directly by the deployment of AVs.
\subsection{Method}
\label{sec:Qualitative}
Qualitative research is a social science methodology that collects non-numeric, value-laden data. Although qualitative research is not generally known in computer science, outside of a few specialties, it is a useful approach for addressing the ethical and social implications of technology.
In this study, we conducted semi-structured one-on-one interviews. This method is well suited to exploratory studies and helps gather rich and meaningful data that (1) future research can build on, and (2) could provide decision makers with a broader understanding of the phenomena under investigation as they make decisions that will impact the stakeholders \citep{Given-08:qualitative}.
This method enables interviewers to maintain focus on key research questions while facilitating a free exchange between the interviewer and interviewee. Interviewers followed a script to guide the conversation but varied the conversation as needed. Specifically, this approach enables the interviewer to ask relevant follow-up questions and the interviewee feels free to expound on anything they feel is important for the interviewer to know, producing as much information as possible from the drivers within a restricted time.
Each interview was transcribed. Data analysis was carried out concurrently with data collection to enable the integration of information from each step of the process. Codebook development and coding of responses followed an iterative process. We developed codes via abductive analysis, a form of qualitative content analysis that combines elements of both induction and deduction \citep{Timmermans+Tavory-12:qualitative}.
Our strategy employed five analytic stages, namely: (1) familiarization with the data through reading of interview transcripts; (2) identification of a thematic framework that reflects the ideas discussed; (3) indexing the data, i.e., identifying patterns across the transcripts; (4) charting the data, i.e., comparing data across identified patterns; and (5) mapping and interpreting the data, i.e., making sense of the data holistically \citep{Ritchie+Spencer-02:qualitative}.
\subsection{Drivers as Stakeholders}
\label{sec:Drivers}
By engaging with affected individuals, we hope to ensure that the voices of this important group of stakeholders are heard during agenda setting as well as during research on AV deployment, consistent with the expectations of social justice. Recognizing and probing how stakeholders are affected is an important component of any ``power analysis'' \citep[p.~19]{Sandler-14:technology-ethics} of technological changes in society (i.e., recognizing who is empowered and disempowered by changes).
Accordingly, we interviewed drivers who had at least two years of paid driving experience. We report on nine interviews with professional truck drivers and semi-professional Uber or Lyft drivers. Our goal was to identify and analyze perceptions among these stakeholders concerning relevant challenges of deploying AVs.
\subsection{Implications of AVs Investigated}
\label{sec:Claims}
Previous studies on this topic are sparse and largely focused on experts. For instance, \citet{Pettigrew+Fritschi+Norman-18:AV-implications} investigated societal implications of AVs by conducting interviews of expert stakeholders from government at multiple levels, trade unions, law, technology firms, AV manufacturing, academia, and other areas. Thus, although they interviewed stakeholders in the AV deployment process, those stakeholders did not represent professional and semi-professional drivers, whose livelihoods could be radically affected by AVs.
In contrast, we investigated potential implications of AVs on this key stakeholder and society at large \emph{through the eyes of the driver community}. We focused on the claims shown in Table~\ref{tab:claims}. Though these claims are based on previous findings \citep[pp.~4--7]{Pettigrew+Fritschi+Norman-18:AV-implications}, we approach them from the ground-level perspective of drivers.
\begin{table}[htb]
\centering
\caption{Claims about social aspects of AVs that we evaluate.}
\label{tab:claims}
\begin{tabular}{p{0.21\columnwidth} p{0.7\columnwidth}}
\toprule
\textsl{Transportation industry} & AVs' significant impact on transportation is inevitable\\
\textsl{Drivers' expectations} & Employers should be straightforward about changes and options toward the potentially affected stakeholders\\
\textsl{Reskilling} & Reskilling is manageable due to the decade-plus lead time (this, we note, is generous as technological changes sometimes occur more quickly than expected)\\
\textsl{Responsibility to respond} & Responsibility for dealing with these issues falls across all elements of society (government, business, drivers)\\
\textsl{Driving as a profession} & To the extent that driving jobs are ``unsatisfying and potentially unhealthy'' their elimination could be a positive development so long as other opportunities are available\\
\bottomrule
\end{tabular}
\end{table}
\subsection{Study Logistics}
Approval was obtained from the Institutional Review Board of the authors' institution (IRB approval no.\ 20276) before interviews were conducted. No personally identifying information was collected during the interviews. Participants were recruited via study flyers. The interviewees verbally indicated their willingness to participate at the start of audio recording. Each interviewee received a \$60 gift card as compensation for their time.
\begin{table*}[htb]
\centering
\caption{Interview questions.}
\label{tab:S1-questions}
\begin{tabular}{l l p{0.75\textwidth}}\toprule
\textbf{Category} & Q\# & Question or prompt\\\midrule
Situation & Q1 & ``Please reflect on morally problematic situations concerning driving decisions from your own experience.''\\%\cmidrule(l){2-3}
& Q2 & ``Was there an occasion where your split-second decision prevented a crash?''\\%\cmidrule(l){2-3}
& Q3 & ``Could you describe what exactly happened and what could have happened?''\\\midrule
Self-driving & Q4 & ``What is your opinion of self-driving technology?''\\%\cmidrule(l){2-3}
& Q5 & ``Is it fair?''\\%\cmidrule(l){2-3}
& Q6 & ``How soon do you think it (self-driving technology) might be implemented?''\\%\cmidrule(l){2-3}
& Q7 & ``Do you have plans for your career if that turns out to be sooner than anticipated?''\\\midrule
Professional driving & Q8 & ``Do you like your current job?''\\%\cmidrule(l){2-3}
& Q9 & ``Do you feel meaningfully connected with others by performing your job duties?''\\%\cmidrule(l){2-3}
& Q10 & ``Is your job overall a positive or negative experience for you?''\\
\bottomrule
\end{tabular}
\end{table*}
In total, nine interviews were conducted: four with professional truck drivers (identified as TR in the quotes below), one female and three males, and five with semi-professional Uber or Lyft drivers (all male, identified as UL in the quotes below). The TR drivers were professional, typically older than the UL drivers, who were mostly students supplementing their income by driving (hence, ``semi-professional''). The interviews were conducted from December 2019 through March 2020 by the corresponding author or by undergraduate research assistants trained by the corresponding author. The undergraduate research assistants received an orientation to the study protocol, as well as training in ethics, qualitative research methods, and interview administration.
The number of interviews we conducted is sufficient, as six interviews is generally recognized in the qualitative research literature as leading to 80\% of thematic saturation \citep{Guest+06:how-many-interviews,Kuzel-99:sampling-qualitative}. Interviews varied in length from 30 to 70 minutes and were all digitally recorded. We transcribed them and applied the method detailed in Section~\ref{sec:Qualitative}.
The questions asked and topics discussed are shown in Table~\ref{tab:S1-questions}. Since professional driving was described by experts (not drivers) as ``unsatisfying and potentially unhealthy'' \citep[p.~5]{Pettigrew+Fritschi+Norman-18:AV-implications} we also asked drivers to elaborate on their feelings about the job, whether they felt meaningfully connected to others socially, and whether they perceived professional driving is overall a positive or negative experience.
\section{Results and Discussion}
\label{sec:Results}
\begin{table*}[htb]
\centering
\caption{Comments on inevitability and fairness of automation.}
\label{tab:S2-fairness}
\begin{tabular}{l p{0.80\textwidth}}\toprule
\textbf{Respondent ID} & \textbf{Comments}\\\midrule
UL-1/Allen & ``I'm in a position where I don't rely on this income. I use it to supplement things.''\newline
It [Uber driving] is a great college job, but not a good career. It is a very, very sad career that often puts drivers and passengers in danger.''\newline
``As someone that is relatively comfortable with defending themselves, can drive, work late hours, and is in college it is a great job. But if you are out of college, in your 30's, and you do this for a full-time job, it would be an unimaginable hell.''\\%\midrule
UL-2/Bob & ``can't wait for cars to be fully autonomous''\newline
``it would definitely wipe out a lot of jobs but also\ldots we need to look toward the future\ldots and start seeing what industries are going to be created''\newline
``I definitely have plans outside of driving if it becomes obsolete sooner than expected.''\\%\midrule
UL-3/Charley & ``a lot of people would be put out of a job because of that [AVs]\ldots but that's also\ldots a sad result of capitalism in my opinion.''\newline
``I've always used Lyft as a supplement to my income\ldots so I'm not too worried with that\ldots but I'm sure there will be a lot of people out there who will be negatively affected by it [AVs].''\\%\midrule
UL-4/Dan & ``AVs are lazy, they would make us dependent and not use our whole brain.'' \\%\midrule
UL-5/Earl & ``AVs are not fair from the standpoint of Lyft, Uber, Truck, Taxicab drivers, or food delivery drivers\ldots because that's income for them\ldots there are consequences for that degree of automation.''\\%\midrule
TR-1/Anthony & ``it's going to happen to one degree or another [AVs] but I don't know as if the driver will ever fully be replaced''\newline
``the smaller companies, with a few exceptions, will still rely on traditionally driven vehicles\ldots the first generation [of AVs] will start going to the resale market'' \\%\midrule
TR-2/Bill & ``To a lot of people, it's not gonna be fair, but to those who stay on top of the things they are going on\ldots it shuts a lot of doors, but it opens a lot more.''\newline
``I think we'll lose a lot of people\ldots and yeah, they gonna hurt''\newline
``I don't expect it to impact my career at all''\\%\midrule
TR-3/Christina & ``it [AVs] would put a lot of drivers out of jobs\ldots I don't think it's right.''\newline
``I think it's very important to be able to connect to all of them [the people in warehouses] instead of the machine doing all of it.''\\%\midrule
TR-4/David & ``It [AVs] would suck to take a lot of drivers off the road.''\\
\bottomrule
\end{tabular}
\end{table*}
Our results include the drivers' perceptions about their jobs as well as how AVs might impact their livelihoods and society at large. Overall, the drivers expressed ethical concerns about the effects AVs could have on society, specifically on non-professional drivers and passengers. The UL drivers lamented automation's weakness in human interaction (see Table~\ref{tab:S2-fairness}). Frequently mentioned were \emph{conversation}, \emph{human intuition}, and \emph{human knowledge} as benefits that society would lose with AVs. The TR drivers were generally more concerned about the \emph{welfare} of the truck-driving community and the \emph{potential harm} AVs could cause to their livelihoods. In addition, the TR drivers expressed concerns about the \emph{decision-making process of AVs}. Nevertheless, there was some variation in TR driver responses. For instance, some stated that automation could be integrated into the industry with drivers learning to adapt to the new technology, whereas others stated that automation offered too many risks---to both drivers and society---and that, as a result, it would be morally wrong to introduce further automation (see TR-3).
Some researchers have asserted that driving jobs are unsatisfactory, boring and unhealthy \citep{Pettigrew+Fritschi+Norman-18:AV-implications,Viscelli-18:driverless}, but our results paint a picture of TR respondents who see the ``joy of driving'' and at the very least, UL drivers who value the supplemental income. Altogether, the respondents viewed their job as a net positive for themselves and society, and many were against full automation---some were even opposed to partial automation (see TR-3). Many of the respondents realized that automation was likely coming. However, almost all thought it was at least 10 years away. None had given serious thought to alternative careers, should automation come sooner than they anticipated.
\begin{table*}[htb]
\centering
\caption{Comments on effects on the community and society at large.}
\label{tab:S3-effects}
\begin{tabular}{l p{0.80\textwidth}}\toprule
\textbf{Respondent ID} & \textbf{Comments}\\\midrule
UL-1/Allen & ``self-driving\ldots cars\ldots[are] not a solution to transportation issues. What \emph{is} a solution is public transportation.''\newline
``the gig-economy is just a band-aid on the lack of a social net in this country''\newline
``when you have an autonomous vehicle, you gain some efficiency improvements\ldots but you lose the human factor\ldots like chatting with the driver about lunch recommendations\ldots or these specific things that warrant a tip''\newline
``If you gave people that infrastructure with the improvements of efficiency with autonomous buses, I think more people would use it and we'd all be better off.''\\%\midrule
UL-2/Bob & ``it [AVs] could eliminate the need for stoplights because everyone would know what they were doing''\newline
``industries are going to be created that way, we can create new jobs with technology''\newline
``generally fair, you just have to determine which outcome causes the least amount of total harm''\newline
``self-driving technologies are a type of philanthropy it almost seems''\\%\midrule
UL-3/Charley & ``it [AVs] would make a much wider impact on the economy than a lot of people realize''\newline
``I could see how it could be unfair, but that's also capitalism''\\%\midrule
UL-4/Dan & ``we're going to have a whole bunch of people on the road that really don't even know how to drive''\newline
``There are so many people who utilize Uber as their therapist''\\%\midrule
UL-5/ Earl & ``self-driving\ldots needs to be looked at more than the excitement of just pulling off an incredible technology. And how it impacts people and society, before you start jumping on bandwagons.'' \\%\midrule
TR-1/Anthony & ``in the urban environment there's plenty of things, whether it's a pedestrian that is walking or something that's just stopped in the street\ldots what does it do when it senses [this]''\\%\midrule
TR-2/Bill & NA\\%\midrule
TR-3/Christina & ``Who's going to be accountable for all the mistakes that they [AVs] will do?\ldots Who is going to be responsible [if an accident occurs?]''\newline
``I would not let my kids ride on a school bus without a physical driver\ldots[which] they tried''\newline
``automation at every level will create many many problems that we are not equipped to deal with''\\%\midrule
TR-4/David & NA\\
\bottomrule
\end{tabular}
\end{table*}
Table~\ref{tab:S2-fairness} is a selection of personal or group-related ethical concerns stated by each respondent, which can be read as concerns regarding automation or AVs. Notable examples are \emph{loss of jobs} and \emph{loss of safety} for passengers, as the driver is in a position to intervene or call the authorities if one passenger seems to be threatened by another passenger. This contrasts with Table~\ref{tab:S3-effects}, where general-level concerns from each respondent were listed specifically in regard to society or some larger community. This was done to see not just to what extent these issues were a concern, but also to express the range of concerns.
\subsection{Micro-Ethics: Safety For or From Others}
\label{sec:Micro}
\begin{table*}[htb]
\centering
\caption{Comments on initial problems and ``bugs'' with AV technology.}
\label{tab:S4-problems}
\begin{tabular}{l p{0.80\textwidth}}\toprule
\textbf{Respondent ID} & \textbf{Comments}\\\midrule
UL-1/Allen & ``others on the road may not understand that your fancy new Tesla may not have all of the bugs worked out''\\%\midrule
UL-2/Bob & ``there will be growing pains with that, but that's all of technology for all of human history has growing pains. But that's not something that scares me''\\%\midrule
UL-3/Charley & ``I think there's just so many factors [for AVs] on the road that I wouldn't want to be on the front end on it--because there's going to be a lot of mistakes made\ldots it's that first phase where they bring it in, I think it's a phase they're trying to get around''\newline
``they don't know what the bugs are in the system until they happen'' \\%\midrule
UL-4/Dan & ``what happens when someone comes from the side of you and your car doesn't know? What happens if your car malfunctioned? \ldots there is no way you're going to tell me that my car can drive itself and still be alert like I am''\newline
``Why weren't your feet on the floor? Why are you not paying attention? I think it's [AV] just a crutch''\newline
``what happens when shit malfunctions and things gotta get real? And you don't have anybody to blame or anything?'' \\%\midrule
UL-5/ Earl & ``I would be more inclined to wait until they've got more time working out the bugs, I'm weary of it [AV].''\newline
``a lot of that would have to be debugged or tried/experimented with.''\newline
``[Self-driving tech] needs to be looked at more than the excitement of just pulling off an incredible technology''\\%\midrule
TR-1/Anthony & ``I think the technology can all sense all that stuff, but it's just a matter of what does it so when it senses it? Does it shut down completely\ldots It's just a lot of testing that I think has to take place before it's widely accepted''\\%\midrule
TR-2/Bill & NONE\\%\midrule
TR-3/Christina & ``What if the truck in front of me blows out a tire and it's all kinds of debris on the road? What is the computer going to know to do in that situation\ldots?''\newline
``there was an incident where they [an AV] killed someone who was crossing the street''\newline
``A Tesla was in an accident and the car completely shut down; you couldn't open the window. He [the driver] burned alive in the car''\\%\midrule
TR-4/David & NONE\\
\bottomrule
\end{tabular}
\end{table*}
Note that Table~\ref{tab:S4-problems} is a gauge for the level of awareness of each respondent, not a qualitative difference in definition of problems with AV technology.
Concerning micro-ethics, and specifically safety concerns, we present summaries of responses in Table~\ref{tab:S4-problems}, citing safety concerns regarding AVs, especially with regard to ``technical bugs'' or ``growing pains.'' It also shows that respondents are, generally, well aware of a variety of important social and ethical issues regarding the deployment of AV technology.
\subsection{Macro-Ethics: Society and Workforce}
\label{sec:Macro}
The UL drivers perceived that AV technology would be deployed in 10 years, and they had concerns regarding safety, social, and ethical issues. Besides job losses, they were concerned that early deployment could create many other problems. Respondents pointed out that some new technologies might work in experiments with controlled environments, but the results could be considerably different outside those limits. Pedestrians, for instance, were seen as being negatively affected by automation; drivers stated that human intuitions were better suited to pedestrian interactions than AVs. As one of the UL drivers stated, ``People have run in front of my car while I'm having a conversation and it's hard to balance that. [\ldots] They might not really take that [a car is autonomous] into account and that could lead to accidents'' (UL-1). This driver raised his concerns regarding pedestrians and industry over-reach emphatically: ``There's no coincidence that Google Waymo is testing their technology in Scottsdale. The city planning there is very grid-like and it's also very hot, during the day, which is the only time that Google Waymo is active, so there's less pedestrians, and less bicyclists out there which is another danger, so in places that have a grid plan, [it is easier to implement] [\ldots], but places like Boston it'll be [difficult]. Because there's a lot of city streets and curves and stuff like that [\ldots]. But, that doesn't mean companies won't try it'' (UL-1).
The intuition to protect others and react instinctively to avoid accidents is not the only human characteristic that UL drivers felt automation could not replicate. They believed social practices would also be affected. Another UL driver noted that ``most people do want to talk [\ldots even] with a random stranger [\ldots]. For them [the passengers] it's definitely probably going to be a little more soul-crushing to get into the car and see a box making the moves instead of you, or someone who can at least try to put on a friendly face'' (UL-3). Other social changes include lack of agency. As noted by this UL driver, ``I think we're going to get into a generation where\ldots it makes us lazy and\ldots dependent'' (UL-4). Semi-professional drivers felt that the \emph{dehumanization of driving} would be a loss caused by AV technology.
The TR drivers expressed much greater skepticism about AVs. They perceived that AVs may be deployed in 10 to 20 years and identified a range of concerns. Some of these concerns were similar to those of the UL drivers, yet their concerns were often more nuanced. As one TR driver stated, ``I don't know as if the driver will ever be fully replaced. [\ldots] I think for interstate travel where it's highways and open roads, that's gonna happen first but when you're in the city and suburbs and there's more variability [\ldots] on an interstate highway you don't have pedestrians that you have to worry about'' (TR-1).
Some of the TR drivers were vehemently opposed to the AV technology. TR driver Christina was highly critical, declaring that: ``I think it [AV Tech] would be wrong, and it would put a lot of people in danger'' (TR-3). There is some reluctance among TR drivers critical toward AVs to even estimate when the technology might be widely adopted. In her words, ``I hope never. I don't know if it will be. I hope not soon'' (TR-3). Her criticism of automation is specific and based on concrete experiences with automation technology: ``The robots load you. The robots put the things in your trailer. Everything was done by robots. Until delivery when you find out they didn't put the slab right or they forgot something else. [\ldots] before you used to be able to go inside and look through the load to have it how you like it to be loaded, now the machine goes in and loads you'' (TR-3). The drivers seem to base their negative opinion of AV technology on personal experiences with automation and highly publicized tragedies involving self-driving technology: ``What is going to happen when [\ldots] you let that computer drive you? [\ldots] A Tesla was in an accident and the car completely shut down; you couldn't open the window. [\ldots] He burned completely. The car was on fire and the fire department couldn't take him out, because it was all electrical and it all shut down. So, I think that is completely wrong'' (TR-3).
Other professional drivers offered a more mixed assessment with detailed or nuanced estimates. TR driver Bill estimated that ``the first 10 years are gonna be driver and technology both together\ldots then I'm sure at one point we will see it to where it will completely take over trucking'' (TR-2). Some drivers even provided state specific estimates: ``I'd say 10-15 years maybe, in the right places, in the right states. Like I'd see them driving around in Texas or Nebraska, but Colorado or Wyoming where there's white out conditions? I don't see them dealing with that'' (TR-4).
An overarching concern was that \emph{their voices would not be heard at any stage of public policy deployment}. As one TR driver noted, ``When you have people sitting there making rules like that, who have never been in the truck---it's just mind boggling how they get away with this?'' (TR-4).
\subsection{Evaluating the Initial Claims}
\label{sec:Evaluating}
The results of our interviews help assess the claims introduced in Section~\ref{sec:Claims}. Does our interview data confirm, disconfirm, or neither confirm nor disconfirm the relevant claim? In sum, the claims about the \textsl{transportation industry} and \textsl{drivers' expectations} are confirmed; the claims about \textsl{reskilling} and \textsl{responsibility to respond} are neither confirmed nor disconfirmed; the claim about \textsl{driving as a profession} is disconfirmed.
\begin{description}
\item[Transportation industry] \emph{AVs significant impact on transportation is inevitable: \textbf{Confirmed}}. For TR respondents, this is confirmed by all except Christina (TR-3), who hopes it does not ever come. UL respondents agree wholeheartedly that AVs will inevitably impact the way in which the transportation system is managed.
\item[Drivers' expectations] \emph{Employers should be straightforward about changes and options toward the potentially affected stakeholders: \textbf{Confirmed}}. For TR respondents, this is the case across the board. They referenced the electronic logbook regulations that had recently been put into place, with which they were frustrated, and they all spoke of the importance of communication in regard to changes in the industry and associated regulations. UL drivers unanimously agreed as well, although it was less of a concern for them than for TR drivers. Two UL drivers were interested in the recent California law that established ``gig-economy workers'' as employees, rather than as contractors. The two respondents, UL-1, and UL-4, were on opposite sides of opinion regarding the law, but both agreed that greater consideration for workers was necessary. UL-4 expressed outright distaste for the communication (or lack thereof) received from Uber about new changes to Uber's policy. UL-1 expressed contempt for Uber and Lyft, calling them both ``slimy'' companies. The other UL drivers all anticipated changes coming and expressed their approval for better communication with drivers.
\item[Reskilling] \emph{Reskilling is manageable due to the decade-plus lead time: \textbf{Neither}}. This claim is marginally correct. Half of the TR respondents agreed with this claim, whereas two others (TR-3 and TR-4) were either dismissive of all automation, or frustrated that so many jobs would be lost, respectively. The UL drivers viewed the advancement of AVs more positively, and were less likely to care about reskilling, as they used the income merely to supplement their livelihood. To the UL drivers, it was seen mostly as a necessary, but unfortunate, step into the future.
\item[Responsibility to respond] \emph{Responsibility for dealing with these issues falls across all elements of society (government, business, drivers): \textbf{Neither}}. The discussion of responsibility varied across respondents, and no clear trend was observed.
\item[Driving as a profession] \emph{To the extent that driving jobs are ``unsatisfying and potentially unhealthy'' their elimination could be a positive development so long as other opportunities are available: \textbf{Disconfirmed}}. Not one driver from either category said this was the case. Each said that driving for them was a net positive, and that they enjoyed what they did. Some interesting caveats were brought forth by UL-1. This respondent gives stipulations, such as being a college student, who is comfortable defending themselves, but says that driving for Uber as a full-time job and only source of income would be an ``unimaginable hell.'' However, every professional driver found their job to be positive, despite the challenges, even going as far to say that technology and the elimination of their jobs would, in fact, be a negative. Overall, our interviewee data disconfirms the idea that drivers found their jobs unsatisfying or unhealthy.
\end{description}
\section{Conclusions and Directions}
\label{sec:Conclusions}
We claimed that stakeholders' perspectives constitute important data in a full assessment of the social and ethical impact of AVs. Toward that end, we engaged with professional and semi-professional drivers using a semi-structured interview methodology. These drivers are aware of and concerned with both micro and macro-ethical issues in relation to AVs, both of which motivate this investigation and which society should actively address. By considering drivers, we obtained results that disconfirm some of the findings of studies of experts.
We view the interviewee data analyzed in this paper as informing the thought process of parties in Rawls' original position, particular in later stages as specific social and economic policies are developed. The perspectives and opinions of drivers (and stakeholders in other relevant fields impacted by AI) are clearly morally relevant to the kinds of principles that should be devised in the original position. To be clear, we are not suggesting that our drivers were, or should be, placed behind a veil of ignorance during interviews.
Instead, we suggest that drivers' lived experience and perspectives should be part of a general basis of knowledge about the effects of AVs on transportation and the economy, because this better informs the design of principles and polices to govern deployments of AV technology. On the most practical level, decision makers should pay close attention to these perspectives as they develop policies. A reasonable principle of fairness requires it.
Additional studies that focus on diverse stakeholder groups, such as pedestrians, public transport users, and people with mobility limitations, are necessary to fully align AV technology's deployment with values of the public at large. Doing so can improve the ethical standing of our policies and increase public confidence that values of the relevant stakeholders are being respected.
Our findings raise interesting challenges for further investigation. Along the theoretical dimension is advancing theories of justice and fairness in light of the permeation of AI into society. Along the practical dimension are identifying the elements of work processes that would be lost due to automation. The future is promising for AI and for deeper analysis of the ethics of AI from the perspectives of Science and Technology Studies.
|
1,108,101,563,305 | arxiv | \section{Introduction}
This work is a sequel to \cite{ZS2}-\cite{ZS4}. In \cite{ZS4}, we asserted that the lower bounds of modular representation dimensions stated in the abstract are attainable under the assumption that the associated complex finite $W$-superalgebras admit one- or two-dimensional representations. In the present paper, we certify the assumption in the case of minimal nilpotent elements.
\subsection{} A finite $W$-algebra $U(\ggg,e)$ is a certain associative algebra associated with a complex semisimple Lie algebra $\mathfrak{g}$ and a nilpotent element $e\in{\ggg}$. The study of finite $W$-algebras can be traced back to Kostant's work in the case when $e$ is regular \cite{Ko}, then a further study was done by Lynch in the case when $e$ is arbitrary even nilpotent element (cf. \cite{Ly}). Premet developed finite $W$-algebras in full generality in \cite{P2}. On his way to proving the celebrated Kac-Weisfeiler conjecture for Lie algebras of reductive groups in \cite{P1}, Premet first constructed the modular version of finite $W$-algebras in \cite{P2}. By means of a complicated but natural ``admissible'' procedure, the finite $W$-algebras over the field of complex numbers were introduced in \cite{P2}, which shows that they are filtered deformations of the coordinate rings of Slodowy slices.
Aside from the advances in finite $W$-algebras over complex numbers, the modular theory of finite $W$-algebras has also developed excitingly. It is worth noting that in \cite{P7} Premet proved that if the $\bbc$-algebra $U(\ggg,e)$ has a one-dimensional representation, then under the assumption $p\gg 0$ for the ground algebraically closed field $\bbk$ of positive characteristic $p$, the reduced enveloping algebra $U_\chi(\ggg_\bbk)$ of the modular counterpart $\ggg_\bbk$ of $\ggg$ possesses an irreducible module of dimension $d(e)$ (where $\chi$ is the linear function on $\ggg_{{\bbk}}$ corresponding to $e$, and $d(e)$ is half of the dimension of the orbit $G_\bbk\cdot \chi$ for the simple, simply connected algebraic group $G_\bbk$ with $\ggg_\bbk=\Lie(G_\bbk)$), which is a lower bound predicted by the Kac-Weisfeiler conjecture mentioned above.
The existence of one-dimensional representations for $U(\ggg,e)$ associated with $\ggg=\text{Lie}(G)$ of a simple algebraic group $G$ over $\mathbb{C}$ was conjectured by Premet, and confirmed in the classical cases by Losev in \cite[Theorem 1.2.3(1)]{L3} (see also \cite[\S6]{L1}). Goodwin-R\"{o}hrle-Ubly \cite{GRU} proved that the finite $W$-algebras associated with exceptional Lie algebras $E_6,E_7,F_4,G_2$, or $E_8$ with $e$ not rigid, admit one-dimensional representations (see also \cite{P7}). Finally Premet solved this problem completely in \cite{P9}.
\subsection{}
The theory of finite $W$-superalgebras was developed in the same time. In the work of Sole and Kac \cite{SK}, finite $W$-superalgebras were defined in terms of BRST cohomology under the background of vertex algebras and quantum reduction. The theory of finite $W$-superalgebras for the queer Lie superalgebras over an algebraically closed field of characteristic $p>2$ was first introduced and discussed by Wang and Zhao in \cite{WZ2}, then studied by Zhao over the field of complex numbers in \cite{Z2}. The topics on finite $W$-superalgebras attracted many researchers, and the structure theory of $W$-superalgebras is developed in various articles (cf. \cite{BBG2}, \cite{BBG3}, \cite{Peng2}, \cite{Peng3}, \cite{PS} and \cite{PS2}, $\sl{etc.}$).
In mathematical physics, $W$-(super)algebras are divided into four types: classical affine, classical finite, quantum affine, and quantum finite $W$-(super)algebras. These types of algebras are endowed with Poisson vertex algebras, Poisson algebras, vertex algebras, and associative algebras structures, respectively. In the present paper, finite $W$-superalgebras will be referred to the so-called quantum finite $W$-(super)algebras.
Apart from the ones associated with principal nilpotent elements, the most elementary examples of finite $W$-superalgebras are those ones corresponding to (even) minimal nilpotent elements of a given Lie superalgebra $\ggg$. For the counterpart associated with Lie algebras, Premet described generators of finite $W$-algebras associated with minimal nilpotent elements in \cite{P3}. Under the background of vertex algebras and quantum reduction, similar results for classical affine $W$-algebras can be found in \cite{SKV} and \cite{suh2}
For the case of basic Lie superalgebras, the related study associated with (even) minimal nilpotent elements is made mainly in the context of vertex operators and quantum reduction (see \cite{AKFPP}, \cite{KRW}, \cite{KW}, {\sl{etc}}.).
Recently, Suh described the generators of a classical affine $W$-(super)algebra associated with a minimal nilpotent element explicitly in \cite{suh}, and also the ones of quantum finite $W$-superalgebras analogously. It is worthwhile reminding that there are some errors in the presentation of generators and their relations of finite $W$-superalgebras in \cite{suh}. In the present paper, we will rewrite these generators and their commutators in an analogue of Premet's strategy of finite $W$-algebras case in \cite{P3}.
\subsection{}
In \cite{WZ}, the authors initiated the study of modular representations of basic Lie superalgebras over an algebraically closed field of positive characteristic, formulating the super Kac-Weisfeiler property for those Lie superalgebras as well as presenting the definition of modular $W$-superalgebras.
\subsection{}
Based on Premet's and Wang-Zhao's work as mentioned above, in \cite{ZS2} we presented the PBW theorem for the finite $W$-superalgebras over $\bbf$ ($\bbf=\bbc$ or $\bbk$ with characteristic $p\gg0$), which shows that the construction of finite $W$-superalgebras can be divided into two cases by virtue of the so-called judging parity of the dimension of certain specific subspace for basic Lie superalgebra $\mathfrak{g_\bbf}$. The situation of finite $W$-superalgebras is significantly different from that of finite $W$-algebras at the odd judging parity.
To be explicit, for a given complex basic classical Lie superalgebra ${\ggg}={\ggg}_{\bar0}+{\ggg}_{\bar1}$ and a nilpotent element $e\in\ggg_\bz$ (thereby a linear function $\chi$ in $\ggg^*_\bz$. see \S\ref{0.1.1}),
one has a so-called $\chi$-admissible algebra $\mmm$ (see (\ref{admissible alg})).
Consider a generalized Gelfand-Graev ${\ggg}$-module associated with $\chi$
$$Q_\chi:=U({\ggg})\otimes_{U(\mathfrak{m})}{\bbc}_\chi,$$
where ${\bbc}_\chi={\bbc}1_\chi$ is a one-dimensional $\mathfrak{m}$-module such that $x.1_\chi=\chi(x)1_\chi$ for all $x\in\mathfrak{m}$.
A finite $W$-superalgebra $U(\ggg,e)$ is by definition equal to $(\text{End}_{\ggg}Q_{\chi})^{\text{op}}$ which is isomorphic to $Q_{\chi}^{\text{ad}\,{\mmm}}$ (see \cite[Theorem 2.12]{ZS2}),
where $Q_{\chi}^{\text{ad}\,{\mmm}}$ is the invariant subalgebra of $Q_{\chi}$ under the adjoint action of ${\mmm}$. A PBW theorem of $U(\ggg,e)$ (see \cite[Theorem 0.1]{ZS2}) shows that the structure of $U(\ggg,e)$ is crucially dependent on the parity of a discriminant number $\mathsf{r}$ (the meaning of this notation can be seen in the above of (\ref{extend admiss alg})). The parity of $\mathsf{r}$ is therefore called the judging parity.
\subsection{} When we turn to finite-dimensional representations of finite $W$-superalgebras over complex numbers, their minimal dimensions will be crucial to small representations of modular Lie superalgebras.
Under an assumption on the minimal dimensions of representations for complex finite $W$-superalgebrs, we proved in \cite[Theorem 1.6]{ZS4} the accessibility of lower-bounds of dimensions for modular representations of basic Lie superalgebras (see the next subsection). Such an assumption is also predicted to be true, as a conjecture listed below (as an analogy of Premet's work on finite $W$-algebras).
\begin{conj}\label{conjectureold}(\cite{ZS4})
Let ${\ggg}$ be a basic Lie superalgebra over ${\bbc}$. Then the following statements hold:
\begin{itemize}
\item[(1)] when $\sfr$ is even, the finite $W$-superalgebra $U({\ggg},e)$ affords a one-dimensional representation;
\item[(2)] when $\sfr$ is odd, the finite $W$-superalgebra $U({\ggg},e)$ affords a two-dimensional representation.
\end{itemize}
\end{conj}
\iffalse
For the ordinary finite $W$-algebra counterpart of the above issue, Premet proposed a conjecture that finite $W$-algebras over ${\bbc}$ affords a one-dimensional representation, and which has already been confirmed (cf. \cite{P7} and \cite{P9}).
However, when we turn to the study of the finite $W$-superalgebra case, the tool available is very limited.
\fi
For the case ${\ggg}$ is of type $A(m,n)$, Conjecture \ref{conjectureold} was confirmed in \cite[Proposition 4.7]{ZS4}, which was accomplished by conversion from the verification of the attainableness of lower-bounds of modular dimensions for basic Lie superalgebras of the same type by some direct computation; see \cite{ZS3} for more details. For the case ${\ggg}$ is of type $B(0,n)$ with $e$ being a regular nilpotent element in ${\ggg}$, we certified Conjecture \ref{conjectureold} in \cite[Proposition 5.8]{ZS4}. In the present paper, we will certify this conjecture for minimal nilpotent elements.
\subsection{}\label{1.5} Let us recall the lower bounds of dimensions in modular representation for basic Lie superalgebras. Let $(\cdot,\cdot)$ be a bilinear form on ${\ggg}_{\bbk}$ which is induced from that on ${\ggg}$, and $\chi\in({\ggg}_{\bbk})^*_{\bar0}$ be the nilpotent $p$-character of ${\ggg}_{\bbk}$ corresponding to $\bar e\in ({\ggg}_{\bbk})_{\bar0}$ such that $\chi(\bar y)=(\bar e,\bar y)$ for any $\bar y\in{\ggg}_{\bbk}$; where $\bar{e}=e\otimes1$ is obtained from $e\in\ggg$ by ``reduction modulo $p$" .
Set $d_0=\text{dim}\,({\ggg}_{\bbk})_{\bar 0}-\text{dim}\,({\ggg}_{\bbk}^{\bar e})_{\bar 0}$ and $d_1=\text{dim}\,({\ggg}_{\bbk})_{\bar 1}-\text{dim}\,({\ggg}_{\bbk}^{\bar e})_{\bar 1}$, where ${\ggg}_{\bbk}^{\bar e}$ denotes the centralizer of $\bar e$ in ${\ggg}_{\bbk}$. For any real number $a\in\mathbb{R}$, let $\lfloor a\rfloor$ denote the least integer upper bound of $a$. In \cite[Theorem 5.6]{WZ}, Wang-Zhao showed that the dimension of any irreducible representation of $\ggg_{\bbk}$ is divisible by the number $p^{\frac{d_0}{2}}2^{\lfloor\frac{d_1}{2}\rfloor}$. With this number, we partially answered the question whether there exist modules of dimensions equal to such a number (see \cite[Theorems 1.5 and 1.6]{ZS4})
\iffalse
Theorem as follows.
\begin{theorem}\label{intromain-33}(\cite{ZS4})
Let ${\ggg}_{\bbk}$ be a basic Lie superalgebra over ${\bbk}=\overline{\mathbb{F}}_p$, and let $\chi\in({\ggg}_{\bbk})^*_{\bar0}$ be a nilpotent $p$-character, with respect to the element $\bar e\in(\ggg_\bbk)_\bz$ as described above. If the corresponding finite $W$-superalgebra $U({\ggg},e)$ over ${\bbc}$ affords a one-dimensional (resp. two-dimensional) representation when $\sfr$ is even (resp. odd), then for $p\gg0$ the reduced enveloping algebra $U_\chi({\ggg}_{\bbk})$ admits irreducible representations of dimension $p^{\frac{d_0}{2}}2^{\lfloor\frac{d_1}{2}\rfloor}$.
\end{theorem}
For the general case, let $\xi\in({\ggg}_{\bbk})^*_{\bar0}$ be any $p$-character of ${\ggg}_{\bbk}$ corresponding to an element $\bar x\in ({\ggg}_{\bbk})_{\bar0}$ such that $\xi(\bar y)=(\bar x,\bar y)$ for all $\bar y\in{\ggg}_{\bbk}$. Set $d'_0=\text{dim}\,({\ggg}_{\bbk})_{\bar 0}-\text{dim}\,({\ggg}_{\bbk}^{\bar x})_{\bar 0}$ and $d'_1=\text{dim}\,({\ggg}_{\bbk})_{\bar 1}-\text{dim}\,({\ggg}_{\bbk}^{\bar x})_{\bar 1}$, where ${\ggg}_{\bbk}^{\bar x}$ denotes the centralizer of $\bar x$ in ${\ggg}_{\bbk}$. A more general formulation can be presented as belo
\begin{theorem}\label{intromain-22}(\cite{ZS4})
Let ${\ggg}_{\bbk}$ be a basic Lie superalgebra over ${\bbk}=\overline{\mathbb{F}}_p$, and let $\xi\in({\ggg}_{\bbk})^*_{\bar0}$. If Conjecture \ref{conjectureold} establishes for all the basic Lie superalgebras over $\bbc$ excluding type $D(2,1;a)$ with $a\notin\overline{\mathbb{Q}}$, then for $p\gg0$ the reduced enveloping algebra $U_\xi({\ggg}_{\bbk})$ admits irreducible representations of dimension $p^{\frac{d'_0}{2}}2^{\lfloor\frac{d'_1}{2}\rfloor}$.
\end{theorem}
\fi
\vskip0.3cm
The main purpose of the present paper is to certify Conjecture \ref{conjectureold} for the case when $e$ is an (even) minimal nilpotent element, and thereby to show the accessibility of the lower bounds of dimensions in the modular representations of basic Lie superalgebras in this case (see the forthcoming Theorem \ref{intromainminnimalf}).
\subsection{} Recall that in \cite[Remark 70]{W}, Wang introduced another definition of finite $W$-superalgebra $W'_\chi:=Q_\chi^{\text{ad}\,{\mmm}'}$, where $\mmm'$ is the so-called extended $\chi$-admissible algebra which is either a one-dimensional extension of $\mmm$, or $\mmm$ itself, dependent on the judging parity (see (\ref{extend admiss alg})).
We will call it a refined $W$-superalgebra in the present paper. By definition, the refined $W$-superalgebra $W'_\chi$ is a subalgebra of the finite $W$-superalgebra $U(\ggg,e)$.
In this paper, we will take use of the refined $W$-superalgebras, instead of our original finite $W$-superalgebras, which enables us to unify the related statements.
\subsection{} Let us introduce the main results in the present paper.
We first establish an isomorphism (Proposition \ref{important}) between the refined $W$-superalgebra $W_\chi'$ and the quantum finite $W$-superalgebra $W^{\text{fin}}(\ggg,e)$ introduced by Suh in \cite[Definition 4.3]{suh}. This enables us to take $W^{\text{fin}}(\ggg,e)$ as the refined $W$-superalgebra $W_\chi'$ for the following discussions.
Then we focus on the case with $e$ being a minimal nilpotent element in basic Lie superalgebra $\ggg$. We call a refined $W$-superalgebra minimal when the defining nilpotent element associated with $W_\chi'$ is minimal in $\ggg_\bz$. We can present a set of generators of minimal $W$-superalgebras as below, correcting and reformulating the ones in \cite[Propositions 5.3 and 5.4]{suh}.
\begin{prop}\label{ge}
Let $e$ be a minimal nilpotent element in $\ggg$. Suppose $v\in\ggg^e(0)$, $w\in\ggg^e(1)$, $C$ is a central element of $W_\chi'$, and set $s=\text{dim}\,\ggg(-1)_{\bar0}$ and $\sfr=\text{dim}\,\ggg(-1)_{\bar1}$. Then the followings are free generators of the refined $W$-superalgebra $W_\chi'$:
\begin{equation*}
\begin{split}
\Theta_v=&(v-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}z_\alpha[z_\alpha^*,v])\otimes1_\chi,\\
\Theta_w=&(w-\sum\limits_{\alpha\in S(-1)}z_\alpha[z_\alpha^*,w]+\frac{1}{3}(\sum\limits_{\alpha,\beta\in S(-1)}z_\alpha z_\beta[z_\beta^*,[z_\alpha^*,w]]-2[w,f]))\otimes1_\chi,\\
C=&(2e+\frac{h^2}{2}-(1+\frac{s-\sfr}{2})h+\sum\limits_{i\in I}(-1)^{|i|}a_ib_i
+2\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[e,z_\alpha^*]z_\alpha)\otimes1_\chi,
\end{split}
\end{equation*}where $\{z_\alpha^*\mid\alpha\in S(-1)\}$ and $\{z_\alpha\mid\alpha\in S(-1)\}$ are dual bases of $\ggg(-1)$ with respect to $\langle\cdot,\cdot\rangle=(e,[\cdot,\cdot])$, and $\{a_i\mid i\in I\}$ and $\{b_i\mid i\in I\}$ are dual bases of $\ggg^e(0)$ with respect to $(\cdot,\cdot)$.
\end{prop}
The proof of Proposition \ref{ge} will be given in \S\ref{3.1.3}. Then the commutators between the generators are presented as below.
\begin{theorem}\label{maiin1}
The minimal $W$-superalgebra is generated by the Casimir element $C$ and the subspaces $\Theta_{\ggg^e(i)}$ for $i=0,1$, as described in Proposition \ref{ge}, subject to the following relations:
\begin{itemize}
\item[(1)] $[\Theta_{v_1},\Theta_{v_2}]=\Theta_{[v_1,v_2]}$ for all $v_1, v_2\in\ggg^e(0)$;
\item[(2)] $[\Theta_{v},\Theta_{w}]=\Theta_{[v,w]}$ for all $v\in\ggg^e(0)$ and $w\in\ggg^e(1)$;
\item[(3)] $[\Theta_{w_1},\Theta_{w_2}]=\frac{1}{2}([w_1,w_2],f)(C-\Theta_{\text{Cas}}-c_0)-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*,w_2]^{\sharp}}\\ \hspace*{2.2cm} -(-1)^{|w_1||w_2|}\Theta_{[w_2,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_1]^{\sharp}})$
for all $w_1, w_2\in\ggg^e(1)$;
\item[(4)] $[C,W_\chi']=0$.
\end{itemize}
In (3), the meaning of the notation $\sharp$ will be explained in \eqref{xh}, and the constant $c_0$ is decided by the following equation:
\begin{equation*}
\begin{split}
c_0([w_1,w_2],f)=&\frac{1}{12}\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha||w_1|+|\beta||w_1|+|\alpha||\beta|}\otimes[[z_\beta,[z_\alpha,w_1]],[z_\beta^*,[z_\alpha^*,w_2]]]\\
&-\frac{3(s-\sfr)+4}{12}([w_1,w_2],f),
\end{split}
\end{equation*}
where $s=\text{dim}\,\ggg(-1)_{\bar0}$ and $\sfr=\text{dim}\,\ggg(-1)_{\bar1}$.
\end{theorem}
In virtue of these results, Proposition \ref{minimal nilpotent} says that minimal $W$-superalgebra affords a two-sided ideal of codimension one. As an immediate consequence, we obtain the main result as below
\begin{theorem}\label{intromainminnimalf}
Let ${\ggg}_{\bbk}$ be a basic Lie superalgebra over ${\bbk}=\overline{\mathbb{F}}_p$, and let $\chi\in({\ggg}_{\bbk})^*_{\bar0}$ be a nilpotent $p$-character, with respect to a minimal nilpotent element $e\in(\ggg_\bbk)_\bz$. If $p\gg0$, then the reduced enveloping algebra $U_\chi({\ggg}_{\bbk})$ admits irreducible representations of dimension $p^{\frac{d_0}{2}}2^{\lfloor\frac{d_1}{2}\rfloor}$.
\end{theorem}
The proof of the above theorems will be fulfilled in \S\ref{5.2.4} and \S\ref{5.2.5} respectively.
\subsection{}
The paper is organized as follows. In \S\ref{Backgrounds}, some basics on Lie superalgebras and finite $W$-superalgebras are recalled. In \S\ref{crw}, we first study the construction of refined reduced $W$-superalgebra $(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}$ over $\bbk$, and then reformulate the PBW theorem for refined $W$-superalgebra $W_\chi'$ over $\mathbb{C}$. In the new setting-up of refined $W$-superalgebras, we refine the conjecture \cite[Conjecture 1.3]{ZS4} in \S\ref{Refined W-superalgebras and the dimensional lower bounds}. We first introduce Conjecture \ref{conjecture22}, which is irrelevant to the judging parity we mentioned above. Under the assumption of Conjecture \ref{conjecture22}, we show that the lower bounds of dimensions in the modular representations of basic Lie superalgebras are attainable. In the end of \S\ref{Refined W-superalgebras and the dimensional lower bounds}, we introduce a variation on the definition of refined $W$-superalgebras, which will be applied in the next section. \S\ref{structure} is the main body of the present paper. In the first part of \S\ref{structure}, we introduce the explicit expression of the generators of lower Kazhdan degree for finite $W$-superalgebras associate with arbitrary nilpotent elements over $\mathbb{C}$, and also the commutators between them. Then in the second part of \S\ref{structure}, we completely determine the structure of minimal $W$-superalgebras. In the meantime, we complete the proof of Proposition \ref{ge} in \S\ref{3.1.3}, Theorem \ref{maiin1} in \S\ref{5.2.4} (modulo Proposition \ref{1commutator}), and Theorem \ref{intromainminnimalf} in \S\ref{5.2.5}. The concluding and lengthy section \S\ref{proof} will be devoted to the proof of Proposition \ref{1commutator} leading to Theorem \ref{maiin1} (3) by lots of computation,
which is postponed there from \S\ref{structure}.
\subsection{}
Throughout we work with the field of complex numbers ${\bbc}$, or the algebraically closed field ${\bbk}=\overline{\mathbb{F}}_p$ of positive characteristic $p$ as the ground field.
Let ${\bbz}_+$ be the set of all the non-negative integers in ${\bbz}$, and denote by ${\bbz}_2$ the residue class ring modulo $2$ in ${\bbz}$. A superspace is a ${\bbz}_2$-graded vector space $V=V_{\bar0}\oplus V_{\bar1}$, in which we call elements in $V_{\bar0}$ and $V_{\bar1}$ even and odd, respectively. Write $|v|\in{\bbz}_2$ for the parity (or degree) of $v\in V$, which is implicitly assumed to be ${\bbz}_2$-homogeneous. We will use the notation $\text{dim}V=\text{dim}V_{\bar0}+\text{dim}V_{\bar1}$. All Lie superalgebras ${\ggg}$ will be assumed to be finite-dimensional.
We consider vector spaces, subalgebras, ideals, modules, and submodules $etc.$ in the super sense throughout the paper.
\section{Preliminaries}\label{Backgrounds}
In this section, we will recall some knowledge on basic classical Lie superalgebras and finite $W$-(super)algebras for use in the sequel. We refer the readers to \cite{CW}, \cite{K} and \cite{K2} for Lie superalgebras, and \cite{P2}, \cite{P3}, \cite{P7}, \cite{W}, \cite{ZS2} and \cite{ZS4} for finite $W$-(super)algebras.
\subsection{Basic Lie superalgebras}
Following \cite[\S1]{CW}, \cite[\S2.3-\S2.4]{K}, \cite[\S1]{K2} and \cite[\S2]{WZ}, we recall the list of basic classical Lie superalgebras over $\bbf$ for $\bbf=\bbc$ or $\bbf=\bbk$.
These Lie superalgebras, with even parts being Lie algebras of reductive algebraic groups, are simple over $\bbf$ (the
general linear Lie superalgebras, though not simple, are also included), and they admit an even non-degenerate supersymmetric invariant bilinear form in the following sense.
\begin{defn}\label{form}
Let $V=V_{\bar0}\oplus V_{\bar1}$ be a $\mathbb{Z}_2$-graded space and $(\cdot,\cdot)$ be a bilinear form on $V$.
\begin{itemize}
\item[(1)] If $(a,b)=0$ for any $a\in V_{\bar0}, b\in V_{\bar1}$, then $(\cdot,\cdot)$ is called even.
\item[(2)] If $(a,b)=(-1)^{|a||b|}(b,a)$ for any homogeneous elements $a,b\in V$, then $(\cdot,\cdot)$ is called supersymmetric.
\item[(3)] If $([a,b],c)=(a,[b,c])$ for any homogeneous elements $a,b,c\in V$, then $(\cdot,\cdot)$ is called invariant.
\item[(4)] If one can conclude from $(a,V)=0$ that $a=0$, then $(\cdot,\cdot)$ is called non-degenerate.
\end{itemize}
\end{defn}
Note that when $\bbf=\bbk$ is a field of characteristic $p>0$, there are restrictions on $p$, as shown for example in \cite[Table 1]{WZ}. So we have the following list
\vskip0.3cm
\begin{center}\label{Table 1}
({\sl{Table 1}}): basic classical Lie superalgebras over $\bbk$
\vskip0.3cm
\begin{tabular}{ccc}
\hline
$\frak{g}_\bbk$ & $\ggg_{\bar 0}$ & Restriction of $p$ when $\bbf=\bbk$\\
\hline
$\frak{gl}(m|n$) & $\frak{gl}(m)\oplus \frak{gl}(n)$ &$p>2$ \\
$\frak{sl}(m|n)$ & $\frak{sl}(m)\oplus \frak{sl}(n)\oplus \bbk$ & $p>2, p\nmid (m-n)$ \\
$\frak{osp}(m|n)$ & $\frak{so}(m)\oplus \frak{sp}(n)$ & $p>2$ \\
$\text{D}(2,1,\bar a)$ & $\frak{sl}(2)\oplus \frak{sl}(2)\oplus \frak{sl}(2)$ & $p>3$ \\
$\text{F}(4)$ & $\frak{sl}(2)\oplus \frak{so}(7)$ & $p>15$ \\
$\text{G}(3)$ & $\frak{sl}(2)\oplus \text{G}_2$ & $p>15$ \\
\hline
\end{tabular}
\end{center}
\vskip0.3cm
Throughout the paper, we will simply call all $\ggg_\bbf$ listed above {\sl{``basic Lie superalgebras"}} for both $\bbf=\bbc$ and $\bbf=\bbk$.
\subsection{Finite $W$-superalgebras over the field of complex numbers}\label{background}
\subsubsection{}\label{0.1.1}
Let ${\ggg}$ be a basic Lie superalgebra over ${\bbc}$, and $\mathfrak{h}$ be a typical Cartan subalgebra of ${\ggg}$. Let $\Phi$ be a root system of ${\ggg}$ relative to $\mathfrak{h}$ whose simple root system $\Delta=\{\alpha_1,\cdots,\alpha_l\}$ is distinguished (cf. \cite[Proposition 1.5]{K2}). By \cite[\S3.3]{FG} we can choose a Chevalley basis $B=\{e_\gamma\mid\gamma\in\Phi\}\cup\{h_\alpha\mid\alpha\in\Delta\}$ of ${\ggg}$ excluding the case $D(2,1;a)$ with $a\notin{\bbz}$ (in the case $D(2,1;a)$ with $a\notin\bbz$ being an algebraic number, one needs to adjust the definition of Chevalley basis by changing $\bbz$ to the $\bbz$-algebra generated by $(a)$, in the range of construction constants; see \cite[\S3.1]{Gav}). Let ${\ggg}_{\bbz}$ denote the Chevalley ${\bbz}$-form in ${\ggg}$ and $U_{\bbz}$ the Kostant ${\bbz}$-form of $U({\ggg})$ associated with $B$. Given a ${\bbz}$-module $V$ and a ${\bbz}$-algebra $A$, we write $V_A:=V\otimes_{\bbz}A$.
Let $G$ be an algebraic supergroup with $\Lie(G)=\ggg$, and let $G_\ev$ be a subgroup scheme of $G$ such that $G_\ev$ is an ordinary connected reductive group with $\Lie(G_\ev)=\ggg_\bz$. Denote the corresponding super Harish-Chandra pair by $(G_\ev,\ggg)$. For a given nilpotent element $e\in{\ggg}_{\bar0}$, by Dynkin-Kostant theory one can further assume that $e$ is in $({\ggg}_{\bbz})_{\bar{0}}$ up to an $\text{Ad}\,G_{\text{ev}}$-action. Choose $f,h\in({\ggg}_\mathbb{Q})_{\bar{0}}$ such that $(e,h,f)$ is an $\mathfrak{sl}_2$-triple in ${\ggg}$. Let $(\cdot,\cdot)$ be an even nondegenerate supersymmetric invariant bilinear form, under which the Chevalley basis $B$ of ${\ggg}$ take values in $\mathbb{Q}$, and $(e,f)=1$. Define $\chi\in{\ggg}^{*}$ by letting $\chi(x)=(e,x)$ for all $x\in{\ggg}$.
A commutative ring $A$ is called {\sl admissible} if $A$ is a finitely generated ${\bbz}$-subalgebra of ${\bbc}$, $(e,f)\in A^{\times}(=A\backslash \{0\})$ and all bad primes of the root system of ${\ggg}$ and the determinant of the Gram matrix of ($\cdot,\cdot$) relative to a Chevalley basis of ${\ggg}$ are invertible in $A$. It is clear by the definition that every admissible ring is a Noetherian domain. Given a finitely generated ${\bbz}$-subalgebra $A$ of ${\bbc}$, denote by $\text{Specm}\,A$ the maximal spectrum of $A$. It is well known that for every element $\mathfrak{P}\in\text{Specm}\,A$, the residue field $A/\mathfrak{P}$ is isomorphic to $\mathbb{F}_{q}$, where $q$ is a $p$-power depending on $\mathfrak{P}$. We denote by $\Pi(A)$ the set of all primes $p\in\mathbb{N}$ that occur in this way, and the set $\Pi(A)$ contains almost all primes in $\mathbb{N}$. We denote by ${\ggg}_A$ the $A$-submodule of ${\ggg}$ generated by the Chevalley basis $B$.
Let ${\ggg}(i)=\{x\in{\ggg}\mid[h,x]=ix\}$, then ${\ggg}=\bigoplus_{i\in{\bbz}}{\ggg}(i)$. By $\mathfrak{sl}_2$-theory, all subspaces ${\ggg}(i)$ are defined over $\mathbb{Q}$. Also, $e\in{\ggg}(2)_{\bar{0}}$ and $f\in{\ggg}(-2)_{\bar{0}}$. Define a symplectic (resp. symmetric) bilinear form $\langle\cdot,\cdot\rangle$ on the ${\bbz}_2$-graded subspace ${\ggg}(-1)_{\bar{0}}$ (resp. ${\ggg}(-1)_{\bar{1}}$) by $\langle x,y\rangle:=(e,[x,y])=\chi([x,y])$ for all $x,y\in{\ggg}(-1)_{\bar0}~(\text{resp.}\,x,y\in{\ggg}(-1)_{\bar1})$.
There exist bases $\{u_1,\cdots,u_{s}\}$ of ${\ggg}(-1)_{\bar0}$ and $\{v_1,\cdots,v_\sfr\}$ of ${\ggg}(-1)_{\bar1}$ contained in ${\ggg}_\mathbb{Q}:={\ggg}_A\otimes_{A}\mathbb{Q}$ such that $\langle u_i, u_j\rangle =i^*\delta_{i+j,s+1}$ for $1\leqslant i,j\leqslant s$, where $i^*=\left\{\begin{array}{ll}-1&\text{if}~1\leqslant i\leqslant \frac{s}{2};\\ 1&\text{if}~\frac{s}{2}+1\leqslant i\leqslant s\end{array}\right.$, and $\langle v_i,v_j\rangle=\delta_{i+j,\sfr+1}$ for $1\leqslant i,j\leqslant \sfr$.
We can introduce the so-called ``$\chi$-admissible algebra" as below
\begin{align}\label{admissible alg}
\mathfrak{m}:=\bigoplus_{i\leqslant -2}{\ggg}(i)\oplus{\ggg}(-1)^{\prime}
\end{align}
with ${\ggg}(-1)^{\prime}={\ggg}(-1)^{\prime}_{\bar0}\oplus{\ggg}(-1)^{\prime}_{\bar1}$, where ${\ggg}(-1)^{\prime}_{\bar0}$ is the ${\bbc}$-span of $u_{{s\over 2}+1},\cdots,u_{s}$ and
${\ggg}(-1)^{\prime}_{\bar1}$ is the ${\bbc}$-span of $v_{\frac{\sfr}{2}+1},\cdots,v_\sfr$ (resp. $v_{\frac{\sfr+3}{2}},\cdots,v_\sfr$) when $\sfr:=\text{dim}\,{\ggg}(-1)_{\bar{1}}$ is even (resp. odd), then $\chi$ vanishes on the derived subalgebra of $\mathfrak{m}$. Define ${\ppp}:=\bigoplus_{i\geqslant 0}{\ggg}(i)$. We also have an extended $\chi$-admissible algebra as below
\begin{align}\label{extend admiss alg}
\mathfrak{m}^{\prime}:=\left\{\begin{array}{ll}\mathfrak{m}&\text{if}~\sfr~\text{is even;}\\
\mathfrak{m}\oplus {\bbc}v_{\frac{\sfr+1}{2}}&\text{if}~\sfr~\text{is odd.}\end{array}\right.
\end{align}
Write ${\ggg}^e$ for the centralizer of $e$ in ${\ggg}$ and denote by $d_i:=\text{dim}\,{\ggg}_i-\text{dim}\,{\ggg}^e_i$ for $i\in{\bbz}_2$, then \cite[Theorem 4.3]{WZ} shows that $\sfr$ and $d_1$ always have the same parity. This parity is a crucial factor deciding the structure of finite $W$-superalgebras (cf. \cite[Theorem 4.5]{ZS2}).
After enlarging $A$ one can assume that ${\ggg}_A=\bigoplus_{i\in{\bbz}}{\ggg}_A(i)$, and each ${\ggg}_A(i):={\ggg}_A\cap{\ggg}(i)$ is freely generated over $A$ by a basis of the vector space ${\ggg}(i)$. Then $\{u_1,\cdots,u_{s}\}$ and $\{v_1,\cdots,v_\sfr\}$ are free basis of $A$-modules ${\ggg}_A(-1)_{\bar0}$ and ${\ggg}_A(-1)_{\bar1}$, respectively. It is obvious that
$\mathfrak{m}_A:={\ggg}_A\cap\mathfrak{m}$, $\mathfrak{m}^{\prime}_A:={\ggg}_A\cap\mathfrak{m}^{\prime}$ and ${\ppp}_A:={\ggg}_A\cap{\ppp}$ are free $A$-modules and direct summands of ${\ggg}_A$. Moreover, one can assume $e,f\in({\ggg}_A)_{\bar0}$ after enlarging $A$ possibly; $[e,{\ggg}_A(i)]$ and $[f,{\ggg}_A(i)]$ are direct summands of ${\ggg}_A(i+2)$ and ${\ggg}_A(i-2)$ respectively, and ${\ggg}_A(i+2)=[e,{\ggg}_A(i)]$ for each $i\geqslant -1$ by $\mathfrak{sl}_2$-theory.
\subsubsection{}\label{2.2.2}
Define a generalized Gelfand-Graev ${\ggg}$-module associated with $\chi$ by $$Q_\chi:=U({\ggg})\otimes_{U(\mathfrak{m})}{\bbc}_\chi,$$
where ${\bbc}_\chi={\bbc}1_\chi$ is a one-dimensional $\mathfrak{m}$-module such that $x.1_\chi=\chi(x)1_\chi$ for all $x\in\mathfrak{m}$. The super structure of $Q_\chi$ is dependent on the parity of $\bbc_\chi$, which is indicated to be even hereafter.
Define the {\sl finite $W$-superalgebra over $\mathbb{C}$} by $$U({\ggg},e):=(\text{End}_{\ggg}Q_{\chi})^{\text{op}},$$
where $(\text{End}_{\ggg}Q_{\chi})^{\text{op}}$ denotes the opposite algebra of the endomorphism algebra of ${\ggg}$-module $Q_{\chi}$.
Let $I_\chi$ denote the ${\bbz}_2$-graded ideal in $U({\ggg})$ generated by all $x-\chi(x)$ with $x\in\mathfrak{m}$. The fixed point space $(U({\ggg})/I_\chi)^{\ad\,\mmm}$ carries a natural algebra structure given by $(x+I_\chi)\cdot(y+I_\chi):=(xy+I_\chi)$ for all $x,y\in U({\ggg})$. Then $Q_\chi\cong U({\ggg})/I_\chi$ as ${\ggg}$-modules via the ${\ggg}$-module map sending $1+I_\chi$ to $1_\chi$, and $Q_{\chi}^{\ad\,\mmm}\cong U({\ggg},e)$ as $\bbc$-algebras. Any element of $U({\ggg},e)$ is uniquely determined by its effect on the generator $1_\chi\in Q_\chi$, and the canonical isomorphism between $U({\ggg},e)$ and $Q_{\chi}^{\ad\,\mmm}$ is given by $u\mapsto u(1_\chi)$ for any $u\in U({\ggg},e)$. In what follows we will often identify $Q_\chi$ with $U({\ggg})/I_\chi$ and $U({\ggg},e)$ with $Q_{\chi}^{\ad\,\mmm}$.
Let $w_1,\cdots, w_c$ be a basis of $\ggg$ over $\bbc$. Let $U({\ggg})=\bigcup_{i\in{\bbz}}\text{F}_iU({\ggg})$ be a filtration of $U({\ggg})$, where $\text{F}_iU({\ggg})$ is the ${\bbc}$-span of all $w_1\cdots w_c$ with $w_1\in{\ggg}(j_1),\cdots,w_c\in{\ggg}(j_c)$ and $(j_1+2)+\cdots+(j_c+2)\leqslant i$. This filtration is called {\sl Kazhdan filtration}. The Kazhdan filtration on $Q_{\chi}$ is defined by $\text{F}_iQ_{\chi}:=\pi(\text{F}_iU({\ggg}))$ with $\pi:U({\ggg})\twoheadrightarrow U({\ggg})/I_\chi$ being the canonical homomorphism, which makes $Q_{\chi}$ into a filtered $U({\ggg})$-module. Then there is an induced Kazhdan filtration
$\text{F}_i U({\ggg},e)$ on the subspace $U({\ggg},e)=Q_{\chi}^{\ad\,\mmm}$ of $Q_{\chi}$ such that $\text{F}_j U({\ggg},e)=0$
unless $j\geqslant0$.
Choose a basis $x_1,\cdots,x_l,x_{l+1},\cdots,x_m\in({\ppp}_A)
_{\bar{0}}, y_1,\cdots, y_q, y_{q+1}, \cdots,y_n\in({\ppp}_A)_{\bar{1}}$ of the free $A$-module ${\ppp}_A=\bigoplus_{i\geqslant 0}{\ggg}_A(i)$ such that
(a) $x_i\in{\ggg}_A(k_i)_{\bar{0}}, y_j\in{\ggg}_A(k'_j)_{\bar{1}}$, where $k_i,k'_j\in{\bbz}_+$ with $1\leqslant i\leqslant m$ and $1\leqslant j\leqslant n$;
(b) $x_1,\cdots,x_l$ is a basis of $({\ggg}_A)^e_{\bar{0}}$ and $y_1,\cdots,y_q$ is a basis of $({\ggg}_A)^e_{\bar{1}}$;
(c) $x_{l+1},\cdots,x_m\in[f,({\ggg}_A)_{\bar{0}}]$ and $ y_{q+1},\cdots,y_n\in[f,({\ggg}_A)_{\bar{1}}]$.
For $k\in\mathbb{Z}_+$, define
\begin{equation*}
\begin{split}
\mathbb{Z}_+^k:=&\{(i_1,\cdots,i_k)\mid i_j\in\mathbb{Z}_+\},\\
\Lambda_k:=&\{(i_1,\cdots,i_k)\mid i_j\in{\bbz}_+,~0\leqslant i_j\leqslant p-1\},\\
\Lambda'_k:=&\{(i_1,\cdots,i_k)\mid i_j\in\{0,1\}\}
\end{split}
\end{equation*}
with $1\leqslant j\leqslant k$. For $\mathbf{i}=(i_1,\cdots,i_k)$ in $\mathbb{Z}_+^k$, $\Lambda_k$ or $\Lambda'_k$, set $|\mathbf{i}|=i_1+\cdots+i_k$. For any real number $a\in\mathbb{R}$, let $\lceil a\rceil$ denote the largest integer lower bound of $a$, and $\lfloor a\rfloor$ the least integer upper bound of $a$. Given $(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in{\bbz}^m_+\times\Lambda'_n\times{\bbz}^{\frac{s}{2}}_+\times\Lambda'_{\lfloor\frac{\sfr}{2}\rfloor}$, let $x^\mathbf{a}y^\mathbf{b}u^\mathbf{c}v^\mathbf{d}$ denote the monomial $x_1^{a_1}\cdots x_m^{a_m}y_1^{b_1}\cdots y_n^{b_n}u_1^{c_1}\cdots u_{\frac{s}{2}}^{c_{\frac{s}{2}}}v_1^{d_1}\cdots v_{\lfloor\frac{\sfr}{2}\rfloor}^{d_{\lfloor\frac{\sfr}{2}\rfloor}}$ in $U({\ggg})$. Set $Q_{\chi,A}:=U({\ggg}_A)\otimes_{U(\mathfrak{m}_A)}A_\chi$, where $A_\chi=A1_\chi$. It is obvious that $Q_{\chi,A}$ is a ${\ggg}_A$-stable $A$-lattice in $Q_{\chi}$ with $\{x^\mathbf{a}y^\mathbf{b}u^\mathbf{c}v^\mathbf{d}\otimes1_\chi\mid(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in{\bbz}^m_+\times\Lambda'_n\times{\bbz}^{\frac{s}{2}}_+\times\Lambda'_{\lfloor\frac{\sfr}{2}\rfloor}\}$
being a free basis. Given $(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in{\bbz}_+^m\times\Lambda'_n\times{\bbz}_+^{\frac{s}{2}}\times\Lambda'_{\lfloor\frac{\sfr}{2}\rfloor}$, set
\begin{equation*}
\begin{split}
|(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})|_e:=&\sum_{i=1}^ma_i(k_i+2)+\sum_{i=1}^nb_i(k'_i+2)+\sum_{i=1}^{\frac{s}{2}}c_i+\sum_{i=1}^{\lfloor\frac{\sfr}{2}\rfloor}d_i,\\
\text{wt}(\bar x^{\mathbf{a}}\bar y^\mathbf{b}\bar u^\mathbf{c}\bar v^\mathbf{d}):=&(\sum\limits_{i=1}^mk_ia_i)+(\sum\limits_{i=1}^nk'_ib_i)-|\mathbf{c}|-|\mathbf{d}|,
\end{split}
\end{equation*} which are called the $e$-degree and the weight of $x^{\mathbf{a}}y^\mathbf{b}u^\mathbf{c}v^\mathbf{d}$.
For any non-zero element $h\in Q_{\chi}^{\ad\,\mmm}$, write$$h=(\sum\limits_{|(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})|_e\leqslant n(h)}\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}
x^{\mathbf{a}}y^\mathbf{b}u^\mathbf{c}v^\mathbf{d})\otimes1_\chi,$$ where $n(h)$ is the highest $e$-degree of the terms in the linear expansion of $h$, and $\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}\neq0$ for at least one $(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})$ with $|(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})|_e=n(h)$.
For $k\in\mathbb{Z}_+$, put $\Lambda^{k}_{h}=\{(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\mid\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}\neq0\mbox{ and }|(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})|_e=k\}$ and set
\begin{align}\label{MaxLambda}
\Lambda^{\text{max}}_{h}:=
\{ (\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in\Lambda^{n( h)}_{h}\mid
\text{wt}(x^{\mathbf{a}}y^\mathbf{b} u^\mathbf{c}v^\mathbf{d}) \mbox{ takes its maximum value}\}.
\end{align}
This maximum value mentioned in (\ref{MaxLambda}) will be denoted by $N(h)$.
\subsection{Finite $W$-superalgebras in positive characteristic}\label{2.3}
\subsubsection{}\label{2.3.1}
Pick a prime $p\in\Pi(A)$ and denote by ${\bbk}=\overline{\mathbb{F}}_p$ the algebraic closure of $\mathbb{F}_p$. Since the bilinear form $(\cdot,\cdot)$ is $A$-valued on ${\ggg}_A$, it induces a bilinear form on the Lie superalgebra ${\ggg}_{\bbk}\cong{\ggg}_A\otimes_A{\bbk}$. For $x\in{\ggg}_A$, set $\bar{x}:=x\otimes1$, an element of ${\ggg}_{\bbk}$. To simplify notation we identify $e,f,h$ with the nilpotent elements $\bar{e}=e\otimes1,~\bar{f}=f\otimes1$ and $\bar{h}=h\otimes1$ in ${\ggg}_{\bbk}$, and $\chi$ with the linear function $(\bar e,\cdot)$ on ${\ggg}_{\bbk}$.
The Lie superalgebra ${\ggg}_{\bbk}$ carries a natural $p$-mapping $x\mapsto x^{[p]}$ for all $x\in({\ggg}_{\bbk})_{\bar0}$. For any $\xi\in({\ggg}_{\bbk})_{\bar{0}}^*$ we denote by $J_\xi$ the two-sided ideal of $U({\ggg}_{\bbk})$ generated by the even central elements $\{\bar x^p-\bar x^{[p]}-\xi(\bar x)^p\mid\bar x\in({\ggg}_{\bbk})_{\bar{0}}\}$. Then the quotient algebra $U_\xi({\ggg}_{\bbk}):=U({\ggg}_{\bbk})/J_\xi$ is called the reduced enveloping algebra with $p$-character $\xi$. We often regard $\xi\in{\ggg}_{\bbk}^*$ by letting $\xi(({\ggg}_{\bbk})_{\bar{1}})=0$.
\subsubsection{}\label{2.3.2}
For $i\in{\bbz}$, set ${\ggg}_{\bbk}(i):={\ggg}_A(i)\otimes_A{\bbk}$ and put $\mathfrak{m}_{\bbk}:=\mathfrak{m}_A\otimes_A{\bbk}$, then $\mathfrak{m}_{\bbk}$ is a restricted subalgebra of $\mathfrak{g}_{\bbk}$. Denote by $\mathfrak{m}'_{\bbk}:=\mathfrak{m}'_A\otimes_A{\bbk}$ and $\mathfrak{p}_{\bbk}:=\mathfrak{p}_A\otimes_A{\bbk}$. Due to our assumptions on $A$, the elements $\bar{x}_1,\cdots,\bar{x}_l$ and $\bar{y}_1,\cdots,\bar{y}_q$ form bases of the centralizer $({\ggg}^e_{\bbk})_{\bar{0}}$ and $({\ggg}^e_{\bbk})_{\bar{1}}$ of $e$ in ${\ggg}_{\bbk}$, respectively. \cite[\S4.1]{WZ} showed that the subalgebra $\mathfrak{m}_{\bbk}$ is $p$-nilpotent, and the linear function $\chi$ vanishes on the $p$-closure of $[\mathfrak{m}_{\bbk},\mathfrak{m}_{\bbk}]$. Set $Q_{\chi,{\bbk}}:=U({\ggg}_{\bbk})\otimes_{U(\mathfrak{m}_{\bbk})}{\bbk}_\chi$, where ${\bbk}_\chi=A_\chi\otimes_{A}{\bbk}={\bbk}1_\chi$. Clearly, ${\bbk}1_\chi$ is a one-dimensional $\mathfrak{m}_{\bbk}$-module with the property $\bar x.1_\chi=\chi(\bar x)1_\chi$ for all $\bar x\in\mathfrak{m}_{\bbk}$, and it is obvious that $Q_{\chi,{\bbk}}\cong Q_{\chi,A}\otimes_A{\bbk}$ as ${\ggg}_{\bbk}$-modules.
Set the ${\ggg}_{\bbk}$-module $Q_{\chi}^\chi:=Q_{\chi,{\bbk}}/J_\chi Q_{\chi,{\bbk}}$. Then we can define the {\sl reduced $W$-superalgebra} by $U_\chi({\ggg}_{\bbk},e):=(\text{End}_{{\ggg}_{\bbk}}Q_{\chi}^\chi)^{\text{op}}$. It follows from \cite[Proposition 2.21]{ZS2} that $U_\chi({\ggg}_{\bbk},e)\cong(Q_\chi^\chi)^{\text{ad}\,{\mmm}_{\bbk}}$ as ${\bbk}$-algebras.
Given $(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in\Lambda_m\times\Lambda'_n\times\Lambda_{\frac{s}{2}}\times\Lambda'_{\lfloor\frac{\sfr}{2}\rfloor}$, let $\bar x^\mathbf{a}\bar y^\mathbf{b}\bar u^\mathbf{c}\bar v^\mathbf{d}$ denote the monomial $\bar x_1^{a_1}\cdots\bar x_m^{a_m}\bar y_1^{b_1}\cdots\bar y_n^{b_n}\bar u_1^{c_1}\cdots\bar u_{\frac{s}{2}}^{c_{\frac{s}{2}}}\bar v_1^{d_1}\cdots\bar v_{\lfloor\frac{\sfr}{2}\rfloor}^{d_{\lfloor\frac{\sfr}{2}\rfloor}}$ in $U({\ggg}_{\bbk})$. Denote by $|(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})|_e$ and $\text{wt}(\bar x^{\mathbf{a}}\bar y^\mathbf{b}\bar u^\mathbf{c}\bar v^\mathbf{d})$ be the $e$-degree and the weight of $\bar x^{\mathbf{a}}\bar y^\mathbf{b}\bar u^\mathbf{c}\bar v^\mathbf{d}$ as defined in \S\ref{2.2.2}, respectively.
For any non-zero element $\bar h\in (Q_\chi^\chi)^{\text{ad}\,{\mmm}_{\bbk}}$ we let $n(\bar h)$, $\Lambda^{k}_{\bar h}$, $\Lambda_{\bar h}^{\text{max}}$ and $N(\bar h)$ have the same meaning as in \S\ref{2.2.2}.
\section{The structure theory of refined $W$-superalgebras}\label{crw}
In this section we will first introduce the definition of refined reduced $W$-superalgebras associated with a
basic Lie superalgebra $\ggg$ over $\bbk$, an algebraically closed field of prime characteristic $p$; and also refined $W$-superalgebras over $\mathbb{C}$. Then the structure theory of these algebras is studied. We mainly follow Premet's strategy on finite $W$-algebras \cite[\S3-\S4]{P2}, and also the method applied by Zeng-Shu on finite $W$-superalgebras \cite[\S3-\S4]{ZS2}, with a few modifications.
\subsection{The structure theory of refined reduced $W$-superalgebras over $\bbk$}\label{rrw}
This subsection is devoted to the refined reduced $W$-superalgebras over $\bbk$.
\begin{defn}\label{rewcc} Define the refined reduced $W$-superalgebra over $\bbk$ by
$$(Q_\chi^{\chi})^{\text{ad}\,{\mmm}'_\mathds{k}}:=(Q_{\chi,{\bbk}}/J_\chi Q_{\chi,{\bbk}})^{\text{ad}\,{\mmm}'_\mathds{k}}
\equiv\{\bar{y}\in Q_{\chi,{\bbk}}/J_\chi Q_{\chi,{\bbk}} \mid [a,y]\in J_\chi Q_{\chi,{\bbk}}, \forall a\in{\mmm}'_\mathds{k}\},$$
and $\bar{y}_1\cdot\bar{y}_2:=\overline{y_1y_2}$ for all $\bar{y}_1,\bar{y}_2\in (Q_\chi^{\chi})^{\text{ad}\,{\mmm}'_\mathds{k}}$.
\end{defn}
Retain the notations as in \S\ref{2.3}.
Now we will discuss the structure of refined reduced $W$-superalgebra over $\bbk$. First note that
\begin{lemma} \label{hw}
Let $\bar h\in (Q_\chi^{\chi})^{\text{ad}\,{\mmm}'_\mathds{k}}\backslash\{0\}$ and $(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in \Lambda^{\text{max}}_{\bar h}$. Then $\mathbf{a}\in\Lambda_{l}\times\{\mathbf{0}\}, \mathbf{b}\in\Lambda'_{q}\times\{\mathbf{0}\}$, $\mathbf{c}=\mathbf{0}$ and $\mathbf{d}=\mathbf{0}$.
\end{lemma}
\begin{proof}
Since $(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}$ is a subalgebra of the reduced $W$-superalgebra $U_\chi(\ggg_\mathds{k},e)\cong(Q_\chi^\chi)^{\text{ad}\,{\mmm}_\mathds{k}}$, it follows from \cite[Lemma 3.3]{ZS2} that $\mathbf{a}\in\Lambda_{l}\times\{\mathbf{0}\},~\mathbf{b}\in\Lambda'_{q}\times\{\mathbf{0}\}$ and $\mathbf{c}=\mathbf{0}$. Moreover, the sequence $\mathbf{d}$ satisfies:
(1) $\mathbf{d}=\mathbf{0}$ when $\sfr=\dim\ggg_{\bbk}(-1)_{\bar{1}}$ is even;
(2) $\mathbf{d}\in\{\mathbf{0}\}_{\frac{\sfr-1}{2}}\times\Lambda'_{1}$ when $\sfr=\dim\ggg_{\bbk}(-1)_{\bar{1}}$ is odd.
For the case when $\sfr$ is even, we have ${\mmm}'_\mathds{k}={\mmm}_\mathds{k}$ by definition, then $(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}=(Q_\chi^\chi)^{\text{ad}\,{\mmm}_\mathds{k}}\cong U_\chi(\ggg_\mathds{k},e)$, and the lemma readily follows from (1). So we just need to consider the case when $\sfr$ is odd. Our arguments mainly follow the proof of \cite[Lemma 3.3]{ZS2}, with some modifications. Now we proceed by steps.
Step 1: Denote by $\mathbf{e}_{\frac{\sfr+1}{2}}:=(\{\mathbf{0}\}_{\frac{\sfr-1}{2}},1)$.
We claim that $(\Lambda_{l}\times\{\mathbf{0}\},\Lambda'_{q}\times\{\mathbf{0}\},\mathbf{0},\mathbf{e}_{\frac{\sfr+1}{2}})\notin\Lambda^{\text{max}}_{\bar h}$ for any $\bar h\in(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}$.
Suppose the contrary. Let $(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in\Lambda^{d'}_{\bar h}$ where $d'\in\mathbb{Z}_+$. It follows from \cite[Lemma 3.1]{ZS2} and \cite[(3.7)]{ZS2} that
\begin{equation}\label{rhow}
(\bar v_{\frac{\sfr+1}{2}}\otimes\bar1_\chi)\cdot(\bar x^{\mathbf{a}}\bar y^\mathbf{b}\bar u^\mathbf{c}\bar v^\mathbf{d}\otimes\bar 1_\chi)=\sum_{\mathbf{i}\in\Lambda_{m}}\sum_{\mathbf{j}\in\Lambda'_{n}}\left(\begin{array}{@{\hspace{0pt}}c@{\hspace{0pt}}} \mathbf{a}\\ \mathbf{i}\end{array}\right)\bar x^{\mathbf{a}-\mathbf{i}}\bar y^{\mathbf{b}-\mathbf{j}}\cdot[\bar v_{\frac{\sfr+1}{2}}\bar x^{\mathbf{i}}\bar y^{\mathbf{j}}]\cdot \bar u^\mathbf{c}\bar v^\mathbf{d}\otimes\bar 1_\chi,
\end{equation}
where $\mathbf{a}\choose\mathbf{i}$$=\prod\limits_{l'=1}^m$$a_{l'}\choose i_{l'}$ and $$[\bar v_{\frac{\sfr+1}{2}}\bar x^{\mathbf{i}}\bar y^{\mathbf{j}}]=k_{1,b_1,j_1}\cdots k_{n,b_n,j_n}(-1)^{|\mathbf{i}|}(\text{ad}\,\bar y_n)^{j_n}\cdots(\text{ad}\,\bar y_1)^{j_1}(\text{ad}\,\bar x_m)^{i_m}\cdots(\text{ad}\,\bar x_1)^{i_1}(\bar v_{\frac{\sfr+1}{2}}),$$
in which the coefficients $k_{1,b_1,j_1},\cdots,k_{n,b_n,j_n}\in {\bbk}$ (note that $b_1,\cdots,b_n,j_1,\cdots,j_n\in\{0,1\}$) are defined by
$$k_{t',0,0}=1, k_{t',0,1}=0, k_{t',1,0}=(-1)^{1+j_1+\cdots+j_{{t'}-1}}, k_{t',1,1}=(-1)^{j_1+\cdots+j_{{t'}-1}}$$
with $1\leqslant {t'}\leqslant n$ ($j_0$ is interpreted as $0$). Moreover, the summation on the right side of \eqref{rhow} runs through all $(\mathbf{i},\mathbf{j})\in\Lambda_m\times\Lambda'_n$ such that $[\bar v_{\frac{\sfr+1}{2}}\bar x^{\mathbf{i}}\bar y^{\mathbf{j}}]$ is nonzero and $\text{wt}([\bar v_{\frac{\sfr+1}{2}}\bar x^{\mathbf{i}}\bar y^{\mathbf{j}}])\geqslant -1$.
Step 2: First note that
$[\bar v_{\frac{\sfr+1}{2}},\bar v_i]\otimes\bar1_\chi=1\otimes\chi([\bar v_{\frac{\sfr+1}{2}},\bar v_i])\bar1_\chi=0$ for all $1\leqslant i\leqslant \frac{\sfr-1}{2}$.
We continue the arguments case by case, according to the values of $\text{wt}([\bar v_{\frac{\sfr+1}{2}}\bar x^{\mathbf{i}}\bar y^{\mathbf{j}}])$ with respect to summation parameters $(\mathbf{i},\mathbf{j})\in \Lambda_m\times\Lambda'_n$.
(Case 1) wt$([\bar v_{\frac{\sfr+1}{2}}\bar x^{\mathbf{i}}\bar y^{\mathbf{j}}])\geqslant 0$.
We have $|\mathbf{i}|+|\mathbf{j}|\geqslant 1$. By Step 2(Case 1) in the proof of \cite[Lemma 3.3]{ZS2} we know that $\bar x^{\mathbf{a}-\mathbf{i}}\bar y^{\mathbf{b}-\mathbf{j}}\cdot[\bar v_{\frac{\sfr+1}{2}}\bar x^{\mathbf{i}}\bar y^{\mathbf{j}}]\cdot\bar u^\mathbf{c}\bar v^\mathbf{d}\otimes\bar 1_\chi$ is a linear combination of $\bar x^{\mathbf{i'}}\bar y^{\mathbf{j'}}\bar u^\mathbf{c}\bar v^\mathbf{d}\otimes\bar 1_\chi$ with
$$\text{wt}(\bar x^{\mathbf{i'}}\bar y^{\mathbf{j'}}\bar u^\mathbf{c}\bar v^\mathbf{d})=-1+\text{wt}(\bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\bar u^\mathbf{c}\bar v^\mathbf{d}),$$
and \[
\text{deg}_e(\bar x^{\mathbf{i'}}\bar y^{\mathbf{j'}}\bar u^\mathbf{c}\bar v^\mathbf{d})
\leqslant1+d'-2(|\mathbf{i}|+|\mathbf{j}|).
\]
(Case 2) wt$([\bar v_{\frac{\sfr+1}{2}}\bar x^{\mathbf{i}}\bar y^{\mathbf{j}}])=-1$. By Step 2(Case 2) in the proof of \cite[Lemma 3.3]{ZS2}, the vector $\bar x^{\mathbf{a}-\mathbf{i}}\bar y^{\mathbf{b}-\mathbf{j}}\cdot[\bar v_{\frac{\sfr+1}{2}}\bar x^{\mathbf{i}}\bar y^{\mathbf{j}}]\cdot \bar u^\mathbf{c}\bar v^\mathbf{d}\otimes\bar 1_\chi$ is a linear combination of $\bar x^{\mathbf{a}-\mathbf{i}}\bar y^{\mathbf{b}-\mathbf{j}}\bar u^\mathbf{i'}\bar v^\mathbf{j'}\otimes\bar 1_\chi$ with $|\mathbf{i'}|=|\mathbf{c}|\pm1, \mathbf{j'}=\mathbf{d}$, or $\mathbf{i'}=\mathbf{c}, |\mathbf{j'}|=|\mathbf{d}|\pm1$.
(a) If $|\mathbf{i'}|=|\mathbf{c}|+1, \mathbf{j'}=\mathbf{d}$, or $\mathbf{i'}=\mathbf{c}, |\mathbf{j'}|=|\mathbf{d}|+1$, then $|\mathbf{i}|+|\mathbf{j}|\geqslant 1$,
$$\text{wt}(\bar x^{\mathbf{a}-\mathbf{i}}\bar y^{\mathbf{b}-\mathbf{j}}\bar u^\mathbf{i'}\bar v^\mathbf{j'})=-1+\text{wt}(\bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\bar u^\mathbf{c}\bar v^\mathbf{d}),$$
and\[
\text{deg}_e(\bar x^{\mathbf{a}-\mathbf{i}}\bar y^{\mathbf{b}-\mathbf{j}}\bar u^\mathbf{i'}\bar v^\mathbf{j'})=1+d'-2(|\mathbf{i}|+|\mathbf{j}|).
\]
(b) If $|\mathbf{i'}|=|\mathbf{c}|-1, \mathbf{j'}=\mathbf{d}$, or $\mathbf{i'}=\mathbf{c}, |\mathbf{j'}|=|\mathbf{d}|-1$, then
$$\text{wt}(\bar x^{\mathbf{a}-\mathbf{i}}\bar y^{\mathbf{b}-\mathbf{j}}\bar u^\mathbf{i'}\bar v^\mathbf{j'})
=1+\text{wt}(\bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\bar u^\mathbf{c}\bar v^\mathbf{d}),$$
and\[\text{deg}_e(\bar x^{\mathbf{a}-\mathbf{i}}\bar y^{\mathbf{b}-\mathbf{j}}\bar u^\mathbf{i'}\bar v^\mathbf{j'})=-1+d'-2(|\mathbf{i}|+|\mathbf{j}|).
\]
For concluding our arguments, we adopt an auxiliary endomorphism.
For $i,j\in\mathbb{Z}$, take $\pi_{ij}$ to be an endomorphism of $Q_\chi^\chi$ defined via
\begin{align}\label{pi}
\pi_{ij}(\bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\bar u^\mathbf{c}\bar v^\mathbf{d}\otimes\bar1_\chi)=\begin{cases}\bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\bar u^\mathbf{c}\bar v^\mathbf{d}\otimes\bar 1_\chi &\mbox{ if }\text{deg}_e(\bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\bar u^\mathbf{c}\bar v^\mathbf{d})=i\\ &\mbox{ and } \text{wt}(\bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\bar u^\mathbf{c}\bar v^\mathbf{d})=j;\\0 &\mbox{ otherwise}.
\end{cases}
\end{align}
Step 3:
Now we proceed to complete the arguments by reducing contradictions in Step 1. Since $\bar h\in(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}$ and $\bar v_{\frac{\sfr+1}{2}}\in{\mmm}'_\mathds{k}$, then we have
\begin{equation}\label{mh}
[\bar v_{\frac{\sfr+1}{2}}\otimes\bar1_\chi,\bar h]=0
\end{equation}
by the definition. On the other hand, by \cite[(3.12)]{ZS2} we have
\begin{equation}\label{exten}
\begin{split}
&[\bar v_{\frac{\sfr+1}{2}}\otimes\bar1_\chi,\bar h]=(\bar v_{\frac{\sfr+1}{2}}\otimes\bar 1_\chi)\cdot\bar h-(-1)^{|\bar h|}\bar h\cdot(\bar v_{\frac{\sfr+1}{2}}\otimes\bar 1_\chi)\\
=&(\sum\limits_{(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})
\in\Lambda_{\bar h}^{n(\bar h)}}\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}(
\sum\limits_{i=1}^m(-1)^{\sum\limits_{j=1}^nb_j}\bar{a}_i\bar x^{\mathbf{a}-\mathbf{e}_i}\bar y^{\mathbf{b}}\cdot[\bar v_{\frac{\sfr+1}{2}},\bar x_i]\cdot \bar u^\mathbf{c}\bar v^\mathbf{d}\\
&+\sum\limits_{i=1}^n(-1)^{1+\sum\limits_{j=1}^ib_j}\bar x^{\mathbf{a}}\bar y^{\mathbf{b}-\mathbf{e}_i}\cdot[\bar v_{\frac{\sfr+1}{2}},\bar y_i]\cdot \bar u^\mathbf{c}\bar v^\mathbf{d})\\
&+\sum\limits_{(\mathbf{a},\mathbf{b},\mathbf{c},d_1,\cdots,d_{\frac{\sfr-1}{2}},0)
\in\Lambda_{\bar h}^{n(\bar h)}}\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},d_1,\cdots,d_{\frac{\sfr-1}{2}},0}(-1)^{\sum\limits_{j=1}^nb_j+\sum\limits_{j=1}^{\frac{r-1}{2}}d_{j}} \bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\cdot\bar u^{\mathbf{c}}\bar v_1^{d_1}\cdots\bar v_{\frac{\sfr-1}{2}}^{d_{\frac{\sfr-1}{2}}}\cdot v_{\frac{\sfr+1}{2}}\\
&+\sum\limits_{(\mathbf{a},\mathbf{b},\mathbf{c},d_1,\cdots,d_{\frac{\sfr-1}{2}},1)
\in\Lambda_{\bar h}^{n(\bar h)}}\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},d_1,\cdots,d_{\frac{\sfr-1}{2}},1}
(-1)^{\sum\limits_{j=1}^nb_j+\sum\limits_{j=1}^{\frac{r-1}{2}}d_{j}} \bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\cdot\bar u^{\mathbf{c}}\bar v_1^{d_1}\cdots\bar v_{\frac{\sfr-1}{2}}^{d_{\frac{\sfr-1}{2}}}v_{\frac{\sfr+1}{2}}\cdot v_{\frac{\sfr+1}{2}}\\&+
\sum\limits_{|(\mathbf{i},
\mathbf{j},\mathbf{k},\mathbf{l})|_e\leqslant n(\bar h)-2}\beta_{\mathbf{i},\mathbf{j},\mathbf{k},\mathbf{l}}\cdot\bar x^{\mathbf{i}}\bar y^{\mathbf{j}}\bar u^\mathbf{k}\bar v^\mathbf{l})\otimes\bar1_\chi\\
&-(\sum\limits_{(\mathbf{a},\mathbf{b},\mathbf{c},d_1,\cdots,d_{\frac{\sfr-1}{2}},0)
\in\Lambda_{\bar h}^{n(\bar h)}}\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},d_1,\cdots,d_{\frac{\sfr-1}{2}},0}
(-1)^{\sum\limits_{j=1}^nb_j+\sum\limits_{j=1}^{\frac{r-1}{2}}d_{j}}
\bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\cdot\bar u^{\mathbf{c}}\bar v_1^{d_1}\cdots\bar v_{\frac{\sfr-1}{2}}^{d_{\frac{\sfr-1}{2}}}\cdot v_{\frac{\sfr+1}{2}}\\
&+\sum\limits_{(\mathbf{a},\mathbf{b},\mathbf{c},d_1,\cdots,d_{\frac{\sfr-1}{2}},1)
\in\Lambda_{\bar h}^{n(\bar h)}}\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},d_1,\cdots,d_{\frac{\sfr-1}{2}},1}
(-1)^{\sum\limits_{j=1}^nb_j+\sum\limits_{j=1}^{\frac{r-1}{2}}d_{j}+1}
\bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\cdot\bar u^{\mathbf{c}}\bar v_1^{d_1}\cdots\bar v_{\frac{\sfr-1}{2}}^{d_{\frac{\sfr-1}{2}}}v_{\frac{\sfr+1}{2}}\cdot
\\& v_{\frac{\sfr+1}{2}}+\sum\limits_{|(\mathbf{i},
\mathbf{j},\mathbf{k},\mathbf{l})|_e\leqslant n(\bar h)-2}\beta_{\mathbf{i},\mathbf{j},\mathbf{k},\mathbf{l}}\cdot\bar x^{\mathbf{i}}\bar y^{\mathbf{j}}\bar u^\mathbf{k}\bar v^\mathbf{l})\otimes\bar1_\chi,
\end{split}
\end{equation}
where at least one $\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},d_1,\cdots,d_{\frac{\sfr-1}{2}},1}\neq0$ by our assumption in Step 1.
It is obvious that
\begin{equation*}
2\bar v_{\frac{\sfr+1}{2}}^2\otimes\bar1_\chi=[\bar v_{\frac{\sfr+1}{2}},\bar v_{\frac{\sfr+1}{2}}]\otimes
\bar1_\chi=1\otimes\chi([\bar v_{\frac{\sfr+1}{2}},\bar v_{\frac{\sfr+1}{2}}])\bar1_\chi=1\otimes\bar1_\chi,
\end{equation*}thus \eqref{exten} equals
\begin{equation}\label{exten2}
\begin{split}
[\bar v_{\frac{\sfr+1}{2}}\otimes\bar1_\chi,\bar h]
=&(\sum\limits_{(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})
\in\Lambda_{\bar h}^{n(\bar h)}}\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}(
\sum\limits_{i=1}^m(-1)^{\sum\limits_{j=1}^nb_j}\bar{a}_i\bar x^{\mathbf{a}-\mathbf{e}_i}\bar y^{\mathbf{b}}\cdot[\bar v_{\frac{\sfr+1}{2}},\bar x_i]\\
&\cdot \bar u^\mathbf{c}\bar v^\mathbf{d}+\sum\limits_{i=1}^n(-1)^{1+\sum\limits_{j=1}^ib_j}\bar x^{\mathbf{a}}\bar y^{\mathbf{b}-\mathbf{e}_i}\cdot[\bar v_{\frac{\sfr+1}{2}},\bar y_i]\cdot \bar u^\mathbf{c}\bar v^\mathbf{d}
\\&+\sum\limits_{|(\mathbf{a},
\mathbf{b},\mathbf{c},\mathbf{d})|_e=n(\bar h)-1}\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}
(-1)^{\sum\limits_{j=1}^nb_j+\sum\limits_{j=1}^{\frac{r-1}{2}}d_{j}} \bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\cdot\bar u^{\mathbf{c}}\bar v_1^{d_1}\cdots\bar v_{\frac{\sfr-1}{2}}^{d_{\frac{\sfr-1}{2}}}\\
&+\sum\limits_{|(\mathbf{i},
\mathbf{j},\mathbf{k},\mathbf{l})|_e\leqslant n(\bar h)-2}\beta_{\mathbf{i},\mathbf{j},\mathbf{k},\mathbf{l}}\cdot\bar x^{\mathbf{i}}\bar y^{\mathbf{j}}\bar u^\mathbf{k}\bar v^\mathbf{l})\otimes\bar1_\chi.
\end{split}
\end{equation}
When $[\bar v_{\frac{\sfr+1}{2}}\otimes\bar1_\chi,\bar h]$ is written as a linear combination of the canonical basis of $Q_\chi^\chi$, it is immediate from \eqref{exten2} and the arguments in Step 2 that the terms with $e$-degree $n(\bar h)-1$ and weight $N(\bar h)+1$ only occur as in (Case 2)(b) with wt$([\bar v_{\frac{\sfr+1}{2}}\bar x^{\mathbf{i}}\bar y^{\mathbf{j}}])=-1$. By \eqref{exten2} we have
\[\begin{array}{ll}
&\pi_{n(\bar h)-1,N(\bar h)+1}([\bar v_{\frac{\sfr+1}{2}}\otimes\bar1_\chi,\bar h])\\
=&\sum\limits_{|(\mathbf{a},
\mathbf{b},\mathbf{c},\mathbf{d})|_e=n(\bar h)-1}\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}
(-1)^{\sum\limits_{j=1}^nb_j+\sum\limits_{j=1}^{\frac{r-1}{2}}d_{j}} \bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\cdot\bar u^{\mathbf{c}}\bar v_1^{d_1}\cdots\bar v_{\frac{\sfr-1}{2}}^{d_{\frac{\sfr-1}{2}}}\otimes\bar1_\chi\\
\neq&0,
\end{array}\]
which contradicts to \eqref{mh}. This is to say, $(\Lambda_{l}\times\{\mathbf{0}\},\Lambda'_{q}\times\{\mathbf{0}\},\mathbf{0},
\mathbf{e}_{\frac{\sfr+1}{2}})\in\Lambda^{\text{max}}_{\bar h}$ is not possible. The proof is completed.
\end{proof}
In the following we will introduce a basis of the refined reduced $W$-superalgebra $(Q_\chi^{\chi})^{\text{ad}\,{\mmm}'_\mathds{k}}$ over $\mathds{k}$. We mainly follow Premet's strategy on finite $W$-algebras in \cite[Proposition 3.3]{P2}, with a few modifications.
\begin{prop}\label{reduced basis} For any $(\mathbf{a},\mathbf{b})\in\Lambda_l\times\Lambda'_q$ there is $\bar h_{\mathbf{a},\mathbf{b}}\in (Q_\chi^{\chi})^{\text{ad}\,{\mmm}'_\mathds{k}}$ such that $\Lambda_{\bar h_{\mathbf{a},\mathbf{b}}}^{\text{max}}=\{(\mathbf{a},\mathbf{b})\}$. The vectors $\{\bar h_{\mathbf{a},\mathbf{b}}\mid(\mathbf{a},\mathbf{b})\in\Lambda_l\times\Lambda'_q\}$ form a basis of $(Q_\chi^{\chi})^{\text{ad}\,{\mmm}'_\mathds{k}}$ over $\mathds{k}$.
\end{prop}
\begin{proof}
As $(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}\cong U_\chi(\ggg_\mathds{k},e)$ when $\sfr$ is even, the proposition follows from \cite[Proposition 3.5(1)]{ZS2} in this case.
Now we will consider the case when $\sfr$ is odd. For $k\in\mathbb{Z}_+$ let $H^k$ denote the ${\bbk}$-linear span of all $0\neq \bar h\in (Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}$ with $n(\bar h)\leqslant k$.
Given $(a,b)\in\mathbb{Z}_+^2$, let $H^{a,b}$ denote the subspace of $(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}$ spanned by $H^{a-1}$ and all $\bar h\in (Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}$ with $n(\bar h)=a,~N(\bar h)\leqslant b$. Order the elements in $\mathbb{Z}_+^2$ lexicographically. By construction, $H^{a,b}\subseteq H^{c,d}$ whenever $(a,b)\prec(c,d)$. Note that $(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}$ is finite-dimensional, then $(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}$ has basis $B:=\bigcup_{(i,j)}B_{i,j}$ such that $n(\mu)=i,~N(\mu)=j$ whenever $\mu\in B_{i,j}$.
For $i,j\in\mathbb{Z}$, take $\pi_{ij}$ to be an endomorphism of $Q_\chi^\chi$ defined via
\begin{align*}\label{pi}
\pi_{ij}(\bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\bar u^\mathbf{c}\bar v^\mathbf{d}\otimes\bar1_\chi)=\begin{cases}\bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\bar u^\mathbf{c}\bar v^\mathbf{d}\otimes\bar 1_\chi &\mbox{ if }\text{deg}_e(\bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\bar u^\mathbf{c}\bar v^\mathbf{d})=i\\ &\mbox{ and } \text{wt}(\bar x^{\mathbf{a}}\bar y^{\mathbf{b}}\bar u^\mathbf{c}\bar v^\mathbf{d})=j;\\0 &\mbox{otherwise}.
\end{cases}
\end{align*}
Define the linear map $\pi_B: (Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}\longrightarrow Q_\chi^\chi$ via $\pi_B(\mu)=\pi_{i,j}(\mu)$ for any $\mu\in B_{i,j}$. It follows from Lemma \ref{hw} that $\pi_B$ maps $(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}$ into the subspace of $U_\chi({\ggg}_{\bbk}^e)\otimes\bar1_\chi$ of $Q_\chi^\chi$. By construction, $\pi_B$ is injective.
By the same discussion as in \cite[Proposition 4.9]{ZS2}, one can verify that
\begin{equation}\label{mvm'}
(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}=[\bar v_{\frac{\sfr+1}{2}}\otimes\bar1_\chi,(Q_\chi^\chi)^{\text{ad}\,{\mmm}_\mathds{k}}].
\end{equation}
Let $((Q_\chi^\chi)^{\text{ad}\,{\mmm}_\mathds{k}})^{\text{ad}\,\bar v_{\frac{\sfr+1}{2}}}$ denote the invariant subspace of $(Q_\chi^\chi)^{\text{ad}\,{\mmm}_\mathds{k}}$ under the adjoint action of $\bar v_{\frac{\sfr+1}{2}}$. Obviously we have $$((Q_\chi^\chi)^{\text{ad}\,{\mmm}_\mathds{k}})^{\text{ad}\,\bar v_{\frac{\sfr+1}{2}}}=(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}$$ by definition, and we can conclude from \eqref{mvm'} that
\begin{equation*}
\begin{split}
\text{dim}\,(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}=&\text{dim}\,[\bar v_{\frac{\sfr+1}{2}}\otimes\bar1_\chi,(Q_\chi^\chi)^{\text{ad}\,{\mmm}_\mathds{k}}]=\text{dim}\,(Q_\chi^\chi)^{\text{ad}\,{\mmm}_\mathds{k}}
-\text{dim}\,((Q_\chi^\chi)^{\text{ad}\,{\mmm}_\mathds{k}})^{\text{ad}\,\bar v_{\frac{\sfr+1}{2}}}\\
=&\text{dim}\,(Q_\chi^\chi)^{\text{ad}\,{\mmm}_\mathds{k}}
-\text{dim}\,(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}},
\end{split}
\end{equation*}i.e.,
\begin{equation}\label{half}
\text{dim}\,(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}=\frac{1}{2}\text{dim}\,
(Q_\chi^\chi)^{\text{ad}\,{\mmm}_\mathds{k}}.
\end{equation}
Recall that in the proof of \cite[Proposition 3.5]{ZS2}, we showed that $$\text{dim}\,
(Q_\chi^\chi)^{\text{ad}\,{\mmm}_\mathds{k}}=p^{\text{dim}\,({\ggg}_{\bbk}^e)_{\bar{0}}}\cdot2^{\text{dim}\,({\ggg}_{\bbk}^e)_{\bar{1}}+1},$$ then it follows from \eqref{half} that $$\text{dim}\,(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}=p^{\text{dim}\,({\ggg}_{\bbk}^e)_{\bar{0}}}\cdot2^{\text{dim}\,({\ggg}_{\bbk}^e)_{\bar{1}}}
=\text{dim}\,U_\chi({\ggg}_{\bbk}^e)\otimes\bar1_\chi.$$
Thus $\pi_B: (Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}\longrightarrow U_\chi({\ggg}_{\bbk}^e)\otimes\bar1_\chi$ is a linear isomorphism. For $(\mathbf{a},\mathbf{b})=(a_1,\cdots,a_l;b_1,\cdots,b_q)\in\Lambda_l\times\Lambda'_q$ set $$h_{\mathbf{a},\mathbf{b}}=\pi_B^{-1}(\bar x_1^{a_1}\cdots \bar x_l^{a_l}\bar y_1^{b_1}\cdots \bar y_q^{b_q}\otimes\bar1_\chi).$$ By the bijectivity of $\pi_B$ and the PBW theorem of $U_\chi({\ggg}_{\bbk}^e)$, the vectors $h_{\mathbf{a},\mathbf{b}}$ with $(\mathbf{a},\mathbf{b})\in\Lambda_l\times\Lambda'_q$ form a basis of $(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}$, while from the definition of $\pi_B$ it follows that $\Lambda_{\bar h_{\mathbf{a},\mathbf{b}}}^{\text{max}}=\{(\mathbf{a},\mathbf{b})\}$ for any $(\mathbf{a},\mathbf{b})\in\Lambda_l\times\Lambda'_q$.
\end{proof}
As an immediate consequence of Proposition~\ref{reduced basis}, we have
\begin{corollary}\label{rg}
There exist even elements $\theta_1,\cdots,\theta_l\in (Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}_{\bar0}$ and odd elements
$\theta_{l+1},\cdots,\theta_{l+q}\in (Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}_{\bar1}$ such that
\begin{itemize}
\item[(a)]\begin{equation*}
\begin{split}
\theta_k=&(\bar x_k+\sum\limits_{\mbox{\tiny $\begin{array}{c}|\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}|_e=m_k+2,\\|\mathbf{a}|
+|\mathbf{b}|+|\mathbf{c}|+|\mathbf{d}|\geqslant 2\end{array}$}}\lambda^k_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}\bar x^{\mathbf{a}}
\bar y^{\mathbf{b}}\bar u^{\mathbf{c}}\bar v^{\mathbf{d}}\\&+\sum\limits_{|\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}|_e<m_k+2}
\lambda^k_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}\bar x^{\mathbf{a}}
\bar y^{\mathbf{b}}\bar u^{\mathbf{c}}\bar v^{\mathbf{d}})\otimes\bar 1_\chi,
\end{split}
\end{equation*}
where $\bar x_k\in{\ggg}^e_{\bbk}(m_k)_{\bar0}$ for $1\leqslant k\leqslant l$.
\item[(b)]\begin{equation*}
\begin{split}
\theta_{l+k}=&(\bar y_k+\sum\limits_{\mbox{\tiny $\begin{array}{c}|\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}|_e=n_k+2,\\|\mathbf{a}|
+|\mathbf{b}|+|\mathbf{c}|+|\mathbf{d}|\geqslant 2\end{array}$}}\lambda^k_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}\bar x^{\mathbf{a}}
\bar y^{\mathbf{b}}\bar u^{\mathbf{c}}\bar v^{\mathbf{d}}\\&+\sum\limits_{|\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}|_e<n_k+2}
\lambda^k_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}\bar x^{\mathbf{a}}
\bar y^{\mathbf{b}}\bar u^{\mathbf{c}}\bar v^{\mathbf{d}})\otimes\bar 1_\chi,
\end{split}
\end{equation*}
where $\bar y_k\in{\ggg}^e_{\bbk}(n_k)_{\bar1}$ for $1\leqslant k\leqslant q$.
\end{itemize}
All the coefficients $\lambda^k_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}$ above are in ${\bbk}$. Moreover, $\lambda^k_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}=0$ if $(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})$ is such that $a_{l+1}=\cdots=a_m=b_{q+1}=\cdots=b_n=c_1=\cdots=c_s=d_1=\cdots=d_{\lfloor\frac{\sfr}{2}\rfloor}=0$.
\end{corollary}
Set
\[\bar Y_i:=\left\{
\begin{array}{ll}
\bar x_i&\text{if}~1\leqslant i\leqslant l;\\
\bar y_{i-l}&\text{if}~l+1\leqslant i\leqslant l+q,
\end{array}
\right.
\]
and assume that the homogenous element $\bar Y_i$ is in $\ggg_{\bbk}(m_i)$ for $1\leqslant i\leqslant l+q$.
Since $(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}$ is a subalgebra of the reduced $W$-superalgebra $U_\chi(\ggg_\mathds{k},e)$,
we can substitute the generators $\{\theta_1,\cdots,\theta_{l+q}\}$ of $U_\chi(\ggg_\mathds{k},e)$ in \cite[Corollary 3.6(1)]{ZS2} for the ones in Corollary~\ref{rg}, and all the results in \cite[Theorem 3.7]{ZS2} remain true. What is more, the same discussion as in \cite[Remark 3.8(2)]{ZS2} shows that there exist super-polynomials $\bar F_{ij}$ of $l+q$ indeterminants over $\bbk$ ($1\leqslant i<j\leqslant l+q$, or $l+1\leqslant i=j\leqslant l+q$) with the first $l$ indeterminants being even, and the others being odd, such that
\begin{align*}\label{explainforeq}
[\theta_i,\theta_j]=\bar F_{i,j}(\theta_1,\cdots,\theta_{l+q}),\quad i,j=1,\cdots,l+q.
\end{align*}
Moreover, the Lie bracket relations in the Lie superalgebra $\ggg_\bbk^e$ with
$$[\bar Y_i,\bar Y_j]=\sum\limits_{k=1}^{l+q}\alpha_{ij}^k\bar Y_k$$ imply
$$\bar F_{ij}(\theta_1,\cdots,\theta_{l+q})\equiv\sum\limits_{k=1}^{l+q}\alpha_{ij}^k\theta_k+\bar q_{ij}(\theta_1,\cdots,\theta_{l+q})\quad(\mbox{mod }F_{m_i+m_j+1}(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}),$$ where $\bar q_{ij}\in\mathds{k}[X_1,\cdots,X_l;X_{l+1},\cdots,X_{l+q}]$ is a super-polynomial in $l+q$ invariables whose constant term and linear part are both zero, and $F_{m_i+m_j+1}(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}$ denotes the component of Kazhdan filtration of $(Q_\chi^\chi)^{\text{ad}\,{\mmm}'_\mathds{k}}$ with degree $m_i+m_j+1$.
\subsection{The PBW structure theory of refined $W$-superalgebras over $\mathbb{C}$}\label{PBWCC}
Recall that in \cite[Remark 70]{W} Wang introduced what we call refined $W$-superalgebras as follows.
\begin{defn}(\cite{W})\label{rewcc} Let $\ggg$ be a basic Lie superalgebra over $\mathbb{C}$. Define the refined $W$-superalgebra over $\bbc$ by$$W'_\chi:=(U({\ggg})/I_\chi)^{\text{ad}\,{\mmm}'}\cong Q_\chi^{\text{ad}\,{\mmm}'}
\equiv\{\bar{y}\in U({\ggg})/I_\chi \mid [a,y]\in I_\chi, \forall a\in{\mmm}'\},$$
and $\bar{y}_1\cdot\bar{y}_2:=\overline{y_1y_2}$ for all $\bar{y}_1,\bar{y}_2\in W'_\chi$.
\end{defn}
In this subsection we will concentrate on the structure of refined $W$-superalgebras over $\mathbb{C}$. First note that
\begin{lemma}\label{hwc2}
Let $h\in W_\chi'\backslash\{0\}$ and $(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in \Lambda^{\text{max}}_h$. Then $\mathbf{a}\in\mathbb{Z}_+^{l}\times\{\mathbf{0}\},~\mathbf{b}\in\Lambda'_{q}\times\{\mathbf{0}\}$, $\mathbf{c}=\mathbf{0}$, $\mathbf{d}=\mathbf{0}$.
\end{lemma}
\begin{proof}
The proof is the same as proof of Lemma \ref{hw} but apply \cite[Lemma 4.3]{ZS2} in place of \cite[Lemma 3.3]{ZS2}, thus will be omitted.
\end{proof}
Recall that $\{x_1,\cdots,x_l\}$ and $\{y_1,\cdots,y_q\}$ are the $\mathbb{C}$-basis of ${\ggg}^e_{\bar{0}}$ and ${\ggg}^e_{\bar{1}}$, respectively. Set
\begin{equation}\label{Y_i}
Y_i:=\left\{
\begin{array}{ll}
x_i&\text{if}~1\leqslant i\leqslant l;\\
y_{i-l}&\text{if}~l+1\leqslant i\leqslant l+q.
\end{array}
\right.
\end{equation}
Assume that $Y_i$ belongs to ${\ggg}(m_i)$ for $1\leqslant i\leqslant l+q$.
In virtue of the structure theory of $(Q_\chi^{\chi})^{\text{ad}\,{\mmm}'_\mathds{k}}$ over $\mathds{k}$ obtained in \S\ref{rrw}, the same discussion as in \cite[Theorem 4.5, Theorem 4.7]{ZS2} shows that
\begin{theorem}\label{PBWC} The following PBW structural statements for $W'_\chi$ hold.
\begin{itemize}
\item[(1)] There exist homogeneous elements $\Theta_1,\cdots,\Theta_{l}\in (W'_\chi)_{\bar0}$ and $\Theta_{l+1},\cdots,\Theta_{l+q}\in (W'_\chi)_{\bar1}$ such that
\begin{equation*}
\begin{split}
\Theta_k=&(Y_k+\sum\limits_{\mbox{\tiny $\begin{array}{c}|\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}|_e=m_k+2,\\|\mathbf{a}|
+|\mathbf{b}|+|\mathbf{c}|+|\mathbf{d}|\geqslant 2\end{array}$}}\lambda^k_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}x^{\mathbf{a}}
y^{\mathbf{b}}u^{\mathbf{c}}v^{\mathbf{d}}\\&+\sum\limits_{|\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}|_e<m_k+2}\lambda^k_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}x^{\mathbf{a}}
y^{\mathbf{b}}u^{\mathbf{c}}v^{\mathbf{d}})\otimes1_\chi
\end{split}
\end{equation*}
for $1\leqslant k\leqslant l+q$, where $\lambda^k_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}\in\mathbb{Q}$, and $\lambda^k_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}=0$ if $a_{l+1}=\cdots=a_m=b_{q+1}=\cdots=b_n=c_1=\cdots=c_s=
d_1=\cdots=d_{\lfloor\frac{\sfr}{2}\rfloor}=0$.
\item[(2)] The monomials $\Theta_1^{a_1}\cdots\Theta_l^{a_l}\Theta_{l+1}^{b_1}\cdots\Theta_{l+q}^{b_{q}}$ with $a_i\in\mathbb{Z}_+,~b_j\in\Lambda'_1$ for $1\leqslant i\leqslant l$ and $1\leqslant j\leqslant q$ form a basis of $W'_\chi$ over $\mathbb{C}$.
\item[(3)] For $i,j$ satisfying $1\leqslant i<j\leqslant l+q$ and $l+1\leqslant i=j\leqslant l+q$, there exist super-polynomials
$F_{ij}\in\mathbb{Q}[X_1,\cdots,X_{l+q}]$ such that $$[\Theta_i,\Theta_{j}]=F_{ij}(\Theta_1,\cdots,\Theta_{l+q}).$$ Moreover, if the elements $Y_i,~Y_j\in {\ggg}^e$ satisfy $[Y_i,Y_j]=\sum\limits_{k=1}^{l+q}\alpha_{ij}^kY_k$ in ${\ggg}^e$, then
\begin{equation}\label{Thetaa2}
\begin{array}{llllll}
&F_{ij}[X_1,\cdots,X_{l+q}]&\equiv&\sum\limits_{k=1}^{l+q}\alpha_{ij}^k\Theta_k+q_{ij}(\Theta_1,\cdots,\Theta_{l+q})&(\mbox{mod }\text{F}_{m_i+m_j+1}W_\chi'),
\end{array}
\end{equation}where $q_{ij}$ is a super-polynomial in $l+q$ variables in $\mathbb{Q}$ whose constant term and linear part are zero, and $\text{F}_{m_i+m_j+1}W_\chi'$ denotes the component of Kazhdan filtration of $W_\chi'$ with degree $m_i+m_j+1$.
\item[(4)] The refined $W$-superalgebra $W'_\chi$ is generated by the $\mathbb{Z}_2$-homogeneous elements $\Theta_1,\cdots,\Theta_{l}\in (W'_\chi)_{\bar0}$ and $\Theta_{l+1},\cdots,\Theta_{l+q}\in (W'_\chi)_{\bar1}$ subject to the relations in \eqref{Thetaa2}
with $1\leqslant i<j\leqslant l+q$ and $l+1\leqslant i=j\leqslant l+q$.
\end{itemize}
\end{theorem}
As a direct corollary of Theorem~\ref{PBWC}, we have
\begin{corollary}\label{isogr}
There is an isomorphism between $\mathbb{C}$-algebras $$\text{gr}\,(W_\chi')\cong S(\ggg^e),$$ where $\text{gr}\,(W_\chi')$ denotes the graded algebra of $W_\chi'$ under Kazhdan grading, and $S(\ggg^e)$ is the supersymmetric algebra on $\ggg^e$.
\end{corollary}
\begin{rem}
It is worth noting that in the case when $\sfr$ is even, we have $W_\chi'\cong U(\ggg,e)$ as $\mathbb{C}$-algebras, and Corollary \ref{isogr} is just \cite[Theorem 0.1(1)]{ZS2}. But for the case when $\sfr$ is odd, $W_\chi'$ is a proper subalgebra of $U(\ggg,e)$, and the consequence of Corollary \ref{isogr} is completely different from that of \cite[Theorem 0.1(2)]{ZS2}.
\end{rem}
\section{Refined $W$-superalgebras and the dimensional lower bounds for the modular representations of basic Lie superalgebras}\label{Refined W-superalgebras and the dimensional lower bounds}
In virtue of the results on refined $W$-superalgebras in \S\ref{crw}, in this section we will refine \cite[Conjecture 1.3]{ZS4}, then discuss the accessibility of the dimensional lower bounds for the modular representations of basic Lie superalgebras based on this conjecture. In the final part we will introduce an equivalent definition of refined $W$-superalgebras, which will be applied to describe the structure of minimal $W$-superalgebras in the next section.
\subsection{The super Kac-Weisfeiler property}
Recall that the dimensional lower bounds for the irreducible representations of $U_\xi({\ggg}_{\bbk})$ were predicted by Wang-Zhao in \cite[Theorem 5.6]{WZ} as below:
\begin{prop}(\cite{WZ})\label{wzd2}
Let ${\ggg}_{\bbk}$ be a basic Lie superalgebra over ${\bbk}=\overline{\mathbb{F}}_p$, assuming that the prime $p$ satisfies the restriction imposed in \cite[Table 1]{WZ}. Let $\xi$ be arbitrary $p$-character in $({\ggg}_{\bbk})^*_{\bar0}$, corresponding to an element $\bar x\in ({\ggg}_{\bbk})_{\bar0}$ such that $\xi(\bar y)=(\bar x,\bar y)$ for any $\bar y\in{\ggg}_{\bbk}$. Set $d_0=\text{dim}\,({\ggg}_{\bbk})_{\bar 0}-\text{dim}\,({\ggg}_{\bbk}^{\bar x})_{\bar 0}$ and $d_1=\text{dim}\,({\ggg}_{\bbk})_{\bar 1}-\text{dim}\,({\ggg}_{\bbk}^{\bar x})_{\bar 1}$, where ${\ggg}_{\bbk}^{\bar x}$ denotes the centralizer of $\bar x$ in ${\ggg}_{\bbk}$. Denote by $\lfloor\frac{d_1}{2}\rfloor$ the least integer upper bound of $\frac{d_1}{2}$. Then the dimension of every $U_\xi({\ggg}_{\bbk})$-module $M$ is divisible by $p^{\frac{d_0}{2}}2^{\lfloor\frac{d_1}{2}\rfloor}$.
\end{prop}
A natural question is whether or not there exist some irreducible modules of $U_\xi(\ggg_\bbk)$ with dimension equaling $p^{\frac{d_0}{2}}2^{\lfloor\frac{d_1}{2}\rfloor}$. Conjecture \ref{conjectureold} on the minimal dimensional representations of the complex finite $W$-superalgebra $U({\ggg},e)$ was raised in \cite{ZS4}.
Under the assumption of the conjecture, the lower bounds of the dimensions given in Proposition \ref{wzd2} are correspondingly to be indicated accessible for $p\gg0$ in \cite[Theorem 1.5]{ZS4}.
The conjecture in the general case is still open, out of scope we solved in \cite[Proposition 4.7, Proposition 5.8]{ZS4}.
\iffalse
For the confirmation of Conjecture \ref{conjecture}, we demonstrated the conjecture for some special cases in \cite[Proposition 4.7, Proposition 5.8]{ZS4}. But for the general case, the conjecture is far from being solved; see \cite[\S5.1]{ZS4} for more details.
\fi
\subsection{A conjecture on the representation of refined $W$-superalgebras}
In virtue of the structure theory of refined $W$-superalgebras in \S\ref{crw}, we can reformulate Conjecture \ref{conjectureold} as follows.
\begin{conj}\label{conjecture22}
Let ${\ggg}$ be a basic Lie superalgebra over ${\bbc}$, then the refined $W$-superalgebra $W_\chi'$ affords a one-dimensional representation.
\end{conj}
Now we make some explanation on the above. Let ${\ggg}$ be a basic Lie superalgebra over $\mathbb{C}$. For the case with $\sfr$ being odd, we can obtain the following result.
\begin{prop}\label{intromain}
Assume $\sfr$ is odd. If the refined $W$-superalgebra $W_\chi'$ affords a one-dimensional representation, then the corresponding finite $W$-superalgebra $U({\ggg},e)$ admits a two-dimensional irreducible representation.
\end{prop}
\begin{proof}
As the refined $W$-superalgebra $W_\chi'$ affords a one-dimensional representation, it is immediate that $Q_\chi^{\text{ad}\,{\mmm}'}=W_\chi'$ has a two-sided ideal of codimensional $1$. Denote this ideal by $I$. In the following we will prove that $I\oplus I(v_{\frac{\sfr+1}{2}}\otimes1_\chi)$ is a two-sided ideal of codimensional $2$ in $Q_\chi^{\text{ad}\,{\mmm}}\cong U({\ggg},e)$.
For any $\mathbb{Z}_2$-homogeneous element $h\in Q_\chi^{\text{ad}\,{\mmm}'}$, both $h\cdot I$ and $I\cdot h$ are contained in $I$, then $h\cdot I(v_{\frac{\sfr+1}{2}}\otimes1_\chi)\subseteq I(v_{\frac{\sfr+1}{2}}\otimes1_\chi)$. Let $h'$ be a $\mathbb{Z}_2$-homogeneous element in $I$, then $$h'(v_{\frac{\sfr+1}{2}}\otimes1_\chi)\cdot h=(-1)^{|h|}h'h (v_{\frac{\sfr+1}{2}}\otimes1_\chi)+h'[v_{\frac{\sfr+1}{2}}\otimes1_\chi,h].$$ Since $h'h(v_{\frac{\sfr+1}{2}}\otimes1_\chi)\in I(v_{\frac{\sfr+1}{2}}\otimes1_\chi)$, and $[v_{\frac{\sfr+1}{2}}\otimes1_\chi,h]=0$ by the definition of $Q_\chi^{\text{ad}\,{\mmm}'}$, we have $h'(v_{\frac{\sfr+1}{2}}\otimes1_\chi)\cdot h\in I(v_{\frac{\sfr+1}{2}}\otimes1_\chi)$, which yields $I(v_{\frac{\sfr+1}{2}}\otimes1_\chi)\cdot h\subseteq I(v_{\frac{\sfr+1}{2}}\otimes1_\chi)$ by the arbitrary of $h'$. Summing up, we have
\begin{equation}\label{eq0}
Q_\chi^{\text{ad}\,{\mmm}'}\cdot(I\oplus I(v_{\frac{\sfr+1}{2}}\otimes1_\chi))\subseteq I\oplus I(v_{\frac{\sfr+1}{2}}\otimes1_\chi),\quad(I\oplus I(v_{\frac{\sfr+1}{2}}\otimes1_\chi))\cdot Q_\chi^{\text{ad}\,{\mmm}'}\subseteq I\oplus I(v_{\frac{\sfr+1}{2}}\otimes1_\chi).
\end{equation}
Now consider the element $v_{\frac{\sfr+1}{2}}\otimes1_\chi\in Q_\chi^{\text{ad}\,{\mmm}}\backslash Q_\chi^{\text{ad}\,{\mmm}'}$.
Let $h$ be arbitrary $\mathbb{Z}_2$-homogeneous element in $I\subseteq Q_\chi^{\text{ad}\,{\mmm}'}$. Note that $$v_{\frac{\sfr+1}{2}}^2\otimes1_\chi=\frac{1}{2}
[v_{\frac{\sfr+1}{2}},v_{\frac{\sfr+1}{2}}]\otimes1_\chi=\frac{1}{2}\otimes1_\chi,$$
and $[v_{\frac{\sfr+1}{2}}\otimes1_\chi,h]=0$ by the definition of $Q_\chi^{\text{ad}\,{\mmm}'}$,
then we have
\begin{equation*}
\begin{split}
(v_{\frac{\sfr+1}{2}}\otimes1_\chi)\cdot h=&[v_{\frac{\sfr+1}{2}}\otimes1_\chi,h]+(-1)^{|h|}h (v_{\frac{\sfr+1}{2}}\otimes1_\chi)\\
=&(-1)^{|h|}h(v_{\frac{\sfr+1}{2}}\otimes1_\chi)\in I(v_{\frac{\sfr+1}{2}}\otimes1_\chi),\\
(v_{\frac{\sfr+1}{2}}\otimes1_\chi)\cdot h(v_{\frac{\sfr+1}{2}}\otimes1_\chi)
=&[v_{\frac{\sfr+1}{2}}\otimes1_\chi,h](v_{\frac{\sfr+1}{2}}\otimes1_\chi)+(-1)^{|h|}h(v_{\frac{\sfr+1}{2}}^2\otimes1_\chi)\\
=&(-1)^{|h|}[v_{\frac{\sfr+1}{2}}\otimes1_\chi,[v_{\frac{\sfr+1}{2}}\otimes1_\chi,h]]-(-1)^{|h|}(v_{\frac{\sfr+1}{2}}\otimes1_\chi)\cdot\\
&[v_{\frac{\sfr+1}{2}}\otimes1_\chi,h]+(-1)^{|h|}h(v_{\frac{\sfr+1}{2}}^2\otimes1_\chi)\\
=&\frac{(-1)^{|h|}}{2}h\in I,
\end{split}\end{equation*}
which shows that \begin{equation}\label{eq1}
(v_{\frac{\sfr+1}{2}}\otimes1_\chi)\cdot(I\oplus I(v_{\frac{\sfr+1}{2}}\otimes1_\chi))\subseteq I\oplus I(v_{\frac{\sfr+1}{2}}\otimes1_\chi).\end{equation}
On the other hand, we have
\begin{equation*}
\begin{split}
h\cdot(v_{\frac{\sfr+1}{2}}\otimes1_\chi)\in&I(v_{\frac{\sfr+1}{2}}\otimes1_\chi),\\
h(v_{\frac{\sfr+1}{2}}\otimes1_\chi)\cdot(v_{\frac{\sfr+1}{2}}\otimes1_\chi)=&h(v_{\frac{\sfr+1}{2}}^2\otimes1_\chi)
=\frac{h}{2}\in I,\end{split}\end{equation*}
i.e., \begin{equation}\label{eq2}(I\oplus I(v_{\frac{\sfr+1}{2}}\otimes1_\chi))\cdot(v_{\frac{\sfr+1}{2}}\otimes1_\chi)\subseteq I\oplus I(v_{\frac{\sfr+1}{2}}\otimes1_\chi).\end{equation}
Since the $\mathbb{C}$-algebra $Q_\chi^{\text{ad}\,{\mmm}}$ is generated by $v_{\frac{\sfr+1}{2}}\otimes1_\chi$ and the elements in $Q_\chi^{\text{ad}\,{\mmm}'}$, by \eqref{eq0}, \eqref{eq1} and \eqref{eq2} we can conclude that $I\oplus I(v_{\frac{\sfr+1}{2}}\otimes1_\chi)$ is a two-sided ideal of $U({\ggg},e)$.
Moreover, it is immediate from the PBW theorem of $Q_\chi^{\text{ad}\,{\mmm}}$ in \cite[Theorem 4.5(2)]{ZS2} that $Q_\chi^{\text{ad}\,{\mmm}}/(I\oplus I(v_{\frac{\sfr+1}{2}}\otimes1_\chi))\cong \bbc_\chi\oplus\bbc (v_{\frac{\sfr+1}{2}}\otimes1_\chi)$ as $\bbc$-vector spaces, then $I\oplus I(v_{\frac{\sfr+1}{2}}\otimes1_\chi)$ is codimensional $2$ in $U({\ggg},e)\cong Q_\chi^{\text{ad}\,{\mmm}}$. Thus the $\mathbb{C}$-algebra $U({\ggg},e)$ admits a two-dimensional irreducible representation, completing the proof.
\end{proof}
Recall that for the case when $\sfr$ is even, we have $W_\chi'=U(\ggg,e)$ by the definition. Then
in virtue of Proposition \ref{intromain}, we can refine Conjecture \ref{conjectureold} as Conjecture \ref{conjecture22}.
It is worth noting that in contrast to Conjecture \ref{conjectureold}, the assumption in Conjecture \ref{conjecture22} is irrelevant to the parity of $\sfr$.
\subsection{The minimal dimensional representations of reduced enveloping algebras}
We will consider the minimal dimensional representations of reduced enveloping algebra of a
basic Lie superalgebra in this subsection.
Let ${\ggg}_{\bbk}$ be a basic Lie superalgebra over ${\bbk}=\overline{\mathbb{F}}_p$. Set $d_0=\text{dim}\,({\ggg}_{\bbk})_{\bar 0}-\text{dim}\,({\ggg}_{\bbk}^{e})_{\bar 0}$ and $d_1=\text{dim}\,({\ggg}_{\bbk})_{\bar 1}-\text{dim}\,({\ggg}_{\bbk}^{e})_{\bar 1}$, where ${\ggg}_{\bbk}^{e}$ denotes the centralizer of $e$ in ${\ggg}_{\bbk}$.
Denote by $\lfloor\frac{d_1}{2}\rfloor$ the least integer upper bound of $\frac{d_1}{2}$. The main result of this section is the following theorem.
\begin{theorem}\label{intromain-2}
Let ${\ggg}_{\bbk}$ be a basic Lie superalgebra over ${\bbk}=\overline{\mathbb{F}}_p$, and let $\chi\in({\ggg}_{\bbk})^*_{\bar0}$ be the $p$-character of ${\ggg}_{\bbk}$ corresponding to a nilpotent element $e\in(\ggg_\bbk)_\bz$ such that $\chi(\bar y)=(e,\bar y)$ for any $\bar y\in{\ggg}_{\bbk}$. If Conjecture \ref{conjecture22} establishes for the refined $W$-superalgebra $W_\chi'$ associated with (${\ggg},e)$ over $\mathbb{C}$, then for $p\gg0$ the reduced enveloping algebra $U_\chi({\ggg}_{\bbk})$ admits irreducible representations of dimension $p^{\frac{d_0}{2}}2^{\lfloor\frac{d_1}{2}\rfloor}$.
\end{theorem}
\begin{proof}
The theorem readily follows from proposition \ref{intromain} and \cite[Theorem 1.6]{ZS4}.
\end{proof}
As for the general case, let $\xi\in({\ggg}_{\bbk})^*_{\bar0}$ be any $p$-character of ${\ggg}_{\bbk}$ corresponding to an element $\bar x\in ({\ggg}_{\bbk})_{\bar0}$ such that $\xi(\bar y)=(\bar x,\bar y)$ for any $\bar y\in{\ggg}_{\bbk}$. Let $d'_0=\text{dim}\,({\ggg}_{\bbk})_{\bar 0}-\text{dim}\,({\ggg}_{\bbk}^{\bar x})_{\bar 0}$ and $d'_1=\text{dim}\,({\ggg}_{\bbk})_{\bar 1}-\text{dim}\,({\ggg}_{\bbk}^{\bar x})_{\bar 1}$, where ${\ggg}_{\bbk}^{\bar x}$ denotes the centralizer of $\bar x$ in ${\ggg}_{\bbk}$. Then we have
\begin{theorem}\label{intromain-3}
Let ${\ggg}_{\bbk}$ be a basic Lie superalgebra over ${\bbk}=\overline{\mathbb{F}}_p$, and let $\xi\in({\ggg}_{\bbk})^*_{\bar0}$. If all the refined $W$-superalgebras associated with the basic Lie superalgebras over $\bbc$ excluding type $D(2,1;a)$ with $a\notin\overline{\mathbb{Q}}$ afford one-dimensional representations, then for $p\gg0$ the reduced enveloping algebra $U_\xi({\ggg}_{\bbk})$ admits irreducible representations of dimension $p^{\frac{d'_0}{2}}2^{\lfloor\frac{d'_1}{2}\rfloor}$.
\end{theorem}
\subsection{A variation on the definition of refined $W$-superalgebras}\label{variation}
In this part we will introduce a variation on the definition of refined $W$-superalgebras.
Let $\ggg$ be a basic Lie superalgebra over $\mathbb{C}$, then we have ${\ggg}=\bigoplus_{i\in{\bbz}}{\ggg}(i)$, where ${\ggg}(i)=\{x\in{\ggg}\mid[h,x]=ix\}$. Set $\mathfrak{n}:=\bigoplus_{i\leqslant-1}{\ggg}(i)$ to be a nilpotent subalgebra of $\ggg$.
Let $I^{\text{fin}}$ be the ideal of $U(\ggg)$ generated by the elements $\{x-\chi(x) \mid x\in\mathfrak{n}\}$, and let $Q^{\text{fin}}_\chi:=U(\ggg)/I^{\text{fin}}$ be an induced $\ggg$-module. In \cite[Definition 4.3]{suh} Suh introduced the quantum finite $W$-superalgebra $W^{\text{fin}}(\ggg,e)$ as follows.
\begin{defn}(\cite{suh})\label{quafi}
Define the quantum finite $W$-superalgebra $W^{\text{fin}}(\ggg,e)$ associated with $\ggg$ and $e$ by
$$W^{\text{fin}}(\ggg,e):=(Q^{\text{fin}}_\chi)^{\text{ad}\,\mathfrak{n}},$$
where $(Q^{\text{fin}}_\chi)^{\text{ad}\,\mathfrak{n}}$ denotes the invariant subspace of $Q^{\text{fin}}_\chi$ under the adjoint action of $\mathfrak{n}$, and the associative product of $W^{\text{fin}}(\ggg,e)$ is defined by $$(x+I^{\text{fin}})\cdot(y+I^{\text{fin}}):=xy+I^{\text{fin}}.$$
\end{defn}
In \cite[(3.25)]{suh}, Suh introduced a grading on $Q^{\text{fin}}_\chi$. Let $w_1,\cdots, w_c$ be a basis of $\ggg$ over $\bbc$. Let $U({\ggg})=\bigcup_{i\in{\bbz}}\text{F}_i^pU({\ggg})$ be a filtration of $U({\ggg})$, where $\text{F}_i^pU({\ggg})$ is the ${\bbc}$-span of all $w_1\cdots w_c$ with $w_1\in{\ggg}(j_1),\cdots,w_c\in{\ggg}(j_c)$ and $(j_1+1)+\cdots+(j_c+1)\leqslant i$. The corresponding filtration on $Q_{\chi}^{\text{fin}}$ is defined by $\text{F}_i^pQ_{\chi}^{\text{fin}}:=\pi(\text{F}_i^pU({\ggg}))$ with $\pi:U({\ggg})\twoheadrightarrow U({\ggg})/I^{\text{fin}}$ being the canonical homomorphism, which makes $Q_{\chi}^{\text{fin}}$ into a filtered $U({\ggg})$-module. This grading is called $p$-grading.
Suppose $\{v_\alpha\}_{\alpha\in J}$ is a basis of $\ggg^e$ such that $v_\alpha\in\ggg(j_\alpha)$, and let $S=\{v_\alpha+a_\alpha\mid\alpha\in J\}$ be a subset of $W^{\text{fin}}(\ggg,e)$ such that the $p$-grading of $a_\alpha$ is smaller than that of $v_\alpha$. In \cite[Proposition 4.7]{suh} Suh showed that $S$ is a set of free generators of $W^{\text{fin}}(\ggg,e)$. Moreover, the proofs of \cite[Proposition 3.12]{suh} and \cite[Proposition 4.7]{suh} showed that the quantum finite $W$-superalgebra version of \cite[Proposition 3.10]{suh} is also valid, then we have
\begin{prop}\label{Suh}
There is an isomorphism between $\mathbb{C}$-algebras $$\text{gr}^p\,(W^{\text{fin}}(\ggg,e))\cong S(\ggg^e),$$ where $\text{gr}^p\,(W^{\text{fin}}(\ggg,e))$ denotes the graded algebra of $W^{\text{fin}}(\ggg,e)$ under $p$-grading, and
$S(\ggg^e)$ is the supersymmetric algebra on $\ggg^e$.
\end{prop}
\begin{rem}
It is worth noting that in \cite{suh} the quantum finite $W$-superalgebras are introduced in the context of vertex algebras and quantum reduction, and the nilpotent element chosen there is $f$, but not $e$. What is more, the original grading on $\ggg$ applied there was under the action of $\text{ad}\,\frac{h}{2}$, while the grading we used is under the action of $\text{ad}\,h$. Moreover, the $p$-grading on $Q^{\text{fin}}_\chi$ introduced there is also a little different from Kazhdan grading. Although there are so many differences, one can find that they are essentially the same, i.e., all the discussions there still go through in our settings. Therefore, the isomorphism between $\mathbb{C}$-algebras $\text{gr}\,(W^{\text{fin}}(\ggg,e))$ and $S(\ggg^e)$ in Proposition~\ref{Suh} can also be established under Kazhdan grading, i.e.,
\end{rem}
\begin{prop}\label{p-kazhdan}
There is an isomorphism between $\mathbb{C}$-algebras $$\text{gr}\,(W^{\text{fin}}(\ggg,e))\cong S(\ggg^e),$$ where $\text{gr}\,(W^{\text{fin}}(\ggg,e))$ denotes the graded algebra of $W^{\text{fin}}(\ggg,e)$ under Kazhdan grading, and
$S(\ggg^e)$ is the supersymmetric algebra on $\ggg^e$.
\end{prop}
As an immediate corollary of Corollary~\ref{isogr} and Proposition~\ref{p-kazhdan}, we have
\begin{prop}\label{p-kazhdan2}
The graded algebra of finite $W$-superalgebra $\text{gr}\,(W_\chi')$ is isomorphism to the graded algebra of the quantum finite $W$-superalgebras $\text{gr}\,(W^{\text{fin}}(\ggg,e))$ under Kazhdan grading.
\end{prop}
As $\ggg^e$ is finite-dimensional as a vector space, it is readily from Proposition \ref{p-kazhdan2} that
\begin{theorem}\label{important}
There is an isomorphism between Kazhdan filtered algebras
$$W_\chi'\cong W^{\text{fin}}(\ggg,e).$$
\end{theorem}
In virtue of Theorem \ref{important}, in the following we will take the quantum finite $W$-superalgebra $W^{\text{fin}}(\ggg,e)$ in Definition \ref{quafi} as the refined $W$-superalgebra $W_\chi'$ introduced in Definition \ref{rewcc} , which will cause no confusion.
\begin{rem}
In fact, we can generalize the definition of refined $W$-superalgebras as follows.
Fix an isotropic subspace $\mathfrak{l}$ of $\ggg(-1)$ with respect to $\langle\cdot,\cdot\rangle$, and let $\mathfrak{l}'=\{x\in\ggg(-1)\mid\langle x,\mathfrak{l}\rangle=0\}$. Obviously we have $\mathfrak{l}\subseteq \mathfrak{l}'$. Let $\mathfrak{m}_\mathfrak{l}$ and $\mathfrak{m}'_\mathfrak{l}$ be nilpotent subalgebras of $\ggg$ defined by
$${\mmm}_\mathfrak{l}=\bigoplus_{i\leqslant -2}{\ggg}(i)\oplus \mathfrak{l},\qquad {\mmm}'_{\mathfrak{l}}=\bigoplus_{i\leqslant -2}{\ggg}(i)\oplus \mathfrak{l}',$$ then we have ${\mmm}_\mathfrak{l}\subseteq{\mmm}'_{\mathfrak{l}}$.
The linear function $\chi\in\ggg^*$ restricts to a character on ${\mmm}_\mathfrak{l}$, and denote by $\mathbb{C}_\mathfrak{l}$ the corresponding $1$-dimensional representation of ${\mmm}_\mathfrak{l}$. Define the generalized Gelfand-Graev module associated with $\mathfrak{l}$ by $Q_\mathfrak{l}=U({\ggg})\otimes_{U({\mmm}_\mathfrak{l})}\mathbb{C}_\mathfrak{l}=U({\ggg})/I_\mathfrak{l}$, where $I_\mathfrak{l}$ is the ${\bbz}_2$-graded ideal of $U({\ggg})$ generated by all $x-\chi(x)$ with $x\in\mathfrak{m}_\mathfrak{l}$.
Define the $W$-superalgebra associated with isotropic subspace $\mathfrak{l}$ by
$$W_\mathfrak{l}:=(U({\ggg})/I_\mathfrak{l})^{\text{ad}\,{\mmm}'_\mathfrak{l}}\cong\{\bar x\in U({\ggg})/I_\mathfrak{l}\mid
[a,x]\in I_\mathfrak{l},~\forall a\in{\mmm}'_{\mathfrak{l}}\},$$ where $\bar x$ is the coset of $x\in U(\ggg)$, and the multiplication is given by $\bar x_1\cdot\bar x_2=\overline{x_1x_2}$ for $\bar x_1\, \bar x_2\in W_\mathfrak{l}$.
It is easy to observe that the refined $W$-superalgebra $W'_\chi$ in Definition \ref{rewcc} is the case with $\mathfrak{l}$ being chosen as a Lagrangian subspace of $\ggg(-1)$, and the algebra $W^{\text{fin}}(\ggg,e)$ in Definition \ref{quafi}
corresponds to the case with $\mathfrak{l}=0$.
Recall that in \cite[Theorem 4.1]{GG} Gan-Ginzburg showed that the definition of finite $W$-algebras is independent of the choice of an isotropic subspaces $\mathfrak{l}\subseteq\ggg'(-1)$ associated with a complex semisimple Lie algebra $\ggg'$.
For the super case, one may wonder whether this conclusion is also valid for the algebra $W_\mathfrak{l}$. In fact, for the case when $\sfr=\dim\ggg(-1)_{\bar{1}}$ is even,
Zhao has proved in \cite[Remark 3.11]{Z2} that the definition of the $W$-superalgebra $W_\mathfrak{l}$ is indeed independent of the choice of isotropic subspaces. For the case with $\sfr$ being odd, one needs to check whether the discussions in \cite[Lemma 3.9, Propostion 3.10, Theorem 4.6]{suh} go through for the $\mathbb{C}$-algebra $W_\mathfrak{l}$. If they are true, then Theorem \ref{important} follows just as a corollary.
\end{rem}
\section{The structure theory of minimal $W$-superalgebras}\label{structure}
This section is the main part of the paper. A refined $W$-superalgebra $W_\chi'$ will be called a minimal one if it is associated with a minimal nilpotent element in $\ggg$. In this section we will describe the explicit expression of the generators of minimal $W$-superalgebras over $\mathbb{C}$, and also the commutators between them. These generators and their relations completely determine the structure of minimal $W$-superalgebras. In virtue of these results, we will show that the dimensional lower bounds for the modular representations of reduced enveloping of basic Lie superalgebras associated with minimal nilpotent elements are accessible, completing the proof of Theorem \ref{intromainminnimalf}.
In \cite[Proposition 5.3]{suh} Suh has introduced the free generators of $W_\chi'$, and also the commutators between these generators in \cite[Proposition 5.4]{suh}. However, there are some errors in Suh's proof, thus also the second equation of (5.12) and the last equation of (5.17) in the results obtained by Suh. Moreover, the proof there is too brief to get the point. In order to determine the minimal dimension for the representations of reduced enveloping algebra $U_\chi(\ggg_{\mathds{k}})$, we must give an accurate description on the construction of corresponding refined $W$-superalgebra $W_\chi'$. In this section we will correct these errors and give the detailed calculation, mainly following Premet's strategy on finite $W$-algebras \cite[\S4]{P3}, with a few modifications.
This section is divided into two parts. The first part will be devoted to the general situation, where we consider refined $W$-superalgebras associated with arbitrary nilpotent elements, and the minimal $W$-superalgebras case will be dealt with in the second part.
Throughout this section, we will take $W^{\text{fin}}(\ggg,e)$ in Definition \ref{quafi} as the refined $W$-superalgebra $W_\chi'$ by Theorem \ref{important}.
\subsection{The case of arbitrary nilpotent elements}
In this part we will give the explicit formulae for the generators of lower Kazhdan degree for refined $W$-superalgebras associated with arbitrary nilpotent elements, and also some commutators between them.
\subsubsection{}\label{general}
Let ${\ggg}$ be a basic Lie superalgebra over ${\bbc}$ and $\mathfrak{h}$ be a typical Cartan subalgebra of ${\ggg}$. Let $\Phi$ be a root system of ${\ggg}$ relative to $\mathfrak{h}$ whose simple root system $\Delta=\{\alpha_1,\cdots,\alpha_l\}$ is distinguished. Let $\{e,h,f\}$ be an $\mathfrak{sl}_2$-triple in ${\ggg}$ such that
$$[e,f]=h\in\mathfrak{h},\quad[h,e]=2e,\quad[h,f]=-2f.$$
Normalize the invariant bilinear form $(\cdot,\cdot)$ on $\ggg$ by the condition $(e,f)=1$. This entails $(h,h)=2$.
Recall that there exists a symplectic (resp. symmetric) bilinear form $\langle\cdot,\cdot\rangle$ on the ${\bbz}_2$-graded subspace ${\ggg}(-1)_{\bar{0}}$ (resp. ${\ggg}(-1)_{\bar{1}}$) given by $\langle x,y\rangle=(e,[x,y])=\chi([x,y])$ for all $x,y\in{\ggg}(-1)_{\bar0}~(\text{resp.}\,x,y\in{\ggg}(-1)_{\bar1})$. Set $s=\text{dim}\,\ggg(-1)_{\bar0}$ (note that $s$ is an even number), and $\sfr=\text{dim}\,\ggg(-1)_{\bar1}$. In \S\ref{0.1.1} we choose $\mathbb{Z}_2$-homogenenous bases $\{u_1,\cdots,u_{s}\}$ of ${\ggg}(-1)_{\bar0}$ and $\{v_1,\cdots,v_\sfr\}$ of ${\ggg}(-1)_{\bar1}$ contained in ${\ggg}$ such that $\langle u_i, u_j\rangle =i^*\delta_{i+j,s+1}$ for $1\leqslant i,j\leqslant s$, where $i^*=\left\{\begin{array}{ll}-1&\text{if}~1\leqslant i\leqslant \frac{s}{2};\\ 1&\text{if}~\frac{s}{2}+1\leqslant i\leqslant s\end{array}\right.$, and $\langle v_i,v_j\rangle=\delta_{i+j,\sfr+1}$ for $1\leqslant i,j\leqslant \sfr$.
Set $z_\alpha:=u_\alpha$ for $1\leqslant\alpha\leqslant s$, and $z_{\alpha+s}:=v_\alpha$ for $1\leqslant\alpha\leqslant\sfr$. Let $\{z_\alpha\mid\alpha\in S(-1)\}$ denote the union of $\{z_\alpha\mid\alpha\in S(-1)_{\bar0}\}$ and $\{z_{\alpha+s}\mid\alpha\in S(-1)_{\bar1}\}$, which is a base of ${\ggg}(-1)$. Denote by $A_e$ the associative algebra generated by $z_\alpha$ with $\alpha\in S(-1)$ subject to the relations given above.
Set $z_\alpha^*:=\alpha^\natural z_{s+1-\alpha}$ for $1\leqslant\alpha\leqslant s$, where $\alpha^\natural=\left\{\begin{array}{ll}1&\text{if}~1\leqslant \alpha\leqslant \frac{s}{2};\\ -1&\text{if}~\frac{s}{2}+1\leqslant \alpha\leqslant s\end{array}\right.$, and $z_{\alpha+s}^*:=z_{\sfr+1-\alpha+s}$ for $1\leqslant\alpha\leqslant \sfr$. Then $\{z_\alpha^*\mid\alpha\in S(-1)\}$ is dual base of $\{z_\alpha\mid\alpha\in S(-1)\}$ such that $\langle z^*_\alpha,z_\beta\rangle =\delta_{\alpha,\beta}$ for $\alpha,\beta\in S(-1)$. From the assumption above we can conclude that $\langle z_\alpha,z_\beta\rangle\neq0$ if and only if $z_\alpha=\mathbb{C}z_\beta^*$ and $z_\beta=\mathbb{C}z_\alpha^*$, and $\langle z_\alpha,z_\beta\rangle=0$ otherwise. Moreover, $\langle z_\alpha,z_\beta\rangle=1$ if and only if $z_\alpha=z_\beta^*$ and $z_\beta=-(-1)^{|\alpha|}z_\alpha^*$. The same conclusion also establishes for the pair $(z_\alpha^*,z_\beta^*)$. We can further assume that $\{z_\alpha\mid\alpha\in S(-1)\}$ and $\{z_\alpha^*\mid\alpha\in S(-1)\}$ are root vectors for $\mathfrak{h}$.
In the following discussion, for any $\alpha\in S(-1)$ we will denote the parity of $z_\alpha$ by $|\alpha|$ for simplicity. It is straightforward that $z_\alpha$ and $z_\alpha^*$ have the same parity.
\subsubsection{}
Let $Y_1,\cdots,Y_{l+q}$ be a basis of $\ggg^e$ as defined in \eqref{Y_i}. Recall that in Theorem \ref{PBWC} we choose $\Theta_{k}$ for $1\leqslant k\leqslant l+q$ as the generators of the refined $W$-superalgebra, with $Y_k$ being the leading term. In what follows we will introduce explicit formulae for the generators $\Theta_k$ of lower Kazhdan degree for a refined $W$-superalgebra $W_\chi'$.
For the case with $v\in\ggg^e(0)$, Suh introduced in \cite[Proposition 5.3]{suh} the corresponding generators $\Theta_v$ of $W_\chi'$ as follows:
\begin{prop}(\cite{suh})\label{v}
If $v\in\ggg^e(0)$ then it can be assumed that
\begin{equation*}
\Theta_v=(v-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}z_\alpha[z_\alpha^*,v])\otimes1_\chi.
\end{equation*}
\end{prop}
For the generators of $W_\chi'$ introduced in Proposition \ref{v}, Suh introduced the commutators between them in \cite[Proposition 5.4]{suh}, i.e.,
\begin{prop}(\cite{suh})\label{v1v2}
Let $v_1, v_2$ be elements in $\ggg^e(0)$. The commutator between the generators introduced in Proposition \ref{v} is
\begin{equation}
[\Theta_{v_1},\Theta_{v_2}]=\Theta_{[v_1,v_2]}.
\end{equation}
\end{prop}
For the case with $w\in\ggg^e(1)$, Suh introduced the corresponding generators $\Theta_w$ of $W_\chi'$ in \cite[Proposition 5.3]{suh}. However, there are some errors in the proof there. Now we will put forward a new set of generators.
\begin{prop}\label{w}
If $w\in\ggg^e(1)$ then the generator $\Theta_w\in W_\chi'$ has the following property:
\begin{equation*}
\Theta_w=(w-\sum\limits_{\alpha\in S(-1)}z_\alpha[z_\alpha^*,w]+\frac{1}{3}(\sum\limits_{\alpha,\beta\in S(-1)}z_\alpha z_\beta[z_\beta^*,[z_\alpha^*,w]]-2[w,f]))\otimes1_\chi.
\end{equation*}
\end{prop}
\begin{proof}
By the definition of $W_\chi'$, it is enough to show that $[z_\gamma^*,\Theta_w]=0$ for any $\gamma\in S(-1)$. First note that
\begin{equation}\label{wxyuv}
\begin{split}
&\sum\limits_{\alpha\in S(-1)}[z_\gamma^*,z_\alpha[z_\alpha^*,w]]\otimes1_\chi\\
=&\sum\limits_{\alpha\in S(-1)}[z^*_\gamma,z_\alpha][z_\alpha^*,w]\otimes1_\chi
+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||\gamma|}z_\alpha[z_\gamma^*,[z_\alpha^*,w]]\otimes1_\chi\\
=&[z_\gamma^*,w]\otimes1_\chi+[f,[z_\gamma^*,w]]\otimes1_\chi+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||\gamma|}z_\alpha[z_\gamma^*,[z_\alpha^*,w]]\otimes1_\chi\\
=&[z_\gamma^*,w]\otimes1_\chi-[z_\gamma^*,[w,f]]\otimes1_\chi+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||\gamma|}z_\alpha[z_\gamma^*,[z_\alpha^*,w]]\otimes1_\chi.
\end{split}
\end{equation}
Recall that $[z_\alpha^*,z_\beta]=\delta_{\alpha,\beta}f$ for all $\alpha,\beta\in S(-1)$, and $\chi(f)=(e,f)=1$, then we have
\begin{equation}\label{3.5}
\begin{split}
&\sum\limits_{\alpha,\beta\in S(-1)}[z_\gamma^*,z_\alpha z_\beta[z_\beta^*,[z_\alpha^*,w]]]\otimes1_\chi\\
=&\sum\limits_{\alpha,\beta\in S(-1)}[z_\gamma^*,z_\alpha]z_\beta[z_\beta^*,[z_\alpha^*,w]]\otimes1_\chi
+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha||\gamma|}z_\alpha[z_\gamma^*,z_\beta][z_\beta^*,[z_\alpha^*,w]]\otimes1_\chi\\
&+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{(|\alpha|+|\beta|)|\gamma|}z_\alpha z_\beta[z_\gamma^*,[z_\beta^*,[z_\alpha^*,w]]]\otimes1_\chi\\
=&\sum\limits_{\beta\in S(-1)}z_\beta[z_\beta^*,[z_\gamma^*,w]]\otimes1_\chi+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||\gamma|}z_\alpha[z_\gamma^*,[z_\alpha^*,w]]\otimes1_\chi\\
&+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{(|\alpha|+|\beta|)|\gamma|}z_\alpha z_\beta[z_\gamma^*,[z_\beta^*,[z_\alpha^*,w]]]\otimes1_\chi.
\end{split}
\end{equation}
Since $\langle z_\beta^*,z_\gamma^*\rangle\neq0$ if and only if $z_\beta=z_\gamma^*$ and $z_\gamma=-(-1)^{|\beta|}z_\beta^*$, and $\langle z_\beta^*,z_\gamma^*\rangle=1$ in this case, then
\begin{equation}\label{3.6}
\begin{split}
&\sum\limits_{\beta\in S(-1)}z_\beta[z_\beta^*,[z_\gamma^*,w]]\otimes1_\chi\\=&\sum\limits_{\beta\in S(-1)}(z_\beta[[z_\beta^*,z_\gamma^*],w]+(-1)^{|\beta||\gamma|}z_\beta[z_\gamma^*,[z_\beta^*,w]])\otimes1_\chi\\
=&-z_\gamma^*[w,f]\otimes1_\chi+\sum\limits_{\beta\in S(-1)}(-1)^{|\beta||\gamma|}z_\beta[z_\gamma^*,[z_\beta^*,w]]\otimes1_\chi.
\end{split}
\end{equation}
Moreover, it is obvious that $z=\sum\limits_{\alpha\in S(-1)}[z_\alpha^*,z]z_\alpha$ for all $z\in \ggg(-1)$,
thus
\begin{equation}\label{3.7}
\begin{split}
&\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{(|\alpha|+|\beta|)|\gamma|}z_\alpha z_\beta[z_\gamma^*,[z_\beta^*,[z_\alpha^*,w]]]\otimes1_\chi\\
=&\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{(|\alpha|+|\beta|)|\gamma|}(z_\alpha z_\beta[[z_\gamma^*,z_\beta^*],[z_\alpha^*,w]]+(-1)^{|\beta||\gamma|}z_\alpha z_\beta[z_\beta^*,[z_\gamma^*,[z_\alpha^*,w]]])\otimes1_\chi\\
=&\sum\limits_{\alpha\in S(-1)}(-(-1)^{|\alpha||\gamma|}z_\alpha z_\gamma^*[f,[z_\alpha^*,w]]+(-1)^{|\alpha||\gamma|}z_\alpha[z_\gamma^*,[z_\alpha^*,w]])\otimes1_\chi\\
=&\sum\limits_{\alpha\in S(-1)}((-1)^{|\alpha||\gamma|}z_\alpha z_\gamma^*[z_\alpha^*,[w,f]]+(-1)^{|\alpha||\gamma|}z_\alpha[z_\gamma^*,[z_\alpha^*,w]])\otimes1_\chi\\
=&\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||\gamma|}([z_\alpha, z_\gamma^*][z_\alpha^*,[w,f]]+(-1)^{|\alpha||\gamma|}z_\gamma^*z_\alpha[z_\alpha^*,[w,f]]+z_\alpha[z_\gamma^*,[z_\alpha^*,w]])\otimes1_\chi\\
=&-[z_\gamma^*,[w,f]]\otimes1_\chi+z_\gamma^*[w,f]\otimes1_\chi+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||\gamma|}z_\alpha[z_\gamma^*,[z_\alpha^*,w]]\otimes1_\chi.
\end{split}
\end{equation}
By \eqref{wxyuv}, \eqref{3.5}, \eqref{3.6} and \eqref{3.7} we conclude that $[z_\gamma^*,\Theta_w]=0$ for any $\gamma\in S(-1)$, completing the proof.
\end{proof}
\begin{rem}
It is worth noting that similar result as in Proposition \ref{w} has been obtained by Suh in \cite[Proposition 5.3]{suh}. However, by careful inspection one can find that there are some errors in the first equation of \cite[(5.13)]{suh} and also the equation \cite[(5.16)]{suh}. Thus there is an error in the last part of the second equation of (5.12) in \cite[Proposition 5.3]{suh}, which is a little different from the consequence we obtained in Proposition \ref{w}.
\end{rem}
\subsubsection{}
Let $v\in\ggg^e(0)$ and $w\in\ggg^e(1)$. To determine the commutator between $\Theta_{v}$ and $\Theta_{w}$ defined in Proposition \ref{v} and Proposition \ref{w} respectively, we need to introduce an automorphism on $W_\chi'$.
Similar to the Lie algebra case in \cite[\S2.1]{GG}, we can define a linear action of $\mathbb{C}^*$ on $\ggg$. Given $\mathfrak{sl}_2$-triple $e,h,f\in\ggg_{\bar0}$, consider
the Lie algebra homomorphism $\mathfrak{sl}_2(\mathbb{C})\rightarrow\ggg_{\bar0}$ defined by
$$\left(\begin{array}{cc} 0 & 1\\0 & 0\end{array}\right)\mapsto e,
\qquad \left(\begin{array}{@{\hspace{0pt}}c@{\hspace{8pt}}c@{\hspace{0pt}}} 1 & 0\\0 & -1\end{array}\right)\mapsto h,
\qquad \left(\begin{array}{cc} 0 & 0\\1 & 0\end{array}\right)\mapsto f.$$
This Lie algebra homomorphism exponentiates to a rational homomorphism $$\tilde{\gamma}:~SL_2\rightarrow G_{\text{ev}}\rightarrow G.$$ We put
$$\gamma:\mathbb{C}^*\rightarrow G,\qquad \gamma(t)=\tilde{\gamma}
\left(\begin{array}{@{\hspace{0pt}}c@{\hspace{8pt}}c@{\hspace{0pt}}} t & 0\\0 & t^{-1}\end{array}\right),
\qquad \forall\,t\in\mathbb{C}^*.$$
Define $\sigma:=\text{Ad}\gamma(-1)$, an element of order $\leqslant2$ in $\text{Ad}\,G_{\text{ev}}$. Since the grading $\ggg=\bigoplus_{i\in \mathbb{Z}}\ggg(i)$ is obtained under the action of $\text{ad}\,h$, then for $x\in\ggg(i)$ we have $\sigma(x)=(-1)^{|i|}x$. Obviously $\sigma$ acts on $U(\ggg)$ as algebra automorphism. As $\sigma$ preserves the left ideal $I_\chi$ of $U(\ggg)$, it then acts on $Q_\chi$. As $\sigma$ preserves $\mathfrak{n}$ too, it acts on $W_\chi'\cong Q_\chi^{\text{ad}\,\mathfrak{n}}$ as algebra automorphism. One can easily conclude that
\begin{equation}\label{edegree}
\sigma(x^\mathbf{a}y^\mathbf{b}u^\mathbf{c}v^\mathbf{d}\otimes1_\chi)=(-1)^{|(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})|_e}
x^\mathbf{a}y^\mathbf{b}u^\mathbf{c}v^\mathbf{d}\otimes1_\chi
\end{equation}
for all $(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in{\bbz}_+^m\times\Lambda'_n\times{\bbz}_+^{s}\times\Lambda'_{\sfr}$.
We continue to consider the elements $\Theta_{v}$ and $\Theta_{w}$ for $v\in\ggg^e(0)$ and $w\in\ggg^e(1)$. In fact, we have the following result.
\begin{prop}\label{vw}
Let $v$ be an element in $\ggg^e(0)$ and $w$ be an element in $\ggg^e(1)$. Then the commutator between $\Theta_{v}$ and $\Theta_{w}$ in Proposition \ref{v} and Proposition \ref{w} is
\begin{equation}
[\Theta_{v},\Theta_{w}]=\Theta_{[v,w]}.
\end{equation}
\end{prop}
\begin{proof}
In virtue of \eqref{edegree}, we have $\sigma(\Theta_{v})=\Theta_{v}$ and $\sigma(\Theta_{w})=-\Theta_{w}$ by the definition of $\Theta_{v}$ and $\Theta_{w}$. Set $\Theta:=[\Theta_{v},\Theta_{w}]-\Theta_{[v,w]}$, an element in $W_\chi'$. Note that
\begin{equation*}
\begin{split}
\sigma(\Theta)=&\sigma([\Theta_{v},\Theta_{w}]-\Theta_{[v,w]})\\=&\sigma(\Theta_{v})\sigma(\Theta_{w})-(-1)^{|v||w|}
\sigma(\Theta_{w})\sigma(\Theta_{v})-\sigma(\Theta_{[v,w]})\\
=&-\Theta_{v}\Theta_{w}+(-1)^{|v||w|}\Theta_{w}\Theta_{v}+\Theta_{[v,w]}\\
=&-([\Theta_{v},\Theta_{w}]-\Theta_{[v,w]})\\=&-\Theta.
\end{split}
\end{equation*}
On the other hand, it follows from Theorem \ref{PBWC}(3) that $\Theta$ is a super-polynomial in $\Theta_{x_i}$ with $x_i\in\ggg^e(0)$. From all the discussion above, one knows that this super-polynomial must be zero. So $[\Theta_{v},\Theta_{w}]=\Theta_{[v,w]}$ necessarily holds, completing the proof.
\end{proof}
\begin{rem}
In fact, one can also prove Proposition \ref{vw} by computing the commutator between $\Theta_{v}$ and $\Theta_{w}$ for $v\in\ggg^e(0)$ and $w\in\ggg^e(1)$ directly. However, that proof is much complicate than the one we have applied above.
\end{rem}
\subsection{The case of minimal $W$-superalgebras}
Now we come to the case of minimal $W$-superalgebras. In this subsection the structure and the existence of $1$-dimensional representations of minimal $W$-superalgebras will be determined.
\subsubsection{}\label{3.1.1}
Retain the notations as in \S\ref{general}. A root $-\theta$ is called minimal if it is even and there exists an additive function $\varphi:\Phi\rightarrow\mathbb{R}$ such that $\varphi_{\mid{\Phi}}\neq0$ and $\varphi(\theta)>\varphi(\eta)$ for all $\eta\in\Phi\backslash\{\theta\}$. It is obvious that a minimal root $-\theta$ is the lowest root of one of the simple components of $\ggg_{\bar0}$ (in the ordering defined by $\varphi$).
Fix a minimal root $-\theta$ of $\ggg$. We may choose root vectors $e:=e_\theta$ and $f:=e_{-\theta}$ such
that $$[e,f]=h\in\mathfrak{h},\quad[h,e]=2e,\quad[h,f]=-2f.$$Then $e$ is a minimal nilpotent element in $\ggg$. Due to the minimality of $-\theta$, the eigenspace decomposition of $\text{ad}\,h$ gives rise to a short $\mathbb{Z}$-grading
$$\ggg=\ggg(-2)\oplus\ggg(-1)\oplus\ggg(0)\oplus\ggg(1)\oplus\ggg(2).$$
Moreover, $\ggg(2)=\mathbb{C}e, \ggg(-2)=\mathbb{C}f$.
We thus have a bijective correspondence between minimal gradings (up to an automorphism of $\ggg$) and minimal roots (up to the
action of the Weyl group). Furthermore, one has
$$\ggg^e=\ggg(0)^\sharp\oplus\ggg(1)\oplus\ggg(2),$$where $\ggg^e$ denotes the centralizer of $e$ in $\ggg$, and $\ggg(0)^\sharp=\{x\in\ggg(0)\mid[x,e]=0\}$.
Note that $\ggg(0)^\sharp$ is the centralizer of the triple $\{e,f,h\}$ by $\mathfrak{sl}_2$-theory.
Moreover, $\ggg(0)^\sharp$ is the orthogonal complement to $\mathbb{C}h$ in $\ggg(0)$, and coincides with the image of the Lie
superalgebra endomorphism
\begin{equation}\label{xh}
\sharp: \ggg(0)\rightarrow\ggg(0), x\mapsto x-\frac{1}{2}(h,x)h.
\end{equation} Obviously $\ggg(0)^{\sharp}$ is an ideal of codimensional $1$ in the Levi subalgebra $\ggg(0)$.
\subsubsection{The proof of Proposition \ref{ge}}\label{3.1.3}
Let $C$ denote the Casimir element of $U(\ggg)$ corresponding to the invariant form $(\cdot,\cdot)$. Since $C$ induces a $\ggg$-endomorphism of $Q_\chi$, it can be viewed as a central element of $W_\chi'$. Now we will determine the explicit formula of $C$.
Let $\{a_i\mid i\in I\}$ and $\{b_i\mid i\in I\}$ be dual bases of $\ggg^e(0)$ with respect to the restriction of the invariant form $(\cdot,\cdot)$ to $\ggg^e(0)$, then
$$\{e,h,f\}\cup\{a_i\mid i\in I\}\cup\{[e,z_\alpha^*]\mid\alpha\in S(-1)\}\cup\{z_\alpha\mid \alpha\in S(-1)\}$$
and
$$\{f,\frac{h}{2},e\}\cup\{b_i\mid i\in I\}\cup\{z_\alpha\mid \alpha\in S(-1)\}\cup\{(-1)^{|\alpha|}[e,z_\alpha^*]\mid\alpha\in S(-1)\}$$
are dual bases of $\ggg$ with respect to $(\cdot,\cdot)$. Since $a_i$ and $b_i$ have the same parity for given $i\in I$, we will denote this parity by $|i|$. Recall that if $z_\alpha=z_\beta^*$ for some $\alpha,\beta\in S(-1)$, then the assumption on dual bases of $\ggg(-1)$ in \S\ref{general} implies that $z_\alpha^*=-(-1)^{|\beta|}z_\beta$. In virtue of this, we have\begin{equation}\label{zalphazalpha}
\sum\limits_{\alpha\in S(-1)}[[z_\alpha,e],z_\alpha^*]\otimes1_\chi=-\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[[z_\alpha^*,e],z_\alpha]\otimes1_\chi.\end{equation}Recall that $s=\text{dim}\,\ggg(-1)_{\bar0}$ and $\sfr=\text{dim}\,\ggg(-1)_{\bar1}$. Since
\begin{equation}\label{3.9}
\begin{split}
\sum\limits_{\alpha\in S(-1)}[z_\alpha,[e,z_\alpha^*]]\otimes1_\chi
=&\sum\limits_{\alpha\in S(-1)}([[z_\alpha,e],z_\alpha^*]+[e,[z_\alpha,z_\alpha^*]])\otimes1_\chi\\
=&-\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}([[z_\alpha^*,e],z_\alpha]
+[e,[z_\alpha^*,z_\alpha]])\otimes1_\chi\\=&-\sum\limits_{\alpha\in S(-1)}[z_\alpha,[e,z_\alpha^*]]\otimes1_\chi+(\sfr-s)h\otimes1_\chi,
\end{split}
\end{equation} where the second equation in \eqref{3.9} follows from \eqref{zalphazalpha}. Then we have $$\sum\limits_{\alpha\in S(-1)}[z_\alpha,[e,z_\alpha^*]]\otimes1_\chi=\frac{(\sfr-s)}{2}h\otimes1_\chi.$$
As a consequence,
\begin{equation}\label{C}
\begin{split}
C=&(2e+\frac{h^2}{2}-h+\sum\limits_{i\in I}(-1)^{|i|}a_ib_i+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[e,z_\alpha^*]z_\alpha
+\sum\limits_{\alpha\in S(-1)}z_\alpha[e,z_\alpha^*])\otimes1_\chi\\
=&(2e+\frac{h^2}{2}-h+\sum\limits_{i\in I}(-1)^{|i|}a_ib_i+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[e,z_\alpha^*]z_\alpha
+\sum\limits_{\alpha\in S(-1)}[z_\alpha,[e,z_\alpha^*]]\\
&+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[e,z_\alpha^*]z_\alpha)\otimes1_\chi\\
=&(2e+\frac{h^2}{2}-(1+\frac{s-\sfr}{2})h+\sum\limits_{i\in I}(-1)^{|i|}a_ib_i
+2\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[e,z_\alpha^*]z_\alpha)\otimes1_\chi.
\end{split}
\end{equation}
Combining Proposition \ref{v}, Proposition \ref{w} and the discussion in \S\ref{3.1.1}, we now obtain a generating set (as an associative superalgebra) of a refined $W$-superalgebra associated with a minimal nilpotent element as presented in Proposition \ref{ge}.
\iffalse
\begin{prop}\label{generator}
Let $e$ be a minimal nilpotent element in $\ggg$. Suppose $v\in\ggg^e(0)$, $w\in\ggg^e(1)$, $C$ is a central element of $W_\chi'$, and set $s=\text{dim}\,\ggg(-1)_{\bar0}$ and $\sfr=\text{dim}\,\ggg(-1)_{\bar1}$. Then the followings are free generators of the refined $W$-superalgebra $W_\chi'$:
\begin{equation*}
\begin{split}
\Theta_v=&(v-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}z_\alpha[z_\alpha^*,v])\otimes1_\chi,\\
\Theta_w=&(w-\sum\limits_{\alpha\in S(-1)}z_\alpha[z_\alpha^*,w]+\frac{1}{3}(\sum\limits_{\alpha,\beta\in S(-1)}z_\alpha z_\beta[z_\beta^*,[z_\alpha^*,w]]-2[w,f]))\otimes1_\chi,\\
C=&(2e+\frac{h^2}{2}-(1+\frac{s-\sfr}{2})h+\sum\limits_{i\in I}(-1)^{|i|}a_ib_i
+2\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[e,z_\alpha^*]z_\alpha)\otimes1_\chi,
\end{split}
\end{equation*}where $\{z_\alpha^*\mid\alpha\in S(-1)\}$ and $\{z_\alpha\mid\alpha\in S(-1)\}$ are dual bases of $\ggg(-1)$ with respect to $\langle\cdot,\cdot\rangle$, and $\{a_i\mid i\in I\}$ and $\{b_i\mid i\in I\}$ are dual bases of $\ggg^e(0)$ with respect $(\cdot,\cdot)$.
\end{prop}
\fi
Now we can easily make an account of Proposition \ref{ge} as below.
\begin{proof}
The proposition follows from Theorem \ref{PBWC}, Proposition \ref{v}, Proposition \ref{w}, \eqref{C} and the discussion in \S\ref{3.1.1}.
\end{proof}
\subsubsection{}
In this part, we make further preparation for Theorem \ref{maiin1}, by determining the commutators between the generators of $W_\chi'$ given in Proposition \ref{ge}.
Recall that in \S\ref{3.1.1} and \S\ref{3.1.3} we showed that $\ggg^e(0)=\ggg(0)^\sharp$ is an ideal of codimensional $1$ in the Levi subalgebra $\ggg(0)$ of $\ggg$, with $\{a_i\mid i\in I\}$ and $\{b_i\mid i\in I\}$ being dual bases of $\ggg^e(0)$ with respect to the restriction of the invariant form $(\cdot,\cdot)$ to $\ggg^e(0)$. Now let $C_0:=\sum_{i\in I}a_ib_i$ be the corresponding Casimir element of $U(\ggg^e(0))$, and set $\Theta_{\text{Cas}}:=\sum_{i\in I}(-1)^{|i|}\Theta_{a_i}\Theta_{b_i}$ to be an element of $W_\chi'$. Although $\Theta_{\text{Cas}}$ is not central in $W_\chi'$, we have the following result.
\begin{prop}\label{vcommutate}
The element $\Theta_{\text{Cas}}$ commutes with all operators $\Theta_v$ for $v\in\ggg^e(0)$.
\end{prop}
\begin{proof}
Since $\{a_i\mid i\in I\}$ and $\{b_i\mid i\in I\}$ are dual bases of $\ggg^e(0)$ with respect to the invariant form $(\cdot,\cdot)$, we have
\begin{equation}\label{3.46}
\begin{split}
[a_i,v]=&\sum\limits_{j\in I}([a_i,v],b_j)a_j=\sum\limits_{j\in I}(a_i,[v,b_j])a_j\\
=&-\sum\limits_{j\in I}(-1)^{|v||j|}(a_i,[b_j,v])a_j,\\
[b_i,v]=&\sum\limits_{j\in I}(a_j,[b_i,v])b_j
\end{split}
\end{equation}for any $i\in I$.
Then one can obtain from \eqref{3.46} and Proposition \ref{v1v2} that
\begin{equation}\label{3.47}
\begin{split}
&\Big[\sum\limits_{i\in I}(-1)^{|i|}\Theta_{a_i}\Theta_{b_i},\Theta_v\Big]\\
=&\sum\limits_{i\in I}((-1)^{(|v|+1)|i|}[\Theta_{a_i},\Theta_v]\Theta_{b_i}+(-1)^{|i|}\Theta_{a_i}[\Theta_{b_i},\Theta_v])\\
=&\sum\limits_{i\in I}((-1)^{(|v|+1)|i|}\Theta_{[{a_i},v]}\Theta_{b_i}+(-1)^{|i|}\Theta_{a_i}\Theta_{[{b_i},v]})\\
=&-\sum\limits_{i,j\in I}(-1)^{|v|(|i|+|j|)+|i|}(a_i,[b_j,v])\Theta_{a_j}\Theta_{b_i}
+\sum\limits_{i,j\in I}(-1)^{|i|}(a_j,[b_i,v])\Theta_{a_i}\Theta_{b_j}\\
=&-\sum\limits_{i,j\in I}(-1)^{|v|(|i|+|j|)+|j|}(a_j,[b_i,v])\Theta_{a_i}\Theta_{b_j}
+\sum\limits_{i,j\in I}(-1)^{|i|}(a_j,[b_i,v])\Theta_{a_i}\Theta_{b_j}.
\end{split}
\end{equation}
As the bilinear form $(\cdot,\cdot)$ is even, the first term of the last equation in \eqref{3.47} is nonzero if and only if
\begin{equation*}
|v|(|i|+|j|)+|j|=(|i|+|j|)^2+|j|=|i|+|j|+|j|=|i|.
\end{equation*}
Combining this with \eqref{3.47}, we have
$$\Big[\sum\limits_{i\in I}(-1)^{|i|}\Theta_{a_i}\Theta_{b_i},\Theta_v\Big]=0,$$completing the proof.
\end{proof}
Now we can calculate the remaining commutators of the generators of a minimal $W$-superalgebra $W_\chi'$ in Proposition \ref{ge}.
First note that the Casimir element $C$ is a central element of $W_\chi'$, then we have
\begin{equation}\label{casimircom}
[C,W_\chi']=0.
\end{equation}
Then we have finished the arguments for all the items of relations in Theorem \ref{maiin1} except (3). Next we deal with the exceptional item.
\begin{prop}\label{1commutator}
Let $w_1, w_2\in\ggg^e(1)$. Then the following relation holds in $W_\chi'$:
\begin{equation}
\begin{split}
[\Theta_{w_1},\Theta_{w_2}]=&\frac{1}{2}([w_1,w_2],f)(C-\Theta_{\text{Cas}}-c_0)-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}\\
&-(-1)^{|w_1||w_2|}\Theta_{[w_2,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_1]^{\sharp}}),
\end{split}
\end{equation}
where $c_0\in\mathbb{C}$ is a constant depending on $\ggg$. To be explicit,
the constant $c_0$ satisfies the following equation:
\begin{equation*}
\begin{split}
c_0([w_1,w_2],f)=&\frac{1}{12}\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha||w_1|+|\beta||w_1|+|\alpha||\beta|}\otimes[[z_\beta,[z_\alpha,w_1]],[z_\beta^*,[z_\alpha^*,w_2]]]\\
&-\frac{3(s-\sfr)+4}{12}([w_1,w_2],f),
\end{split}
\end{equation*}
where $s=\text{dim}\,\ggg(-1)_{\bar0}$ and $\sfr=\text{dim}\,\ggg(-1)_{\bar1}$.
\end{prop}
Since the proof of Proposition \ref{1commutator} is rather lengthy, we will postpone it till \S\ref{proof}.
\begin{rem}
Compared with the result obtained by Suh in \cite[Proposition 5.4]{suh}, one can find that there are subtle differences between the last equation of (5.17) obtained by Suh in \cite{suh} and the one we introduced in Theorem \ref{maiin1}(3), i.e., the term $-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}$ in Theorem \ref{maiin1}(3) did not appear in \cite[Proposition 5.4]{suh}. One reason may lie in the fact that the expression in the second equation of \cite[(5.12)]{suh} is error. On the other hand, from the detailed calculation in the proof of \eqref{3.57}, one will observe that without the term we mentioned above, the term $\frac{1}{4}\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_2|}[[w_1,z_\alpha],[w_2,z_\alpha^*]]\otimes1$ (which is not a constant in general) cannot be eliminated by other terms.
\end{rem}
We are now in a position to prove the main results of the present paper.
\subsubsection{The Proof of Theorem \ref{maiin1}} \label{5.2.4}
This theorem readily follows from Proposition \ref{ge}, Proposition \ref{v1v2}, Proposition \ref{vw}, \eqref{casimircom}, Proposition \ref{1commutator}, and finally Theorem \ref{PBWC}.
\subsubsection{The Proof of Theorem \ref{intromainminnimalf}}\label{5.2.5}
In virtue of the results we obtained above, we will show that Conjecture \ref{conjecture22} is true for minimal $W$-superalgebras, and the lower bounds for the representations of basic Lie superalgebras with minimal nilpotent $p$-characters in Theorem \ref{intromain-2} are accessible. We mainly follow the method used in \cite[Corollary 4.1]{P3} for the finite $W$-algebra case.
Let $\ggg$ be a basic Lie superalgebra over $\mathbb{C}$, and $e$ be a minimal nilpotent element in $\ggg$. Denote by $\ggg^e$ the centralizer of $e$ in $\ggg$. Set $\{x_1,\cdots,x_m\}$, $\{y_1,\cdots,y_n\}$, $\{u_1,\cdots,u_{s}\}$ and $\{v_1,\cdots,v_t\}$ to be bases of $\ggg^e(0)_{\bar0}$, $\ggg^e(0)_{\bar1}$, $\ggg^e(1)_{\bar0}$ and $\ggg^e(1)_{\bar1}$, respectively. Let $W_\chi'$ be the minimal $W$-superalgebra associated with the pair ($\ggg$, $e$). Denote by $(W_\chi')^+$ the $\mathbb{C}$-span of the monomials
$$\Theta_{x_1}^{i_1}\cdots\Theta_{x_m}^{i_m}\cdot
\Theta_{y_1}^{j_1}\cdots\Theta_{y_n}^{j_n}\cdot
\Theta_{u_1}^{k_1}\cdots\Theta_{u_{s}}^{k_{s}}\cdot
\Theta_{v_1}^{l_1}\cdots\Theta_{v_t}^{l_t}\cdot(C-c_0)^q$$
with $i_r, k_r, q\in\mathbb{Z}_+$ and $j_r, l_r\in\{0,1\}$ such that $\sum i_r+\sum j_r+\sum k_r+\sum l_r+q\geqslant1$. It follows from Theorem \ref{PBWC}(2) that $(W_\chi')^+$ is a subspace of codimensional $1$ in $W_\chi'$. We now prove that $(W_\chi')^+$ is a two-sided ideal of $W_\chi'$, i.e.,
\begin{prop}\label{minimal nilpotent}
The subspace $(W_\chi')^+$ is a two-sided ideal of codimension $1$ in the refined $W$-superalgebra $W_\chi'$ associated with minimal nilpotent element $e$. Moreover, if $\ggg$ is not of type $A(m|n)$ with $m, n\geqslant0$, then $(W_\chi')^+$ is the only ideal of codimensional $1$ in $W_\chi'$.
\end{prop}
\begin{proof}
We first show that $h\cdot h'\in (W_\chi')^+$ for all $h, h'\in (W_\chi')^+$. Since $C-c_0$ is in the center of $W_\chi'$, we have $(C-c_0)\cdot (W_\chi')^+\subseteq (W_\chi')^+$. Therefore, we just need to prove that $\Theta_x\cdot (W_\chi')^+\subseteq(W_\chi')^+$ for all $x\in\ggg^e(0)\cup\ggg^e(1)$.
For the case with $x\in\ggg^e(0)$, we can conclude from Proposition \ref{v1v2} that the span of all $\Theta_{x_1}^{i_1}\cdots\Theta_{x_m}^{i_m}\cdot\Theta_{y_1}^{j_1}\cdots\Theta_{y_n}^{j_n}$ with $i_r\in\mathbb{Z}_+, j_r\in\{0,1\}$ and $\sum i_r+\sum j_r\geqslant1$ is stable under the left multiplications of $\Theta_{x}$ by $x\in\ggg^e(0)$. Now we assume that $x\in\ggg^e(1)$. Since the monomials $\Theta_{x_1}^{i_1}\cdots\Theta_{x_m}^{i_m}\cdot
\Theta_{y_1}^{j_1}\cdots\Theta_{y_n}^{j_n}\cdot
\Theta_{u_1}^{k_1}\cdots\Theta_{u_{s}}^{k_{s}}\cdot
\Theta_{v_1}^{l_1}\cdots\Theta_{v_t}^{l_t}\cdot(C-c_0)^q$ with $i_r, k_r\in\mathbb{Z}_+$ and $j_r, l_r\in\{0,1\}$ form a basis of the refined $W$-superalgebra $W_\chi'$, by Theorem \ref{PBWC} and Proposition \ref{vw} it suffices to prove that $\Theta_{x}\cdot\Theta_{u_1}^{k_1}\cdots\Theta_{u_{k_M}}^{k_M}\cdot
\Theta_{v_1}^{l_1}\cdots\Theta_{v_N}^{l_N}\in (W_\chi')^+$ for all $k_1,\cdots,k_M\in\{1,\cdots,s\}$, $l_1,\cdots,l_N\in\{1,\cdots,t\}$ and all $x\in\ggg^e(1)$. In fact, in view of Proposition \ref{vw} and Proposition \ref{1commutator} we can reach this conclusion by induction on $M$ and $N$.
Let $\ggg$ be not of type $A(m|n)$ with $m, n\geqslant0$, then it follows from \cite[Table 1-Table 3]{KW} that $\ggg^e(0)$ is a semisimple Lie (super)algebra (which is denoted by $\ggg^\natural$ in the settings there). Let $I$ be any ideal of codimensional $1$ in the refined $W$-superalgebra $W_\chi'$. One can easily verify that $[x,y]=xy-(-1)^{|x||y|}yx\in I$ for all $x, y\in W_\chi'$. Since $\ggg^e(0)=[\ggg^e(0),\ggg^e(0)]$ by the above remark, Proposition \ref{v1v2} implies that $\Theta_x\in I$ for $x\in\ggg^e(0)$. Moreover, \cite[Table 1-Table 3]{KW} also showed that $\ggg^e(1)=\ggg(1)$ (which is denoted by $\ggg_{\frac{1}{2}}$ there) is an irreducible $\ggg^e(0)$-module, then we have $[\ggg^e(0),\ggg^e(1)]=\ggg^e(1)$. Thus $\Theta_x\in I$ for all $x\in \ggg^e(1)$ by Proposition \ref{vw}. As $I$ is a subalgebra of $W_\chi'$ containing $\ggg^e(0)\cup\ggg^e(1)$, Proposition \ref{1commutator} implies that $C-c_0$ also in $I$. All the above shows that $(W_\chi')^+\subseteq I$ by definition. Since the codimension of $(W_\chi')^+$ is $1$ in $W_\chi'$, we conclude that $I=(W_\chi')^+$, as desired.
\end{proof}
It is immediate from Proposition \ref{minimal nilpotent} that Conjecture \ref{conjecture22} is true for minimal $W$-superalgebras.
As a direct corollary of Theorem \ref{intromain-2} and Proposition \ref{minimal nilpotent}, we complete the proof of Theorem \ref{intromainminnimalf}.
\iffalse
\begin{theorem}\label{intromainminnimal}
Let ${\ggg}_{\bbk}$ be a basic Lie superalgebra over ${\bbk}=\overline{\mathbb{F}}_p$, and let $\chi\in({\ggg}_{\bbk})^*_{\bar0}$ be a nilpotent $p$-character, with respect to a minimal nilpotent element $e\in(\ggg_\bbk)_\bz$. If $p\gg0$, then the reduced enveloping algebra $U_\chi({\ggg}_{\bbk})$ admits irreducible representations of dimension $p^{\frac{d_0}{2}}2^{\lfloor\frac{d_1}{2}\rfloor}$.
\end{theorem}
\fi
\section{The proof of Proposition \ref{1commutator}}\label{proof}
This section is contributed to the proof of Proposition \ref{1commutator}. We mainly follow Premet's strategy on finite $W$-algebras \cite[\S4]{P3}, with a few modifications.
In the first two subsections, we will make some necessary preparation. To simplify calculation, in \S\ref{improve} we rewrite generators of minimal $W$-superalgebras appearing in Proposition \ref{ge} in another way. Then in \S\ref{6.2} we will present relations of the generators by lots of computations. The final proof of Proposition \ref{1commutator} will be given in \S\ref{6.4}.
Retain all the notations as in \S\ref{structure} throughout this section.
\subsection{}\label{improve} In this subsection, we first express the generators $\Theta_w$ with $w\in\ggg^e(1)$ of $W_\chi'$ in Proposition \ref{ge} in some more computable way, which will be fulfilled in (\ref{rewrite}) via Proposition \ref{translate}.
Let us begin with the following equation
\begin{equation}\label{zzwzzw}
\sum\limits_{\alpha\in S(-1)}[z_\alpha,[z_\alpha^*,w]]\otimes1_\chi=-\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[z_\alpha^*,[z_\alpha,w]])\otimes1_\chi,
\end{equation} then we have
\begin{equation*}\label{zzw}
\begin{split}
\sum\limits_{\alpha\in S(-1)}[z_\alpha,[z_\alpha^*,w]]\otimes1_\chi
&=\sum\limits_{\alpha\in S(-1)}([[z_\alpha,z_\alpha^*],w]+(-1)^{|\alpha|}[z_\alpha^*,[z_\alpha,w]])\otimes1_\chi\\
&=((s-\sfr)[w,f]-\sum\limits_{\alpha\in S(-1)}[z_\alpha,[z_\alpha^*,w]])\otimes1_\chi,
\end{split}
\end{equation*} thus $\sum\limits_{\alpha\in S(-1)}[z_\alpha,[z_\alpha^*,w]]\otimes1_\chi=\frac{(s-\sfr)}{2}[w,f]\otimes1_\chi$.
It follows that
\begin{equation*}
\begin{split}
\sum\limits_{\alpha\in S(-1)}z_\alpha[z_\alpha^*,w]\otimes1_\chi&=\sum\limits_{\alpha\in S(-1)}([z_\alpha,[z_\alpha^*,w]]+(-1)^{|w||\alpha|+|\alpha|}[z_\alpha^*,w]z_\alpha)\otimes1_\chi\\
&=\Big(\frac{(s-\sfr)}{2}[w,f]-\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[w,z_\alpha^*]z_\alpha\Big)\otimes1_\chi.
\end{split}
\end{equation*}
Define \begin{equation}\label{varphi}
\varphi_w:=\frac{1}{3}(\sum\limits_{\alpha,\beta\in S(-1)}z_\alpha z_\beta[z_\beta^*,[z_\alpha^*,w]]-\frac{3(s-\sfr)+4}{2}[w,f]),\end{equation}
then we can write
\begin{equation}\label{anotherw}
\Theta_w=(w+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[w,z_\alpha^*]z_\alpha+\varphi_w)\otimes1_\chi.
\end{equation}
Now we come to the case with $w_1, w_2\in\ggg^e(1)$. To simplify calculation, we need the following settings.
Since each vector $h\in W_\chi'$ can be uniquely expressed as $h=\sum\limits_{(\mathbf{i},\mathbf{j})\in\mathbb{Z}_+^{s}
\times\Lambda'_{\sfr}}x_\mathbf{i}y_\mathbf{j}u^\mathbf{i}v^\mathbf{j}\otimes1_\chi$ with $x_\mathbf{i}y_\mathbf{j}\in U(\mathfrak{p})$, one can define a natural linear injection
\begin{equation*}
\begin{array}{llll}
\mu:&W_\chi'&\longrightarrow&U(\mathfrak{p})\otimes A_e^{\text{op}}\\
&h&\mapsto&\sum\limits_{(\mathbf{i},\mathbf{j})\in\mathbb{Z}_+^{s}\times\Lambda'_{\sfr}}x_\mathbf{i}y_\mathbf{j}\otimes u^\mathbf{i}v^\mathbf{j},
\end{array}
\end{equation*}where $A_e^{\text{op}}$ denotes the opposite algebra of $A_e$ as in \S\ref{general}.
Obviously the mapping $\mu$ is injective. Moreover, we have the following result.
\begin{prop}\label{translate}
The map $\mu: W_\chi' \longrightarrow U(\mathfrak{p})\otimes A_e^{\text{op}}$ is an algebra homomorphism.
\end{prop}
\begin{proof}
Denote by $\mathscr{Z}$ the linear span of all $u^\mathbf{i}v^\mathbf{j}$ with $(\mathbf{i},\mathbf{j})\in\mathbb{Z}_+^{s}\times\Lambda'_{\sfr}$. Then we can identify $\mathscr{Z}$ with the span of the left regular representation of $A_e$ via $u^\mathbf{i}v^\mathbf{j}\otimes1_\chi\mapsto u^\mathbf{i}v^\mathbf{j}$. Let $\rho_\chi$ denote the left regular representation of $U(\mathfrak{n})$ in $\text{End}(\mathscr{Z})$. Recall that we have assumed that $\{z_\alpha\mid\alpha\in S(-1)\}=\{u_\alpha\mid\alpha\in S(-1)_{\bar0}\}\bigcup\{v_{\alpha}\mid\alpha\in S(-1)_{\bar1}\}$ in \S\ref{general}. As $\mathfrak{g}(-1)\subseteq\mathfrak{n}$, and $\mathfrak{g}(i)\subseteq\text{Ker}\,\chi$ for all $i\leqslant-3$, the definition of $W_\chi'$ and induction on $k$ show that
\begin{equation}\label{uvz}
\rho_\chi(z_1\cdots z_k)(h)=(-1)^{|h|(|z_1|+\cdots+|z_k|)}\sum\limits_{(\mathbf{i},\mathbf{j})\in\mathbb{Z}_+^{s}
\times\Lambda'_{\sfr}}x_\mathbf{i}y_\mathbf{j}\cdot\rho_\chi(u^\mathbf{i}v^\mathbf{j}\cdot z_1\cdots z_k)(1_\chi)
\end{equation}
for all $z_1,\cdots,z_k\in\mathfrak{g}(-1)$.
Let $h'$ be another element in $W_\chi'$ such that $h'=\sum\limits_{(\mathbf{i},\mathbf{j})\in\mathbb{Z}_+^{s}
\times\Lambda'_{\sfr}}
x'_\mathbf{i}y'_\mathbf{j}(u')^{\mathbf{i}}(v')^{\mathbf{j}}\otimes1_\chi$ with $x'_\mathbf{i}y'_\mathbf{j}\in U(\mathfrak{p})$, then it follows from \eqref{uvz} that \[
\begin{split}
h\cdot h'=&\sum\limits_{(\mathbf{i},\mathbf{j})}
\rho_\chi(x_\mathbf{i}y_\mathbf{j})\cdot\rho_\chi(u^{\mathbf{i}}v^{\mathbf{j}})(h')\\
=&\sum\limits_{(\mathbf{i},\mathbf{j})}\sum\limits_{(\mathbf{k},\mathbf{l})}(-1)^{|h'||\mathbf{j}|}x_\mathbf{i}y_\mathbf{j}\cdot
x'_\mathbf{k}y'_\mathbf{l}\cdot\rho_\chi((u')^\mathbf{k}(v')^\mathbf{l}\cdot u^{\mathbf{i}}v^{\mathbf{j}})\otimes1_\chi,
\end{split}\]
thus
\begin{equation*}
\mu(h\cdot h')=\sum\limits_{(\mathbf{i},\mathbf{j})}\sum\limits_{(\mathbf{k},\mathbf{l})}(-1)^{|h'||\mathbf{j}|}x_\mathbf{i}y_\mathbf{j}\cdot
x'_\mathbf{k}y'_\mathbf{l}\otimes(u')^\mathbf{k}(v')^\mathbf{l}\cdot u^{\mathbf{i}}v^{\mathbf{j}}.
\end{equation*}
It remains to note that the map $u^\mathbf{i}v^\mathbf{j}\otimes1_\chi\mapsto u^\mathbf{i}v^\mathbf{j}$ mentioned above identifies $\rho_\chi(U(\mathfrak{n}))$ with the image of $A_e$ in its left regular representation, then the proof is completed.
\end{proof}
In virtue of Proposition \ref{translate}, we can rewrite the generators of $W_\chi'$ in Proposition \ref{ge} and \eqref{anotherw} as follows:
\begin{equation}\label{rewrite}
\begin{split}
\Theta_v=&v\otimes1-\sum\limits_{\alpha\in S(-1)}\frac{1}{2}\otimes z_\alpha[z_\alpha^*,v],\\
\Theta_w=&w\otimes1+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[w,z_\alpha^*]\otimes z_\alpha+1\otimes\varphi_w,\\
C=&2e\otimes1+\frac{h^2}{2}\otimes1-(1+\frac{s-r}{2})h\otimes1+\sum\limits_{i\in I}(-1)^{|i|}a_ib_i\otimes1\\
&+2\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[e,z_\alpha^*]\otimes z_\alpha,
\end{split}
\end{equation} where $\{z_\alpha^*\mid\alpha\in S(-1)\}$ and $\{z_\alpha\mid\alpha\in S(-1)\}$ are dual bases of $\ggg(-1)$ with respect to $\langle\cdot,\cdot\rangle$, $\{a_i\mid i\in I\}$ and $\{b_i\mid i\in I\}$ be dual bases of $\ggg^e(0)$ with respect $(\cdot,\cdot)$, and $\varphi_w=\frac{1}{3}(\sum\limits_{\alpha,\beta\in S(-1)}z_\alpha z_\beta[z_\beta^*,[z_\alpha^*,w]]-\frac{3(s-\sfr)+4}{2}[w,f])$.
\subsection{}\label{6.2} In this subsection, we will present the relations of the generators described just above, which will be given in (\ref{3.57}). The process will be long and tedious.
First, notice that in $U(\mathfrak{p})\otimes A_e^{\text{op}}$ we have
\begin{equation}\label{aop}
\begin{split}
[a\otimes f,b\otimes g]&=(-1)^{|b||f|+|f||g|}ab\otimes gf-(-1)^{(|a|+|f|)(|b|+|g|)+|a||g|+|f||g|}ba\otimes fg\\
&=(-1)^{|f|(|b|+|g|)}[a,b]\otimes gf-(-1)^{|b|(|a|+|f|)}ba\otimes[f,g]
\end{split}
\end{equation}
for all $a,b\in U(\mathfrak{p})$ and $f,g\in A_e$.
For all $w_1,w_2\in\ggg^e(1)$, we will compute the commutators between $\Theta_{w_1}$ and $\Theta_{w_2}$ in this subsection.
\subsubsection{}
Keeping \eqref{aop} in mind, we have
\begin{equation}\label{theta12}
\begin{split}
[\Theta_{w_1},\Theta_{w_2}]
=&[w_1\otimes1+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[w_1,z_\alpha^*]\otimes z_\alpha+1\otimes\varphi_{w_1},w_2\otimes1\\
&+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[w_2,z_\alpha^*]\otimes z_\alpha+1\otimes\varphi_{w_2}]\\
=&[w_1,w_2]\otimes1+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|+|\alpha||w_2|}[[w_1,z_\alpha^*],w_2]\otimes z_\alpha\\
&+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[w_1,[w_2,z_\alpha^*]]\otimes z_\alpha\\
&+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||w_2|}[[w_1,z_\alpha^*],[w_2,z_\beta^*]]\otimes z_\beta z_\alpha\\&-\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\beta||w_1|+|w_1||w_2|}[w_2,z_\beta^*][w_1,z_\alpha^*]\otimes[z_\alpha,z_\beta]\\
&-\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|+|\alpha||w_1|+|w_1||w_2|}[w_2,z_\alpha^*]\otimes[\varphi_{w_1},z_\alpha]\\
&-\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[w_1,z_\alpha^*]\otimes[z_\alpha,\varphi_{w_2}]-1\otimes[\varphi_{w_1},\varphi_{w_2}]\\
=&[w_1,w_2]\otimes1+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[[w_1,w_2],z_\alpha^*]\otimes z_\alpha\\
&+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||w_2|}[[w_1,z_\alpha^*],[w_2,z_\beta^*]]\otimes z_\beta z_\alpha\\
&+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|+|\alpha||w_1|+|w_1||w_2|}[w_2,z_\alpha^*][w_1,z_\alpha]\otimes1\\
&-\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}
[w_2,z_\alpha]\otimes[z_\alpha^*,\varphi_{w_1}]+\sum\limits_{\alpha\in S(-1)}[w_1,z_\alpha]\otimes[z_\alpha^*,\varphi_{w_2}]\\
&-1\otimes[\varphi_{w_1},\varphi_{w_2}].
\end{split}
\end{equation}
For any $y\in\ggg^e(1)$ and $\alpha\in S(-1)$, it follows from the definition of $\varphi_y$ and \eqref{3.5}, \eqref{3.6}, \eqref{3.7} that
\begin{equation}\label{zw}
\begin{split}
[z_\alpha^*,\varphi_y]=&[z_\alpha^*,\frac{1}{3}(\sum\limits_{\beta,\gamma\in S(-1)}z_\beta z_\gamma[z_\gamma^*,[z_\beta^*,y]]-\frac{3(s-\sfr)+4}{2}[y,f])]\\
=&\frac{\sfr-s-2}{2}[z_\alpha^*,[y,f]]+\sum\limits_{\beta\in S(-1)}(-1)^{|\alpha||\beta|}z_\beta[z_\alpha^*,[z_\beta^*,y]].
\end{split}
\end{equation}
It is worth noting that for any $u\in\ggg(-1)$, we have
\begin{equation}\label{zuz}
u=\sum\limits_{\alpha\in S(-1)}[z_\alpha^*,u]z_\alpha=-\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[z_\alpha,u]z_\alpha^*.
\end{equation} One can conclude from \eqref{zzwzzw}, \eqref{zw} and \eqref{zuz} that
\begin{equation}\label{xzzy}
\begin{split}
&\sum\limits_{\alpha\in S(-1)}[x,z_\alpha]\otimes[z_\alpha^*,\varphi_y]\\
=&\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha||\beta|}[x,z_\alpha]\otimes z_\beta[z_\alpha^*,[z_\beta^*,y]]+\frac{\sfr-s-2}{2}\sum\limits_{\alpha\in S(-1)}[x,z_\alpha]\otimes[z_\alpha^*,[y,f]]\\
=&\sum\limits_{\alpha,\beta,\gamma\in S(-1)}(-1)^{|\alpha||\beta|}[x,z_\alpha]\otimes z_\beta [z_\gamma^*,[z_\alpha^*,[z_\beta^*,y]]]z_\gamma\\&+\frac{\sfr-s-2}{2}\sum\limits_{\alpha\in S(-1)}[x,z_\alpha]\otimes[z_\alpha^*,[y,f]]\\
=&\sum\limits_{\alpha,\beta,\gamma\in S(-1)}(-1)^{|\alpha||\beta|}[x,z_\alpha]\otimes z_\beta [[z_\gamma^*,z_\alpha^*],[z_\beta^*,y]]z_\gamma\\
&+\sum\limits_{\alpha,\beta,\gamma\in S(-1)}(-1)^{|\alpha|(|\beta|+|\gamma|)}[x,z_\alpha]\otimes z_\beta[z_\alpha^*,[z_\gamma^*,[z_\beta^*,y]]]z_\gamma\\&+\frac{\sfr-s-2}{2}\sum\limits_{\alpha\in S(-1)}[x,z_\alpha]\otimes[z_\alpha^*,[y,f]]\\
=&\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||y|+|\alpha|}[x,z_\alpha^*]\otimes[y,f]z_\alpha\\
&+\sum\limits_{\beta,\gamma\in S(-1)}(-1)^{(|\beta|+|\gamma|)(1+|y|)}[x,[z_\gamma^*,[z_\beta^*,y]]]\otimes z_\beta z_\gamma\\
&+\frac{\sfr-s-2}{2}\sum\limits_{\alpha\in S(-1)}[x,z_\alpha]\otimes[z_\alpha^*,[y,f]]\\
=&-\sum\limits_{\alpha\in S(-1)}[x,z_\alpha]\otimes z_\alpha^*[y,f]+\sum\limits_{\beta,\gamma\in S(-1)}(-1)^{(|\beta|+|\gamma|)(1+|y|)}[x,[z_\gamma^*,[z_\beta^*,y]]]\otimes z_\beta z_\gamma\\
&+\frac{\sfr-s}{2}\sum\limits_{\alpha\in S(-1)}[x,z_\alpha]\otimes[z_\alpha^*,[y,f]]\\
=&-\sum\limits_{\alpha\in S(-1)}[x,z_\alpha]\otimes z_\alpha^*[y,f]+\sum\limits_{\beta,\gamma\in S(-1)}(-1)^{(|\beta|+|\gamma|)(1+|y|)}[x,[z_\gamma^*,[z_\beta^*,y]]]\otimes z_\beta z_\gamma\\
&+\frac{\sfr-s}{2}[x,[y,f]]\otimes1
\end{split}
\end{equation} for all $x,y\in\ggg^e(1)$.
It is worth noting that the assumption in \S\ref{general} shows that the following relations hold in $U(\ggg)$:
\begin{equation}\label{zz*}
\begin{array}{lllll}
\sum\limits_{\alpha\in S(-1)_{\bar0}}z_\alpha z_\alpha^*&=-\sum\limits_{\alpha\in S(-1)_{\bar0}}z_\alpha^* z_\alpha&\equiv -\frac{\text{dim}\,\ggg(-1)_{\bar0}}{2}&=-\frac{s}{2}&(\text{mod}\,I_\chi),\\
\sum\limits_{\alpha\in S(-1)_{\bar1}}z_\alpha z_\alpha^*&=\sum\limits_{\alpha\in S(-1)_{\bar1}}z_\alpha^* z_\alpha&\equiv \frac{\text{dim}\,\ggg(-1)_{\bar1}}{2}&=\frac{\sfr}{2}&(\text{mod}\,I_\chi).
\end{array}
\end{equation}
Now we consider the second term of the last equation in \eqref{xzzy}.
In virtue of \eqref{zz*}, we can obtain
\begin{equation}\label{wzzw}
\begin{split}
&\sum\limits_{\alpha,\beta\in S(-1)}((-1)^{(|\alpha|+|\beta|)(1+|w_2|)}[w_1,[z_\beta^*,[z_\alpha^*,w_2]]]\otimes z_\alpha z_\beta\\
&-(-1)^{(|\alpha|+|\beta|)(1+|w_1|)+|w_1||w_2|}[w_2,[z_\beta^*,[z_\alpha^*,w_1]]]\otimes z_\alpha z_\beta)\\
=&\sum\limits_{\alpha,\beta\in S(-1)}((-1)^{(|\alpha|+|\beta|)(1+|w_2|)}([[w_1,z_\beta^*],[z_\alpha^*,w_2]]\otimes z_\alpha z_\beta\\
&+(-1)^{|\beta||w_1|}[z_\beta^*,[w_1,[z_\alpha^*,w_2]]]\otimes z_\alpha z_\beta)\\
&-(-1)^{(|\alpha|+|\beta|)(1+|w_1|)+|w_1||w_2|}([[w_2,z_\beta^*],[z_\alpha^*,w_1]]\otimes z_\alpha z_\beta\\
&+(-1)^{|\beta||w_2|}[z_\beta^*,[w_2,[z_\alpha^*,w_1]]]\otimes z_\alpha z_\beta))\\
=&\sum\limits_{\alpha,\beta\in S(-1)}(-(-1)^{|\alpha|+|\beta|+|\alpha||w_2|}[[w_1,z_\alpha^*],[w_2,z_\beta^*]]\otimes z_\beta z_\alpha\\
&+(-1)^{|\alpha|+|\beta|+|\alpha||\beta|}[[w_1,[w_2,z_\alpha^*]],z_\beta^*]\otimes z_\alpha z_\beta\\
&-(-1)^{|\alpha|+|\beta|+|\alpha||\beta|+|\alpha||w_2|}[[w_1,z_\alpha^*],[w_2,z_\beta^*]]\otimes z_\alpha z_\beta\\
&+(-1)^{|\alpha|+|\beta|+|\alpha||\beta|+|\alpha||w_2|}[[[w_1,z_\alpha^*],w_2],z_\beta^*]\otimes z_\alpha z_\beta)\\
=&-2\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||w_2|}[[w_1,z_\alpha^*],[w_2,z_\beta^*]]\otimes z_\beta z_\alpha\\
&+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_2|}[[w_1,z_\alpha],[w_2,z_\alpha^*]]\otimes1\\
&+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||\beta|}([w_1,w_2],f)[e,[z_\alpha^*,z_\beta^*]]\otimes z_\alpha z_\beta\\
&+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}([w_1,w_2],f)[[e,z_\beta^*],z_\alpha^*]\otimes z_\alpha z_\beta\\
=&-2\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||w_2|}[[w_1,z_\alpha^*],[w_2,z_\beta^*]]\otimes z_\beta z_\alpha\\
&+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_2|}[[w_1,z_\alpha],[w_2,z_\alpha^*]]\otimes1+\frac{s-\sfr}{2}([w_1,w_2],f)h\otimes1\\
&+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}([w_1,w_2],f)[[e,z_\beta^*],z_\alpha^*]\otimes z_\alpha z_\beta.
\end{split}
\end{equation}
As \begin{equation*}\begin{split}
&[w_1,[w_2,f]]\otimes1-(-1)^{|w_1||w_2|}[w_2,[w_1,f]]\otimes1\\=&[[w_1,w_2],f]\otimes1=([w_1,w_2],f)[e,f]\otimes1=([w_1,w_2],f)h\otimes1,
\end{split}\end{equation*}
combining this with \eqref{theta12}, \eqref{xzzy} and \eqref{wzzw} we have
\begin{equation}\label{5.22}
\begin{split}
[\Theta_{w_1},\Theta_{w_2}]=&([w_1,w_2],f)(e\otimes1+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[e,z_\alpha^*]\otimes z_\alpha\\
&+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}[[e,z_\beta^*],z_\alpha^*]\otimes z_\alpha z_\beta)\\&-\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||w_2|}[[w_1,z_\alpha^*],[w_2,z_\beta^*]]\otimes z_\beta z_\alpha\\&+\sum\limits_{\alpha\in S(-1)}
(-1)^{|\alpha||w_2|}[w_1,z_\alpha][w_2,z_\alpha^*]\otimes1+\frac{s-\sfr}{2}([w_1,w_2],f)h\otimes1\\
&+\frac{\sfr-s}{2}[w_1,[w_2,f]]\otimes1-(-1)^{|w_1||w_2|}\frac{\sfr-s}{2}[w_2,[w_1,f]]\otimes1\\
&-\sum\limits_{\alpha\in S(-1)}[w_1,z_\alpha]\otimes z_\alpha^*[w_2,f]\\
&+\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}[w_2,z_\alpha]\otimes z_\alpha^*[w_1,f]-1\otimes[\varphi_{w_1},\varphi_{w_2}]\\
=&([w_1,w_2],f)(e\otimes1+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[e,z_\alpha^*]\otimes z_\alpha\\
&+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}[[e,z_\beta^*],z_\alpha^*]\otimes z_\alpha z_\beta)\\
&-\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||w_2|}[[w_1,z_\alpha^*],[w_2,z_\beta^*]]\otimes z_\beta z_\alpha\\
&+\sum\limits_{\alpha\in S(-1)}
(-1)^{|\alpha||w_2|}[w_1,z_\alpha][w_2,z_\alpha^*]\otimes1-\sum\limits_{\alpha\in S(-1)}[w_1,z_\alpha]\otimes z_\alpha^*[w_2,f]\\
&+\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}[w_2,z_\alpha]\otimes z_\alpha^*[w_1,f]-1\otimes[\varphi_{w_1},\varphi_{w_2}].
\end{split}
\end{equation}
\subsubsection{}
For all $x\in\ggg^e(1)$, we have defined the Lie superalgebra endomorphism $\sharp: \ggg(0)\rightarrow\ggg(0)$ in \eqref{xh} by
\begin{equation}\label{3.25}
[x,z_\alpha]^{\sharp}=[x,z_\alpha]-\frac{1}{2}(h,[x,z_\alpha])h.
\end{equation}
Moreover, we have
\begin{equation}\label{3.26}
\begin{split}
\sum\limits_{\alpha\in S(-1)}(h,[x,z_\alpha^*])z_\alpha=&\sum\limits_{\alpha\in S(-1)}(e,[f,[x,z_\alpha^*]])z_\alpha
=\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||x|}\langle z_\alpha^*,[x,f] \rangle z_\alpha\\
=&(-1)^{|x|}[x,f]
\end{split}
\end{equation}
and \begin{equation}\label{3.27}
\sum\limits_{\alpha\in S(-1)}(h,[x,z_\alpha])z_\alpha^*=-\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}(h,[x,z_\alpha^*])z_\alpha=-[x,f].
\end{equation}
As $[w_2,z_\alpha]^{\sharp}\in\ggg^e(0)$, it follows from \eqref{rewrite}, \eqref{zz*} and \eqref{3.25} that
\begin{equation}\label{wzsharp}
\begin{split}
\Theta_{[w_2,z_\alpha]^{\sharp}}=&[w_2,z_\alpha]^{\sharp}\otimes1-\sum\limits_{\beta\in S(-1)}\frac{1}{2}\otimes z_\beta[z_\beta^*,[w_2,z_\alpha]^{\sharp}]\\
=&[w_2,z_\alpha]\otimes1-\frac{1}{2}(h,[w_2,z_\alpha])h\otimes1\\&-\sum\limits_{\beta\in S(-1)}\frac{1}{2}\otimes z_\beta[z_\beta^*,([w_2,z_\alpha]-\frac{1}{2}(h,[w_2,z_\alpha])h)]\\
=&[w_2,z_\alpha]\otimes1-\frac{1}{2}(h,[w_2,z_\alpha])h\otimes1\\&-\sum\limits_{\beta\in S(-1)}\frac{1}{2}\otimes z_\beta[z_\beta^*,[w_2,z_\alpha]]+\frac{1}{4}\sum\limits_{\beta\in S(-1)}(h,[w_2,z_\alpha])\otimes z_\beta z_\beta^*\\
=&[w_2,z_\alpha]\otimes1-\frac{1}{2}(h,[w_2,z_\alpha])h\otimes1\\&-\sum\limits_{\beta\in S(-1)}\frac{1}{2}\otimes z_\beta[z_\beta^*,[w_2,z_\alpha]]+\frac{\sfr-s}{8}(h,[w_2,z_\alpha]).
\end{split}
\end{equation}
The same discussion as in \eqref{wzsharp} shows that
\begin{equation}\label{zwsharp}
\begin{split}
\Theta_{[z_\alpha^*, w_1]^{\sharp}}=&[z_\alpha^*, w_1]\otimes1-\frac{1}{2}(h,[z_\alpha^*, w_1])h\otimes1-\sum\limits_{\beta\in S(-1)}\frac{1}{2}\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_1]]\\&+\frac{\sfr-s}{8}(h,[z_\alpha^*, w_1]).
\end{split}
\end{equation}
Taking \eqref{wzsharp} and \eqref{zwsharp} into account, we can deduce that
\begin{equation}\label{3.30}
\begin{split}
&\sum\limits_{\alpha\in S(-1)}\Theta_{[w_2,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_1]^{\sharp}}\\
=&\sum\limits_{\alpha\in S(-1)}(([w_2,z_\alpha]\otimes1-\frac{1}{2}(h,[w_2,z_\alpha])h\otimes1-\frac{1}{2}\sum\limits_{\beta\in S(-1)}1\otimes z_\beta[z_\beta^*,[w_2,z_\alpha]]\\
&+\frac{\sfr-s}{8}(h,[w_2,z_\alpha]))([z_\alpha^*, w_1]\otimes1-\frac{1}{2}(h,[z_\alpha^*, w_1])h\otimes1\\
&-\frac{1}{2}\sum\limits_{\beta\in S(-1)}1\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_1]]+\frac{\sfr-s}{8}(h,[z_\alpha^*, w_1])))\\
=&\sum\limits_{\alpha\in S(-1)}[w_2,z_\alpha][z_\alpha^*, w_1]\otimes1-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(h,[z_\alpha^*, w_1])[w_2,z_\alpha]h\otimes1\\
&-\frac{1}{2}\sum\limits_{\alpha,\beta\in S(-1)}[w_2,z_\alpha]\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_1]]+\frac{\sfr-s}{8}\sum\limits_{\alpha\in S(-1)}(h,[z_\alpha^*, w_1])[w_2,z_\alpha]\otimes1\\
&-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(h,[w_2,z_\alpha])[z_\alpha^*, w_1]h\otimes1+\frac{1}{4}\sum\limits_{\alpha\in S(-1)}(h,[w_2,z_\alpha])(h,[z_\alpha^*, w_1])h^2\otimes1\\
&+\frac{1}{4}\sum\limits_{\alpha,\beta\in S(-1)}(h,[w_2,z_\alpha])h\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_1]]\\
&+\frac{s-\sfr}{16}\sum\limits_{\alpha\in S(-1)}(h,[z_\alpha^*, w_1])(h,[w_2,z_\alpha])h\otimes1\\&
-\frac{1}{2}\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{(|w_1|+|\alpha|)(|w_2|+|\alpha|)}[z_\alpha^*, w_1]\otimes z_\beta[z_\beta^*,[w_2,z_\alpha]]\\
&+\frac{1}{4}\sum\limits_{\alpha,\beta\in S(-1)}(h,[z_\alpha^*, w_1])h\otimes z_\beta[z_\beta^*,[w_2,z_\alpha]]\\
&+\frac{1}{4}\sum\limits_{\alpha,\beta,\gamma\in S(-1)}(-1)^{(|w_1|+|\alpha|)(|w_2|+|\alpha|)}\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_1]]z_\gamma[z_\gamma^*,[w_2,z_\alpha]]\\
&+\frac{s-\sfr}{16}\sum\limits_{\alpha,\beta\in S(-1)}(h,[z_\alpha^*, w_1])\otimes z_\beta[z_\beta^*,[w_2,z_\alpha]]
+\frac{\sfr-s}{8}\sum\limits_{\alpha\in S(-1)}(h,[w_2,z_\alpha])[z_\alpha^*, w_1]\otimes1\\
&+\frac{s-\sfr}{16}\sum\limits_{\alpha\in S(-1)}(h,[w_2,z_\alpha])(h,[z_\alpha^*, w_1])h\otimes1\\
&+\frac{s-\sfr}{16}\sum\limits_{\alpha,\beta\in S(-1)}(h,[w_2,z_\alpha])\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_1]]\\
&+\frac{(s-\sfr)^2}{64}\sum\limits_{\alpha\in S(-1)}(h,[w_2,z_\alpha])(h,[z_\alpha^*, w_1]).
\end{split}
\end{equation}
Now we will deal with the terms in \eqref{3.30}. First note that
\begin{equation}\label{3.31}
\begin{split}
&\sum\limits_{\alpha\in S(-1)}[w_2,z_\alpha][z_\alpha^*, w_1]\otimes1
=-\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_1|}[w_2,z_\alpha][w_1,z_\alpha^*]\otimes1\\
=&\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_1|+|\alpha|+(|w_1|+|\alpha|)(|w_2|+|\alpha|)}(-[[w_1,z_\alpha],[w_2,z_\alpha^*]]+[w_1,z_\alpha][w_2,z_\alpha^*])\otimes1\\
=&-\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_2|+|w_1||w_2|}[[w_1,z_\alpha],[w_2,z_\alpha^*]]\otimes1\\
&+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_2|+|w_1||w_2|}[w_1,z_\alpha][w_2,z_\alpha^*]\otimes1.
\end{split}
\end{equation}
Moreover, it follows from \eqref{zz*}, \eqref{3.26} and \eqref{3.27} that
\begin{equation}\label{3.32}
\begin{split}
&\frac{1}{4}\sum\limits_{\alpha,\beta\in S(-1)}(h,[w_2,z_\alpha])h\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_1]]
+\frac{1}{4}\sum\limits_{\alpha,\beta\in S(-1)}(h,[z_\alpha^*, w_1])h\otimes z_\beta[z_\beta^*,[w_2,z_\alpha]]\\
=&\frac{1}{4}\sum\limits_{\alpha,\beta\in S(-1)}h\otimes z_\beta[z_\beta^*,[(h,[w_2,z_\alpha])z_\alpha^*, w_1]]\\
&-\frac{1}{4}\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha||w_1|}h\otimes z_\beta[z_\beta^*,[w_2,(h,[w_1,z_\alpha^*])z_\alpha]]\\
=&-\frac{1}{4}\sum\limits_{\alpha\in S(-1)}h\otimes z_\alpha[z_\alpha^*,[[w_2,f], w_1]]
-\frac{1}{4}\sum\limits_{\alpha\in S(-1)}h\otimes z_\alpha[z_\alpha^*,[w_2,[w_1,f]]]\\
=&\frac{1}{4}(-1)^{|w_1||w_2|}\sum\limits_{\alpha\in S(-1)}h\otimes z_\alpha[z_\alpha^*,[[w_1,w_2],f]]\\
=&\frac{1}{4}(-1)^{|w_1||w_2|}([w_1,w_2],f)\sum\limits_{\alpha\in S(-1)}h\otimes z_\alpha[z_\alpha^*,h]\\
=&\frac{1}{4}(-1)^{|w_1||w_2|}([w_1,w_2],f)\sum\limits_{\alpha\in S(-1)}h\otimes z_\alpha z_\alpha^*
=\frac{\sfr-s}{8}(-1)^{|w_1||w_2|}([w_1,w_2], f)h\otimes1.
\end{split}
\end{equation}
By the same discussion as in \eqref{3.32} we can obtain that
\begin{equation}
\begin{split}
&-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(h,[z_\alpha^*, w_1])[w_2,z_\alpha]h\otimes1-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(h,[w_2,z_\alpha])[z_\alpha^*, w_1]h\otimes1\\
=&-\frac{(-1)^{|w_1||w_2|}}{2}([w_1,w_2],f)h^2\otimes1,
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&\frac{\sfr-s}{8}\sum\limits_{\alpha\in S(-1)}(h,[z_\alpha^*, w_1])[w_2,z_\alpha]\otimes1+\frac{\sfr-s}{8}\sum\limits_{\alpha\in S(-1)}(h,[w_2,z_\alpha])[z_\alpha^*, w_1]\otimes1\\
=&\frac{\sfr-s}{8}(-1)^{|w_1||w_2|}([w_1,w_2], f)h\otimes1,
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&\frac{1}{4}\sum\limits_{\alpha\in S(-1)}(h,[w_2,z_\alpha])(h,[z_\alpha^*, w_1])h^2\otimes1
=\frac{1}{4}\sum\limits_{\alpha\in S(-1)}(h,[(h,[w_2,z_\alpha])z_\alpha^*, w_1])h^2\otimes1\\
=&\frac{1}{4}(-1)^{|w_1||w_2|}([w_1,w_2], f)h^2\otimes1,
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&\frac{s-\sfr}{16}(\sum\limits_{\alpha\in S(-1)}(h,[z_\alpha^*, w_1])(h,[w_2,z_\alpha])h+\sum\limits_{\alpha\in S(-1)}(h,[w_2,z_\alpha])(h,[z_\alpha^*, w_1])h)\otimes1\\
=&\frac{s-\sfr}{8}(-1)^{|w_1||w_2|}([w_1,w_2],f)h\otimes1,
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&\frac{s-\sfr}{16}(\sum\limits_{\alpha,\beta\in S(-1)}(h,[z_\alpha^*, w_1])\otimes z_\beta[z_\beta^*,[w_2,z_\alpha]]+\sum\limits_{\alpha,\beta\in S(-1)}(h,[w_2,z_\alpha])\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_1]])\\
=&\frac{s-\sfr}{16}\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}([w_1,w_2],f)\otimes z_\alpha z_\alpha^*=-\frac{(s-\sfr)^2}{32}(-1)^{|w_1||w_2|}([w_1,w_2],f),
\end{split}
\end{equation}
\begin{equation}\label{3.38}
\frac{(s-\sfr)^2}{64}\sum\limits_{\alpha\in S(-1)}(h,[w_2,z_\alpha])(h,[z_\alpha^*, w_1])=\frac{(s-\sfr)^2}{64}(-1)^{|w_1||w_2|}([w_1,w_2],f).
\end{equation}
On the other hand, it follows from the procedure of \eqref{xzzy} that
\begin{equation}\label{3.39}
\begin{split}
&\sum\limits_{\alpha,\beta\in S(-1)}[w_2,z_\alpha]\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_1]]\\
=&\sum\limits_{\alpha,\beta\in S(-1)}[w_2,z_\alpha]\otimes z_\beta[[z_\beta^*,z_\alpha^*], w_1]+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha||\beta|}[w_2,z_\alpha]\otimes z_\beta[z_\alpha^*,[z_\beta^*,w_1]]\\
=&-\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[w_2,z_\alpha^*]\otimes z_\alpha[f,w_1]+[w_2,[w_1,f]]\otimes1-\sum\limits_{\alpha\in S(-1)}[w_2,z_\alpha]\otimes z_\alpha^*[w_1,f]\\
&+\sum\limits_{\beta,\gamma\in S(-1)}(-1)^{(|\beta|+|\gamma|)(1+|w_1|)}[w_2,[z_\gamma^*,[z_\beta^*, w_1]]]\otimes z_\beta z_\gamma\\
=&-2\sum\limits_{\alpha\in S(-1)}[w_2,z_\alpha]\otimes z_\alpha^*[w_1,f]+[w_2,[w_1,f]]\otimes1\\
&+\sum\limits_{\beta,\gamma\in S(-1)}(-1)^{(|\beta|+|\gamma|)(1+|w_1|)}[w_2,[z_\gamma^*,[z_\beta^*, w_1]]]\otimes z_\beta z_\gamma.
\end{split}
\end{equation}
By the same discussion as in \eqref{3.39}, one can conclude that
\begin{equation}\label{3.40}
\begin{split}
&\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{(|w_1|+|\alpha|)(|w_2|+|\alpha|)}[z_\alpha^*, w_1]\otimes z_\beta[z_\beta^*,[w_2,z_\alpha]]\\
=&(-1)^{|w_1||w_2|}\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|}[w_1,z_\alpha^*]\otimes z_\beta[z_\beta^*,[z_\alpha,w_2]]\\
=&-(-1)^{|w_1||w_2|}\sum\limits_{\alpha,\beta\in S(-1)}[w_1,z_\alpha]\otimes z_\beta[z_\beta^*,[z_\alpha^*,w_2]]\\
=&2\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}[w_1,z_\alpha]\otimes z_\alpha^*[w_2,f]-(-1)^{|w_1||w_2|}[w_1,[w_2,f]]\otimes1\\
&-\sum\limits_{\beta,\gamma\in S(-1)}(-1)^{(|\beta|+|\gamma|)(1+|w_2|)+|w_1||w_2|}[w_1,[z_\gamma^*,[z_\beta^*, w_2]]]\otimes z_\beta z_\gamma.
\end{split}
\end{equation}
Combining \eqref{3.39} with \eqref{3.40}, it is immediate from \eqref{wzzw} that
\begin{equation}\label{3.41}
\begin{split}
&\sum\limits_{\alpha,\beta\in S(-1)}[w_2,z_\alpha]\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_1]]\\
&+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{(|w_1|+|\alpha|)(|w_2|+|\alpha|)}[z_\alpha^*, w_1]\otimes z_\beta[z_\beta^*,[w_2,z_\alpha]]\\
=&-2\sum\limits_{\alpha\in S(-1)}[w_2,z_\alpha]\otimes z_\alpha^*[w_1,f]+2\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}[w_1,z_\alpha]\otimes z_\alpha^*[w_2,f]\\
&-(-1)^{|w_1||w_2|}([w_1,w_2],f)h\otimes1-\frac{s-\sfr}{2}(-1)^{|w_1||w_2|}([w_1,w_2],f)h\otimes1\\
&+2\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||w_2|+|w_1||w_2|}[[w_1,z_\alpha^*],[w_2,z_\beta^*]]\otimes z_\beta z_\alpha\\
&-\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_2|+|w_1||w_2|}[[w_1,z_\alpha],[w_2,z_\alpha^*]]\otimes1\\
&-([w_1,w_2],f)\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|w_1||w_2|}[[e,z_\beta^*],z_\alpha^*]\otimes z_\alpha z_\beta.
\end{split}
\end{equation}
Now we can deduce from \eqref{3.30}-\eqref{3.38} and \eqref{3.41} that
\begin{equation}\label{3.42}
\begin{split}
&\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}\Theta_{[w_2,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_1]^{\sharp}}\\
=&\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_2|}[w_1,z_\alpha][w_2,z_\alpha^*]\otimes1-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_2|}[[w_1,z_\alpha],[w_2,z_\alpha^*]]\otimes1\\
&-\frac{1}{2}([w_1,w_2],f)h^2\otimes1+\frac{\sfr-s}{8}([w_1,w_2],f)h\otimes1+\frac{1}{4}([w_1,w_2],f)h^2\otimes1\\
&+\frac{\sfr-s}{8}([w_1,w_2],f)h\otimes1+\frac{s-\sfr}{8}([w_1,w_2],f)h\otimes1-\frac{(s-\sfr)^2}{32}([w_1,w_2],f)\\
&+\frac{(s-\sfr)^2}{64}([w_1,w_2],f)+\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}[w_2,z_\alpha]\otimes z_\alpha^*[w_1,f]\\
&-\sum\limits_{\alpha\in S(-1)}[w_1,z_\alpha]\otimes z_\alpha^*[w_2,f]+\frac{1}{2}([w_1,w_2],f)h\otimes1+\frac{s-\sfr}{4}([w_1,w_2],f)h\otimes1\\
&-\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||w_2|}[[w_1,z_\alpha^*],[w_2,z_\beta^*]]\otimes z_\beta z_\alpha\\
&+\frac{1}{2}([w_1,w_2],f)\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}[[e,z_\beta^*],z_\alpha^*]\otimes z_\alpha z_\beta\\
&+\frac{1}{4}\sum\limits_{\alpha,\beta,\gamma\in S(-1)}(-1)^{|\alpha||w_1|+|\alpha||w_2|+|\alpha|}\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_1]]z_\gamma[z_\gamma^*,[w_2,z_\alpha]]\\
=&\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_2|}[w_1,z_\alpha][w_2,z_\alpha^*]\otimes1-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_2|}[[w_1,z_\alpha],[w_2,z_\alpha^*]]\otimes1\\
&-\frac{1}{4}([w_1,w_2],f)h^2\otimes1+\frac{s-\sfr}{8}([w_1,w_2],f)h\otimes1-\frac{(s-\sfr)^2}{64}([w_1,w_2],f)\\
&+\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}[w_2,z_\alpha]\otimes z_\alpha^*[w_1,f]-\sum\limits_{\alpha\in S(-1)}[w_1,z_\alpha]\otimes z_\alpha^*[w_2,f]\\
&+\frac{1}{2}([w_1,w_2],f)h\otimes1-\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||w_2|}[[w_1,z_\alpha^*],[w_2,z_\beta^*]]\otimes z_\beta z_\alpha\\
&+\frac{1}{2}([w_1,w_2],f)\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}[[e,z_\beta^*],z_\alpha^*]\otimes z_\alpha z_\beta\\
&+\frac{1}{4}\sum\limits_{\alpha,\beta,\gamma\in S(-1)}(-1)^{|\alpha||w_1|+|\alpha||w_2|+|\alpha|}\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_1]]z_\gamma[z_\gamma^*,[w_2,z_\alpha]].
\end{split}
\end{equation}
Interchanging the roles of $w_1$ and $w_2$ in \eqref{3.42}, we have
\begin{equation}\label{3.43}
\begin{split}
&\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}\\
=&\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_1|+|w_1||w_2|}[w_2,z_\alpha][w_1,z_\alpha^*]\otimes1\\
&-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_1|+|w_1||w_2|}[[w_2,z_\alpha],[w_1,z_\alpha^*]]\otimes1\\
&-\frac{1}{4}(-1)^{|w_1||w_2|}([w_2,w_1],f)h^2\otimes1\\
&+\frac{s-\sfr}{8}(-1)^{|w_1||w_2|}([w_2,w_1],f)h\otimes1-\frac{(s-\sfr)^2}{64}(-1)^{|w_1||w_2|}([w_2,w_1],f)\\
&+\sum\limits_{\alpha\in S(-1)}[w_1,z_\alpha]\otimes z_\alpha^*[w_2,f]-\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}[w_2,z_\alpha]\otimes z_\alpha^*[w_1,f]\\
&+\frac{1}{2}(-1)^{|w_1||w_2|}([w_2,w_1],f)h\otimes1\\
&-\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||w_1|+|w_1||w_2|}[[w_2,z_\alpha^*],[w_1,z_\beta^*]]\otimes z_\beta z_\alpha\\
&+\frac{1}{2}(-1)^{|w_1||w_2|}([w_2,w_1],f)\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}[[e,z_\beta^*],z_\alpha^*]\otimes z_\alpha z_\beta\\
&+\frac{1}{4}\sum\limits_{\alpha,\beta,\gamma\in S(-1)}(-1)^{|\alpha||w_1|+|\alpha||w_2|+|\alpha|+|w_1||w_2|}\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_2]]z_\gamma[z_\gamma^*,[w_1,z_\alpha]]\\
=&-\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_2|}[w_1,z_\alpha][w_2,z_\alpha^*]\otimes1\\
&-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_2|}[[w_1,z_\alpha],[w_2,z_\alpha^*]]\otimes1+\frac{1}{4}([w_1,w_2],f)h^2\otimes1\\
&-\frac{s-\sfr}{8}([w_1,w_2],f)h\otimes1
+\frac{(s-\sfr)^2}{64}([w_1,w_2],f)+\sum\limits_{\alpha\in S(-1)}[w_1,z_\alpha]\otimes z_\alpha^*[w_2,f]\\
&-\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}[w_2,z_\alpha]\otimes z_\alpha^*[w_1,f]-\frac{1}{2}([w_1,w_2],f)h\otimes1\\
&+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||w_2|}[[w_1,z_\alpha^*],[w_2,z_\beta^*]]\otimes z_\beta z_\alpha\\
&-\frac{1}{2}([w_1,w_2],f)\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}[[e,z_\beta^*],z_\alpha^*]\otimes z_\alpha z_\beta\\
&+\frac{1}{4}\sum\limits_{\alpha,\beta,\gamma\in S(-1)}(-1)^{(|\alpha|+|w_1|)(|\alpha|+|w_2|)}\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_2]]z_\gamma[z_\gamma^*,[w_1,z_\alpha]].
\end{split}
\end{equation}
In \eqref{3.43} we used the fact that
\begin{equation}
\begin{split}
&-\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||w_1|+|w_1||w_2|}[[w_2,z_\alpha^*],[w_1,z_\beta^*]]\otimes z_\beta z_\alpha\\
=&-\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\beta||w_1|+|w_1||w_2|}[[w_2,z_\beta^*],[w_1,z_\alpha^*]]\otimes z_\alpha z_\beta\\
=&\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||w_2|}[[w_1,z_\alpha^*],[w_2,z_\beta^*]]\otimes z_\beta z_\alpha\\
&+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||\beta|+|\alpha||w_2|}[[w_1,z_\alpha^*],[w_2,z_\beta^*]]\otimes [z_\alpha,z_\beta]\\
=&\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|+|\alpha||w_2|}[[w_1,z_\alpha^*],[w_2,z_\beta^*]]\otimes z_\beta z_\alpha\\
&-\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha||w_2|}[[w_1,z_\alpha],[w_2,z_\alpha^*]]\otimes1.
\end{split}
\end{equation}
We now combine \eqref{5.22} with \eqref{3.42} and \eqref{3.43} to deduce that
\begin{equation}\label{3.45}
\begin{split}
&[\Theta_{w_1},\Theta_{w_2}]+\frac{1}{2}\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}\Theta_{[w_2,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_1]^{\sharp}}\\
=&([w_1,w_2],f)((e+\frac{h^2}{4}-(\frac{1}{2}+\frac{(s-\sfr)}{8})h)\otimes1+\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[e,z_\alpha^*]\otimes z_\alpha\\
&+\frac{1}{2}\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}[[e,z_\alpha^*],z_\beta^*]\otimes z_\beta z_\alpha)\\
&+\frac{1}{8}\sum\limits_{\alpha,\beta,\gamma\in S(-1)}(-1)^{(|\alpha|+|w_1|)(|\alpha|+|w_2|)}\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_2]]z_\gamma[z_\gamma^*,[w_1,z_\alpha]]\\
&-\frac{1}{8}\sum\limits_{\alpha,\beta,\gamma\in S(-1)}(-1)^{|\alpha||w_1|+|\alpha||w_2|+|\alpha|}\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_1]]z_\gamma[z_\gamma^*,[w_2,z_\alpha]]\\
&+\frac{(s-\sfr)^2}{64}([w_1,w_2],f)-1\otimes[\varphi_{w_1},\varphi_{w_2}].
\end{split}
\end{equation}
\subsubsection{}
Recall that in \eqref{C} we introduced $C$, which is the central element of $W_\chi'$, and showed that the element $\Theta_{\text{Cas}}=\sum_{i\in I}(-1)^{|i|}\Theta_{a_i}\Theta_{b_i}$
commutes with all operators $\Theta_v$ for $v\in\ggg^e(0)$ in Proposition \ref{vcommutate}, where $\{a_i\mid i\in I\}$ and $\{b_i\mid i\in I\}$ are dual bases of $\ggg^e(0)$ with respect to the restriction of the invariant form $(\cdot,\cdot)$ to $\ggg^e(0)$. Now we will calculate $C-\Theta_{\text{Cas}}$ in this part.
Since $[z_\beta^*,b_i],\,[z_\beta^*,a_i]\in\ggg(-1)$ for any $\beta\in S(-1)$ and $i\in I$, it is immediate from \eqref{zuz} that
\begin{equation}\label{3.48}
\begin{split}
[z_\beta^*,b_i]=&\sum\limits_{\alpha\in S(-1)}[z_\alpha^*,[z_\beta^*,b_i]]z_\alpha=\sum\limits_{\alpha\in S(-1)}(e,[z_\alpha^*,[z_\beta^*,b_i]])z_\alpha,\\
[z_\beta^*,a_i]=&\sum\limits_{\alpha\in S(-1)}[z_\alpha^*,[z_\beta^*,a_i]]z_\alpha=\sum\limits_{\alpha\in S(-1)}(e,[z_\alpha^*,[z_\beta^*,a_i]])z_\alpha.
\end{split}
\end{equation}
As the even bilinear form $(\cdot,\cdot)$ is invariant and $\mathbb{C}h$ is orthogonal to $\ggg(0)^\sharp$ with respect to $(\cdot,\cdot)$, it follows from \eqref{3.48} that
\begin{equation}\label{3.49}
\begin{split}
&\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}[[e,z_\alpha^*],z_\beta^*]^\sharp\otimes z_\beta z_\alpha\\
&=\sum\limits_{\alpha,\beta\in S(-1),i\in I}(-1)^{|i|}([[e,z_\alpha^*],z_\beta^*]^\sharp,b_i)a_i\otimes z_\beta z_\alpha\\
&=\sum\limits_{\alpha,\beta\in S(-1),i\in I}(-1)^{|i|}a_i\otimes z_\beta (e,[z_\alpha^*,[z_\beta^*,b_i]])z_\alpha\\
&=\sum\limits_{\beta\in S(-1),i\in I}(-1)^{|i|}a_i\otimes z_\beta[z_\beta^*,b_i].
\end{split}
\end{equation}
Interchanging the roles of $\{a_i\}$ and $\{b_i\}$, we can obtain
\begin{equation}\label{3.50}
\begin{split}
&\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}[[e,z_\alpha^*],z_\beta^*]^\sharp\otimes z_\beta z_\alpha\\
=&\sum\limits_{\alpha,\beta\in S(-1),i\in I}(-1)^{|i|}(a_i,[[e,z_\alpha^*],z_\beta^*]^\sharp)b_i\otimes z_\beta z_\alpha\\
=&\sum\limits_{\alpha,\beta\in S(-1),i\in I}(-1)^{|i|}(a_i,[[e,z_\alpha^*],z_\beta^*])b_i\otimes z_\beta z_\alpha\\
=&\sum\limits_{\alpha,\beta\in S(-1),i\in I}b_i\otimes z_\beta(e,[z_\alpha^*,[z_\beta^*,a_i]])z_\alpha\\
=&\sum\limits_{\beta\in S(-1),i\in I}b_i\otimes z_\beta[z_\beta^*,a_i],
\end{split}
\end{equation}
where the third equation comes from the supersymmetry of the invariant bilinear form $(\cdot,\cdot)$.
On the other hand, it can be observed that
\begin{equation}\label{3.51}
\begin{split}
&\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}[[e,z_\alpha^*],z_\beta^*]^\sharp\otimes z_\beta z_\alpha\\
=&\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}([[e,z_\alpha^*],z_\beta^*]\otimes z_\beta z_\alpha-\frac{1}{2}(h,[[e,z_\alpha^*],z_\beta^*])h\otimes z_\beta z_\alpha)\\
=&\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}([[e,z_\alpha^*],z_\beta^*]\otimes z_\beta z_\alpha-\frac{1}{2}(-1)^{|\alpha||\beta|}([[h,z_\beta^*],z_\alpha^*],e)h\otimes z_\beta z_\alpha)\\
=&\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}[[e,z_\alpha^*],z_\beta^*]\otimes z_\beta z_\alpha+\frac{1}{2}\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}([[h,z_\beta^*],z_\alpha],e)h\otimes z_\beta z_\alpha^*\\
=&\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}[[e,z_\alpha^*],z_\beta^*]\otimes z_\beta z_\alpha+\frac{(s-\sfr)}{4}h\otimes 1,
\end{split}
\end{equation}
where the last equation in \eqref{3.51} follows from \eqref{zz*} and the fact that
\begin{equation*}
\begin{split}
&\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}([[h,z_\beta^*],z_\alpha],e)h\otimes z_\beta z_\alpha^*\\
=&-\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}([z_\beta^*,z_\alpha],e)h\otimes z_\beta z_\alpha^*\\
=&-\sum\limits_{\alpha\in S(-1)}h\otimes z_\alpha z_\alpha^*=\frac{(s-\sfr)}{2}h\otimes1.
\end{split}
\end{equation*}
As a result, it follows from \eqref{3.49}, \eqref{3.50} and \eqref{3.51} that
\begin{equation}\label{3.52}
\begin{split}
&\sum\limits_{\beta\in S(-1),i\in I}((-1)^{|i|}a_i\otimes z_\beta[z_\beta^*,b_i]+b_i\otimes z_\beta[z_\beta^*,a_i])\\&=2\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}[[e,z_\alpha^*],z_\beta^*]\otimes z_\beta z_\alpha+\frac{(s-\sfr)}{2}h\otimes 1.
\end{split}
\end{equation}
As $\mathbb{C}h$ is orthogonal to $\ggg(0)^\sharp$ with respect to $(\cdot,\cdot)$, \eqref{3.48} yields
\begin{equation}\label{3.53}
\begin{split}
&\sum\limits_{\alpha,\beta\in S(-1),i\in I}1\otimes z_\alpha[z_\alpha^*,b_i]z_\beta[z_\beta^*,a_i]\\
=&\sum\limits_{\alpha,\beta,\gamma,\delta\in S(-1),i\in I}1\otimes z_\alpha(e,[z_\gamma^*,[z_\alpha^*,b_i]])z_\gamma z_\beta(e,[z_\delta^*,[z_\beta^*,a_i]])z_\delta\\
=&\sum\limits_{\alpha,\beta,\gamma,\delta\in S(-1),i\in I}([[e,z_\delta^*],z_\beta^*],a_i)([[e,z_\gamma^*],z_\alpha^*],b_i)\otimes z_\alpha z_\gamma z_\beta z_\delta\\
=&\sum\limits_{\alpha,\beta,\gamma,\delta\in S(-1),i\in I}([[e,z_\delta^*],z_\beta^*]^\sharp,a_i)([[e,z_\gamma^*],z_\alpha^*]^\sharp,b_i)\otimes z_\alpha z_\gamma z_\beta z_\delta\\
=&\sum\limits_{\alpha,\beta,\gamma,\delta\in S(-1),i\in I}([[e,z_\delta^*],z_\beta^*]^\sharp,([[e,z_\gamma^*],z_\alpha^*]^\sharp,b_i)a_i)\otimes z_\alpha z_\gamma z_\beta z_\delta\\
=&\sum\limits_{\alpha,\beta,\gamma,\delta\in S(-1)}([[e,z_\delta^*],z_\beta^*]^\sharp,[[e,z_\gamma^*],z_\alpha^*]^\sharp)\otimes z_\alpha z_\gamma z_\beta z_\delta.
\end{split}
\end{equation}
Since $a_i$ and $b_i$ are in $\ggg^e(0)$ for all $i\in I$, we can conclude from \eqref{rewrite}, \eqref{3.52} and \eqref{3.53} that
\begin{equation}\label{3.54}
\begin{split}
&\Theta_{\text{Cas}}=\sum\limits_{i\in I}(-1)^{|i|}\Theta_{a_i}\Theta_{b_i}\\
=&\sum\limits_{i\in I}(-1)^{|i|}(a_i\otimes1-\frac{1}{2}\otimes\sum\limits_{\alpha\in S(-1)}z_\alpha[z_\alpha^*,a_i])(b_i\otimes1-\frac{1}{2}\otimes\sum\limits_{\alpha\in S(-1)}z_\alpha[z_\alpha^*,b_i])\\
=&\sum\limits_{i\in I}(-1)^{|i|}a_ib_i\otimes1-\frac{1}{2}\sum\limits_{\alpha\in S(-1),i\in I}((-1)^{|i|}a_i\otimes z_\alpha[z_\alpha^*,b_i]+b_i\otimes z_\alpha[z_\alpha^*,a_i])\\
&+\frac{1}{4}\sum\limits_{\alpha,\beta\in S(-1),i\in I}z_\alpha[z_\alpha^*,b_i]z_\beta[z_\beta^*,a_i]\\
=&\sum\limits_{i\in I}(-1)^{|i|}a_ib_i\otimes1-\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}[[e,z_\alpha^*],z_\beta^*]\otimes z_\beta z_\alpha-\frac{(s-\sfr)}{4}h\otimes1\\
&+\frac{1}{4}\sum\limits_{\alpha,\beta,\gamma,\delta\in S(-1)}([[e,z_\delta^*],z_\beta^*]^\sharp,[[e,z_\gamma^*],z_\alpha^*]^\sharp)\otimes z_\alpha z_\gamma z_\beta z_\delta.
\end{split}
\end{equation}
Recall that in \eqref{rewrite} we introduced a central element of $W_\chi'$:
\begin{equation*}
\begin{split}
C=&2e\otimes1+\frac{h^2}{2}\otimes1-(1+\frac{s-\sfr}{2})h\otimes1+\sum\limits_{i\in I}(-1)^{|i|}a_ib_i\otimes1\\
&+2\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[e,z_\alpha^*]\otimes z_\alpha.
\end{split}
\end{equation*}
In view of \eqref{3.54} we have
\begin{equation}\label{3.56}
\begin{split}
C-\Theta_{\text{Cas}}=&(2e+\frac{h^2}{2}-(1+\frac{(s-\sfr)}{4})h)\otimes1+2\sum\limits_{\alpha\in S(-1)}(-1)^{|\alpha|}[e,z_\alpha^*]\otimes z_\alpha\\
&+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha|+|\beta|}[[e,z_\alpha^*],z_\beta^*]\otimes z_\beta z_\alpha\\
&-\frac{1}{4}\sum\limits_{\alpha,\beta,\gamma,\delta\in S(-1)}([[e,z_\delta^*],z_\beta^*]^\sharp,[[e,z_\gamma^*],z_\alpha^*]^\sharp)\otimes z_\alpha z_\gamma z_\beta z_\delta.
\end{split}
\end{equation}
We finally combine \eqref{3.45} and \eqref{3.56} to deduce that
\begin{equation}\label{3.57}
\begin{split}
&[\Theta_{w_1},\Theta_{w_2}]+\frac{1}{2}\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}\Theta_{[w_2,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_1]^{\sharp}}\\
&-\frac{1}{2}([w_1,w_2],f)(C-\Theta_{\text{Cas}})\\
=&\frac{1}{8}\sum\limits_{\alpha,\beta,\gamma\in S(-1)}(-1)^{(|\alpha|+|w_1|)(|\alpha|+|w_2|)}\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_2]]z_\gamma[z_\gamma^*,[w_1,z_\alpha]]\\
&-\frac{1}{8}\sum\limits_{\alpha,\beta,\gamma\in S(-1)}(-1)^{|\alpha||w_1|+|\alpha||w_2|+|\alpha|}\otimes z_\beta[z_\beta^*,[z_\alpha^*, w_1]]z_\gamma[z_\gamma^*,[w_2,z_\alpha]]\\
&+\frac{1}{8}([w_1,w_2],f)\sum\limits_{\alpha,\beta,\gamma,\delta\in S(-1)}([[e,z_\delta^*],z_\gamma^*]^\sharp,[[e,z_\beta^*],z_\alpha^*]^\sharp)\otimes z_\alpha z_\beta z_\gamma z_\delta\\
&+\frac{(s-\sfr)^2}{64}([w_1,w_2],f)-1\otimes[\varphi_{w_1},\varphi_{w_2}],
\end{split}
\end{equation}
where $\varphi_{w_1}$ and $\varphi_{w_2}$ are defined as in \eqref{varphi}, taking $w_1$ and $w_2$ in place of $w$, respectively.
By a direct but rather lengthy calculation, we can obtain that the right hand of \eqref{3.57} equals
\begin{equation}\label{3.58}
\begin{split}
&-\frac{1}{24}\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha||w_1|+|\beta||w_1|+|\alpha||\beta|}\otimes[[z_\beta,[z_\alpha,w_1]],[z_\beta^*,[z_\alpha^*,w_2]]]\\
&+\frac{3(s-\sfr)+4}{24}([w_1,w_2],f).
\end{split}
\end{equation}
As \eqref{3.58} is computed by brute force, the completely elementary yet tedious proof will be omitted.
\subsection{}\label{6.4} Now we are in a position to give the proof of Proposition \ref{1commutator}. The arguments will be divided into two steps, according to whether or not $\mathfrak{g}=\mathfrak{sl}(2|2)/\mathbb{C}I$.
(Step 1) Set \begin{equation}\label{bw}
\begin{split}
B(w_1,w_2):=&[\Theta_{w_1},\Theta_{w_2}]-\frac{1}{2}([w_1,w_2],f)(C-\Theta_{\text{Cas}})+\frac{1}{2}\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}\\
&-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}\Theta_{[w_2,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_1]^{\sharp}},
\end{split}
\end{equation}
which is an element in $W_\chi'$. Moreover, the discussion in \eqref{3.57} shows that $B(w_1,w_2)=1\otimes b(w_1,w_2)$ for some
$b(w_1,w_2)\in A_e$. In conjunction with Lemma \ref{hwc2} this shows that $b(w_1,w_2)\in\mathbb{C}$ for all $w_1, w_2\in\ggg^e(1)$. Then
\begin{equation*}
\begin{array}{lcll}
b:&\ggg^e(1)\times\ggg^e(1)&\rightarrow&\mathbb{C}\\
&(w_1,w_2)&\mapsto &b(w_1,w_2)
\end{array}\end{equation*} is a bilinear form on $\ggg^e(1)$. Moreover, it is immediate from \eqref{3.57} and \eqref{3.58} that this bilinear form is even, i.e., $b(w_1,w_2)=0$ if $w_1$ and $w_2$ have different parities. In the following discussion we will show that this bilinear form is invariant under the adjoint action of $\ggg^e(0)_{\bar0}$, i.e., for any $v\in\ggg^e(0)_{\bar0}$, we will prove
\begin{equation}\label{5.}
b([w_1,v],w_2)-b(w_1,[v,w_2])=0.
\end{equation}
First note that
\begin{equation}\label{invariantform}
\begin{split}
&b([w_1,v],w_2)-b(w_1,[v,w_2])\\
=&[\Theta_{[{w_1},v]},\Theta_{w_2}]-[\Theta_{w_1},\Theta_{[v,{w_2}]}]-\frac{1}{2}([[{w_1},v],w_2],f)(C-\Theta_{\text{Cas}})\\
&+\frac{1}{2}([w_1,[v,{w_2}]],f)(C-\Theta_{\text{Cas}})
+\frac{1}{2}\sum\limits_{\alpha\in S(-1)}\Theta_{[[{w_1},v],z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}\\
&-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*,[v,{w_2}]]^{\sharp}}
-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}\Theta_{[w_2,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*,[{w_1},v]]^{\sharp}}\\
&+\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}\Theta_{[[v,{w_2}],z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_1]^{\sharp}}.
\end{split}
\end{equation}
In virtue of Proposition \ref{vw}, we have
\begin{equation}\label{key1}
\begin{split}
[\Theta_{[w_1,v]},\Theta_{w_2}]-[\Theta_{w_1},\Theta_{[v,w_2]}]
=&[[\Theta_{w_1},\Theta_v],\Theta_{w_2}]
-[\Theta_{w_1},[\Theta_v,\Theta_{w_2}]]\\
=&[[\Theta_{w_1},\Theta_{w_2}],\Theta_v].
\end{split}
\end{equation}
Moveover, it is immediate from $v\in\ggg^e$ that
\begin{equation}\label{key2}
\begin{split}
-([[{w_1},v],w_2],f)+([w_1,[v,{w_2}]],f)=&-([[w_1,w_2],v],f)\\
=&-([w_1,w_2],f)([e,v],f)=0.
\end{split}
\end{equation}
For the remaining terms in \eqref{invariantform}, first note that
\begin{equation}\label{thegoal}
\begin{split}
&\sum\limits_{\alpha\in S(-1)}\Theta_{[[w_1,v],z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}-\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*,[v,w_2]]^{\sharp}}\\
=&\sum\limits_{\alpha\in S(-1)}\Theta_{[[w_1,z_\alpha],v]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}+\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,[v,z_\alpha]]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}\\
&-\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[[z_\alpha^*,v],w_2]^{\sharp}}-\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[v,[z_\alpha^*,w_2]]^{\sharp}}.
\end{split}
\end{equation}
For any $\alpha\in S(-1)$, it is worth noting that both $[v,z_\alpha]$ and $[z_\alpha^*,v]$ are in $\ggg(-1)$, then we have
\begin{equation}\label{extend}
[v,z_\alpha]=\sum\limits_{\beta\in S(-1)}\langle z_\beta^*,[v,z_\alpha]\rangle z_\beta,\qquad
[z_\alpha^*,v]=-\sum\limits_{\beta\in S(-1)}(-1)^{|\beta|}\langle z_\beta,[z_\alpha^*,v]\rangle z_\beta^*.
\end{equation}
We claim that the bilinear form $\langle\cdot,\cdot\rangle$ is $\ggg^e(0)_{\bar0}$-invariant. In fact, for any $v\in\ggg^e(0)_{\bar0}$ and $\alpha, \beta\in S(-1)$, we have
\begin{equation}\label{angle}
\begin{split}
\langle[z_\alpha,v],z_\beta\rangle=&(e,[[z_\alpha,v],z_\beta])
=(e,[z_\alpha,[v,z_\beta]])+(e,[[z_\alpha,z_\beta],v])\\
=&(e,[z_\alpha,[v,z_\beta]])-(e,[v,[z_\alpha,z_\beta]])\\
=&(e,[z_\alpha,[v,z_\beta]])-([e,v],[z_\alpha,z_\beta])\\
=&(e,[z_\alpha,[v,z_\beta]])=\langle z_\alpha,[v,z_\beta]\rangle.
\end{split}
\end{equation}
Taking \eqref{extend} and \eqref{angle} into account, one can conclude that
\begin{equation}\label{3.66angle}
\begin{split}
&\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,[v,z_\alpha]]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}-\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[[z_\alpha^*,v],w_2]^{\sharp}}\\
=&\sum\limits_{\alpha,\beta\in S(-1)}\langle z_\beta^*,[v,z_\alpha]\rangle \Theta_{[w_1,z_\beta]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}+\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\beta|}\langle z_\beta,[z_\alpha^*,v]\rangle \Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\beta^*,w_2]^{\sharp}}\\
=&\sum\limits_{\alpha,\beta\in S(-1)}\langle z_\alpha^*,[v,z_\beta]\rangle \Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\beta^*, w_2]^{\sharp}}-\sum\limits_{\alpha,\beta\in S(-1)}\langle [z_\alpha^*,v],z_\beta\rangle \Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\beta^*,w_2]^{\sharp}}\\
=&\sum\limits_{\alpha,\beta\in S(-1)}\langle z_\alpha^*,[v,z_\beta]\rangle \Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\beta^*, w_2]^{\sharp}}-\sum\limits_{\alpha,\beta\in S(-1)}\langle z_\alpha^*,[v,z_\beta]\rangle \Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\beta^*,w_2]^{\sharp}}\\
=&0.
\end{split}
\end{equation}
For any $w\in\ggg^e(1)$ and $\alpha\in S(-1)$, it follows from the definition of ${\sharp}$ in \S\ref{3.1.1} that $[w,z_\alpha]^{\sharp}\in\ggg^e(0)$, then we have
\begin{equation*}
\begin{split}
[[w,z_\alpha]^{\sharp},v]^{\sharp}=&[[w,z_\alpha]^{\sharp},v]-\frac{1}{2}(h,[[w,z_\alpha]^{\sharp},v])h\\
=&[[w,z_\alpha]^{\sharp},v]-\frac{1}{2}([h,[w,z_\alpha]],v)h\\
=&[[w,z_\alpha]^{\sharp},v].
\end{split}
\end{equation*}
As $v\in\ggg^e(0)_{\bar0}$, and $\mathbb{C}h$ is orthogonal to $\ggg^e(0)$ with respect to $(\cdot,\cdot)$, then
\begin{equation*}
[[w,z_\alpha],v]^{\sharp}=[([w,z_\alpha]-\frac{1}{2}(h,[w,z_\alpha])h),v]^{\sharp}=[[w,z_\alpha]^{\sharp},v]^{\sharp}=[[w,z_\alpha]^{\sharp},v].
\end{equation*}
By the same discussion we have $[v,[z_\alpha^*,w_2]]^{\sharp}=[v,[z_\alpha^*,w_2]^{\sharp}]$.
In view of Proposition \ref{v1v2} this yields
\begin{equation}\label{thethird}
\begin{split}
&\sum\limits_{\alpha\in S(-1)}\Theta_{[[w_1,z_\alpha],v]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}-\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[v,[z_\alpha^*,w_2]]^{\sharp}}\\
=&\sum\limits_{\alpha\in S(-1)}\Theta_{[[w_1,z_\alpha]^{\sharp},v]}\Theta_{[z_\alpha^*, w_2]^{\sharp}}-\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[v,[z_\alpha^*,w_2]^{\sharp}]}\\
=&\sum\limits_{\alpha\in S(-1)}[\Theta_{[w_1,z_\alpha]^{\sharp}},\Theta_v]\Theta_{[z_\alpha^*, w_2]^{\sharp}}-\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,z_\alpha]^{\sharp}}[\Theta_v,\Theta_{[z_\alpha^*,w_2]^{\sharp}}]\\
=&\sum\limits_{\alpha\in S(-1)}[\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}},\Theta_v].
\end{split}
\end{equation}
Combining \eqref{thegoal} with \eqref{3.66angle} and \eqref{thethird}, we obtain
\begin{equation}\label{reachthegoal1}
\sum\limits_{\alpha\in S(-1)}\Theta_{[[w_1,v],z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}-\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*,[v,w_2]]^{\sharp}}
=\sum\limits_{\alpha\in S(-1)}[\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}},\Theta_v].
\end{equation}
Interchanging the roles of $w_1$ and $w_2$ in \eqref{reachthegoal1}, we have
\begin{equation}\label{reachthegoal2}
\begin{split}
&\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}\Theta_{[[v,w_2],z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_1]^{\sharp}}-\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}\Theta_{[w_2,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*,[w_1,v]]^{\sharp}}\\
=&-(-1)^{|w_1||w_2|}\sum\limits_{\alpha\in S(-1)}[\Theta_{[w_2,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_1]^{\sharp}},\Theta_v].
\end{split}
\end{equation}
Moreover, it is worth noting that $[C-\Theta_{\text{Cas}},\Theta_v]=0$ by Proposition \ref{vcommutate}.
As an immediate consequence of \eqref{invariantform}, \eqref{key1}, \eqref{key2}, \eqref{reachthegoal1} and \eqref{reachthegoal2}, we have
\begin{equation*}
\begin{split}
&b([w_1,v],w_2)-b(w_1,[v,w_2])\\
=&[[\Theta_{w_1},\Theta_{w_2}],\Theta_v]+\frac{1}{2}\sum\limits_{\alpha\in S(-1)}[\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}},\Theta_v]\\
&-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}[\Theta_{[w_2,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_1]^{\sharp}},\Theta_v]\\
=&[([\Theta_{w_1},\Theta_{w_2}]-\frac{1}{2}([w_1,w_2],f)(C-\Theta_{\text{Cas}})+\frac{1}{2}\sum\limits_{\alpha\in S(-1)}\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}\\
&-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(-1)^{|w_1||w_2|}\Theta_{[w_2,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_1]^{\sharp}}),\Theta_v]\\
=&[1\otimes b(w_1,w_2),\Theta_v]=0,
\end{split}
\end{equation*}
where the last equation comes from the fact that $b(w_1,w_2)\in\mathbb{C}$. We complete the proof of \eqref{5.}.
On the other hand, Kac-Roan-Wakimoto gave a description of the $\ggg(0)^\sharp$-module $\ggg(1)$ in \cite[Proposition 4.1]{KRW} (see also \cite[Table 1-Table 3]{KW}), which showed that except the case with $\ggg=\mathfrak{sl}(2|2)/\mathbb{C}I$ (where $I$ denotes the unitary matrix), either $\ggg^e(1)$ is an irreducible $\ggg(0)^\sharp$-module, or $\ggg^e(1)\cong M\oplus M^*$ with $M$ and $M^*$ being irreducible $\ggg(0)^\sharp$-modules such that $M\ncong M^*$ (note that the grading on $\ggg$ applied there was under the action of $\text{ad}\,\frac{h}{2}$, thus $\ggg_{\frac{1}{2}}$ there is just $\ggg(1)$ in our case). If just consider the even part of $\ggg(0)^\sharp$, one can readily conclude from \cite[Table 1-Table 3]{KW} that $\ggg(0)^\sharp_{\bar0}$-module $\ggg(1)$ has the same property as described above,
except the case $\mathfrak{sl}(2|2)/\mathbb{C}I$.
Since $\ggg(0)^\sharp_{\bar0}=\ggg^e(0)_{\bar0}$ and $\ggg(1)=\ggg^e(1)$ in this situation, from all the discussion above one can conclude that $b=c_0([\cdot,\cdot],f)$ for some $c_0\in\mathbb{C}$ except $\ggg=\mathfrak{sl}(2|2)/\mathbb{C}I$.
(Step 2) Now we turn to the case when $\ggg=\mathfrak{sl}(2|2)/\mathbb{C}I$. In this case, it follows from \cite[Table 2]{KW}
that the $\ggg^e(0)_{\bar0}$-module $\ggg^e(1)$ is isomorphic to $\mathfrak{sl}_2$-module $\mathbb{C}^2\oplus\mathbb{C}^2$, thus the discussion in the end of (Step 1) can not be applied. So we need to calculate the value of \eqref{3.58} with $w_1,w_2\in\ggg^e(1)$.
Recall that $\mathfrak{sl}(2|2)\subseteq\mathfrak{gl}(2|2)$ consists of $4\times4$ matrices in the following $(2|2)$-block form
\begin{center}
$\begin{matrix}
\begin{pmatrix} A & B\\C & D \end{pmatrix}
\end{matrix}$,
\end{center}
where $A, B, C, D$ are all $2\times2$ matrices, and $\text{tr}\,A-\text{tr}\,D=0$. Then $\ggg$ is a quotient of $\mathfrak{sl}(2|2)$ by the scalars of unitary matrix $I$. Denote by $e_{\bar i\bar j}, e_{\bar ij}, e_{i\bar j}, e_{ij}\in\mathfrak{gl}(2|2)$ the matrix with $1$ in $ij$-entry of $A, B, C, D$ respectively, and $0$ other entries.
It is a direct consequence from PBW theorem that $\ggg$ has a basis
\begin{equation*}
\begin{split}
&h=e_{\bar1\bar1}-e_{\bar2\bar2}, H_1=e_{\bar2\bar2}+e_{11}, H_2=e_{11}-e_{22},\\
&e_{\bar1\bar2}, e_{\bar2\bar1}, e_{\bar11},e_{1\bar1},e_{\bar12}, e_{1\bar2}, e_{\bar21}, e_{2\bar1}, e_{\bar22}, e_{2\bar2}, e_{12}, e_{21}.
\end{split}
\end{equation*}
Consider the $\mathfrak{sl}_2$-triple $(e,h,f)$ with $e=e_{\bar 1\bar 2}, h=e_{\bar 1\bar 1}-e_{\bar 2\bar 2}, f=e_{\bar 2\bar 1}$. It is obvious that $e$ is a minimal nilpotent element in $\ggg$. Set $str(\cdot,\cdot)$ to be the nondegenerate supersymmetric invariant bilinear form on $\ggg$, then we have $(e,f)=\frac{1}{2}(h,h)=1$.
Let ${\ggg}(i)=\{x\in{\ggg}\mid[h,x]=ix\}$, then ${\ggg}=\bigoplus_{i\in{\bbz}}{\ggg}(i)$. It can be observed that
\begin{equation*}
\begin{array}{rlrl}
h, H_1, H_2, e_{12}, e_{21}&\in\ggg(0)_{\bar0},&&\\
e_{\bar11}, e_{\bar12}, e_{1\bar2}, e_{2\bar2}&\in\ggg(1)_{\bar1},&e_{\bar1\bar2}&\in\ggg(2),\\
e_{\bar21}, e_{\bar22}, e_{1\bar1}, e_{2\bar1}&\in\ggg(-1)_{\bar1},&e_{\bar2\bar1}&\in\ggg(-2).
\end{array}
\end{equation*}
As
\begin{equation*}
h+2H_1, H_2, e_{12}, e_{21}, e_{\bar1\bar2}, e_{\bar11}, e_{\bar12}, e_{1\bar2}, e_{2\bar2}
\end{equation*}
constitute a basis of $\ggg^e$, by Proposition \ref{v1v2} and Proposition \ref{vw} we can choose
\begin{equation*}
h+2H_1, e_{12}, e_{21}\in\ggg(0)_{\bar0},\qquad e_{\bar11}, e_{1\bar2}\in\ggg(1)_{\bar1}
\end{equation*}
as the generators of $\ggg^e$. Note that
\begin{equation*}1\otimes e_{\bar21}^2=\frac{1}{2}\otimes[e_{\bar21},e_{\bar21}]=0,\end{equation*}
and by the same discussion we have
\begin{equation*}
1\otimes e_{\bar22}^2=1\otimes e_{1\bar1}^2=1\otimes e_{2\bar1}^2=0.
\end{equation*}
Moreover, $e_{\bar21}$ and $e_{\bar22}$ are dual bases of $e_{1\bar1}$ and $e_{2\bar1}$ with respect to $(e,[\cdot,\cdot])$.
Let $w_1, w_2$ be any of $e_{\bar11}$ and $e_{1\bar2}$. By case-by-case calculations one can obtain that
\begin{equation*}
\sum\limits_{\alpha,\beta\in S(-1)}(-1)^{|\alpha||w_1|+|\beta||w_1|+|\alpha||\beta|}[[z_\beta,[z_\alpha,w_1]],[z_\beta^*,[z_\alpha^*,w_2]]]=4([w_1,w_2],f).
\end{equation*}
Taking $ s=0, \sfr=4$ into account, it is immediate from \eqref{3.57} and \eqref{3.58} that
\begin{equation*}
\begin{split}
[\Theta_{w_1},\Theta_{w_2}]=&\frac{1}{2}([w_1,w_2],f)(C-\Theta_{\text{Cas}}-1)-\frac{1}{2}\sum\limits_{\alpha\in S(-1)}(\Theta_{[w_1,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_2]^{\sharp}}\\
&-(-1)^{|w_1||w_2|}\Theta_{[w_2,z_\alpha]^{\sharp}}\Theta_{[z_\alpha^*, w_1]^{\sharp}}).
\end{split}
\end{equation*}
Thus $c_0=1$ in this case.
Summing up both results in Steps (1) and (2), and also \eqref{3.57}, \eqref{3.58}, we complete the proof of Proposition \ref{1commutator}.
\begin{rem}
For the Lie algebra version of Proposition \ref{1commutator}, Premet made a crucial use of the machinery of associated varieties and Joseph ideal for Lie algebras, by which Premet calculated the exact values of $c_0$ for each type of simple Lie algebras, respectively. Unfortunately, we are in lack of such a powerful tool for Lie superalgebras. On the other hand, it is still a hard work to compute the exact values of $c_0$ in Proposition \ref{1commutator} with \eqref{3.58}. Since our finial result in Theorem \ref{intromainminnimalf} does not have much to do with the exact values of $c_0$, we give up the calculation of $c_0$ in the present paper.
\end{rem}
\vskip0.2cm
{\bf Acknowledgements} \; The authors got much help from Suh, who explained some results in her paper \cite{suh}. The authors express great thanks to her.
|
1,108,101,563,306 | arxiv | \section{Introduction}
Metamaterials and metasurfaces have recently emerged as promising platforms for manipulating the polarization state of electromagnetic waves because of their compactness, flexibility, and easy integration \cite{FPC2010PolMM_rev,RPP2016Metasurface_Rev,IEEEPJ2018LPC,IEEEAccess2019dual,JPD2020broadband}. Half-wave plates (HWPs) and quarter-wave plates (QWPs) are two key devices for realizing polarization conversion among two orthogonal polarization states, and the left- and right-handed circular polarization (LCP and RCP) states. In the terahertz regime, metamaterial- or metasurface-based HWPs/QWPs are of particular interest because conventional approaches based on birefringence or total internal reflection effects are usually bulky, narrowband, and sometimes lossy.
Over the years, metamaterial-based terahertz polarization converters have evolved from narrow-band \cite{NJP2012THzMMPol,APL2013THzMMPol} to broadband \cite{Science2013HWP-THz,APL2014HWP-THZ,LPR2014QWP-THZ,OL2016Gbroadband,OE2017Gbroadband,EPL2017HWPQWPBroad}. However, the functionalities of these devices, which are made of metals or dielectrics, cannot be dynamically tuned. Recently, dynamically switchable HWPs/QWPs based on metamaterials or metasurfaces incorporating tunable materials, such as liquid crystal, vanadium dioxide (VO$_2$), or graphene, have attracted increasing attentions. Based on metamaterials incorporating liquid crystals, Vasi{\'c} \cite{Nanotech2017TunableLC} and Ji {\sl et al.} \cite{OE2018SwitchQWP} respectively proposed electrically tunable terahertz polarization converters, of which the bandwidths are only 0.056~THz and 0.35~THz, respectively. Based on VO$_2$-metal metasurfaces, Wang {\sl et al.} \cite{SCIREP2015QWPVO2,IEEEPJ2016QWPVO2} demonstrated terahertz QWPs with switchable single or multiple operation frequencies. Nakata {\sl et al.} \cite{PRA2019QWP-THz} demonstrated a switchable QWP operating at the specific frequency of 0.617~THz by designing a VO$_2$-based anisotropic checkerboard metasurface. Zhao {\sl et al.} \cite{CPL2020QWP2HWP} proposed a switchable terahertz metamaterial that can be switched between an HWP and a QWP in the spectral band of 2.09--2.27~THz, corresponding to a relatively narrow bandwidth of 0.18~THz. Quite recently, some of the authors \cite{OE2020HWP2QWP} proposed a metal-VO$_2$ metamaterial for achieving broadband switchable terahertz HWP/QWP, the bandwidth of which covers 0.66--1.40 THz, corresponding to a relative bandwidth of 71.8\%. We further proposed a novel design philosophy making use of the transition from the overdamped to the underdamped resonance, and designed a VO$_2$-metal hybrid metasurface for achieving broadband dynamically switchable HWP/QWP in the spectral band of 0.8--1.2~THz \cite{arXiv2021HWPQWPBroad}.
Compared with liquid crystals and VO$_2$, graphene have many unique characteristics, including ultra-thin thickness (only 0.33~nm), low loss, and continuous and flexible tunability via electric biasing \cite{ACSN2012Gtune}, chemical doping \cite{JMC2011Gdoping}, or optical pumping \cite{JAP2007GOpump}. Zhang {\sl et al.} \cite{OE2015QWP2Linear-THZ} and Tavakol {\sl et al.} \cite{PTL2019TunableQWP} respectively proposed switchable QWPs based on graphene metamaterials, and showed that the polarization state of the output wave can be dynamically switched among linear, left- and right-handed polarization by changing the graphene chemical potential. Zhang {\sl et al.} \cite{OSAC2018QWP2HWP} proposed functional switch from a QWP to an HWP within 4.80--5.10~THz based on a graphene metasurface. The corresponding bandwidth is only 0.3~THz and the relative bandwidth is only 6\%. Guan {\sl et al.} \cite{OL2019HWP2QWP_GSi} propose a hybrid graphene-dielectric metasurface for realizing switchable HWP/QWP operating at the single frequency of 1~THz. Qi {\sl et al.} \cite{OE2020HWPQWPG} also proposed a graphene-based high-efficiency switchable HWP/QWP operating at the specific frequency of 15.96~THz. Quite recently, Zhang {\sl et al.} \cite{OE2020HWPQWPGbroad} proposed a hybrid graphene-metal metasurface for achieving terahertz HWP/QWP in the frequency range of 1.38--1.72~THz. However, the corresponding bandwidth is 0.34~THz and the relative bandwidth is only 22\%. Therefore, it remains challenging to design broadband switchable QWPs/HWPs based on graphene metasurfaces, greatly hindering their practical applications.
In this work, we propose a graphene-metal hybrid metasurface for achieving broadband switchable terahertz HWPs/QWPs. The metasurface unit cell is composed of a gold stripe and a graphene stripe on a thick gold film sandwiched by a dielectric spacer. We will show that the function of the proposed metasurface can be switched from a broadband HWP with polarization conversion ratio (PCR) exceeding 97\% in the band of 0.7--1.3~THz, to a broadband QWP with ellipticity over 0.92 in the range of 0.78--1.33~THz. Therefore, these two switchable functionalities share the same spectral band of 0.78--1.3~THz, corresponding to a strikingly broad bandwidth of 0.52~THz and relative bandwidth of 50\%, which is much larger than those of graphene-based metasurface HWPs/QWPs in the literature \cite{OSAC2018QWP2HWP,OL2019HWP2QWP_GSi,OE2020HWPQWPG,OE2020HWPQWPGbroad}.
\section{Simulation setup}
Figure~\ref{fig:schem} illustrates the proposed metasurface acting as a broadband and dynamically switchable terahertz HWP/QWP. The metasurface unit cell consists of a gold short stripe and a graphene long stripe, which are perpendicular to each other and placed obliquely of 45$^\circ$ with respect to the $x$-axis. These graphene-gold hybrid stripes stand on a thick gold film sandwiched by a polyimide spacer, forming a metal-insulator-metal configuration. The metasurface can be fabricated using the state-of-the-art top-down microfabrication processes. A thick gold film is first deposited onto a silicon substrate, followed by spin-coating of a polyimide film of designed thickness. A monolayer graphene layer is then transferred on the top, and patterned into two-dimensional array of stripes using photolithography and reactive ion etching. Finally, the gold stripe array can be fabricated via photolithography, thermal evaporation, and liftoff processes.
\begin{figure}[htp]
\centering
\includegraphics[width=\linewidth]{FIG1.pdf}
\caption{Schematics of the proposed graphene-metal hybrid metasurface. (a) 3D view of the metasurface, which can function as an HWP converting the incident linearly $y$-polarized terahertz wave into $x$-polarized wave when the Fermi level of graphene $E_{\rm F}$ is 0~eV, or as a QWP converting the incident $y$-polarized wave into right-handed circularly polarized wave when $E_{\rm F}=1$~eV. (b) Top view of the unit cell with $\Lambda=120~\mu$m, $W_{\rm Au}=30~\mu$m, $W_{\rm G}=20~\mu$m, $L=90~\mu$m, $h_{\rm PI}=38~\mu$m, and $h_{\rm Au}=200~$nm.}
\label{fig:schem}
\end{figure}
The operation principle of the proposed metasurface is as follows. When the Fermi level of graphene $E_{\rm F}$ is pinned at 0~eV, the graphene conductivity is as low as the dielectric material for the terahertz wave. In this scenario, the metasurface unit cell is effectively composed of the gold stripe, and thus the metasurface works as an HWP which can convert the incident linear polarization state into the orthogonal linear polarization state. When the graphene Fermi level is tuned to $E_{\rm F}=1$~eV, which can be done via external stimulus, such as electric biasing \cite{ACSN2012Gtune} or optical pumping \cite{JAP2007GOpump}, the graphene conductivity is as high as metal-like material. In this case, the metasurface acts as a QWP that converts the linearly polarized incident wave into circularly polarized wave.
The polarization-dependent reflection amplitude and phase spectra of the proposed metasurface were numerically simulated using the frequency-domain solver in CST Microwave Studio. The unit cell boundary conditions were applied in the $x$ and $y$ directions, and the open boundary condition was used in the $z$ direction. Tetrahedral meshes with adaptive refinement process were applied. In all the simulations, gold was modelled using the lossy metal model with conductivity of $4.56 \times 10^7$~S/m, and the built-in material permittivity was adopted for polyimide.
The surface conductivity of graphene $\sigma_{\rm G}$ can be described by the Kubo equation with intraband and interband contributions~\cite{GraphenePerformance2014}, that is
\begin{equation}
\sigma_{\rm G}=\sigma_{\rm intra}+\sigma_{\rm inter}\,. \label{eq:sigmaG}
\end{equation}
According to Pauli exclusion principle, the interband contribution can be safely neglected compared with the intraband contribution for low terahertz frequencies at room temperature~\cite{Graphene2007}. Thus the expression of the graphene conductivity can be simplified into~\cite{GraphenePlasmonics2011}
\begin{equation}
\sigma_{\rm G} \approx \sigma_{\rm intra}=\frac{-i e^{2}k_{\rm B}T}{\pi\hbar^{2}(\omega+2i\tau)}\left[\frac{E_{\rm F}}{k_{\rm B}T}+2 \ln(e^{\frac{E_{\rm F}}{k_{\rm B}T}}+1)\right]\,.
\label{eq:sigmaG2}
\end{equation}
Here $e$ is the charge of an electron, $k_{\rm B}$ is the Boltzmann constant, $T$ is the temperature, $\omega$ is the light frequency, $\tau$ is the carrier relaxation time from the impurities in graphene, and $\hbar$ is the Planck constant.
\section{Results and discussion}
\subsection{HWP function}
Figures~\ref{fig:SpecHWP}(a)(b) depict the simulated reflection amplitude and phase spectra of the proposed metasurface when the graphene Fermi level is set to be 0~eV, and under the incidence of $u$- or $v$-polarized terahertz wave. Results show that $|r_{uu}|$ and $|r_{vv}|$ are almost equal to each other, and are close to 0.9 within the frequency range of 0.7--1.3~THz. Meanwhile, their phase differences are approximately equal to $-180^{\circ}$, {\sl i.e.}, $\Delta\Phi=\Phi_{vv}-\phi_{uu}=-180^{\circ}$. Therefore, the graphene-metal hybrid metasurface with $E_{\rm F}=0$~eV can act as a broadband and efficient HWP.
\begin{figure}[htp]
\centering
\includegraphics[width=\linewidth]{FIG2.pdf}
\caption{Simulated reflection (a) amplitude $|r|$ (b) phase $\Phi$ spectra of the proposed hybrid metasurface with $E_{\rm F}=0$~eV under $u$- and $v$-polarized incidences. (c) Calculated reflection amplitude spectra for the co-polarization $|r_{yy}|$ and cross-polarization $|r_{xy}|$. (d) Calculated PCR spectra under $y$-polarized incidence. Blue regions in (b)--(d) indicate the broad operation band of the obtained HWP.}
\label{fig:SpecHWP}
\end{figure}
\begin{figure*}[hbtp]
\centering
\includegraphics[width=14cm]{FIG4.pdf}
\caption{Surface current distributions (arrows for directions and colors for strengths) on the gold and graphene stripes (1$^{\rm st}$ and 3$^{\rm rd}$ columns) and on the gold film (2$^{\rm nd}$ and 4$^{\rm th}$ columns) for three resonant frequencies of (a)(b) 0.74~THz, (c)(d) 0.99~THz, (e)(f) 1.23~THz when the graphene Fermi level is $E_{\rm F}=0$~eV. The 1$^{\rm st}$ and 2$^{\rm nd}$ columns are for the $u$-polarized incidence, and the 3$^{\rm rd}$ and 4$^{\rm th}$ columns are for the $v$-polarized incidence. The black arrows indicate the dominant current directions.}
\label{fig:FieldHWP}
\end{figure*}
In order to evaluate the performance of the obtained HWP under the $y$-polarized incidence, we calculate the co-polarized and cross-polarized reflection amplitude spectra, $|r_{yy}|$ and $|r_{xy}|$, and the spectra of the polarization conversion ratio (PCR), which can be calculated by
\begin{eqnarray}
{\rm PCR} & = & |r_{xy}|^2/(|r_{xy}|^2+|r_{yy}|^2)\,.
\label{Eq:PCR}
\end{eqnarray}
Figure~\ref{fig:SpecHWP}(c) shows that the cross-polarized reflection amplitude $|r_{xy}|$ is larger than 0.9, whereas the co-polarized reflection amplitude $|r_{yy}|$ is smaller than 0.1 in the range of 0.7--1.3~THz. For $|r_{yy}|$, there exist three dips locating at 0.74~THz, 0.99~THz and 1.23~THz. As a result, the calculated PCRs are larger than 0.97 within the broad spectral band of 0.7--1.3~THz. This corresponds to a relative bandwidth of $\Delta f/f_0 = 60\%$ with the central frequency locating at $f_0=1$~THz. These results suggest that the incident $y$-polarized terahertz wave can be efficiently converted to the $x$-polarized wave by the reflective metasurface with a broad operation bandwidth and high polarization conversion ratios. In other words, the proposed metasurface can work as a broadband and efficient HWP if the graphene Fermi level is 0~eV.
The broadband performance of the obtained HWP can be understood from the surface current distributions on the gold and graphene stripes and on the bottom gold film. Figure~\ref{fig:FieldHWP} depicts the surface current distributions under the $u$- and $v$-polarized incidences at the three resonant frequencies of $f=0.74$~THz, 0.99~THz, and 1.23~THz, for which $|r_{yy}|$ shows dips, as shown in Figure~\ref{fig:SpecHWP}(c). Figures~\ref{fig:FieldHWP}(a)(b) show that at 0.74~THz, the surface currents on the gold stripe are weak under the $u$-polarized incidence, but strong under the $v$-polarized incidence. These strong surface currents have opposite direction compared to those on the bottom gold film, producing a magnetic resonance under the $v$-polarized incidence. Similarly, at 0.99~THz and 1.23~THz, Figures~\ref{fig:FieldHWP}(c)(e) show that magnetic resonances are also generated under the $u$-polarized incidence, because the surface currents on the gold stripe have opposite direction to those on the gold film. Figure~\ref{fig:FieldHWP}(d) shows that for the $v$-polarized incidence at 0.99~THz, the surface currents on the gold stripe are strong and have clear flow direction, whereas those on the gold film have counter-propagating directions and almost cancel each other. At 1.23~THz, the surface currents on the gold stripe and on the gold film are parallel to each other, resulting in an equivalent electric resonance, as shown by Figure~\ref{fig:FieldHWP}(f). Therefore, the broadband performance of the obtained HWP should originate from the superposition of these multiple resonances.
\subsection{QWP function}
We now tune the graphene Fermi level to $E_{\rm F}=1$~eV. Figure~\ref{fig:SpecQWP}(a) shows that the reflection amplitudes $|r_{uu}|$ and $|r_{vv}|$ are close to 0.9 for frequencies above 0.80~THz. Therefore, $|r_{yy}|$ and $|r_{xy}|$ are also close to each other, and their phase differences are approximate to $-270^\circ$ within the broadband frequency range of 0.78--1.33~THz, as shown in Figures~\ref{fig:SpecQWP}(b)(c). These results suggest that the proposed metasurface with $E_{\rm F}=1$~eV now acts as a terahertz QWP.
\begin{figure}[htp]
\centering
\includegraphics[width=\linewidth]{FIG3.pdf}
\caption{Simulated reflection (a)(b) amplitude $|r|$ and (c) phase spectra of the proposed hybrid metasurface with $E_{\rm F}=1$~eV. (d) The calculated ellipticity under linearly $y$-polarized incidence. Blue regions in (b)--(d) indicate the broad operation band of the obtained QWP.}
\label{fig:SpecQWP}
\end{figure}
The performance of the obtained QWP can be quantified using the ellipticity defined as $\chi \equiv S_3/S_0$, where the Stokes parameters $S_0$ and $S_3$ are expressed as \cite{LPR2014QWP-THZ},
\begin{eqnarray}\label{Eq:S}
S_0 &=& |r_{xy}|^2+|r_{yy}|^2\,,\\
S_3 &=& 2|r_{xy}||r_{yy}|\sin(\Delta\Phi)\,.
\end{eqnarray}
Here $\Delta\Phi=\Delta\Phi_{yy}-\Delta\Phi_{xy}$ means the phase difference between co-polarization and cross-polarization reflections. When $\chi$ equals to 1 or $-1$, the polarization state of the reflective terahertz wave is LCP or RCP, respectively. Figure~\ref{fig:SpecQWP}(d) shows that $\chi>0.92$ over the broad frequency range of 0.78--1.33~THz. This result means that the graphene-metal hybrid metasurface with $E_{\rm F}=1$~eV now functions as a broadband and efficient QWP that converts the incident linear $y$ polarization into the RCP.
\begin{figure*}[htp]
\centering
\includegraphics[width=14cm]{FIG5.pdf}
\caption{Similar to Figure~\ref{fig:FieldHWP} but for (a)(b) 0.7~THz, (c)(d)0.97~THz, and (e)(f) 1.23~THz when $E_{\rm F}=1$~eV.}
\label{fig:CurrentQWP}
\end{figure*}
To understand the broadband performance of the obtained QWP, we also plot the surface current distributions on the gold and graphene stripes and on the bottom gold film. Here we use the three frequencies of 0.7~THz, 1.23~THz and 0.97~THz, which corresponds to two peaks and a dip in the $|r_{xy}|$ spectra, respectively, as shown by Figure~\ref{fig:SpecQWP}(b). Figures~\ref{fig:CurrentQWP}(a)(b) show that at 0.7~THz, the surface currents on the gold and graphene stripes have the opposite direction compared with those on the gold film, forming magnetic resonances for both the $u$-polarized and $v$-polarized incidences. For the $u$-polarized incidence at 0.97~THz, and for both the $u$-polarized and $v$-polarized incidences at 1.23~THz, the surface currents the gold and graphene stripes and on the gold film have the same directions, resulting equivalent electric resonances, as shown by Figures~\ref{fig:CurrentQWP}(c)(e)(f). For the $v$-polarized incidence at 0.97~THz, Figure~\ref{fig:CurrentQWP}(d) shows that the surface currents on the gold and graphene stripes are strong and have well defined direction, whereas those on the gold film are weak and have counter-propagating directions. Therefore, the broadband performance of the obtained QWP should also originate from the superposition of multiple resonances.
\section{Conclusions}
In conclusion, we have proposed a broadband switchable terahertz HWP/QWP based on a graphene-metal hybrid metasurface. Simulation results have shown that, by varying the Fermi energy of graphene from 0~eV to 1~eV, the function of the reflective metasurface can be switched from an HWP, which converts the incident $y$-polarized terahertz wave into $x$-polarized wave, to a QWP that converts the $y$-polarized wave into right-handed circularly polarized wave. The polarization conversion ratio of the obtained HWP exceeds 97\% over a wide band ranging from 0.7~THz to 1.3~THz, and the ellipticity of the obtained QWP is larger than 0.92 over 0.78--1.33~THz. Therefore, we have shown that the proposed metasurface can function either as an HWP or as a QWP in the spectral band of 0.78--1.3~THz. This corresponds to a strikingly broad bandwidth of 0.52~THz and relative bandwidth of 50\%, which is much larger than the literature on graphene-based metasurface HWPs/QWPs. The broadband properties have been explained with the superposition of multiple resonances. We expect the designed broadband and dynamically switchable terahertz HWP/QWP will find applications in terahertz polarization-dependent systems of sensing, imaging, and telecommunications.
\section*{Acknowledgments}
This work was supported by the Shenzhen Research Foundation (JCYJ20180507182444250).
\bibliographystyle{unsrt}
|
1,108,101,563,307 | arxiv | \section{INTRODUCTION}
It is generally accepted that relativistic outflows from the core regions of
radio-loud active galactic
nuclei (AGN) are responsible for many of the
interesting phenomena that are observed in these sources.
Relativistic motion of synchrotron emitting plasma will result in
the Doppler boosting of the synchrotron radiation from these outflows
(discussed by many authors, e.g. Marscher 1987).
In addition, relativistic outflows would
readily explain the apparent superluminal motion observed on VLBI (Very Long
Baseline Interferometry) scales (e.g. Pearson \& Zensus 1987).
The Doppler factor is a key quantity in deriving intrinsic
properties of an AGN. It is well known that the unbeamed
flux density of a radiating source is related to
the observed flux density
by the Doppler factor.
When
combined with the apparent motion in the plane of the sky,
the Doppler factor can be used to estimate the Lorentz factor
and the viewing angle of an outflow,
assuming the motion responsible for the Doppler boosting
has the same speed as the motion of radio features in the plane of the sky
(Ghisellini et al. 1993; Daly, Guerra, \& G\"uijosa 1997). The
values of these Lorentz factors
and viewing angles are
significant in
understanding AGN physics and are relevant to
orientation unified
models of AGN.
There are several methods for computing the
Doppler factor. One method, called
the inverse Compton Doppler factor, is
derived by assuming the observed X-ray flux is caused by inverse Compton
scattering of synchrotron photons off the radiating particles (e.g.
Marscher 1987). A second method, called the equipartition
Doppler factor, is derived assuming that the
sources are at or near equipartition of energy between radiating
particles and magnetic field (Readhead 1994). Other
methods (not discussed further here)
include those based on variability
measurements (Rees 1967) or jet/counter jet flux ratios (Conway 1982).
Ghisellini et al. (1993; hereafter GPCM93) compute
the inverse Compton Doppler factor for a sample of 105 sources, using
data compiled from the literature. The data
compiled by GPCM93 were used by G\"uijosa \& Daly (1996) to compute
equipartition Doppler factors.
These sets of inverse Compton Doppler factors and equipartition
Doppler factors are used here to compute
the Lorentz factors, viewing angles, and
intrinsic properties of outflows from these sources.
Updated estimates of both inverse Compton and equipartition Doppler factors
for a sample of 100 sources
are discussed in \S \ref{ssec:GPCM}; intrinsic brightness
temperatures computed for both sets of Doppler factors are discussed
in \S
\ref{ssec:temp}.
The reliability of Doppler factors
and the errors assigned to these estimates
are discussed in \S \ref{ssec:rely} and
\S \ref{ssec:err}
respectively. Estimates of the intrinsic
luminosity density and intrinsic luminosity are examined in \S
\ref{ssec:lum}. The computations
of the bulk Lorentz factor and
the viewing angle
are discussed in \S \ref{sec:anres}. A subsample of sources with estimates
of Doppler factors and apparent motion in the plane of the
sky is introduced in \S \ref{ssec:VC}, and a set of solutions for
the bulk Lorentz factor and viewing angle are computed
in \S \ref{ssec:solve} for this subsample.
In \S \ref{ssec:comm}, these results are discussed
in the context of orientation
unified models. A general discussion follows in \S
\ref{sec:disc}.
\section{A SAMPLE OF DOPPLER FACTOR ESTIMATES \label{sec:est}}
\subsection{Previous and Updated Doppler Factor Estimates \label{ssec:GPCM}}
GPCM93 assemble the relevant data to compute the inverse Compton
Doppler factor for
those AGN with VLBI core size data available in the literature circa
1986-92. The five BL Lacs without known redshifts are excluded from
the sample examined here, since inverse Compton and equipartition
Doppler factor estimates depend on
redshift (see eqs. \ref{eq:dic} \& \ref{eq:deq}). This leaves 100 sources
in this sample which fall into the following classifications (GPCM93):
32 BL Lacerate objects (BL Lac),
53 core-dominated quasars (CDQ), 11
lobe-dominated quasars (LDQ), and 9 radio galaxies (RG). The CDQs are
further classified on the basis of their optical polarization
into 24 core-dominated high-polarization quasars (CDHPQ),
22 core-dominated low-polarization quasars (CDLPQ), and 7
core-dominated
quasars without polarization information (CDQ-NPI).
A detailed discussion of these classifications can be found in GPCM93.
Both the inverse Compton Doppler factor, $\delta_{IC}$, and the equipartition
Doppler factor,
$\delta_{eq}$, can be computed using
the data compiled by GPCM93 (see the Appendix for the relevant formulae).
By assuming that the observed radio frequencies and radio
flux densities are the
true peak radio frequencies and radio flux densities, GPCM93 use the data
to compute $\delta_{IC}$. This approximation is made
since the multi-frequency observations and analysis of the radio spectra
necessary to obtain true peak frequencies and flux densities are available
for only a small fraction of these
sources (e.g. Marscher \& Broderick 1985, Unwin et al. 1994).
G\"uijosa \& Daly (1996) recompute $\delta_{IC}$ taking into account
corrections to the angular size and flux density neglected by GPCM93 (see
Appendix),
and compute $\delta_{eq}$ with the same assumptions
for this sample.
Both $\delta_{eq}$ and $\delta_{IC}$ are recomputed in this study assuming
a deceleration parameter of $q_o=0.05$ (i.e. $\Omega_o=0.1$,
$\Omega_{\Lambda}=0$) and $h=0.75$, and using
updated VLBI data for four sources (1101+384, 0615+820, 1039+811, \&
1150+812)
from Xu et al. (1995); here $\Omega_o$ is
the mean mass density relative to the critical value, $\Omega_{\Lambda}$ is
the normalized cosmological constant, and Hubble's constant
is parameterized in the usual way:
$H_o = 100\;h\mbox{ km s}^{-1}\mbox{ kpc}^{-1}$.
When computing $\delta_{eq}$ and $\delta_{IC}$ for each source,
we assume an optically thin spectral index $\alpha = 0.75$ ($S_{\nu}
\propto\nu^{-\alpha}$) and a spherical geometry, as was done by GPCM93
and G\"uijosa \& Daly (1996) (see eqs. \ref{eq:dic} \& \ref{eq:deq}).
Note that
$\delta_{eq}$ is changed by
only a few percent relative to the values computed assuming an Einstein
de-Sitter universe ($\Omega_o=1.0$, $\Omega_{\Lambda}=0$)
and $h=1$, while $\delta_{IC}$
is independent of these assumptions.
Following GPCM93,
upper bounds on angular sizes and lower bounds on redshifts are taken as
detections in this section,
but are treated as bounds when computing
bulk Lorentz factors and viewing angles
in \S \ref{ssec:solve}.
Figures \ref{fig:deleq}a \& \ref{fig:deleq}b show $\delta_{eq}$ and
$\delta_{IC}$ respectively, computed as described in the Appendix,
as functions of $(1+z)$.
In these figures and throughout, solid circles represent BL Lacs,
solid diamonds represent CDHPQs, solid squares represent CDLPQs,
solid triangles represent CDQ-NPIs, open diamonds represent LDQs,
and open squares represent RGs.
The error estimates for $\delta_{eq}$ and $\delta_{IC}$
are discussed in \S \ref{ssec:err}.
The trend in redshift is easily understood in terms of the effect of
Doppler boosting on the observed flux density and the flux limited nature
of this sample. Only at low redshift are we able to observe those sources
with lower Doppler factors, and
the sample is dominated by higher Doppler
factors at all redshifts. It is also apparent
that most LDQs and RGs have Doppler factors
less than 1, and most CDQs have Doppler factors greater than one. BL Lacs have
a much wider range of Doppler factors, covering the entire range of the whole
sample.
Figure \ref{fig:dophist} shows the distribution of
the equipartition Doppler factors,
$\delta_{eq}$ (dotted line), and the inverse Compton Doppler factors,
$\delta_{IC}$ (dashed line), for all 100
sources in the sample; the horizontal axis
is expressed in terms of $\log\delta$. Note that the
distributions of $\delta_{eq}$ and $\delta_{IC}$ are similar, and
both have a range of a few orders of magnitude. In addition,
the peaks of both
distributions are between 5 and 10.
Figures \ref{fig:dophistcl}a-f show the distribution
of $\delta_{eq}$ (dotted line) and $\delta_{IC}$ (dashed line) for each
class of AGN described above.
For BL Lacs, the distributions
range over a few orders of magnitude in $\delta$, with a peak around
$\delta\simeq 5$ (Figure \ref{fig:dophistcl}a). CDHPQs have
$\delta_{eq}$ and $\delta_{IC}$ that cover a smaller range of values than BL
Lacs, and peak noticeably at about $\delta \simeq 10$ (Figure
\ref{fig:dophistcl}b), while the
distribution of $\delta_{eq}$ and $\delta_{IC}$ for CDLPQs have a similar
range as CDHPQs, and peak somewhere around $\delta \simeq 5$ (Figure
\ref{fig:dophistcl}c).
CDQ-NPIs have the same range of
$\delta_{eq}$ and $\delta_{IC}$ as
CDHPQs and CDLPQs
(Figure \ref{fig:dophistcl}d). The distributions of $\delta_{eq}$ and
$\delta_{IC}$ for LDQs are wider than those for RGs, but both classes
peak at $\delta<1$ (see Figures \ref{fig:dophistcl}e \& \ref{fig:dophistcl}f).
The classification 3C216 as a LDQ should be examined closely since
it has relatively large values for $\delta_{eq}$ and $\delta_{IC}$
(61 and 33 respectively).
The separation of quasars
into CDQ and LDQ was done by GPCM93 on
the basis of the core dominance parameter
computed using fluxes that were K-corrected to 5 Ghz. Recent
observation by Reid et al. (1995) at 5 GHz give a core dominance
parameter slightly greater than 1.0, and this would place 3C216
between the LDQs and CDQs in terms of the core dominance parameter.
Most LDQs
in this sample exhibit a triple morphology, consisting of a core with a
flat radio spectrum and two lobes with steep radio spectra and have
angular extents of about 10"-30"
(e.g. \cite{hoo}). 3C216 shows a triple morphology, but has an angular
extent of only about 1.7". It
could be argued that 3C216 should be reassigned as a CDQ, but in this
paper we keep this source in the LDQ sample but make a special note of it.
\subsubsection{Brightness Temperatures \label{ssec:temp}}
Both the equipartition Doppler factor and inverse Compton Doppler factor,
$\delta_{eq}$ and $\delta_{IC}$, are used here to compute
the intrinsic peak
brightness temperature for each source in the sample (see the
Appendix for relevant
formulae).
Histograms of the observed brightness temperature, intrinsic brightness
temperature base on $\delta_{eq}$, and intrinsic brightness
temperature base on $\delta_{IC}$ ($T_{Bo}$, $T_{Bi}(eq)$,
and $T_{Bi}(IC)$)
are shown in Figure \ref{fig:temphist} for all 100 sources, and in
Figures \ref{fig:temphistcl}a-f for each class of AGN.
Table \ref{tb:bri} lists the mean and the standard deviation of the
mean of
$T_{Bo}$, $T_{Bi}(eq)$, and $T_{Bi}(IC)$ for the full sample and each
class.
The observed peak brightness
temperatures span a few orders of magnitude even within some individual
classes of AGN.
In contrast, there is a sharp peak in the distribution
of the estimates of
intrinsic brightness temperature, $T_{Bi}(eq)$ and $T_{Bi}(IC)$,
and both are centered around about
$7.5\times10^{10}\hbox{ K}$. These two
estimates of the intrinsic brightness temperature agree well with each
other, which
is not surprising given the strong correlation between
$\delta_{eq}$ and $\delta_{IC}$ described above and
by G\"uijosa \& Daly (1996).
It is interesting to note that the mean $T_{Bi}(IC)$
for all classes (except CDQ-NPIs) are consistent within $1\sigma$,
while the LDQs and RGs have larger $T_{Bi}(eq)$
than the other
classes at about the $2\sigma$ level. These two classes have values of
$R=\delta_{eq}/\delta_{IC}$ less than one (G\"uijosa \& Daly 1996),
which could possibly be due
to a systematic underestimate of $\delta_{eq}$ by about 75\%.
Such an underestimate of $\delta_{eq}$ could account for the larger
$T_{Bi}(eq)$ found for LDQs and RGs.
The standard
deviations of $T_{Bi}(eq)$ and $T_{Bi}(IC)$ are $0.27\times 10^{11}$ K and
$0.33\times 10^{11}$ K respectively.
It should be noted that $T_{Bi}(eq)\propto S_{op}^{0.1} \nu_{op}^{0.3}
\theta_d^{0.3}$ and $T_{Bi}(IC)\propto S_{op}^{0} \nu_{op}^{-0.7}
\theta_d^{-0.4}$ which are weak dependences on observable,
much weaker than the dependences of $T_{Bo}$ on the same parameters.
If the range of values
for the observables that go into $T_{Bi}(eq)$ and $T_{Bi}(IC)$
are not too large, then their
narrow distributions can be explained by weak dependences on observable
parameters. The range of $\theta_d$ contributes the
most to the range of $T_{Bi}(eq)$ and $T_{Bi}(IC)$, since $\nu_{op}$ is
fixed at discrete values corresponding to the frequencies
of observations, and $S_{op}$ is virtually insignificant
in computing $T_{Bi}(eq)$ and $T_{Bi}(IC)$.
If the narrow ranges of $T_{Bi}(eq)$
and $T_{Bi}(IC)$ are due to some physical cause and not the range in
$\theta_d$, then randomly reassigning the angular sizes used to compute
$T_{Bi}(eq)$ and $T_{Bi}(IC)$ should widen the distributions of each.
When the angles are reassigned randomly, the means are quite similar
($0.85\times10^{11}$ K and $0.79\times10^{11}$ K respectively),
and the standard
deviations are $0.26\times 10^{11}$ K and
$0.43\times 10^{11}$ K respectively.
Each width is not significantly affected by
randomly reassigning angular sizes, which suggests that the narrow range
of $T_{Bi}(eq)$ and $T_{Bi}(IC)$ obtained here
is primarily due to the range of observables, and are probably not
due to the physical state of the sources.
Readhead (1994) points out that if the intrinsic brightness temperatures
were cutoff due to the inverse Compton catastrophe
(i.e. the radiation
lifetime plays a role in the observed cutoff),
one would expect the intrinsic
brightness temperatures of compact radio sources to
cluster near $10^{11.5}\hbox{ K}$.
Both estimates of intrinsic brightness temperatures for this sample
peak below $10^{11.5}\hbox{ K}$ which agrees with the results of Readhead
(1994). This suggests that the inverse Compton
catastrophe does not play a role in the observed brightness temperature
distribution, or if the peaks are set by the inverse Compton catastrophe,
a factor of about 4 has not been accounted
for in computing the unbeamed brightness temperatures.
\subsection{Reliability of $\delta_{eq}$ and $\delta_{IC}$
as Estimators of
the Doppler Factor \label{ssec:rely}}
Previous results indicate that $\delta_{eq}\simeq\delta_{IC}$ for this sample
(G\"uijosa \& Daly 1996). To further test this conclusion, the effect of
randomly reassigning one of the observable quantities used to compute
$\delta_{eq}$ and $\delta_{IC}$ is examined. Since both Doppler factors depend
strongly on angular size ($\delta_{eq}\propto\theta_d^{-2.3}$ and $\delta_{IC}
\propto\theta_d^{-1.6}$),
the angular sizes used to compute $\delta_{eq}$
and $\delta_{IC}$ were randomly reassigned between sources within each
class, and the mean values are shown in Table \ref{tb:test}.
Note that the means
of $\delta_{eq}$ and $\delta_{IC}$ tend to
systematically increase from their true
values for all classes when the angular sizes are reassigned.
The fractional uncertainties in the mean values
of $\delta_{eq}$ and $\delta_{IC}$ are of order
20\% to 30\%, so that the mean $\delta_{eq}$ and
$\delta_{IC}$ are significant at about the $3\sigma$ to $5\sigma$ level
(G\"uijosa \& Daly 1996).
In contrast, the ``random'' mean values are significant at $2 \sigma$ or less
(i.e. their uncertainties are
greater than 50\%, Table \ref{tb:test}).
This suggests that these Doppler factor estimate are reliable.
\subsection{Errors for the Doppler Factor Estimates \label{ssec:err}}
The main source of error for both Doppler factor estimates is the
assumption that the radio flux density and frequency used to compute
the Doppler factor are the peak
radio flux density and peak frequency.
Thus, an estimate of errors for this data set should focus on
systematic errors instead of statistical ones.
A dependence of observed angular sizes
on frequency occurs if the
synchrotron-emitting plasma has gradients in magnetic field and particle
density (Marscher 1977, 1987), but since the model
considered here is that of a uniform
sphere, the errors associated with this frequency dependence of
angular size are
neglected.
The computed errors are found by assuming all sources have the same
fractional errors on the peak frequency, $\nu_{op}$;
this uncertainty is likely of order
100\% and dominates all other sources of error, as discussed in
G\"uijosa \& Daly (1996). It is assumed that the
flux density has an error which is related to the spectral index
between the observed frequency and the peak frequency, $\alpha_{op}$,
where $S_{op}=S_{o}(\nu_{op}/\nu_{o})^{-\alpha_{op}}$ and
$S_{o}$ and $\nu_{o}$ are the observed flux density and frequency;
$\alpha_{op}$
should not to be confused with the optically thin spectral index, $\alpha$.
In terms of VLBI observed properties,
$\delta_{eq}\propto S_{op}^{1.1}\nu_{op}^{-2.3}\theta_d^{-2.3}$ and
$\delta_{IC}\propto S_{op}^{1.0}\nu_{op}^{-1.3}\theta_d^{-1.6}$,
assuming $\alpha=0.75$.
These two Doppler factor estimates can be related to
the ratio of the peak frequency to the observed frequency,
$\delta_{eq}\propto (\nu_{op}/\nu_{o})^{-1.1\alpha_{op}-2.3}$
and $\delta_{IC}\propto
(\nu_{op}/\nu_{o})^{-1.0\alpha_{op}-1.3}$.
One finds that
\begin{equation}
{{\sigma\delta_{eq}}\over{\delta_{eq}}}=
(1.1\alpha_{op}+2.3)\left({{\sigma\nu_{op}}\over{\nu_{op}}}
\right)\qquad \mbox{and} \qquad
{{\sigma\delta_{IC}}\over{\delta_{IC}}}=
(1.0\alpha_{op}+1.3)\left({{\sigma\nu_{op}}\over{\nu_{op}}}
\right),
\label{eq:deqerr}
\end{equation}
where $\sigma\delta_{eq}$, $\sigma\delta_{IC}$, and $\sigma\nu_{op}$ are
the error estimates for $\delta_{eq}$, $\delta_{IC}$,
and $\nu_{op}$.
An estimate of $\sigma\nu_{op}/\nu_{op}$ is needed to assign errors for
the Doppler factor estimates using equation (\ref{eq:deqerr}).
A reasonable value of
$\sigma\nu_{op}/\nu_{op}$ is that which gives a reduced $\chi^2$ of
1.0 when fitting $R\equiv\delta_{eq}/\delta_{IC}$ to unity.
Since $R\propto S_{op}^{0.1}\nu_{op}^{-1.0}\theta_d^{-0.7}$ for $\alpha=0.75$,
it can be shown that
$R\propto
(\nu_{op}/\nu_{o})^{-0.1\alpha_{op}-1.0}$.
The error on the ratio $R$,
denoted $\sigma R$, can be related to
$\sigma\nu_{op}/\nu_{op}$ by
\begin{equation}
{{\sigma R}\over{R}}=(0.1\alpha_{op}+1.0)
\left({{\sigma\nu_{op}}\over{\nu_{op}}}\right).
\label{eq:Rerr}
\end{equation}
Note that if $\sigma\nu_{op}/\nu_{op}$ and $\alpha_{op}$ are the same
for all sources, then $\sigma R/R$ is the same for all
sources.
The value of $\sigma R/R=0.61$
gives a reduced $\chi^2$ of 1.0 when fitting $R$ to a constant equal to
unity for all sources.
A value of $\alpha_{op}=0.4$ is assumed for all
sources since radio sources with
observed optically
thin spectral indices for their cores have spectral indices in the range
of about 0.5 to 1, and the spectral index must go to zero towards the
peak. A lower value of $\alpha_{op}$ could be used, and
this will decrease the errors slightly on $\delta_{eq}$ and
$\delta_{IC}$. For $\alpha_{op}=0.4$ and $\sigma R/R=0.61$, equation
(\ref{eq:Rerr}) gives $\sigma\nu_{op}/\nu_{op}=0.59$, and one finds that
$\sigma\delta_{eq}/\delta_{eq}=1.6$ and
$\sigma\delta_{IC}/\delta_{IC}=1.0$ using equation (\ref{eq:deqerr}).
These error estimates for $\delta_{eq}$ and $\delta_{IC}$
are adopted throughout this paper.
Some objects in this sample have been observed at multiple frequencies, and
spectra are available in the literature. The observed frequencies
used to compute the Doppler factors for eight of the sources
studied here are compared to the peak frequencies
of the integrated spectra available in Xu et al. (1995), in order to check
if our estimate of $\sigma\nu_{op}/\nu_{op}$ is reasonable. It should
be noted that the integrated spectra include components besides the
cores, and that the core component is used to
determine the Doppler factor (see \S \ref{sec:disc}). Here
we approximate that the
peak of the integrated spectra corresponds to the peak of the core.
On this basis,
four sources have $\sigma\nu_{op}/\nu_{op}$ between 0 and 0.5, and the other
four have $\sigma\nu_{op}/\nu_{op}$ between 0.5 and 4, which would indicate
that the value of $\sigma\nu_{op}/\nu_{op}=0.59$ adopted for the whole sample
is quite reasonable.
\subsection{Luminosity Densities and Luminosities\label{ssec:lum}}
The uncorrected luminosity density is given by
\begin{equation}
L_{\nu_{un}}=4 \pi (a_or)^2 (1+z) S_o,
\label{eq:oblum}
\end{equation}
where $S_o$ is the observed flux density, $z$ is the source redshift,
$(a_or)$ is the coordinate distance to the source, and $\nu_{un}$ is
frequency as would be measured in the local Hubble frame of the source without
correcting for Doppler boosting ($\nu_{un}=\nu_o (1+z)$ where $\nu_o$ is the
observed frequency).
The flux density corrected for Doppler boosting of
a uniform spherical source, $S_i$, is given by $S_i=S_o/\delta^3$, where
$\delta$ is the Doppler factor.
The corrected flux density is the flux
density that would be received by an observer
moving with the same velocity with
respect to the Hubble flow as the radiating source.
Substituting $S_i$ for $S_o$ in equation (\ref{eq:oblum}) gives the
Doppler corrected, or intrinsic, luminosity density:
\begin{equation}
L_{\nu i}=4 \pi (a_or)^2 {{1+z}\over{\delta^3}} S_o
\label{eq:inlum}
\end{equation}
where $\nu_i=\nu_o (1+z) / \delta$.
Substituting both estimates of the Doppler factor $\delta_{eq}$ and
$\delta_{IC}$ into equation (\ref{eq:inlum}) gives two sets of intrinsic
luminosity densities which shall be referred to as $L_{\nu i}(eq)$ and
$L_{\nu i}(IC)$ respectively.
It should be noted that equations (\ref{eq:oblum}) and (\ref{eq:inlum})
are the luminosity densities at frequencies that differ from the
observing frequencies, which are
$\nu_{un}=\nu_o(1+z)$ and $\nu_i=\nu_o(1+z)/\delta$
respectively.
Thus, the uncorrected luminosity can
be approximated as $L_{un}\approx L_{\nu_{un}}
\nu_o(1+z)$, and the intrinsic luminosity can be approximated as
\begin{equation}
L_{i}\approx L_{\nu i}\nu_o(1+z)/\delta.
\label{eq:inlumint}
\end{equation}
Note
that luminosities computed in this manner
are crude approximations that may be
systematically biased by the approximated peak frequencies, yet still
give order of magnitude estimates.
Figures \ref{fig:oblumz} \& \ref{fig:oblumintz} show $L_{\nu_{un}}$ and
$L_{un}$ as functions of $(1+z)$. The flux-limited nature
of this sample is evident in these two figures, and it is quite obvious
that selection effects play a role in the composition of this sample.
The intrinsic luminosity densities, $L_{\nu i}(eq)$,
are plotted versus the
rest frame frequency $\nu_{i}(eq)$ in
Figure \ref{fig:lumeq}, and $L_{i}(eq)$ versus $\nu_{i}(eq)$ is plotted
in Figure \ref{fig:luminteq}. By
fitting all the data plotted in Figures \ref{fig:lumeq} and
\ref{fig:luminteq}, one finds that $L_{\nu i}(eq)\propto \nu_i(eq)^{2.3\pm0.1}$
with a reduced $\chi^2$ equal to 4.6 and a correlation coefficient $r=0.91$,
and $L_{i}(eq)\propto \nu_i(eq)^{3.3\pm0.1}$ with a reduced $\chi^2$ equal
to 4.0 and a correlation coefficient $r=0.96$. Both of these power law
fits are very significant since $r$ is close to unity. Figures
\ref{fig:lumz} and
\ref{fig:lumintz} show $L_{\nu i}(eq)$ and $L_{i}(eq)$ respectively
as functions of $(1+z)$; fitting all these data, one finds that
$L_{\nu i}(eq)\propto(1+z)^{2.5\pm1.6}$ with a reduced $\chi^2$ equal to 1.3
and a correlation coefficient $r=0.16$, and
$L_{i}(eq)\propto (1+z)^{0.9\pm2.2}$ with a reduced $\chi^2$ equal to
1.4 and a correlation coefficient $r=0.04$.
Similar results are found for $L_{\nu i}(IC)$ and $L_{i}(IC)$.
Figures \ref{fig:lumeq} \& \ref{fig:luminteq} show a trend in
$L_{\nu i}(eq)$ and $L_{i}(eq)$ with $\nu_{i}(eq)$
that can be explained
by the strong dependence of these quantities on the Doppler
factor and a systematic effect introduced by
assuming the observed frequency is the peak frequency. Equations
(\ref{eq:inlum}) \& (\ref{eq:deq}) indicate
that $L_{\nu i}(eq)\propto{\delta_{eq}}^{-3}\propto \nu_{op}^{6.9}$
and that $\nu_i(eq)\propto\nu_{op}{\delta_{eq}}^{-1}\propto \nu_{op}^{3.3}$,
for $\alpha=0.75$.
Since it is assumed that
$\nu_{op}=\nu_{o}$ when computing $\delta_{eq}$,
the offset of $\nu_{o}/\nu_{op}$ from unity will cause $L_{\nu i}(eq)$
to be offset by a factor of $(\nu_{o}/\nu_{op})^{6.9}$ and $\nu_i(eq)$
to be offset by a factor of $(\nu_{o}/\nu_{op})^{3.3}$ from their actual
values. One would expect to
find a systematic effect that would increase the range of $L_{\nu
i}(eq)$ and $\nu_i(eq)$ such that $L_{\nu i}(eq)\propto\nu_i(eq)^{2.1}$.
In fact, one obtains $L_{\nu i}(eq)\propto \nu_i(eq)^{2.3\pm0.1}$ by
fitting the data in Figure \ref{fig:lumeq}. A similar calculation
gives the prediction that $L_{i}(eq)\propto\nu_i(eq)^{3.1}$, and the
fits of the data in Figure \ref{fig:luminteq} give $L_{i}
(eq)\propto \nu_i(eq)^{3.3\pm0.1}$. The spread over such a large range
of values for $L_{\nu i}(eq)$, $L_{i}(eq)$, and $\nu_{i}(eq)$ can thus be
explained by this systematic effect. On the other hand,
the errors in Figures
\ref{fig:lumeq} \& \ref{fig:luminteq} are quite large, and
all the $L_{\nu i}(eq)$ and $L_{i}(eq)$
are still consistent with each other.
It is interesting to note that
$I_{\nu}(eq)\propto\nu_{i}(eq)^{2.2\pm0.1}$ and
$I_{\nu}(IC)\propto\nu_{i}(IC)^{2.0\pm0.1}$ for this sample,
where $I_{\nu}\propto
L_{\nu i}/{\theta_d}^2$. This dependence would be expected if
these sources were intrinsically similar, have black a body spectrum, and
are observed in the
Rayleigh-Jeans size of the black body
spectrum. It should be noted that the small dispersion in intrinsic brightness
temperatures described above (\S \ref{ssec:GPCM}) and the fact that
$I_{\nu}$ is approximately proportional to $\nu^2$ go hand in hand.
Table \ref{tb:lum} lists the mean and
standard deviation of the mean for the logarithmic values of $L_{\nu
o}$, $L_{o}$, $L_{\nu i}(eq)$, $L_{i}(eq)$, $L_{\nu i}(IC)$, and
$L_{i}(IC)$, for each class of AGN. Comparing $L_{\nu i}(eq)$ to
$L_{\nu i}(IC)$ and $L_{i}(eq)$ to $L_{i}(IC)$,
agreement
for each class of source is found within about $2\sigma$.
Ranges of
intrinsic luminosity densities and intrinsic luminosities
for all classes of AGN are $10^{32.4\;
to\;33.1}\mbox{ergs s}^{-1}\mbox{Hz}^{-1}$ and $10^{42\;
to\;43}
\mbox{ergs s}^{-1}$.
\section{DISENTANGLING RELATIVISTIC EFFECTS IN RADIO-LOUD
AGN\label{sec:anres}}
The effects of relativistic motion can be described by two
quantities:
the bulk Lorentz factor of the outflow, $\Gamma$, and the angle
subtended by
the outflow direction and the line of sight to the observer, $\phi$.
Note that the bulk Lorentz factor is determined by the speed of the
outflow relative to the speed of light, $\beta$, by the familiar
equation $\Gamma=1/ \sqrt{1-\beta^2}$.
Two observable quantities
which are combinations of $\Gamma$ and $\phi$ are
the Doppler factor, $\delta$, and the apparent
speed projected onto the plane
of the sky relative to the speed of light, $\beta_{app}$.
These are expressed in terms of $\beta$ and $\phi$ as
\begin{equation}
\delta={\sqrt{1-\beta^2}\over 1-\beta\cos\phi},
\label{eq:delta}
\end{equation}
and
\begin{equation}
\beta_{app}={\beta\sin\phi\over1-\beta\cos\phi}.
\label{eq:betapp}
\end{equation}
Equations (\ref{eq:delta}) and (\ref{eq:betapp})
can be used to solve for $\Gamma$ and $\phi$ in terms of
$\delta$ and $\beta_{app}$, and it can be shown that
\begin{equation}
\Gamma={\beta_{app}^2+\delta^2+1\over 2\delta}
\label{eq:gamma}
\end{equation}
and
\begin{equation}
\tan\phi={2\beta_{app}\over \beta_{app}^2+\delta^2-1}
\label{eq:phi}
\end{equation}
(see GPCM93; Daly, Guerra, \& G\"uijosa 1996).
Thus, $\Gamma$ and $\phi$ can be estimated
for relativistic outflows in AGN, by combining
observational estimates of $\delta$ and $\beta_{app}$.
Here $\delta_{eq}$ and $\delta_{IC}$ are used to estimate $\delta$.
\subsection{Apparent Bulk Flow Versus Pattern Flow \label{ssec:pat}}
It has been suggested that the pattern velocities, the velocities of
radio
features observed in VLBI proper motion studies, are
greater than the bulk velocities, the velocities
responsible for the observed Doppler boosting
(see, for example, Lind \& Blandford 1985).
This issue was addressed by GPCM93, who conclude that the data are
consistent
with pattern velocities equaling bulk velocities for the sources in
their
sample with observed superluminal motion.
Vermeulen \& Cohen (1994, hereafter VC94) examine
the equivalence of pattern velocities and bulk velocities in
detail using Monte Carlo simulations of a simple model where
the angle between the direction of outflow and the line
of sight is allowed to vary randomly, but the bulk Lorentz factor
is the same in every jet. They find that the pattern
Lorentz factor, $\Gamma_p$, would have to be
about twice the bulk Lorentz factor, $\Gamma_b$,
in order to be consistent with the data from the GPCM93 sample. They
point
out, however, that for a broad
distribution of bulk Lorentz factors the data are easily consistent
with
the $\Gamma_p\simeq\Gamma_b$.
It is assumed, here and throughout, that $\Gamma_p\simeq\Gamma_b$, and any
departures from this equality are of order unity.
\section{ESTIMATES OF BULK LORENTZ FACTORS AND VIEWING
ANGLES\label{sec:sol}}
\subsection{A Subsample with $\beta_{app}$ Compiled\label{ssec:VC}}
VC94 compile a list of 66 radio sources with
multi-epoch VLBI observations of the internal proper motions. Table 1 in VC94
lists $\beta_{app}$ for the features in these radio sources. There are 43
sources that overlap the GPCM93 and the VC94 samples, and these sources are
examined below in order to estimate $\Gamma$ and $\phi$
(\S \ref{ssec:solve}); preliminary results were presented by Daly, Guerra,
\& G\"uijosa (1996).
Table \ref{tb:in}
lists the 43 sources that overlap the GPCM93 and VC94 samples. Column
(1) lists the IAU Name, column (2) lists the common name if it exists,
Column (3) gives the redshift listed by VC94, Column (4) lists the
$\beta_{app}$ estimates from VC94, except for 0108+388 where the
estimate is based on more recent observations by Taylor, Readhead, \& Pearson
(1996).
When multiple components
are listed by VC94, the weighted mean is taken for $\beta_{app}$.
Figure \ref{fig:betapp} shows $\beta_{app}$ versus $(1+z)$ for the 43 sources
in the overlapping sample.
Columns (5) and
(6) give estimates of $\delta_{eq}$ and $\delta_{IC}$, respectively,
which are discussed in \S \ref{ssec:GPCM} above.
\subsection{Values for $\Gamma$ and $\phi$\label{ssec:solve}}
The values of $\beta_{app}$, $\delta_{eq}$, and $\delta_{IC}$
listed in Table \ref{tb:in} can be used with equations (\ref{eq:gamma})
and (\ref{eq:phi}) to produce two sets of estimates for $\Gamma$ and
$\phi$. In the cases where only bounds on $\beta_{app}$ or $\delta$
are available, bounds on
$\Gamma$ and $\phi$ are computed.
Table \ref{tb:out} lists the solutions for $\Gamma$ and $\phi$ using the
equipartition Doppler factor, called $\Gamma_{eq}$ and
$\phi_{eq}$, and using the inverse Compton Doppler factor, called
$\Gamma_{IC}$ and $\phi_{IC}$. Columns (1) and (2) list the IAU name and
the common name respectively, Column (3) gives the redshift listed by
VC94, Columns (4) and (5) list $\Gamma_{eq}$ and $\phi_{eq}$
respectively, and columns (6) and (7) list $\Gamma_{IC}$ and $\phi_{IC}$
respectively. Both sets of solutions for $\Gamma$ and $\phi$ agree
within errors
for any given source in Table \ref{tb:out}.
Table \ref{tb:med} lists the median values of
$\Gamma_{eq}$, $\phi_{eq}$, $\Gamma_{IC}$, and $\phi_{IC}$ for each
class of source, excluding those values which are upper or lower bounds.
The median
values of the equipartition set ($\Gamma_{eq}$ and $\phi_{eq}$) and the
inverse Compton set ($\Gamma_{IC}$ and $\phi_{IC}$) of estimates agree
well, except for the median $\Gamma_{eq}$ and $\Gamma_{IC}$
of the LDQs which are a factor of about two greater in the equipartition
case. Note that excluding 3C216 from the LDQs would not significantly
change the
median values for this category.
Figures \ref{fig:gamtheeq}a \& \ref{fig:gamtheeq}b show $\Gamma_{eq}$
versus
$\phi_{eq}$ with and without errors respectively,
and Figures \ref{fig:gamtheic}a \& \ref{fig:gamtheic}b show $\Gamma_{IC}$
versus $\phi_{IC}$ with and without errors respectively.
The figures without errors are
shown so that the symbols are more visible to the reader and
bounds on $\Gamma$ and $\phi$ are denoted by arrows in these figures.
Figures \ref{fig:zeq}a \& \ref{fig:zeq}b show $\phi_{eq}$ and $\Gamma_{eq}$ as
functions of $(1+z)$ respectively,
and Figures \ref{fig:zic}a \& \ref{fig:zic}b show $\phi_{IC}$ and
$\Gamma_{IC}$ as functions of $(1+z)$ respectively.
The median values for both sets of estimates of $\Gamma$ and $\phi$
make it possible to compare different classes of AGN in the
context of
orientation unified models (see discussion in \S \ref{ssec:comm}).
A detailed discussion of each set of $\Gamma$ and $\phi$ estimates
follows in \S\S \ref{ssec:soleq} \& \ref{ssec:solic}.
\subsubsection{Solutions Using the Equipartition Doppler
Factor \label{ssec:soleq}}
The eight BL Lacs studied here span a relatively large range of $\Gamma_{eq}$;
one source has a value of $\Gamma_{eq}=1.4\pm0.2$, and two sources
have $\Gamma_{eq}$ between 40 and 100, although these two sources
have large errors on $\Gamma_{eq}$.
The remaining five BL Lacs
have $\Gamma_{eq}$ between 3 and 6.
The values of $\phi_{eq}$ for BL Lacs span a range from about
$0^{\circ}$ to
about $50^{\circ}$, with six of the eight
BL Lacs having $\phi_{eq}\les20^{\circ}$.
Seven of the
eight CDHPQs have $\phi_{eq}\les12^{\circ}$, and at least
five of these sources
have $\phi_{eq}\les6^{\circ}$.
Three out of eight CDHPQs have
$\Gamma_{eq}$ between 3 and 10, and two
CDHPQs have
$\Gamma_{eq}$ between 20 and 40. Three sources have limits on
$\Gamma_{eq}$ and $\phi_{eq}$ instead of values (see \S
\ref{ssec:GPCM}). Two of these sources with bounds, 1156+295
(4C 29.45) and 2230+114 (CTA 102),
stand out in Figures \ref{fig:gamtheeq}a,b.
The bounds
on $\Gamma_{eq}$ for these two sources, $\Gamma_{eq}<405$ for 1156+295
and $\Gamma_{eq}>2.8$ for 2230+114, still allow these sources to
have Lorentz factor similar to other CDHPQs.
Five CDLPQs have
$\Gamma_{eq}$ between 5 and 10, and
three have $\Gamma_{eq}$ between 15 and 30. The other two sources
only have bounds on $\Gamma_{eq}$. Eight out of ten
CDLPQs have
$\phi_{eq}\les14^{\circ}$.
The three CDQ-NPIs all have $\Gamma_{eq}<9$, and
have quite different values for $\phi_{eq}$. The sources in this small
category in reality belong to the CDHPQs or CDLPQs, and may contain
both types.
Five LDQs have $\Gamma_{eq}$ between 10 and 50. These
values of $\Gamma_{eq}$ are larger than the typical values
for the other classes of AGN studied
here. One source has $\Gamma_{eq}$ less than 10 (4C 21.35,
$\Gamma_{eq}=2.8\pm2.9$), while another has a lower bound
$\Gamma_{eq}>3$.
Only one LDQ has $\phi_{eq}$ less than
$15^{\circ}$ (3C216), while the other six LDQs have
$\phi_{eq}$ between $15^{\circ}$ and $41^{\circ}$.
LDQs have the larger values for $\phi_{eq}$ than most of the BL Lacs,
CDQs (HP, LP, and NPI) studied here.
RGs have the largest values and widest range of
$\phi_{eq}$ of all the categories. Four out of seven RGs have
$45^{\circ}<\phi_{eq}<135^{\circ}$,
which suggests that these sources have
outflows that typically lie close to the plane of the sky (see \S
\ref{ssec:comm}). RGs also have the lowest $\Gamma_{eq}$ of all the
categories, with at least four RGs having $\Gamma_{eq}<2$. Note that
the radio galaxies in this sample are
compact steep spectrum sources or FRI; there are
no FRII radio galaxies in this sample.
\subsubsection{Solutions Using the Inverse Compton Doppler
Factor\label{ssec:solic}}
The results for $\Gamma_{IC}$ and $\phi_{IC}$ are quite similar to those
for $\Gamma_{eq}$ and $\phi_{eq}$, and agree well within errors. This
agreement is not surprising considering the correlation and
agreement found between
$\delta_{IC}$ and $\delta_{eq}$ (see \S \ref{ssec:GPCM}).
Five out of eight BL Lacs have
$\Gamma_{IC}$ between 3 and 5, while the total range
for all BL Lacs is between 1.5 and 60.
Seven BL Lacs have $\phi_{IC}<20^{\circ}$.
Five out of eight CDHPQs have $\Gamma_{IC}$
between 8 and 16, while the other three have various
upper and lower bounds.
These same five CDHPQs all have
$\phi_{IC}<10^{\circ}$.
Six out of ten CDLPQs have $\Gamma_{IC}$ between 3 and 11,
and two have $\Gamma_{IC}$ around 20. Eight CDLPQs have
$\phi_{IC}<14^{\circ}$.
Five out of seven LDQs have $\Gamma_{IC}$ between 10 and
30, and five LDQs have $\phi_{IC}$ between $15^{\circ}$
and $45^{\circ}$. As in the equipartition case only one LDQ has
$\phi_{IC}$ less than $15^{\circ}$.
Three out of seven RGs have $\phi_{IC}$ between
$45^{\circ}$ and $90^{\circ}$, and three RGs have
$\phi_{IC}>90^{\circ}$. At least six RGs have $\Gamma_{eq}<2$.
\subsection{Implications for Orientation
Unified Models of AGN
\label{ssec:comm}}
The results presented \S \ref{ssec:solve} are consistent with an
orientation unification scheme for radio-loud AGN (e.g.
Antonucci 1993).
Common orientation models have
a torus or disk of material that absorbs or obscures
the broad absorption lines and
other non-stellar radiation
emitted from the
nuclear region of the AGN, and the classification as a radio galaxy occurs
if the torus intersects the line of sight from the nuclear region to the
observer. The radio jet axis in radio-loud AGN is thought to be more or
less perpendicular to the plane of the torus or disk.
In such models (e.g. Barthel 1989), RGs, LDQs, and CDQs, are
viewed with different
angles with respect to the jet axis ($\phi$, which is referred
to as the viewing angle): radio galaxies have the
largest angles ($\phi\ges45^{\circ}$), LDQs have smaller angles
($20^{\circ}\;_\sim^<\;\phi\les45^{\circ}$), and CDQs have the smallest angles
($\phi\les20^{\circ}$).
The RGs, LDQs, and CDQs in overlap of the GPCM93 and VC94 samples have
estimates of $\phi$ consistent with this orientation unification
scheme.
The RGs in this sample have a median $\phi_{eq}$ of $110^{\circ}$ and a
median $\phi_{IC}$ of $81^{\circ}$, which suggests that they have
jets that lie close
to the plane of the sky.
LDQs typically have a median $\phi_{eq}$ of $26^{\circ}$
and a median $\phi_{IC}$
of $25^{\circ}$.
CDLPQs have a median $\phi_{eq}$ of $7^{\circ}$
and a median $\phi_{IC}$ of $6^{\circ}$, while CDHPQs have a median
$\phi_{eq}$ of $4^{\circ}$
and a median $\phi_{IC}$ of $3^{\circ}$.
The median values for CDQ-NPIs are insignificant since only two sources
are used to compute them.
It is interesting to note that CDHPQs and CDLPQs have similar median
$\phi_{eq}$ and $\phi_{IC}$. This suggests that it is intrinsic
differences between these two classes of AGN (not
viewing angles) that determine the amount of optical polarization observed.
The BL Lacs in this sample cover a wide range of
$\Gamma$ and $\phi$, but have viewing angles typically
less than
$45^{\circ}$.
An extension of the orientation model described above assigns BL Lacs to
the parent population of FRI (edge-darkened, \cite{fr})
RGs, where BL Lacs are seen
at smaller viewing angles and radio selected BL Lacs have smaller angles
than X-ray selected BL Lacs (see Padovani \& Urry 1992).
The sample of BL Lacs examined
here are
almost all radio selected. Mkn 421 (1101+384) is the only X-ray
selected BL Lac in this sample, but
does not stand out against the other BL Lacs in terms of its $\Gamma$ or
$\phi$ estimates.
The BL Lacs in this sample
have a median $\phi_{eq}$ of $12^{\circ}$
and a median
$\phi_{IC}$ of $14^{\circ}$. These results are consistent with the
those found by
Kollgaard et al. (1996), who estimate the viewing angles for samples of
radio selected BL Lacs, X-ray selected BL Lacs, and RGs by comparing the
core enhancement relative to the more diffuse radio emission for each
class of AGN. Assuming a constant bulk Lorentz factor,
they find that their samples of
radio selected BL Lacs, X-ray selected BL
Lacs, and RGs
have average viewing angles of
$10^{\circ}$, $20^{\circ}$, and $60^{\circ}$
respectively.
CDQs in this sample are centered around a $\Gamma\approx9$. If it is
assumed that the distribution of bulk Lorentz factors for all classes of
radio-loud AGN is centered around $\Gamma\approx9$,
then one might naively expect the mean or median $\Gamma$ for each of
class of AGN to similar. Selection biases likely
play a role in the composition of the samples studied, and
systematic effects can affect the values that can be obtained from the data.
LDQs have median $\Gamma_{eq}$ and $\Gamma_{IC}$ that are larger than
those for the CDQs, and these two estimates are different almost by a
factor of two, yet the mean ratio of $R=\delta_{eq}/\delta_{IC}$ is only
about 0.76 for LDQs. This could be caused by the
sensitivity of the estimates of
$\Gamma$ to the precise values
of $\beta_{app}$ and $\delta$ when these to parameters are close to one
(see eq. \ref{eq:gamma}), which is the case for this sample of LDQs.
A decrease of $\delta$ away from one will cause the corresponding
$\Gamma$
estimate to increase. If the $\delta$ estimates of LDQs in this sample
are systematically
underestimated, then a noticeable systematic increase in the $\Gamma$ will
result, which could be caused by an overestimate of the peak
frequency (see eqs. \ref{eq:deq} and \ref{eq:dic}). It is not clear
whether this is one of the causes of the larger $\Gamma$ for LDQs.
The lower $\Gamma$ for RGs compared other classes of AGN can be
explained as follows. RGs will appear in this sample if
there cores are bright enough to have VLBI core size data, which occurs
if the sources are nearby (like M87 and NGC 6251) or have very
bright compact cores. In particular, FRII radio galaxies are missing
from this sample since a majority of their cores are not significantly
Doppler boosted and they make difficult targets for VLBI.
In the unification scheme described above, RGs have viewing angles
around $90^{\circ}$. Approximating $\phi\approx90^{\circ}$, one finds
that $\delta\approx\Gamma^{-1}$ from equation (\ref{eq:delta}). Thus,
$\Gamma\approx\delta^{-1}$, and
those RGs with smaller $\Gamma$
will appear in this
sample since these RGs will have larger $\delta$ and brighter cores. Even if
the true distribution of
$\Gamma$ for RGs peaked at around 10, we may only be able to observe
RGs with $\Gamma$ from 1 to 2. It should be noted that
RGs in this sample have values of
$\Gamma$ near unity which are
in agreement with other evidence that Doppler boosting
is not significant in these sources (e.g. Taylor, Readhead, \& Pearson 1996).
There are two sources that seem to have
viewing angles greater than $90^{\circ}$ based on equipartition and inverse
Compton Doppler factors. One, 2352+495, has a lower bound on the viewing
angle, the other, 0710+439, is greater than $90^{\circ}$ at around $3\sigma$.
This would suggest that some RGs have observed outflows that
are moving away from the observer, and that if there is an outflow moving
toward the observer it must be intrinsically fainter.
BL Lacs have lower values of $\Gamma$ than the CDQs, with a median
$\Gamma_{eq}$ of 4.2 and a median $\Gamma_{IC}$ of 5. Since the BL Lacs
studied in this paper are almost exclusively radio selected, it would be
interesting to compare $\Gamma$ and $\phi$ estimates for a sample of
X-ray selected BL Lacs. Kollgaard et al. (1996) find that the core
enhancements they observe for BL Lacs and RGs, require that $\Gamma>4.5$,
assuming a constant bulk Lorentz factor
for all these sources. The estimates of $\Gamma$ found here are barely
consistent with $\Gamma>4.5$,
but a distribution of Lorentz factors may loosen the constraints placed
by Kollgaard et al. (1996).
\section{DISCUSSION\label{sec:disc}}
The two estimates of the Doppler factor for the sample
of radio-loud AGN
discussed in \S\ref{ssec:GPCM} agree with
each other on average and provide a means of estimating the intrinsic
properties of these sources (e.g. brightness temperatures, luminosities)
and the parameters that describe the kinematics of the relativistic bulk
flow of the radio emitting plasma. Although the results here are based
on rough
estimates of the Doppler factor, the average or typical
properties of
different classes of AGN can be compared.
Care should be taken when using
the estimates of Doppler factors and
intrinsic properties computed here
since systematic errors contribute most of the
uncertainty. These systematic errors arise
from the simplification that the observed frequencies correspond to the peak
of the core emission. Another concern with this sample is completeness.
The sample examined here was
defined by GPCM93 as those which had VLBI core size data
in the literature at that point in time. The sources that enter this sample
will have VLBI cores with
higher surface brightness than those excluded from the sample;
this will favor the brighter (more highly Doppler boosted) and more compact
(smaller angular size)
cores. GPCM93 warn that this sample
is not complete in any statistical sense, and that biases may have a large
effect on the content of this sample.
Sources in the first and second
Caltech-Jodrell Bank VLBI surveys (CJ1 and CJ2) form a complete flux-limited
sample of
almost 400 compact radio sources with VLBI measured core sizes
(see Henstock et al. 1995, Xu et al. 1995) and some multi-frequency data.
Doppler factor estimates for sources from these samples
are currently under investigation, including the use
of spectra to estimate the peak frequency of
core components in order to more accurately compute Doppler
factors (\cite{prep}). Large
samples such as these
could be used to estimate the Doppler factors and intrinsic properties
of different classes of AGN with greater confidence and statistical
weight.
The intrinsic brightness temperatures for this
sample are estimated using two sets of Doppler factor estimates,
$\delta_{eq}$ and $\delta_{IC}$, and
both histograms of these values are centered around $7.5\times10^{10}\mbox{
K}$ (see \S \ref{ssec:GPCM}).
The intrinsic brightness temperatures are
similar for different classes of AGN, despite the fact that the
observed brightness
temperatures are different for different classes of AGN.
The center of the distribution of intrinsic brightness temperatures
is a factor of 4
lower than what would be expected if the ``inverse Compton
catastrophe'' played a major role in the observed brightness temperature
cutoff, as noted previously by Readhead (1994). This suggests that either a
factor of about 4 has not been accounted for in these estimates (e.g. a
geometric factor), or that a physical process other than the ``inverse Compton
catastrophe'' is limiting the
intrinsic brightness temperature.
Estimates of the intrinsic
luminosity density and luminosity depend strongly on the Doppler factor
($L_{\nu i}\propto\delta^{-3}$ and $L_{i}\propto\delta^{-4}$), and systematic
errors are quite large for these estimates.
The large scatter of $L_{\nu i}$ and $L_{i}$ is a cause for concern
since it can be attributed to systematic errors introduced by assuming
the observed frequencies
correspond to the actual peak frequencies
of these sources. In any event, these
estimates can be compared to the Eddington luminosity for a supposed
massive central object. The Eddington luminosity is
$L_E=1.3\times10^{44}M_6\mbox{ ergs s}^{-1}$ where $M_6$ is the mass of
the central object in units of $10^6 M_{\odot}$ which gives a
range from $10^{44}\mbox{ to }10^{46}\mbox{ ergs s}^{-1}$
for central objects with $M_6=1
\mbox{ to }100$. The mean intrinsic luminosity for the sources examined here
is around $10^{42}\mbox{ to }10^{43}\mbox{ ergs s}^{-1}$, while
only five objects (3 BL Lacs, 1 LDQ, and 1 RG)
have intrinsic luminosities greater than $10^{46}\mbox{ ergs s}^{-1}$
by more than $1\sigma$. This is consistent with the radio
luminosity produced being 0.1 to 1 percent of the Eddington luminosity for
a central compact object with mass $10^6\mbox{ to }10^8 M_{\odot}$.
A subsample of 43 sources have observed proper motions in
the plane of the sky and were used by VC94 and others to
compute $\beta_{app}$. The
values of $\beta_{app}$ for each of these sources
can be combined with the Doppler factor
estimates to solve for the bulk Lorentz factor, $\Gamma$, and the viewing
angle, $\phi$ (see \S
\ref{sec:anres}). The estimates using the data compiled here
are consistent with the orientations
unification scheme for AGN (discussed in \S \ref{ssec:comm}),
where CDQs have the smallest $\phi$ ($<15^{\circ}$),
LDQs have larger $\phi$ ($15^{\circ}<\phi<45^{\circ}$), and
RGs have the largest
$\phi$ ($>45^{\circ}$); note that the RGs in this sample are compact steep
spectrum sources or FRI.
Selection biases and systematic errors may play a role in these
estimates of $\Gamma$ and $\phi$, in particular for LDQs and RGs
(see \S \ref{ssec:comm}).
The RGs in this sample have lower
$\Gamma$ than the other classes of AGN, which is
consistent with the parent population having
outflows that lie close to the plane of the sky (i.e. $\phi\sim
90^{\circ}$) and the flux-limited nature of the sample studied. The
LDQs are in a regime where small systematic errors in
$\delta$ can strongly affect $\Gamma$ estimates.
The estimates of $\phi$ and $\Gamma$ for the BL Lacs in this sample are
consistent with the results of Kollgaard et al. (1996) for radio selected
BL Lacs (see \S \ref{ssec:comm}).
Their estimates of $\phi$ are based on the core enhancement of radio
selected BL Lacs, X-ray selected BL Lacs, and RGs.
A sample of
X-ray selected BL Lacs with
estimates of $\phi$ and $\Gamma$ computed as in \S \ref{ssec:solve} would
give a consistency check for
both methods.
Marscher (1977)
discusses the effect of gradients in magnetic field and
density with radius on the relationships between the observed quantities
and the important physical parameters in a nonuniform source.
The effect on $\delta_{eq}$ and $\delta_{IC}$ of including
inhomogeneities is constant
multiplicative factor from 0.7 to 1.4, and
the the functional dependence on observable quantities is
identical to the homogeneous case; the form of the inhomogeneity
and the parameter choices are discussed by Marscher (1977). It should be noted
that the intrinsic brightness temperatures will be changed by
the same constant factor, and that inhomogeneities do not account
for the factor of 4 in brightness temperature
that would allow the inverse Compton catastrophe to play a major role in
the brightness temperature distribution (\S \ref{ssec:GPCM}).
The spherical model for the radio cores of AGN, which is used throughout
this paper to compute Doppler factors, may not be as
realistic as those models which consider a jet-like geometry.
The expression
for $\delta_{IC}$ in the jet-like case, which is derived by GPCM93, is
\begin{equation}
\delta_{IC}(jet)=\left[\delta_{IC}(sph)\right]^{(4+2\alpha)/(3+2\alpha)},
\end{equation}
where $\delta_{IC}(sph)$ is the expression given by equation
(\ref{eq:dic}). For a value
of $\alpha=0.75$, $\delta_{IC}(jet)=\delta_{IC}(sph)^{1.2}$ which
is a power law only slightly greater than unity.
The expression for $\delta_{eq}$ in the jet-like case is
\begin{equation}
\delta_{eq}(jet)=\left[\delta_{eq}(sph)\right]^{(13+2\alpha)/(9+2\alpha)}
(\sin\phi)^{2/(9+2\alpha)}
\left({{\theta_a}\over{\theta_b}}\right)^{1/(9+2\alpha)},
\label{eq:jet}
\end{equation}
where $\delta_{eq}(sph)$ is given by equation (\ref{eq:deq}),
$\phi$ is the angle subtended by the direction of the outflow and the line
of sight, and $(\theta_a/\theta_b)$ is the ratio of the angular sizes
of the major and minor axes (see Appendix for the derivation
of this equation). For $\alpha=0.75$,
$\delta_{eq}(jet)\propto (\sin \phi)^{0.19}$, and $\delta_{eq}(jet)\propto
(\theta_a/\theta_b)^{0.09}$,
which are very weak dependences. The relation between the
jet-like and spherical cases of the equipartition Doppler factor is
$\delta_{eq}(jet)\propto\delta_{eq}(sph)^{1.4}$ which is
a power-law
close to unity.
The expressions for inverse Compton and equipartition
Doppler factors in the jet-like case appear to be
quite similar to those in the spherical case. Thus, approximating a
spherical geometry for simplicity can still
provide useful Doppler factor estimates even if the true geometry of
these outflows
is more jet-like.
\acknowledgments
The authors would like to thank Tom Herbig,
Alan Marscher, Tony Readhead, Larry Rudnick,
David Schramm, Steve Unwin, Lin Wan, and
Greg Wellman for
helpful discussions.
This work was supported in part by the US National Science
Foundation, by an NSF Graduate
Fellowship, the Independent College Fund
of New Jersey, and by a grant from W. M. Wheeler III.
|
1,108,101,563,308 | arxiv | \subsubsection{#1}\vspace{-3\baselineskip}\color{black}\medskip{\noindent \bf \thesubsubsection. #1.}}
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\section{Introduction}
5G wireless networks are expected to provide high rates, low latency and flexible constructions through an ultra-dense deployment of small cells \cite{Gupta2015}. The cloud radio access network (C-RAN) emerges as a promising cellular architecture to satisfy these requirements \cite{Wu2015}. It moves the baseband signal processing to a centralized baseband unit (BBU) pool, and distributed remote radio heads (RRHs) are responsible for capturing signals and forwarding them to the BBU. RRHs are connected to the BBU via fronthaul links. These links are traditionally optical fibers with high capacity and zero interference \cite{Poor2015}. However, optical fiber is not ubiquitous and its deployment can be expensive. Free space optical (FSO) communication provides an attractive alternative that maintains comparable advantages as optical fiber \cite{Alzenad2018, Ahmed2018}. More importantly, it is cost-efficient and flexible in implementation. On the other hand, FSO links are sensitive to channel characteristics and may be significantly impacted by factors like weather and turbulence. Various models are put forth to characterize the FSO channel, based on which different techniques are developed to reduce channel effects \cite{Andrews2005, Gao2017, Gao2018, zhang2019ergodicity}.
To mitigate the channel dependent degradation, cooperative transmissions has been proposed in FSO networks. With a certain transmit power budget, adaptive powers are assigned based on the CSI to maximize the channel capacity \cite{ Zhou2013, Zhou2015}. For the relayed communication, joint relay selection and power adaptation are proposed to minimize the outage probability or maximize the network throughout \cite{Chatzidiamantis2013, Hassan2018, Hassan2017}. However, the aforementioned works employ relaxations to simplify problem models leading to approximate solutions, and are computationally expensive to implement with respect to instantaneous channel state changes. These issues motivate the application of machine learning due to its low complexity and potential for model-free implementation. Deep learning in particular has been leveraged for resource allocation problems in wireless radio frequency domain \cite{Sun2018, Xu2017, Eisen2019}. In \cite{Gao2019}, we proposed the deep neural network to solve the optimal power allocation for WDM radio on free space optical (RoFSO) systems. Deep neural networks are limited, however, in their ability to generalize over varying network topologies.
As an extension of convolutional neural networks, graph neural networks (GNNs) have been used to model and analyze data collected from networks and has achieved signifigant successes in many learning tasks \cite{Henaff2015, Gama2019, Eisen2020, Zhan2020}. With the use of graph convolutional filters, GNNs exploit the underlying irregular structure of network data in a manner that results in lower computation complexity, less parameters for training, and generalization capabilities relative to traditional deep neural networks. As such, the GNN is considered as a suitable candidate learning over network scenarios such as the FSO fronthaul network and related communication networks.
In this paper, we consider the resource allocation problem in the FSO fronthaul network of C-RAN architecture. RRHs transmit optical signals to intermediate aggregation nodes (ANs) through FSO links, and the latter forward aggregated signals to the BBU through high speed fiber. Based on the CSI, different powers are assigned to different RRHs and an optimal AN is selected at each RRH to maximize the objective function, subject to power limitation and data congestion constraints. The formulated optimization problem is non-convex with complicated constraints (Section II). We introduce the graph neural network (GNN) to parameterize the resource allocation policy and translate the problem into a statistical learning problem (Section III). The GNN is shown to be permutation equivariant in the sense that relabelling RRHs or ANs in network results in the same optimal policy. The primal-dual learning algorithm is developed to train the GNN solving the learning problem (Section IV). With the use of policy gradient, its implementation is completely model-free that avoids the error induced by the system model inaccuracy. Numerical simulations are present to show significant performance of the proposed GNN learning algorithm (Section V).
\section{Problem Formulation}\label{sec_problem}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth , height=0.45\linewidth, trim=20 20 20 20]{FSOnetwork.pdf}
\caption{The FSO fronthaul network. }
\label{fig_network}
\end{figure}
We consider the free space optical fronthaul network consists of remote radio heads, aggregation nodes and the baseband unit. RRHs transmit optical signals through free space to selected ANs. ANs then aggregate received signals and forward them to the BBU through optical fiber. The RRHs are distributed remotely in the space around ANs and equipped with multiple apertures pointing towards ANs, such that each RRH can possibly transmit signals to all ANs. The ultra-wide field of view (UWFOV) beam acquisition is available in ANs, through which each AN can receive optical signals from different RRHs simultaneously. See Fig. 1 for details of the proposed FSO fronthaul network. The resource allocation problem considered here comprises the power adaptation and the AN selection. Based on the CSI, different powers are assigned to different RRHs and each RRH selects a best AN for signal transmission, in order to maximize the objective function. The exact objective can be adjusted according to practical situations.
Assume there are $N$ RRHs, $M$ ANs and one BBU in a FSO fronthaul network. The CSI between RRHs and ANs is represented by the matrix $\bbH \in \mathbb{R}^{N \times M}$, where $[\bbH]_{nm} = h_{nm}$ is the CSI between $n$-th RRH and $m$-th AN for all $n=1,\ldots,N$ and $m=1,\ldots,M$. The assigned power and the selected AN for $n$-th RRH is based on the observed CSI $\bbH$ via a power assignment policy $P_n(\bbH)$ and an AN selection policy $\bbalpha_n(\bbH)=[\alpha_{n1}(\bbH),\ldots,\alpha_{nM}(\bbH)]^\top$. Here $\alpha_{nm}(\bbH) \in \{ 0,1 \}$ is the indicator taking one if $m$-th AN is selected by $n$-th RRH and zero if not, such that $\sum_{m=1}^M \alpha_{nm}(\bbH) \le 1$. In addition to channel states, separate state variables $\bbx \in \mathbb{R}^{N+M}$ represent the status of RRHs and ANs, such as equipment conditions of RRHs and ANs. Given the collection of resource allocations $\bbP(\bbH)= [P_1(\bbH),\ldots,P_N(\bbH)]$ and $\bbDelta(\bbH)=[\bbalpha_1(\bbH),\ldots,\bbalpha_M(\bbH)]$ with network states $\bbH$ and $\bbx$, a channel capacity $C_{nm}(\bbx, \bbH,\bbP(\bbH), \bbDelta(\bbH))$ is achieved between $n$-th RRH and $m$-th AN. Since the FSO channel is a fading process with coherence time on the order of milliseconds, we consider it as an ergodic and i.i.d block fading process. The instantaneous CSI tends to vary fast, so as the instantaneous channel capacity. Therefore, a long term average $\mathbb{E}\left[ C_{nm}(\bbH,\bbP(\bbH)) \right]$ is the more meaningful metric. We then consider the objective function as the weighted sum of channel capacities over RRHs
\begin{equation}
\sum_{n=1}^N \omega_n \sum_{m=1}^M \mathbb{E}\left[ C_{nm}(\bbx,\bbH,\bbP(\bbH),\bbDelta(\bbH)) \right],
\end{equation}
where $\bbomega = [\omega_1,\ldots,\omega_N]^\top$ is the weight vector representing priorities of different RRHs and the expectation $\mathbb{E}[\cdot]$ is with respect to the probability distribution of the CSI $\bbH$.
Three constraints are considered. Assuming RRHs are connected to a common power supply, the first is the expected total power limitation
\begin{equation}
\mathbb{E} \left[ \sum_{n=1}^N P_n(\bbH) \right] \le P_t.
\end{equation}
The second is motivated by the safety concern. To avoid possible danger in optical beam transmission, we limit the peak power that can be allocated on single optical signal
\begin{equation}\label{eq:safe}
0 \le P_n(\bbH) \le P_s,~\forall n=1,...,N.
\end{equation}
The third is to prevent the data congestion at ANs. Specifically, we require the incoming sum-capacity at each AN less than or equal to the capacity of optical fiber
\begin{equation}
\sum_{n=1}^N \mathbb{E}\left[ C_{nm}(\bbH,\bbP(\bbH),\bbDelta(\bbH)) \right] \le C_t,~\forall m=1,\ldots,M.
\end{equation}
Together, we formulate the resource allocation problem in the FSO fronthaul network as follows
\begin{alignat}{3} \label{eq_problem111}
\mathbb{P}&:= \max_{\bbP, \bbDelta} \ \sum_{n=1}^N \omega_n \sum_{m=1}^M \mathbb{E}\left[ C_{nm}(\bbx,\bbH,\bbP(\bbH),\bbDelta(\bbH)) \right] , \\
& \st \quad \mathbb{E} \left[ \sum_{n=1}^N P_n(\bbH) \right] \le P_t,~ 0 \le P_n(\bbH) \!\le\! P_s, \nonumber \\
& \quad \quad\quad \sum_{m=1}^M \!\alpha_{nm}(\bbH)\leq1,\forall n=1,...,N, \nonumber \\
& \quad\!\! \sum_{n\!=\!1}^N\! \mathbb{E}\left[ C_{nm}(\bbx,\bbH,\bbP(\bbH),\bbDelta(\bbH)) \right] \!\le\! C_t,\forall m\!=\!1,...,M. \nonumber
\end{alignat}
While some theoretical models exist for the channel distribution, e.g. \cite{Andrews2005}, and the channel capacity for FSO fronthaul networks, e.g. \cite{Hassan2017}, we stress that the above problem is given without any particular system model. Pre-existing models may not be accurate in practical systems, leading to inevitable errors for model-based algorithms. We therefore propose a ``model-free'' approach in this paper in which no explicit knowledge or form of any of the models in \eqref{eq_problem111} is assumed.
\section{Graph Neural Networks}\label{sec_gnn}
\begin{figure}
\centering
\includegraphics[width=0.35\textwidth, trim=0 0 0 0]{graph_structure1.pdf}
\caption{The weighted bipartite graph representation of an FSO fronthaul network with $N=5$ RRHs and $M=2$ ANs. The RRHs (grey nodes) are connected only to ANs (blue nodes) and vice versa through FSO links. The edge weight between nodes is the CSI of the associated link.}
\label{fig_gnn}
\end{figure}
With unknown objective function and complicated constraints, the optimization problem \eqref{eq_problem111} is challenging. We pursue to solve it with a learning procedure. In particular, the resource allocation policies are functions of the CSI $\bbH$. By introducing a parameterization $\bbtheta \in \mathbb{R}^p$, we use a function family $\bbPhi(\bbH, \bm{\theta})$ to model $\bbP(\bbH)$ and $\bbDelta(\bbH)$ as
\begin{equation} \label{eq_gnn}
\begin{split}
[\bbP(\bbH), \bbDelta(\bbH)] = \bbPhi(\bbH, \bm{\theta}).
\end{split}
\end{equation}
Substituting \eqref{eq_gnn} into \eqref{eq_problem111}, we translate it into a statistical learning problem. The goal here becomes to learn the optimal function $\bbPhi(\bbH, \bm{\theta}^*)$ with optimal parameters $\bbtheta^*$, which maximizes the objective function. This alternative learning problem can be solved without system models but only depending on system observations, as we demonstrate in Section \ref{primaldual}.
For the learning parameterization $\bbPhi(\bbH, \bm{\theta})$, we introduce the graph neural network (GNN). The GNN is well-know for its ability to exploit network structures to process data, which makes it a suitable candidate in our case. By interpreting RRHs and ANs as nodes and links between them as edges, the FSO fronthaul network can be abstracted as a weighted bipartite graph $\ccalG$. The graph representation matrix $\bbS$ captures the structure and the channel state information over network, which is defined as
\begin{equation}
\bbS = \left(
\begin{array}{cc}
\bb0^{N \times M} & \bbH\\
\bbH^\top & \bb0^{M\times N}\\
\end{array}
\right).
\end{equation}
See Fig. \ref{fig_gnn} for details of the graph structure. The graph signal $\bbx = [x_1,\ldots,x_{N+M}]^\top$ is defined on the top of nodes, where $x_i$ is the signal value of $i$-th node indicating the status of RRH or AN.
The key component constituting the GNN is the graph convolutional filter, which processes the graph signal $\bbx$ based on the graph matrix $\bbS$. Recall the graph shift operation $\bbS \bbx$ assigns to node $i$ the aggregated information $[\bbS \bbx]_i$ from its neighbors scaled by edge weights. It generalizes the signal shifting from the time domain to the graph domain and plays an important role in defining the graph convolution. Shifting $\bbx$ for $k$ times yields the signal $\bbS^k \bbx$, indicating $k$ node exchanges over graph. With a set of parameters $\theta_0,\ldots,\theta_K$, we aggregate $K$ shifted signals to obtain the higher-level feature
\begin{equation} \label{eq_gf}
\begin{split}
\bbu = \sum_{k=0}^K \theta_k \bbS^k \bbx := \bbG(\bbS)\bbx
\end{split}
\end{equation}
where $\bbG(\bbS)$ is defined as the graph filter. Observe that $\bbG(\bbS)$ exploits the node information up to a neighborhood of radius $K$ and thus contains higher-level features generated from a more complete picture of the graph. If particularizing $\ccalG$ as the line graph where each node is a time instant and $\bbx$ is the signal sampled over time, the filter output $\bbG(\bbS)\bbx$ reduces to the traditional convolution. We can then think of \eqref{eq_gf} as the generalization of the convolution for graph signals.
The GNN is defined as a concatenation of layers, each of which consists of a bank of graph filters followed by a pointwise nonlinearity. To be more precise, at layer $\ell = 1,...,L$, we have $F$ input features $\{ x^{g}_{\ell-1} \}_{g=1}^{F}$. These features are processed by graph filters $\{ \bbG_{\ell}^{fg}(\bbS) \}_{fg}$ to produce higher-level features $\{ \bbu_{\ell}^{fg} \}_{fg}$ as in \eqref{eq_gf}. We aggregate the latter over the index $g$ and then apply the nonlinearity $\sigma(\cdot)$ to get the $f$-th output feature
\begin{equation} \label{eqn:GNN}
\bbx_{\ell}^{f} \!=\! \sigma \left( \sum_{g=1}^{F} \bbu_\ell^{fg}\! \right)
\!=\! \sigma \left( \sum_{g=1}^{F}\!\! \bbG_{\ell}^{fg}(\bbS) \bbx_{\ell\!-\!1}^{g} \!\right)\!,\forall f\!=\!1,...,F.
\end{equation}
The input of the GNN is the input of $1$-st layer $\bbx_0^1 = \bbx$ and the output of the GNN is the output of $L$-th layer $\bbx^1_L$. The learning parameters $\bbtheta$ are filter parameters $\{ \theta_{\ell 0}^{fg},\ldots,\theta_{\ell K}^{fg} \}_{\ell f g}$. We then denote
\begin{equation} \label{eqn:GNN1}
\bbPhi(\bbx,\bbS,\bbtheta) = \bbx_L^1
\end{equation}
as the non-linear map of the GNN on the graph matrix $\bbS$ with parameters $\bbtheta$. The number of GNN parameters is independent of the network size, making it computationally efficient for
training compared with the dense deep neural network.
\subsection{Permutation Equivariance}
With graph convolutional filters, the GNN accounts for the network structure in its parameterization. One key property obtained by doing this is the permutation equivariance, corresponding to that the resource allocation policy of \eqref{eq_problem111} is permutation equivariant. In particular, we define the permutation matrix $\bbPi$ as
\begin{equation} \label{eq_permutation}
\begin{split}
\bbPi \in \{ 0,1 \}^{(N+M) \times (N+M)}: \bm{\Pi} \bm{1} = \bm{1}, \quad \bbPi^\top \bm{1} = \bm{1}.
\end{split}
\end{equation}
Put simply, the permuted vector $\bbPi \bbx$ reorders the entries of $\bbx$ and the permuted matrix $\bbPi^\top \bbS \bbPi$ reorders the columns and rows of $\bbS$. We give the following theorem to formally state this property according to \cite{Gama2019}.
\begin{theorem}\label{theorem1}
Consider the GNN $\bbPhi(\bbx,\bbS,\bbtheta)$ with the graph signal $\bbx$, graph matrix $\bbS$ and parameters $\bbtheta$. For any permutation $\bbPi$, it holds that
\begin{equation} \label{eq_permutation2}
\begin{split}
\bbPhi(\bbPi \bbx,\bbPi\bbS \bbPi^\top ,\bbtheta) = \bbPi \bbPhi(\bbx,\bbS,\bbtheta).
\end{split}
\end{equation}
\end{theorem}
Theorem \ref{theorem1} establishes that the GNN with a permutation of underlying graph $\bbPi^\top \bbS \bbPi$ and input signal $\bbPi \bbx$ generates an equally permuted output. In the context of our case, it indicates that reordering RRHs or ANs in the FSO network indices an associated reordered resource allocation policy with policy parameters $\bbtheta$ unchanged. This property follows the intuition of resource allocation problems since the labelling of network nodes is arbitrary which should be reflected in our learning parameterization $\bbPhi(\bbx,\bbS,\bbtheta)$. We remark that the deep neural network does not satisfy the permutation equivariance without considering the graph structure in its architecture.
With the use of GNN, we represent \eqref{eq_gnn} as
\begin{equation} \label{eq_gnn11}
\begin{split}
\!\bbPhi(\bbx,\!\bbS,\!\bbtheta)\!=\![\bbPhi_{\bbP}(\bbx,\!\bbS,\!\bbtheta),\! \bbPhi_{\bbDelta}(\bbx,\!\bbS,\!\bbtheta)]
\end{split}
\end{equation}
where $\bbPhi_{\bbP}(\bbx,\!\bbS,\!\bbtheta)=[\Phi_{\bbP 1}(\bbx,\!\bbS,\!\bbtheta),\ldots,\Phi_{\bbP K}(\bbx,\!\bbS,\!\bbtheta)]^\top=\bbP(\bbH)$ are assigned powers and $\bbPhi_{\bbDelta}(\bbx,\!\bbS,\!\bbtheta)=\bbDelta(\bbH)$ are selected ANs. With $\ccalP = [0,P_s]^N$ the space satisfying the peak power constraint \eqref{eq:safe} and $\Lambda$ the space restricting only one AN is selected by one RRH, we require $\bbtheta$ belongs to the set $\Theta = \{ \bbtheta | \bbPhi_{\bbP}(\bbx,\!\bbS,\!\bbtheta) \in \ccalP, \bbPhi_{\bbDelta}(\bbx,\!\bbS,\!\bbtheta) \in \Lambda \}$. The optimization problem \eqref{eq_problem111} is then translated into the following learning problem
\begin{alignat}{3} \label{eq_problem2}
\mathbb{P}:= & \max_{\bbtheta \in \Theta} \ && \sum_{n=1}^N \omega_n \sum_{m=1}^M \mathbb{E}\left[ C_{nm}(\bbH,\bbPhi(\bbx,\bbS,\bbtheta)) \right] , \\
& \st && \mathbb{E} \left[ \sum_{n=1}^N \Phi_{Pn}(\bbx,\bbS,\bbtheta) \right] \le P_t, \nonumber \\
& &&\! \sum_{n\!=\!1}^N\! \mathbb{E}\left[ C_{nm}(\bbH,\bbPhi(\bbx,\bbS,\bbtheta)) \right] \!\le\! C_t,\forall m\!=\!1,...,M. \nonumber
\end{alignat}
Note both the graph matrix $\bbS$ and the graph signal $\bbx$ are input variables of the GNN, which vary across time reflecting the instantaneous CSI of the FSO network and the status of RRHs and ANs. Our goal is to learn optimal GNN parameters $\bbtheta^* \in \Theta$ that maximize the objective and satisfy constraints.
\section{Primal Dual learning algorithm}\label{primaldual}
Consider the alternative problem \eqref{eq_problem2}. We develop a model-free primal-dual learning algorithm to train the graph neural network. With multiple constraints, it is straightforward to consider working in the dual domain. By introducing the non-negative dual variables $\bblambda = [\lambda_1,\ldots,\lambda_{M+1}] \in \mathbb{R}^{M+1}$, the Lagrangian of the problem is given by
\begin{align} \label{eq_lagran}
\mathcal{L}(\bbtheta,\bblambda) &= \sum_{n=1}^N \omega_n \sum_{m=1}^M \mathbb{E}\left[ C_{nm}(\bbH,\bbPhi(\bbx,\bbS,\bbtheta)) \right] \nonumber\\
&+ \lambda_1 \left( P_t - \mathbb{E}\left[ \sum_{n=1}^N \Phi_{Pn}(\bbx,\bbS,\bbtheta)\right] \right) \\
&+ \sum_{m=1}^{M} \lambda_{m+1} \left( C_t - \sum_{n=1}^N\mathbb{E}\left[ C_{nm}(\bbH,\bbPhi(\bbx,\bbS,\bbtheta)) \right] \right)\!\nonumber
\end{align}
where constraints in \eqref{eq_problem2} are reinterpreted as weighted penalties in \eqref{eq_lagran}. The associated dual problem is to search for optimal $\bbtheta$ and $\bblambda$ that make a so-called ``saddle point'', i.e. respectively maximize and minimize the Lagrangian,
\begin{equation} \label{eq_dualprob}
\begin{split}
\mathbb{D} = \min_{\bblambda} \mathcal{D}(\lambda)=\min_{\bblambda} \max_{\bbtheta\in \Theta} \mathcal{L}(\bbPhi(\bbx,\bbS,\bbtheta),\bblambda).
\end{split}
\end{equation}
The null duality gap $\mathbb{D} - \mathbb{P}=0$ is achieved for convex optimization problems, in which we can work on the dual problem \eqref{eq_dualprob} instead without loss of optimality. The optimization problem \eqref{eq_problem2} is not necessarily convex with unknown system models, i.e., unknown objective function and constraints can be non-convex. However, for the learning parameterization $\bbPhi(\bbH, \bm{\theta})$ with strong function approximation ability, the duality gap of \eqref{eq_problem2} can be sufficiently small close to null \cite{Gao2019, Eisen2019}.
We develop a primal-dual learning algorithm to solve the dual poblem \eqref{eq_dualprob}, which then solves the primal problem \eqref{eq_problem2} as well. The primal-dual algorithm updates both the primal variables $\bbtheta$ and the dual variables $\bblambda$ iteratively with gradient descents, searching for the saddle point $(\bbtheta^*, \bblambda^*)$ that is maximal w.r.t. the primal variables and minimal w.r.t. the dual variables. Note that this saddle point is local because of the non-convexity, whose influence can be mitigated by methods such as performing algorithm multiple times to find the best solution. In particular, the algorithm is divided into two steps at each iteration:
(1) \emph{Primal step.} At $k$-th iteration, let $\bbtheta^k$ and $\bblambda^k$ be current primal variables and dual variables. We update the primal variables as
\begin{align} \label{eq_priup1}
&\bm{\theta}^{k+1} = \bm{\theta}^{k} + \delta^k \nabla_{\bm{\theta}} \mathcal{L}(\bm{\theta}^{k},\bblambda^{k}).
\end{align}
where $\delta^k>0$ is the step-size at iteration $k$. Observe that this update needs the system model to compute the gradients of the Lagrangian defined in \eqref{eq_lagran}. As we do not assume this to be available due to unknown system models, we resolve this issue using the policy gradient method common in reinforcement learning settings \cite{Sutton2000}. In particular, this approach considers the resource allocation policy $\bbPhi(\bbx,\bbS,\bbtheta)$ as samples drawn from a predetermined probability distribution $\pi_{\bbx,\bbS,\bm{\theta}}(\bbPhi)$ parameterized by the output of the GNN. With the use of the likelihood ratio identity, we can express the function gradient that takes
the form of $\nabla_{\bbtheta} \mathbb{E} [f(\bbH,\bbPhi(\bbx,\bbS,\bbtheta))]$ as
\begin{equation} \label{eq_grad}
\begin{split}
\nabla_{\bbtheta} \mathbb{E} [f(\bbH,\!\bbPhi(\bbx,\!\bbS,\!\bbtheta))] \!=\! \mathbb{E}[f(\bbH,\!\bbPhi) \nabla_{\bm{\theta}}\! \log\! \pi_{\bbx,\bbS,\bm{\theta}}(\bbPhi)]
\end{split}
\end{equation}
where $\bbPhi$ is the random sample drawn from the distribution $\pi_{\bbx,\bbS,\bm{\theta}}(\bbPhi)$ that is determined by $(\bbx,\bbS,\bm{\theta})$. As such, we can represent the gradient of the Lagrangian in \eqref{eq_priup1} as
\begin{align} \label{eq_polgrad}
&\nabla_{\bm{\theta}} \mathcal{L}(\bm{\theta},\bblambda) \!\!=\!\! \mathbb{E}\! \left[\! \left(\! \sum_{n=1}^N \!\!\omega_n \!\!\!\sum_{m=1}^M \! C_{nm}(\bbH,\bbPhi) \right.\!+\! \lambda_1^k\! \left(\!\! P_t \!-\!\! \sum_{n\!=\!1}^N \Phi_{Pn}\!\! \right) \right.\nonumber\\
&\left.\!+\! \sum_{m=1}^{M} \!\!\lambda_{m+1}^k\! \left(\! C_t\! -\! \sum_{n=1}^N\!C_{nm}(\bbH,\!\bbPhi)\!\right)\! \nabla_{\bm{\theta}}\! \log \pi_{\bbx,\bbS,\bm{\theta}}(\bbPhi) \right]\!\!
\end{align}
where $\mathbb{E}[\cdot]$ is approximated by sampling $N$ realizations from $\pi_{\bbx,\bbS,\bm{\theta}}(\bbPhi)$ and taking the average. It should be emphasized that \eqref{eq_polgrad} is model-free as the gradient of $\log \pi_{\bbx,\bbS,\bm{\theta}}(\bbPhi)$ can be computed given the distribution $\pi_{\bbx,\bbS,\bm{\theta}}(\bbPhi)$, while capacity values $C_{nm}(\bbH,\bbPhi)$ and $\Phi_{Pn}$ can be observed in the system. Furthermore, appropriate distributions shall be selected for $\pi_{\bbx,\bbS,\bm{\theta}}(\bbPhi)$ to satisfy the feasibility condition $\bbtheta \in \Theta$.
(2) \emph{Dual step.} With the obtained $\bm{\theta}^{k+1}$, the update of dual variables $\bblambda$ takes the form
\begin{align} \label{eq_dualup1}
\lambda_1^{k+1} &=\left[ \lambda_1^{k} - \eta^k \left( P_t -\mathbb{E} \left[ \sum_{n\!=\!1}^N\!\! \Phi_{Pn}(\bbx,\!\bbS,\!\bbtheta) \right] \right) \right]_+, \\
\lambda_{m+1}^{k+1} &\!\!=\!\!\left[\! \lambda_{m+1}^{k} \!-\! \eta^k \!\! \left(\!\! C_t \!-\!\mathbb{E}\! \left[ \sum_{n\!=\!1}^N\!\!C_{nm\!}(\bbH,\!\bbPhi(\bbx,\!\bbS,\!\bbtheta)\!)\! \right]\! \right)\! \right]_+ \label{eq_dualup2}
\end{align}
for all $m=1,\ldots,M$, where $\eta^k$ is the step-size and $[\cdot]_+$ is the non-negativity operator due to the definition of $\bblambda$. The dual update can be implemented with system observations $\Phi_{Pn}(\bbx,\bbS,\bbtheta)$ and $C_{nm}(\bbH,\bbPhi(\bbx,\bbS,\bbtheta))$, such that it is also model-free with no need of system models. As in the primal update, the expectation $\mathbb{E}[\cdot]$ can be approximated with the average of $S$ samples of $\bbH$.
The primal-dual update performed iteratively provides a model-free approach towards solving for the GNN parameters $\bbtheta$ and corresponding dual variables $\bblambda$ in problem \eqref{eq_problem2}. We summarize the whole training method in Algorithm 1.
{\linespread{0.9}
\begin{algorithm}[b] \begin{algorithmic}[1]
\STATE \textbf{Input:} Initial primal and dual variables $\bbtheta^0, \bblambda^0$
\FOR [main loop]{$k = 0,1,2,\hdots$}
\STATE Draw CSI samples $\{ \bbH \}$ of batch size $S$, and compute the allocated resources $\{ \bbPhi \}$ according to the GNN $\bbPhi(\bbx,\bbS,\bbtheta)$ and the policy distribution $\pi_{\bbx,\bbS,\bm{\theta}^k}(\bbPhi)$
\STATE Obtain channel capacity observations $C_{nm}(\bbx, \bbH,\!\bbPhi)$ with current allocated resources $\{ \!\bbPhi\! \}$ and the CSI $\{ \!\bbH\! \}$
\STATE Compute the policy gradient $\nabla_{\bm{\theta}} \mathcal{L}(\bm{\theta}^k,\bblambda^k)$ by \eqref{eq_polgrad}
\STATE Update the primal variable by \eqref{eq_priup1} \\
$\bm{\theta}^{k+1} = \bm{\theta}^{k} + \delta^k\nabla_{\bm{\theta}} \mathcal{L}(\bm{\theta}^k,\bblambda^k) \nonumber $
\STATE Update the dual variable by \eqref{eq_dualup1}-\eqref{eq_dualup2}\\
$\bblambda^{k+1} = \left[ \bblambda^{k} - \eta^k \nabla_{\bblambda} \mathcal{L}(\bm{\theta}^{k+1},\bblambda^k) \right]_+ \nonumber $
\ENDFOR
\end{algorithmic}
\caption{GNN primal-dual learning algorithm}\label{alg:learning} \end{algorithm}}
\section{Simulation Results}\label{sec_numerical_results}
In this section, we present simulation results to corroborate our theory. We compare the GNN primal-dual learning policy with the baseline policy, i.e., equal power assignment and random AN selection, to show its strong performance.
\begin{figure}[t]
\centering
\includegraphics[width=0.47\linewidth, height=1\textheight, keepaspectratio]{25objective.pdf}
\quad
\includegraphics[width=0.47\linewidth, height=1\textheight, keepaspectratio]{25constraint.pdf}
\quad
\caption{The performance of the GNN policy and the baseline policy for $5$ RRHs and $2$ ANs with $P_t = 1.5 {\rm W}$ and $P_s = 0.5 {\rm W}$: (left) the objective value; (right) the constraint value.}\label{fig_simple_results}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.47\linewidth, height=1\textheight, keepaspectratio]{25objectivelarge.pdf}
\quad
\includegraphics[width=0.47\linewidth, height=1\textheight, keepaspectratio]{25objectivemore6.pdf}
\quad
\caption{The performance of the GNN policy and the baseline policy under different network scenarios: (left) power limitations with $P_t = 3 {\rm W}$ and $P_s = 1 {\rm W}$; (right) the network with $10$ RRHs and $4$ ANs.}\label{fig_simple_results}
\end{figure}
We consider a FSO fronthaul network as in Fig. 1. RRHs and ANs are distributed uniformly at random at locations $\bbr_n \in [-5{\rm km}, 5{\rm km}]^2$ and $\bba_m \in [-1{\rm km}, 1{\rm km}]^2$, and the weight vector $\bbomega$ is drawn randomly from zero to one. Since we are doing numerical simulations not physical experiments, the CSI samples $\{ \bbH \}$ and corresponding channel capacities $\{ C_{nm}(\bbx, \bbH,\!\bbPhi)\}$ cannot be observed. We use system models in \cite{Hassan2017} to compute these observation but keep in mind that the GNN learning policy works in a model-free manner. The GNN architecture is with $L=8$ layers, each of which contains $F=1$ graph filter of order $K=5$ followed by the ReLU nonlinearity $\sigma(\cdot) = [\cdot]_+$. The final layer passes through a sigmoid function to normalize the outputs and the latter is used as parameters of the policy distribution $\pi_{\bbx,\bbS,\bm{\theta}}(\bbPhi)$. The truncated Gaussian and categorical distributions are selected to satisfy the feasibility condition $\bbtheta \in \Theta$. The GNN is trained with the primal-dual learning algorithm, where the ADAM optimizer and geometrically decaying step-sizes are used for the primal update and the dual update, respectively. We point out this implementation is completely model-free, requiring only system observations $C_{nm}(\bbH,\bbPhi(\bbx,\bbS,\bm{\theta}))$ in practice.
We first simulate on a small network with $N=5$ RRHs and $M=2$ ANs. The limitations are $P_t = 1.5 {\rm W}$, $P_s = 0.5 {\rm W}$ and $C_s = 20$. Fig. 3 (left) shows the performance, i.e., the weighted sum-capacity, achieved by the GNN learning policy and the baseline policy. We see that the learning process of the GNN converges as the training iteration increases. The GNN outperforms the baseline as we expected, and we emphasise that this performance improvement is obtained without explicit knowledge of capacity function models. The constraint values are shown in Fig. 3 (right). Both the power limitation and the data congestion constraints are satisfied as the learning process converges. This implies the feasibility of the optimal solution generated by the learned GNN.
We then consider the GNN learning policy under different system scenarios; namely, larger power limitations $P_t = 3 {\rm W}$ and $P_s=1 {\rm W}$ (Fig. 4 (left)) and the larger network with $N=10$ and $M=4$ (Fig. 4 (right)). In general, the GNN maintains good performance for both cases. Specifically, we observe that the performance improvement of the GNN is emphasised compared with Fig. 3. This is because with larger allowed powers or at a larger network, the GNN gains more space to manipulate the resource allocation and thus better exhibits its learning capacity. Moreover, since the number of GNN parameters does not scale with the size of networks, either the training or the implementation of the GNN keeps computationally efficient for large networks. However, model-based algorithms will face a more complicated problem with more expensive computations.
We now evaluate the permutation equivariance of the GNN. For the FSO network with $N=5$ RRHs and $M=2$ ANs, we consider two permutations $\bbPi_1 \bbx = [x_3, x_4, x_5, x_2, x_1, x_6, x_7]^\top$ and $\bbPi_2 \bbx = [x_2, x_1, x_5, x_4, x_3, x_7, x_6]^\top$, i.e., relabelling RRHs and ANs. Table I shows the expected objective value over $100$ samples for the original and two permuted network scenarios. The results show that the same GNN learned from original network performs well for two permuted networks, verifying the permutation equivariance proposed in our theory. The small differences among three cases are because $100$ CSI samples are drawn from the probability distribution randomly.
\begin{table}[h] \label{table2}
\begin{center}
\caption{Performance of the GNN learning policy for permuted networks.}
\begin{tabular}{|l|l|l|l| p{2cm}|}
\hline
& Objective value \\ \hline
Original network & 21.397 \\ \hline
Network permutation $\bbPi_1$ & 21.405 \\ \hline
Network permutation $\bbPi_2$ & 21.381 \\ \hline
\end{tabular} \vspace{-.55cm}
\end{center}
\end{table}
\section{Conclusion} \label{sec_con}
We consider the optimal resource allocation in FSO fronthaul networks. The optimization problem takes the form of constrained statistical learning, where the resource allocation policy can be parameterized with the graph neural network. The GNN accounts for the network structure in its parameterization and thus exhibits the permutation equivariance, showing that it can achieve same performance on reordered FSO networks. We further develop a primal-dual learning algorithm to train the GNN, the implementation of which is model-free without requiring information of system models. This property is essentially important for FSO networks, in which cases sophisticated optical systems may be difficult to model or modelled inaccurately, leading to the performance degradation of model-based algorithms. Numerical simulations demonstrated the GNN is an effective parameterization for learning resource allocation policies and outperforms the baseline policy significantly. In the near future, we will extend the GNN learning algorithm to more FSO resource allocation scenarios.
\bibliographystyle{IEEEbib}
|
1,108,101,563,309 | arxiv | \section{Introduction}
The decisive requirement for the functioning of any complex system ranging from the subcellular level of biological organisms
to globe-spanning man-made structures is the transportation of matter and information. From theoretical point of view, these stochastic transport phenomenon are an intriguing example of multi-particle systems addressing far from equilibrium processes. The extensive organisation of interconnected linelike pathways for transport mechanisms forms a network like structure. However, the study of complex frameworks still remains a major challenge in the field of physics and cellular biology. In this direction, the investigation of relatively simpler topologies is a crucial step towards understanding the complex network systems.
In statistical physics, lattice gas exclusion processes have gained much popularity to model the active stochastic motion of particles along a one-dimensional lane \cite{schutz2000exactly,spitzer1991interaction,spohn2012large}. Specifically, totally asymmetric simple exclusion process (TASEP) has been a paradigmatic model to study the motion of self-propelled particles in one preferred direction subjected to excluded volume interactions \cite{derrida1992exact,parmeggiani2003phase}. It was originally introduced in the context of RNA polymerization by ribosomes \cite{schutz2000exactly,macdonald1968kinetics,macdonald1969concerning}. Since then these models have further stimulated a lot of fundamental research including vehicular traffic, intracellular transport, surface growth, transport in ion channels, etc \cite{macdonald1968kinetics,mallick2011some,schadschneider2000statistical,helbing2001traffic,chou2011non}. In terms of TASEP, numerous simpler topologies like junctions, tree like structures, structureless links have been extensively analysed in context of traffic flow and biological transportation \cite{neri2011totally,embley2009understanding,basu2010asymmetric,neri2013exclusion}.
However, the collective behaviour of particles on elementary structures of networks is still a subject of comprehensive discussion.
In the perspective of modelling generic features of transportation processes on lane-based systems, studies abound in the literature analysing the topology of intersecting lanes \cite{fouladvand2004optimized,thunig2016braess,crociani2016multiscale}.
Owing to an extensive body of TASEP models, quantitative characterisation of two crossing roads with parallel update rules in a closed geometry has been well examined \cite{ishibashi1996phase,ishibashi2001phase,ishibashi2001phase1}. Recently, a variant topology considering ``figure-of-eight" network of two intersecting TASEPs has been studied with periodic conditions following random sequential update rules \cite{braess2005paradox}. This prototype provides an insight to the Braess paradox that explains counter-intuitive situation in which
adding an edge to a road network leads to a user optimum with higher travel times for all network users \cite{pas1997braess}. However, the dynamics of two intersecting lanes with open boundary conditions revealed a much interesting phenomenon known as spontaneous symmetry breaking (SSB) \cite{yuan2008spontaneous,tian2020totally}. This displays the occurrence of macroscopic asymmetric stationary states under the symmetric microscopic dynamical rules. The``bridge model" was the pioneer model to exhibit SSB where two species of particles are allowed to move in opposite direction on a single lane TASEP \cite{evans1995spontaneous,evans1995asymmetric}. Since then this aspect has been of specific interest studied in detail utilising variants of TASEP assimilating various additional processes \cite{muhuri2011phase,popkov2008spontaneous,pronina2007spontaneous,sharma2017phase,verma2018far}. Such models have been thoroughly investigated in the vicinity of unlimited supply of particles.
In recent years, much more generalised versions of TASEP model have been contemplated where the particles are injected from a finite reservoir of particles. Such models have shown a wide applicability in many physical and biological systems such as protein synthesis, movement of motor proteins, ``parking garage problems", vehicular traffic \cite{adams2008far,ha2002macroscopic,blasius2013recycling}. To explore such scenario few studies based on single as well as multi-lane TASEP prototypes have been conducted where the dynamics compete for limited number of particles \cite{verma2018limited,greulich2012mixed,adams2008far,cook2009feedback,cook2009competition}. This generalisation reveals a non-trivial behavior of system including the extension of ``shock phase" that leads to traffic jam like situations on lanes. For this phase domain-wall approach provides a powerful theoretical technique to incorporate fluctuations and accurately calculate the stationary properties of the system \cite{cook2009competition}.
\begin{figure*}
\centering
\includegraphics[width = 0.43\textwidth]{fig1a}
\includegraphics[width = 0.24\textwidth]{fig1b}
\caption{\label{fig1}Schematic representation of two intersected lanes in a reservoir with finite number of particles. Green and blue coloured lane represents lane $L$ and $T$ respectively with $i=1,2,\hdots,N$ labelled sites. The intersected site $k=N/2$ is highlighted in red colour that can occupy any of the particle arriving from lane $L$ or $T$. If this site accommodates particle from $L(T)$ it can jump to the unoccupied site $k+1$ of same lane with unit rate. The entry and exit rate of particles in either lane is $\alpha^*$ and $\beta$. Lane $L(T)$ is divided into two parts left segment $L_1(T_1)$ and right segment $L_2(T_2)$. The particles can leave $L_1(T_1)$ with effective exit rate $\beta_{eff,L}(\beta_{eff,T})$ and enter $L_2(T_2)$ with effective entry rate $\alpha_{eff,L}(\alpha_{eff,T})$.}
\end{figure*}
In this work, we present a unifying picture for the emergence of shock and symmetry breaking on two-intersecting lanes coupled to a reservoir of finite number of particles. We attempt to provide an insight to mechanisms including transportation processes where the particles can conquer a jam like situation. We treat this model as a two-lane coupled system with an inhomogeneity for which we exploit the idea of effective rates and domain wall theory to analyse the interplay of intersected site and finite reservoir. We attempt to answer few important questions like (1) Does symmetry breaking prevail in a system of intersected lanes under the effect of finite resources? (2) Are there non-trivial effects on the qualitative and quantitative behaviour on phase diagram with variation of number of particles in the system? We also provide a fundamental brief by considering appropriate limiting cases to visualise the steady-state characteristics of the system.
\section{Model Definition and Dynamical Rules}
This section intends to elaborate a minimalistic model of two-lane transport intersecting at a special site. The extreme ends of both the lanes are coupled to a single reservoir having finite number of identical particles denoted by $N_r$. The total number of particles $(N_{tot})$ in the system remain constant at any instant of time. The two lanes are labelled as $L$ and $T$ assuming each lane to be composed of $i=1,2,\hdots,N$ sites with special site at $k=N/2$ common to both the lanes as shown in Fig. \eqref{fig1}. We assume this site far away from boundaries to probe the effect of intersected lanes on the overall dynamics of the system. The transition rules are in accordance to random sequential update rules. Each site including intersected site obeys hard core exclusion principle that allows each site to occupy atmost one particle. \par
We presume particles in both the lanes to move in one preferred direction from left to right.
A particle is allowed to enter the vacant site $1$ of any of the two lanes from the reservoir with effective intrinsic rate $\alpha^{*}$ depending on the reservoir density given by,
\begin{equation}
\alpha^{*}=\alpha f(N_r)
\end{equation}
where $\alpha$ is entry rate for the case with infinite number of particles.
It is reasonable to adopt a monotonic increasing function satisfying $f(0)=0$ and $f(N_{tot})=1$. This means smaller the number of particles in the reservoir lower the effective intrinsic rate of particles in the two lanes. And, the enhanced particle content in the reservoir leads to greater rush of particles in the two lanes. Based on these arguments, the simplified choice for $f(N_r)$ is,
\begin{equation}
f(N_r)=\dfrac{N_r}{N_{tot}}
\end{equation}
that implies the effective intrinsic rate given by,
\begin{equation}\label{alphaeff}
\alpha^{*}=\alpha \dfrac{N_r}{N_{tot}}.
\end{equation}
This relation implies that the entry rate of particles is directly proportional to the free concentration of particles in the reservoir as long as it is not too crowded. The choice of function is generic and suits well to imitate biological as well as vehicular transport processes \cite{adams2008far,ha2002macroscopic,blasius2013recycling}. For either lane, the exit rate of particles is independent of the number of particles present in the reservoir. A particle at site $N$ can escape with constant rate $\beta$ back to the reservoir from where it is free to rejoin any lane.
In the bulk of each lane, a particle seeks to jump to the adjacent vacant site of the same lane with unit rate and are not allowed to switch their lanes. However, the intersection of two lanes at site $k$ distinguishes it from a homogeneous two-lane TASEP model. Since, site $k$ is shared by both the lanes, any of the particle approaching from lane $L$ or $T$ can occupy this site at any instant of time. A particle at site $k-1$ of lane $L(T)$ can jump to intersected site $k$ with unit rate if found empty. Further, if site $k$ is occupied with the particle arriving from lane $L$ or $T$, it is allowed to jump to the unoccupied site $k+1$ of its own respective lane with rate 1. Here, the particles are not allowed to change their lanes even after jumping from the intersected site. For the proposed model, particles of both the lanes interact only at intersected site because a particle at site $k-1$ of lane $L(T)$ compete to find an empty site $k$.
It has been noticed that for the case of infinite number of particles, the considered topology of lanes induces a non-trivial effect on the qualitative behaviour of system in terms of symmetry breaking \cite{tian2020totally}. However, in the previous work, the existence of phase regimes specially for asymmetric phase is calculated numerically.
\section{Theoretical Framework}
The present model can be viewed as a variant of TASEP model with two incoming segments reaching a junction site and diverging into two outgoing segments \cite{embley2009understanding}. This represents a network of $2\times 2$ segments provided the particles at site $k$ are distinguishable and are constrained to jump to the next site of specific lane. When these lanes do not intersect and there are infinite number of particles, the proposed model reduces to a two lane homogeneous TASEP model.
\subsection{Brief discussion of results for two-lane homogeneous TASEP model with infinite reservoir}
In literature, mean-field approximation has been discerned to provide exact results for homogeneous single-lane TASEP model with infinite reservoir \cite{derrida1992exact,chou2011non,kolomeisky1998phase}. It has been found that there exists three distinct stationary phases: low density (LD), high density (HD) and maximal current (MC) \cite{derrida1992exact,chou2011non,kolomeisky1998phase}. Various network TASEP models have been extensively explored utilising mean-field theory \cite{pronina2005theoretical,neri2013exclusion,neri2011totally,embley2009understanding,basu2010asymmetric,braess2005paradox}. This approximation ignores all possible correlation in the system and assumes the occupancy of two consecutive sites independent of each other.
Based on this approach, it has been deliberated that for equal entry and exit rates of two uncoupled lanes ($L$ and $T$), there exist three dynamic regimes namely LD:LD, HD:HD and MC:MC \cite{derrida1992exact}. The bulk density in each phase remains equal for both the lanes, leading to the existence of symmetric phases. In the notation, first part of `:' denotes the state of lane $L$, and second part describes the phase manifested by lane $T$. The description of theoretically calculated density profiles, particle currents and existence of phases is summarised in table \eqref{simple}.
In the continuum limit, steady-state particle current in the bulk ($1<i<N$) of lane $L$ and $T$ denoted by $J_L$ and $J_T$ respectively, is written as,
\begin{equation}
J_L=\rho(1-\rho),\qquad J_T=\sigma(1-\sigma)
\end{equation}
where $\rho$ and $\sigma$ denotes the average bulk density in lane $L$ and $T$ respectively.
And, the current at site $1$ and $N$ of each lane is given by,
\begin{eqnarray}
J_L^1=\alpha(1-\rho^1),\qquad J_L^N=\rho^N\beta,\\
J_T^1=\alpha(1-\sigma^1),\qquad J_T^N=\sigma^N\beta
\end{eqnarray}
where $\rho^1(\sigma^1)$ and $\rho^N(\sigma^N)$ represents the average density of particles at site $1$ and $N$ of lane $L(T)$ respectively.
\begin{table*}[!htb]
\caption{\label{simple}Summary of results for a two uncoupled lanes TASEP model with finite reservoir \cite{greulich2012mixed}. When $\mu\rightarrow\infty$, $\alpha^*\rightarrow\alpha$, the results converge to that for model with infinite reservoir of particles \cite{derrida1992exact}. Here, $x_w$ denotes the position of domain wall.}
\begin{center}
\renewcommand{\arraystretch}{2}
\resizebox{0.9\textwidth}{!}{
\begin{tabular}{||c|c||c|c|c|c|c||}
\hline \hline
&Phase Region&$\rho^1=\sigma^1$&$\rho=\sigma$&$\rho^N=\sigma^N$&Current($J_L=J_T$)&$\alpha^*$\\
\hline \hline
LD:LD&$\alpha^*<\min\{\beta,1/2\}$&$\alpha^*$&$\alpha^*$&$\dfrac{\alpha^*(1-\alpha^*)}{\beta}$&$\alpha^*(1-\alpha^*)$&$\alpha\left(1-\dfrac{1}{\mu}\right)$\\
\hline
HD:HD&$\beta<\min\{\alpha^*,1/2\}$&$1-\dfrac{\beta(1-\beta)}{\alpha^*}$&$1-\beta$&$1-\beta$&$\beta(1-\beta)$&$\alpha\left(1-\dfrac{2(1-\beta)}{\mu}\right)$\\
\hline
MC:MC&$1/2<\min\{\alpha^*,\beta\}$&$1-\dfrac{1}{4\alpha^*}$&$\dfrac{1}{2}$&$\dfrac{1}{4\beta}$&$\dfrac{1}{4}$&$\alpha\left(1-\dfrac{1}{\mu}\right)$\\
\hline
S:S&$\alpha^*=\beta,~\beta<1/2$&$\alpha^*$&$x_w\alpha^*+(1-\beta)(1-x_w)$&$1-\beta$&$\alpha^*(1-\alpha^*)=\beta(1-\beta)$&$\dfrac{1}{x_w}\left(\dfrac{\mu}{2\alpha}(\alpha-\beta)-(1-\beta)(1-x_w)\right)$\\
\hline \hline
\end{tabular}
}
\end{center}
\end{table*}
\subsection{Bulk dynamics: Intersection of two lanes}\label{intersectedsection}
The intersection of two lanes at a special site introduces an inhomogeneity in the system. For this, we divide each lane into two segments, left segment $L_1$, $T_1$: $i=1,2,\hdots,k-1$ and right segment $L_2$, $T_2$: $i=k+1,k+2,\hdots,N$ coupled at special site $k$.
The two segments of both the lanes are properly integrated by determining the effective rate of particles. We define, particles in lane $L$ and $T$ can leave their respective left segment with effective exit rate $\beta_{eff,L}$ and $\beta_{eff,T}$ respectively. Similarly, the particles can enter the right segment of lane $L$ and $T$ with effective entry rate $\alpha_{eff,L}$ and $\alpha_{eff,T}$ respectively. The average density of particles at site $k-1$, $k$ and $k+1$ of lane $L$ is written as $\rho_1^{k-1}$, $\rho^{k}$ and $\rho_2^{k+1}$. Similarly, for lane $T$, to represent these densities $\rho$ is replaced by $\sigma$.
We denote the particle current induced in each lane as $J_{L_j}$ and $J_{T_j}$ where $j=1$ (left segment) and $j=2$ (right segment).
The stationary current arguments in both the lanes leads to equal current in two segments of each lane, read as,
\begin{equation}\label{current}
J_{L_1}=J_{L_2}\qquad \text{and} \qquad J_{T_1}=J_{T_2}
\end{equation}
that implies
\begin{equation}\label{eqcur}
\rho_1^b(1-\rho_1^b)=\rho_2^b(1-\rho_2^b),\quad \text{and} \quad\sigma_1^b(1-\sigma_1^b)=\sigma_2^b(1-\sigma_2^b)
\end{equation}
where $\rho_j^b$ and $\sigma_j^b$ denotes the bulk density in two segments of lane $L$ and $T$ respectively.
The above equations in Eqn. \eqref{eqcur} further specifies that the bulk density satisfies,
\begin{eqnarray}
\rho_1^b=\rho_2^b,\quad \text{or} \quad \rho_1^b+\rho_2^b=1,\\
\sigma_1^b=\sigma_2^b,\quad \text{or} \quad \sigma_1^b+\sigma_2^b=1.
\end{eqnarray}
In each lane, the condition of current continuity suggests that the exit current of left segment is equal to the current passing from site $k-1$ to $k$, given by,
\begin{eqnarray}
\rho_1^{k-1}~\beta_{eff,L}&=&\rho_1^{k-1}~(1-\rho^{k}-\sigma^{k}),\\
\sigma_1^{k-1}~\beta_{eff,T}&=&\sigma_1^{k-1}~(1-\rho^{k}-\sigma^{k}).
\end{eqnarray}
that results in,
\begin{equation}\label{ebeta}
\beta_{eff,L}=\beta_{eff,T}=1-\rho^{k}-\sigma^{k}=\beta_{eff}~(\text{say}).
\end{equation}
Also, the current entering into the right segment is equal to current passing from site $k$ to $k+1$, written as,
\begin{eqnarray}
(1-\rho_2^{k+1})~\alpha_{eff,L}&=&\rho^{k} ~(1-\rho_2^{k+1}),\\
(1-\sigma_2^{k+1})~\alpha_{eff,T}&=&\sigma^{k} ~(1-\sigma_2^{k+1})
\end{eqnarray}
that leads to,
\begin{eqnarray}\label{inter}
\alpha_{eff,L}=\rho^{k},\\
\alpha_{eff,T}=\sigma^{k}.
\end{eqnarray}
Our main aim is to calculate the effective rates and average density of particles in $L$ and $T$ including at intersected site. The explicitly computed effective rates helps to determine the stationary properties of the system. When there are infinite number of particles in the system, it has been observed that the intersection of lanes assists the phenomenon of symmetry breaking. The appearance of two symmetric and one asymmetric phase has been reported \cite{tian2020totally}.
\subsection{Boundary dynamics: Lanes connected to finite reservoir of particles}
The extreme ends of two intersected lanes are coupled to a finite reservoir of particles that governs the effective intrinsic rate of particles into the lanes as given in Eqn. \eqref{alphaeff}. Now, the total number of particles in the system can be written as,
\begin{equation}
N_{tot}=N_r+N_L+N_T
\end{equation}
where $N_L$ and $N_T$ signifies the number of particles in lane $L$ and $T$ respectively.
Since, each lane is divided in two segments, the average density of particles in each lane is the sum of average density of particles in the respective left and right segment. Since, in the continuum limit, the spatial variables are rescaled as $x=i/N$, therefore, we can write the average density of particles in lane $L$ and $T$ as,
\begin{eqnarray}
\int_0^{1}\rho ~dx &=&\int_0^{1/2}\rho_1^b ~dx+\int_{1/2}^{1}\rho_2^b~ dx,\\
\int_0^{1}\sigma ~dx &=&\int_0^{1/2}\sigma_1^b ~dx+\int_{1/2}^{1}\sigma_2^b ~dx
\end{eqnarray}
respectively, where $\int_0^{1}\rho~dx=\dfrac{N_L}{N}$ and $\int_0^{1}\sigma~dx=\dfrac{N_T}{N}$.
Further, the effective intrinsic rate in Eqn. \eqref{alphaeff} can be written as,
\begin{eqnarray}
\alpha^* &=&\alpha-\dfrac{\alpha N}{N_{tot}}\left(\int_0^{1/2}(\rho_1^b+\sigma_1^b)~ dx+\int_{1/2}^{1}(\rho_2^b+\sigma_2^b)~ dx\right),\qquad
\end{eqnarray}
that implies
\begin{eqnarray}\label{finite}
\alpha^* &=&\alpha-\dfrac{\alpha}{\mu}\left(\int_0^{1/2}(\rho_1^b+\sigma_1^b)~ dx+\int_{1/2}^{1}(\rho_2^b+\sigma_2^b)~ dx\right),\qquad
\end{eqnarray}
where $\mu=\dfrac{N_{tot}}{N}$. As already observed in table \eqref{simple} that the density of particles is distinct in each phase, therefore the effective intrinsic rate $\alpha^*$ alters correspondingly.
For the case when two lanes do not intersect, we retrieve a two-lane homogeneous model coupled to a finite reservoir of particles. Therefore, the expression for effective intrinsic rate in Eqn. \eqref{finite} reduces to \cite{verma2018limited},
\begin{equation}
\alpha^* =\alpha\left(1-\dfrac{1}{\mu}\int_0^{1}(\rho+\sigma)~ dx
\right).
\end{equation}
Here, in addition to existing distinct stationary phases for homogeneous two-lane model with infinite reservoir, a new symmetric coexistence LD-HD phase namely shock (S) phase has been observed \cite{greulich2012mixed}.
A shock phase is characterised by an existing discontinuity in the bulk connected by a low-density to high-density regime. In literature, a well-known approach `` domain-wall theory" has been deployed to estimate the position of this domain-wall \cite{de1989microscopic,kolomeisky1998phase,verma2018limited}. The basic idea of this theory is to assume a sharp shock in the density profile located at any site between the two regimes. The estimation of locating this shock at a particular site helps to evaluate the overall density profile of particles. This shock is situated anywhere between [0,1] and its position is denoted by $x_w$ throughout the paper. The conditions for the existence of stationary phases with finite reservoir have been reviewed in table \eqref{simple}.
In the next section, we discuss the conditions for the existence of different phase regimes in ($\alpha,\beta$) plane for the proposed inhomogeneous model of intersecting lanes. The phase boundaries are obtained theoretically utilising the framework adopted in section \eqref{intersectedsection} along with the concept of domain-wall theory. In addition, we present the convergence of our theoretical results to the limiting case of intersected lanes with infinite reservoir \cite{tian2020totally}.
\begin{figure*}[!htb]
\subfigure[\label{001}$\mu=0.001$]{\includegraphics[width = 0.3\textwidth]{fig2a}}
\subfigure[\label{05}$\mu=0.5$]{\includegraphics[width = 0.3\textwidth]{fig2b}}
\subfigure[\label{1}$\mu=1$]{\includegraphics[width = 0.3\textwidth]{fig2c}}\\
\subfigure[\label{12}$\mu=1.2$]{\includegraphics[width = 0.3\textwidth]{fig2d}}
\subfigure[\label{15}$\mu=1.5$]{\includegraphics[width = 0.3\textwidth]{fig2e}}
\subfigure[\label{2}$\mu=2$]{\includegraphics[width = 0.3\textwidth]{fig2f}}
\caption{\label{Sphases} Stationary phase diagrams for increasing parameter $\mu=N_{tot}/N$ notified in sub-captions of (a-f). In the limiting case $\mu\rightarrow\infty$, the phase diagram converges to the two intersected lane model with infinite particles \cite{tian2020totally}. The white coloured regions represent symmetric phases, whereas, the coloured regions denote asymmetric phases. The black circles are the simulated results. The phase boundaries are computed
within an estimated error of less than $1\%$.}
\end{figure*}
\section{Stationary system states}\label{phases}
In this section we elaborate the qualitative and quantitative behaviour of stationary phase diagrams depending on three controlling parameters $(\alpha,\beta,\mu)$. As already discussed, we have divided each lane into two segments the possible phases are labelled as A-B:C-D where A and B describes the phase in left and right segment of lane $L$. Similarly, C and D symbolises the phase manifested in $T_1$ and $T_2$. In addition, a phase is characterised as a symmetric phase if the particle density in $L_1(L_2)$ is equal to density in $T_1(T_2)$ (i.e. $\rho_1^b=\sigma_1^b$ and $\rho_2^b=\sigma_2^b$). Otherwise, the phase is indicated as a asymmetric phase and is labelled in \textit{italics}.
For the proposed model, each segment can exhibit four possible stationary states LD, HD, MC or S. Therefore, the maximum possible number of stationary phases in each lane is $2^4=16$. However, the existence of an ample number of phases is prohibited due to various restrictions. For example, from Eqn. \eqref{current} it can be easily realised that for either lane the possibility of having MC phase in any of the segments and the LD, HD or S in the other segment can be discarded because these phases support different particle currents. Moreover, both the segments cannot exhibit MC phase simultaneously. This is because if left segment shows average density 1/2, the inhomogeneous dynamical rules does not allow the right segment to achieve maximal current.
Also, due to interaction of particles at site $k$, the right segments of both the lanes cannot exhibit HD phase simultaneously. Since, a particle at site $k$ is restricted to jump only in the same lane. As a consequence, when any of particle arriving from lane $L$ or $T$ resides on the intersected site, the particle on site $k-1$ of other lane has to wait till the particle at $k^{th}$ site jumps to the site of its own respective lane. Owing to this fact, a asymmetric phase has been observed in ref.\cite{tian2020totally}.
Now we theoretically investigate the conditions of existence of phases, phase boundaries for the proposed model with varying entry and exit rates. We provide explicit expressions for the density profiles, phase boundaries and position of shock in terms of $\mu$.
\subsection{Symmetric Phases}
We now discuss the occurrence of different symmetric phases and aim to calculate the effective rates and densities to determine the phase boundaries. As discussed, for symmetric phase the following conditions hold i.e.,
\begin{eqnarray}\label{symcon1}
\rho_1^b=\sigma_1^b,\qquad \rho_2^b=\sigma_2^b,
\end{eqnarray}
that also implies,
\begin{eqnarray}
\label{symcon2}
\alpha_{eff,L}=\alpha_{eff,T}=\alpha_{eff}, \quad \rho^{k}=\sigma^{k}.
\end{eqnarray}
Without any loss of generality, we can thus analyse the dynamics of particles in any one lane (say $L$). The same results are pertinent for the other lane $T$. Hence, Eqn. \eqref{finite} reduces to,
\begin{equation}\label{finitesym}
\alpha^* =\alpha-\dfrac{2\alpha}{\mu}\left(\int_0^{1/2}\rho_1^b~dx+\int_{1/2}^{1}\rho_2^b~dx\right).
\end{equation}
To determine the conditions of existence of different phases we need to compute effective rates $\alpha^*$, $\alpha_{eff}$ and $\beta_{eff}$. Depending on the phase we can also determine the densities, $\rho_1^b$, $\rho_2^b$, $\rho^{k}$, $\rho_1^{k-1}$ and $\rho_2^{k+1}$.
\subsubsection{LD-LD:LD-LD Phase}
In this phase, we assume both the segments of lane $L$ to exhibit low-density phase. Each homogeneous segment is entry-dominated for which the conditions of existence are given by,
\begin{eqnarray}
\alpha^*<\min\{\beta_{eff},1/2\},\qquad
\alpha_{eff}<\min\{\beta,1/2\}.
\end{eqnarray}
The density of particles in the bulk of each segment is given by,
\begin{eqnarray}\label{int}
\rho^1=\rho_1^b=\alpha^* \quad and \quad \rho_2^{k+1}=\rho_2^b=\alpha_{eff}.
\end{eqnarray}
Since, current is equal for both the segments Eqn. \eqref{current} results in,
\begin{eqnarray}
\alpha^* &=&\alpha_{eff}.
\end{eqnarray}
Solving Eqn. \eqref{finitesym}, we have,
\begin{eqnarray}
\alpha^* =\alpha-\dfrac{2\alpha}{\mu}\int_0^{1}\alpha^*~dx,
\end{eqnarray}
that leads to
\begin{eqnarray}\label{alphald}
\alpha^* =\dfrac{\alpha \mu}{\mu+2\alpha}.
\end{eqnarray}
The density of particles arriving from lane $L$ at intersected site is given by Eqn. \eqref{inter}, $\rho^{k}=\alpha^*$.
The effective exit rate with which a particle leaves from left segment given in Eqn. \eqref{ebeta} reduces to,
\begin{equation}
\beta_{eff}=1-2\alpha^*.
\end{equation}
In addition, stationary current argument at site $k-1$ implies that the bulk current in right segment is equal to the current entering into it that yields,
\begin{eqnarray}
\rho_1^{k-1}=\dfrac{\alpha^*(1-\alpha^*)}{1-2\alpha^*}.
\end{eqnarray}
The conditions of existence for this phase thus reduces to,
\begin{eqnarray}
\alpha^*<\min\{\beta,1/3\}.
\end{eqnarray}
For the case when $\mu\rightarrow\infty$, the expression for effective intrinsic rate in Eqn. \eqref{alphald} reduces to $\alpha^*=\alpha$. As a result, we retrieve the phase boundaries for the model with infinite number of particles given by $\alpha<\min\{\beta,1/3\}$.
\subsubsection{HD-LD:HD-LD Phase}
We assume in this phase for each lane, left segment to exhibit high density phase and right segment in low density phase. Correspondingly, the homogeneous left segment is exit-dominated, whereas, right segment is entry dominated. This phase is determined by,
\begin{eqnarray}
\beta_{eff}<\min\{\alpha^*,1/2\},\qquad
\alpha_{eff}<\min\{\beta,1/2\}.
\end{eqnarray}
The density of particles in the bulk of each segment is given by,
\begin{eqnarray}
\rho_1^b=\rho_1^{k-1}=1-\beta_{eff} \quad and \quad \rho_2^b=\rho_2^{k+1}=\alpha_{eff}.
\end{eqnarray}
The condition of constant current in Eq. \eqref{current} leads to,
\begin{eqnarray}
\beta_{eff} &=&\alpha_{eff}.
\end{eqnarray}
Clearly, Eqn. \eqref{inter} provides the density of particles at site $k$ as $\rho^{k}=\beta_{eff}$. Now, plugging these values in Eqn. \eqref{ebeta} we obtain,
\begin{eqnarray}
\beta_{eff}=1-2\beta_{eff},
\end{eqnarray}
implying,
\begin{eqnarray}
\beta_{eff}=\dfrac{1}{3}.
\end{eqnarray}
Further, the effective intrinsic rate given in Eqn. \eqref{finitesym} is obtained as follows,
\begin{eqnarray}
\alpha^* = \alpha-\dfrac{2\alpha}{\mu}\left(\int_0^{1/2}(1-\beta_{eff})~dx+\int_{1/2}^1\alpha_{eff}\right),
\end{eqnarray}
that yields,
\begin{eqnarray}\label{muhlhl}
\alpha^* =\alpha\left(\dfrac{\mu-1}{\mu}\right).
\end{eqnarray}
From the above equation we conclude that this phase exists only when $\mu>1$. The conditions of existence reduces to,
\begin{equation}\label{conhl}
1/3<\min\{\alpha^*,\beta\}.
\end{equation}
In the limiting case, as $\mu\rightarrow \infty$, the effective intrinsic rate in Eqn. \eqref{muhlhl} reduces to $\alpha$. Correspondingly, the conditions that favor the existence of this phase reduces to $1/3<\min\{\alpha,\beta\}$ for the model with infinite number of particles \cite{tian2020totally}.
\subsubsection{S-LD:S-LD Phase}
For this phase, the particles in $L_1$ exhibit shock phase i.e. a part of segment is in LD phase and the rest is in HD phase. Whereas, the right segment shows low-density phase. The existence of this phase is determined by following conditions,
\begin{eqnarray}
\alpha^*=\beta_{eff},~~\beta_{eff}<1/2,\qquad
\alpha_{eff}<\min\{\beta,1/2\}.
\end{eqnarray}
For this phase, in left segment the site $1$ and $k-1$ is entry and exit-dominated respectively. The density in $L_1$ is written as,
\begin{eqnarray}
\rho^1&=&\alpha^*,\\
\rho_1^{k-1} &=& 1-\beta_{eff},\\
\int_0^{1/2}\rho_1^b~dx &=&\int_0^{x_w}\alpha^*~dx+\int_{x_w}^{1/2}(1-\beta_{eff})~dx.
\end{eqnarray}
Similarly, the density in $L_2$ is given by,
\begin{equation}
\rho_2^b=\rho_2^{k+1}=\alpha_{eff}.
\end{equation}
Also, the current is constant in both the segments as given in Eqn. \eqref{current} that leads to,
\begin{equation}\label{rates}
\alpha^*=\beta_{eff}=\alpha_{eff}.
\end{equation}
From Eqn. \eqref{ebeta} we obtain the effective exit rate of particles from left segment as,
\begin{equation}
\beta_{eff}=\frac{1}{3}.
\end{equation}
Hence, Eqn. \eqref{finitesym} yields,
\begin{equation}
\begin{split}
\alpha^* =\alpha-\dfrac{2\alpha}{\mu}\left(\int_0^{x_w}\alpha^*~dx+\int_{x_w}^{1/2}(1-\beta_{eff})~dx\right.\\+\left.\int_{1/2}^{1}\alpha^*~dx\right),
\end{split}
\end{equation}
that reduces to,
\begin{eqnarray}
\alpha^* =\alpha\left(\dfrac{3(\mu-1)+2x_w-1}{3\mu}\right).
\end{eqnarray}
Since, $\alpha^*=\frac{1}{3}$ from Eqn. \eqref{rates}, the shock position is given by,
\begin{equation}\label{ssld}
x_w=\dfrac{3}{2}\left(\dfrac{\mu}{3\alpha}-\mu+1\right).
\end{equation}
As the shock position is bounded between $0<x_w<1/2$ that provides one of the condition for existence of this phase. This $x_w$ depends on the parameter $\mu$ and $\alpha$. For a fixed value of $\mu$, as $\alpha$ increases shock travels to left of the lattice on left segments of lane $L$. Hence, this phase exists when,
\begin{equation}
0<x_w<1/2,\qquad \beta\geq 1/3.
\end{equation}
For a system with infinite number of particles $\mu\rightarrow\infty$, we can clearly see that $x_w\rightarrow\infty$, as a result this phase cease to exist and converges to HD-LD:HD-LD phase \cite{tian2020totally}.
\subsubsection{LD-S:LD-S Phase}
In each lane, the density of particles in left segment is in low density phase, while, in right segment particles are in shock phase. The conditions that support the existence of this phase are,
\begin{eqnarray}
\alpha^{*}<\min\{\beta_{eff},1/2\},\qquad
\alpha_{eff}=\beta,~~\beta<1/2.
\end{eqnarray}
Since, the right segment is entry-dominated, we can write the density at site $k+1$, $\rho_2^{k+1}=\alpha_{eff}$.
By the current constancy condition from Eqn. \eqref{current}, the rates are given by,
\begin{equation}
\alpha^*=\alpha_{eff}=\beta.
\end{equation}
The density of particles in left and right segment is given by,
\begin{eqnarray}
\rho_1^b &=& \rho^1=\alpha^*,\qquad \rho_2^{k+1}=\alpha_{eff},\\
\int_{1/2}^1 \rho_2^b ~dx &=&\int_{1/2}^{x_w} \alpha_{eff} ~dx + \int_{x_w}^1 (1-\beta) ~dx,
\end{eqnarray}
respectively.
As a result, solving Eq. \eqref{finitesym} provides the effective intrinsic rate given by,
\begin{eqnarray}
\beta &=& \alpha-\dfrac{2\alpha}{\mu}\left(\int_0^{1/2} \beta ~dx-\int_{1/2}^{x_w} \beta ~dx-\int_{x_w}^{1} (1-\beta) ~dx\right),\qquad
\end{eqnarray}
that implies
\begin{eqnarray}\label{slds}
x_w &=& \dfrac{\beta\mu-\alpha(\mu+2\beta-2)}{2\alpha(1-2\beta)}.
\end{eqnarray}
For this phase to exist the shock travels within the range $1/2<x_w<1$. Also, from Eqn. \eqref{ebeta} the density at $k^{th}$ site is $\rho^k=\alpha_{eff}=\beta$. Hence, the effective exit rate is given as,
\begin{eqnarray}
\beta_{eff} &=& 1-2\beta.
\end{eqnarray}
In addition, stationary current argument at site $k-1$ implies that the bulk current in right segment is equal to the current entering into it that yields,
\begin{eqnarray}
\rho_1^{k-1}=\dfrac{\alpha^*(1-\alpha^*)}{1-2\beta}.
\end{eqnarray}
The conditions of this phase to exist are,
\begin{equation}
1/2<x_w<1, \qquad \beta\leq 1/3.
\end{equation}
For $\mu\rightarrow \infty$, the computed expression for the position of shock converges to $\infty$. This shows that this symmetric phase vanishes when there are infinite number of particles and tends to LD-LD:LD-LD phase \cite{tian2020totally}.
\begin{figure*}[!htb]
\centering
\subfigure[\label{lsls}$\alpha=0.5,~\beta=0.2$]{\includegraphics[width = 0.45\textwidth]{fig3a}}
\subfigure[\label{ldld}$\alpha=0.1,~\beta=0.8$]{\includegraphics[width = 0.45\textwidth]{fig3b}}\\
\subfigure[\label{hdldhdld}$\alpha=2.5,~\beta=0.8$]{\includegraphics[width = 0.45\textwidth]{fig3c}}
\subfigure[\label{slsl}$\alpha=1.5,~\beta=0.8$]{\includegraphics[width = 0.45\textwidth]{fig3d}}
\caption{\label{densym}Density profiles attributed to symmetric phases for $(\alpha,\beta)$ notified in sub-captions of (a-d) with parameter $\mu=1.2$. Solid lines represent theoretical results, while markers denote the simulated results.}
\end{figure*}
\begin{figure*}[!htb]
\centering
\subfigure[\label{hdhdhdld}]{\includegraphics[width = 0.35\textwidth]{fig4a}}
\subfigure[\label{hishl}]{\includegraphics[width = 0.4\textwidth]{fig4b}}\\
\subfigure[\label{shd}]{\includegraphics[width = 0.35\textwidth]{fig4c}}
\subfigure[\label{hisshd}]{\includegraphics[width = 0.4\textwidth]{fig4d}}
\caption{\label{denasym}(Left) Density profiles attributed to asymmetric phases for $\alpha=2.5,~\beta=0.2$ in (a) and (c) with parameter $\mu=1.5$ and $\mu=1.2$ respectively. Such parameters are chosen to show how for a particular value of $\alpha,~\beta$, considering different values of $\mu$, two different asymmetric phases exist. Solid lines represent theoretical results, while markers denote the simulated results. (Right) Particle-density histograms (b) and (d) corresponding to (a) and (c) respectively are plotted to probe the phenomenon of symmetry breaking through Monte Carlo simulations.}
\end{figure*}
\subsection{Asymmetric Phases}
For asymmetric phase, the characteristics of both the lanes are different. So, we need to consider densities of both the lanes $L$ and $T$ in our further calculations.
The specifications that support the extant of asymmetric phases are,
\begin{eqnarray}
\rho_1^b\neq\sigma_1^b,\qquad\text{or}\qquad \rho_2^b\neq\sigma_2^b,
\end{eqnarray}
that can lead to
\begin{eqnarray}
\alpha_{eff,L}\neq\alpha_{eff,T}, \quad \rho^{k}\neq\sigma^{k}.
\end{eqnarray}
We aim to explicitly calculate the effective rates for both the lanes and densities in bulk as well as at intersected site of particles moving in each lane.
\subsubsection{\textit{S-HD:HD-LD} Phase}
We presume particles in left segment of lane $L$ to exhibit shock phase and right segment to manifest high density phase. While, the particles in left and right segment of lane $T$ display high and low density phase respectively. This phase exists when the boundary controlling parameters satisfies the following conditions described in table \eqref{sh}.
\begin{table}[!htb]
\centering
\caption{\label{sh} Conditions for existence of asymmetric \textit{S-HD:HD-LD} Phase}
\begin{tabular}{|c||c||c|}
\hline
& $L$ & $T$\\
\hline
\hline
Left segment & $\alpha^*=\beta_{eff},~~ \beta_{eff}<1/2$ & $\beta_{eff}<\min\{\alpha^*,1/2\}$\\
Right segment & $\beta<\min\{\alpha_{eff},1/2\}$ & $\alpha_{eff,T}<\min\{\beta,1/2\}$.\\
\hline
\end{tabular}
\end{table}\\
The density in bulk of two segments of lane $L$ is given by,
\begin{eqnarray}
\rho_1^{k-1} &=& 1-\beta_{eff},\\
\int_0^{1/2}\rho_1^b~dx &=&\int_0^{x_w}\alpha^*~dx+\int_{x_w}^{1/2}(1-\beta_{eff})~dx,\\
\rho_2^b &=& 1-\beta.
\end{eqnarray}
Next, the density in the bulk of left and right segment of lane $T$ is given by,
\begin{eqnarray}
\label{asymh1}
\sigma_1^{k-1}=\sigma_1^b &=& 1-\beta_{eff},\\\label{asymh2}
\sigma_2^{k+1}=\sigma_2^b &=& \alpha_{eff,T},
\end{eqnarray}
respectively. Since, current is constant in each segment of $L$, from Eqn. \eqref{eqcur} we have,
\begin{equation}
\alpha^*(1-\alpha^*)=\beta_{eff}(1-\beta_{eff})=\beta(1-\beta)
\end{equation}
and similarly for lane $T$, we can write,
\begin{equation}
\beta_{eff}(1-\beta_{eff})=\alpha_{eff,T}(1-\alpha_{eff,T}).
\end{equation}
This yields,
\begin{equation}
\alpha^*=\alpha_{eff,T}=\beta
\end{equation}
and these values are further substituted in Eqn. \eqref{finite} to evaluate the shock position in $L_1$.
\begin{equation}
\begin{split}
\beta =\alpha-\dfrac{\alpha}{\mu}\left(\int_0^{x_w}\beta ~dx+\int_{x_w}^{1/2}(1-\beta)~dx\right.\\ +\left.\int_0^{1}(1-\beta)~dx+\int_{1/2}^1 \beta ~dx\right).
\end{split}
\end{equation}
The shock position $x_w$ is obtained as,
\begin{equation}\label{sshd}
x_w=\dfrac{2\mu(\beta-\alpha)+\alpha(3-2\beta)}{2\alpha(1-2\beta)}
\end{equation}
that lies in [0,1/2]. From Eqn. \eqref{ebeta}, plugging $\sigma^{k}=\alpha_{eff,T}=\beta$ we have,
\begin{eqnarray}
\beta_{eff}=1-\rho^{k}-\beta,
\end{eqnarray}
that provides the density of particles arriving from lane $L$ at intersected site, given by,
\begin{eqnarray}
\rho^{k}= 1-2\beta.
\end{eqnarray}
This gives the effective entry rate of particles in $L_2$, $\alpha_{eff,L}=1-2\beta$. Hence, this phase exists for,
\begin{equation}
0<x_w<1/2, \qquad \beta\leq 1/3
\end{equation}
Under these conditions, one can easily obtain that this phase exists only when $\mu>1$.
However, for $\mu\rightarrow\infty$, $x_w\rightarrow\infty$ that implies this phase vanishes in the limiting case of infinite number of particles and converges to HD-HD:HD-LD phase \cite{tian2020totally}.
\subsubsection{\textit{HD-HD:HD-LD} Phase}
Without any loss of generality, in this phase, we assume both the segments of lane $L$ exhibit high density, whereas, left and right segment of $T$ display high and low density respectively. This phase exists when boundary controlling parameters satisfy the conditions presented in table \eqref{hh}.
\begin{table}[!htb]
\centering
\caption{\label{hh} Conditions for existence of asymmetric \textit{HD-HD:HD-LD} Phase}
\begin{tabular}{|c||c||c|}
\hline
& $L$ & $T$\\
\hline
\hline
Left segment & $\beta_{eff}<\min\{\alpha^{*},1/2\}$ & $\beta_{eff}<\min\{\alpha^{*},1/2\}$\\
Right segment & $\beta_{eff}<\min\{\alpha_{eff,L},1/2\}$ & $\alpha_{eff}<\min\{\beta,1/2\}$.\\
\hline
\end{tabular}
\end{table}\\
The density in each segment is given by,
\begin{eqnarray}
\rho_1^b=\rho_1^{k-1}=1-\beta_{eff} \quad && \text{and} \quad \rho_2^b=1-\beta,\\
\sigma_1^b=\sigma_1^{k-1}=1-\beta_{eff} \quad && \text{and} \quad \sigma_2^b=\sigma_2^{k+1}=\alpha_{eff,T}.
\end{eqnarray}
Since, the current is constant in each lane given in Eqn. \eqref{current} yields,
\begin{eqnarray}
\beta_{eff} &=& \beta,\\
\beta_{eff} &=& \alpha_{eff,T},
\end{eqnarray}
that implies,
\begin{equation}
\alpha_{eff,T} = \beta=\sigma^{k},
\end{equation}
giving density of particles from lane $T$ at intersected site.
The effective intrinsic rate is calculated utilising Eqn. \eqref{finite} that leads to,
\begin{eqnarray}
\alpha^* &=& \alpha\left(\dfrac{2\mu+2\beta-3}{2\mu}\right).
\end{eqnarray}
From Eqn. \eqref{ebeta} we have,
\begin{eqnarray}
\beta_{eff}=1-\rho^{k}-\beta,
\end{eqnarray}
that gives particle density of particles from lane $L$ as,
\begin{eqnarray}
\rho^{k} = 1-2\beta.
\end{eqnarray}
As a result the existence of conditions for this phase reduce to,
\begin{equation}\label{conhlhl}
\dfrac{\alpha(3-2\mu)}{2(\alpha-\mu)}<\beta, \quad \beta\leq 1/3.
\end{equation}
Combing these conditions, we can easily conclude that this phase exists when $\mu>1.1$.
Here, in the limiting case $\mu\rightarrow\infty$, the conditions of existence reduces to $\alpha<\min\{\beta,1/3\}$, that for model with two intersected lanes and infinite number of particles \cite{tian2020totally}.
However, it is notable that not all combinations for asymmetric phases exist in the proposed model. For instance, suppose lane $L$ to exhibit high density in both the segments and lane $T$ to display low density in both the segments. The conditions of existence for this HD:HD-LD:LD phase are given by,
\begin{eqnarray}
\text{Lane $L$}\quad\beta<\{\alpha^*,1/2\},\\
\text{Lane $T$}\quad\alpha^*<\{\beta,1/2\}
\end{eqnarray}
which contradict each other. Similarly, other asymmetric phases cease to exist because either the conditions disagree or no values of $(\alpha,\beta)$ satisfy these conditions.
\begin{figure*}[!htb]
\centering
\subfigure[\label{25_08}]{\includegraphics[width = 0.4\textwidth,height=5.5cm]{fig5a}}
\subfigure[\label{2_02}]{\includegraphics[width = 0.4\textwidth,height=5.5cm]{fig5b}}
\caption{\label{effmu}Phase transitions with respect to $\mu$ for fixed boundary controlling parameters (a) $(\alpha,\beta)=(2.5,0.8)$ that presents transitions within symmetric phases LD-LD:LD-LD $\longrightarrow$ S-LD:S-LD $\longrightarrow$ HD-LD:HD-LD. (b) $(\alpha,\beta)=(2,0.2)$ illustrates phase transitions from symmetric to asymmetric phases LD-S:LD-S $\longrightarrow$ \textit{S-HD:HD-LD} $\longrightarrow$ \textit{HD-HD:HD-LD}. Since, for symmetric phase density profiles are same for both the lane, we plot for only one lane (say $L$). The solid lines represent theoretical results, while, markers denote the simulated outcomes.}
\end{figure*}
\section{Results and Discussions}
In this section, we exploit the results discussed in section \eqref{phases} to address the behaviour of system in ($\alpha,\beta$) plane depending on total number of particles in the system.
We aim to investigate the effect of finite resources in terms of $\mu=\frac{N_{tot}}{N}$ on the complex dynamical properties of the system. We observe qualitative as well as quantitative non-trivial effects on the topology of phase schema especially in terms of symmetry breaking and shock dynamics. To validate our theoretical outcomes we perform Monte Carlo simulations for system size $N=1000$. The computer simulations are carried out for $2 \times 10^9$ time steps and initial $5\%$ of the time steps are scraped to ensure the occurrence of steady state. The average densities in both the lanes are computed by considering time averages over an interval of $10N$. We observe that the theoretically computed density profiles, phase boundaries, shock positions match well with the simulations.
\subsection{Phase boundaries: Effect of $\mu$}
We have theoretically computed the existence of distinct stationary phases in terms of $\mu$ where we analysed that the symmetry of the system persists for $\mu\leq 1$. However, for $\mu>1$ the symmetry of the system is disrupted and asymmetric phases appears in the stationary phase diagram. So, to understand the effect of finite resources we elaborate two different cases and possible phase transitions originating in the system by monitoring the propagation of domain wall for varying boundary controlling parameters.
\subsubsection{$\mu\leq1$}
When there are very less number of particles in the system i.e., $\mu \approx 0$, only one symmetric phase namely LD-LD:LD-LD appears in the entire phase regime as presented in Fig. \eqref{001} for $\mu=0.001$. This can be easily realised by a simple argument that due to scarcity of particles in the reservoir less number of particles are allowed to enter any of the lane, leading to low-density phase in each segment.
Also, for lower values of $\beta$ the particles tend to accumulate at the right end of each lane. As a consequence, the boundary layer at the right boundary enters the bulk leading to the emergence of boundary induced shock in the right segment. Thus, a symmetric phase namely LD-S:LD-S phase appears in the phase diagram for $\mu>0$ as shown in Fig. \eqref{05} for $\mu=0.05$. The phase boundary between these two phases is obtained from Eqn. \eqref{slds} given by,
\begin{equation}
\alpha=\dfrac{\beta\mu}{\mu-2\beta},\quad\beta\leq 1/3.
\end{equation}
With significant increase in $\mu$ no qualitative changes are observed in the system except the shifting of phase boundaries resulting in the shrinkage of LD-LD:LD-LD phase and expansion of LD-S:LD-S phase. However, beyond a critical value $\mu_{C_1}$, due to interaction of particles at the intersected site a bulk induced shock emerges in the left segment of each lane resulting in a new symmetric regime S-LD:S-LD as shown in Fig. \eqref{1}. This critical value is computed from Eqn. \eqref{ssld} that yields $\mu_{C_1}=2/3$ and,
\begin{equation}
\alpha=\dfrac{\mu}{3\mu_{C_1}-2},\quad\beta\geq 1/3.
\end{equation}
However, for $\mu_{C_1}<\mu\leq\mu_{C_2}$ no new phase appears in the phase diagram, only quantitative changes are observed.
\subsubsection{$\mu>1$ (Symmetry Breaking)}
Distinctively, beyond a critical value $\mu_{C_2}$ in addition to the emergence of a new symmetric phase the system experiences rich topological changes with the occurrence of symmetry breaking phenomenon. As soon $\mu>\mu_{C_2}$, a new symmetric HD-LD:HD-LD phase emerges next to S-LD:S-LD phase resulting in the shrinkage of observed symmetric till now as shown in Fig. \eqref{12}. From Eqn. \eqref{muhlhl}, we observe that this phase exists for boundary controlling parameters satisfying,
\begin{equation}\label{bounsl}
\alpha=\dfrac{\mu_{C_2}}{3(\mu_{C_2}-1)},\quad\beta\geq 1/3.
\end{equation}
that yields the critical value $\mu_{C_2}=1$. In addition, due to interaction of particles at intersected site asymmetric \textit{S-HD:HD-LD} phase emerges in $(\alpha,\beta)$ plane for which the phase boundary is obtained from Eqn. \eqref{sshd} that leads to,
\begin{equation}
\alpha=\dfrac{\beta\mu_{C_2}}{\mu_{C_2}-1},\quad\beta\leq 1/3.
\end{equation}
This is because with increase in number of particles, the boundary induced shock in LD-S:LD-S phase absorbs the incoming particles and travel towards the left side of each lane. As soon it reaches the intersected site, the particle at intersected site totally blocks the flow of particles resulting in HD-LD phase in one lane (say lane $L$). For other lane ($T$) the boundary induced shock crosses the intersected site and shifts to the left segment.
\begin{figure*}[!htb]
\centering
\subfigure[\label{spls}$\beta=0.2$]{\includegraphics[width = 0.35\textwidth,height=5.5cm]{fig6a}}
\subfigure[\label{spsl}$\beta=0.8$]{\includegraphics[width = 0.35\textwidth]{fig6b}}
\caption{\label{spsym}Symmetric Phases: Movement of shock with increasing values of $\alpha$ for a chosen $\mu=1.2$ in (a) LD-S:LD-S phase and (b) S-LD:S-LD for $\beta=0.2$ and $0.8$ respectively. As already discussed, the symmetric phase satisfies Eqn. \eqref{symcon1} and \eqref{symcon2}. Therefore, without any loss of generality, we plot the density profile of particles of only one lane (say $L$). The same results are pertinent for the other lane $T$. The solid, dashed lines represent the theoretical computed results. While, markers denote the simulated results.}
\end{figure*}
\begin{figure*}[!htb]
\centering
\includegraphics[width = 0.35\textwidth]{fig7}
\caption{\label{spsh}Asymmetric Phase: Movement of shock with increasing values of $\alpha$ for $\mu=1.2$ and $\beta=0.2$ in \textit{S-HD:HD-LD} phase. One of the lane portrays HD-LD phase for chosen parameters marked with purple markers and solid line. However, the other lane manifests S-HD phase for which the shock position varies with $\alpha$. As $\alpha$ increases the domain wall sweeps to the left of the lane. The solid, dashed lines represent the theoretical computed results and, markers denote the simulated results.}
\end{figure*} On further increasing $\mu$ after a crucial value $\mu_{C_3}$ an additional asymmetric phase \textit{HD-HD:HD-LD} emerges in the phase schema (as prescribed in Fig. \eqref{15}) for,
\begin{equation}
\alpha=\dfrac{2\beta\mu_{C_3}}{2(\mu_{C_3}+\beta)-3},\quad\beta\leq 1/3,
\end{equation}
and crucial value is obtained as $\mu_{C_3}=1.1$. This happens because as soon a particle enters the lane for entry rate satisfying Eqn. \eqref{conhlhl}, it is quickly assimilated by the segment that exhibits shock in \textit{S-HD:HD-LD} phase. Beyond this crucial value $\mu_{C_3}$, we observe only quantitative alterations in the phase diagram as presented in Fig. \eqref{2} for $\mu=2$. Even for finite value of $\mu$ the phase diagram of proposed model converges to that of two intersected TASEP model in ref.\cite{tian2020totally}. However, in the limiting case $\mu\rightarrow \infty$, it is clearly evident from Eqns. \eqref{ssld}, \eqref{slds} and \eqref{sshd} that the boundary and bulk induced shock disappears, and we retrieve phase diagram for two intersected lanes with infinite number of particles.\\
\subsection{Density Profiles and Phase Transitions}
The density profiles attributed to symmetric and asymmetric stationary phases are presented in Fig. \eqref{densym} and \eqref{denasym} respectively are found to agree well with Monte Carlo simulations. However, the shock present in Fig. \eqref{lsls} and \eqref{slsl} becomes sharper for increasing values of $N$. Due to intersection of lanes, the density profile of LD-S:LD-S, LD-LD:LD-LD, \textit{HD-HD:HD-LD} and \textit{S-HD:HD-LD} phase admits a kink at the intersected site as clearly visible in Fig. \eqref{lsls}, \eqref{ldld}, \eqref{hdhdhdld} and \eqref{shd} respectively.\\
\indent To further inspect a deeper insight in the phenomenon of SSB, we probe particle density histograms $P(\rho_1,\rho_2)$ and $P(\sigma_1,\sigma_2)$, where $\rho_j$ and $\sigma_j$ are instantaneous particle densities on segment $j$ ($j=1,2$). For $\alpha=2.5$ and $\beta=0.8$, we present typical density histogram for asymmetric phases \textit{S-HD:HD-LD} and \textit{HD-HD:HD-LD} with $\mu=1.2$ and $\mu=1.5$ respectively. One can clearly see in Fig. \eqref{hishl} that the peaks in distributions are achieved for $\rho_1=\rho_2>1/2$ and $\sigma_1>1/2$, $\sigma_2<1/2$ that corresponds to \textit{HD-HD:HD-LD} phase. Fig. \eqref{hisshd} demonstrates that the peak occurs for $0<\rho_1<1$, $\rho_2>1/2$ and $\sigma_1>1/2$, $\sigma_2<1/2$ portraying \textit{S-HD:HD-LD} phase.\\
In order to visualise phase transitions with respect to $\mu$ we chose particular values of $\alpha,\beta$ and plot Fig. \eqref{effmu}. For chosen parameters $\alpha=2.5$ and $\beta=0.8$, in Fig. \eqref{25_08} we portray transitions within symmetric phases LD-LD:LD-LD $\longrightarrow$ S-LD:S-LD $\longrightarrow$ HD-LD:HD-LD. When $\mu=0.5$ the particles exhibit symmetric LD-LD:LD-LD phase with a kink in the density profile at intersected site. With increase in $\mu=1.2$, the density at intersected site increases as also evident from Eqn. \eqref{int}, that indicates the existence of symmetric S-LD:S-LD phase. Further increasing $\mu=1.5$, the bulk induced shock transforms into HD regime and leads to the occurrence of symmetric HD-LD:HD-LD phase. Similarly, Fig. \eqref{2_02} illustrates the phase transitions from symmetric to asymmetric phases LD-S:LD-S $\longrightarrow$ \textit{S-HD:HD-LD} $\longrightarrow$ \textit{HD-HD:HD-LD} for $\alpha=2$ and $\beta=0.2$. When $\mu=1$ particles manifest boundary induced shock in right segment of both the lanes displaying symmetric LD-S:LD-S phase. Increasing $\mu=1.2$, due to availability of ample number of particles this shock stabilises and transforms into asymmetric \textit{S-HD:HD-LD} phase which further converges to asymmetric \textit{HD-HD:HD-LD} phase for $\mu=1.5$.
To summarize, there exist maximum six stationary phases in the overall system including four symmetric and two asymmetric phases. The complexity of phase diagram shows non-monotonic behaviour with increasing values of $\mu$. Initially, there exists only one phase while in the midrange dynamics becomes complex and six phases are observed. Besides, the intuitive observations of the effect of finite resources, the appearance and disappearance of phases is examined from theoretically computed phase boundaries. In the limiting case $\mu\rightarrow \infty$, we have explained in section \eqref{phases}, how effective intrinsic rate $\alpha^*$ approaches to $\alpha$ in each phase. Consequently, topological structure of phase schema is modified and the number of phases drastically reduces from six to three \cite{tian2020totally}.
\subsection{Shock Dynamics}\label{Shock}
In the above sections, we have seen that due to finite number of particles in the system, two types of shock emerges in the dynamical regimes. It has been observed that a bulk induced shock exists in symmetric S-LD:S-LD phase. Whereas, a boundary induced shock persists in a symmetric LD-S:LD-S as well as a asymmetric \textit{S-HD:HD-LD} phase. The shock emerging in the right segment of any lane is boundary induced, while, that appearing in left segment might be boundary or bulk induced. Here, to discuss the phase transitions arising due to propagation of shock in ($\alpha-\beta$) plane we fixed parameter and vary boundary controlling parameters $(\alpha,\beta)$.
Beginning with the boundary induced shock in symmetric LD-S:LD-S phase appearing for $\beta<1/3$, the shock lies in the right segment of both the lanes determined by Eqn. \eqref{slds}. For $\beta=0.2$, we plot Fig. \eqref{spls} where one can notice that as $\alpha$ increases shock propagates toward the left side of right segment. This means, if more number of particles are allowed to enter the lattice for a significant choice of $\mu$, the right segment incorporates these particles tending towards a high dense region. As soon this wall reaches the intersected site, due to interactions other lane is forced to exhibit low density of particles. Thus, next to this phase, with increasing $\alpha$, a asymmetric phase \textit{S-HD:HD-LD} appears in the steady-state phase diagram. For this phase, the position of boundary induced shock lies within the range $[0,1/2]$ and can be computed using Eqn. \eqref{sshd}. With increase in $\alpha$, the particles are absorbed by the segment specifying S phase. As a consequence, shock moves toward the left of the lane as shown in Fig. \eqref{spsh} for $\beta=0.2$ and beyond a critical value of $\alpha$, the left lane exhibits HD indicating the appearance of asymmetric \textit{HD-HD:HD-LD} phase.
For $\beta>1/3$, a bulk induced shock in symmetric S-LD:S-LD phase appears in the phase schema. In this phase the particles in left segment of both the lanes portray a discontinuity. The explicit expression for the location of this wall is computed in Eqn. \eqref{ssld} that suggests for a fixed $\beta$, as more number of particles are permitted to enter, the shock in both the lanes sweeps to the left of the lane. This can be easily noticed from Fig. \eqref{spsl} where we can clearly see that as $\alpha$ increases for $\beta=0.8$, the HD part of S increases and after a crucial value of $\alpha$ given in Eqn. \eqref{bounsl} shock vanishes. As a result, left segments of both the lanes attain HD phase leading to the occurrence of symmetric HD-LD:HD-LD phase.
\section{Conclusion}
In this work, we have studied a specific variant of network TASEP model with two intersected lanes, a class of minimal models for transportation phenomenon. The two extreme ends of each lane are connected to a reservoir having finite number of particles. The intersection of lanes introduces an inhomogeneity in the system that is suitably dealt by considering effective entry and exit rates. Even though, the particles interact at intersected site, mean-field approximation works well to theoretically investigate various crucial steady-state properties of the system such as density profiles, phase transitions and phase boundaries. The theoretical predictions are validated through extensive Monte Carlo simulations.
We extensively probe the effect of finite resources on the phenomenon of spontaneous symmetry breaking, since the same persists for infinite number of particles. With an increase in number of particles, crucial qualitative and quantitative changes in the topology of phase diagram are observed. The symmetry of phase schema is preserved till the total number of particles do not exceed the total number of sites. However, as soon as more number of particles are available than the number of sites, the symmetry of the phase diagram is disrupted. The interaction of particles at intersected site is responsible for the symmetry breaking phenomenon. There exist maximum six possible stationary regimes in the system including four symmetric and two asymmetric phases. The exact number and location of phases depend on the number of particles in the system. The existence of asymmetric phases is explored by delineating density histograms using computer simulations. In addition, a symmetric phase exhibits a bulk and boundary induced shock in each lane. Whereas, a asymmetric phase manifests a boundary induced shock in one of the lanes due to interactions of particles at intersected site. The density in the shock profile is estimated by employing domain-wall approach. We explicitly calculate the phase boundaries to determine dynamic regimes and location of different phases. Also, by monitoring the movement of shock we describe the phase transitions as more particles are allowed to enter.
The proposed study can be extended to analyse more complex dynamics in the network of intersected lanes when inter-lane switching of particles is allowed in the bulk as well as while jumping from the intersected site. Also, several generalisations of the model can be explored in future by considering various other processes such as the interplay with the non-conserving dynamics, extended particle size, particle-particle interactions etc..
\section*{Acknowledgement}
AK Gupta acknowledges support from DST-SERB, Govt. of India (Grant CRG/2019/004669).
|
1,108,101,563,310 | arxiv | \section{Introduction}
Terahertz (THz) imaging is an emerging sensing technology with a great potential for hidden object imaging, contact-free analysis, non-destructive testing and stand-off detection in various application fields, including semi-conductor industry, biological and medical analysis, material and quality control, safety and security~\cite{chan2007imaging,Jansen2010,siegel2002terahertz}. The physically interpretable quantities relevant to the aforementioned applications, however, cannot always be measured directly. Instead, in THz imaging systems, each pixel contains implicit information about such quantities, making the \textit{inverse problem} of inferring these physical quantities a challenging problem with high practical relevance.
As we will discuss in Sec.~\ref{sec:THz}, at each pixel location $\vec{x}$ the relation between the desired (unknown) parameters $\vec{\overline{p}}(\vec{x})=(\hat{e}(\vec{x}),\sigma(\vec{x}), \mu(\vec{x}),\phi(\vec{x}))\in \mathbb{R}^4$, i.e., the electric field amplitude $\hat{e}$, the position of the surface $\mu$, the width of the reflected pulse $\sigma$, and the phase $\phi$, and the actual measurements $g(\vec{x})\in \mathbb{R}^{n_z}$ can be modelled via the equation $g(\vec{x},z) = (f_{\hat{e},\sigma,\mu,\phi}(z_i))_{i \in \{1,\hdots,n_z\}} + \text{noise}$, where
\begin{eqnarray}
\label{eq:thzEquation}
& & f_{\hat{e},\sigma,\mu,\phi}(z) = \hat{e} \sinc\left(\sigma (z-\mu)\right) \exp\left(-i(\omega z - \phi)\right), \\
& & \sinc(t) = \begin{cases} \cfrac{\sin(\pi t)}{\pi t} & t \neq 0, \\
1 & t = 0,
\end{cases}
\end{eqnarray}
and $(z_i)_{i \in \{1,\hdots,n_z\}}$ is a device-dependent sampling grid $z_{grid}$.
More details of the THz model are described in~\cite{wong2019computational}. Thus, the crucial step in THz imaging is the solution of optimization problem of the form
\begin{equation}
\label{eq:thzOptimization}
\min_{\hat{e},\sigma,\mu,\phi} \quad \text{Loss}(f_{\hat{e},\sigma,\mu,\phi}(z_{grid}), g(\vec{x})),
\end{equation}
at each pixel $\vec{x}$, possibly along with additional regularizers on the unknown parameters.
Even with simple choices of the loss function such as an $\ell^2$-squared loss, the resulting fitting problem is highly nonconvex and global solutions become rather expensive. Considering that the number $(n_x\cdot n_y)$ of pixels, i.e., of optimization problem \eqref{eq:thzOptimization} to be solved, typically is in the order of hundred thousands to millions, even local first order or quasi-Newton methods become quite costly: For example, running the build-in Trust-Region solver of MATLAB\textsuperscript{\textregistered} to reconstruct a $446 \times 446$ THz image takes over 170 minutes.
In this paper, we propose to train a neural network to solve the per-pixel optimization problem \eqref{eq:thzOptimization} directly. We formulate the training of the network as a model-based autoencoder (AE), which allows us to train the corresponding network with real data in an unsupervised way, i.e., without ground truth. We demonstrate that the resulting optimization network yields parameters $(\hat{e},\sigma,\mu,\phi)$ that result in only slightly higher losses than actually running an optimization algorithm, despite the advantage of being more than 140 times faster. Moreover, we demonstrate that our network can serve as an excellent initialization scheme for classical optimizers. By using the network's prediction as a starting point for a gradient-based optimizer, we obtain lower losses and converge more than 2x faster than classical optimization approaches, while benefiting from all theoretical guarantees of the respective minimization algorithm.
This paper is organized as follows: Sec.~\ref{sec:THz} gives more details on how THz imaging systems work. Sec.~\ref{sec:related} summarizes the related work on learning optimizers, machine learning for THz imaging techniques, and model-based autoencoders. Sec.~\ref{sec:ourArchi} describes model-based AEs in contrast to classical supervised learning approaches in detail, before Sec.~\ref{sec:implementation} summarizes our implementation. Sec.~\ref{sec:experiments} compares the proposed approaches to classical (optimization-based) reconstruction techniques in terms of speed and accuracy before Sec.~\ref{sec:conclusions} draws conclusions.
\section{THz Imaging Systems}
\label{sec:THz}
There are several approaches to realizing THz imaging, e.g. femtosecond laser based scanning system~\cite{cooper2011thz,hu1995imaging}, synthetic aperture systems~\cite{mcclatchey2001time,ding2013thz}, and hybrid systems \cite{kahl2012}.
A typical approach to THz imaging is based on the Frequency Modulated Continuous Wave (FMCW) concept~\cite{ding2013thz}, which uses active frequency modulated THz signals to sense reflected signals from the object. The reflected energy and phase shifts due to the signal path length make 3D THz imaging possible.
In Figure \ref{fig:thz_geometry}, the setup of our electronic FMCW-THz 3D imaging system is shown. More details on the THz imaging system are described in \cite{ding2013thz}.
\begin{figure}[thb]
\centering
\includegraphics[width=0.7\textwidth]{THz_geometry.pdf}
\caption{THz 3D imaging geometry. Both transmitter (Tx) and receiver (Rx) are mounted on the same platform. The imaging unit, consisting of Tx, Rx and optical components, are moved along the x and y direction using stepper motors and linear stages. This imaging unit takes a depth profile of the object at each lateral position, in order to acquire a full THz 3D image.}
\label{fig:thz_geometry}
\end{figure}
In this paper, we denote by $g_t(\vec{x},t)$ the measured demodulated time domain signal of the reflected electric field amplitude of the FMCW system at lateral position $\vec{x}\in \mathbb{R}^2$.
In FMCW radar signal processing, this continuous wave temporal signal is converted into frequency domain by a Fourier transform~\cite{munson1989signal,skolnik1970radar}.
Since the linear frequency sweep has a unique frequency at each spatial position in $z$-direction, the converted frequency domain signal directly relates to the spatial azimuth ($z$-direction) domain signal
\begin{equation}
\label{eq:thzEquation_fft}
g_c(\vec{x},z) = \mathcal{F}\{g_{t}(\vec{x},t)\}.
\end{equation}
The resulting 3D image $g_c \in \mathbb{C}^{n_x \times n_y \times n_z}$ is complex data in the spatial domain, representing per-pixel complex reflectivity of THz energy. The quantities $n_x$, $n_y$, $n_z$ resemble the discretization in vertical, horizontal and depth-direction, respectively. Equivalently, we may represent $g_c$ by considering the real and imaginary parts as two separate channels, resulting a 4D real data tensor $g \in \mathbb{R}^{n_x \times n_y \times n_z \times 2}$.
Since the system is calibrated by amplitude normalization with respect to an ideal metallic reflector, a rectangular frequency signal response is ensured for the FMCW frequency dependance~\cite{ding2013thz}.
After the FFT in~\eqref{eq:thzEquation_fft}, the $z$-direction signal envelope is an ideal $\sinc$ function as continuous spatial signal amplitude, giving rise to the physical model given in~\eqref{eq:thzEquation} in the introduction.
In~\eqref{eq:thzEquation}, the electric field amplitude $\hat{e}$ is the reflection coefficient for the material, which is dependent on the complex dielectric constant of the material and helps to identify and classify materials. The depth position $\mu$ is the position at which maximum reflection occurs, i.e., the position of the surface reflecting the THz energy. $\sigma$ is the width of the reflected pulse, which includes information on the dispersion characteristics of the material. The phase $\phi$ of the reflected wave depends on the ratio of real to imaginary parts of the dielectric properties of the material. Thus, the parameters $\vec{\overline{p}}=(\hat{e},\sigma, \mu,\phi)$ contain important information about the geometry as well as the material of the imaged object, which is of interest in a wide variety of applications.
\section{Related Work}
\label{sec:related}
Due to the revolutionary success (convolutional) neural networks have had on computer vision problems over the last decade, researchers have extended the fields of applications of neural networks significantly. A particularly interesting concept is to learn the solution of complex, possibly nonconvex, optimization problems. Different lines of research have considered directly learning the optimizer itself, e.g. modelled as a recurrent neural network~\cite{Andrychowicz2016LearningTL}, or rolling out optimization algorithms and learning the incremental steps, e.g. in the form of parameterized proximal operators in \cite{Kobler2017VariationalNC}. Further hybrid approaches include optimization problems in the networks' architecture, e.g. \cite{Amos2017OptNetDO}, or combining optimizers with networks that have been trained individually \cite{Chang17,Meinhardt17}. The recent work of Moeller et al.~\cite{energyDissipation19} trains a network to predict descent directions to a given energy in order to give provable convergence results on the learned optimizer.
Objectives similar to the one arising in the training of our model-based AEs are considered, for instance, for solving inverse problems with deep image priors~\cite{Ulyanov2018DeepIP} or deep decoders~\cite{Heckel2019DeepDC}. These works, however, consider the input to the networks being fixed random noise and have to solve an optimization problem for the networks weights for each inverse problems, such that they are regularization-by-parametrization approaches rather than learned optimizers.
The most related prior work is the 3D face reconstruction network from Tewari~et~al\onedot\cite{tewari17MoFA}. They aimed at finding a semantic code vector from a given facial image such that feeding this code vector into a rending engine yields an image similar to the input image itself. While this problem had been addressed using optimization algorithms a long time ago \cite{Blanz99} (also known under the name of analysis-by-synthesis approaches), the approach by Tewari~et~al\onedot\cite{tewari17MoFA} replaced the optimizer with a neural network and kept the original cost function to train the network in an unsupervised way. The resulting structure resembles an AE in which the decoder fixed to the forward model and was therefore coined model-based AE. As we will discuss in the next section, the idea of model-based AEs generalizes far beyond 3D face reconstruction and can be used to boost the THz parameter identification problem significantly.
Finally, a recent work has exploited deep learning techniques in Terahertz imaging in~\cite{long2019terahertz}, but the considered application of super-resolving the THz amplitude image by training a convolutional neural network on synthetically blurred images is not directly related to our proposed approach.
\section{A Model-Based Autoencoder for THz Image Reconstruction}
\label{sec:ourArchi}
Let us denote the THz input data by $g \in \mathbb{R}^{n_x \times n_y \times n_z \times 2}$, and consider our four unknown parameters $(\hat{e},\sigma,\mu,\phi)$ to be $\mathbb{R}^{n_x \times n_y}$ matrices, allowing each parameter to change at each pixel. Under slight abuse of notation we can interpret all operations in~\eqref{eq:thzEquation} to be pointwise and again identify complex values with two real values in order to have $f_{\hat{e},\sigma,\mu,\omega,\phi}(z_{grid}) \in \mathbb{R}^{n_x \times n_y \times n_z \times 2}$, where $z_{grid} = (z_i)_{i \in \{1, \hdots, n_z\}}$ denotes the depth sampling grid. Concatenating all four matrix valued parameters into a single parameter tensor $\vec{P} \in \mathbb{R}^{n_y \times n_x \times 4}$, our goal can be formalized as finding $\vec{P}$ such that $f_{\vec{P}}(z_{grid})\approx g$.
A classical supervised machine learning approach to problems with known forward operator is illustrated in Figure \ref{fig:classicalArchi} for the example of THz image reconstruction: The explicit forward model $f$ is used to simulate a large set of images $g$ from known parameters $P$ which can subsequently be used as training data for predicting $P$ via a neural network $\mathcal{G}(g;\theta)$ depending on weights $\theta$. Such supervised approaches with simulated training data are frequently used in other image reconstruction areas, e.g. super resolution \cite{dong2014learning,kim2016accurate}, or image deblurring \cite{nah2017deep,schuler2016learning}. The accuracy of networks trained on simulated data, however, crucially relies on precise knowledge of the forward model and the simulated noise. Slight deviations thereof can significantly degrade a network performance as demonstrated in \cite{DarmstadtDenoising}, where deep denoising networks trained on Gaussian noise were outperformed by BM3D when applied to realistic sensor noise.
Instead of pursuing the supervised learning approach described above, we replace $\vec{\overline{p}}=(\hat{e},\sigma, \mu, \phi)$ in the optimization approach \eqref{eq:thzOptimization} by a suitable network $\mathcal{G}(g;\theta)$ that depends on the raw input data $g$ and learnable parameters $\theta$, that can be trained in an \textit{unsupervised} way \textit{on real data}. Assuming we have multiple examples $g^k$ of THz data, and choosing the loss function in \eqref{eq:thzOptimization} as an $\ell^2$-squared loss, gives rise to the unsupervised training problem
\begin{equation}
\label{eq:training}
\min_\theta \sum_{\text{training examples }k} \|f_{\mathcal{G}(g^k;\theta)}(z_{grid}) - g^k\|_F^2.
\end{equation}
As we have illustrated in Figure \ref{fig:netArchi}, this training resembles an AE architecture: The input to the network is data $g^k$ which gets mapped to parameters $P$ that -- when fed into the model function $f$ -- ought to reproduce $g^k$ again.
Opposed to the straight forward supervised learning approach, the proposed approach \eqref{eq:training} has two significant advantages
\begin{itemize}
\item It allows us to train the network in an \textit{unsupervised} way, i.e., on real data, and therefore learn to deal with measurement-specific distortions.
\item The cost function in \eqref{eq:training} implicitly handles the scaling of different parameters, and therefore circumvents the problem of defining meaningful cost functions on the parameter space: Simple parameter discrepancies such as $\|P_1 - P_2\|_2^2$ for two different parameters sets $P_1$ and $P_2$ largely depend on the scaling of the individual parameters and might even be meaningless, e.g. for cyclic parameters such as the phase offset $\phi$.
\end{itemize}
\begin{figure}[tb]
\centering
\includegraphics[width=1\textwidth]{classicalArchitecture.pdf}
\caption{Classical supervised learning strategy with simulated data: The forward model $f_P$ (e.g. from \eqref{eq:thzEquation}) is used to simulate data $g$, which can subsequently be fed into a network to be trained to reproduce the simulation parameters in a supervised way.}
\label{fig:classicalArchi}
\end{figure}
\begin{figure}[thb]
\centering
\includegraphics[width=1\textwidth]{autoencArchitecture.pdf}
\caption{A model-based AE for THz image reconstruction: The input data $g$ is fed into a network $\mathcal{G}$ whose parameters $\theta$ are trained in such a way that feeding the network's prediction $\mathcal{G}(g;\theta)$ into a model function $f$ again reproduces the input data $g$. Such an architecture resembles an AE with a learnable encoder and a model-based decoder and allows an unsupervised training on real data.}
\label{fig:netArchi}
\end{figure}
\section{Encoder Network Architecture and Training}
\label{sec:implementation}
\subsection{Data Preprocessing}
\label{sec:preproc}
As illustrated in the plot of the magnitude of an exemplary measured THz signal shown in Figure \ref{fig:exemplarySignal}, the THz energy is mainly focused in the main lobe and first side-lobes of the $\sinc$ function. Because the physical model remains valid in close proximity of the main lobe only, we preprocess the data to reduce the impressively large range of $12600$ measurements per pixel. We, therefore, crop out 91 measurements per pixel centered around the main lobe, whose position is related to the object distance and to the parameter $\mu$. Details of the cropping window are described in~\cite{wong2019computational}. We represent the THz data in a 4D real tensor $g \in \mathbb{R}^{n_x \times n_y \times n_z \times 2}$, where $n_x = n_y = 446$, and $n_z$ is the size of the cropping window, i.e. $91$ in our case.
\begin{figure}[tb]
\centering
\includegraphics[width=0.5\textwidth]{signal_magnitude-eps-converted-to.pdf}
\caption{Magnitude of a sample point of measured THz signal. The main lobe and major side-lobes are included in the grid window, which is colored in gray.}
\label{fig:exemplarySignal}
\end{figure}
\subsection{Encoder Architecture and Training}
For the encoder network $\mathcal{G}(g;\theta)$ we pick a spatially decoupled architecture using $1\times 1$ convolutions on $g$ only, leading to a signal-by-signal reconstruction mechanism that allows a high level of parallelism and therefore maximizes the reconstruction speed on a GPU. The specific architecture (illustrated in Figure~\ref{fig:encoder}) applies a first set of convolutional filters on the real and imaginary part separately, before concatenating the activations, and applying three further convolutional filters on the concatenated structure. We apply batch-normalization (BN)~\cite{ioffe2015batch} after each convolution and use leaky rectified linear units (LeReLU)~\cite{glorot2011deep} as activations. Finally, a fully connected layer reduces the dimension to the desired size of four output parameters per pixel. To ensure that the amplitude is physically meaningful, i.e., non-negative, we apply an absolute value function on the first component. Interestingly, this choice compared favorably to a plain rectified linear unit when the network is trained.
\begin{figure}[htb]
\centering
\includegraphics[width = 1\textwidth]{networkArchitecture.pdf}
\caption{Architecture of encoding network $\mathcal{G}(g;\theta)$ that predicts the parameters: At each pixel the real and imaginary part is extracted, convolved, concatenated and processed via three convolutional and 1 fully connected layer. To obtain physically meaningful (non-negative) amplitudes, we apply an absolute value function to the first component. }
\label{fig:encoder}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth]{loss_db-eps-converted-to.pdf}
\caption{
\protect The average losses of the training and validation sets over 1200 epochs on a decibel scale illustrate that there is almost no generalization gap between training and validation.
}
\label{fig:training}
\end{figure}
We train our model optimizing \eqref{eq:training} using the Adam optimizer~\cite{kingma2014adam} on $80\%$ of the $446 \times 446$ pixels from a real (measured) THz image for 1200 epochs. The remaining $20\%$ of the pixels serve as a validation set. The batch size is set to $4096$. The initial learning rate is set to $0.005$, and is reduced by a factor of 0.99 every 20 epochs. Figure~\ref{fig:training} illustrates the decay of the training and validation losses over 1200 epochs. As we can see, the validation loss nicely resembles the training loss with almost no generalization gap.
\section{Numerical Experiments}
\label{sec:experiments}
\begin{figure}[thb]
\centering
\subfloat[\label{fig:thz_object_metalpcb}]{\includegraphics[width=0.25\textwidth]{USAF_Target_Metal_PCB.jpg}}
\hspace{0.1\textwidth}
\subfloat[\label{fig:thz_object_stepchart}]{\includegraphics[width=0.25\textwidth]{Step_Chart_ver_003_view_3.jpg}}
\caption{Objects of evaluated datasets \protect\subref{fig:thz_object_metalpcb} MetalPCB dataset
\protect\subref{fig:thz_object_stepchart} StepChart dataset
}
\label{fig:thz_object}
\end{figure}
We evaluate the proposed model-based AE on two datasets, which are acquired using the setup described in Sec.~\ref{sec:THz}, namely the \emph{MetalPCB} dataset and the \emph{StepChart} dataset.
The MetalPCB dataset is measured by a nearly planar copper target etched on a circuit board (Figure~\ref{fig:thz_object_metalpcb}),
which includes \emph{metal} and \emph{PCB} material regions,
in the standard size scale of USAF target \emph{MIL-STD-150A}~\cite{standard1959photographic}.
After the preprocessing described in Sec.~\ref{sec:preproc}, the MetalPCB dataset has $446 \times 446 \times 91$ sample points.
The StepChart dataset is based on an aluminum object (Figure~\ref{fig:thz_object_stepchart}) with sharp edges to evaluate the distance measurement accuracy using a 3D object.
The StepChart dataset has $113 \times 575 \times 91$ sample points after preprocessing.
In order to evaluate the optimization quality on different materials and structures,
MetalPCB dataset is evaluated in regions:
\emph{PCB region} is a local region that contains PCB material only,
\emph{Metal region} is a local region that contains copper material only,
and \emph{All region} is the entire image area.
Similarly, the StepChart dataset is evaluated by 3 regions:
\emph{Edge region} is the region that contains physical edges,
\emph{Steps region} is the center planar region of each steps,
and \emph{All region} is the entire image area. This segmentation is done, because the THz measurements of the highly specular aluminum target results in strong multi-path interference artifacts at the edges that should be investigated separately.
The proposed model-based AE is trained on the MetalPCB dataset only, while the parameter inference is made for both the MetalPCB and StepChart datasets.
This cross-referencing between two datasets can verify whether the proposed AE method is modelling the physical behavior of the system without overfitting to a specific dataset or recorded material.
To compare with the classical optimization methods, the parameters are estimated using the Trust-Region Algorithm (TRA)~\cite{coleman1996interior}, which is implemented in MATLAB{\textsuperscript{\textregistered} }. The TRA optimization requires a proper definition of the parameter ranges. Furthermore, it is very sensitive with respect to the initial parameter set. We, therefore, carefully select the initial parameters by sequentially estimating them from the source data (see~\cite{wong2019computational} for more details). Still, the optimization may result in a parameter set with significant loss values; see Sec.~\ref{sec:experiments.quality}.
The trained encoder network is independent of any initialization scheme as it tries to directly predict optimal parameters from the input data. While the network alone gives remarkably good results with significantly lower runtimes than the optimization method, there is no guarantee that the network's predictions are critical points of the energy to be minimized. This motivates the use of the encoder network as an initialization scheme to the TRA, specifically because the TRA guarantees the monotonic decrease of objective function such that using the TRA on top of the network can only improve the results. We abbreviate this approach to \emph{AE+TRA} for the rest of this paper.
To fairly compare all three approaches, the optimization time of TRA and the inference time of the AE are both recorded by an Intel\textsuperscript{\textregistered} i7-8700K CPU computation, while the AE is trained on a NVIDIA\textsuperscript{\textregistered} GTX 1080 GPU.
The PyTorch source code is available at \url{https://github.com/tak-wong/THz-AutoEncoder}.
\subsection{Loss and timing}
\label{sec:experiments.loss}
\begin{table}[htb]
\renewcommand{\arraystretch}{1.1}
\centering
\caption{Loss and timing enhancement based on the proposed model-based AE}
\label{tab:loss}
\begin{tabular}{|l|l|r|r|r|}
\hline
Dataset (Region) & Measurement & TRA & AE & AE+TRA \\ \hline
MetalPCB (All) & Average Loss & 693.9 & 886.3 & \textbf{442.2} \\
MetalPCB (PCB) & Average Loss & \textbf{589.0} & 872.6 & \textbf{589.0} \\
MetalPCB (Metal) & Average Loss & 519.6 & 446.1 & \textbf{115.7} \\
StepChart (All) & Average Loss & 3815.1 & 5148.3 & \textbf{3675.3} \\
StepChart (Edges) & Average Loss & 4860.4 & 6309.1 & \textbf{2015.7} \\
StepChart (Steps) & Average Loss & 1152.5 & 2015.7 & \textbf{1150.3} \\ \hline
MetalPCB & Training time (sec.) & \textbf{none} & {9312.8} & {9312.8} \\
MetalPCB & Run time (sec.) & 10391.2 & $^{\dagger}$\textbf{73.5} & $^*$4854.7 \\
StepChart & Run time (sec.) & 3463.9 & $^{\dagger}$\textbf{22.8} & $^*$1712.4 \\ \hline
\multicolumn{5}{l}{\footnotesize{$^{\dagger}$ Inference time}} \\
\multicolumn{5}{l}{\footnotesize{$^*$ Run time is the sum of AE inference and TRA optimization time}} \\
\end{tabular}
\end{table}
In Table~\ref{tab:loss}, the average loss in \eqref{eq:training} and the timing are shown for the Trust-Region Algorithm (TRA), the Autoencoder (AE) and the joint AE+TRA approaches, respectively. We can see that the proposed encoder network achieves a lower average loss than the TRA method in the metal region of the MetalPCB dataset, it yields higher average losses than the TRA on both datasets.
It is encouraging to see that although the AE was trained on the MetalPCB dataset, the relative performance in comparison to the TRA does not decay too significantly when changing to an entirely unseen data set with a different material, with the AE loss being $21.7\%$ and $25.9\%$ higher than the TRA loss on the MetalPCB and StepChart data sets, respectively. If such a sacrifice in accuracy is acceptable, the speed-up in runtime is tremendous with the AE being over 140 times faster than the TRA (for both methods being evaluated on a CPU). Note that even the sum of training and inference time are smaller for the proposed AE than the runtime of the TRA on the MetalPCB dataset.
Interestingly, the combined AE+TRA approach of initializing the TRA with the encoder network's prediction leads to better losses than the TRA alone in all regions. Additionally, the AE-initialized TRA converged more than 2 times faster due to the stopping criterion being reached earlier.
We note that the losses of all approaches are significantly higher for the StepCart data set than they are for the MetalPCB.
This is because the aluminum StepChart object (Figure~\ref{fig:thz_object_stepchart}) has a more complex physical structure than the MetalPCB object, which results in a mixture of scattered THz pulses by multi-path interference effects in all object regions. Incorporating such effects in the reflection model of \eqref{eq:thzEquation} could therefore be an interesting aspect of future research for improving the explainability of the measured data with the physical model.
\subsection{Quality Assessment of THz Images}
\label{sec:experiments.quality}
\begin{figure}[!h]
\centering
\subfloat[\label{fig:intensity_source}]{\includegraphics[width=0.49\textwidth]{intensity_source.pdf}}
\hfill
\subfloat[\label{fig:intensity_hybrid}]{\includegraphics[width=0.49\textwidth]{intensity_hybrid.pdf}}
\vfill
\subfloat[\label{fig:line_intensity}]{\includegraphics[width=0.49\textwidth]{line_intensity-eps-converted-to.pdf}}
\hfill
\subfloat[\label{fig:line_loss}]{\includegraphics[width=0.49\textwidth]{line_loss-eps-converted-to.pdf}}
\caption{Comparison of the THz intensity for the MetalPCB dataset:
\protect\subref{fig:intensity_source} intensity image extracted from the source data without any model-based processing (in red: the pixel line for plots \protect\subref{fig:line_intensity} and \protect\subref{fig:line_loss});
\protect\subref{fig:intensity_hybrid} image extracted by the proposed AE+TRA approach (in red: the pixel line for plots \protect\subref{fig:line_intensity} and \protect\subref{fig:line_loss});
\protect\subref{fig:line_intensity} plot of the intensity extracted along the horizontal line in the copper region;
\protect\subref{fig:line_loss} plot of the per-pixel loss by TRA, AE, and AE+TRA approaches along the horizontal line in the copper region.}
\label{fig:intensity_comparison}
\end{figure}
In THz imaging, the \emph{intensity} image $I$ that is equal to the squared amplitude, i.e. $I=\hat{e}^2$ is the most important criteria for quality assessment.
Note that the intensity could be inferred directly from the data by considering that \eqref{eq:thzEquation} yields
\begin{equation}
\label{eq:thzEquation_intensity}
f_{\hat{e},\sigma,\mu,\phi}(\mu) \cdot f^*_{\hat{e},\sigma,\mu,\phi}(\mu) = \hat{e}^2 \cdot {\sinc}^2(0) = \hat{e}^2 = I
\end{equation}
where $f^*$ is the complex conjugate of $f$.
As we illustrate in Figure \ref{fig:intensity_comparison}, the model-based approach is not only capable of extracting all relevant parameters, i.e., $\hat{e}$, $\mu$, $\sigma$ and $\phi$, but, compared to values directly extracted from the source data, the resulting intensity $I$ is more homogeneous in homogeneous material regions. The homogeneity of the directly extracted intensity results from the very low depth of field of THz imaging systems in general, combined with the slight non-planarity of the MetalPCB target. As depicted in Figure~\ref{fig:line_intensity}, the intensity variations along the selected line in the homogeneous copper region are reduced using the three model-based methods, i.e. TRA, AE, and AE+TRA. However, due to the crucial selection of the initial parameters (see discussion at the beginning of Sec.~\ref{sec:experiments}), the TRA optimization results exhibit significant amplitude fluctuations and loss values (Figure \ref{fig:line_loss}) in the two horizontal sub-regions $x\in[150,200]$ and $x>430$. The proposed AE and AE+TRA methods, however, deliver superior results with respect to the main quality measure applied in THz imaging, i.e. to the intensity homogeneity and the loss in model fitting. Still, the AE approach shows very few extreme loss values, while the AE+TRA method's loss values are consistently low along the selected line in the homogeneous copper region.
\section{Conclusions and Future Work}
\label{sec:conclusions}
In this paper, we propose a model-based autoencoder for THz image reconstruction.
Comparing to a classical Trust-Region optimizer, the proposed autoencoder gets within $25\%$ margin to the objective value of the optimizer, while being more than 140 times faster. Using the network's prediction as an initialization to a gradient-based optimization scheme improves the result over a plain optimization scheme in terms of objective values while still being two times faster. We believe that these are very promising results for training optimizers/initialization schemes for parameter identification problems in general by exploiting the idea of model-based autoencoders for unsupervised learning.
Future research will include exploiting spatial information during the reconstruction as well as considering joint parameter identification and reconstruction problems such as denoising, sharpening, and super-resolving parameter images such as the amplitude images shown in Figure \ref{fig:intensity_hybrid}.
\section*{Acknowledgement}
This is a pre-print of a conference proceeding article published in German Conference on Pattern Recognition. The final authenticated version is available online at: \href{https://doi.org/10.1007/978-3-030-33676-9\_7}{https://doi.org/10.1007/978-3-030-33676-9\_7}
\bibliographystyle{unsrt}
|
1,108,101,563,311 | arxiv | \section{Introduction}
\label{sec:intro}
The ATLAS and CMS collaborations discovered~\cite{HIGG-2012-27,CMS-HIG-12-028} a particle consistent with the
Standard Model (SM)~\cite{Glashow:1961tr,Weinberg:1967tq,Salam:1968rm} Higgs boson~\cite{Englert:1964et,Higgs:1964pj,Guralnik:1964eu,Higgs:1966ev,Kibble:1967sv} in
2012.
Several properties of this particle, such as its coupling strengths, spin and charge-parity (CP) quantum numbers, were
studied with 7 and \SI{8}{\TeV} center-of-mass energy ($\sqrt s$) proton--proton
collision data delivered by the Large Hadron Collider (LHC) in 2011 and 2012,
respectively, referred to as \enquote{\RunOne{}}.
These results rely predominantly on studies of the bosonic decay modes~\cite{HIGG-2014-06,CMS-HIG-14-009,HIGG-2013-17,CMS-HIG-14-018}
and have not shown any significant deviations from the SM expectations.
The coupling of the Higgs boson to the fermionic sector has
been established with the observation of the \Htt decay mode with a
signal significance of 5.5$\sigma$ from a combination of ATLAS and CMS results~\cite{HIGG-2013-32,CMS-HIG-13-033,HIGG-2015-07}
using LHC Run-1 data.
A measurement performed by the CMS Collaboration with Run-2 data at $\sqrt s=\SI{13}{\TeV}$ reached a significance of 4.9$\sigma$
using \SI{35.9}{\per\fb} of integrated luminosity and 5.9$\sigma$ combined with data from \RunOne~\cite{CMS-HIG-16-043}.
While the Higgs-boson coupling to
other fermions such as top quarks~\cite{HIGG-2018-13,CMS_ttH_Observation} and bottom quarks~\cite{ATLAS_Hbb_obs,CMS_Hbb_obs} have been observed,
only upper limits exist on its coupling to muons ~\cite{HIGG-2016-10,CMS_Hmm}
and the \Htautau decay mode has been the only accessible leptonic decay
mode. It was also used to
constrain CP violation in the production via vector-boson fusion (\VBF)~\cite{HIGG-2015-06}
and is unique in that it provides sensitivity to CP violation in the Higgs-boson coupling to leptons~\cite{HtautauCP_theory}.
This paper presents cross-section times branching-fraction measurements of Higgs bosons that decay into a pair of \Pgt{}-leptons in proton--proton (\pp) collisions at $\sqrt{s}=\SI{13}{\TeV}$ using data collected by the ATLAS experiment in 2015 and 2016,
corresponding to an integrated luminosity of \SI{36.1}{\per\fb}.
All combinations of leptonic ($\Pgt \to \Pl \Pgn \Pagn$ with $\Pl = \Pe,
\Pgm$) and hadronic ($\Pgt \to \text{hadrons} \ \Pgn$) \Pgt decays
are considered.\footnote{Throughout this paper
the inclusion of charge-conjugate decay modes is implied.
The symbol \Pl is used to denote electrons and muons, also referred to as \enquote{light leptons}.}
The corresponding three analysis channels are denoted by $\tlep\tlep$,
$\tlep\thad$ and $\thad\thad$ and are composed of different dominant
backgrounds. While \Ztt is a dominant background in all channels, the
relative contributions from other backgrounds from top-quark and
other vector-boson decays, as well as from misidentified leptonic or hadronic \Pgt decays, vary considerably between the channels.
Two analysis categories
are defined that are predominantly sensitive to Higgs bosons produced via \VBF and
gluon--gluon fusion (\ggF). A maximum-likelihood fit is performed on data
using distributions of the reconstructed di-\Pgt mass in signal regions (SRs), simultaneously with event
yields from control regions (CRs) that are included to constrain
normalizations of major backgrounds estimated from simulation.
The dominant and irreducible \Ztt background is
estimated from simulation. This is different from the search for \Htt
decays in Run~1~\cite{HIGG-2013-32}, which used the embedding
technique~\cite{HIGG-2014-09}. A reliable modeling of this background is therefore
of crucial importance for this analysis. Validation regions (VRs) based on \Zll events are studied, but not included in the fit,
to verify as precisely as possible the modeling of the \Ztt background.
The paper is organized as follows. \Cref{sec:detector} describes the
ATLAS detector. This is followed in \cref{sec:samples}
by a description of the dataset and Monte Carlo (MC) simulated samples employed
by this measurement. \Cref{sec:reco} details the reconstruction
of particles and jets.
The event selection for each channel and event category as well as signal, control and validation regions are
discussed in \cref{sec:selection}. Background estimation techniques
and the systematic uncertainties of the analysis are described in
\cref{sec:background,sec:systematics}, respectively.
The signal extraction procedure and the results of the Higgs cross-section measurements in the
\Htt decay mode are presented in \cref{sec:results}.
\Cref{sec:conclusions} gives the conclusions.
\section{The ATLAS detector}
\label{sec:detector}
The ATLAS experiment~\cite{ATLASDetector} at the LHC is a multipurpose particle
detector with a forward--backward symmetric cylindrical geometry and
a near-$4\pi$ coverage in solid angle.\footnote{
The ATLAS Collaboration uses a right-handed coordinate system with its origin at
the nominal interaction point (IP) in the center of the detector
and the $z$-axis along the beam pipe. The $x$-axis points from the IP to the
center of the LHC ring, and the $y$-axis points upwards. Cylindrical coordinates
$(r,\phi)$ are used in the transverse plane, $\phi$ being the azimuthal angle
around the $z$-axis. The pseudorapidity is defined in terms of the polar angle
$\theta$ as $\eta=-\ln\tan(\theta/2)$. Angular distance
is measured in units of $\Delta R \equiv \sqrt{(\Delta\eta)^2 + (\Delta\phi)^2} $.}
It consists of an inner tracking detector
surrounded by a thin superconducting solenoid,
electromagnetic and hadron calorimeters, and a muon spectrometer.
The inner tracking detector covers the pseudorapidity range $|\eta| < 2.5$.
It consists of a silicon pixel detector, which
has an additional innermost layer (positioned at a radial distance of \SI{3.3}{\cm} from the beam line)
that was installed after Run~1~\cite{IBL,IBL2}, and a
silicon microstrip detector surrounding the pixel detector, both
covering $|\eta| < 2.5$, followed by a transition radiation straw-tube tracker
covering $|\eta| < 2$.
The inner tracking detector is immersed in a \SI{2}{\tesla} axial magnetic field provided
by the solenoid.
Lead/liquid-argon (LAr) sampling calorimeters provide electromagnetic (EM) energy measurements
with high granularity.
A hadron (steel/scintillator-tile) calorimeter covers the central pseudorapidity range ($|\eta| < 1.7$).
The endcap and forward regions are instrumented with LAr calorimeters
for both the EM and hadronic energy measurements up to $|\eta| = 4.9$.
The muon spectrometer surrounds the calorimeters and is based on three large air-core
toroidal superconducting magnets with eight coils each.
The field integral of the toroids ranges between \num{2.0} and \SI{6.0}{\tesla\metre}
across most of the detector.
The muon spectrometer includes a system of precision tracking chambers and fast detectors for triggering.
Events are selected using a two-level trigger system. The
first-level trigger is implemented in hardware and uses a subset of the
detector information to filter events that are then processed by a software-based
high-level trigger. This further reduces the average recorded collision rate to
approximately \SI{1}{\kilo\hertz}.
\section{Data and simulation samples}
\label{sec:samples}
The data used in this analysis were taken from \pp collisions at the LHC where
proton bunches are collided every \SI{25}{\ns} at $\sqrt{s} = \SI{13}{\TeV}$.
A combination of several triggers for single light leptons, two light leptons and two hadronically decaying \Pgt{}-leptons were used
to record the data for the analysis, depending on the analysis channel (see \cref{sec:presel}).
After data quality requirements, the samples used for this measurement
consist of \SI{3.2}{\ifb} of data recorded in 2015, with an average of 14 interactions per
bunch crossing, and \SI{32.9}{\ifb} recorded in 2016, with an average of 25 interactions
per bunch crossing.
Samples of signal and background processes were simulated using various MC generators as summarized in \cref{tab:MCSamples}.
The signal contributions considered include the following four processes for Higgs-boson production at the LHC: \ggF, \VBF
and associated production of a Higgs boson with a vector boson (\VH) or with a top--antitop quark pair (\ttH)
where all decay modes for the \Htt process are included.
Other Higgs production processes such as associated production with a bottom--antibottom quark pair and with a single top quark are found to be negligible.
Higgs decays into $\PW\PW$ are considered background and simulated similarly for these production processes.
The mass of the Higgs boson was assumed to be \SI{125}{\GeV}~\cite{HIGG-2014-14}.
\begin{table}
\caption{Monte Carlo generators used to describe all signal and background processes together with the
corresponding PDF set and the model of parton showering, hadronization and underlying event (UEPS).
In addition, the order of the total cross-section calculation is given.
The total cross section for \VBF production is calculated at approximate-NNLO QCD. More details are given in the text.}
\label{tab:MCSamples}
\centering
\vspace{2mm}
\small
\begin{tabular}{lllll}
\toprule
Process & Monte~Carlo generator & PDF & UEPS & Cross-section order \\
\midrule
\ggF & \powhegbox v2 & PDF4LHC15 NNLO & \pythiaEightTwo & N$^3$LO QCD + NLO EW \\
\VBF & \powhegbox v2 & PDF4LHC15 NLO & \pythiaEightTwo & $\sim$NNLO QCD + NLO EW \\
\VH & \powhegbox v2 & PDF4LHC15 NLO & \pythiaEightTwo & NNLO QCD + NLO EW \\
\ttH & MG5\_aMC@NLO v2.2.2 & \textsc{NNPDF30LO} & \pythiaEightTwo & NLO QCD + NLO EW \\
\midrule
$\PW/\Zjets$ & \sherpa 2.2.1 & \textsc{NNPDF30NNLO} & \sherpa 2.2.1 & NNLO \\
$VV$/$V\Pggx$ & \sherpa 2.2.1 & \textsc{NNPDF30NNLO} & \sherpa 2.2.1 & NLO \\
\ttbar & \powhegbox v2 & CT10 & \pythiaSix & NNLO+NNLL \\
$\PW\Pqt$ & \powhegbox v1 & CT10F4 & \pythiaSix & NLO \\
\bottomrule
\end{tabular}
\end{table}
Higgs production by \ggF was simulated with the \powhegbox v2~\cite{Powheg_Nason,Powheg_Frixione,Powheg_Alioli,Powheg_Bagnaschi} NNLOPS program~\cite{Hamilton:2013fea} at next-to-leading-order (NLO) accuracy in quantum chromodynamics (QCD) using the MiNLO approach~\cite{Hamilton:2015nsa}, and reweighted to next-to-next-to-leading order (NNLO) in QCD in the Higgs rapidity.
The \VBF and \VH production processes were simulated at NLO accuracy in QCD using \powhegbox with the MiNLO approach.
The \ttH production process was simulated with \MCatNLO v2.2.2~\cite{McAtNLO} at NLO accuracy in QCD.
For these signal samples, the simulation was interfaced to the \pythiaEightTwo~\cite{Pythia8} model of parton showering, hadronization and underlying event (UEPS).
To estimate the impact of UEPS uncertainties, the \ggF, \VBF and \VH samples were also simulated with the \herwig~\cite{HerwigPP,Herwig7} UEPS model.
The \textsc{PDF4LHC15}~\cite{Butterworth:2015oua} parameterization of the parton
distribution functions (PDFs) was used for these production processes.
The
\textsc{AZNLO}~\cite{STDM-2012-23} set of tuned parameters was used, with the
\textsc{CTEQ6L1}~\cite{PDF-CTEQ6L1} PDF set, for the modeling of
non-perturbative effects.
For the \ttH production process the \textsc{NNPDF30LO}~\cite{PDF-NNPDF30} PDF parametrization was used in the matrix element and the \textsc{NNPDF23LO}~\cite{PDF-NNPDF} PDF parametrization for the UEPS model with the \textsc{A14}~\cite{ATL-PHYS-PUB-2014-021} set of tuned parameters for the modeling of non-perturbative effects.
\textsc{Photos++} version 3.52~\cite{PhotosPP} was used for QED emissions from electroweak (EW) vertices and charged leptons.
The overall normalization of the \ggF process is taken from a
next-to-next-to-next-to-leading-order (N$^3$LO)
QCD calculation with NLO EW corrections
included~\cite{Anastasiou:2015ema,Anastasiou:2016cez,Actis:2008ug,Anastasiou:2008tj}.
Production by \VBF is normalized to an approximate-NNLO QCD cross
section with NLO EW corrections included~\cite{Ciccolini:2007jr,Ciccolini:2007ec,Bolzoni:2010xr}.
The \VH samples are normalized to cross sections calculated at NNLO in QCD,
with NLO EW corrections included~\cite{Brein:2003wg,Altenkamp:2012sx,Denner:2011id}.
The \ttH process is normalized to a cross section calculated at NLO in QCD with
NLO EW corrections applied~\cite{ttH_NLO1,ttH_NLO2,ttH_NLO3, ttH_NLO4,ttH_NLO5,ttH_NLO6}.
Background samples of EW production of $\PW/\PZ$ bosons from \VBF, $\PW/\PZ$-boson production with associated jets and diboson production processes were simulated
with the \sherpa 2.2.1~\cite{Sherpa} generator.
Matrix elements were calculated using the Comix~\cite{Gleisberg:2008fv} and
OpenLoops~\cite{Cascioli:2011va} matrix-element generators and merged
with the \sherpa UEPS model~\cite{Schumann:2007mg} using the
\textsc{ME+PS@NLO} prescription~\cite{Hoeche:2012yf}.
For \PW and \PZ production with associated jets the matrix elements were calculated for up to two partons at NLO and four partons at LO precision.
Their inclusive cross sections are normalized to NNLO calculations from \textsc{Fewz}~\cite{Melnikov:2006kv,Anastasiou:2003ds}.
In particular, the dominant \Ztt background is estimated using these simulations of \PZ-boson production.
For diboson production, the matrix elements were calculated for up to one additional parton at NLO and up to three additional partons at LO precision.
For all samples the \textsc{NNPDF30NNLO}~\cite{PDF-NNPDF30} PDF set was used together with
the \sherpa UEPS model.
The impact of UEPS uncertainties, and other modeling uncertainties such as LO/NLO precision comparison for leading jets, on the main background from \Ztt is studied in
an alternative sample which was simulated using \MCatNLO 2.2.2~\cite{McAtNLO} at leading order interfaced to the \pythiaEightOne UEPS model. The A14 set of tuned parameters~\cite{ATL-PHYS-PUB-2014-021} was used together with the \textsc{NNPDF23LO} PDF set~\cite{PDF-NNPDF}.
For the generation of \ttbar production, the \powhegbox
v2~\cite{Powheg_Nason,Powheg_Frixione,Powheg_Alioli,Alioli:2011as}
generator with the \textsc{CT10} PDF sets in the matrix element
calculations was used.
The predicted \ttbar cross section was calculated with the \textsc{Top++}2.0 program to NNLO in perturbative QCD, including soft-gluon resummation to next-to-next-to-leading-log order~\cite{Czakon:2011xx}.
Single top-quark production of $\PW\Pqt$ was simulated
using the \powhegbox v1~\cite{Alioli:2009je,Re:2010bp} generator.
This generator uses the four-flavor scheme for the NLO matrix-element calculations together
with the fixed four-flavor PDF set \textsc{CT10F4}.
For all top-quark production processes, top-quark spin correlations
were preserved, using
MadSpin~\cite{Madspin} for the t-channel.
The parton shower, hadronization, and the
underlying event were simulated using \pythiaSix~\cite{Pythia6} with the \textsc{CTEQ6L1} PDF set and the
corresponding Perugia 2012 set of tuned parameters~\cite{Perugia2012}.
The top mass was assumed to be \SI{172.5}{\GeV}.
The EvtGen v.1.2.0 program~\cite{EvtGen} was used for the properties of \Pqb- and \Pqc-hadron decays.
For all samples, a full simulation of the ATLAS detector
response~\cite{ATLAS_Simulation} using the \textsc{Geant4}
program~\cite{GEANT4} was performed.
The effect of multiple \pp interactions in the same and neighboring bunch
crossings (pileup) was included by overlaying minimum-bias events simulated
with \pythiaEightOne using the MSTW2008LO PDF~\cite{MSTW2008LO}
and the A2~\cite{A2tune} set of tuned parameters
on each generated signal and background event.
The number of overlaid events was chosen such that the distribution of the average
number of interactions per \pp bunch crossing in the simulation matches that
observed in data.
\section{Object reconstruction}
\label{sec:reco}
Electron candidates are reconstructed from energy deposits in the
electromagnetic calorimeter associated with a charged-particle track
measured in the inner detector. The electron candidates
are required to pass the \enquote{loose} likelihood-based identification
selection of Refs.~\cite{PERF-2013-05,ATLAS-CONF-2016-024},
to have transverse momentum
$\pt > \SI{15}{\GeV}$ and to be in the fiducial volume of the
inner detector, $\abseta<2.47$. The transition region between the barrel
and endcap calorimeters ($1.37<\abseta<1.52$) is excluded.
The trigger efficiency for single electrons selected in the analysis ranges between 90\% and 95\%~\cite{TRIG-2016-01}.
Electron candidates are ignored if they share their reconstructed track with a muon candidate defined below
or if their angular distance from a jet is within $0.2 < \Delta R < 0.4$.
Muon candidates are constructed by matching an inner detector track
with a track reconstructed in the muon spectrometer~\cite{PERF-2015-10}.
The muon candidates are required to have $\pt > \SI{10}{\GeV}$
and $\abseta<2.5$ and to pass the \enquote{loose} muon identification requirements of Ref.~\cite{PERF-2015-10}.
The trigger efficiency for single muons selected in the analysis is close to 80\% (70\%) in the barrel in the 2016 (2015) dataset and 90\% in the endcaps~\cite{TRIG-2016-01}.
Muon candidates are ignored if their angular distance from a jet is $\Delta R < 0.4$ with the following exceptions:
If $\Delta R < 0.2$ or the muon track is associated with the jet, and if the jet has either
less than three tracks or less than twice the transverse momentum of the
muon candidate, the jet is removed instead.
This recovers efficiency for muons that radiate a hard bremsstrahlung photon in the calorimeter.
In the \tll and \tlhad signal regions, events are selected only if the selected electron and muon candidates satisfy their respective \enquote{medium} identification criteria.
The reconstruction and identification efficiency for muons with the \enquote{medium} identification requirement has been measured in \Zmm events~\cite{PERF-2015-10}. It is well above 98\% over the full phase space, except for $|\eta| < 0.1$ where the reconstruction efficiency is about 70\%.
The combined identification and reconstruction efficiency for \enquote{medium} electrons ranges from 80\% to 90\% in the \pT range of \SIrange{10}{80}{\GeV} as measured in \Zee events~\cite{ATLAS-CONF-2016-024}.
In addition, the electrons and muons must satisfy the \enquote{gradient} isolation criterion, which requires that there are no additional high-\pT tracks in a cone around the track and no significant energy deposits in a cone around the calorimeter clusters of the object after correcting for pileup. The size of the respective cones depends on the \pT of the light lepton.
This isolation requirement rejects about 10\% of light leptons for low \pT and less than 1\% for $\pT > \SI{60}{\GeV}$~\cite{PERF-2015-10,ATLAS-CONF-2016-024}.
Jets are reconstructed from topological clusters in the calorimeter using
the anti-$k_t$ algorithm~\cite{AntiKT,PERF-2014-07}, with a radius parameter
value $R = 0.4$, and have $\pt > \SI{20}{\GeV}$ and $\abseta < 4.9$.
To reject jets from pileup a \enquote{Jet Vertex Tagger} (JVT)~\cite{PERF-2014-03}
algorithm is used for jets with $\pt < \SI{50}{\GeV}$ and $\abseta < 2.4$. It
employs a multivariate technique that relies on jet-tracking and calorimeter-cluster-shape variables
to determine the likelihood that the jet originates from pileup.
Similarly, pileup jets in the forward region are suppressed with a \enquote{forward JVT}~\cite{PERF-2016-06}
algorithm, relying in this case only on calorimeter-cluster-shape variables, which is applied to all jets with $\pt < \SI{50}{\GeV}$ and $\abseta > 2.5$.
In the pseudorapidity range $\abseta<2.5$, $\Pqb$-jets are selected
using a multivariate algorithm~\cite{PERF-2012-04,ATL-PHYS-PUB-2016-012}.
A working point is chosen that corresponds to an efficiency of approximately 85\% for $\Pqb$-jets and rejection factors of 2.8 and 28 for $\Pqc$-jets and light-flavor jets, respectively, in simulated \ttbar events.
A jet is ignored if it is within $\Delta R = 0.2$ of an electron or hadronically decaying \Pgt candidate.
Leptonic \Pgt decays are reconstructed as electrons and muons. The reconstruction of the visible decay products of hadronic \Pgt decays (\tauhadvis)~\cite{ATLAS-CONF-2017-029} starts with a reconstructed jet
that has $\pt > \SI{10}{\GeV}$ and $|\eta| < 2.5$.
As in the case of electron reconstruction the transition region between the barrel and endcap calorimeters is excluded.
To discriminate \tauhadvis from jets initiated by light-quarks or gluons, an identification algorithm using multivariate techniques is applied to \tauhadvis candidates.
They have to pass the \enquote{loose} identification requirement of Ref.~\cite{ATLAS-CONF-2017-029}.
In addition, the \tauhadvis candidates are required to have $\pt > \SI{20}{\GeV}$, to have one or three associated tracks and an absolute electric charge of one.
Their energy is reconstructed by multivariate regression techniques using information about the associated tracks and calorimeter clusters, as well as the average number of collisions recorded.
The trigger efficiency per \tauhadvis selected in the analysis is 95\% and 85\% for 1-prong and 3-prong \Pgt{}-leptons, respectively~\cite{ATLAS-CONF-2017-061}.
The \tauhadvis candidates are ignored if they are within $\Delta R = 0.2$ of a muon or electron candidate
or if they have a high likelihood score of being an electron~\cite{ATLAS-CONF-2016-024}.
The requirement on the likelihood score corresponds to a \tauhadvis efficiency measured in \Ztt decays of 95\%~\cite{ATLAS-CONF-2017-029}.
In the \tlhad signal regions, events are selected only if the \tauhadvis candidate passes the \enquote{medium} identification requirement, corresponding to an efficiency of 55\% and 40\% for real 1-prong and 3-prong \tauhadvis, respectively~\cite{ATLAS-CONF-2017-029}. In addition,
if a 1-prong \tauhadvis candidate and an electron candidate are selected, a dedicated multivariate algorithm to reject electrons misidentified as \tauhadvis is applied to suppress \Zee events.
In the \thadhad signal regions, both selected \tauhadvis candidates have to fulfill the \enquote{tight} identification requirement, which corresponds to a selection efficiency of 45\% for real 1-prong \tauhadvis and 30\% for real 3-prong \tauhadvis~\cite{ATLAS-CONF-2017-029}.
The missing transverse momentum vector is calculated as the
negative vectorial sum of the \pt of the fully calibrated and
reconstructed physics objects~\cite{Aaboud:2018tkc}.
This procedure includes a soft
term, which is calculated from the inner detector tracks that originate
from the vertex associated with the hard-scattering process and that are not associated with any
of the reconstructed objects.
The missing transverse momentum (\met) is defined as the magnitude of this vector.
The Higgs-boson candidate is reconstructed from the visible decay products of the \Pgt{}-leptons
and from the \met, which is assumed to originate from the final-state neutrinos.
The di-\Pgt invariant mass (\mMMC) is determined using the missing-mass calculator (MMC)~\cite{MMC}.
The standard deviation of the reconstructed di-\Pgt mass is \SI{17.0}{\GeV}, \SI{15.3}{\GeV} and \SI{14.7}{\GeV} for signal events selected in the \tll, \tlhad and \thadhad channels, respectively.
The \pt of the Higgs-boson candidate (\pTH) is computed as the vector sum of the transverse momenta
of the visible decay products of the \Pgt{}-leptons and the missing transverse momentum vector.
\section{Event selection and categorization}
\label{sec:selection}
In addition to data quality criteria that ensure that the detector was functioning
properly, events are rejected if they contain reconstructed jets
associated with
energy deposits that can arise from hardware problems, beam-halo events
or cosmic-ray showers. Furthermore, events are required to have at least one reconstructed primary vertex with at least
two associated tracks with $\pT > \SI{0.5}{\GeV}$, which rejects non-collision events originating from cosmic rays or beam-halo
events.
The primary vertex is chosen as the \pp vertex candidate with the highest sum of the squared transverse momenta of all associated tracks.
The triggers and event selection for the three analysis channels are described in \cref{sec:presel}.
Selected events are categorized into exclusive signal regions, with enhanced signal-to-background
ratios. In addition, control regions are defined where a specific background is dominant, and thus a CR
facilitates the adjustment of the simulated prediction of a background contribution to match the observed data.
The signal and control regions are included in the fit described in \cref{sec:results}.
They are described in \cref{sec:categories} together with validation regions (VRs) used to validate the simulation of the dominant \Zjets background.
\subsection{Event selection}
\label{sec:presel}
Depending on the trigger, transverse momentum requirements are applied to selected electron, muon, and
\tauhadvis~candidates.
They are summarized in \cref{tab:triggers} and their per-object efficiencies are given in \cref{sec:reco}.
Due to the increasing luminosity and the different
pileup conditions, the \pT thresholds of the triggers were increased during
data-taking in 2016, which is taken into account in the \pT requirements of the event selection.
In the \tll channel, the triggers for multiple light leptons are used only if the
highest-\pT light lepton does not pass the corresponding single-light-lepton trigger
\pT requirement. This ensures that each trigger selects an exclusive set of events.
\begin{table}
\caption{Summary of the triggers used to select events for the
three analysis channels during 2015 and 2016
data-taking and the corresponding \pt requirements applied in the
analysis.
For the electron+muon trigger the first number corresponds to the electron \pT requirement, the
second to the muon \pT requirement.
For the \thadhad channel, at least one high-\pT jet in addition to the two \tauhadvis candidates is required for the 2016 dataset (see \cref{sec:presel}).}
\label{tab:triggers}
\centering
\vspace{2mm}
\begin{tabular}{llcc}
\toprule
Analysis & Trigger & \multicolumn{2}{c}{Analysis \pt requirement [\GeV]} \\
channel & & 2015 & 2016 \\
\midrule
\multirow{2}{*}{\tll \& \tlhad} & Single electron & 25 & 27 \\
& Single muon & 21 & 27 \\
\midrule
\multirow{3}{*}{\tll} & Dielectron & 15 / 15 & 18 / 18 \\
& Dimuon & 19 / 10 & 24 / 10 \\
& Electron+muon & 18 / 15 & 18 / 15 \\
\midrule
\thadhad & Di-\tauhadvis & 40 / 30 & 40 / 30 \\
\bottomrule
\end{tabular}
\end{table}
All channels require the exact number of identified \enquote{loose}
leptons, i.e.\ electrons, muons and \tauhadvis, as defined in \cref{sec:reco}, corresponding to their respective final state.
Events with additional \enquote{loose} leptons are rejected.
The two leptons are required to be of opposite charge and they have to fulfill the \pT requirements of the respective trigger shown in \cref{tab:triggers}.
The selected \tauhadvis in the \tlhad channel is required to have $\pT > \SI{30}{\GeV}$.
The event selection for the three analysis channels is summarized in \cref{t:selection}.
Only events with $\met > \SI{20}{\GeV}$ are selected to reject events without neutrinos.
In the \tll channel with two same-flavor (SF) light leptons this requirement is further tightened to suppress the large \Zll background.
For the same reason, requirements are tightened on the invariant mass of two light leptons (\mll) and a requirement is introduced on the \met calculated only from the physics objects without the soft track term (\METHPTO).
Requirements on the angular distance between the visible decay products of the two selected \Pgt{}-lepton decays (\dRtt) and their
pseudorapidity difference ($|\detatt|$) are applied in all channels to reject non-resonant background events.
Requirements are applied to the fractions of the \Pgt{}-lepton momenta carried by each visible decay product
$x_i=p^\text{vis}_i/\left(p^\text{vis}_i+p^\text{miss}_i\right)$, where $p^\text{vis}_i$ and $p^\text{miss}_i$ are the visible and missing momenta of the $i$th \Pgt lepton, ordered in descending \pT, calculated in the collinear
approximation~\cite{CollinearMass}, to suppress events with \met that is incompatible with a di-\Pgt decay.
Low transverse mass (\mT), calculated from \MET and the momentum of the selected light lepton,
is required in the \tlhad channel to reject events with leptonic \PW decays.
A requirement on the di-\Pgt mass calculated in the collinear approximation (\mcoll) of $\mcoll > m_{\PZ}-\SI{25}{\GeV}$ is introduced in the \tll channel to suppress events from \Zll and to ensure orthogonality between this measurement and the measurement of $\HWW\to\Pl\Pgn\Pl\Pgn$~\cite{HIGG-2013-13}, which has a similar final state.
All channels require at least one jet ($j_1$) with $\pTjl > \SI{40}{\GeV}$ to select Higgs bosons produced by VBF and to suppress background from \Ztt events when selecting Higgs bosons produced through \ggF.
Since 2016 the di-\tauhadvis first-level trigger requires a jet with $\pT > \SI{25}{\GeV}$ calibrated at trigger level with $|\eta| < 3.2$ in addition to the two \tauhadvis candidates.
In the \thadhad channel the jet \pt requirement is thus raised to $\pTjl > \SI{70}{\GeV}$ to achieve uniform trigger selection efficiency as a function of \pTjl. The trigger efficiency for the additional jet ranges from 95\% to 100\% for these requirements.
In the \tll and \tlhad channels, the top-quark background is suppressed by requiring that no jet with $\pT>\SI{25}{\GeV}$ is tagged as a $\Pqb$-jet.
\begin{table}
\caption{Summary of the event selection requirements for the three analysis channels
that are applied in addition to the respective lepton \pT requirements listed in \cref{tab:triggers}.
\METHPTO is an alternative \met calculated only from the physics objects without the soft-track term.
The transverse mass (\mT) is calculated from \MET and the momentum of the selected light lepton.
The visible momentum fractions $x_1$ and $x_2$ of the respective \Pgt{}-lepton and the collinear di-\Pgt mass (\mcoll) are calculated in the collinear approximation~\cite{CollinearMass}.}
\label{t:selection}
\centering
\vspace{2mm}
\small
\begin{tabular}{c|c c c}
\toprule
\multicolumn{2}{c}{\tll} & \tlhad & \thadhad \\
$\Pe\Pe$/$\Pgm\Pgm$ & $\Pe\Pgm$ & & \\
\midrule
\multicolumn{2}{c}{$N^\text{loose}_{\Pe/\Pgm}=2$, $N^\text{loose}_{\tauhadvis}=0$} & $N^\text{loose}_{\Pe/\Pgm}=1$, $N^\text{loose}_{\tauhadvis}=1$ & $N^\text{loose}_{\Pe/\Pgm}=0$, $N^\text{loose}_{\tauhadvis}=2$ \\
\multicolumn{2}{c}{$\Pe$/$\Pgm$ : Medium, gradient iso.} & $\Pe$/$\Pgm$ : Medium, gradient iso. & \\
\multicolumn{2}{c}{} & \tauhadvis : Medium & \tauhadvis : Tight \\
\multicolumn{2}{c}{Opposite charge} & Opposite charge & Opposite charge \\
\multicolumn{2}{c}{$\mcoll > m_{\PZ}-\SI{25}{\GeV}$} & $\mT < \SI{70}{\GeV}$ & \\
$30<\mll<\SI{75}{\GeV}$ & $30<\mll<\SI{100}{\GeV}$ & & \\
$\met > \SI{55}{\GeV}$ & $\met > \SI{20}{\GeV}$ & $\met > \SI{20}{\GeV}$ & $\met > \SI{20}{\GeV}$ \\
$\METHPTO > \SI{55}{\GeV}$ & & & \\
\multicolumn{2}{c}{$\dRll < 2.0$} & $\dRlt < 2.5$ & $0.8 < \dRtt < 2.5$ \\
\multicolumn{2}{c}{$|\detall| < 1.5$} & $|\detalt| < 1.5$ & $|\detatt| < 1.5$ \\
\multicolumn{2}{c}{$0.1<x_1<1.0$} & $0.1<x_1<1.4$ & $0.1<x_1<1.4$ \\
\multicolumn{2}{c}{$0.1<x_2<1.0$} & $0.1<x_2<1.2$ & $0.1<x_2<1.4$ \\
\multicolumn{2}{c}{$\pTjl > \SI{40}{\GeV}$} & $\pTjl > \SI{40}{\GeV}$ & $\pTjl > \SI{70}{\GeV}, |\eta_{j_1}| < 3.2$ \\
\multicolumn{2}{c}{$N_\text{\Pqb-jets} = 0$} & $N_\text{\Pqb-jets} = 0$ & \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Signal, control and validation regions}
\label{sec:categories}
To exploit signal-sensitive event topologies, a \enquote{\VBF} and a \enquote{boosted} analysis category are defined without any overlap in phase space.
The \VBF category targets events with a Higgs boson produced by \VBF and is characterized by the presence of a second high-\pT jet ($\pTjsl > \SI{30}{\GeV}$).
In addition, the two jets are required to be in opposite hemispheres of the detector with a large pseudorapidity separation of $|\detajj| > 3$ and their invariant mass (\mjj) is required to be larger than \SI{400}{\GeV}.
The selected leptons are required to have $\eta$-values that lie between those of the two jets (\enquote{central leptons}).
Although this category is dominated by \VBF production, it also includes significant contributions from \ggF production, amounting to up to 30\% of the total expected Higgs-boson signal.
The boosted category targets events with Higgs bosons produced through \ggF with additional recoiling jets, which is motivated by the harder \pT-spectrum of the \Htt signal compared to the dominant background from \Ztt.
It contains all events with $\pTH > \SI{100}{\GeV}$ that do not pass the \VBF selection.
In addition to events from \ggF, the boosted categories contain sizable contributions from \VBF and \VH production of 10--20\% of the expected signal.
Events that pass the event selection, detailed in \cref{t:selection}, but do not fall into the \VBF or boosted categories, are not used in the analysis.
Using \pTH, \dRtt and \mjj, the \VBF and boosted categories, referred to as \enquote{inclusive} categories, are split further into 13 exclusive signal regions with different signal-to-background ratios to improve the sensitivity.
\Cref{t:categorydef} summarizes the analysis categories and signal region definitions. \Cref{fig:bkgcomp} illustrates the expected signal and background composition in the signal and control regions of all analysis channels.
\Cref{fig:SRmodelling} compares for each analysis channel the observed distributions with predictions, as resulting from the fit described in \cref{sec:results}, for \pTH in the boosted inclusive categories, and for \mjj in the \VBF inclusive categories.
The observed data agree within the given uncertainties with the background expectation described in \cref{sec:background} for all distributions.
\begin{table}
\begin{center}
\caption{Definition of the \VBF and boosted analysis categories and of their respective signal regions (SRs).
The selection criteria, which are applied in addition to those described in \cref{t:selection}, are listed for each channel.
The \VBF high-\pTH SR is only defined for the \thadhad channel, resulting in a total of seven \VBF SRs and six boosted SRs.
All SRs are exclusive and their yields add up to those of the corresponding \VBF and boosted inclusive regions.}
\label{t:categorydef}
\centering
\vspace{2mm}
\small
\begin{tabular}{cc|c|c|c|c}
\toprule
\multicolumn{2}{c|}{Signal Region} & Inclusive & \tll & \tlhad & \thadhad \\
\midrule
\parbox[t]{12pt}{\multirow{6}{*}{\rotatebox[origin=c]{90}{\VBF\quad}}} & \multirow{2}{*}{High-\pTH} &
\multirow{6}{*}{\begin{tabular}[c]{c}\rule{0pt}{\normalbaselineskip+9pt}$\pTjsl>\SI{30}{\GeV}$\\
$|\detajj|>3$ \\ $\mjj>\SI{400}{\GeV}$ \\ $\eta_{j_1}\cdot\eta_{j_2}<0$ \\ Central leptons \end{tabular}}
& \multicolumn{2}{c|}{\multirow{2}{*}{---}} & $\pTH>\SI{140}{\GeV}$ \\
& & & \multicolumn{2}{c|}{} & $\dRtt<1.5$ \\[2pt]
\cline{2-2} \cline{4-6}
\rule{0pt}{\normalbaselineskip} & \multirow{2}{*}{Tight} & & $\mjj>\SI{800}{\GeV}$ & $\mjj>\SI{500}{\GeV}$ & Not \VBF high-\pTH \\[2pt]
& & & & $\pTH>\SI{100}{\GeV}$ & $\mjj>(1550 - 250\cdot |\detajj|)\,\GeV$ \\[2pt]
\cline{2-2} \cline{4-6}
\rule{0pt}{\normalbaselineskip} & \multirow{2}{*}{Loose} & & \multicolumn{2}{c|}{\multirow{2}{*}{Not \VBF tight}} & Not \VBF high-\pTH \\
& & & \multicolumn{2}{c|}{} & and not \VBF tight \\
\midrule
\parbox[t]{12pt}{\multirow{3}{*}{\rotatebox[origin=c]{90}{Boosted\enspace}}} & \multirow{2}{*}{High-\pTH} & \multirow{3}{*}{\begin{tabular}[c]{c}\rule{0pt}{\normalbaselineskip+1pt}Not \VBF\\$\pTH>\SI{100}{\GeV}$\end{tabular}} & \multicolumn{3}{c}{$\pTH>\SI{140}{\GeV}$} \\
& & & \multicolumn{3}{c}{$\dRtt<1.5$} \\[2pt]
\cline{2-2} \cline{4-6}
\rule{0pt}{\normalbaselineskip} & Low-\pTH & & \multicolumn{3}{c}{Not boosted high-\pTH} \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{fig1}
\caption{Expected signal and background composition in 6 control regions (CRs) and the 13 signal regions (SRs) used in the analysis.}
\label{fig:bkgcomp}
\end{figure}
\begin{figure}[htbp]
\centering
\subfloat[]{\includegraphics[width=0.33\textwidth]{fig2a} }
\subfloat[]{\includegraphics[height=0.33\textwidth]{fig2b} }
\subfloat[]{\includegraphics[height=0.33\textwidth]{fig2c} } \\
\subfloat[]{\includegraphics[width=0.33\textwidth]{fig2d} }
\subfloat[]{\includegraphics[height=0.33\textwidth]{fig2e} }
\subfloat[]{\includegraphics[height=0.33\textwidth]{fig2f} }
\caption{Comparisons between data and predictions as computed by the fit of (top) the \pt of the Higgs-boson candidate (\pTH) in the boosted inclusive category and (bottom) the invariant mass of the two highest-\pT jets (\mjj) in the VBF inclusive category for (left) the \tll channel, (center) the \tlhad channel and (right) the \thadhad channel.
The ratios of the data to the background model are shown in the lower panels.
The observed Higgs-boson signal ($\mu = 1.09$) is shown with the solid red line.
Entries with values that would exceed the $x$-axis range are shown in the last bin of each distribution.
The size of the combined statistical, experimental and theoretical uncertainties in the background is indicated by the hatched bands.}
\label{fig:SRmodelling}
\end{figure}
Six control regions are defined to constrain the normalization of the dominant backgrounds in regions of phase space where their purity is high.
Their definitions are summarized in \cref{tab:cr}.
Two \Zll CRs, which are both more than 90\% pure in \Zll events, are defined by applying the same selection as for the
SF \tll \VBF and boosted inclusive regions, respectively, but with the \mll
requirement modified to $80<\mll<\SI{100}{\GeV}$.
The top-quark background is characterized by the presence of \Pqb-jets.
Four separate top CRs are defined by inverting the \Pqb-jet veto in the inclusive \VBF and boosted categories for each of the \tll and \tlhad channels.
The top CRs in the \tll channel are about 80\% pure in top-quark events.
For the top CRs in the \tlhad channel, the requirement of $\mT < \SI{70}{\GeV}$ is replaced by $\mT>\SI{40}{\GeV}$ to further enhance the purity to about 70\% in the \VBF top CR and about 60\% in the boosted top CR.
No such control regions are defined for the \thadhad channel since the top and \Zll backgrounds are negligible in this case.
One validation region is defined for each signal region (\enquote{\Ztt VRs}) to validate
the event yields and kinematic distributions of simulated \Ztt events.
The \Ztt VRs are composed of \Zll events with kinematics similar to the \Ztt
background in the respective signal regions.
This is achieved by starting with an event selection that is
based on the SF \tll channel preselection with
the following differences that account for the
selection of light leptons instead of decay products from \Pgt{}-leptons:
The \mcoll, \met and \METHPTO requirements are dropped and the \mll requirement is inverted to
$\mll>\SI{80}{\GeV}$.
The other requirements on \Pgt{}-lepton decays are replaced with requirements on the two light leptons.
In particular, the requirements on \pTH are substituted by the \pt of the \PZ boson computed from the \pT of the light leptons (\pTZ).
Requirements on jets are unchanged since they define the shape of most kinematic distributions for \PZ-boson production similarly in the SRs and the \Ztt VRs.
More than 99\% of the selected events are from \Zll in all \Ztt VRs.
\begin{table}
\caption{Definitions of the six control regions (CRs) used to constrain the \Zll and top backgrounds to the event yield in data in the \tll and \tlhad channels.
\enquote{SF} denotes a selection of same-flavor light leptons.}
\label{tab:cr}
\centering
\vspace{2mm}
\begin{tabular}{ll}
\toprule
Region & Selection \\
\midrule
\tll \VBF \Zll CR & \tll VBF incl. selection, $80<\mll<\SI{100}{\GeV}$, SF \\
\tll boosted \Zll CR & \tll boosted incl. selection, $80<\mll<\SI{100}{\GeV}$, SF \\
\tll \VBF top CR & \tll VBF incl. selection, inverted $\Pqb$-jet veto \\
\tll boosted top CR & \tll boosted incl. selection, inverted $\Pqb$-jet veto \\
\tlhad \VBF top CR & \tlhad VBF incl. selection, inverted $\Pqb$-jet veto, $\mT > \SI{40}{\GeV}$ \\
\tlhad boosted top CR & \tlhad boosted incl. selection, inverted $\Pqb$-jet veto, $\mT > \SI{40}{\GeV}$ \\
\bottomrule
\end{tabular}
\end{table}
\section{Background estimation}
\label{sec:background}
The final-state topologies of the three analysis channels have different background
compositions, which necessitates different strategies for the background estimation. In each SR,
the expected number of background events and the associated kinematic distributions are
derived from a mixture of data-driven methods and simulation.
Background contributions with \tauhadvis, with prompt light leptons and with light leptons
from \Pgt{}-lepton decays are estimated from simulation.
If their contribution is significant, their normalization is constrained by the observed event yields in CRs.
For smaller contributions of this type, their normalization is entirely taken from the
theoretical cross sections with the precision in QCD listed in \cref{tab:MCSamples}.
This includes di-boson processes and a small contribution from EW production of \PW/\PZ bosons from \VBF.
Contributions from light- and heavy-flavor jets that are misidentified as
prompt, light leptons or \tauhadvis are estimated using data-driven methods.
They are labeled as \enquote{fake-\Pl{}} and \enquote{fake-\tauhadvis{}} backgrounds, respectively,
and collectively as \enquote{misidentified \Pgt{}}, throughout this paper.
The contamination from \HWW decays is treated
as a background in the \tll channel, while it is negligible in other channels.
For the background sources that have
their normalization constrained using data, \cref{tab:sf} shows the
normalization factors and their uncertainties obtained from the fit (see \cref{sec:results}).
For simulated backgrounds, the factors compare the background
normalizations with
values determined from their theoretical cross sections. The
normalization factor for the data-driven fake-\tauhadvis background
scales the event yield of the template of events that fail the
opposite-charge requirement (see \cref{sec:bkgFakes}).
The \Ztt normalization is constrained by data in the \mMMC
distributions of the signal regions.
Systematic uncertainties are the dominant contribution to the normalization factor uncertainties.
\begin{table}
\caption{Normalization factors for backgrounds that have their
normalization constrained using data
in the fit, including all statistical and systematic uncertainties described in \cref{sec:systematics},
but without uncertainties in total simulated cross sections extrapolated to the selected phase space.
Systematic uncertainties are the dominant contribution to the normalization factor uncertainties.
Also shown are the
analysis channels to which the normalization factors are applied.
}
\label{tab:sf}
\centering
\vspace{2mm}
\begin{tabular}{llll}
\toprule
Background & Channel & \multicolumn{2}{c}{Normalization factors} \\
& & \multicolumn{1}{c}{\VBF} & \multicolumn{1}{c}{Boosted} \\
\midrule
\Zll (CR) & \tll & $0.88^{+0.34}_{-0.30}$ & $1.27^{+0.30}_{-0.25}$ \\
Top (CR) & \tll & $1.19\pm0.09$ & $1.07\pm0.05$ \\
Top (CR) & \tlhad & $1.53^{+0.30}_{-0.27}$ & $1.13\pm0.07$ \\
Fake-\tauhadvis (data-driven) & \thadhad & \multicolumn{2}{c}{$1.12\pm0.12$} \\
\Ztt (fit in each SR) & \tll,\tlhad,\thadhad & $1.04^{+0.10}_{-0.09}$ & $1.11\pm0.05$ \\
\bottomrule
\end{tabular}
\end{table}
\subsection{\Ztt background validation}
\label{sec:bkgZtautau}
The Drell--Yan process $\pp \to \PZ/\Pggx \to \Pgt\Pgt$ is a dominant irreducible background in
all analysis categories and contributes between 50\% and 90\% of the total background depending
on the signal region. The separation between the Drell--Yan and the \Htt signal processes
is limited by the $\mMMC$ resolution.
The modeling of this important background is validated using \Ztt VRs that consist of \Zll events.
In \cref{fig:bkg-Zvalid}, the observed distributions of several variables
are compared with simulation normalized to the event yield in data.
The selected observables correspond to either variables correlated with \mMMC (\pTll and \pTlsl), or to major variables used for categorization (\pTZ, $\Delta R_{\Pl\Pl}$, \detajj and \mjj), or to variables to which different requirements are applied in each decay channel (\pTjl).
Generally, the \sherpa simulation describes the shape of data distributions within the experimental and theoretical uncertainties (see \cref{sec:systematics}),
with the exception of a slight trend in the ratio of data to simulation as a function of \detajj and \mjj shown in \cref{fig:bkg-Zvalid}.
These trends have no impact on the modeling of \mMMC.
Reweighting the simulation with the observed \mjj distribution, which is an important variable for \VBF categorization, has a negligible impact on the measurement.
In the fit, the normalization of the \Ztt background is correlated across the decay channels and
constrained by data in the \mMMC distributions of the signal regions associated with the boosted
and \VBF categories, independently.
As shown in \cref{tab:sf}, it is constrained to $\pm$5\% in the boosted category
and to $\pm$9\% in the \VBF category.
The relative acceptance of events among the signal regions within a category is validated by applying the corresponding event-selection criteria to the \Ztt VRs.
The expected relative acceptance from simulation agrees with data within uncertainties for all regions.
\Cref{fig:CBA_m-tautau_channel,fig:CBA_m-tautau} show the good modeling of the \Ztt \mMMC distribution in all signal regions.
Additional uncertainties in the relative acceptances
and in the shape of the \mMMC distributions in the signal regions
are evaluated from theoretical and experimental uncertainties
described in \cref{sec:systematics}.
\begin{figure}[htbp]
\centering
\subfloat[]{\includegraphics[width=0.33\textwidth]{fig3a} }
\subfloat[]{\includegraphics[width=0.33\textwidth]{fig3b} }
\subfloat[]{\includegraphics[width=0.33\textwidth]{fig3c} } \\
\subfloat[]{\includegraphics[width=0.33\textwidth]{fig3d} }
\subfloat[]{\includegraphics[width=0.33\textwidth]{fig3e} }
\subfloat[]{\includegraphics[width=0.33\textwidth]{fig3f} } \\
\subfloat[]{\includegraphics[width=0.33\textwidth]{fig3g} }
\subfloat[]{\includegraphics[width=0.33\textwidth]{fig3h} }
\subfloat[]{\includegraphics[width=0.33\textwidth]{fig3i} }
\caption{Observed and expected distributions in the \Ztt validation
regions (VRs) corresponding to (a)--(d) the \tlhad \VBF inclusive
category and (e)--(i) the \tlhad boosted inclusive category. Shown
are, in the respective region: (a) the pseudorapidity separation ($|\detajj|$)
and (b) the invariant mass (\mjj) of the two highest-\pT
jets; (c) and (e) the \pT of the di-lepton system (\pTZ); (d) and
(g) the \pT of the highest-\pT jet (\pTjl); (f) the angular
distance between the light leptons ($\Delta R_{\Pl\Pl}$); (h) the \pT
of the highest-\pT light lepton (\pTll); and (i) the \pT of the
second-highest-\pT light lepton (\pTlsl).
The predictions in these validation regions are not computed by the fit, but
are simply normalized to the event yield in data.
The size of the combined statistical, experimental and theoretical uncertainties is indicated by the hatched bands.
The ratios of the data to the background model are shown in the lower panels together with the theoretical uncertainties in the \sherpa simulation of \Zll, which are indicated by the blue lines.}
\label{fig:bkg-Zvalid}
\end{figure}
\subsection{\Zll background}
\label{sec:bkgZll}
Decays of \PZ bosons into light leptons are a significant background for
the \tll and \tlhad channels, where mismeasured \met can bias the reconstructed \mMMC
of light-lepton pairs towards values similar to those expected for the signal.
The observed event yields in the \Zll CRs constrain the normalization of simulated \Zll
events in the \tll channel to $\pm$40\% in the \VBF category
and to $\pm$25\% in the boosted category, as shown in \cref{tab:sf}.
The good modeling of the \mMMC distribution in the \tll \VBF \Zll CR is shown in \cref{fig:CRmodelling}(a).
In other channels, the contribution from \Zll events
is normalized
to its theoretical cross section.
In the \tlhad channel, \Zll background contributes primarily through \Zee decays where an electron is misidentified as a \tauhadvis~candidate.
Due to the dedicated electron veto algorithm applied to selected 1-prong \tauhadvis candidates (see \cref{sec:presel}), this background is small.
This and other backgrounds from light leptons misidentified as \tauhadvis
in this channel are estimated from simulation, with the
probability for electrons misidentified as \tauhadvis~candidates scaled to match that observed in data~\cite{ATLAS-CONF-2017-029}.
\begin{figure}[tbp]
\centering
\subfloat[]{\includegraphics[width=0.33\textwidth]{fig4a} }
\subfloat[]{\includegraphics[width=0.33\textwidth]{fig4b} }
\subfloat[]{\includegraphics[width=0.33\textwidth]{fig4c} }
\caption{For the control regions (CRs) defined in \cref{sec:selection}, comparisons between data and predictions as computed by the fit for the reconstructed di-\Pgt invariant mass ($\mMMC$). Shown are (a) the \tll \VBF \Zll control region (CR), (b) the \tll boosted top CR and (c) the \tlhad \VBF top CR.
Entries with values that would exceed the $x$-axis range are shown in the last bin of each distribution.
The size of the combined statistical, experimental and theoretical uncertainties in the background is indicated by the hatched bands.
The ratios of the data to the background model are shown in the lower panels.}
\label{fig:CRmodelling}
\end{figure}
\subsection{Top-quark background}
\label{sec:bkgTop}
The production of \ttbar pairs or single top quarks is a significant background (\enquote{top background})
for the \tll and \tlhad channels, due to the production of prompt light leptons with
associated \met in the top-quark decay chain $\Pqt \to \PW\Pqb,\, \PW \to \Pl\nu,\Pgt\nu$.
Events where a selected \Pgt{}-lepton decay product is misidentified, are estimated using data-driven methods that are
discussed in \cref{sec:bkgFakes}. The remaining top background is estimated from simulation.
In the \tll and \tlhad channels the normalization of simulated top background is additionally constrained by the
absolute event yields in their respective top CRs to $\pm$30\% in the \tlhad \VBF top CR
and less than $\pm$10\% in the other top CRs, as shown in \cref{tab:sf}.
Figures~\ref{fig:CRmodelling}(b) and \ref{fig:CRmodelling}(c) show \mMMC distributions in the \tll boosted top CR and the \tlhad \VBF top CR, respectively.
\subsection{Backgrounds from misidentified \Pgt}
\label{sec:bkgFakes}
Apart from the small contribution from light leptons misidentified as
\tauhadvis described in \cref{sec:bkgZll}, hadronic jets can be misidentified
as \tauhadvis, electrons and muons.
These sources of background contribute up to half of the total background, depending
on the signal region, and are estimated with
data-driven techniques. Since the background sources depend on the event
topology, specific methods are applied to each individual channel.
In the \tll channel, the
main sources of the fake-\Pl background are multijets, \PW bosons in association with jets, and semileptonically decaying \ttbar events.
All these background sources are treated together.
Fake-\Pl regions are defined in data by requiring that the light lepton with the second-highest \pt does not satisfy
the \enquote{gradient} isolation criterion.
This is referred to as \enquote{inverted} isolation. In addition, if the light lepton is an electron, its identification criteria
are relaxed to \enquote{loose}.
Fake-\Pl templates are
created from these samples by subtracting top and \Zll
backgrounds that produce real light leptons, estimated from simulation. The
normalization of each template is then scaled by a factor that
corrects for the inverted-isolation requirement.
These correction factors are computed for each combination of lepton flavor
from events that pass the \tll selection but have same-charge light leptons,
subtracting simulated top and \Zll backgrounds.
Fake-\Pl background in the top-quark CRs is estimated following the same procedure.
Systematic uncertainties in the shape and
normalization of the fake-\Pl background in the \tll channel
depend on the \pT of the second-highest-\pt lepton and are estimated as follows.
A closure test of the background estimate is
performed using events where the leptons are required to have the same charge
and yields an uncertainty ranging between 20\% and 65\%. An uncertainty in the
heavy-flavor content is estimated by using isolation correction factors
that are computed from samples selected with inverted $\Pqb$-jet requirements.
This uncertainty is as large as 50\%.
Minor contributions come from the uncertainty in the fractional
composition of the fake-\Pl background in top-quark decays, multijet
events and \PW-boson production.
In the \tlhad channel, a \enquote{fake-factor} method is used to derive
estimates for fake-\tauhadvis events,
composed mainly of multijet events and
\PW-boson production in association with jets.
A fake-factor is defined as the ratio of the number of events where the highest-\pt jet is identified
as a \enquote{medium} \tauhadvis~candidate to the number of events with a highest-\pt jet that passes
a very loose \tauhadvis~identification but fails the \enquote{medium} one.
Fake-factors
depend on the \pT and track multiplicity of the \tauhadvis~candidate and
on the type of parton initiating the jet.
Therefore, they are computed depending on the \pt and the track multiplicity,
in both quark-jet-dominated \enquote{\PW-enhanced} and gluon-jet-dominated \enquote{multijet-enhanced} regions.
The \PW-enhanced regions are defined by inverting the $\mT<\SI{70}{\GeV}$ requirement and the multijet-enhanced regions are defined
by inverting the light-lepton isolation, relative to the inclusive boosted and \VBF selections.
Backgrounds from \PZ-boson production with associated jets and semileptonically decaying \ttbar have fake-factors similar to those
found in backgrounds from \PW bosons, and their contributions are negligible.
The fake-factors are in the range 0.15--0.25 for 1-prong and 0.01--0.04 for 3-prong \tauhadvis.
To obtain the fake-\tauhadvis background estimate for the signal regions, these fake-factors are first weighted by the multijets-to-\PW fraction.
The weighted fake-factors are then applied to events in regions
defined by the selections of the corresponding signal regions, except that the highest-\pt \tauhadvis
candidate passes a very loose \tauhadvis identification and fails the \enquote{medium} one (\enquote{anti-ID} regions).
The relative multijet contribution in each anti-ID region is estimated from the yield of events that fail the light-lepton isolation requirement,
multiplied by a factor that corrects for this requirement.
The multijet contribution varies by more than 50\% and
depends on the lepton \pT and on the $\Delta\phi$ between \tauhadvis and \met.
The good agreement between data and background estimates is shown in \cref{fig:bkg-fake}(a)
for the main discriminant of the analysis, \mMMC, in the boosted \PW-enhanced region.
The dominant contribution to the uncertainties in the fake-\tauhadvis background
in the \tlhad channel originates from the
statistical uncertainty in the individual fake-factors of up to
10\% in the boosted signal regions and up to 35\% in the VBF
signal regions. Minor contributions originate from the statistical
uncertainty in the anti-ID regions and uncertainties in the fractional
size of the multijet contribution to the fake-\tauhadvis background.
\begin{figure}[tbp]
\centering
\subfloat[]{\includegraphics[height=0.25\textheight]{fig5a} }
\subfloat[]{\includegraphics[height=0.25\textheight]{fig5b} }
\caption{Observed distributions and predictions computed by the fit for (a) \mMMC in the \PW-enhanced region of the \tlhad boosted inclusive category,
and (b) $\Delta\eta$ between the two \tauhadvis, for events in the boosted low-\pTH signal region (SR) of the \thadhad channel.
Entries with values that would exceed the $x$-axis range are shown in the last bin of each distribution.
The size of the combined statistical, experimental and theoretical uncertainties in the background is indicated by the hatched bands.
The ratios of the data to the background model are shown in the lower panels.}
\label{fig:bkg-fake}
\end{figure}
In the \thadhad channel, the multijet background is modeled using a template extracted
from data that pass the signal-region selections, but where the \tauhadvis~candidates
are allowed to have two tracks and required to fail the opposite-charge requirement
(nOC region).
The contribution of events with true \Pgt{}-leptons from other SM processes is subtracted
from this template using simulation. The template is then reweighted using scale factors
dependent on the difference in $\phi$ between the \tauhadvis candidates (\dphitt).
These scale factors are derived
by comparing the template from an nOC selection with a region
obtained by requiring the \tauhadvis pair to have opposite charge and the second-highest-\pt \tauhadvis
to fail the \enquote{tight} but pass the \enquote{medium} identification requirements.
As the yield of events that pass these identification requirements is small, the scale factors are derived from events
that pass the \thadhad selection with looser $\detatt$ and $\dRtt$ requirements to gain statistical power.
The normalization of the multijet background is constrained in the fit by data in the \mMMC distribution
in the signal regions.
For this, a normalization factor is defined and it is correlated across all \thadhad signal regions.
\Cref{fig:bkg-fake}(b) shows good agreement between data and background
predictions in the distribution of $\Delta\eta$ between the
two \tauhadvis, which has a quite different shape for the multijets than for the \Ztt process.
In this figure, events are selected that pass the \thadhad boosted low-\pTH selection.
Contributions from other backgrounds, such as \PW with associated jets, range from 2\% to 5\% in the \thadhad SRs.
The event yield of the multijet background in the \thadhad channel is
constrained by data to $\pm$15\% in the signal regions as shown in \cref{tab:sf}.
The dominant contribution to the uncertainties that affect the \mMMC shape
originates from the statistical uncertainties in the
\dphitt scale factors
and amounts to 8\%. The systematic
uncertainty in these scale factors is
estimated by comparing them with scale factors computed from the nOC
region and a CR defined by requiring opposite-charge \tauhadvis to pass
\enquote{loose} but not \enquote{medium} identification. Minor contributions arise from
the uncertainty in the extrapolation from the nOC requirement and the uncertainty
from the subtraction of simulated backgrounds. The combination of these uncertainties
leads to a total variation in the \mMMC template shape by at most 10\% between bins.
\section{Systematic uncertainties}
\label{sec:systematics}
The expected signal and background yields in the various signal and control regions
as well as the shape of the \mMMC distributions in the signal regions are affected by systematic uncertainties.
These are discussed below, grouped into three categories: theoretical uncertainties in signal, theoretical uncertainties in background, and experimental uncertainties.
The uncertainties in backgrounds from misidentified \Pgt{}-leptons, which are estimated using data-driven techniques, are discussed in \cref{sec:bkgFakes}.
The effects of all uncertainties are included in the fit model described in \cref{sec:results}.
\subsection{Theoretical uncertainties in signal}
\label{sec:systematics_theory_signal}
The procedures to estimate the uncertainty in the Higgs production cross sections
follow the recommendations by the LHC Higgs Cross Section Working
Group~\cite{Dittmaier:2011ti}.
They are briefly summarized below.
Uncertainties are evaluated separately for their impact on the total cross section,
their impact on the acceptance in different SRs, and on the shape of the \mMMC distribution in each SR.
The cross section of \ggF production in association with an exclusive number of additional jets has large uncertainties from higher-order QCD corrections~\cite{deFlorian:2016spz}.
In this analysis, the boosted and \VBF categories almost exclusively select \ggF events with one and two additional jets, respectively.
To take this effect into account, nine uncertainty sources are included.
Four sources account for uncertainties in the jet multiplicities due to missing higher-order corrections: Two sources account for yield uncertainties and two sources account for migration uncertainties of zero to one jets and one to at least two jets in the event, respectively, using the STWZ~\cite{Stewart:2013faa} and BLPTW~\cite{Stewart:2013faa,Liu:2013hba,Boughezal:2013oha} predictions as an input.
Three uncertainty sources parameterize modeling uncertainties in the Higgs-boson \pT, two of which encapsulate the migration uncertainty between the intermediate and high-\pT regions of events with at least one jet, and one which encapsulates the treatment of the top-quark mass in the loop corrections, where the difference between the LO and NLO predictions is taken as an uncertainty due to missing higher-order corrections.
Two sources account for the acceptance uncertainties of \ggF production in the \VBF phase space from selecting exactly two and at least three jets, respectively. Their size is estimated using an extension of the Stewart--Tackmann method~\cite{Stewart:2011cf,Gangal:2013nxa}.
The resulting acceptance uncertainties from these nine sources range from 1\% to 10\%, with the dominant uncertainties due to the modeling of the Higgs \pT distribution in all SRs, to the scale variations in the boosted SRs, and to the acceptance uncertainties in the \VBF signal regions.
For \VBF and \VH production cross sections, the
uncertainties due to missing higher-order QCD corrections are
estimated by varying the factorization and renormalization scales by
factors of two around the nominal scale.
The resulting uncertainties in the total cross section are below 1\% for \VBF and $\PW\PH$
production and below 5\% for $\PZ\PH$ production.
The uncertainties in the acceptance in the different SRs are about
1\% for \VBF production in all categories.
For \VH production the relative acceptance uncertainty ranges between $-10\%$ and $+20\%$
in \VBF SRs. It is below 10\% in boosted SRs.
Uncertainties related to the simulation of the underlying event, hadronization and parton shower for all signal samples are estimated by comparing the acceptance when using the default UEPS model from \pythiaEightTwo with an alternative UEPS model from \herwig.
The resulting acceptance uncertainties range from 2\% to 26\% for \ggF production and from 2\% to 18\% for \VBF production, depending on the signal region.
The PDF uncertainties are estimated using 30 eigenvector variations and two \alphas variations
that are evaluated independently relative to the default PDF set \textsc{PDF4LHC15}~\cite{Butterworth:2015oua}.
The total uncertainty due to these variations is 5\% or less depending on the SR and the Higgs production mode.
Finally, an uncertainty in the \Htt decay branching ratio
of 1\%~\cite{Dittmaier:2011ti} affects the signal rates.
All sources of theoretical uncertainties in the signal expectation are correlated across SRs.
\subsection{Theoretical uncertainties in backgrounds}
\label{sec:systematics_theory_background}
Uncertainties from missing higher-order corrections, the PDF parameterization and the UEPS modelling are also considered for the dominant \Ztt background.
The UEPS modelling uncertainties are estimated by comparing with an alternative \Ztt sample as described in \cref{sec:samples}.
Since its overall normalization is constrained separately in the \VBF and boosted SRs, variations due to these uncertainties are considered in the event migration within an analysis channel, in the \mMMC shape and in the relative change in acceptance between the three analysis channels.
These variations are treated as uncorrelated between the \VBF and boosted SRs.
In addition, the first two types of variations are treated as uncorrelated between the three analysis channels.
This treatment accounts for the differences in the corresponding event selections.
The largest uncertainties are due to the CKKW matching~\cite{ckkw} and are evaluated as a function of the number of true jets and the \PZ-boson \pT.
They vary between 1\% and 5\% depending on the SR.
The uncertainty in the measured cross section for electroweak \PZ production with two associated jets~\cite{STDM-2016-09} is found to be small compared to the other uncertainties in \PZ-boson production.
The top-quark background normalization in the \tll and \tlhad channels as well as the \Zll background normalization in the \tll channel are constrained by data in dedicated CRs.
All other simulated background contributions are normalized to their Monte~Carlo prediction.
For all simulated background contributions, other than \Ztt, no theoretical uncertainties are considered, as their impact is small compared to the uncertainties in the dominant backgrounds from \Ztt and misidentified leptons.
\subsection{Experimental uncertainties}
Experimental systematic uncertainties result from uncertainties in
efficiencies for triggering, object reconstruction and identification,
as well as from uncertainties in the energy scale and resolution of
jets, \tauhadvis, light leptons and \met.
These uncertainties affect both the event
yields and the shape of the \mMMC.
The dominant experimental
uncertainties in the final result are related to jet and \tauhadvis
reconstruction.
The impact of the electron- and muon-related
uncertainties~\cite{PERF-2015-10,ATL-PHYS-PUB-2015-041,TRIG-2016-01}
on the measurement are small.
Uncertainties in the integrated
luminosity affect the number of predicted signal and background events,
with the exception of processes that are normalized to data, see \cref{tab:sf}.
This uncertainty is 2.1\% for the combined 2015+2016
dataset. It is derived using a methodology similar to that detailed in
Ref.~\cite{DAPR-2013-01}, and using the LUCID-2 detector for the baseline
luminosity measurements~\cite{Avoni:2018iuv}, from a calibration of the luminosity
scale using $x$-$y$ beam-separation scans.
The uncertainties of the
\tauhadvis~identification efficiency are in the range of 2--4.5\%
for the reconstruction efficiency~\cite{ATL-PHYS-PUB-2015-045},
3--14\% for the trigger efficiency (depending on the \tauhadvis
\pt), 5--6\% for the identification efficiency and 3--14\%
for the rate at which an electron is misidentified as
\tauhadvis (depending on the \tauhadvis
$\eta$)~\cite{ATLAS-CONF-2017-029}.
The uncertainties of the $\Pqb$-tagging efficiencies are measured in
dedicated calibration analyses~\cite{PERF-2012-04} and are decomposed
into uncorrelated components.
Uncertainties in the efficiency to pass the JVT and forward JVT requirements are also considered~\cite{ATLAS-CONF-2014-018,PERF-2016-06}.
Simulated events are corrected for
differences in these efficiencies between data and simulation and the
associated uncertainties are propagated through the analysis.
The uncertainties of the \tauhadvis energy
scale~\cite{ATLAS-CONF-2017-029} are determined
by fitting the \PZ-boson mass in \Ztt events, reconstructed using the visible \Pgt decay products.
The precision amounts to 2--3\%, which is dominated by the
uncertainty of background modeling.
Additional uncertainties based on the modeling of the calorimeter response to single particles are added for \tauhadvis with $\pt>\SI{50}{\GeV}$~\cite{PERF-2013-06}.
The jet energy scale and its uncertainty are derived by combining
information from test-beam data, LHC collision data and
simulation~\cite{PERF-2016-04}. The uncertainties from these
measurements are factorized into eight
principal components.
Additional uncertainties that are considered are related to jet flavor, pileup
corrections, $\eta$-dependence, and high-\pT jets, yielding a total of
20 independent sources. The uncertainties amount to 1--6\% per jet,
depending on the jet \pT. The jet energy resolution
uncertainties~\cite{ATL-PHYS-PUB-2015-015} are divided into 11
independent components and amount to 1--6\%.
Since systematic uncertainties of the energy scales of all objects
affect the reconstructed \met, this is recalculated after each
variation is applied. The scale uncertainty of \met due to the energy
in the calorimeter cells not associated with physics objects is also
taken into account~\cite{Aaboud:2018tkc}. The uncertainty of the
resolution of \met arises from the energy resolution uncertainties of
each of the \met terms and the modeling of pileup and its effects on the soft term (see \cref{sec:reco}).
\section{Results}
\label{sec:results}
Maximum-likelihood fits are performed on data to extract parameters of
interest that probe \Htt production with increasing granularity. Firstly, a
single parameter is fitted to measure the total cross section of the
\Htt production processes.
Then, a two-parameter cross-section fit is
presented separating the \ggF and \VBF production processes. Finally,
a three-parameter fit is performed to measure \ggF production
cross sections in two exclusive regions of phase space.
For the small contribution from \HWW decays, the measurements assume
the SM predictions for production cross section and branching ratio.
A probability model is constructed that describes the \mMMC
distributions in the 13 signal regions and the event yields in 6 control regions.
The latter are included to constrain the normalizations of the dominant backgrounds.
Each signal region is modeled by a product of Poisson distributions, where each such distribution describes the expected event count in intervals of \mMMC.
Each control region is modeled by a single Poisson distribution that
describes the total expected event count in that region.
Signal and background predictions depend on systematic uncertainties, which are parameterized as nuisance parameters and are constrained
using Gaussian or log-normal probability distributions.
The latter are used for normalization factors (see \cref{tab:sf}) to ensure that they are
always positive.
The dependence of the predictions on nuisance parameters related to systematic uncertainties is modeled with an interpolation
approach between yields obtained at different fixed systematic uncertainty settings.
A smoothing procedure is applied to remove occasional large local fluctuations in the \mMMC distribution templates, which encode
systematic uncertainties of some background processes in certain regions.
For the measurements,
all theoretical uncertainties are included, except those related to
the respective measured signal cross sections, and are correlated as described in \cref{sec:systematics_theory_signal}.
The experimental uncertainties are fully correlated across categories and
the background modeling uncertainties are generally uncorrelated, with the exception of the
normalization factors as described in \cref{sec:background}.
Estimates of the parameters of interest and the confidence intervals are calculated with
the profile likelihood ratio~\cite{Cowan:2010js} test statistic, whereas the test statistic $\tilde{q}_0$~\cite{Cowan:2010js}
is used to compute the significances of the deviations from the background-only hypothesis.
The observed (expected) significance of the signal excess relative to the background-only hypothesis computed from the
likelihood fit is 4.4 (4.1) standard deviations, compatible with a SM Higgs boson with a mass
$m_{\PH}=\SI{125}{\GeV}$.
This result is combined with the result of the search for \Htt using data at 7 and \SI{8}{\TeV} center-of-mass energies~\cite{HIGG-2013-32}.
The combined observed (expected) significance amounts to 6.4 (5.4) standard deviations.
In this combination, all nuisance parameters are treated as uncorrelated between the two analyses.
In particular, the dominant \Ztt background is estimated differently, as mentioned in \cref{sec:intro}.
The parameter ${\POI\equiv\sigma_{\PH} \cdot \mathcal{B}(\Htt)}$ is
fitted, where $\sigma_{\PH}$ is the total cross section of the
considered Higgs-boson production processes \ggF, \VBF, \VH and \ttH, and
where $\mathcal{B}(\Htt)$ is the \Htt branching fraction.
For this measurement, the relative contributions from the various Higgs production processes are assumed as predicted by the SM
and the uncertainties related to the predicted total signal cross section are excluded.
The measured value of \POI is
$\num[round-mode=figures,round-precision=3]{3.76558372941}\,^{+\num[round-mode=figures,round-precision=2]{0.601597224288}}_{-\num[round-mode=figures,round-precision=2]{0.591224858352}}\,\text{(stat.)}\,^{+\num[round-mode=figures,round-precision=2]{0.874736193936}}_{-\num[round-mode=figures,round-precision=2]{0.74335289208}}\,\text{(syst.)}\,\si{\pb}$,
consistent with the SM prediction, $\POIpred=\num[round-mode=figures,round-precision=3]{3.457455312} \pm \num[round-mode=figures,round-precision=2]{0.131919772196}\,\si{\pb}$~\cite{Dittmaier:2011ti}.
The signal strength \POImu is
defined as the ratio of the measured signal yield to the
Standard Model expectation. It is computed by the fit
described above, including uncertainties in the predicted signal
cross section, and is evaluated to be
$1.09\,^{+0.18}_{-0.17}\,\text{(stat.)}\,^{+0.26}_{-0.22}\,\text{(syst.)}\,^{+0.16}_{-0.11}\,\text{(theory syst.)}$.
\Cref{t:yields_ll,t:yields_lh,t:yields_hh} summarize the expected signal and background yields computed by the fit in each signal region for the \POI measurement.
The signal event yields are given separately for each production
process of relevance. Within the uncertainties, good agreement is observed
between the data and the predicted sum of signal and background contributions, for
a SM Higgs boson of mass $m_{\PH}=\SI{125}{\GeV}$ with the
measured value of \POI reported above.
\begin{table}
\caption{Observed event yields and predictions as computed by the fit in the \tll signal regions. Uncertainties include statistical and systematic components.}
\label{t:yields_ll}
\centering
\vspace{2mm}
\begin{tabular}{
S[table-number-alignment=right, table-figures-integer=3, table-figures-decimal=1]@{$\,\pm\,$}
S[table-number-alignment=left, table-figures-integer=2, table-figures-decimal=1]
S[table-number-alignment=right, table-figures-integer=3, table-figures-decimal=1]@{$\,\pm\,$}
S[table-number-alignment=left, table-figures-integer=2, table-figures-decimal=1]
S[table-number-alignment=right, table-figures-integer=4, table-figures-decimal=1]@{$\,\pm\,$}
S[table-number-alignment=left, table-figures-integer=2, table-figures-decimal=1]
S[table-number-alignment=right, table-figures-integer=4, table-figures-decimal=1]@{$\,\pm\,$}
S[table-number-alignment=left, table-figures-integer=2, table-figures-decimal=1]
}
\toprule
& \multicolumn{4}{c}{\tll \VBF} & \multicolumn{4}{c}{\tll boosted} \\
\cmidrule(lr){2-5}
\cmidrule(lr){6-9}
& \multicolumn{2}{c}{Loose} & \multicolumn{2}{c}{Tight} & \multicolumn{2}{c}{Low-\pTH} & \multicolumn{2}{c}{High-\pTH} \\
\midrule
\Ztt & 151 & 13 & 107 & 12 & 2977 & 90 & 2687 & 64 \\
\Zll & 15.1 & 4.9 & 20.3 & 6.6 & 360 & 54 & 236 & 31 \\
Top & 33.0 & 6.4 & 25.1 & 4.5 & 321 & 50 & 189 & 29 \\
\VV & 11.8 & 2.2 & 10.7 & 1.5 & 194.1 & 8.5 & 195.3 & 8.8 \\
Misidentified \Pgt & 18.3 & 9.6 & 9.6 & 4.8 & 209 & 92 & 80 & 35 \\
\ggF, \HWW & 1.2 & 0.2 & 1.4 & 0.3 & 11.8 & 2.6 & 16.4 & 1.7 \\
\VBF, \HWW & 1.7 & 0.2 & 4.1 & 0.5 & 2.9 & 0.3 & 2.9 & 0.3 \\
\midrule
\ggF, \Htt & 2.6 & 0.9 & 1.8 & 0.9 & 34.4 & 9.2 & 33.8 & 9.5 \\
\VBF, \Htt & 5.3 & 1.5 & 11.3 & 3.0 & 7.7 & 2.1 & 8.2 & 2.3 \\
\WH, \Htt & \multicolumn{2}{l}{$<0.1$} & \multicolumn{2}{l}{$<0.1$} & 2.5 & 0.7 & 3.1 & 0.9 \\
\ZH, \Htt & \multicolumn{2}{l}{$<0.1$} & \multicolumn{2}{l}{$<0.1$} & 1.3 & 0.4 & 1.6 & 0.4 \\
\ttH, \Htt & \multicolumn{2}{l}{$<0.1$} & 0.1 & 0.1 & 1.5 & 0.5 & 1.2 & 0.4 \\
\midrule
Total background & 232 & 13 & 178 & 12 & 4075 & 61 & 3408 & 54 \\
Total signal & 8.0 & 2.2 & 13.2 & 3.5 & 47 & 12 & 48 & 12 \\
\midrule
Data & \multicolumn{2}{l}{237} & \multicolumn{2}{l}{188} & \multicolumn{2}{l}{4124} & \multicolumn{2}{l}{3444} \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}
\caption{Observed event yields and predictions as computed by the fit in the \tlhad signal regions. Uncertainties include statistical and systematic components.}
\label{t:yields_lh}
\centering
\vspace{2mm}
\begin{tabular}{
S[table-number-alignment=right, table-figures-integer=3, table-figures-decimal=1]@{$\,\pm\,$}
S[table-number-alignment=left, table-figures-integer=2, table-figures-decimal=1]
S[table-number-alignment=right, table-figures-integer=3, table-figures-decimal=1]@{$\,\pm\,$}
S[table-number-alignment=left, table-figures-integer=2, table-figures-decimal=1]
S[table-number-alignment=right, table-figures-integer=4, table-figures-decimal=1]@{$\,\pm\,$}
S[table-number-alignment=left, table-figures-integer=2, table-figures-decimal=1]
S[table-number-alignment=right, table-figures-integer=4, table-figures-decimal=1]@{$\,\pm\,$}
S[table-number-alignment=left, table-figures-integer=2, table-figures-decimal=1]
}
\toprule
& \multicolumn{4}{c}{\tlhad \VBF} & \multicolumn{4}{c}{\tlhad boosted} \\
\cmidrule(lr){2-5}
\cmidrule(lr){6-9}
& \multicolumn{2}{c}{Loose} & \multicolumn{2}{c}{Tight} & \multicolumn{2}{c}{Low-\pTH} & \multicolumn{2}{c}{High-\pTH} \\
\midrule
\Ztt & 178 & 18 & 323 & 21 & 4187 & 92 & 5347 & 82 \\
\Zll & 10.0 & 3.0 & 12.7 & 3.1 & 130 & 37 & 115 & 16 \\
Top & 5.8 & 1.6 & 17.9 & 4.6 & 121 & 20 & 57 & 10 \\
Misidentified \Pgt & 103 & 16 & 101 & 15 & 1895 & 80 & 605 & 29 \\
Other backgrounds & 4.0 & 1.6 & 9.3 & 1.9 & 115.0 & 7.8 & 129.0 & 8.8 \\
\midrule
\ggF, \Htt & 3.8 & 1.1 & 7.1 & 1.9 & 62 & 16 & 66 & 22 \\
\VBF, \Htt & 7.6 & 2.2 & 24.7 & 6.8 & 11.9 & 3.4 & 14.0 & 4.0 \\
\WH, \Htt & \multicolumn{2}{l}{$<0.1$} & 0.1 & 0.0 & 3.9 & 1.1 & 5.4 & 1.4 \\
\ZH, \Htt & \multicolumn{2}{l}{$<0.1$} & \multicolumn{2}{l}{$<0.1$} & 1.8 & 0.5 & 2.8 & 0.7 \\
\ttH, \Htt & \multicolumn{2}{l}{$<0.1$} & \multicolumn{2}{l}{$<0.1$} & 0.1 & 0.0 & 0.2 & 0.1 \\
\midrule
Total background & 301 & 17 & 463 & 21 & 6448 & 81 & 6253 & 80 \\
Total signal & 11.5 & 3.2 & 32.0 & 8.2 & 80 & 20 & 89 & 26 \\
\midrule
Data & \multicolumn{2}{l}{318} & \multicolumn{2}{l}{496} & \multicolumn{2}{l}{6556} & \multicolumn{2}{l}{6347} \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}
\caption{Observed event yields and predictions as computed by the fit in the \thadhad signal regions. Uncertainties include statistical and systematic components.}
\label{t:yields_hh}
\vspace{2mm}
\centering
\begin{tabular}{
S[table-number-alignment=right, table-figures-integer=3, table-figures-decimal=1]@{$\,\pm\,$}
S[table-number-alignment=left, table-figures-integer=1, table-figures-decimal=1]
S[table-number-alignment=right, table-figures-integer=3, table-figures-decimal=1]@{$\,\pm\,$}
S[table-number-alignment=left, table-figures-integer=2, table-figures-decimal=1]
S[table-number-alignment=right, table-figures-integer=3, table-figures-decimal=1]@{$\,\pm\,$}
S[table-number-alignment=left, table-figures-integer=2, table-figures-decimal=1]
S[table-number-alignment=right, table-figures-integer=4, table-figures-decimal=1]@{$\,\pm\,$}
S[table-number-alignment=left, table-figures-integer=3, table-figures-decimal=1]
S[table-number-alignment=right, table-figures-integer=4, table-figures-decimal=1]@{$\,\pm\,$}
S[table-number-alignment=left, table-figures-integer=2, table-figures-decimal=1]
}
\toprule
& \multicolumn{6}{c}{\thadhad \VBF} & \multicolumn{4}{c}{\thadhad boosted} \\
\cmidrule(lr){2-7}
\cmidrule(lr){8-11}
& \multicolumn{2}{c}{Loose} & \multicolumn{2}{c}{Tight} & \multicolumn{2}{c}{High-\pTH} & \multicolumn{2}{c}{Low-\pTH} & \multicolumn{2}{c}{High-\pTH} \\
\midrule
\Ztt & 67.3 & 9.2 & 100 & 12 & 141 & 12 & 3250 & 130 & 3582 & 82 \\
Misidentified \Pgt & 45.0 & 5.4 & 96.4 & 9.2 & 20.0 & 2.9 & 1870 & 140 & 364 & 53 \\
Other backgrounds & 4.4 & 1.4 & 11.6 & 1.7 & 4.4 & 0.7 & 281 & 21 & 109.9 & 9.2 \\
\midrule
\ggF, \Htt & 1.1 & 0.4 & 2.0 & 0.7 & 3.5 & 1.0 & 41 & 11 & 48 & 14 \\
\VBF, \Htt & 1.4 & 0.5 & 6.4 & 1.8 & 11.2 & 3.0 & 9.0 & 3.4 & 10.7 & 2.9 \\
\WH, \Htt & \multicolumn{2}{l}{$<0.1$} & \multicolumn{2}{l}{$<0.1$} & \multicolumn{2}{l}{$<0.1$} & 3.3 & 0.9 & 4.4 & 1.2 \\
\ZH, \Htt & \multicolumn{2}{l}{$<0.1$} & \multicolumn{2}{l}{$<0.1$} & \multicolumn{2}{l}{$<0.1$} & 2.4 & 0.7 & 2.9 & 0.8 \\
\ttH, \Htt & \multicolumn{2}{l}{$<0.1$} & \multicolumn{2}{l}{$<0.1$} & \multicolumn{2}{l}{$<0.1$} & 1.6 & 0.5 & 1.9 & 0.5 \\
\midrule
Total background & 116.7 & 9.4 & 208 & 12 & 165 & 12 & 5401 & 78 & 4057 & 64 \\
Total signal & 2.6 & 0.8 & 8.6 & 2.4 & 14.9 & 3.8 & 57 & 15 & 68 & 18 \\
\midrule
Data & \multicolumn{2}{l}{121} & \multicolumn{2}{l}{220} & \multicolumn{2}{l}{179} & \multicolumn{2}{l}{5455} & \multicolumn{2}{l}{4103} \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}
\caption{Summary of different sources of uncertainty in decreasing
order of their impact on \POI.
Their observed
and expected fractional (\%) impacts, both computed by the fit, are given, relative to the \POI value.
Experimental uncertainties in reconstructed objects combine efficiency and energy/momentum scale and resolution uncertainties.
Background statistics includes the bin-by-bin statistical uncertainties in the simulated backgrounds as well as statistical uncertainties in misidentified \Pgt backgrounds, which are estimated using data.
Background normalization describes the combined impact of all background normalization uncertainties.
}
\label{tab:systs}
\centering
\vspace{2mm}
\begin{tabular}{l%
S[table-number-alignment=right, table-figures-integer=2, table-figures-decimal=1, table-space-text-pre=$+$, retain-explicit-plus]@{$\,/\,$}
S[table-number-alignment=left, table-figures-integer=2, table-figures-decimal=1, table-space-text-pre=$-$]
S[table-number-alignment=right, table-figures-integer=2, table-figures-decimal=1, table-space-text-pre=$+$, retain-explicit-plus]@{$\,/\,$}
S[table-number-alignment=left, table-figures-integer=2, table-figures-decimal=1, table-space-text-pre=$-$]}
\toprule
Source of uncertainty & \multicolumn{4}{c}{Impact $\Delta\sigma/\POI$ [\%]} \\
& \multicolumn{2}{c}{Observed} & \multicolumn{2}{c}{Expected} \\
\midrule
Theoretical uncert. in signal & +13.4 & -8.7 & +12.0 & -7.8 \\
Background statistics & +10.8 & -9.9 & +10.1 & -9.7 \\
Jets and \met & +11.2 & -9.1 & +10.4 & -8.4 \\
Background normalization & +6.3 & -4.4 & +6.3 & -4.4 \\
Misidentified \Pgt & +4.5 & -4.2 & +3.4 & -3.2 \\
Theoretical uncert. in background & +4.6 & -3.6 & +5.0 & -4.0 \\
Hadronic \Pgt decays & +4.4 & -2.9 & +5.5 & -4.0 \\
Flavor tagging & +3.4 & -3.4 & +3.0 & -2.3 \\
Luminosity & +3.3 & -2.4 & +3.1 & -2.2 \\
Electrons and muons & +1.2 & -0.9 & +1.1 & -0.8 \\
\midrule
Total systematic uncert. & +23 & -20 & +22 & -19 \\
Data statistics & \multicolumn{2}{l}{$\qquad\pm 16$} & \multicolumn{2}{l}{$\qquad\pm 15$}\\
Total & +28 & -25 & +27 & -24 \\
\bottomrule
\end{tabular}
\end{table}
\Cref{tab:systs} shows a summary of the dominant uncertainties in
\POI, grouped by their respective sources. \Cref{fig:nuisance-parameters} shows the systematic
uncertainties with the largest impact, together with a comparison with their
nominal values used as input to the fit.
In both the table and the figure the shown uncertainties are
ranked by their fractional impact on the measurement of \POI.
To compute the impact for each
nuisance parameter, a separate fit is performed again with the parameter fixed to its
fitted value, and the resulting uncertainty in \POI
is subtracted from the uncertainty obtained in the original fit via variance subtraction.
The dominant uncertainties
are related to the limited number of events in the simulated samples,
the missing higher-order QCD corrections to the signal
process cross sections, the jet energy resolution, the \tauhadvis
identification and the normalizations of the \Ztt and \Zll backgrounds.
\Cref{fig:nuisance-parameters} also shows that in most cases the fitted parameters are in agreement
with the nominal values,
except for the uncertainties related to jet
energy resolution and scale.
In the case of real di-\Pgt events, the distribution of \mMMC is sensitive to the jet-related uncertainties because
selected di-\Pgt events in the \VBF and boosted categories are characterized by one or more high-\pt jets that
recoil against the two \Pgt{}-leptons.
The main contributions to \met are thus the neutrinos in the \Pgt{}-lepton decays and the impact of the jet energy resolution when projected onto the \met direction.
Applying both the jet energy resolution and scale uncertainties causes a shift in the mean jet \pT, which therefore translates directly into a shift of the reconstructed \met.
This, in turn, translates into a shift of the reconstructed \mMMC that is constrained by data in the region of the \Ztt mass peak.
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig6}
\end{center}
\caption{Fractional impact of systematic uncertainties in \POI as computed by
the fit. The systematic uncertainties are listed in decreasing
order of their impact on \POI on the $y$-axis. The hatched
blue and open blue boxes show the variations of \POI
referring to the
top $x$-axis (impact), as described in the text.
The filled circles, referring to the bottom $x$-axis,
show the pulls of the fitted nuisance parameters, i.e.\ the
deviations of the fitted parameters $\hat\theta$ from their
nominal values $\theta_0$, normalized to their nominal
uncertainties $\Delta\theta$. The black lines show the
uncertainties of the nuisance parameters resulting from the fit.
Several sources of uncertainties such as the jet energy scale and resolution as well as the \Pqb-mistag rate are described by their principal components in the fit.
The open circles, also referring to the bottom $x$-axis, show the
values of the fitted \Ztt and \Zll normalization factors in the
boosted category as listed in \cref{tab:sf}.
Their uncertainties do not include uncertainties in total simulated cross sections extrapolated to the selected phase space.
}
\label{fig:nuisance-parameters}
\end{figure}
Results of the fit when only the data of an individual channel or of an individual category are used,
are shown in \cref{fig:x-section_summary}. Also shown is the result from the
fit and the uncertainty in \POIpred.
All results are consistent with the SM expectations.
The simple combination of the individual fit results does not agree exactly with
the result of the combined fit because the values of the nuisance
parameters are different.
The \mMMC distributions in all signal regions with background predictions adjusted by the likelihood fit are shown in \cref{fig:MMC-signal_vbf,fig:MMC-signal_boosted} in the Appendix.
The \mMMC distributions for the predicted signal plus background are compared with the data in \cref{fig:CBA_m-tautau_channel}, separately for the combined signal regions of \thadhad, \tlhad and \tll analysis channels, and in \cref{fig:CBA_m-tautau}, separately for the combined \VBF and the combined boosted signal regions.
A weighted combination of the \mMMC distributions in all signal regions is shown in \cref{fig:CBA_m-tautau_weighted}.
The events are weighted by a factor of $\ln(1+S/B)$ which enhances the events compatible with the signal hypothesis.
Here, $S/B$ is the expected signal-to-background ratio in the corresponding signal region.
\begin{figure}[htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\textwidth]{fig7a} }
\subfloat[]{\includegraphics[width=0.49\textwidth]{fig7b} }
\caption{The measured values for
\POI when only the data of (a) individual channels
or (b) individual categories are used.
Also shown is the result from the
combined fit. The total $\pm{}1\sigma$ uncertainty in the measurement is indicated by
the black error bars, with the individual contribution from the
statistical uncertainty in blue. The theory uncertainty
in the predicted signal cross section is shown by the yellow band.}
\label{fig:x-section_summary}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.32\textwidth]{fig8a}
\includegraphics[width=0.32\textwidth]{fig8b}
\includegraphics[width=0.32\textwidth]{fig8c}
\caption{Distributions of the reconstructed di-\Pgt invariant mass ($\mMMC$) for the sum of
(left) all \tll, (center) all \tlhad and (right) all \thadhad signal regions (SRs).
The bottom panels show the differences between observed data events and expected background
events (black points).
The observed Higgs-boson signal ($\mu = 1.09$) is shown with the solid red line.
Entries with values that would exceed the $x$-axis range are shown in the last bin of each distribution.
The signal and background predictions are determined in the likelihood fit.
The size of the combined statistical, experimental and theoretical uncertainties in the background is indicated by the hatched bands.}
\label{fig:CBA_m-tautau_channel}
\end{figure}
\begin{figure}[htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\textwidth]{fig9a}}
\subfloat[]{\includegraphics[width=0.49\textwidth]{fig9b}}
\caption{Distribution of the reconstructed di-\Pgt invariant mass ($\mMMC$) for the sum of
(a) all \VBF and (b) all boosted signal regions (SRs).
The bottom panels show the differences between observed data events and expected background
events (black points).
The observed Higgs-boson signal ($\mu = 1.09$) is shown with the solid red line.
Entries with values that would exceed the $x$-axis range are shown in the last bin of each distribution.
The signal and background predictions are determined in the likelihood fit.
The size of the combined statistical, experimental and theoretical uncertainties in the background is indicated by the hatched bands.}
\label{fig:CBA_m-tautau}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.49\textwidth]{fig10}
\caption{Distribution of the reconstructed di-\Pgt invariant mass ($\mMMC$) for the sum of
all signal regions (SRs).
The contributions of the different SRs are weighted by a factor of $\ln(1+S/B)$, where $S$ and $B$ are the expected numbers of signal and background events in that region, respectively.
The bottom panel shows the differences between observed data events and expected background
events after applying the same weights (black points).
The observed Higgs-boson signal ($\mu = 1.09$) is shown with the solid red line.
Entries with values that would exceed the $x$-axis range are shown in the last bin of each distribution.
The signal and background predictions are determined in the likelihood fit.
The size of the combined statistical, experimental and theoretical uncertainties in the background is indicated by the hatched bands.}
\label{fig:CBA_m-tautau_weighted}
\end{figure}
\Cref{fig:x-section_summary} illustrates that the \VBF and boosted categories provide good sensitivity, respectively, to \VBF and \ggF Higgs-boson production.
A two-parameter fit is therefore performed to determine the cross sections of these production processes by exploiting the sensitivity offered by the use of the event categories in the analysis of the three channels.
Two cross-section parameters \xsecVBF and \xsecggF are introduced
and the data are fitted to these parameters, separating
the vector-boson-mediated
\VBF process from the fermion-mediated \ggF process, while the contributions from other Higgs production
processes are set to their predicted SM values.
The two-dimensional $68\%$ and $95\%$ confidence level
(CL) contours in the plane of \xsecVBF and \xsecggF
are shown in \cref{fig:2D-contour}. The best-fit values are
$\xsecVBF = \num[round-mode=figures,round-precision=2]{0.2800453818} \pm \num[round-mode=figures,round-precision=1]{0.09}\,\text{(stat.)}\,^{+\num[round-mode=figures,round-precision=2]{0.106}}_{-\num[round-mode=figures,round-precision=1]{0.093}}\,\text{(syst.)}\,\si{\pb}$
and
$\xsecggF = \num[round-mode=figures,round-precision=2]{3.096016028} \pm \num[round-mode=figures,round-precision=2]{1.0}\,\text{(stat.)}\,^{+\num[round-mode=figures,round-precision=2]{1.63}}_{-\num[round-mode=figures,round-precision=2]{1.258}}\,\text{(syst.)}\,\si{\pb}$,
in agreement with the predictions from the Standard Model of
$\xsecpredVBF=\num[round-mode=figures,round-precision=3]{0.23721}\pm 0.006\,\si{\pb}$
and
$\xsecpredggF=\num[round-mode=figures,round-precision=3]{3.0469}\pm\num[round-mode=figures,round-precision=2]{0.128994749633}\,\si{\pb}$~\cite{Dittmaier:2011ti}. The two
results are strongly anti-correlated (correlation coefficient of
$-52$\%), as can be seen in \cref{fig:2D-contour}.
The \ggF signal provides enough events to measure \ggF cross
sections in mutually exclusive regions of the \ggF phase space. Two
\ggF regions are defined by particle-level events with at least one
jet where a jet is required to have $\pT>\SI{30}{\GeV}$: events with a
Higgs-boson \pT of $60<\pTHtruth<\SI{120}{\GeV}$ and events with
$\pTHtruth>\SI{120}{\GeV}$.
A cross-section parameter for each of the two \ggF regions is
introduced, along with a parameter for \VBF production in an inclusive
region, and a combined three-parameter fit is performed using the
event categories in the analysis of the three channels. The
particle-level definitions of all three phase-space regions closely follow
the framework of simplified template cross
sections~\cite{deFlorian:2016spz} where the Higgs-boson rapidity $y_H$
is required to satisfy $|y_H|<2.5$.
The \ggF and \VBF production cross sections outside the respective particle-level region requirements are set to the measured values reported above.
Cross sections of other Higgs-boson production processes are set to their SM values.
\Cref{tab:stxs} shows the resulting cross sections along with the
SM predictions in the respective particle-level region.
The measurements in all regions have a precision similar to that of
the inclusive ggF and VBF measurements reported above.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=11cm]{fig11}
\end{center}
\caption{Likelihood contours
for the combination of all channels in the
(\xsecggF, \xsecVBF) plane. The 68\% and 95\% CL contours are shown as dashed
and solid lines, respectively, for $m_{\PH}=\SI{125}{\GeV}$. The SM
expectation is indicated by a plus symbol and the best fit to
the data is shown as a star.}
\label{fig:2D-contour}
\end{figure}
\begin{table}
\caption{Measurement of the \VBF and \ggF production cross sections
in three mutually exclusive regions of phase space
of particle-level events.
The number of jets $N_\text{jets}$ in \ggF events comprises all jets with $\pT>\SI{30}{\GeV}$.
The cross section of \ggF events that
fail the particle-level requirements of the two \ggF regions is set to the measured \xsecggF value.
Results are shown along with
the SM predictions in the respective particle-level regions.
The definitions of the regions closely follow the framework of simplified
template cross sections~\cite{deFlorian:2016spz}.}
\label{tab:stxs}
\centering
\vspace{2mm}
\begin{tabular}{lc%
S[table-number-alignment=right, table-figures-integer=1, table-figures-decimal=2]@{$\,\pm\,$}
S[table-number-alignment=right, table-figures-integer=1, table-figures-decimal=2]@{$\,\text{(stat.)} \pm \,$}
S[table-number-alignment=right, table-figures-integer=1, table-figures-decimal=2]@{$\,\text{(syst.)}\ \ \ $}
S[table-number-alignment=right, table-figures-integer=1, table-figures-decimal=2]@{$\,\pm\,$}
S[table-number-alignment=right, table-figures-integer=1, table-figures-decimal=2]}
\toprule
Process & Particle-level selection & \multicolumn{3}{c}{$\sigma$ [pb]} & \multicolumn{2}{c}{$\sigma^\text{SM}$ [pb]} \\
\midrule
\ggF & $N_\text{jets} \geq 1$, $60<\pTHtruth<\SI{120}{\GeV}$, $|y_H|<2.5$ & 1.79 & 0.53 & 0.74 & 0.40 & 0.05 \\
\ggF & $N_\text{jets} \geq 1$, $\pTHtruth>\SI{120}{\GeV}$, $|y_H|<2.5$ & 0.12 & 0.05 & 0.05 & 0.14 & 0.03 \\
\VBF & $|y_H|<2.5$ & 0.25 & 0.08 & 0.08 & 0.22 & 0.01 \\
\bottomrule
\end{tabular}
\end{table}
\FloatBarrier
\section{Conclusions}
\label{sec:conclusions}
A measurement of total production cross sections of the Higgs boson
in proton--proton collisions is presented in the
\Htautau decay channel. The analysis was performed using
\SI{36.1}{\per\fb} of data recorded by the ATLAS experiment at the LHC
at a center-of-mass energy of $\sqrt{s}=\SI{13}{\TeV}$.
All combinations of leptonic and hadronic \Pgt
decays were considered. An excess of events over the expected
background from other Standard Model processes was found with an
observed (expected) significance of 4.4 (4.1) standard deviations.
Combined with results using data taken
at $\sqrt s$ of~7 and \SI{8}{\TeV}, the observed (expected) significance amounts to 6.4 (5.4)
standard deviations and constitutes an observation of \Htt decays by the ATLAS experiment.
Using the data taken at $\sqrt{s}=\SI{13}{\TeV}$, the $\pp \to \Htt$ total cross section
is measured to be
$\num[round-mode=figures,round-precision=3]{3.76558372941}\,^{+\num[round-mode=figures,round-precision=2]{0.601597224288}}_{-\num[round-mode=figures,round-precision=2]{0.591224858352}}\,\text{(stat.)}\,^{+\num[round-mode=figures,round-precision=2]{0.874736193936}}_{-\num[round-mode=figures,round-precision=2]{0.74335289208}}\,\text{(syst.)}\,\si{\pb}$,
for a Higgs boson of mass \SI{125}{\GeV}.
A two-dimensional fit was performed to
separate the vector-boson-mediated \VBF process from the
fermion-mediated \ggF process.
The cross sections of the Higgs boson decaying into two \Pgt leptons are measured to be
$\xsecVBF = \num[round-mode=figures,round-precision=2]{0.2800453818} \pm \num[round-mode=figures,round-precision=1]{0.09}\,\text{(stat.)}\,^{+\num[round-mode=figures,round-precision=2]{0.106}}_{-\num[round-mode=figures,round-precision=1]{0.093}}\,\text{(syst.)}\,\si{\pb}$ and
$\xsecggF = \num[round-mode=figures,round-precision=2]{3.096016028} \pm \num[round-mode=figures,round-precision=2]{1.0}\,\text{(stat.)}\,^{+\num[round-mode=figures,round-precision=2]{1.63}}_{-\num[round-mode=figures,round-precision=2]{1.258}}\,\text{(syst.)}\,\si{\pb}$, respectively, for the two production processes.
Similarly, a three-dimensional fit was performed in the framework of simplified template cross sections.
Results are reported for the \VBF cross section in an inclusive phase space and
\ggF cross sections in two exclusive regions of phase space
defined by particle-level requirements on the Higgs-boson \pT.
All measurements are consistent with SM predictions.
\clearpage
\section*{Acknowledgements}
\input{Acknowledgements}
\printbibliography
\end{document}
|
1,108,101,563,312 | arxiv | \section{Introduction}
\label{intro}
The spectroscopic study of light-quark baryons ($N^*$, $\Delta^*$, $\Lambda^*$, $\Sigma^*$)
remains a central issue in the hadron physics.
Since the excitation spectra of light-quark baryons are an embodiment of
the dynamics of confinement in Quantum Chromodynamics (QCD),
their precise determination through analyzing reaction data is essential to
test QCD in the nonperturbative domain.
In this regard, several attempts to link the (real) energy spectrum of QCD in the finite box to
the complex pole masses of actual hadron resonances have been being made recently
(see, e.g., Refs.~\cite{dudek,wu,molina,lee}).
A particular complexity exists in extracting the light-quark baryons from reaction data.
This originates from a peculiar character of the light-quark baryons:
they are broad and highly overlapping resonances.
The light-quark baryons are thus strongly correlated with
each other and with backgrounds through reaction processes
over the wide energy range,
and the observed shape of cross sections and spin asymmetries
represents such a complicated interference between them.
As a result, the light-quark baryon resonances do not produce any clean isolated peak
in cross sections with a few exceptions.
This makes identification of light-quark baryons from experiments rather difficult,
and cooperative works between experiments and theoretical analyses are indispensable
for the light-quark baryon spectroscopy.
To disentangle complicated correlations between the light-quark baryon resonances,
one needs to perform a partial-wave analysis of various meson-production reactions
comprehensively over the wide energy range.
Here the use of a reaction framework satisfying multichannel unitarity
is particularly important.
In fact, as is well known, multichannel unitarity ensures the conservation of
probabilities in the multichannel reaction processes, and this is crucial for accomplishing
comprehensive analysis of meson-production reactions with various final states consistently within
a single reaction framework.
Also, the multichannel unitarity properly defines
the analytic structure (branch points, cuts, threshold cusps, etc.)
of scattering amplitudes in the complex-energy plane, as required by the scattering theory, which
is essential to extract resonance information from reaction data {\it correctly}.
As an approach that enables such a multichannel analysis,
a dynamical coupled-channels (DCC) model for meson-production reactions in the resonance region
was developed in Ref.~\cite{msl}.
So far, this model has been employed in a series
of works~\cite{jlms07,jlmss07,djlss08,kjlms09,jklmss09,kjlms09-2,sjklms10,knls10,knls13,kamano13}
to analyze the $\pi N$, $\gamma N$, and $e N$ reactions
and study the mass spectrum, structure, and dynamical origins of
nonstrange $N^*$ and $\Delta^*$ resonances.
This model has also been applied to the study of
$\Lambda^*$ and $\Sigma^*$ resonances with strangeness $S=-1$
through a comprehensive analysis of $K^- p$ reactions~\cite{knls14,knls15}.
In addition to these studies related to the light-quark baryon spectroscopy,
we have recently made an application of our DCC model to
neutrino-nucleon reactions in the resonance region~\cite{knls12,nks15}.
The purpose of this contribution is to give an overview of our recent efforts for
these subjects.
\section{ANL-Osaka DCC model}
\label{sec:1}
The basic formula of our DCC model is the coupled-channels integral equation
obeyed by the partial-wave amplitudes of $a \to b$ reactions
that are specified by the total angular momentum ($J$), parity ($P$), and total isospin ($I$)
(here we explain our approach by taking the $N^*$ and $\Delta^*$ sector as an example):
\begin{equation}
T^{(J^P I)}_{b,a} (p_b,p_a;W) =
V^{(J^PI)}_{b,a} (p_b,p_a;W)
+\sum_c \int_C dp_c\,p_c^2 V^{(J^PI)}_{b,c} (p_b,p_c;W) G_c(p_c;W) T^{(J^PI)}_{c,a} (p_c,p_a;W),
\label{lseq}
\end{equation}
with
\begin{eqnarray}
V^{(J^P I)}_{b,a}(p_b,p_a; W)=
v_{b,a}(p_b,p_a)+
Z_{b,a}(p_b,p_a;W)+
\sum_{N^*_n}\frac{\Gamma_{b,N^*_n}(p_b)
\Gamma_{N^*_n,a}(p_a)} {W-M^0_{N^*_n}} .
\label{eq:veq}
\end{eqnarray}
Here the subscripts ($a,b,c$) represent the considered reaction channels (the indices associated
with the total spin and orbital angular momentum of the channels are suppressed);
$p_a$ is the magnitude of the relative momentum for the channel $a$ in the center-of-mass frame; and
$W$ is the total scattering energy.
For the $N^*$ and $\Delta^*$ sector, we have taken into account
the eight channels, $\gamma^{(*)}N$, $\pi N$, $\eta N$, $K\Lambda$,
$K\Sigma$, $\pi\Delta$, $\rho N$, and $\sigma N$\footnote{
Because of the perturbative nature of the electromagnetic interactions,
it is only necessary to solve the coupled-channels equations
in the channel space excluding the $\gamma^{(*)} N$ channel.
Thus the summation in Eq.~(\ref{lseq}) runs over only the hadronic channels.},
where the last three are the quasi two-body channels
that subsequently decay into the three-body $\pi\pi N$ channel.
The Green's functions for the meson($M$)-baryon($B$) channels are given by
$G_c(p_c;W) = 1/[W-E_M(p_c)-E_B(p_c)+i\varepsilon]$ for
$c=\pi N, \eta N, K\Lambda, K\Sigma$, while
$G_c(p_c;W) = 1/[W-E_M(p_c)-E_B(p_c)+\Sigma_{MB}(p_c;W)]$ for
$c=\pi \Delta, \sigma N, \rho N$, where
$E_{\alpha}(p) = \sqrt{m_\alpha^2+p^2}$ is the relativistic single-particle energy
of a particle $\alpha$,
and $\Sigma_{MB}(p_c;W)$ is the self-energy
that produces the three-body $\pi \pi N$ cut.
The transition potential $V_{b,a}$ [Eq.~(\ref{eq:veq})] consists of three pieces.
The first two, $v_{b,a}$ and $Z_{b,a}$, describe the nonresonant processes including only
the ground-state mesons and baryons belonging to each flavor SU(3) multiplet,
and the third one describes the propagation
of the bare $N^*$ states.
In our approach, the unitary transformation method~\cite{msl,sl}
is used to derive the potential $v_{b,a}$ from effective Lagrangians.
With this method, the resulting $v_{b,a}$ becomes energy independent and
its off-shell behavior is specified.
On the other hand, the $Z$-diagram potential, $Z_{b,a}$, is derived using
the projection operator method~\cite{feshbach}.
It also produces the three-body $\pi \pi N$ cut~\cite{msl,knls13},
and implementation of both the $Z$-diagram potential
and the self-energy in the Green's functions
is necessary to maintain the three-body unitarity.
Furthermore, off-shell rescattering effects are also taken
into account properly through the momentum integral in Eq.~(\ref{lseq}),
which are usually neglected in on-shell approaches.
\section{Recent efforts for $N^*$ and $\Delta^*$ spectroscopy with ANL-Osaka DCC model}
\label{sec:nucleon}
Our latest published model for the $N^*$ and $\Delta^*$ sector~\cite{knls13,kamano13} was
constructed by making a simultaneous analysis of
$\pi p \to \pi N, \eta N, K \Lambda, K \Sigma$
and $\gamma p \to \pi N, \eta N, K \Lambda, K \Sigma$.
This contains the data of both unpolarized differential cross sections and polarization
observables up to $W = 2.1$ GeV (up to $W= 2.3$ GeV for $\pi p \to \pi N$),
which results in fitting $\sim 23,000$ data points.
After the completion of this analysis,
we have been mainly proceeding with two subjects for the $N^*$ and $\Delta^*$ spectroscopy:
(1) extraction of the helicity amplitudes for the $\gamma n \to N^*$ transition
from the available $\gamma \textrm{`}n\textrm{'} \to \pi N$ data, and
(2) determination of $Q^2$ dependence of the $p$-$N^*$ and $p$-$\Delta^*$ electromagnetic
transition form factors by analyzing the structure function data for single-pion electroproductions off the proton target.
The latter subject is discussed in Ref.~\cite{tsato},
and in this contribution we focus on presenting the current status of
our $\gamma \textrm{`}n\textrm{'} \to \pi N$ analysis.
There are mainly two reasons for studying meson photo- and electro-production reactions
off the neutron target.
One comes from the fact that the analysis of the data for both proton- and neutron-target
reactions is required to decompose the matrix elements for the electromagnetic currents
into the ones of isoscalar and isovector currents and uniquely determine the isospin structure of
the $\gamma^{(*)} N \to N^*$ transition amplitudes.
Another is because the matrix elements for such ``isospin-decomposed'' currents
are necessary for constructing a model for neutrino-induced reactions, which
will be discussed in Sec.~\ref{sec:neutrino}.
\begin{figure}[t]
\centering
\includegraphics[clip,width=0.48\textwidth]{gnpi-p_dc.eps}
\ \
\includegraphics[clip,width=0.48\textwidth]{gnpi-p_s.eps}
\caption{\label{gn-obs1}
Differential cross sections (left) and photon asymmetries (right)
for
$\gamma \textrm{`}n\textrm{'} \to \pi^- p$.
The numbers shown in each panel are the corresponding total scattering energy $W$ in MeV.
The data are taken from Ref.~\cite{gwu}.
}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[clip,width=0.48\textwidth]{gnpi0n_dc.eps}
\ \
\includegraphics[clip,width=0.48\textwidth]{gnpi0n_s.eps}
\caption{\label{gn-obs2}
Differential cross sections (left) and photon asymmetries (right)
for
$\gamma \textrm{`}n\textrm{'} \to \pi^0 n$.
The numbers shown in each panel are the corresponding total scattering energy $W$ in MeV.
The data are taken from Ref.~\cite{gwu}.
}
\end{figure}
So far, we have performed the fits to the data for $\gamma n \to \pi N$ up to $W\lesssim 2$ GeV.
The data are available for $d\sigma/d\Omega$, $\Sigma$, $T$, and $P$ for $\gamma n \to \pi^- p$,
and $d\sigma/d\Omega$ and $\Sigma$ for $\gamma n \to \pi^0 n$.
This contains $\sim 3,200$ data points.
Some of the results of our fits are presented in Figs.~\ref{gn-obs1} and~\ref{gn-obs2}.
One can see that a reasonably good reproduction of the $\gamma n \to \pi N$ data
has been accomplished for the considered energy region.
\begin{figure}[t]
\centering
\includegraphics[clip,width=0.65\textwidth]{allobs.eps}
\caption{\label{allobs}
Differential cross section and all polarization observables
for $\gamma n \to \pi^- p$ at $W = 1604$ MeV ($E_\gamma =900$ MeV).
The results are predictions from our current DCC model except for $d\sigma/d\Omega$, $\Sigma$, $T$, and $P$.
The data are taken from Ref.~\cite{gwu}.
}
\end{figure}
In Fig.~\ref{allobs}, we present our calculated results for all 16 observables
($d\sigma/d\Omega$ and 15 polarization observables) of
the $\gamma n \to \pi^- p$ reaction at $W = 1604$ MeV.
They are predictions from our current DCC model except for
$d\sigma/d\Omega$, $\Sigma$, $T$, and $P$, whose data displayed in Fig.~\ref{allobs}
were included in our fits to determine the model parameters.
The polarization data are known to provide crucial constraints on
multipole amplitudes~\cite{shkl11}, and
tremendous efforts are now being pursued
at electron- and photon-beam facilities such as JLab, ELSA, and MAMI
to extract polarization observables for the neutron-target
photoproductions through the $\gamma d$ reaction (see, e.g., Ref.~\cite{exp}).
\section{Comprehensive analysis of $K^- p$ reactions and extraction of $\Lambda^*$ and $\Sigma^*$ resonances}
\label{sec:hyperon}
Following the success of our $N^*$ and $\Delta^*$ studies,
we have recently extended our DCC approach to the strangeness $S=-1$ sector
to also explore the $\Lambda^*$ and $\Sigma^*$ hyperon resonances.
In the formulation of the DCC model in the $S=-1$ sector, we have taken into account
seven channels ($\bar K N$, $\pi \Sigma$, $\pi \Lambda$, $\eta \Lambda$,
$K\Xi$, $\pi \Sigma^*$, $\bar K^* N$), where
the $\pi \Sigma^*$ and $\bar K^* N$ channels are the quasi two-body channels
that subsequently decay into the three-body $\pi\pi\Lambda$ and $\pi \bar K N$ channels,
respectively.
The model parameters are then fixed by fitting to all available data of
$K^- p \to \bar K N, \pi \Sigma, \pi \Lambda, \eta \Lambda, K\Xi$
reactions from the threshold up to $W = 2.1$ GeV.
The data contain the total cross section ($\sigma$),
differential cross section ($d\sigma/d\Omega$), and recoil polarization ($P$),
and this results in fitting more than 17,000 data points.
\begin{figure}[t]
\centering
\includegraphics[clip,width=0.7\textwidth]{kmp-tcs.eps}
\caption{
Fitted results for the total cross sections of $K^- p$ reactions~\cite{knls14}.
Red solid (blue dashed) curves are the results of Model A (Model B).
}
\label{fig:kptcs}
\end{figure}
Figure~\ref{fig:kptcs} shows the results of our fits to the total cross sections for all reactions we considered.
Here two curves are plotted in red (Model A) and blue (Model B), respectively.
This is because the existing $K^-p$ reaction data are not sufficient
to determine our model parameters unambiguously, and it allowed us to have
two distinct sets of the model parameters, yet both give almost the same $\chi^2$ value.
We quote Ref.~\cite{knls14} for all the details of our analysis including
the fits to the differential cross sections and polarization observables.
Once the model parameters are determined by fitting to the reaction data,
we can extract various information (mass, width, pole residues,
and branching ratios, etc.) on the $\Lambda^*$ and $\Sigma^*$ resonances
by making an analytic continuation of the calculated scattering amplitudes
in the complex energy plane
(see Refs.~\cite{ssl,ssl2} for a numerical procedure to perform the analytic continuation within our DCC model).
This was done in Ref.~\cite{knls15}, and
the value of the resonance pole masses extracted obtained via such an analytic continuation
is presented in Table~\ref{tab:resmass}.
We searched pole positions of scattering amplitudes in the complex energy region with
$m_{\bar K} + m_N \leq {\rm Re}(W) \leq 2.1$ GeV and
$0 \leq -{\rm Re}(W) \leq 0.2$ GeV, and found
18 resonances (10 $\Lambda^*$ and 8 $\Sigma^*$) in Model A, while
20 resonances (10 $\Lambda^*$ and 10 $\Sigma^*$) in Model B.
The uncertainties assigned for the value of each extracted pole mass was deduced by following a procedure similar to
the one used in Ref.~\cite{shkl11} to determine the uncertainties in extracted multipole amplitudes
for $\gamma p \to K^+ \Lambda$.
Here we did not assign the uncertainty for $J^P=7/2^+$ $\Lambda$ resonance in Model A because
it was turned out to be too large to be meaningful.
\begin{table}[t]
\caption{\label{tab:resmass}
Extracted complex pole masses ($M_R$) for the $\Lambda^*$ and $\Sigma^*$ resonances
found in the energy region above the $\bar K N$ threshold~\cite{knls15}.
The masses are listed as $\bm{(} {\rm Re}(M_R), -{\rm Im}(M_R)\bm{)}$
together with their deduced uncertainties.
The resonance poles are searched in the complex $W$ region with
$m_{\bar K} + m_N \leq {\rm Re}(W) \leq 2.1$ GeV and $0\leq -{\rm Im}(W) \leq 0.2$ GeV,
and all of the resonances listed are located in the complex $W$ Riemann surface
nearest to the physical real $W$ axis.
}
\centering
\begin{tabular}{cccccc}
\hline\noalign{\smallskip}
& \multicolumn{2}{c}{$\Lambda^*$} &
& \multicolumn{2}{c}{$\Sigma^*$} \\[3pt]
& \multicolumn{2}{c}{$M_R$ (MeV)} &
& \multicolumn{2}{c}{$M_R$ (MeV)} \\
\cline{2-3} \cline{5-6}
$J^P(l_{I2J})$ & Model A & Model B &
$J^P(l_{I2J})$ & Model A & Model B \\
\tableheadseprule\noalign{\smallskip}
$1/2^-(S_{01})$&-- &(1512$^{+1}_{-1}$,185$^{+1}_{-2}$) &$1/2^-(S_{11})$&-- &(1551$^{+2}_{-9}$,188$^{+6}_{-1}$)\\
&(1669$^{+3}_{-8}$, 9$^{+9}_{-1}$) &(1667$^{+1}_{-2}$,12$^{+3}_{-1}$) & &(1704$^{+3}_{-6}$, 43$^{+7}_{-2}$) &--\\
$1/2^+(P_{01})$&(1544$^{+3}_{-3}$, 56$^{+6}_{-1}$) &(1548$^{+5}_{-6}$, 82$^{+7}_{-7}$) & & -- &(1940$^{+2}_{-2}$, 86$^{+2}_{-2}$)\\
&-- &(1841$^{+3}_{-4}$, 31$^{+3}_{-2}$) &$1/2^+(P_{11})$&-- &(1457$^{+5}_{-1}$, 39$^{+1}_{-4}$)\\
&(2097$^{+40}_{-1}$, 83$^{+32}_{-6}$) &-- & &(1547$^{+111}_{-59}$, 92$^{+43}_{-39}$)&--\\
$3/2^+(P_{03})$& -- &(1671$^{+2}_{-8}$, 5$^{+11}_{-2}$) & &-- &(1605$^{+2}_{-4}$, 96$^{+1}_{-5}$)\\
&(1859$^{+5}_{-7}$, 56$^{+10}_{-2}$) & -- & &(1706$^{+67}_{-60}$, 51$^{+79}_{-42}$) &--\\
$3/2^-(D_{03})$&(1517$^{+4}_{-4}$, 8$^{+5}_{-4}$) &(1517$^{+4}_{-3}$, 8$^{+6}_{-6}$) & & -- &(2014$^{+6}_{-13}$, 70$^{+14}_{-1}$)\\
&(1697$^{+6}_{-6}$, 33$^{+7}_{-7}$) &(1697$^{+6}_{-5}$, 37$^{+7}_{-7}$) &$3/2^-(D_{13})$&-- &(1492$^{+4}_{-7}$, 69$^{+4}_{-7}$)\\
$5/2^-(D_{05})$&(1766$^{+37}_{-34}$,106$^{+47}_{-31}$)& -- & &(1607$^{+13}_{-11}$,126$^{+15}_{-9}$) &--\\
&(1899$^{+35}_{-37}$, 40$^{+50}_{-17}$)&(1924$^{+52}_{-24}$, 45$^{+57}_{-17}$)& &(1669$^{+7}_{-7}$, 32$^{+5}_{-7}$) &(1672$^{+5}_{-10}$, 33$^{+3}_{-3}$)\\
$5/2^+(F_{05})$&(1824$^{+2}_{-1}$, 39$^{+1}_{-1}$) &(1821$^{+1}_{-1}$, 32$^{+1}_{-1}$) &$5/2^-(D_{15})$&(1767$^{+2}_{-2}$, 64$^{+2}_{-1}$) &(1765$^{+2}_{-1}$, 64$^{+3}_{-1}$)\\
$7/2^+(F_{07})$&(1757, 73) &-- &$5/2^+(F_{15})$&-- &(1695$^{+20}_{-77}$, 97$^{+50}_{-44}$)\\
&-- &(2041$^{+80}_{-82}$,119$^{+57}_{-17}$)& &(1890$^{+3}_{-2}$, 49$^{+2}_{-3}$) &--\\
& & &$7/2^+(F_{17})$&(2025$^{+10}_{-5}$, 65$^{+3}_{-12}$) &(2014$^{+12}_{-1}$,103$^{+3}_{-9}$)\\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[clip,width=0.6\textwidth]{kmpetl-dcs.eps}
\caption{
Differential cross section at $W=1672$ MeV for the $K^- p \to \eta \Lambda$
reaction.
Left (right) panel is the result from Model A (Model B).
Solid curves are the full results, while the dashed curves are the contribution
from the $S_{01}$ partial wave only.
For Model B, the difference between the solid and dashed curves almost comes from
the $P_{03}$ partial wave dominated by the new narrow
$J^P=3/2^+$ $\Lambda$ resonance with $M_R = 1671^{+2}_{-8}-i(5^{+11}_{-2})$ MeV.
}
\label{fig:etldcs}
\end{figure}
We see in Table~\ref{tab:resmass} that the pole mass of 9 resonances (6 $\Lambda^*$ and 3 $\Sigma^*$)
agree well between our two models within the deduced uncertainties (each resonance is aligned in the same row).
These resonances would correspond to $\Lambda(1670)1/2^-$,
$\Lambda(1600)1/2^+$, $\Lambda(1520)3/2^-$, $\Lambda(1690)3/2^-$, $\Lambda(1820)5/2^+$, $\Lambda(1830)5/2^-$
$\Sigma(1670)3/2^-$, $\Sigma(1775)5/2^-$, and $\Sigma(2030)7/2^+$ in the PDG notation~\cite{pdg2014}.
They are assigned as four-star resonances by PDG except for $\Lambda(1600)1/2^+$ with three-star status.
However, within the energy region we are currently interested in,
two more resonances, $\Lambda(1890)3/2^+$ and $\Sigma(1915)5/2^+$, are also rated as four-star by PDG,
but the corresponding resonances are not found in Model B.
This is one example indicating that four-star resonances rated by PDG
using the Breit-Wigner parameters
are not confirmed by the multichannel analyses in
which the resonance parameters are extracted at pole positions~\cite{knls15}.
A noteworthy difference between our two models is the existence of
a new narrow $J^P=3/2^+$ $\Lambda^*$ resonance with $M_R = 1671^{+2}_{-8}-i (5^{+11}_{-2})$ MeV,
which is found only in Model B.
We refer to this resonance as $\Lambda(1671)3/2^+$.
It is located close to the $\eta \Lambda$ threshold and has almost the same $M_R$ value as
the four-star $\Lambda(1670)1/2^-$ resonance
[see the second $J^P(l_{I2J}) = 1/2^- (S_{01})$ resonance in Table~\ref{tab:resmass}].
It is known that the sharp peak of the $K^- p \to \eta \Lambda$ total cross section near the threshold
(see the bottom-right panel of Fig.~\ref{fig:kptcs})
can be explained by the existence of $\Lambda(1670)1/2^-$~\cite{bnl}.
In addition to this, we further find that the new $\Lambda(1671)3/2^+$ resonance seems to be
favored by
the differential cross section data (Fig.~\ref{fig:etldcs}).
Actually, both Models A and B reproduce the $K^- p \to \eta \Lambda$ total cross section equally well, but
Model A with no $\Lambda(1671)3/2^+$ does not reproduce the concave-up behavior of the
differential cross section data well.
\section{Application to neutrino-induced reactions in resonance region}
\label{sec:neutrino}
\begin{figure}[t]
\centering
\includegraphics[clip,width=0.4\textwidth]{kinematics.eps}
\qquad
\includegraphics[clip,width=0.55\textwidth]{neutrino-process.eps}
\caption{
(Left) Kinematical region relevant to neutrino-oscillation experiments.
(Right) Schematic view of neutrino-nucleon reaction processes in resonance region.
}
\label{fig:neutrino1}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[clip,width=0.3\textwidth]{ds_dw_dq2_20.eps}
\qquad
\includegraphics[clip,width=0.3\textwidth]{ds_dw_dq2_21.eps}
\qquad
\includegraphics[clip,width=0.25\textwidth]{neutrino-tcs.eps}
\caption{
(Left) $d\sigma/dWdQ^2$ for $\nu_\mu n \to \mu^- \pi N$ at $E_\nu = 2$ GeV.
(Middle) $d\sigma/dWdQ^2$ for $\nu_\mu n \to \mu^- \pi^+ \pi^- p$ at $E_\nu = 2$ GeV.
(Right) $\sigma$ for $\nu_\mu n \to \mu^- \pi N$ (black curve) and
$\nu_\mu n \to \mu^- \pi \pi N$ (red curve) as a function of $E_\nu$.
See Ref.~\cite{nks15} for the details.
}
\label{fig:neutrino2}
\end{figure}
After the establishment of the neutrino mixing between all three flavors,
major interests in the neutrino physics
are shifting to the determination of the leptonic $CP$ phase and
the neutrino-mass hierarchy (see, e.g., Ref.~\cite{hayato}).
In such neutrino-parameter searches via next-generation neutrino-oscillation experiments,
the lack of knowledge of neutrino-nucleus reaction cross sections
are expected to become one of the major sources of the systematic uncertainties.
Thus the developments of reaction models that give the cross sections
in 10\% or better accuracy are now highly desirable.
Furthermore, the kinematics relevant to the neutrino-oscillation experiments
extends over the quasi-elastic (QE), resonance (RES),
and deep-inelastic-scattering (DIS) regions (see the left panel of Fig.~\ref{fig:neutrino1}),
which are governed by rather different physics mechanisms and theoretical foundations.
These facts lead experimentalists and theorists in different fields to developing a new
collaboration~\cite{kek,kek2} at the J-PARC Branch of KEK Theory Center,
aiming at constructing a unified neutrino-reaction model that covers
all the relevant kinematical regions mentioned above consistently.
We will employ our DCC model as a base model for the RES region.
As a first step towards developing a model for neutrino-nucleus reactions in the RES region,
we have recently constructed a DCC model for neutrino-nucleon ($\nu N$) reactions~\cite{nks15}.
This is actually an important step because it can be used as an input to elementary
processes in nuclear-target reactions.
The right panel of Fig.~\ref{fig:neutrino1} presents
a schematic view of the $\nu N$ reactions in the RES region.
The blue rectangle represents the matrix element for the transition between
the nucleon and meson-baryon ($MB$) states induced by the (``$V-A$''-type) weak currents,
while the red circle represents the rescattering process induced by
the hadronic interactions.
In our DCC model, the rescattering part has been determined through
the analysis of $\pi N$ and $\gamma N$ reaction data.
However, the weak-current matrix elements have to be newly constructed.
The vector-current part can be determined using the data for photon- and electron-induced
meson-production reactions off the proton and neutron targets
(see Sec.~\ref{sec:nucleon} and Ref.~\cite{tsato}), and
in Ref.~\cite{nks15}, the vector-current matrix elements
were determined in the range of $W \leq 2$ GeV and $Q^2 \leq 3$ GeV$^2$.
The axial-current matrix elements are difficult to be determined
because plenty of $\nu N$ and/or $\nu d$ reaction data are required for that purpose, but
the existing data are far from sufficient to constrain our model parameters.
Therefore, at this stage, we determined the axial-current matrix elements
by making use of PCAC hypothesis, and by imposing purely phenomenological
assumptions for their $Q^2$ dependence, for which we will not go into the details here
and just quote Ref.~\cite{nks15}.
With this setup, we can make a prediction for the neutrino-nucleon reactions
in the RES region.
Figure~\ref{fig:neutrino2} shows the results for the charged-current (CC) single- and double-pion
productions predicted from our DCC model.
This is the first-time fully coupled-channels calculation of $\nu N$
reactions beyond the $\Delta(1232)3/2^+$ region.
The contour plot of $d\sigma/dWdQ^2$ for $\nu_\mu n \to \mu^- \pi N$
(the left panel of Fig.~\ref{fig:neutrino2})
show a clear peak due to $\Delta(1232)3/2^+$ and relatively small peak at $W\sim 1.5$
that comes from $N(1535)1/2^-$ and $N(1520)3/2^-$ and so on.
In contrast, for the $\nu_\mu n \to \mu^- \pi^+ \pi^- p$ case,
the contribution from $\Delta(1232)3/2^+$ is barely seen, while higher resonance contributions
are sizable above $W\gtrsim 1.45$ GeV, and their effects on the cross section
reach a rather high $Q^2$, too.
The single-pion productions are known to be a major background in the neutrino-oscillation
experiments at $E_\nu \lesssim 1$ GeV.
However,
above $E_\nu \sim 1$ GeV, the cross section of the double-pion productions
also becomes significant (the right panel of Fig.~\ref{fig:neutrino2})
because of the large branching ratios of higher $N^*$ and $\Delta^*$ resonances
for the decay into the three-body $\pi \pi N$ states.
This implies the need of reliable information on the double-pion productions
for next-generation oscillation experiments using multi-GeV neutrino beams.
\section{Summary}
\label{sec:summary}
We have given an overview of our efforts for the light-quark baryon spectroscopy
through a dynamical coupled-channels analysis of various meson-production reactions
off the nucleon, induced by pion, photon, electron, and anti-kaon beams.
In particular, our recent DCC analysis of $K^- p$ reactions
has successfully extracted parameters associated with $\Lambda^*$ and $\Sigma^*$ resonances
defined by poles of scattering amplitudes, and
has shed light on a possible existence of new
$\Lambda^*$ and $\Sigma^*$ resonances.
However, our results also reveal that the current existing data for $K^- p$ reactions
are still far from sufficient to constrain the extracted partial-wave amplitudes and
$\Lambda^*$ and $\Sigma^*$ resonance parameters unambiguously.
The help of hadron beam facilities such as J-PARC is highly desirable
for further establishing $\Lambda^*$ and $\Sigma^*$ resonances.
Finally, although we did not discuss in this contribution,
our DCC approach has also been applied to meson spectroscopy~\cite{knls11,nkls12}.
We are putting more efforts into this direction, too.
\begin{acknowledgements}
The author thanks T.-S. H. Lee, S. X. Nakamura, and T. Sato for their collaborations.
\end{acknowledgements}
|
1,108,101,563,313 | arxiv | \section{INTRODUCTION}
The Physics community was greatly stimulated by the possibility of supersolidity suggested by the torsional oscillator experiments of Kim and Chan\cite{Kim2004a,Kim2004b}; many groups reported corroborating evidence. But, with the realization that previously unexpected shear modulus behavior was present in the solid\cite{Day2007} the community began to question the supersolid interpretation from several perspectives\cite{Reppy2010}. More recent work has shown that these mechanical effects clearly were dominant and it is now believed that there is little, if any, evidence for a supersolid available from torsional oscillator experiments\cite{Kim2014}. In conceptually different work, studies of the flux of $^4$He that passes through a sample cell filled with solid helium have been carried out\cite{Ray2008a,Ray2009b,Ray2010a,Ray2010c,Vekhov2012,Vekhov2014,Vekhov2014b}. These experiments revealed the dependence of the flux rate on the solid helium temperature, the applied chemical potential difference, $\Delta \mu$, and the $^3$He impurity concentration. They also revealed a dramatic reduction of the flux at a $^3$He concentration-dependent temperature, $T_d$, a universal temperature dependence above $T_d$ and no flux above $T_h \approx$ 630 mK, etc. Some of the UMass data\cite{Vekhov2012,Vekhov2014} were interpreted to be consistent with one-dimensional conductivity\cite{Boninsegni2007} through the solid, a so-called Luttinger-like behavior\cite{DelMaestro2010,DelMaestro2011}.
In our typical experimental arrangement, referred to as the UMass Sandwich\cite{boris-06} (Fig.~\ref{cell_diagram_bw.eps}), a solid helium sample is located between two superfliud-filled Vycor electrodes, which in turn are connected to two superfluid-filled reservoirs. Recently a number of the key experimental results from the UMass group have been confirmed by Cheng et al.\cite{Balibar.APS2015}. Instead of applying $\Delta \mu$ between two liquid helium reservoirs at the ends of two Vycor rods, they mechanically squeezed solid helium in one chamber and observed a pressure gauge response in another solid helium chamber which was separated from the other by superfluid-filled Vycor. Based on the data from both groups, Cheng et al. concluded that the flux rate reduction at $T_d$ is likely limited not by solid helium itself but by $^3$He condensation at the interface between solid helium and liquid helium in the Vycor. The UMass group had suggested that this flux reduction at $T_d$ might be due to the condensation of $^3$He atoms at intersections of dislocation cores or on the dislocations themselves\cite{Corboz2008}. Additional confirmation of evidence for flux through solid helium has been reported recently by Haziot et al.\cite{Chan.APS2015}.
Here we report work that seeks to understand where the limitation of the temperature-dependent mass flux resides for temperatures in the range \emph{above} $T_d$, $T_d < T < T_h$. Is this bottleneck inside the solid helium itself or at the interface between the Vycor and solid helium? We will show evidence that the bottleneck for $T > T_d$ resides in the solid-filled cell itself and not at the interface between the superfluid in the Vycor and the solid helium.
Solid helium samples are typically grown from the nominally pure (0.17 ppm $^3$He impurity\cite{Vekhov2014b}) superfluid by filling the sample cell to near the pressure of the melting curve through a direct-access capillary and subsequently increasing the pressure above the melting curve by use of capillaries connected to the sample cell in series with Vycor rods (porous glass with interconnected pores of diameter about 7~nm) at a constant solid helium temperature, $TC \sim 350$~mK. The Vycor rods are 1.40~mm in diameter and 7.62~cm in length.
In our previous work we created chemical potential differences between two superfluid filled reservoirs to study the resulting flux of helium through the solid-filled cell, Fig.~\ref{cell_diagram_bw.eps}. To explore the bottleneck for $T > T_d$ a modification is made to the experimental arrangement and procedure. Using our standard sample cell\cite{Ray2008a,Ray2009b,Ray2010a,Ray2010b,Vekhov2012}, in this work the top of one Vycor rod, $V2$, is plugged by a high enough temperature to avoid superfluid mass flux through this rod. So, only Vycor rod $V1$ can be a conduit for $^4$He atoms. The temperature, $T1$, of the liquid helium reservoir on the top of $V1$ is kept in the rage of $1.46 - 1.51$~K while the solid helium sample (in the form of a horizontal cylinder of $1.84$~cm$^3$ volume and $4.5$~cm length) has a temperature $TC = 0.1-0.8$~K and pressure $25.9 - 26.4$~bar as measured by two \textit{in situ} capacitance pressure gauges, $C1$ and $C2$, located at the ends of the cylindrical sample cell, 10 mm and 33 mm, respectively, from the end of $V1$ in the solid.
Due to the temperature difference between the solid helium and the liquid helium reservoir, $R1$, the thermo-mechanical effect causes a pressure difference between them, $\Delta P$.
The higher $T1$, the larger is the pressure difference between liquid helium in the reservoir at one end of the Vycor rod and the solid helium sample at the other.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.8\linewidth,keepaspectratio]{cell_diagram_bw.eps}}%
\caption{Schematic diagram of the cell used for
flow experiments. Two capillaries, 1 and 2, go to
liquid reservoirs $R1$ and $R2$ at the top ends of the Vycor rods, $V1$ and $V2$. Capillary 3 enters from the side and is used for initial additions of helium to the cell. Two capacitance
pressure gauges, $C1$ and $C2$, are located on either end of the cell. Gauges $P1$ and $P2$ are outside the
cryostat. Heaters $H1$, $H2$, allow the temperatures of the reservoirs to be controlled. For the work reported here the heater $H2$ is used to elevate the temperature of the top of $V2$ to block the flow of atoms to or from reservoir $R2$. [Reproduced from Fig. 1 in Ref. \cite{Ray2009b}] \label{cell_diagram_bw.eps}}
\end{figure}
Thus, if one wants to change $\Delta P$ and thereby inject (extract) helium atoms into (from) the solid sample, the temperature $T1$ has to be decreased (increased) by some amount, $\delta T$. The response of $C1$ and $C2$ reveals the presence of pressure gradients, if any, and their time-dependent relaxation along the solid helium sample.
There are several possible bottlenecks that might restrict the flow of $^4$He atoms for $T > T_d$: (a) the interface between the helium in the reservoir and the Vycor rod, (b) the Vycor rod, (c) the interface between the superfluid helium in the Vycor rod and the solid helium, (d) the conduction process in solid helium and (e) the dynamics of edge dislocations which are thought to be responsible for the density changes and pressure changes in the solid helium \cite{Soyler2009}.
As was shown in Ref.~\cite{Vekhov2014}, to avoid a bottleneck anywhere in the Vycor rod and at the interface to the superfluid reservoir the temperature $T1$ typically has to be $< 1.49$~K for a low solid helium temperature, $TC$, where the highest flux rates are observed. It was also shown in Ref.~\cite{Vekhov2014} that $T1$ can slightly exceed 1.49~K if $TC > 0.25$~K. We can control the bottleneck due to the Vycor rod by control of $T1$. Next, growth/dissolution of edge dislocations is likely not a bottleneck in the mass flux experiments made earlier \cite{Ray2008a,Ray2009b,Ray2010a,Ray2010b,Vekhov2012,Vekhov2014,Vekhov2014b} because those measurements detected the flux \textit{through} solid helium samples in the presence of a variety of stable pressure gradients inside the solid samples. These various gradients had no effect on the flux. Thus, we are left here to distinguish between two possibilities, (c) and (d). If the mass flux bottleneck for $T > T_d$ is at the interface between the Vycor rod and the solid helium and not in the solid helium itself, then we might expect that no difference in the behavior of $C1$ and $C2$ will be observed when atoms enter or leave the solid through $V1$ in response to changes in $T1$. But, if the bottleneck is in the solid helium sample itself, then we might expect that the time dependent behavior of $C1$ and $C2$ will be different due to the difference in the distances of these gauges from the $V1$ rod.
\begin{figure}[htb]
\centerline{\includegraphics[width=1.2\linewidth,keepaspectratio]{Fig-Procedure386mKabcd-2.eps}}
\caption{(Color online) Procedure of syringe measurements: a) Liquid helium reservoir temperature, $T1$, change to initiate the mass flux to and from the sample cell at $TC = 386$~mK; b) response of $C1$ and $C2$ \textit{in situ} pressure gauges; c) $C1^*$ and $C2^*$ behavior after pressure drift and steady pressure gradient subtraction; d) the difference between $C1^*$ and $C2^*$.}
\label{Fig-Procedure386mK}
\end{figure}
Data that illustrates the procedure of the measurements are shown in
Fig. \ref{Fig-Procedure386mK} for the case of a solid helium temperature, $TC = 386$~mK, midrange in the $T_d < T < T_h$ span in which flux has been previously documented. The time interval between data points is about 13~s. Fig. \ref{Fig-Procedure386mK}~(a) shows a flux-initiating $T1$ temperature change between $1.508 \pm 0.001$ and $1.466 \pm 0.002$~K. One can see that the $T1$ decrease is slower than its increase. We believe that this is due to the thermal impedance between the reservoir $R1$ and the connection to our refrigeration. Increases in $T1$ are prompt because of the proximity of the heater and the reservoir.
The rate of decrease of $T1$ slows at lower solid helium temperatures. During the measurements, the $T1$ temperature change is much faster than the pressure change in solid helium and does not measurably limit the kinetics of this pressure change. Fig. \ref{Fig-Procedure386mK}~(b) presents the response of the \textit{in situ} pressure gauges $C1$ and $C2$. As known from our previous work\cite{Ray2008a,Ray2009b} and can be seen here, $C1$ and $C2$ reveal the presence of a steady pressure difference, $\Delta C_{st}$ (in some samples up to 0.1~bar) in this solid helium sample. This $\Delta C_{st}$ anneals at high enough $TC$ temperatures, usually above 0.7~K. There is a small pressure drift in the solid helium due to a pressure drift in the Vycor filling line which is in turn caused by a level change of liquid helium in the 4K bath of the cryostat. Fig. \ref{Fig-Procedure386mK}~(c) shows the same $C1$ and $C2$ data, but after subtraction of the pressure drift and steady pressure gradient, now denoted $C1^*$ and $C2^*$, where the * designates that these subtractions have taken place. This background subtraction allows a determination of the difference in the kinetics and steady state behavior after the relaxation of the $C1$ and $C2$. Next, for clarity, we introduce the subscript \emph{i} to denote the value of a parameter before an initial change in the reservoir temperature, $T1$; \emph{f} represents the value of a parameter after a new equilibrium is reached.
In the steady state, there is a small difference between $\Delta C1^* = C1^*_f - C1^*_i$ and $\Delta C2^* = C2^*_f - C2^*_i$ (see Fig. \ref{Fig-Procedure386mK}~(c)) after a decrease in $T1$ and mass addition to the solid. $\Delta C1^*$ is always larger than $\Delta C2^*$ and this difference is larger for higher $TC$ temperatures.
The kinetic behaviors of $C1$ and $C2$ are significantly different. The pressure gauge closest to the $V1$ Vycor rod, $C1$, shows a much faster response than the further, $C2$, gauge. This leads to the formation and relaxation of a kinetic pressure difference, $\Delta C^*_k(t) = C1^*(t) - C2^*(t)$. The $C1$ kinetics above $TC \approx 300$~mK has non-monotonic behavior as shown in Fig.~\ref{Fig-Procedure386mK}(c): a fast increase/decrease and then relaxation back to a new equilibrium value. The $C2$ kinetics in turn has a monotonic behavior in the whole range of $TC$ temperatures studied. This non-monotonic $C1$ behavior is likely due to a difference between the mass flux conductivity from $V1$ to $C1$ and to $C2$. The flux from $R1$ to $C1$ is faster then from $R1$ to $C2$, apparently due to the difference in distance between $V1$ and the two detectors.
\begin{figure}[htb]
\centerline{\includegraphics[width=1.1\linewidth,keepaspectratio]{Fig-dCk-3.eps}}%
\caption{(Color online) Kinetic pressure difference, $\Delta C^*_k = C1^* - C2^*$, along a solid sample at different $TC$ temperatures after syringing (a) and withdrawing (b) helium atoms to and from the sample cell, respectively. A pressure drift and stationary pressure gradient in a solid sample are subtracted as a background. For the insert, see text discussion.}
\label{Fig-dCk}
\end{figure}
Based on data of the sort presented in Fig. \ref{Fig-Procedure386mK}(c), the kinetic pressure
gradient $\Delta C^*_k = C1^* - C2^*$ is shown in Fig.~\ref{Fig-dCk} for several $TC$ temperatures. Thus, Fig.~\ref{Fig-dCk}(a) [(b)] shows $\Delta C_k$ after a $T1$ temperature decrease [increase], i.e. after adding [subtracting] helium atoms through the $V1$ Vycor rod. Several features can be noted here: (a) the higher the $TC$, the slower the $\Delta C_k$ development and its further relaxation; (b) the maximum amplitude of $\Delta C_k$ depends non-monotonically on $TC$;
(c) $\Delta C_k$ never relaxes to the initial zero-level, furthermore, the higher $TC$, the larger residual $\Delta C_k^{res}$ value (see Fig.~\ref{Fig-dCk}b, insert).
\begin{figure}[htb]
\centerline{\includegraphics[width=1.0\linewidth,keepaspectratio]{Fig-FCvsT.eps}}%
\caption{(Color online) Temperature dependence of the maximum flux (that is not limited by the Vycor) measured by $C1$ (red data points) and $C2$ (blue data points) during step-wise sample warming (solid data points) and subsequent cooling (open data points); $\delta T = 40$~mK.}
\label{Fig-FCvsT}
\end{figure}
In order to determine a measure of the rate of mass flux from $R1$ through the Vycor rod $V1$ to both \textit{in situ} pressure gauges, $C1$ and $C2$, through the solid helium, the derivatives of the pressures $C1$ and $C2$, $FC = dC/dt$, are taken by means of a three-point algorithm and their maximum values for a constant $T1$ change, $\delta T$, are plotted in Fig.~\ref{Fig-FCvsT} for the same solid helium sample. These data are shown for the case of a $T1$ increase.
One can see that $FC1^{max}$ is several times faster than $FC2^{max}$. This $FC$ flux also monotonically decreases with increasing temperature, which is similar to the temperature dependence found for the mass flux \textit{through} a solid helium filled cell and measured by the pressure gauges on the top of both Vycor rods in previous work \cite{Vekhov2012, Vekhov2014b}. The data here are presented for initial sample warming (solid symbols) and subsequent cooling (open symbols). Before warming, the sample had not been annealed. At the highest temperatures the sample starts to anneal resulting in a flux rate decrease seen on cooling.
\begin{figure}[htb]
\centerline{\includegraphics[width=1.1\linewidth,keepaspectratio]{Fig-1FCvslogT-all.eps}}%
\caption{(Color online) (a) Temperature dependence of the flux resistance, $(FC^{max})^{-1}$, measured by $C1$ (circles) and $C2$ (triangles); $\delta T = 40$~mK. Although the data are limited, the solid lines represent fits to $AT + BT^\alpha$; $\alpha$ is not particularly well determined in the range 5-10. (b) With flux resistance data from many samples from Ref.\cite{Vekhov2014b}, $\alpha = 5.8 \pm 0.3$. The dashed lines in all cases represent a linear temperature dependence. }
\label{Fig-1FCvslogT}
\end{figure}
The data from Fig.~\ref{Fig-FCvsT} can be inverted, $(FC^{max})^{-1}$, to obtain something we might call a flux resistance as shown in Fig.~\ref{Fig-1FCvslogT}(a). When this is done, it appears that there is a crossover in behavior from one conductance regime to another.
We note here that our recent mass flux data \textit{through} many solid samples of various $^3$He concentrations for $T>T_d$ (see Ref.\cite{Vekhov2014b}, Fig.4) demonstrated a universal temperature dependence for the maximum flux, $F$, when normalized, $F/F(T=0.2K)$. That same data can also be presented as a flux resistance, $F(T=0.2K)/F$, as shown in Fig.~\ref{Fig-1FCvslogT}(b). When that is done we see that it demonstrates a very similar behavior. That is, the flux measured \emph{through} many samples also shows this apparent crossover behavior.
It is a question for further study to determine what the origin of the contributions to the temperature dependence of the conductance is that gives rise to this apparent crossover behavior. It is possible that quantum phase slips\cite{Zaikin2008} are responsible, as has been seen for narrow superconducting fibers\cite{Tian2005,Pai2008,Zaikin2008}.
In conclusion we have shown that measured changes in the pressure in a cell filled with solid helium when mass is injected depend on the relative location of the pressure gauges with which the measurements are made. We interpret these measurements to indicate that the bottleneck to the temperature dependent flux \emph{above} the temperature $T_d$ is due to superflow dissipation in the solid sample itself and the flux is not limited by interface effects where superfluid in the Vycor contacts the solid helium; the bottleneck for $T > T_d$ is a bulk phenomena and not a surface effect. Our results support the possibility that the dissipative mass flux seen may be due to the superfluid cores of edge dislocations that carry the flux, but do not conclusively prove it.
We appreciate the early contributions to the apparatus by M.W. Ray and conversations with colleagues in the field, particularly B.V. Svistunov, J. Beamish and M. Chan. This work was supported by the National Science Foundation grant No. DMR 12-05217 and by Trust Funds administered by the Univ. of Mass. Amherst.
|
1,108,101,563,314 | arxiv | \section{Introduction}
Neurological disorders, including Alzheimer's disease (AD) and schizophrenia (SZ), are among the main priorities in the present global health agenda \cite{world2006neurological}. AD is a type of dementia that affects primarily elderly individuals and is characterized by the degeneration of brain tissue, leading to impaired intellectual and social abilities \cite{dolgin2016defeat}. Currently, around 25 million people live with AD \cite{Organization/2021}. In the US, nearly six million individuals are affected by AD, with incidence projected to increase more than two-fold to 13.8 million by 2050 \cite{rodriguez2021machine}. Individuals with SZ have symptoms such as hallucinations, incoherent thinking, delusions, decreased intellectual functioning, difficulty in expressing emotions, and agitation \cite{jahmunah2019automated,gottesman1982schizophrenia}. According to the World Health Organization (WHO), SZ affects around 26 million people worldwide \cite{Organization/20212}.
The base for a successful treatment of AD and SZ is the correct diagnosis. However, in the absence of established clinical tests for neurological disorders, both the diagnosis and the determination of the stage of AD and SZ are based primarily on qualitative interviews, including psychiatric history and current symptoms, and the assessment of behaviour. These observations may be subjective, imprecise, and incomplete \cite{borsboom2013network,Fried2017mental,borsboom2008psychometric}.
To provide a quantitative evaluation of mental disorders, methods based on Magnetic Resonance Imaging (MRI), Computerized Tomography (CT) \cite{jack2009serial}, and Positron Emission Tomography (PET) \cite{ding2019deep,walhovd2010combining} has been used to aid professionals in the diagnostic process \cite{del2018trimage}. However, the use of multiple imaging devices can be expensive to implement and the fusion of images from different devices can have poor quality due to motion artifacts.
\begin{figure*}[t]
\centering
\includegraphics[width=0.99\textwidth]{CNN_new.pdf}
\caption{Illustration of the method for automatic diagnosis of mental disorders based on EEG time series. Time series are collected and the correlation between electrodes are calculated yielding the matrices of connections, which encompass the functional connectivity between brain regions. Finally, the CNN is adjusted to enable the automatic classification of individuals.}
\label{Fig:methodology}
\end{figure*}
To overcome these restrictions, EEG data is a viable candidate to support the diagnosis of SZ and AD \cite{tait2020eeg}. Although EEG has a low spatial resolution, it has a comparatively low cost, good temporal resolution and is easily available in most contexts. Nonetheless, visual analysis of EEG data is time-consuming, requires specialized training, and is error-prone \cite{trambaiolli2011eeg,falk2012eeg,piubelli2021serum}. However, we can consider automatic evaluation of EEG time series using modern classification algorithms, which can help to improve the efficiency and accuracy of AD and SZ diagnosis, as verified in previous works \cite{pineda2020quantile,oh2019deep,ahmadlou2011fractality,buettner2020development}.
Moreover, instead of using raw EEG time series, it is possible to encompass the connections between brain regions by constructing cortical complex networks~\cite{sporns2002network}. In this case, we build cortical networks for healthy and individuals with neurological disorders. To distinguish between them, we use network measures to describe the network structure, as described in a previous work of ours \cite{de2014structure} (see also \cite{costa2007characterization, da2010pattern} for a description of the methodology used in network classification). Therefore, each network is mapped into a $d$ dimensional space, where $d$ is the number of measures adopted for network characterization. This process of building a set of features to represent the input data is called feature engineering. After extracting the network features for the two classes of networks, i.e. healthy and individuals with mental disorders, supervised learning algorithms are adjusted to perform automatic classification. Previous works verified that this approach enables the diagnosis with accuracy higher than $80\%$ in the case of childhood-onset schizophrenia~\cite{de2014structure}.
Although this methodology has been used for many different diseases (e.g. \cite{pineda2020quantile,de2014structure,diykh2017classify}) the performance of the algorithm depends on the measures selected to describe the network structure. The network properties included in the model could represent just a subset of the information necessary to get the best performance of the supervised model. Therefore, the network representation can be incomplete, restricting the accuracy of the classifiers. One possible solution to this problem is the use of a matrix of connections in combination with and deep neural networks~\cite{goodfellow2016deep}, as we show in the present paper. In this case, instead of extracting the network measures, the matrix of connections is considered as input to train a deep neural network. This matrix encodes all the information necessary to represent the network structure and avoid the choice of network measures.
Therefore, we consider the matrix of connections between brain areas and deep neural networks to distinguish individuals with AD and SZ from healthy controls. Other than previous works, where only raw time series are adopted as input for the neural network\cite{acharya2018deep,kashiparekh2019convtimenet,islam2018brain,duneja2019analysis,acharya2018automated,oh2018deep,yildirim2018deep}, we do not ignore the connections between the electrodes used to record the time series. We construct the matrix of connections by using Granger causality, Pearson's and Spearman's correlations \cite{bastos2016tutorial,seth2015granger,bonita2014time}. We verify that this information about the connections is fundamental and improves the classification, compared to the previously mentioned approaches that use only raw EEG time series.
In summary, in this work we achieve the following contributions:
\begin{itemize}
\item We propose a method to classify EEG time series from healthy and patients presenting AD and SZ. With a matrix of connections as input for a tuned Convolutional Neural Network (CNN) model, the accuracy obtained is close to 100 \% for both disorders. Our results are more accurate than those observed in previous works that consider only raw EEG time series, reinforcing the importance of the network structure on the diagnosis of mental disorders.
\item We show that the method to infer the matrices of connections influences the quality of the classification results. For SZ, the Granger causality provides the most accurate classification, whereas, for AD, the Pearson's correlation yields the highest accuracy.
\item Our framework is general and can be used in EEG data from any brain disorder. It allows to determine the best cortical network representation and adjust the CNN to optimize the accuracy.
\end{itemize}
In the next sections, we outline the data set, present the CNN architecture and show our results, comparing them with more common approaches that do not consider the connections between brain areas.
\section{EEG data} \label{Sec:database}
The AD data set considered here is composed of EEG time series recorded at a sampling frequency of 128~Hz and a duration of eight seconds for each individual and at 19 channels ($F_{p1}$, $F_{p2}$, $F_{7}$, $F_{3}$, $F_{z}$, $F_{4}$, $F_{8}$, $T_{3}$, $C_{3}$, $C_{z}$, $C_{4}$, $T_{4}$, $T_{5}$, $P_{3}$, $P_{z}$, $P_{4}$, $T_{6}$, $O_{1}$, and $O_{2}$)~\cite{pineda2020quantile,pritchard1991altered}. The letters F, C, P, O, and T refer to the respective cerebral lobes frontal (F), central (C), parietal (P), occipital (O), and temporal (T). The data is divided into two sets. The first one consists of 24 healthy elderly individuals (control group; aged 72 ± 11 years) who do not have any history of neurological disorders. The second one is made of 24 elderly individuals with AD (aged 69 ± 16 years) diagnosed by the National Institute of Neurological and Communicative Disorders and Stroke, the Alzheimer’s Disease and Related Disorders Association (NINCDS-ADRDA), following the Diagnostic and Statistical Manual of Mental Disorders (DSM)-III-R criteria (\cite{pineda2020quantile,pritchard1991altered}).
The data set used for diagnosis of SZ can be found at \cite{timashev2012analysis}. This data contains 16-channel EEG time series recorded at a sampling frequency of \nolinebreak{128 Hz} over one minute, including $F_{7}$, $F_{3}$, $F_{4}$, $F_{8}$, $T_{3}$, $C_{3}$, $C_{z}$, $C_{4}$, $T_{4}$, $T_{5}$, $P_{3}$, $P_{z}$, $P_{4}$, $T_{6}$, $O_{1}$, and $O_{2}$. Notice that both data set come from studies of 16 common brain regions, with the AD data set having three more regions analyzed. Furthermore, it also includes two sets, (i) one of 39 healthy young individuals (control group; aged 11 to 14 years) and (ii) one of 45 teenagers individuals (aged 11 to 14 years) with symptoms of schizophrenia.
\section{Concepts and Methods}
\label{Sec:concepts}
Our framework to perform the automatic diagnosis of AD and SZ is illustrated in Figure \ref{Fig:methodology}. In a first step, EEG time series, which are free of artifacts, are used to construct the matrices of connections. The strength of the connections between two brain regions is quantified by three different methods: (i) Granger causality test \cite{granger1969investigating}, (ii) the Pearson's ~\cite{benesty2009pearson} and (iii) Spearman's~\cite{lubinski2004introduction} correlation measures. For Granger causality, a statistical hypothesis test is done and p-values are obtained. The matrices are filled with ``1'' if $p < 0.05$ and ``0'' if $p\geq 0.05$. Matrices are calculated for AD data sets (19 EEG channels) and for SZ data sets (16 EEG channels). These matrices are inserted in a CNN to discriminate healthy individuals from individuals diagnosed with AD and SZ (see Figure \ref{Fig:methodology}).
Notice that the use of different methods to infer the brain areas is necessary because there is no general method to infer functional connectivity~\cite{bastos2016tutorial,seth2015granger,bonita2014time, comin2020complex}.
Indeed, choosing the best metric to infer these connections between brain areas is a current challenge in network neuroscience (e.g.~\cite{shandilya2011inferring, lusch2016inferring}).
\subsection{Convolutional Neural Network}
\label{Sec:CNN}
CNN is a type of neural network \cite{millstein2020convolutional} with three types of layers and masked parameters, as proposed in \cite{hubel1962receptive, lopez2002convolutional}.
The convolutional layer performs the mathematical operation called convolution, which is done in more than one dimension at a time. The weights of the artificial neurons are represented by a tensor called kernel (or filter). The outputs from the convolutional layer include the main features from the input data. The convolution process between neurons and kernels produces outputs called feature maps.
The pooling layer reduces the dimensionality and operates similarly to the convolutional layer. The difference is that pooling kernels are weightless and add aggregation functions to their input data, such as a maximum or mean function \cite{lecun1989generalization,lecun2015deep}. The max pooling function is used here to return the highest value within an area of the tensor, which reduces the size of the feature map.
The fully connected layer categorizes input data into different classes, based on an initial set of data used for training. The artificial neurons in the max pooling and fully connected layers are connected, as the output predicts precisely the result of the input EEG data as healthy and unhealthy \cite{oh2019deep}.
Two approaches for the CNN architectures are proposed here, one using a tuning method (CNN$_{tuned}$) and another without this optimization step (CNN$_{untuned}$). Tuning is an optimization method used to find the values of hyperparameters to improve the performance of the CNN model \cite{hutter2015beyond}. Three tuning techniques are used in the present work: (i) random search \cite{bergstra2012random}, (ii) hyper-band \cite{rostamizadeh2017efficient} and (iii) Bayesian optimization \cite{doke2020using}. The traditional way to optimize the hyperparameters is exhaustive searching through a manually specified parameters search space and evaluate all possible combinations of these parameter values. However, this approach has a high computational cost. An alternative method is to select the values of parameters in the search space at random until maximize the objective function (here, this objective function is the maximization of accuracy).
The idea of hyper-band optimizations is to select different possible models (with different hyperparameters values), train them for a time, and discard the worst one at each iteration, until a few combinations remain.
In contrast, Bayesian optimization is a global optimization method that uses the Bayes Theorem to direct the search to find the minimum or maximum of a certain objective function \cite{goodfellow2016deep}.
In the CNN$_{tuned}$ model, the dropout regularization technique is employed to avoid overfitting~\cite{srivastava2014dropout}. The layers and range used for hyperparameters are presented in table \ref{Tab:hyper}. The best CNN$_{tuned}$ architectures tuned for each data set individually are depicted in table \ref{table:tuning-sz-ad}. The CNN$_{untuned}$ model presents fewer layers and therefore lower computational costs. The parameters used in our analysis are described in table \ref{Table:cnn-untuned}.
\begin{table*}[!t]
\centering
\caption{Best hyperparameters and layer configurations obtained for the CNN$_{tuned}$ model.}
\vspace{3mm}
\begin{tabular}{cccc}
\hline
\textbf{Type of Layer} & \textbf{Tuning hyperparameter} & \textbf{Value} \\
\hline
Convolutional &--- &--- \\
\hline
& & [0.00, 0.05, 0.10, 0.15, \\
Convolutional &dropout & 0.20, 0.25, 0.30, \\
& & 0.35, 0.40, 0.45, 0.50] \\
\hline
Convolutional & --- & --- \\
\hline
\hline
Convolutional & number of filters & [32, 64] \\
\hline
\hline
Max Pooling & dropout & [0.00, 0.50, 0.10, 0.15, 0.20] \\
\hline
Flatten & --- & --- \\
\hline
Dense& - units&[32, 64, 96....512]\\
& -activation & [relu, tanh, sigmoid]\\
\hline
Dropout & rate & [0.00, 0.50, 0.10, 0.15, 0.20] \\
\hline
Adam & & $min-value =1e^{-4}$ \\
optimization & learning & $max-value=1e^{-2}$ \\
compile & rate & sampling= LOG \\
\hline
\end{tabular}
\label{Tab:hyper}
\end{table*}
\begin{table*} [!t]
\centering
\caption{The network architecture for the CNN$_{tuned}$ model used in the AD and SZ data sets.}
\vspace{3mm}
\begin{tabular}{ccccc}
\hline
\textbf{Type of Layer} & \textbf{Output Shape (AD)} &
\textbf{Output Shape (SZ)}
&\textbf{Parameter}\\
\hline
Convolutional &(None, 17, 17, 16) &(None, 14, 14, 16) &160 \\
\hline
Convolutional &(None, 15, 15, 16) &(None, 12, 12, 16) &2320\\
\hline
max-pooling &(None, 7, 7, 16) &(None, 6, 6, 16)&0 \\
\hline
dropout & (None, 7, 7, 16)&(None, 6, 6, 16)&0 \\
\hline
Convolutional &(None, 5, 5, 32)&(None, 4, 4, 32) &4640 \\
\hline
Convolutional & (None, 3, 3, 32)& (None, 2, 2, 32)&9248 \\
\hline
max-pooling & (None, 1, 1, 32)& (None, 1, 1, 32)&0 \\
\hline
dropout &(None, 1, 1, 32) & (None, 1, 1, 32)&0 \\
\hline
flatten & (None, 32) & (None, 32)&0\\
\hline
dense & (None, 160) & (None, 160) &5280 \\
\hline
dropout &(None, 160) & (None, 160) &0 \\
\hline
dense &(None, 2) & (None, 2)&3 \\
\hline
\end{tabular}
\label{table:tuning-sz-ad}
\end{table*}
\begin{table*}
\centering
\caption{The network architecture for the CNN$_{untuned}$ model used in the AD and SZ data sets.}
\begin{tabular}{ccccc}
\hline
\textbf{Type of Layer} & \textbf{Output Layer (AD)} & \textbf{Output Layer (SZ)} &\textbf{Kernel} \\ \hline
\textbf{Input Layer} & 19 x 19 x 1 & 16x16x1 &- \\ \hline
\textbf{Convolution} & 18 x 18 x 32 &15 x 15 x 32& 4 \\
\hline
\textbf{Max pooling} & 18 x 18 x 32&15 x 15 x 32 & 2 \\
\hline
\textbf{Convolution} & 17 x17 x 16 & 14 x 14 x 16 & 4 \\
\hline
\textbf{Max pooling} & 17 x17 x 16 & 14 x 14 x16 & 2 \\
\hline
\textbf{Flatten} & 17 x17 x 16 & 3136 & - \\
\hline
\multicolumn{1}{l}{\textbf{Fully connected}} & 10 &10 & - \\
\hline
\textbf{Fully connected} & 1 &1 &- \\ \hline
\end{tabular}
\label{Table:cnn-untuned}
\end{table*}
\subsection{Evaluation}
Since we have a two-class classification problem (negative and positive), we consider the precision and recall measures in the evaluation process \cite{maimon2010data}. Precision (also called specificity) corresponds to the hit rate in the negative class, whereas recall (also called sensitivity) measures how well a classifier can predict positive examples (hit rate in the positive class).
For visualization of these two measures, the receiver operating characteristic (ROC) curve is a common method as it displays the relation between the rate of true positives and false positives. The area below this curve, called area under ROC curve (AUC) has been widely used in classification problems \cite{huang2005using}, mainly for medical diagnoses \cite{ozcift2011classifier,shen2016evolving,tanwani2009guidelines,nanni2010local}. The value of the AUC varies from 0 to 1, where the value of one corresponds to a classification result free of errors. $AUC = 0.5$ indicates that the classifier is not able to distinguish the two classes --- this result is equal to the random choice. Furthermore, we consider the micro average of ROC curve, which computes the AUC metric independently for each class (calculate AUC metric for healthy individuals, class zero, and separately calculate for unhealthy subjects, class one) and then the average is computed considering these classes equally. The macro average is also used in our evaluation, which does not consider both classes equally, but aggregates the contributions of the classes separately and then calculates the average.
\section{Results and discussion}
\begin{figure*}\centering
\subfigure[]{
{\includegraphics[width=0.450\textwidth]{pearson-sick-alzheimer.pdf}}}
\subfigure[]{
{\includegraphics[width=0.450\textwidth]{pearson-healthy-alzheimer.pdf}}}
\caption{Example of matrices of connections calculated with Pearson's correlation for (a) an individual with diagnosed AD and (b) an healthy individual.}
\label{Fig:matrices-pearson}
\end{figure*}
\begin{figure*}[!t]
\centering
\subfigure[]{
{\includegraphics[width=0.450\textwidth]{granger-disease-schizophrenia.pdf}}}
\subfigure[]{
{\includegraphics[width=0.450\textwidth]{granger-healthy-schizophrenia.pdf} }}
\caption {Example of matrices of connections calculated with Granger causality test for (a) an individual with diagnosed SZ and (b) an healthy individual.}
\label{Fig:matrices-granger}
\end{figure*}
We consider the EEG time series described in section~\ref{Sec:database} to construct the matrices of connections for healthy controls and individuals diagnosed with AD and SZ, following the description in section~\ref{Sec:concepts}. These matrices are built by using the Granger causality test, Pearson's and Spearman's correlations measures for both data sets. In Figures \ref{Fig:matrices-pearson} and \ref{Fig:matrices-granger}, some examples of such matrices of connections are showed and differences between them can be noticed visually in both cases.
The matrices of connections are inserted into the CNN by applying the flattening method, which converts the data into a 1-dimensional array that is input to the next layer.
Two CNN architectures are considered, i.e.
CNN$_{tuned}$ and CNN$_{untuned}$, to evaluate the classification. The CNN$_{tuned}$ is obtained by hyperparameter optimization, whereas the CNN$_{untuned}$ is a simpler model, without using the tuning optimization. The evaluation of both models is done by using the area under the ROC curve (AUC). Nested k-fold cross-validation ($k=10$) for model selection, adjustment and evaluation is considered here.
The results for the CNN$_{tuned}$ model is shown in tables \ref{Tab:ad} and~\ref{Tab:sz}, for AD and SZ, respectively. In all the cases, the CNN$_{tuned}$ model can unambiguously distinguish healthy individuals from individuals diagnosed with a brain disorder. The best results with an accuracy close to $100\%$ are obtained for both AD and SZ in the testing set using random search for hyperparameter tuning.
\begin{table*} [!t]
\centering
\caption{Classification results for AD using the CNN$_{tuned}$ model (best results are in bold).}
\vspace{3mm}
\begin{tabular}{cccccccc}
\hline
\textbf{\begin{tabular}[c]{@{}c@{}} Matrices of\\ connections \end{tabular}} & \textbf{Hyperparameter} & \textbf{Sample} & \textbf{Accuracy} & \textbf{Precision}& \textbf{Recall} & \textbf{AUC} \\ \hline
\multirow{6}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Granger\\ causality\end{tabular}}} & \multirow{2}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Random\\ Search\end{tabular}}} & \textbf{Train} & 0.81 & 0.81 & 0.81 & 0.88 \\
& & \textbf{Test} & 0.75 & 0.75 & 0.75 & 0.97 \\
& \multirow{2}{*}{\textbf{hyper-band}} & \textbf{Train} & 0.65 & 0.65 & 0.65 & 0.65 \\
& & \textbf{Test} & 0.75 & 0.75 & 0.75 & 0.97 \\
& \multirow{2}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Bayesian \\ Optimization\end{tabular}}} & \textbf{Train} & 0.68 & 0.68 & 0.68 & 0.82 \\
& & \textbf{Test} & 0.75 & 0.75 & 0.75 & 0.93 \\\hline
\multirow{6}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Pearson's \\ correlation\end{tabular}}} & \multirow{2}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Random\\ Search\end{tabular}}} & \textbf{Train} & \textbf{0.95} & \textbf{0.95} & \textbf{0.95} & \textbf{0.98} \\
& & \textbf{Test} & \textbf{1.00} & \textbf{1.00} & \textbf{1.00} & \textbf{1.00} \\
& \multirow{2}{*}{\textbf{hyper-band}} & \textbf{Train} & 0.86 & 0.86 & 0.86 & 0.90 \\
& & \textbf{Test} & 1.00 & 1.00 & 1.00 & 1.00 \\
& \multirow{2}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Bayesian \\ Optimization\end{tabular}}} & \textbf{Train} & 0.88 & 0.88 & 0.88 & 0.98 \\
& & \textbf{Test} & 1.00 & 1.00 & 1.00 & 1.00 \\\hline
\multirow{6}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Spearman\\ correlation\end{tabular}}} & \multirow{2}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Random\\ Search\end{tabular}}} & \textbf{Train} &0.47 & 0.47 & 0.45 & 0.47 \\
& & \textbf{Test} & 0.75 & 0.75 & 0.75 & 0.75 \\
& \multirow{2}{*}{\textbf{hyper-band}} & \textbf{Train} & 0.47 & 0.47 & 0.47 & 0.45 \\
& & \textbf{Test} & 0.75 & 0.75 & 0.75 & 0.62 \\
& \multirow{2}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Bayesian \\ Optimization\end{tabular}}} & \textbf{Train} & 0.47 & 0.47 & 0.47 & 0.45 \\
& &\textbf{Test} & 0.75 & 0.75 & 0.75 & 0.68 \\
\hline
\end{tabular}
\label{Tab:ad}
\end{table*}
\begin{table*}
\centering
\caption{Classification results for SZ using the CNN$_{tuned}$ model (best results are in bold).}
\vspace{3mm}
\begin{tabular}{cccccccc}
\hline
\textbf{\begin{tabular}[c]{@{}c@{}} Matrices \\ of connections \end{tabular}} & \textbf{Hyperparameter} & \textbf{Sample} & \textbf{Accuracy} & \textbf{Precision}& \textbf{Recall} & \textbf{AUC} \\ \hline
\multirow{6}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Granger\\ causality\end{tabular}}} & \multirow{2}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Random\\ Search\end{tabular}}} & \textbf{Train} & \textbf{0.90} & \textbf{0.90} & \textbf{0.90} & \textbf{0.93} \\
& & \textbf{Test} & \textbf{1.00} & \textbf{1.00} & \textbf{1.00} & \textbf{1.00} \\
& \multirow{2}{*}{\textbf{hyper-band}} & \textbf{Train} & 0.73 & 0.73 & 0.73 & 0.77 \\
& & \textbf{Test} & 0.72 & 0.72 & 0.72 & 0.78
\\
& \multirow{2}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Bayesian \\ Optimization\end{tabular}}} & \textbf{Train} & 0.72 & 0.72 & 0.72 & 0.78 \\
& & \textbf{Test} & 1.00 & 1.00 & 1.00 & 1.00
\\\hline
\multirow{6}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Pearson's \\ correlation\end{tabular}}} & \multirow{2}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Random\\ Search\end{tabular}}} & \textbf{Train} & 0.54 & 0.54 & 0.54 & 0.54 \\
& & \textbf{Test} & 0.50 & 0.50 & 0.50 & 0.50 \\
& \multirow{2}{*}{\textbf{hyper-band}} & \textbf{Train} & 0.54 & 0.54 & 0.54 & 0.54 \\
& & \textbf{Test} & 0.50 & 0.50 & 0.50 & 0.50 \\
& \multirow{2}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Bayesian \\ Optimization\end{tabular}}} & \textbf{Train} & 0.54 & 0.54 & 0.54 & 0.54 \\
& & \textbf{Test} & 0.50 & 0.50 & 0.50 & 0.50 \\\hline
\multirow{6}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Spearman\\ correlation\end{tabular}}} & \multirow{2}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Random\\ Search\end{tabular}}} & \textbf{Train} & 0.53 & 0.53 & 0.53 & 0.53 \\
& & \textbf{Test} & 0.50 & 0.50 & 0.50 & 0.50 \\
& \multirow{2}{*}{\textbf{hyper-band}} & \textbf{Train} & 0.53 & 0.53 & 0.53 & 0.53 \\
& & \textbf{Test} & 0.50 & 0.50 & 0.50 & 0.50 \\
& \multirow{2}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Bayesian \\ Optimization\end{tabular}}} & \textbf{Train} & 0.53 & 0.53 & 0.53 & 0.53 \\
& & \textbf{Test} & 0.50 & 0.50 & 0.50 & 0.50
\\ \hline
\end{tabular}
\label{Tab:sz}
\end{table*}
Concerning the CNN$_{untuned}$ model, the results are shown in tables \ref{Table:cass-ad} and \ref{Table:class-sz} for AD and SZ, respectively. For the AD data set, the best results are found using Pearson's correlation with a test accuracy of $92\%$. Regarding SZ disease, independently of the method used for the construction of the matrices of connections, results are close to the random guessing (see table \ref{Table:class-sz}). Therefore, the CNN$_{tuned}$ model is more accurate for both AD and SZ diagnosis.
\begin{table*}[!t]
\centering
\caption{Classification results for AD using the CNN$_{untuned}$ model (best results are in bold).}
\begin{tabular}{ccccccl}
\hline
\textbf{Matrices of connections} & \textbf{Sample} & \textbf{Accuracy} & \textbf{Precision} & \textbf{Recall} & \multicolumn{1}{l}{\textbf{AUC}} \\ \hline
\multirow{2}{*}{\textbf{Granger causality}} & \textbf{Train} & 0.97 & 0.97 & 0.99 & 0.99 \\
& \textbf{Test} & 0.58 & 0.57 & 0.66 & 0.75 \\
\multirow{2}{*}{\textbf{Pearson's correlation}} & \textbf{Train} & \textbf{0.98} & \textbf{0.99} & \textbf{0.98} & \textbf{0.99} \\
& \textbf{Test} & \textbf{0.92} & \textbf{1.00} & \textbf{0.83} & \textbf{1.00} \\
\multirow{2}{*}{\textbf{Spearman's correlation}} & \textbf{Train} & 0.97 & 0.98 & 0.97 & 0.99 \\
& \textbf{Test} & 0.83 & 1.00 & 0.66 & 1.00 \\ \hline
\end{tabular} \label{Table:cass-ad}
\end{table*}
\begin{table*}[!t]
\centering
\caption{Classification results for SZ using the CNN$_{untuned}$ model.}
\begin{tabular}{ccccccl}
\hline
\textbf{Matrices of connections} & \textbf{Sample} & \textbf{Accuracy} & \textbf{Precision} & \textbf{Recall} & \multicolumn{1}{l}{\textbf{AUC}} \\ \hline
\multirow{2}{*}{\textbf{Granger causality}} & \textbf{Train} & \textbf{0.97} & \textbf{0.97} & \textbf{0.97} & \textbf{0.99} \\
& \textbf{Test} & \textbf{0.52} & \textbf{0.53} & \textbf{0.73} & \textbf{0.55} \\
\multirow{2}{*}{\textbf{Pearson's correlation}} & \textbf{Train} & 0.61 & 0.58 & 1.00 & 0.53 \\
& \textbf{Test} & 0.57 & 0.55 & 1.00 & 0.45 \\
\multirow{2}{*}{\textbf{Spearman's correlation}} & \textbf{Train} & 0.62 & 0.59 & 0.97 & 0.58 \\
& \textbf{Test} & 0.62 & 0.58 & 1.00 & 0.53 \\ \hline
\end{tabular} \label{Table:class-sz}
\end{table*}
Importantly, the overall predictive performance depends on the choice of measure to construct the matrices of connections. In the case of AD, Pearson's correlation provides the best performance in CNN$_{tuned}$ (see table \ref{Tab:ad}). On the other hand, in the case of SZ, Granger causality is superior to the other methods (see table \ref{Tab:sz}). Therefore, there is no general method to infer the connections and obtain the most accurate results. Thus, different methods should be considered to develop an accurate framework for the automatic diagnosis of mental disorders.
For a comparison of our method with the more common approach known from the literature, the classification is performed by applying the raw EEG time series as input for the CNN$_{tuned}$ model (whose performance is the best for both diseases, as discussed before). The results are shown in tables \ref{Table:results-rawAlzheimer} and \ref{Table:results-rawschizophrenia} for AD and SZ, respectively. The accuracy of $75\%$ for AD and $55\%$ for SZ are obtained. This outcome is supported by results available in the literature. Janghel and Rathore~\cite{janghel2021deep} obtained an accuracy of $76\%$ for AD, where the authors did not consider the matrices of connections.
\begin{table}[!t]
\centering
\caption{
Classification results for AD using raw EEG time series and the CNN$_{tuned}$ model.}
\begin{tabular}{cccccc}
\hline
\textbf{Set} & \textbf{Accuracy} & \textbf{Precision} & \textbf{Recall} & \textbf{AUC} \\ \hline
\textbf{Train} & 0.68 & 0.61 & 1.00 & 0.68 \\
\textbf{Test} & 0.75 & 0.66 & 1.00 & 0.75 \\ \hline
\end{tabular}\label{Table:results-rawAlzheimer}
\end{table}
\begin{table}[!t]
\centering
\caption{Classification results for SZ using raw EEG time series and the CNN$_{tuned}$ model.}
\begin{tabular}{cccccc}
\hline
\textbf{Set} & \textbf{Accuracy} & \textbf{Precision} & \textbf{Recall} & \textbf{AUC} \\ \hline
\textbf{Train} & 0.62 & 0.62 & 1.00 & 0.50 \\
\textbf{Test} & 0.55 & 0.55 & 1.00 & 0.50 \\ \hline
\end{tabular}\label{Table:results-rawschizophrenia}
\end{table}
\begin{figure*}
\centering
\subfigure[]{
\includegraphics[width=0.45\textwidth]{pearson-roc-random-research-train.pdf}}
\subfigure[]{
\includegraphics[width=0.45\textwidth]{granger-roc-random-research-schizophrenia-train1}}
\caption{ROC curve obtained from the CNN$_{tuned}$ model. The matrices of connections are constructed by (a) Pearson's correlation for AD disease and (b) Granger causality for individuals diagnosed with SZ.}
\label{Fig:AUC-pearson-granger-schizophrenia-matrix}
\end{figure*}
As we can see, our proposed method based on a matrix of connections provided as input to a CNN allows for more accurate results. This reinforces the importance of using a data set that encompasses the connections between brain regions. Indeed, the network structure is a fundamental ingredient to differentiate healthy individuals from patients presenting neurological disorders, as verified in many papers (e.g.~\cite{de2014structure, lynn2019physics, fallani2011multiple, rodrigues2009structure, antiqueira2010estimating}).
In Figures \ref{Fig:AUC-pearson-granger-schizophrenia-matrix} we show the ROC curve for the best results, i.e. for AD (using Pearson's correlation) and SZ (using Granger causality test), respectively. For AD, the micro and macro-average ROC curve areas are 0.99 and 1.0, respectively, the micro and macro-average ROC curve areas are 0.92 for both cases. For comparison, Figure \ref{Fig:AUC-pearson-granger-schizophrenia} shows the ROC curve for AD and SZ using raw times series, where the micro and macro-average ROC curve areas are 0.75 for AD and around 0.55 for AZ. Comparing these results, we conclude that the use of the matrix of connections provides the most accurate classifications.
\begin{figure*}
\centering
\subfigure[]{
\includegraphics[width=0.45\textwidth]{raw-eeg-alzheimer.pdf}}
\subfigure[]{
\includegraphics[width=0.45\textwidth]{raw-eeg-schizophreinia.pdf}}
\caption{ROC curve obtained from raw EEG time series for (a) individuals diagnosed with AD and (b) individuals with SZ.}
\label{Fig:AUC-pearson-granger-schizophrenia}
\end{figure*}
\section{Conclusion}
In this paper, we propose a method for automatic diagnosis of AD and SZ based on EEG time series and deep learning. We infer the matrix of connections between brain areas following three different approaches, based on Granger causality, Pearson's and Spearman's correlations. These matrices are included in a convolutional neural network, tunned with the random search, hyper-band, and Bayesian optimization. We verify that this approach provides a very accurate classification of patients with AD and SZ diseases. The comparison with the traditional method that considers raw EEG data shows that our method is more accurate, reinforcing the importance of network topology for the description of brain data. Our method is general and can be used for any mental disorder in which EEG times series can be recorded.
A limitation of our analysis is the relatively small data set, although this is common in other studies on disease classification \cite{oh2019deep}. However, even with this restriction, our algorithm worked very well, showing that AD and SZ are associate to changes in brain organization. As future work, we suggest to consider larger data sets and additional information about the patients, like health conditions and age. A method that provides the level of the evolution of the disease is also an interesting topic to be developed from our study.
\section{Acknowledgements}
F.A.R. acknowledges CNPq (grant 309266/2019- 0) and FAPESP (grant 19/23293-0) for the financial support given for this research. A.M.P. acknowlwdges FAPESP (grant 2019/22277-0) for the financial support given this research.
K.R. acknowledges FAPESP grant 2019/26595-7.
C.T. gratefully acknowledges financial support
from the Zentrum für Wisschenschaftliche Services und
Transfer (ZeWiS) Aschaffenburg, Germany.
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